UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS FÍSICAS TESIS DOCTORAL Objetos compactos en teorías basadas en el Ricci: teoría y fenomenología Compact objects in Ricci-based gravity: theory and phenomenology MEMORIA PARA OPTAR AL GRADO DE DOCTORA PRESENTADA POR Mercè Guerrero Román DIRECTOR Diego Rubiera García © Mercè Guerrero Román, 2023 UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS FÍSICAS TESIS DOCTORAL Objetos compactos en teorías basadas en el Ricci: Teoría y Fenomenología Compact objects in Ricci-Based gravity: Theory and Phenomenology MEMORIA PARA OPTAR AL GRADO DE DOCTORA PRESENTADA POR Mercè Guerrero Román DIRECTOR Diego Rubiera García UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS FÍSICAS DEPARTAMENTO DE FÍSICA TEÓRICA TESIS DOCTORAL Objetos compactos en teoŕıas basadas en el Ricci: Teoŕıa y Fenomenoloǵıa Compact objects in Ricci-based gravity: Theory and Phenomenology MEMORIA PARA OPTAR AL GRADO DE DOCTORA PRESENTADA POR Mercè Guerrero Román DIRECTOR Diego Rubiera Garćıa Programa de Doctorado en F́ısica 6 Agradecimientos/acknowledgements Durante estos poco más de tres años he conocido a muchas personas que me han hecho crecer no solamente profesionalmente, sino también personalmente, por eso aunque no os mencione expĺıcitamente he pensado en todos vosotros. Empezando con los doctorandos, tanto los que ya se fueron como los que acaban de entrar. Muchas gracias por todo el tiempo compartido, nuestros debates y las penas compartidas. En especial, me gustaŕıa agradecer a Rita y Adrián, por ser el pilar indispensable desde que empecé aqúı en Madrid y ser un gran apoyo. A mis compañeros de despacho, a todos: Valent́ın, Álvaro y Sara, por compartir tanto conmigo, escucharnos entre dos y aconsejarnos, sin vosotros no seŕıa lo mismo, ¡ojalá hubierais entrado antes! Me falta Eva, mi segunda madre en Madrid, la que me ha acompañado y cuidado como una tercera hija, estaré agradecida eternamente por siempre estar alĺı cuando te necesitaba. A Teo, Lućıa y Alejandro por haberse léıdo algunos trocitos de mi tesis y haberme dado ”feedback”. Siguiendo con los más seniors, ha sido un placer conoceros a todos, durante comidas después de la pandemia, como a los que os habéis unido a la hora del té y me habéis ayudado en el mundo del post-doc. A todos con los que he colaborado (Diego Sáez, Gonzalo y demás), por enseñarme y dar gúıas sobre posibles futuros trabajos o cosas que perd́ıamos mirar. Aunque podŕıa entrar en el párrafo de encima, este párrafo se lo dedico a Diego, por darme esta grand́ısima oportunidad, por enseñarme no solamente F́ısica, sino también cómo funciona este mundillo. Gracias por ser tan organizado y con las ideas tan claras, sin eso me hubiera costado mucho más progresar durante mi tesis. Para terminar, quiero agradecerte todos los consejos y apoyo que me has ido dando durante este tiempo. Following inside the academic world, I would like to thank all the people I have met during my (several) stays in Tartu: Aneta, Laur, Mari, Sebastián, Jorge, Daniel, Rubi, Gerardo, Maria José, Nadja, Maggie, Christian, Margus, Alejandro, Pepe, Adrià and my flat mates, Negar and Darya. Without you Estonia would have never been the same. I want to specially thank Aneta, for not only be my supervisor there, but becoming a friend. Movent-nos una mica més enrere en el temps, no podria oblidar-me dels amics durant els llargs anys a l’Autònoma, Anna, Cristian, Gerard, Javi, Joan i Dani, perquè malgrat les apostes pensant que perdŕıem el contacte, hem seguit quedant (i no sempre ho ha proposat l’Anna). Em queda la Maria mai m’hagués imaginat que els nostres camins es tornarien a creuar a Madrid i creaŕıem nous records juntes aqúı. Falten les meves arrels, la gent de canet, us podria escriure un per un (ja que la llista no es gaire llarga), però tots sabem que aquest poble és una unitat, una famı́lia. Per acabar, vull dedicar-te unes 7 curtes paraules a tu, Dolors, per ser la primera investigadora que he conegut, per motivar-me a continuar en aquest món de la investigació, pel teu suport i, sobretot, per tot el que m’has ajudat des que vam començar a treballar juntes en el meu treball de recerca. Finalment, m’agradaria dedicar aquest últim paràgraf a la meva famı́lia, papa, mama, Rubén heu estat el meu gran suport i els meus fidels defensors des que tinc memòria, perquè sempre m’heu empès a seguir millorant i a no rendir-me, a lluitar pels meus somnis per més inassolible que pensés que fos. Als meus oncles, Encarni i Julián, per recolzar-me i escoltar-me quan els meus pares eren massa durs amb mi i ajudar-me a esbargir-me. Per ser la meva segona famı́lia i estar sempre orgullosos de mi. Als meus cosins, per ensenyar-me la vostra valentia i veure com poc a poc aneu trobant el vostre camı́, veure-us créixer i fer-vos adults. Ha sigut un plaer veure com madureu i seguiu els vostres passos. A tots vosaltres, mil gracies a tots. 8 To my family and myself. Summary Objetos compactos en teoŕıas basadas en el Ricci: Teoŕıa y Fenomenoloǵıa Compact objects in Ricci-Based gravity: Theory and Phenomenology General Relativity (GR) has been the most successful theory to describe gravity up to now, since it is in agreement with many different tests. Despite its achievements, GR also presents some problems that have opened the door to explore other theories beyond it in order to address these issues, for example the fact that it predicts the existence of spacetime singularities at high-energy scales or requires the yet non-detected dark energy/matter for the consistency of the cosmological models. Chapter 1 discusses the different ways one can modify GR and, among all of the different proposals, this Thesis is based on the so-called Ricci-Based Gravities (RBG) a subgroup of gravitational theories inside the metric-affine or Palatini formalism (where the connection and the metric are kept as independent entities). This Thesis is aimed to study the consequences of the RBG on astrophysical objects and find observa- tional discriminators that might allow us to identify such objects as well as to test the theories themselves. A more detailed explanation of the aims and the context of this Thesis can be found in Chapter 2. In order to do it so, we have split the Thesis in two parts; the first one is focused on the theoretical part, and contains the following chapters: • Chapter 3 prepares the ground to understand the next chapters. In particular, it starts by explaining the properties and the different theories inside the RBG, such as f(R) gravity and Eddington- inspired Born-Infeld. Aside from the gravitational theories, the different matter sources used throughout the Thesis are also explained in this chapter together with the geodesic equation within this RBG framework. • Chapter 4 contains one of the aims of this Thesis, to find regular solutions within the RBG and exact axisymmetric solutions that can be tested with the current observational data such as gravitational waves or their direct imaging. In particular, it is checked that at least one branch of the solution of RBG coupled to a non-linear electrodynamics does not have any incomplete geodesics. Additionally, a new tool called mapping that takes advantage of the structure of these gravitational theories is discussed and allows to find exact analytical solutions with rotation. On the other hand, the second part is focused on phenomenology of astrophysical objects, especially, on trying to test such a theories, which is the second main aim of this Thesis. This part is also divided as: • Chapter 5 is aimed to look for qualitative new phenomenology on the optical appearance of compact I object such as asymmetric wormholes with two distinct photon spheres and other background geometries that allowed us built complexity on the analysis of the light ring pattern when an accretion disc is considered around the compact object. • Chapter 6 is devoted to find different observational tests during the early evolution of low-mass stars. In particular, we start from the contraction of the proto-star following the Hayashi track until it reaches the Main Sequence phase where it burns stably hydrogen in its core. At this stage, we can calculate two different limiting masses that can be observed: the minimum main sequence mass and the maximum mass a star can have being fully convective. Both masses and the Hayashi tracks are modified when one considers an extended theory of gravity. Finally, in each chapter we have added a discussion of the main results. Therefore, chapter 7 summarizes the main results of the Thesis and explains the limitations of the analysis carried out here, for example on the modeling of the accretion disc or only assuming static scenarios, and how they can be improved. II Resumen La teoŕıa de la Relatividad General (GR) ha sido la teoŕıa gravitatoria más exitosa hasta ahora, puesto que concuerda con una gran variedad de observaciones y tests. A pesar de su éxito, el hecho de que predice singularidades espaciotemporales a altas enerǵıas o requiere de la materia/enerǵıa oscura aun no detectada para tener modelos cosmológicos consistentes, ha abierto la puerta a explorar otras teoŕıas de gravedad más allá de GR para poder explicar estos problemas. En el caṕıtulo 1 se discute las diferentes maneras en las que uno puede modificar dicha teoŕıa y de todas las opciones, esta tesis está fundada en las llamadas teoŕıas basadas en el Ricci (RBGs). Estas son un subgrupo de teoŕıas gravitatorias dentro del formalismo Palatini o metrico-af́ın (donde la conexión y la métrica se mantienen como identidades distintas). Esta tesis tiene el objetivo se estudiar las consecuencias de las RBG en objetos astrof́ısicos y encontrar, por un lado, discriminadores observacionales que nos puedan ayudar a identificarlos y, por otro lado, proporcionar tests para constreñir esas teoŕıas. En el caṕıtulo 2 se explican más profundamente dichos objetivos y el contexto alrededor de esta tesis. Con el fin de alcanzar dichos objetivos, he dividido la tesis en dos partes; la primera esta principalmente centrada en los aspectos teóricos y contiene los siguientes caṕıtulos: • El capitulo 3 prepara el terreno para entender los siguientes caṕıtulos. Empezando por explicar las propiedades y las diferentes teoŕıas dentro de las RBG, como la f(R) gravedad i la Eddington- inspired Born-Infeld. Aparte de las teoŕıas gravitacionales, también se explican las diferentes fuentes de materia utilizadas a lo largo de la tesis y la ecuación geodésica dentro del marco de las RBG. • El caṕıtulo 4 contiene el primer objetivo de esta tesis para encontrar soluciones dentro de las RBG y las soluciones con rotación exactas que pueden ser testeadas con los datos observacionales actuales como las ondas gravitacionales o su imagen directa. En particular, se puede comprobar que al menos una rama de la solución del acoplo entre la RBG y una electrodinámica no lineal es regular (no tiene ninguna geodésica incompleta). Además, introducimos una nueva herramienta llamada “mapping” que aprovecha la estructura de las ecuaciones de campo de estas teoŕıas y nos permite encontrar soluciones anaĺıticas exactas con rotación. Por otro lado, la segunda parte se centra en la fenomenoloǵıa de los objetos astrof́ısicos, especialmente, en encontrar modelos observacionales que puedan ser comparados con datos y nos permitan testear estas teoŕıas de gravedad. Esta parte también se divide en: III • El caṕıtulo 5 tiene como objetivo buscar fenómenos cualitativamente distintos en la apariencia óptica de estrellas compactas que puedan ser identificados ineqúıvocamente. Por ejemplo, agujeros de gusano asimétricos con dos esferas de fotones distintas u otras geometŕıas que nos permitan construir complejidad en el análisis de los patrones de los anillos luminosos cuando hay un disco de acreción alrededor del objeto compacto. • El caṕıtulo 6 está centrado en encontrar diferentes test observacionales relacionados con la tem- prana evolución de una estrella de masa baja. En particular, empezando por la contracción de la protoestrella siguiendo la llamada “Hayashi track” hasta llegar a la Secuencia Principal en el diagrama de Hertzsprung-Russell (dónde la estrella quema hidrogeno de manera estable). En esta etapa, podemos calcular la mı́nima masa que puede tener una estrella de la Secuencia Principal o la máxima para continuar siendo completamente convectiva. Ambas masas y la Hayashi “track” se modifican cuando se consideran teoŕıas más allá de Relatividad General. Finalmente, el caṕıtulo 7 resume los resultados principales y explica las limitaciones llevadas a cabo, por ejemplo en el modelaje de los discos de acreción o en asumir escenarios estáticos y como se puede mejorar. IV List of publications List of articles and publications (in reverse chronological order) the candidate has participated during this Thesis. 1. M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, D. Sáez-Chillón Gómez, Multiring images of thin accretion disk of a regular naked compact object Published in: Phys.Rev.D 106 (2022) 4, 044070 arXiv: 2205.12147 [gr-qc] 2. M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, D. Sáez-Chillón Gómez Light ring images of double photon spheres in black hole and wormhole spacetimes Published in: Phys.Rev.D 105 (2022) 8, 084057, arXiv: 2202.03809 [gr-qc] 3. M. Guerrero, D. Rubiera-Garcia, A. Wojnar, Pre-main sequence evolution of low-mass stars in Eddington-inspired Born–Infeld gravity Published in: Eur.Phys.J.C 82 (2022) 8, 707, arXiv: 2112.03682 [gr-qc] 4. M. Guerrero, G. Mora-Pérez, G. J. Olmo, E. Orazi and D. Rubiera-Garcia, Charged BTZ-type solutions in Eddington-inspired Born-Infeld gravity Published in: JCAP 11 (2021) 025, arXiv: 2108.09594 [gr-qc] 5. M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, D. Sáez-Chillón Gómez, Shadows and optical appearance of black bounces illuminated by a thin accretion disk Published in: JCAP 08 (2021) 036, arXiv: 2105.15073[gr-qc] 6. M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, Double shadows of reflection-asymmetric wormholes supported by positive energy thin-shells Published in JCAP 04 (2021) 066, arXiv: 2102.00840 [gr-qc] 7. M. Guerrero, D. Rubiera-Garcia and D. Saez-Chillon Gomez, Constant roll inflation in multifield models Published in: Phys.Rev.D 102 (2020) 123528, arXiv: 2008.07260 [gr-qc]. 8. M. Guerrero, G. Mora-Pérez, G. J. Olmo, E. Orazi and D. Rubiera-Garcia, Rotating black holes in Eddington-inspired Born-Infeld gravity: an exact solution Published in: JCAP 07 (2020) 058, arXiv: 2006.00761 [gr-qc]. 9. M. Guerrero and D. Rubiera-Garcia, Nonsingular black holes in nonlinear gravity coupled to Euler-Heisenberg electrodynamics Published in: Phys.Rev.D 102 (2020) 2, 024005, arXiv: 2005.08828 [gr-qc]. Some of the above articles are distributed throughout the Thesis as follows, • Chapter 3 is based on articles 8 and 9. • Chapter 4 is based on articles 1, 2, 5 and 6. • Chapter 5 is based on article 3. V Abbreviations BB Black Bounce BH Black Hole BI Born-Infeld CGS Centimetre–Gram–Second dof degrees of freedom EH Euler-Heisenberg EHT Event Horizon Telescope EiBI Eddington inspired Born-Infeld GR General Relativity HR Hertzsprung–Russell HT Hayashi Track ISCO Innermost Stable Circular Orbit lhs left-hand side LMS Low-Mass Stars MFCM Maximum Fully Convective Mass MMSM Minimum Main Sequence Mass NEC Null Energy Condition NED Non-linear Electrodynamics QED Quantum electrodynamics RBG Ricci-Based Gravities rhs right-hand side RN Reissner-Norström Schw Schwarzschild WH Wormhole Hole Convention • This thesis is mostly based on the Natural System Units where speed of light in vacuum, c, and the vacuum permittivity, 4πϵ0, are set to one. • The Newton constant is also redefined as κ2 ≡ 8πG . (1) • The metric signature is mostly pluses, that is (-,+,+,+). • The convention for the Riemann tensor, the Ricci tensor and the Ricci scalars is Rαµβν ≡ ∂βΓ α µν − ∂νΓ α µβ + ΓασβΓ σ µν − ΓασνΓ σ µβ , (2) Rµν ≡ Rαµαν , (3) R ≡ gµνRµν , (4) R ≡ qµνRµν , (5) VII Abbreviations where gµν is the spacetime metric, whereas qµν is called the Einstenian metric. • In Ch. 5 we have defined R which is the radius of the asymmetric wormhole throat. • In Ch. 6, R is the total radius of the star. Therefore, note the differences between R, R, R and R. • Finally, in the second half of Ch. 6, we change the System of Units, from the natural to the the Centimetre–Gram–Second (CGS). In order to convert from the natural system to the other, one has to multiply by a factor of cγ , where γ is the power of time dimension. VIII 21 21 1 3 Theoretical framework 3.1 Ricci Based Gravities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Palatini f (R) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.3 Eddington-inspired Born-Infeld gravity . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Stress-energy tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Maxwell Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Non-linear Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.3 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.4 Energy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Geodesics in symmetric RBGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Null geodesics in general static, spherically symmetric spacetime . . . . . . . . . . 34 3.3.2 Timelike geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.3 Geodesic completeness and energy conditions . . . . . . . . . . . . . . . . . . . . . 36 4 Applications 39 4.1 Derivation of a RBG spherically symmetric solution . . . . . . . . . . . . . . . . . . . . . 40 4.2 RBGs coupled to Euler-Heisenberg electrodynamics . . . . . . . . . . . . . . . . . . . . . . 41 4.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Properties of the solution: asymptotic behaviour . . . . . . . . . . . . . . . . . . . 44 4.2.3 Properties of the solution: radial function . . . . . . . . . . . . . . . . . . . . . . . 44 Contents Summary I Abbreviations VII 1 Preface 9 2 Introduction 13 I Theory 19 CONTENTS 5.2.2 Accretion disk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 87 5.3.1 Black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3.2 Traversable wormholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2 4.2.4 Properties of the solution: inner behaviour and horizons . . . . . . . . . . . . . . . 46 4.2.5 Properties of the solution: geodesic behaviour and regularity . . . . . . . . . . . . 48 4.3 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.1 Mapping example: EiBI gravity + NED . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3.2 Finding the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4 A rotating black hole solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4.1 Horizons and ergoregions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.2 Metric and curvature divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 II Phenomenology 67 5 Optical appearance of compact objects 69 5.1 Asymmetric thin-shell wormholes with two critical curves . . . . . . . . . . . . . . . . . . 69 5.1.1 Thin-shell formalism in Palatini f (R) gravity . . . . . . . . . . . . . . . . . . . . . 70 5.1.2 Electrovacuum spherically symmetric spacetimes . . . . . . . . . . . . . . . . . . . 71 5.1.3 Traversable wormholes from surgically joined Reissner-Norström spacetimes . . . . 73 5.1.4 Structure of the asymmetric wormhole: horizons and photon spheres . . . . . . . . 77 5.1.5 Double photon sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Imaging model improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.1 Ray tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3 Black Bounce solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The eye of the storm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.4.1 Multi-ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6 Pre-main Sequence evolution of low-mass stars 109 6.1 Non-relativistic stellar structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.1.1 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1.2 Simple photospheric model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1.3 Convective instability - modified Schwarzschild criterion . . . . . . . . . . . . . . . 115 6.2 Hayashi tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3 Minimum main sequence mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4 Fully convective stars on the main sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 CONTENTS 3 7 Conclusion 125 Bibliography 130 A Mathematica® Code 139 A.1 Ray tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.2 Accretion disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 CONTENTS 4 List of Figures 1.1 Diagram of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1 Representation of a light ray trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1 Plot of the dimensionless coordinate z as a function of x with positive model parameters . 45 4.2 Plot of the dimensionless coordinate z as a function of x with negative model parameters 46 4.3 Metric functions for negative values of model parameters . . . . . . . . . . . . . . . . . . . 48 4.4 The affine parameter E ũ(x) versus the radial coordinate for null radial geodesics . . . . . 49 5.1 Parameter space of positive energy density matter, γ̃ > 0, sources and stable (under radial perturbation) configurations, ω2 > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Parameter space of intersection of positive energy densities and radial stability . . . . . . 76 5.3 Representation of the domains with ω2 > 0 (shaded light blue regions) and with γ̃ > 0 . . 77 5.4 Representation of the parameter space regions that have a photon sphere in a traversable wormhole geometry for different values of the charge ratio . . . . . . . . . . . . . . . . . . 80 5.5 Same representation as in Fig. 5.4 but for a relevant value of the charge ratio . . . . . . . 80 5.6 Euclidean embedding of a reflection-asymmetric RN-RN wormhole . . . . . . . . . . . . . 81 5.7 The effective potential on each side of the wormhole throat . . . . . . . . . . . . . . . . . 81 5.8 Pictorial illustration (the colors are exaggerated) of the double shadow as seen from one side of the wormhole throat using celestial coordinates (α, β) . . . . . . . . . . . . . . . . 82 5.9 The effective potential for the Black Bounce geometries with M = 1 as a function of x and different model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.10 The critical impact parameter for the second Black Bounce model as a function of its model parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.11 Ray-tracing of the BH configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.12 The first three transfer functions of Schwarzschild and the extended Black Bounce solutions 92 5.13 The observed luminosity and the optical appearance for the Schwarzschild and the Black Bounce black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.14 Ray-tracing of the traversable WH configurations . . . . . . . . . . . . . . . . . . . . . . . 95 5.15 The three transfer functions of the wormhole configurations . . . . . . . . . . . . . . . . . 95 5.16 The observed intensity profile and optical appearance of the wormholes . . . . . . . . . . 97 5 LIST OF FIGURES 5.17 The effective potential of the Eye of the storm for different model parameters . . . . . . . 99 5.18 Ray-tracing of the Eye of the storm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.19 The transfer functions of the relevant type of emissions for the Eye of the storm . . . . . 101 5.20 The observed intensity (5.48) and its associated multi-ring structure . . . . . . . . . . . . 102 5.21 The optical appearance of the regular naked compact object . . . . . . . . . . . . . . . . . 104 6.1 The Hayashi tracks with different model parameters . . . . . . . . . . . . . . . . . . . . . 118 6.2 The evolution of the Minimum Main Sequence Mass in terms of the model parameter . . 121 6.3 The Maximum Fully Convective Mass in terms of the model parameter . . . . . . . . . . 123 6 List of Tables 5.1 Conditions of the model parameters to have (or not) horizon or photon sphere at the manifold M− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Conditions of the model parameters to do not have horizon but photon sphere at the manifold M+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Classification of the type of object depending on the model parameters . . . . . . . . . . . 90 5.4 Impact parameter range for different types emission for Black Hole configurations . . . . . 90 5.5 Impact parameter range of different types of emission for wormhole configurations . . . . 94 6.1 The MMSM and the parameters {α, γ, ω, δ} . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2 Numerical values for the MFCM using the total bound-free and the free-free opacities . . 123 7 LIST OF TABLES 8 Chapter 1 Preface The idea of gravity was first developed far before it was called as we know it nowadays, when mathe- matics, physics and philosophy were still mixed during the ancient times and tried to answer two main questions: why do object fall and how celestial objects move. The most well-known attempt to describe these phenomena was from the Greek philosopher Aristotle in the 4th century BC, who thought that objects where always moving to their “natural place”: an object would move upwards (like air or fire) or downwards (the “Earth” element) depending on its weight (inner gravitas). With this explanation, he was also capable to describe the sky, as the Earth would be the center of the Universe and all other objects will be naturally placed around it. Thus, stars and planets would be placed on a sphere and have circular trajectories on it. It was not until the 14th century that the main statements of Aristotle started to be rejected little by little, in particular, it was thought that the motion of every object was governed by a momentum that would depend on their velocity and mass (indeed, this idea was previously developed a couple of centuries before by Islamic philosophers). However, the first main change was in 1514 when Copernicus proposed the heliocentric model. Afterwards, Galileo around the 1600s conducted several experiments which allowed him to propose that any object will freely fall at the same acceleration rate independently of their mass in vacuum (without air resistance) and also described the first principle of relativity (Galilean relativity) which stated that Physics must be the same in any inertial frame of reference. At the same time, Kepler proposed his laws of planetary motion, an upgrade of the Copernicus heliocentric model where planets follow an elliptical orbit instead of a circular one and the area speed is constant, in other words, when the planet is closer to the Sun it will go quicker than when it is farther. In 1687, Newton published Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which includes the so-called law of universal gravitation. With this law the existence of Neptune was predicted thanks to the anomalus motion of Uranus. However, in 1859 Le Verrier observed a faster precession of mercury’s perihelion than the one expected by Newtonian physics, which leaded to several attempts to explain it such as a new planet, but all of them end up failing. In 1907, Einstein established the equivalence principle which essentially states that gravity locally has exactly the same effects as acceleration. Together with his special relativity, they planted the seed of 9 CHAPTER 1. PREFACE General Relativity (GR) that culminated with the publication of its field equations in 1915. This theory describes gravity as a non-trivial geometry of spacetime where matter curves spacetime and, at the same time, this geometry affects the trajectory of free-falling objects. In other words, GR gives a relation between matter and curvature. Later on, in 1916, Einstein proposed three tests of GR, now known as the classical tests, • The precession of mercury’s perihelion: General Relativity result was in agreement with the obser- vations of Le Verrier. • The deflection of light by the Sun: it was observed by the expeditions of Dayson and Eddington in 1919, even though there had been two unfruitful expeditions before, in 1912 in Brazil spoiled by the thick clouds covering the sky and, in 1914, in Crimea for the start of World War I [1]. • The gravitational redshift: it was firstly observed in 1525 by Adams using the spectrum of Sirius B, however the results were unreliable as they were contaminated by the light of Sirius A. After this observation, there has many others corroborating such an effect. Moreover, there are also the so-called modern tests such as the Shapiro delay, time dilation with atomic clocks, accurate solar system tests [2], etc. On what regards to cosmology, the expansion of the Universe discovered by Edwin Hubble in 1927 and 1929 is also explained by GR as well as the Cosmological Microwave Background (CMB), among many others. Finally, the strong field limit tests were added to the list, such as the gravitational waves directly detected for the first time by the LIGO collaboration in 2016 [3] or the observation of the Black Hole (BH) at the center of M87 by the Event Horizon Telescope (EHT) collaboration in 2019 [4]. However, everything that glitters is not gold, despite all the passed tests and agreements with obser- vations mentioned above, GR also has several shortcomings. On the one hand, from the cosmological point of view, the observed rotation curve of galaxies did not fit the theoretical models demanding for more matter than the observed. Additionally, a Universe completely filled with matter would be slowing down its expansion due to the attractive character of gravity; however, in the late 1990s, the Supernova Cosmology Project’s results of distant supernovas were completely opposite to what was expected, the universe was accelerating its expansion instead of slowing down due to the attracting behaviour of matter. Both issues could not be explained by only using the GR field equations and the usual matter fields which lead to two distinct ways to solve these problems; the first one is by considering new types of matter fields, whereas the second in modifying the gravitational theory. If one takes the former option, a “dark Universe” appears, consisting in two different components; the dark energy addresses the issue of the accelerated rate of Universe expansion while dark matter the lack of needed matter in galaxies. Both components amount to approximately ∼ 96% of the matter content in the Universe being unknown. The fact that the required matter has to barely interact electromagnetically, the lack of observation of candi- dates together with the tensions between the early and late Universe of ΛCDM model [5] are threatening its viability as well as favouring the alternatives to extend General Relativity. On the other hand, General Relativity also predicts its own demise, the singularities inside the center of BHs and in the Big Bang. The definition of a singular for a general spacetime has been an object 10 of discussion for a long time, since in GR it is not possible to talk about an event without defining a manifold and a metric structure around it. On the other hand, if we take into account the geodesics which are related to particle trajectories, they could get to the singularity in finite affine time, being not extendable (incomplete). Therefore, a spacetime is defined to be singular when it has at least one incomplete geodesic, since the theory loses its predictability there which means that suddenly particles would disappear into the nothingness or appear from nowhere. Thus, one might ask if these shortcomings may be solved when modifying the theory of gravity. However, in order to do it so one needs to know what can actually be changed of GR. Here is where the Lovelock’s Theorem [6, 7] sheds some light since it says that the Einstein-Hilbert action is the only gravitational theory formulated in a four-dimensional Riemannian space whose action solely depends on the metric tensor and has second-order differential equations. In other words, if we want to extend GR, we have to consider at least one of the following options: • A different number of dimensions than four. • Extra fields, such as scalar, vector and tensor fields. • Higher-order derivatives in the field equations. • Abandon locality. Nonetheless, this theorem is restricted to the metric formalism (the action solely depends on the metric tensor), where the compatibility of the metric and the connection is directly assumed; in other words, the connection is the Levi-Civita of the metric. However, these two quantities are not compulsively related since the metric measures infinitesimal distances between points and vector lengths and angles between them via a scalar product, whereas the affine connection is responsible of the parallel transport and defines the covariant derivatives. Therefore, we can go one step further when extending Einstein’s theory of gravity and restore the independence between both structures: this is known as the Palatini or metric-affine formalism. In the metric-affine formalism, the connection is a more general object that can be decomposed in several pieces: the Levi-Civita part of the metric (curvature), contortion and disformation, where the second one is related to the torsion (the antisymetric part of the connection) and the third one to the non- metricity (the covariant derivative of the metric). In order to make this definitions more understandable, let us describe what measures each tensor: • Curvature: the rotation of a vector after being parallel transported along a curve. • Torsion: the deviation from closing a parallelogram of two infinitesimal vectors parallel transported. • Non-metricity: the change of a vector length when it is parallel transported along a curve. Inside the world of metric-affine theories, there is a subgroup of theories called RBG formulated à la Palatini whose Lagrangian density is given by a function of the metric and the Ricci tensor of the independent connection. When one considers the torsion-free case, i.e. only taking the symmetric part 11 CHAPTER 1. PREFACE of the Ricci tensor, these theories yield second-order field equation, guarantees their ghost-free character and does not propagate extra degrees of freedom. This acts as a safeguard of (most of) these theories against getting into conflict with solar system tests and gravitational wave observations so far, while at the same time offering a workable framework to extract new gravitational physics in the strong-field regime. Even if we just consider the torsionless case, one still has a theory with non-metricity and curva- ture. Nevertheless, the fact that in these theories one can always define a new metric that is compatible with the independent connection leads to the conclusion that they are indeed (pseudo-)Riemaniann in terms of the new metric. wormholes Theory Phenomenology Optical appearance (Shadows) Asymmetric wormhole More realistic scenarios Ray tracing Acretion disk model Consists in Black Bounces Eye of the storm Early evolution of low-mass stars Contracting phase Hayashi track Ends Electron degeneracy pressure Hidrogen burning Developement of a radiative core Minimum Main Sequence Mass Max. Mass for fully convective stars Applications RBGs + Euler-Heisenberg electrodynamics Mapping method EiBI Gravity + Born-Infeld electrodynamics Brute force General Formalism Stress-energy tensor Geodesics Ricci Based Gravities (RBGs) General Relativity f ( R) Gravity Eddington-inspired Born-Infeld Gravity Non-linear electrodynamics Fluids Geodesics completeness C an o ne id en tif y th em ? Figure 1.1: Conceptual diagram of the Thesis with the connection between chapters and sections. The red squares represents the chapters, the blue ones are the sections and, lastly, the black corresponds to the subsections. Additionally, they are separated depending on the part they belong, the left are for the theoretical and the right to the phenomenological chapters. 12 Chapter 2 Introduction The theories of gravity describe the non-trivial geometry of spacetime caused by the presence of matter fields and how the observers move in such a geometry. This description can be applied to study the Universe or astrophysical objects. In this Thesis, the attention is focused on the last case where the observational data during the last few years with the gravitational waves detectors and the direct imaging of the central BH of M87 and Sgr A* have boosted the research field. This is so, since future and more they could detect exotic objects coming from the solutions of modified theories of gravity. However, the solutions of these compact objects within GR usually contain a spacetime singularity in its center, meaning that there is at least one incomplete geodesics; on the other hand, those solutions with no singularities requires the violation of energy conditions which in other words translates that exotic matter has to hold for such an object to exist, for example Wormhole Hole (WH)s. On the contrary, this might not be the case in modified gravity where the singularity theorems do not constrain them in an unavoidable way. This Thesis is aimed to investigate the regular character of the solutions within the RBGs applied to compact objects such as BHs or WHs as well as to look for observational fingerprints that might allow to identify such objects or set the ground to constrain our model parameters using astrophysical data. With these objectives in mind, the Thesis has been split into two parts, one theoretical and the other phenomenological. First of all, Ch. 3 is completely devoted to set the theoretical ground. I begin by explaining in detail the Palatini formulation of RBGs, under whose umbrella one can find different theories as GR itself, f(R) and Eddington inspired Born-Infeld (EiBI) [8] gravity, among many others. These theories depart from GR and other metric formulations due to additional contributions to the matter sector translating into a different matter dynamics driven by extra local energy densities. As a consequence, the new corrections coming from the enlargement of the theory are encapsulated in an effective stress-energy tensor which now plays a crucial role. Indeed, if one is in vacuum, the solution boils down to the one of GR (perhaps with a Λ term) meaning that one has exactly the same degrees of freedom propagating as in GR. Moreover, depending on the specific chosen theory of gravity, not all stress-energy tensor will be suitable, for example, f(R) gravity only has access to the trace of such a tensor, this means that Maxwell electrodynamics is not appropriate to couple with such a theory, since its tracelessness would lead to a 13 CHAPTER 2. INTRODUCTION GR solution. We shall be mostly motivated on considering an electromagnetic field; firstly, from the astrophysical point of view, BHs are characterized by three properties, mass, charge and spin/angular momentum. Thus, in order to have only mass, we only have to take into account vacuum and spherically symmetric spacetimes; for spin, vacuum and an axisymmetric solution. However, charges come from a matter source, in particular from electrodynamics. Secondly, from the theoretical point of view, Non-linear Electrodynamics (NED) have been historically used because of their regularizing properties [9–11] and also as an effective description of Quantum electrodynamics (QED). In Ch. 4 we aim to find new regular solutions using the RBGs field equations. Indeed, our first attempt is to solve the field equation for f(R) and EiBI gravities coupled to an Euler Heisenberg electrodynamics [12]. This NED is an effective field theory of Maxwell electrodynamics describing quantum vacuum polarization to one-loop and in the slowly-varying approximation. Rather than making quantitative predictions based on the scales where gravity/matter corrections should presumably appear, our aim is to qualitatively discuss the singularity-avoidance resolution mechanisms within these theories and how they fit within general studies aimed to achieve singularity-avoidance without breaking basic mathematical requirements or getting into contradiction with observations. Even though solving the field equations by brute force is possible, most of the time we will need to make use of approximations or numerical methods as a consequences of additional complexities added to these differential equations, as higher-order derivatives, non-linearity, etc. This is even worst when one wants to consider rotating scenarios. Indeed, testing the Kerr hypothesis on the nature of astrophysical BHs is becoming a hot topic in the investigation on the reliability of General Relativity as compared to its many alternatives thanks to the establishment of new present and future observational devices. However, the low supply of exact rotating solutions available in the literature of modified theories of gravity might prevent to test them using the full power of the observational machinery already available. Therefore, the investigation of rotating BH solutions inside theories of gravity beyond GR demands the development of new strategies. Actually, on what regards to the RBGs, one can take advantage of their field equations structure in the Einstein frame and relate the stress-energy tensor with additional contributions of the theory with an effective one that transforms the RBG equations into the ones of GR. This correspondence between both stress-energy tensors also allows to map the solutions of RBG + the stress-energy tensor with GR + the same type of matter described with a different Lagrangian. The value of these mapping method is of great interest as it transforms the differential form of field equations to an algebraic equation between RBGs and GR. Even if the mapping procedure is mainly meant to help us to find new rotating solutions in RBGs, it is easier to get the basic idea starting from a static, spherically symmetric spacetime at the same time that also allows to check the effectiveness of the method. In order to do it so, we begin by setting our seed solution as GR + Maxwell electrodynamics, meaning that we use it to derive the corresponding solution on the RBG side. In particular, the coupling between GR and Maxwell leads to the well-known Reissner-Norström (RN) solution, a static, charged BH. On the other side, bearing in mind that the type of matter source does not change when using the mapping but its Lagrangian does, the EiBi gravity will 14 be coupled to a NED, in particular, from the mapping equations one can easily check that it corresponds to the Born-Infeld electrodynamics. Thus, one can proceed to find the solution by two means, by brute force or through the mapping and compare both methods as well. Lastly, we follow the same strategy to get the rotating solution, however, we have to carefully adapt it to such a scenario. Despite we have considered different couplings between theories and matter sources, all the relevant solutions have the common feature that they resemble the GR solution far away of the gravitational source, but as one approaches the object, their structures deviate from GR. Apart from WHs there are other objects that have the same properties such as gravastar, boson stars, etc., which are called BH mimickers, since they look like BHs but they do not have an event horizon, they might yield to possible fingerprints in the strong field limit observations, for example the gravitational waves echoes or different shadow images. In this way, we enter to the second part, the phenomenological one; we start by analyzing the optical appearance of several compact objects in Ch. 5. The first attempt was a rather simple analysis. In particular, in Sec. 5.1 we want to look for qualitative new phenomenology that would allow us to easily identify them. Thus, we considered an asymmetric traversable WH with no accretion disk. By traversable we mean that such an object does not have an event horizon, so light rays can cross from one asymptotic region to the other. On the other hand, asymmetric WHs are built using the junction condition formalism in which we surgically cut and joint two different spacetimes, in particular, we are going to couple two RN with different masses and charges. This allows to have two distinct photon sphere (one for each side), i. e. the unstable circular photon orbit (corresponding to the maximum of the effective potential). The idea behind is that since there is no accretion disk, the only light rays that we are going to see are those emitted from relatively far object, such as a companion star. Therefore, such light rays are going to be accumulated (turning more times) near the photon ring region. Then, if we have two photon spheres, it means that in one side we might observed two thin photon rings in one of the two sides of the WH. However, without an accretion disk we cannot know which is the luminosity of the light rings, but also their width are expected to be so small that could barely be observed. Thus, to study the optical appearance of a compact object, one should add an accretion disk which now is going to be the main luminous source. Bearing this in mind, we start our analysis with the ray tracing procedure which consists in tracing back the null geodesics equation from the observer that gets a photon with a certain impact parameter until turning around the object and leaving to the asymptotic region again. With this method we can calculate how many times a photon turns around the object and classify them depending on such a number. Actually, this information is useful when we add the accretion disk. With the data of the ray tracing, we know how many times photons of a certain impact parameter are going to cross the disk or, more appropriately, how many photons are emitted from different radius of the disk and reach us with a certain impact parameter. Indeed, typically after three intersections with the disk, the impact parameter range of the following classified light rays is extremely reduced contributing negligibly to the total luminosity. When it comes on the modeling of the accretion disk, there are two limiting cases: the spherical accretion models where the shadow entirely fills the inner region such that its outer edge precisely coincides 15 CHAPTER 2. INTRODUCTION with the critical curve [13]. On the other hand, geometrically thin accretion models such that the inner edge of the disk extends all the way down to the BH horizon, there will be emission from inside the critical curve up to a certain region whose internal part has been termed as the inner shadow [14]. Particularly, to simplify our model, we consider an optically and geometrically thin disk around our object placed face-on (perpendicular to us). In an optically thin disk, the photons will not be re-absorbed, so the observed luminosity in each intersection photon-disk is going to increase. Moreover, in order to discuss the luminosity of the photon rings, we previously set three emitted intensity profile simulating the evolution of the disk. Starting from the Innermost Stable Circular orbit (ISCO) for time like observers, to the photon ring and ending up to the horizon (if there is) or the WH throat (when is the case). All three models have two features in common, the highest intensity is at the inner edge (where the accretion material is expected to be very hot) and the intensity smoothly decays with the radius. Finally, the fact that we consider geometrically thin disks allows us to see a richer pattern of photon rings, nonetheless, this is a quite strong assumption as accretion disks would have a certain width. Once the new technicalities are introduced, we proceed to study the light ring structure of different static, spherically symmetric solutions used as toy models that may allow to find new phenomenology and prepare the ground for the rotating scenarios. We start by the Black Bounce (BB) [15] solution and its generalization [16] in Sec. 5.3. These family of theories allow us to smoothly transition between the Schwarzschild (Schw) solution, a regular BH and a WH depending on the value of the model parameter. The fact that the first family of solutions has the same photon ring radius as the Schw does not give much room to play yielding to slightly enlargement of the luminous rings luminosities, nevertheless, it is a good starting point to understand better the results lead by this new procedure. Alternatively, the second type of solutions does not only change the photon ring radius, but also one can find a WH with two photon spheres which adds complexity to the luminous ring structure as there will be more rings as compared to the BHs cases. Finally, in Sec. 5.4 we perform the analysis to a geometry called the “eye of the storm” [17]. In this case, the particular shape of the effective potential, which diverges at the center of the solution, enlarges the impact parameter range of those light rays intersecting the disk more than three times making them to contribute non-negligibly to the total luminosity. This is translated to a wealthier pattern of luminous rings. In this geometries, one type of emission can lead to two distinct optical rings. Thus, as one can imagine, it improves the number of rings up to 7, whereas the Schw geometry only has 3 non-negligible rings. Last but not least, we consider other astrophysical objects for which their internal structure and equation of state are better known, stars. Even though in such objects the gravitational interaction is less strong as compared with neutron stars and BHs, modified gravity effects are induced as extra terms to the Poisson equation. This leads to a different stellar structure whose associated macroscopic features can be tested. In particular, Ch. 6 highlights the most important phases of the early evolution of Low-Mass Stars (LMS), within EiBI[18]. The relative simplicity of the stellar structure equations of EiBI gravity is a key feature that has allowed the community to scan its predictions as compared to GR expectations within different types of stars, see e.g. [19–26]. In particular, on its non-relativistic regime, this theory 16 leads to a modified Poisson equation with a single extra free parameter (for a full account of the theory and its phenomenology we refer the reader to [27]). In our analysis of the pre-main sequence evolution of LMS within it, we shall mainly focus on the paths followed by the contracting star, represented by Hayashi tracks [28] on the Hertzsprung-Russell (HR) diagram, and the associated limiting masses by the hydrogen burning as well as the development of a fully convective core, respectively, at the gateway of the main sequence. 17 CHAPTER 2. INTRODUCTION 18 Part I Theory 19 Chapter 3 Theoretical framework In this chapter, we prepare the ground from which we build up the main results of this Thesis. On the one hand, we begin by setting the group of gravitational theories used along the next chapters. Recall that the theories considered here are formulated within the Palatini formalism, for which the metric and the connection are kept as independent entities. On the other hand, the matter fields are fundamental in these theories, since as we show here, the absence of a matter source yields the same field equations as in GR. Thus, we present all the matter fields we use here and their corresponding stress-energy tensors. Finally, we devote a whole section to the geodesics, since they are necessary when discussing the regularity of solutions and also, as they are related to the trajectories of test particles, they will be essential for analyzing the optical structure of compact objects. Additionally, in the sections for the stress-energy tensor and the geodesics, we are going to restrict ourselves to static, spherically symmetric geometry. 3.1 Ricci Based Gravities Inside the Metric-Affine/Palatini formalism there is a intriguing subgroup of gravitational theories called Ricci-Based Gravities where their particularity relies on the specific dependencies of the Lagrangian density. It is built from the contractions between the metric and the Ricci tensor of the independent con- nection. Even though this theories are formulated with the whole decomposition parts of the connection (curvature, torsion and non-metricity), we are considering here the torsion-free case; this means that we only take the symmetric part of the Ricci. This allows the theory to be absent of ghosts instabilities as we will show latter on. However, besides restoring the independence between metric and connection, one needs to go one step further and consider more general actions, where their dynamics now does strongly depart from their metric cousins, leading to new phenomenology on their solutions. Nevertheless, before fully immerse ourselves in the properties as well as our assumptions upon these theories of gravity, let us gradually discover them while writing down their actions and deriving the field equations. 21 CHAPTER 3. THEORETICAL FRAMEWORK Our starting point is the RBGs action defined as Sm = 1 2κ2 ∫ d4x √ −gLG(gµν , Rµν(Γ)) + Sm(gµν , ψm) , (3.1) where κ2 ≡ 8πG is the redefinition of the Newton’s constant, whereas LG is the gravitational Lagrangian which is specifically written in terms of objectMµ ν = gµαRαν with gµν being the spacetime metric (thus, g is its determinant and gµα its inverse) and Rµν(Γ) the Ricci tensor defined in Eq.(3) of the a priori independent affine connection, Γ ≡ Γλµν . Regarding the matter action, Sm = ∫ d4x √ −gLm(gµν , ψm), it is assumed to depend only on the spacetime metric and on a set of matter fields ψm but not on the connection, to ensure the fulfillment of the equivalence principle [29]. Moreover, since we are mainly coupling our theories with bosonic fields of fluids, the minimial coupling does not affect them [30], therefore we can keep just the symmetric part of the Ricci tensor in the action (3.1). Performing independent variation of the action (3.1) with respect to metric and connection yields their respective field equations as ∂LG ∂gµν − 1 2 LGgµν = κ2Tµν , (3.2) ∇Γ α (√ −g ∂LG ∂Rµν ) −∇Γ β (√ −g ∂LG ∂Rµβ ) δνα = 0 , (3.3) where δµν the Kronecker delta and Tµν ≡ − 2√ −g δSm δgµν , (3.4) is the stress-energy tensor of the matter fields. Since the Lagrangian density is a function of the object Mµ ν = gµαRαν , its derivative with respect to gµν will depend on the Ricci tensor and possibly on the spacetime metric too. Thus, from Eq.(3.2) we will get a relation between the spacetime metric, the Ricci and the stress-energy tensors. On the contrary, the second equation might be extremely hard to solve, but if we introduce a new rank-two tensor as √ −qqµν ≡ √ −g ∂LG ∂Rµν , (3.5) the difficulty is remarkably reduced, however this definition only makes sense if we have non-dynamical fields as we are going to show a little bit later. Bearing in mind this definition, if one contracts the α and µ indices in Eq.(3.3), it is trivially seen that ∇Γ β ( √ −qqµν) = 0, yielding to ∇Γ α (√ −qqµβ ) = 0 . (3.6) This result tells us that such metric is compatible with the independent connection once some manipu- lations are made. In particular, expanding the covariant derivatives and contracting with qµβ we find 4 ∂α √ −q + √ −q qµν ∂α qµν − 2 √ −q Γλαλ = 0 , (3.7) 22 3.1. RICCI BASED GRAVITIES and using the relation qµν ∂α q µν = −2∂α ln √ −q, ∂α √ −q − √ −q Γλαλ = ∇Γ α √ −q = 0 . (3.8) Applying the above result into Eq.(3.6), the following is trivially found ∇Γ α q µν = 0 . (3.9) where now the compatibility between this two objects is undoubted as this result implies ∇Γ α qµν = 0, meaning that the connection can be written in terms of this new rank-two tensor and its derivatives. This can be translated as Γαµν being the Levi-Civita of qµν , which proposes the possibility to label the RBGs as (pseudo-)Riemannian geometries in terms of this new metric. The assumption of taking the symmetric part of the Ricci tensor provides the action (3.1) to be invariant under the so-called projective transformations, Γλµν → Γλµν + ξµδ λ ν , (3.10) where ξµ is an arbitrary one-form field associated to the freedom in the parametrization of the affine geodesics. This symmetry prevents our theories to propagate additional degrees of freedom (dof) associ- ated to the connection. This can be seen in the following way: if we had also considered the anti-symmetric part of the Ricci, there would have appear extra terms on the right hand side of Eq.(3.6) proportional to the torsion tensor which would prevent from having the new metric compatibility discussed before and leading to additional propagating fields associated to the connection. Thus, considering the whole Ricci will break such symmetry causing typically new pathologies associated to the connection (see [31] for a deep discussion). Another property of qµν is its inheritance, by construction, of the Ricci tensor index symmetry as can easily been seen form Eq. (3.5). Additionally, in this framework, it is always possible to introduce a deformation matrix, Ωαν , that relates the new and the spacetime metrics as qµν = gµαΩ α ν , (3.11) where the explicit form of Ωµν depends on the particular LG chosen (see Eq. (3.5)), and the metric gµν . Indeed, as we have mentioned before, the density Lagrangian can be written in terms of the matter fields and the spacetime metric so the deformation matrix will have the same dependencies. Additionally, playing with Eqs. (3.2), (3.5), (3.11) and the fact that these theories are constructed using the object Mµ ν lead to the following equation for the Ricci tensor Rµν(q) = 1 |Ω̂|1/2 ( LG 2 δµν + κ2Tµν ) , (3.12) with vertical bars denoting the determinant of the inner matrix and note that here Rµν = qµαRαν . From this equation, we will have a relation between the Ricci, the stress-energy tensor as well as (possibly) the spacetime metric. Hence, one can “redefine” the dependencies of the gravitational Lagrangian and the 23 CHAPTER 3. THEORETICAL FRAMEWORK deformation matrix in terms of the previously mentioned objects. The last feature is worth mentioning is that the field equation may be written in the Einstein-like representation as Gµν(q) = 1 |Ω|1/2 [ κ2Tµν − δµν ( LG + κ2T 2 )] , (3.13) where Gµν(q) = Rµν − 1 2qµνR is the Einstein tensor of qµν and T ≡ gµνTµν is the trace of the stress- energy tensor. From this equation it is easy to check the second-order and ghost-free character of the theory [31, 32], as if someone studies the vacuum case, it boils down to the standard GR solution, since remember that the gravitational Lagrangian can be rewritten in terms of the stress-energy tensor. This is a consequence of the fact that the symmetric description of RBGs introduce new corrections on the curvature associated to the matter fields adding local energy densities. This means that RBGs do not propagate extra dof beyond the two polarizations of the gravitational field of GR and may pass solar system tests provided that the modifications to GR occur in the ultraviolet limit. Again, this is explained by the projective symmetry, which implies that the connection is a non-dynamical field acting as a classical source, causing new interactions on the matter sector. Thus, when we consider a vacuum solution, such corrections disappear and at most an effective cosmological constant remains. Finally, note that the left-hand side (lhs) of this equation is written in terms of the metric compatible with the connection, while the right-hand side of the field equations (3.13) depends on the spacetime metric and energy tensor, a difficulty that we shall sort out later. It is time now to study different theories that belong to the RBGs and we are going to consider as well as employ them to shed a light on the main discussions of above by these particular cases. 3.1.1 General Relativity The first theory inside the RBGs to be analyzed could not be any other than GR. First of all, for its contribution to better understand gravity and its overall success (despite its shortcomings explained in the introduction) and, secondly, because it will be extremely useful in Sec. 4.3 for the mapping procedure, as we use such a theory as a seed to find new solutions in a selected RBG theory. Thus, let us proceed by writing the action as S = 1 2κ2 ∫ d4x √ −ggµνRµν(Γ) + Sm(gµν , ψm) , (3.14) where remember gµν is the inverse of the metric and Rµν is the Ricci tensor of the independent connection. In order to derive the field equations, let us trivially identify the Lagrangian density as LG = gµνRµν = R. Plugging this result into (3.2) and (3.6), the field equations boil down to Rµν − 1 2 R gµν = κ2Tµν , (3.15) ∇Γ µ( √ −ggαβ) = 0 . (3.16) Recall that the second equation corresponds to the compatibility of the metric with the independent con- nection, which means that General Relativity à la Palatini reduces to Einstein’s field equations without 24 3.1. RICCI BASED GRAVITIES imposing the connection to be the Levi-Civita of the metric. Even though we have failed to find new phenomena in the Palatini formulation of General Relativity, this will not be the case for more general actions inside the RBGs since the metric appearing in Eq. (3.16) will not be the spacetime one, gµν , but a new one defined as in (3.5). Thus, with a gravitational Lagrangian different from the Einstein-Hilbert one, the compatibility with the spacetime metric and the connection will be lost. 3.1.2 Palatini f(R) gravity The simplest generalization of the Einstein-Hilbert action one can conceive is considering a function of the Ricci scalar, R = gµνRµν(Γ), as the Lagrangian density, LG = f(R). Historically, these theories became very popular due to the Starobinsky model [33] since it could explain the cosmological problems described in the introduction: the accelerated expansion and structure of the Universe without considering additional dark energy or dark matter fields. Afterwards, it was applied to other topics including compact objects. In addition, their motivation is based on the addition of quadratic or higher-order curvature corrections to the gravitational Lagrangian as a way to address issues regarding high energies regimes. These higher-order corrections may contain new physics at the crossroads of GR and the quantum gravity regimes. In this case, the action is written as S = 1 2κ2 ∫ d4x √ −gf(R) + Sm[gµν , ψ], (3.17) where remember that f is an arbitrary function of the Ricci scalar, R. The gravitational field equations (3.2) and (3.5) for this theory boil down to fRRµν − 1 2 fgµν = κ2Tµν (3.18) ∇Γ α (√ −gfRgµν ) = 0 (3.19) with fR ≡ df/dR. For the sake of comparing the Palatini and metric formulation, let us write the field equation of the later fRRµν − 1 2 fgµν −∇µ∇νfR + gµν□fR = κ2Tµν , (3.20) where the extra terms come as a consequence of the entire dependence of action with the metric. Thus, since the Ricci tensor is a first-order derivative of the connection which at the same time is the Levi-Civita of the metric, this means that the above equation is a fourth-order differential equation, as compared to the second-order of the Palatini formalism. Moreover, fR can be understood as a propagating (scalar) degree of freedom, although, in this case it is non-dynamical. Now, continuing the discussion of the Palatini f(R) properties, from (3.11) the spacetime metric gµν is conformally related to the Einstein frame metric qµν as qµν = fR gµν , (3.21) 25 CHAPTER 3. THEORETICAL FRAMEWORK where the deformation matrix in this particular case is Ωµν = fR δµν . In the Palatini formulation, by taking the trace of the field equation (3.18), one finds that [34] RfR − 2f = κ2T . (3.22) This equation differs from metric formulation of f(R) theories as one could check doing the same pro- cedure in Eq.(3.20). The above equation represents an algebraic relation between the Ricci scalar and the matter fields, R ≡ R(T ), which implies that the curvature can be effectively removed out in terms of the matter fields (being thus closer to the spirit of GR, where R = −κ2T ). Due to this fact and to the equation for the independent connection, Palatini f(R) gravity and its generalizations are sometimes interpreted as GR with extra couplings in the matter fields [30]. Additionally, since the Ricci scalar can be replaced by the trace of the stress-energy tensor, this means that only the non-traceless ones will yield new dynamics as compared to GR. However, instead of taking the trace of Eq. (3.18) one can contract it with the inverse of the spacetime metric leading to Rµν(q) = 1 f2R ( f 2 δµν + κ2Tµν ) , (3.23) where Rµν = qµαRαν and is the particular case of Eq. (3.13). This result is going to be useful when solving the field equations. 3.1.3 Eddington-inspired Born-Infeld gravity This theory is inspired by the Born-Infeld electrodynamics which is based on the principle of finiteness inspired by the upgrading from the Newtonian mechanics to the relativistic and aims to tackle the divergence of a charged point-like self-energy (for a better explanation see Sec. 3.2.2). The action of EiBI gravity can be written as (for a review of this theory see [27]) SEiBI = 1 κ2ϵ ∫ d4x [√ −|gµν + ϵRµν | − λ √ −g ] , (3.24) where ϵ is a parameter with dimensions of length squared. Note that in this case, the action has to be slightly modified to resemble the structure of RBG’s action. Following the definition of the gravitational Lagrangian of RBGs, it corresponds to LG = √ |1 + ϵgµνRµν | − λ ϵ . (3.25) Even though this action might seem difficult to treat, if we check its weak-field limit, |Rµν | ≪ ϵ−1, the theory reduces to SEiBI ≈ ∫ d4x √ −g ( R 2κ2 − 2Λeff ) − ϵ 4κ2 ∫ d4x √ −g ( −R 2 2 +RµνR µν ) +O(2ϵ2) , where the first term corresponds to GR with an effective cosmological constant, Λeff = λ−1 ϵ , so when λ is fixed to one the obtained solutions are asymptotically flat. The higher-order curvature corrections 26 3.2. STRESS-ENERGY TENSOR appearing in the expansion are suppressed by powers of ϵ. Indeed, up to second order, this expansion corresponds to a particular case of quadratic gravity, which the action is written in terms of R and RµνR µν . For EiBI gravity, the Einstein-frame metric appearing in (3.11) is given by qµν ≡ gµν + ϵRµν(Γ) , (3.26) and the field equations boil down to √ −qqµν = √ −g (λgµν − ϵκ2Tµν) , (3.27) ∇ (√ −qqµν ) = 0 , (3.28) where for the first equation we have used the definition of q as well as the Jacobi formula, δ √ −|M̂ | = 1 2 √ −|M̂ |Tr[M̂−1δM̂ ]. From Eq.(3.27), one can check that the deformation matrix Ωµν can be determined via the algebraic expression |Ω̂|1/2(Ωµν)−1 = λ δµν − ϵκ2Tµν . (3.29) Now we can explicitly verify that the deformation matrix depends on the stress-energy tensor as we pointed out before in the RBGs, but in this case not on the metric. On the other hand, the gravitational Lagrangian (defined in Eq.(3.25)) can also be replaced by the deformation matrix and thus by the stress- energy tensor as follows, LG = 2 |Ω̂|1/2 − λ ϵ . (3.30) Here, we have used again the definition of qµν as well as the relation between metrics (3.11). Consequently, the Ricci tensor of (3.12) particularizes to Rµν(q) = 1 |Ω̂|1/2 ( |Ω̂|1/2 − λ ϵ δµν + κ2Tµν ) . (3.31) In this case, as compared to the f(R) gravity, the field equation has access to the full stress-energy tensor leading to a richer dynamics. 3.2 Stress-energy tensor The matter fields play an essential role, since their absence lead to exactly the same field equations as in GR. Thus, let us introduce the matter sources used in the following chapters and their motivations. In particular, since our work is mainly focus in compact objects, the simplest coupling one can conceive are electrodynamics which are responsible of their charge. 3.2.1 Maxwell Electrodynamics Starting by the simplest case, the Maxwell field is described by the following Lagrangian density, LEM = −1 4 FµνF µν , (3.32) 27 CHAPTER 3. THEORETICAL FRAMEWORK where Fµν = ∂µAν−∂νAµ is the field strength tensor and Aµ the vector potential. Thus, its stress-energy tensor can be trivially obtained by Eq.(3.4) as Tµν = FαµF α ν − 1 4 gµνFαβF αβ , (3.33) from this equation is straightforward to check its tracelessness by multiplying the inverse of spacetime metric, gµν , T = gµνTµν = FαµF αµ − 1 4 4FαβF αβ = 0 . (3.34) On the other hand, to find the components of the tensor above, one needs to solve the corresponding field equations which are written as ∇µF µν = 0 . (3.35) For static, spherically symmetric spacetime defined as ds2 = Adt2 +Bdr2 + r2(dθ2 + sin2 θdφ2) , (3.36) where usually A = B−1, but this will not be the case for the RBGs solutions as we are going to show in the following chapter. For electrostatic configurations, the only non-zero component is Ftr ≡ E(r) these equations can be written as E(r) = Q r2 √ AB , (3.37) where Q is an integration constant identified as the electric charge. Finally, one only has to plug this result back to the expression of Tµν and solve the field equations of the chosen theory. 3.2.2 Non-linear Electrodynamics As Maxwell electrodynamics has a linear contribution of the field strength tensor, additional corrections may appear to the Maxwell action motivated from different grounds. On the one hand, the NEDs considered in this Thesis were firstly motivated to address the divergence of a pointlike charged particle self-energy. On the other hand, they have also been used as singularity-regularizators [9–11, 35–43]. Apart from this property, we have an extra motivation to consider them, since taking f(R) theories coupled to the Maxwell electrodynamics yield to the same solutions as the GR ones due to the tracelessness of its stress-energy tensor (recall Eq.(3.34)). However, NED are not generically traceless which allow us to compare the predictions of f(R) and other RBGs on an equal footing. The NED are described by a Lagrangian density Lm = φ(X,Y ) , (3.38) where X = − 1 2FµνF µν and Y = − 1 2FµνF ∗µν are the two electromagnetic field invariants which can be built out of the field strength tensor, Fµν , and its dual F ∗µν = 1 2ϵ µναβFαβ . Similarly to the Maxwell 28 3.2. STRESS-ENERGY TENSOR electrodynamics, the field equations are written as ∇µ(φXF µν + φY F ∗µν) = 0 , (3.39) where φX ≡ ∂φ ∂X and φY ≡ ∂φ ∂Y . For electrostatic configurations, the only non-zero component is Ftr ≡ E(r) (thus Y = 0) and the field equations in any static, spherically symmetric spacetime is Xφ2 X = Q2 r4 , (3.40) where X = E2(AB)−1. The NED stress-energy tensor found by the same means as in the Maxwell case is Tµν = φ(X,Y )δµν − 2(φXF µ αF α ν + φY F µ αF ∗α ν) , (3.41) for electrostatic configurations can be conveniently split into 2× 2 blocks as Tµν =  T1 Î2×2 0̂2×2 0̂2× 2 T2 Î2×2  (3.42) =  (φ− 2XφX) Î2×2 0̂ 2×2 0̂ 2×2 φ Î2×2  , (3.43) where 0̂2×2 and Î2×2 are the 2 × 2 zero and identity matrices, respectively. From this expression, the trace reads T = 4(φ−XφX), which is non-vanishing as long as φ ̸= X (Maxwell electrodynamics). Born-Infeld electrodynamics This electrodynamics was motivated by Infeld and Born to address the self-energy divergence of a pointlike charge. They assumed that all physical quantities should be always finite, modifying non-linearly the Maxwell Lagrangian by adding an upper limit to the electromagnetic field. In particular, they were inspired by the structure and upgrade of the relativistic mechanics from the Galilean transformations. Then, the Born-Infeld (BI) theory of electrodynamics is described by the following density Lagrangian [44] φ(X) = β2 ( 1− √ 1− 2X β2 ) , (3.44) where β is proporional to the upper limit for the electromagnetic field. For this NED model one finds that equation (3.40) can be solved as X = Q2 r4 − r4c , (3.45) where we have introduced the scale r4c ≡ Q2/β2 . (3.46) It is easy to check now that when r → ∞, the Coulomb field is recovered as E ∝ X1/2 → Q/r2, whereas r = 0 leads to E ∝ −Q/r2c . After rewriting this NED model (3.44) in terms of the field equation (3.45), 29 CHAPTER 3. THEORETICAL FRAMEWORK the stress-energy tensor expressions becomes Tµν = 2Q2 r4c  (√ r4−r4c r2 − 1 ) Î2×2 0̂2×2 0̂2×2 ( r2√ r4−r4c − 1 ) Î2×2  . (3.47) It is easily seen that the above expressions recover the Maxwell ones when expanded in series of rc → 0. Also note that, from the above stress-energy tensor, one finds that in the r → 0 limit, the contribution to the energy of the BI field behaves as ∼ ∫ T 0 0 r 2dr ∼ r1/3 → 0, which implies that the total energy associated to electrostatic configurations in EH electrodynamics is finite. Euler-Heisenberg electrodynamics On more physical grounds, it is known that way before the scale where quantum gravity effects are expected to be excited in the innermost region of BHs, the growth of the electric field would induce quan- tum vacuum polarization effects modifying the classical description of Maxwell electrodynamics. In an effective approach, such effects to one loop and in the slowly-varying approximation can be incorporated by adding a quadratic piece in the electromagnetic field invariants to the Maxwell Lagrangian, yielding the so-called Euler-Heisenberg (EH) electrodynamics [12, 45]. The EH is described by the particular function1 φ(X) = X + βX2 . (3.48) The field equations (3.40) for EH electrodynamics can be solved in a exact form as [47] X(z) = 2 3β τ2(z) , (3.49) where we have defined dimensionless coordinate z = r/rc and r4c = 27l2β r 2 q , (3.50) τ(z) = Sinh h(z) , (3.51) h(z) = 1 3 ln [ 1 z2 ( 1 + √ z4 + 1 )] , (3.52) l2β = β/κ2 , (3.53) with l2β the squared NED length and r2q = 2κ2Q2 the squared charge radius. However, to check the weak limit, r → ∞, we should use the NED field equations (3.40) for (electro-) static, spherically symmetric fields E + 2βE3 AB = Q √ AB r2 , (3.54) then in such a limit E = Q/r2, while for r → 0 we have instead E ∝ X1/2 = (Q/2β)1/3r−2/3. 1When considering the effective limit of QED this Lagrangian picks another term in Y , which is vanishing for the electrostatic configurations considered here. In such a limit, β takes the value β = 2α2 45m2 e [46], where me is electron’s mass and α the fine structure constant. However, we shall take β as a free parameter assuming only β > 0. 30 3.2. STRESS-ENERGY TENSOR In terms of this dimensionless variable the stress-energy tensor (3.42) for EH electrodynamics reads Tµν =  −X(1 + 3βX) Î2×2 0̂ 2×2 0̂ 2×2 X(1 + βX) Î2×2  , (3.55) where its components are found upon substitution of (3.49). As in the previous case, one finds that in the r → 0 limit, the contribution to the energy of the EH field behaves as ∼ ∫ T 0 0 r 2dr ∼ r1/3 → 0, which implies that the total energy associated to electrostatic configurations in EH electrodynamics is finite. Moreover, when coupled to the Einstein-Hilbert action of GR, the finite character of the electrostatic solutions of this theory manifests in the fact that, besides configurations with two or a single (degenerate) horizons and naked singularities, typical of the RN solution of GR, there are also configurations with a single non-degenerate horizon (resembling the Schwarzschild BH). However, in all cases, a singular behaviour is found as follows from the geodesic incompleteness of all such solutions (see however [48]). 3.2.3 Fluids Even though we have defined previously the stress-energy tensor from the variation with respect to the action, for those systems with many particles involved, it has no sense to describe each one individually, but consider the continuous description. This is the casefor fluids described by its macroscopic quantities such as the density, pressure, stress, etc. Indeed, they have a huge relevance in cosmology as well as in astrophysics, for example to describe neutron and main sequence stars. Let us begin with the stress-energy tensor of the anisotropic fluid as Tµν = (ρ+ p⊥)u µuν + p⊥δ µ ν + (pr − p⊥)χ µχν , (3.56) where normalized timelike gµνu µuν = −1 and spacelike gµνχ µχν = +1 vectors have been introduced, while ρ is the fluid energy density, pr its pressure in the direction of χµ, and p⊥(r) its tangential pressure in the direction orthogonal to χµ. Using comoving coordinates, the stress-energy tensor can be written as Tµν = diag(−ρ, pr, p⊥, p⊥) . (3.57) Though, we typically use the perfect fluids which are an idealized model since its pressure is isotropic. Thus, its stress-energy tensor is Tµν = (ρ+ p)uµuν + p δµν . (3.58) We are going to use it in Sec. 5.1. Finally, in the particular case of a collapsing star or non-relativistic stars which we are going to consider in Ch. 6, typically it is considered dust or a pressureless fluid since the pressure can be neglected as compared to the density. That is Tdust µ ν = ρ uµuν . (3.59) 31 CHAPTER 3. THEORETICAL FRAMEWORK NEDs as anisotropic fluids The last note about the stress-energy tensor is the relation between NEDs and anisotropic fluids since both stress-energy tensors have te same structure in a static, spherically symmetric spacetime. This fact allows us to work in a more general scope and, indeed, we are going to use it latter on in Sec. 4.3. Recall that spherically symmetric electrostatic fields have a single non-vanishing component Ftr ̸= 0 and can be read off as fluids satisfying the 2x2 blocks structure as pr = −ρ and p⊥ = K(ρ), where the function K(ρ) characterizes (implicitly) the corresponding electrodynamics theory via the identifications [49] φ(X) = K(ρ) , (3.60) φ− 2XφX = −ρ . (3.61) This will be useful when one wants to particularize a general solution derived from an anisotropic fluid to a specific NED. Indeed, this will be applied in Sec. 4.3.1. 3.2.4 Energy conditions Our theory of gravity tells us how the curvature terms are related with the matter fields, however they do not say anything about the structure of the stress-energy tensor. Thus, the Energy Conditions (EC) are coordinate-invariant constrains on such a tensor based on physical reasonable scenarios, for example the positivity of energy density or requesting local causality. For the sake of this Thesis, we are only going to use the following: • The Weak Energy Condition (WEC) demands a non-negative energy density measured by any observed, Tµνu µuν ≥ 0 where uµ is a timelike vector. If we translate it into the conditions for the macroscopic properties of a perfect fluid written in Eq.(3.58), they are ρ+ p ≥ 0 and ρ ≥ 0. • The Null Energy Condition (NEC) is the “null limit version” of WEC, where Tµν l µlν ≥ 0 and lµ are null vectors. In the case of a perfect fluid this means ρ+ p ≥ 0. Thus, the energy density could be negative provided that the positive pressure compensates its value. • The Dominant Energy Condition (DEC) includes the WEC, Tµνu µuν ≥ 0, but also has an additional demand that matter cannot propagate faster that light, i.e. fulfills the local causality. This last requirement can be written as TµνT ν λu µuλ ≤ 0 where T νλu λ is the flux 4-vector and we demand it to be non-spacelike. For perfect fluids, this EC corresponds to ρ ≥ 0 and ρ ≥ |p|. • The Strong Energy Condition (SEC) is extremely tied to the congruence conditions, since it requires gravity to be attractive. Therefore it depends on the theory of gravity as is going to be discussed in Sec. 3.3.3. In particular, for GR, this condition is (Tµν − 1 2Tgµν)u µuν ≥ 0 for a timelike vector uµ. The EC are related, more specifically DEC implies WEC which at the same time contains NEC. Therefore, if NEC is not satisfied, all EC are violated. Those matter fields violating the energy conditions are called exotic matter. 32 3.3. GEODESICS IN SYMMETRIC RBGS 3.3 Geodesics in symmetric RBGs In this section, we are going to derive the geodesic equation in order to use it later on while analyzing its completeness of our RBGs solutions for compact objects and their regularization mechanisms. In addition, we also use them to characterize the critical curves and the optical appearance of a BH mimicker once an accretion disk is defined. Geodesics are those curves xµ(λ) whose tangent vector, tµ = dxµ/dλ, is invariant under parallel transport (autoparallels) defined by the Levi-Civita connection of a metric. This corresponds to the following equation tµ∇µt ν = d2xµ dλ2 + Γµαβ dxα dλ dxβ dλ = 0 . (3.62) where λ is the affine parameter (the proper time for a timelike observer). In our case, as we wrote explicitly in Sec. 3.1, our theory is torsionless and the matter fields are only coupled to the independent metric, consequently the previous equation can be rewritten in terms of such metric and its corresponding Christoffel symbols. As one can already imagine, solving the above equation to find the geodesic, xµ = xµ(λ), can became a nightmare quite easily. Thankfully, such equation can also be obtained through the variational procedure by considering the Lagrangian, L = 1 2 gµν ẋµẋν , (3.63) where dots denote derivatives with respect to an affine parameter, λ. Substituting the above Lagrangian to the Euler-Lagrange equation, one can explicitly check that it leads to Eq.(3.62). For any spherically symmetric spacetime with line element defined in Eq. (3.36), one can assume that the motion takes place in the plane θ = π/2 without loss of generality. Then, the geodesic equations become A(r) ṫ = E , (3.64) 2B(r)r̈ +B′(r)ṙ2 +A′(r)ṫ2 − C ′(r)ϕ̇2 = 0 , (3.65) C(r) ϕ̇ = L . (3.66) Since the Lagrangian does not depend on the coordinates t and ϕ, the first and third equations are constants of motion, also known as Killing symmetries, where E is the total energy and L the angular momentum of a particle per unit of mass. Indeed, if we compute the Hamiltonian, 2H = −E 2 A + ṙ B + L2 C , (3.67) we can also check that it is a constant of motion as it does not explicitly depend on the affine parameter. Actually, we can redefine 2H = k, where k = −1, 0 for timelike and null observers. Therefore, we can rewrite the above equation as ABṙ2 = E2 − V (r) , (3.68) 33 CHAPTER 3. THEORETICAL FRAMEWORK -2 2 4 6 8 10 -1 1 2 3 Figure 3.1: Representation of a light ray trajectory deviated by the gravitational effect of a Schw BH placed at the origin of the reference frame. where we have introduced the effective potential V (r) = A ( −k + L2 C ) . (3.69) Apart of using the geodesics to study the regularity of the solutions, it is also useful to know, for instance, the optical appearance of the astrophysical objects, created by the boarding of light rays. Therefore, let us continue by particularizing the geodesic equation to the null one. 3.3.1 Null geodesics in general static, spherically symmetric spacetime Before digging into the definition of a shadow and photon rings of an object, one should start with the gravitational lensing alone. Consider a light ray starting from spatial infinity approaching to a gravita- tional lens (which basically is the astrophysical object). As the photon gets closer to the gravitational source, due to the spacetime geometry, it begins to deviate from their initial direction until they get to the closest radius and subsequently turn back to spatial infinity again as shows Fig.3.1. When the radius of the turning point reduces, the deflection angle increases, until reaching a critical distance in which light is not able to escape from the object and starts a circular orbit around it. Such a distance is called photon sphere radius, rph. If the radius keep reducing, then the light ray will be dragged into the center of the object2. It is convenient to rewrite Eq.(3.68) in terms of the impact parameter defined as b = L/E, AB L2 ( dr dλ )2 = 1 b2 − Veff ≥ 0 , (3.70) where now the effective potential is Veff = A C . (3.71) Eq.(3.70) describes a one-dimensional trajectory of a photon with impact parameter b governed by a potential Veff . This equation gives us an idea of how close to the object we are and helps us to classify the trajectories depending on the number of turns around the center. Additionally, a light ray only propagates in those regions fulfilling 1/b2 ≥ Veff , and when both quantities are equal, the particle is in 2Note that this is the case for the case of Schw solution, which corresponds to a rather simple effective potential. However, if we consider different potentials, the structure pattern of light rays might be much complex. 34 3.3. GEODESICS IN SYMMETRIC RBGS the turning point, r0, as dr/dλ = 0 there. Moreover, one can obtain the impact parameter value for a particular r0 b = V −1/2 eff (r0) = √ C0 A0 , (3.72) where the subscript (0) means evaluated at r0. If the effective potential has a maximum, there is a radius of closest approach corresponding to the unstable photon orbit (photon sphere) radius, rps and the impact parameter leading to such curve is bc = √ Cps Aps , (3.73) called critical impact parameter. Since this orbit is unstable any small perturbation will make the photon to eventually fall into the BH horizon or escape to the asymptotic infinity. Thus, a photon with an impact parameter arbitrarily close to b ≳ bc will turn a large number of times around the compact object. To calculate rps, we should find the maxima of the potential defined in Eq.(3.71) V ′ eff (rps) = −A(r) C(r) D(r) ∣∣∣∣ r=rps = 0 with D(r) = ( C ′ C − A′ A ) , (3.74) where primes denote derivative respect to radial coordinate. Thus, a photon sphere exists (critical curve) if D(r) = 0. As the maxima of the effective potential are the responsible of the photon spheres, they are a useful tool to study the BH shadow. If one wants to analyze the optical appearance of a compact object illuminated by the light rays passing close by, one has to suitably rewritte the geodesic equation (3.70). As it was mentioned before, this is done by calculating the deflection angle, so the equation must be in terms of the variation of the azimuthal angle ϕ with respect to the radial coordinate, using Eq.(3.66) AB C2 ( dr dϕ )2 = 1 b2 − Veff . (3.75) This equation is going to use in Sec. II for the ray tracing and the discussion of the photon rings and shadows. 3.3.2 Timelike geodesics On the contrary, let us briefly discuss the timelike geodesics. Recall once again the geodesic equation (3.68) which in this case reduces to AB ( dr dλ )2 = E2 −A ( 1 + L2 C ) . (3.76) Usually, the potential has a minimum, that is dV dr = −A C D(r)−A′ = 0 , (3.77) 35 CHAPTER 3. THEORETICAL FRAMEWORK where if there is a radius making the right hand side of this equation zero, and its second derivative positive, such position is known as the Innermost Stable Circular Orbit (ISCO). Conversely to the photon sphere, the fact that this radius corresponds to a minimum of the potential instead of the maximum translates to a stable orbit for timelike observers. We will use this radius afterwords when modelling the accretion discs orbiting around a compact object giving a limit of its inner edge. 3.3.3 Geodesic completeness and energy conditions The singularity theorems prove that under certain conditions of both energy and spacetime, there exist at least one incomplete geodesic, i.e. it cannot be extended to a arbitrarily large values of the affine parameter. However, instead of studding in detail these theorems, we are more interested in the relation between energy conditions and the completeness of geodesics, which changes depending on the theory of gravity. The geodesic congruence is essential in order to study the regularization of any spacetime. Indeed, it is the main ingredient of all the singularity theorems. Thus, let us start by the Raychaudhuri’s equation for timelike observers defined as dθ dτ = −1 3 θ2 − σµνσ µν + ωµνω µν −Rgµν u µuν , (3.78) where uµ is the 4-velocity, Rgµν is the Ricci of the spacetime metric and θ = ∇µu µ , (3.79) σµν = ∇(µuν) − 1 3 θhµν , (3.80) ωµν = ∇[µuν] (3.81) are the expansion scalar, shear and twist tensors respectively. Additionally, hµν is the transverse metric defined as hµν = gµν − uµuν which is purely “spatial”, meaning that is perpendicular to uµ. One can also check that the shear and rotation tensors are also purely spatial. We can now discuss the sign of rhs of Eq.(3.78). The second and third terms fulfill σµνσ µν , ωµνω µν ≥ 0 since they are purely spatial. Additionally, if we assume the geodesic congruence to be hypersurface- orthogonal, then ωµν = 0. Finally, the last term contains the Ricci tensor of the spacetime metric, gµν , however, we have only defined the Ricci of the independent connection. Nonetheless, both Riccis can be translated to an effective stress-energy tensor that contains the matter fields and additional contributions of the modified theories. Thus, in order to get the implications of the energy conditions we should find the expression for Rgµν using the local conformal symmetry written in Eq.(3.11) which for f(R) gravity and GR Rgµνu µuν = 1 fR ( κ2Tµν + f 2 gµν − 3 2fR fR,µ fR,ν + 2fR,µν −gµν□fR 2 ) uµuν , (3.82) RGRµν u µuν = κ2 ( Tµν − 1 2 Tgµν ) uµuν , (3.83) 36 3.3. GEODESICS IN SYMMETRIC RBGS where the two first terms of (3.82) are the Ricci tensor written in (3.23) and the other two come from the transformation between the Ricci tensors of both metrics. Going back to the sign of Rgµνu µuν , if it is positive, then Eq. (3.78) fulfills dθ dτ + 1 3 θ2 ≤ 0 → θ−1 ≥ θ−1 0 + τ 3 . (3.84) Thus, if we consider that the congruence is converging at the beginning, θ0 < 0, we find that at a finite proper time θ → −∞. Even though this is a singularity of the geodesic congruence, this conclusion together with other properties obtained by general arguments are used in the singularity theorems. Finally, the condition Rgµνu µuν ≥ 0 is known as the Timelike Convergence Condition (TCC) when uµ is a timelike vector, whereas if it is null, such condition is called Null Convergence Condition (NCC). If we assume that our matter fields fulfill the NEC, this would mean fR ( Rgµν + 3 2fR ∇µfR∇νfR + 2fR,µν ) lµlν ≥ 0 , (3.85) RGRµν l µlν ≥ 0 , (3.86) where lµ a is a null vector. T he above equation show that NEC in GR implies NCC, whereas for Palatini f(R) gravity, Eq. (3.85) has to be hold, which depends on the particular Lagrangian considered, therefore in general we cannot say that NCC is going to hold. Another way to understand this difference is by recalling two particular cases of geodesics completeness in GR: WHs and bouncing universes. Since both are regular, this implies that one condition of the singularity theorems does not to be hold and in those cases is the violation of the energy conditions. On the contrary, RBGs does not necessarily happen, since the field equation for f(R) gravity is Gµν(g) = 1 fR ( κ2Tµν + f − f2RR 2 gµν + 3 4fR (gµνfR,µ f ,µ R − 2fR,µ fR,ν ) + fR,µν −gµν□fR ) , (3.87) where the right-hand side (rhs) of this equation can be understood as an effective stress-energy tensor that might violate the energies conditions, in particular, the NEC in order to have a regular solution. However, since this tensor in not only related to matter fields but also to gravitational terms, this means that with classical matter sources we could have such geometries if 0 ≤ κ2Tµν l µlν < ( 3 4fR (2fR,µ fR,ν −gµνfR,µ f ,µR ) + gµν□fR − f − f2RR 2 gµν − fR,µν ) lµlν , (3.88) for lµ null vectors. In Ch.4, we find WH structures as a solution of RBGs and NEDs. 37 CHAPTER 3. THEORETICAL FRAMEWORK 38 Chapter 4 Applications The modified theories of gravity were proposed as a manner to face different shortcoming of GR, one of those are for example the singularities appearing in the center of BHs or in the beginning of the Big Bang. In this way, we are aimed to find here regular solutions of compact objects within RBGs. In order to do it so, we begin to use the machinery explained in the previous section to find new solutions and study their properties from the most basic configurations up to more complex but also more observationally appealing. This increase on the complexity comes somewhat naturally when one wants to study more realistic scenarios, for example, rotating BHs. However, a problem might pop up as soon as we try to solve the field equations due to their non-linearity (recall for example Eq. (3.13)). Therefore, building up more attractive solutions is not so straightforward, but requires a new tool that allows us to find new solutions more easily. In particular, we take advantage of the structure of the field equations that allow for an Einstein frame to develop such a tool. The first step consists on finding by brute force the most general solution of a static, spherically symmetric spacetimes for a RBG coupled to a particular matter source whose stress-energy tensor can be divided in 2x2 blocks. This structure includes the electromagnetic field as well as a particular case of an anisotropic fluid. Hence, the following sections will be more pleasant as well as less repetitive to the reader. Afterwords, this result is used for two particular cases; the first one is for f(R) and Born-Infeld gravities coupled to a EH electrodynamics. After finding the solution, we study the regularization mechanisms of each theory and other properties such as the asymptotic behaviour or curvature divergences at their center. For the second case, we use this general deviation to prove that a new tool called mapping presented in Sec. 4.3 actually works and lead to exactly the same solution. This method basically exploits the structure of the field equations to transform their differential character to an algebraic relationship, being much easier to solve. In particular, we start with the assumption of GR + Maxwell electrodynamics as our seed solution and map such configuration to the Eddington-inspired Born-Infeld gravity. The corresponding matter sector coupled to EiBI is the Born-Infeld electrodynamics. After checking that solving by brute force and by the mapping procedure leads to the same result, we use the static case to upgrade to the rotating case and study the obtained solution. 39 CHAPTER 4. APPLICATIONS 4.1 Derivation of a RBG spherically symmetric solution For the moment, let us start with a general derivation of an (electro-)static, spherically symmetric solution that we are going to use in the first two cases. The most general line element for such spacetime in the qµν metric is defined by ds2q = −A(x)e2ψ(x)dt2 + dx2 A(x) + x2dΩ2, (4.1) where ψ(x), A(x) are functions of the radial coordinate, x, which we assume that extends from (−∞,+∞) and dΩ2 = dθ2 + sin2 θdϕ2 is the differential volume element in the unit two-spheres. The next step is to find constraints on the metric functions above, but before that we should set the matter fields. In particular, to work with certain generality, we choose those leading to the 2x2 blocks of Tµν as in Eq. (3.42). Additionally, provided that the gravitational corrections of the RBG theories are contained in a new effective contribution of the matter sector, this is translated to a more involved dependence of the gravitational Lagrangian with such a tensor. Nonetheless, the Ricci tensor appearing on Eq. (3.23) and (3.31) for f(R) and EiBI gravities respectively will inherit the same 2x2 block structure. Then, the Rtt and Rxx components will be the same. Therefore, the best move at this stage is the combination Rtt −Rxx = 0 of the corresponding field equations which allows to set ψ(x) = 0 in (4.1) without loss of generality. However, we have to constrain a little bit more our solution to go forward, that is to define the usual mass ansatz A(x) = 1− 2M(x) x , (4.2) and plug it into the remaining independent component of the field equations as Rθθ = 1 x2 (1−A− xA,x ) = 2M,x x2 , (4.3) where the coma denotes derivative with respect to. Equaling it to the right-hand side of the RBGs field equations (3.12), we find that the mass function satisfies M,x = x2 2|Ω̂|1/2 ( LG 2 + κ2T2 ) . (4.4) Looking at this equation one could think that the problem is solved; first, choose a gravitational La- grangian as well as a particular 2x2 block stress-energy tensor, then solve the differential equation above in order to get the mass function and, finally, plug it back to the line element (4.1), done!. However, things are never so simple, bear in mind that Ω̂ and T̂ are written in terms of metric functions gµν - which is the solution that we are indeed looking for. This means that we must use the relation between metrics (3.11) and write everything in terms of one of them to solve the mass function equation. Let us do it so: the easiest way to proceed is by rewriting everything in terms of the spacetime metric whose line element a priori we set it to be ds2g = −B(r)dt2 + C(r)dx2 + r(x)2dΩ2 , (4.5) 40 4.2. RBGS COUPLED TO EULER-HEISENBERG ELECTRODYNAMICS where this time the radial coordinate has the usual domain range r ∈ [0,+∞). As we referenced before, both metrics are conformally related through the deformation matrix, therefore the components of the metric will change but not the coordinates. Remember that the deformation matrix keeps the same structure as the stress-energy tensor and we will write it as Ωµν =  Ω1Î2×2 0̂ 2×2 0̂ 2×2 Ω2Î2×2  . (4.6) Thus, the spacetime metric functions are related to the ones appearing in (4.1) as r(x) = x√ Ω2 , B(r) = A(x) Ω1 and C(r) = 1 A(x)Ω1 . Note that the former relation allows to move from one radial coordinate to the other. Then, the differential mass function is recast as M,r = r2(Ω2) 1/2 2Ω1 ( 1 + rΩ2,r 2Ω2 )( LG 2 + κ2T2 ) . (4.7) As before, even though we have left everything in terms of the radial function r, since this variable also depends on x, we could also change everything in terms of this latter coordinate depending on which leaves the easier integral. We are now able to find wit (4.7) the function M(x) needed for the metric component A(x), which was defined in (4.2). Finally, one arrives to the solution of the line element (4.5), ds2 = −A(x) Ω1 dt2 + 1 A(x)Ω1 dx2 + r(x)2dΩ2 . (4.8) A quick glance on the above result evidently show the difference between the RBG solution and the Einsteinian metric. The first change is the lost of the property C(x) = B−1(x), as the deformation matrix is dividing in both metric components spoils it. An additional property is that now we have two radial coordinates related which - as we will see in the following sections - may yield to more attractive solutions. From now on, when looking for static, spherically symmetric solutions we will use Eqs. (4.7), (4.2) and (4.8), together with the deformation matrix written in (4.6). 4.2 RBGs coupled to Euler-Heisenberg electrodynamics In Sec. 3.2.2, we have introduced the EH electrodynamics which can be interpreted as an effective expansion into series of QED [45]. Within GR, such a model has a finite energy associated to the system of point-like charges and the existence of new gravitational configurations in terms of the structure of horizons [50–52]. However, these solutions still have singularities in all such configurations. The aim of this part is to verify if the solutions of different RBGs theories coupled to an EH electrodynamics are non-singular and if so to compare the regularization mechanisms of each theory. In particular, such analysis will be performed with the quadratic f(R) and EiBI gravities. The Lagrangian of the former is written as f(R) = R+ αR2 . (4.9) 41 CHAPTER 4. APPLICATIONS and the one of EiBI is defined in Eq.(3.25), respectively. 4.2.1 Solution The first step after all the necessary ingredients are set, i.e. the gravitational theories and the matter sector, is to compute the deformation matrix components (since they play an essential role in the mass function as well as the line element of gµν as shown in Eq. (4.8)). For both theories, such a matrix is going to be diagonal as a consequence of the matter choice. For f(R) gravity, the deformation matrix is defined as Ωµν = fR δ µ ν . Thus, in order to find its components, we should first solve Eq. (3.22) to get an expression for the Ricci scalar. With this result, one finds that the function fR becomes fR = 1 + α̃ τ4(z) , (4.10) where α̃ ≡ 32α/(9l2β) is the redefinition of the f(R) and matter parameters, α and lβ = β/κ2 respectively, whereas τ(z) is the characteristic function of the EH electrodynamics defined in Eq. (3.51). Conversely, in EiBI gravity, the deformation matrix components are found by substituting the stress-energy tensor of EH electrodynamics written in (3.55) into the deformation matrix definition (3.29) Ω1 = λ− l2ϵ τ 2(z) ( 1 + 2 τ2(z) 3 ) , (4.11) Ω2 = λ+ l2ϵ τ 2(z) (1 + 2 τ2(z)) , (4.12) with l2ϵ = 2ϵ /(3 l2β) being the analog of α̃ for f(R) gravity. Remember that λ is related to an effective cosmological constant, in this way we will set it to one in order to have asymptotically flat solutions. After the deformation matrix is determined, the line elements for each theory are easily obtained by having a glance at Eq. (4.8). However, remember that for convenience we redefined the radial coordinate as z = r/rc for the solution of the EH electrodynamics in Sec. (3.2.2). Therefore, the line element will not be written in terms of r, but the new one z. Starting with f(R) gravity, ds2 = −C(z)dt2 + dx2 f2RC(z) + z2(x)dΩ2, (4.13) where we have abused of notation since we introduce an implicit factor rc inside x as x→ x rc, thus x2 = z2fR , (4.14) C(z) = 1 fR ( 1− 1 + δ1G(z) δ2 z f 1/2 R ) , (4.15) where the second term in (4.15) corresponds to the integral of the differential mass function appearing in Eq. (4.7), hereby the new definitions of constants and functions come from Mz = z2 r3c 2 f 3/2 R ( f 2 + κ2φ )( fR + z 2 fR, z ) . (4.16) 42 4.2. RBGS COUPLED TO EULER-HEISENBERG ELECTRODYNAMICS Before integrating the above equation, we separate the constants from the functions of z. This yields to the following definitions of the two main constants characterizing the problem δ1 = r3c 2 l2β rS = (27)3/4 2 rS √ r3q lβ , (4.17) δ2 = rc rS , (4.18) and the function G(z) in (4.15) obtained in terms of its derivative as Gz(z) = z2 f 3/2 R ( f̃ + κ2φ̃ )( fR + z 2 fR, z ) , (4.19) where rS ≡ 2M0 is Schw’s radius, rq the charge radius and the quantities with tilde are multiplied by a factor l2β defined in (3.53). The line element (4.13) with the definitions above is the solution to the problem of electrostatic solutions in quadratic Palatini f(R) gravity coupled to EH electrodynamics, described by the two usual integration constants: the mass M and the electric charge q; and two new scales: the gravity parameter α and the matter one β, all of them are encoded in the two parameters δ1, δ2. Conversely, the solution in EiBI gravity coupled to EH electrodynamics is given by ds2 = −D(z)dt2 + dx2 Ω1D(z) + z2(x)dΩ2, (4.20) which its components relate with the ones of q as x2 = z2Ω2, (4.21) D(z) = A(x) Ω1 = 1 Ω1 ( 1− 1 + γ1J(z) γ2 zΩ 1/2 2 ) . (4.22) Here, as in the case of f(R) gravity, the second term of the metric function comes from the differential mass function which in this case is written as Mz = z2r3c 2 ϵ (Ω2 − 1)Ω2 1/2 ( 1 + zΩ−,z 2 Ω 1/2 2 ) . (4.23) Thus, as happened before we introduce the following definitions γ1 = r3c rS ϵ = 3 2 ( 3 2π )1/4 1 l2ϵ rS √ r3q lβ , (4.24) γ2 = rc rS , (4.25) Jz(z) = z2(Ω2 − 1) Ω 1/2 2 ( 1 + zΩ2,z 2Ω2 ) . (4.26) These two parameters encode the two integration constants, rS and r2q , and the two gravity and model parameters, l2ϵ and l2β , likewise in the f(R) case. 43 CHAPTER 4. APPLICATIONS During the following subsections we are going to study in detail their properties and how the different dependence with the matter sectors affect them. 4.2.2 Properties of the solution: asymptotic behaviour The first property of the solution to be checked is their asymptotic limit, z → ∞, which should tend to General Relativity. One can verify that the deformation matrices components behave in this limit as fR ≈ 1 + α̃ 81z8 , (4.27) Ω1,2 ≈ 1∓ l2ϵ 9z4 . (4.28) At this stage, it is trivial to see that the correction coming form the gravitational sector will appear earlier for EiBI than fR gravity because Ω2 has a lower power in τ(z) as show Eqs. (4.10 - 4.12) and so for z. However, let us go step by step, proceeding with the differential function appearing in the mass definition, Gz = 1 9z2 − 2 243z6 +O ( α z10 ) , (4.29) Jz = l2ϵ 9z2 − ϵ (9 l2ϵ + 4) 486z6 +O ( 1 z10 ) . (4.30) After introducing the above functions in the metric function (4.15) and (4.22), respectively, and reverting back to the original variables leads to C(r) = 1− rS r + r2Q 2r2 − l2βr 4 Q 5r6 +O ( α r10 ) , (4.31) D(r) = 1− rS r + q2 r2 + ϵ rSq 2 2 r5 − (β + 4ϵ) q4 5 r6 +O ( 1 r )10 . (4.32) For both cases, the first three terms correspond to the RN BH of GR as was expected. The fourth term for the f(R) gravity corresponds to a correction from EH electrodynamics which also appears in EiBI now moved to the fifth position and combined with an additional correction of the gravitational sector. Finally, the first gravitational correction for each theory is extremely different, being the fifth order in z for EiBI gravity while for f(R) one has to wait up to tenth order. However, even though such a correction can be neglected far away from the object, let us study what happens in the innermost region of the solution where the new gravitational dynamics encoded in the theory should arise. 4.2.3 Properties of the solution: radial function One of the main features modified close to the center of the solution is the behaviour of the radial function z (x) in (4.14) and (4.21). Remember that the gravitational parameters can have either positive or negative values. Let us start with the positive brunch for both theories, α̃, l2ϵ > 0; in these cases the deformation matrix component relating both radial coordinates does not have zeros. This means that given z ≥ 0, we will never reach x = 0, thus, the radial function z(x) generates two disconnected branches 44 4.2. RBGS COUPLED TO EULER-HEISENBERG ELECTRODYNAMICS -1.0 -0.5 0.0 0.5 1.0 x 0.2 0.4 0.6 0.8 1.0 z(x) (a) α̃ > 0. -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z (x) (b) l2ϵ > 0. Figure 4.1: Plot of the dimensionless coordinate z as a function of x. The green curve represents |α̃| = 0.2, |l2ϵ | = 0.01, the orange |α̃| = 0.5, |l2ϵ | = 0.2 and the purple |α̃| = 1.0, |l2ϵ | = 1. The blue curve represents the case of Maxwell electrodynamics with |l2ϵ | = 0.5 and rq = 0.5. As a comparison, we have plotted the GR case, r2 = x2 (black dashed), for which no such a bounce in the radial function is present. consisting on x < 0 and x > 0 of solutions as depicted in Figs. 4.1. Therefore, the area of the two-spheres can go all the way down to vanishing value. The corresponding solutions are presumably singular and for this reason we shall no longer consider them. Conversely, for α̃, l2ϵ < 0, the deformation matrix components fR and Ω2 will vanish at a certain z = zc with zc = √ 2 a a2 − 1 , (4.33) where we have introduced a new constant, a, that depends on the chosen theory, af = exp { 3ArcSinh ( |α̃|−1/4 )} , (4.34) aBI = exp 3ArcSinh 1 2 √√√√√ |l2ϵ |+ 8 |l2ϵ | − 1   , (4.35) where the first one corresponds to f(R) and the second to EiBI gravity. Despite both cases do not allow an analytic expression for the radial function z = z (x) in its full domain of definition, the fact that z has a minimum at x = 0 (see Fig.4.2) is translated to have a bounded area of the two-spheres S = 4πz2 from below, and the spacetime consist of two patches of the radial function z ∈ [zc,∞) or a single one in terms of the radial coordinate x ∈ (−∞,+∞). The natural interpretation for this bouncing behavior and minimum areal function is that of a WH structure [53], with z = zc representing its throat. The size of the latter grows with the absolute value of the gravity parameters, while it closes when such value goes to zero, corresponding to GR. For the case of EiBI gravity, we have also plotted the radial function when Maxwell electrodynamics is coupled as a matter to compare with our concerning coupling. Even though such a coupling keeps the WH structure, the transition across its throat of the radial functions is not as smooth as when considered EH electrodynamics. On the contrary, this cannot be done with f(R) gravity due to the tracelessness of the Maxwell’s stress-energy tensor, which leads to the exact same field equations as in GR, and that is the RN solution. 45 CHAPTER 4. APPLICATIONS -1.0 -0.5 0.0 0.5 1.0 x 0.2 0.4 0.6 0.8 1.0 z(x) (a) The radial function z(x) for the case α̃ < 0. -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z (x) (b) The radial function z(x) for the case l2ϵ < 0. Figure 4.2: Same notation as in figure 4.1. The WH throat is located at x = 0 (z = zc as defined in Eq.(4.33)). 4.2.4 Properties of the solution: inner behaviour and horizons From now on we shall focus on characterizing the properties of the negative branch, where we have some hope of finding regular BH solutions. Since the deviations as compared to GR solutions are expected to arise in the innermost region, we need to study the behaviour of the main functions there. In this sense, a series expansion of the deformation matrix functions around z = zc (defined in (4.33)) of (4.10) for f(R) gravity and (4.11)- (4.12) for EiBI, yields the result fR = f1(z − zc) +O(z − zc) 2 , (4.36) Ω1 = ω1(zc) +O(z − zc) , (4.37) Ω2 = ω2(zc)(z − zc) +O(z − zc) 2 , (4.38) where we have introduced the constants f1(zc) = 8 coth h(zc) 3 zc √ z4c + 1 , (4.39) ω1(zc) = 2 3 (sech 2h(zc) + 2) , (4.40) ω2(zc) = 4 3 (tanh 2h(zc) + cothh(zc)) zc √ z4c + 1 , (4.41) and we recall that h(z) is defined in Eq.(3.52). As for the functions Gz and Jz, it turns out to be a tougher nut to crack given its involved functional dependence, but it behaves at z = zc as Gz = C1(zc) (z − zc)3/2 +O(z − zc) −1/2 , (4.42) Jz = C2√ z − zc +O(z − zc) 1/2 , (4.43) where C1(z) > 0 is a cumbersome function of the constants of the model and C2 = z3cω 1/2 2 /2. There- 46 4.2. RBGS COUPLED TO EULER-HEISENBERG ELECTRODYNAMICS fore, the above expressions can be plainly integrated as G(z) = −2C̃1(zc) (z − zc)1/2 +O(z − zc) 1/2 , (4.44) J(z) = − 1 γc + 2C2 √ z − zc +O(z − zc) 3/2 , (4.45) where C̃1(zc) > 0 contains all the dependency with zc, conversely γc(zc) > 0 is a constant needed to match the inner and asymptotic expansions of J(z) for the EiBI gravity, and whose explicit dependence on its argument is very complicated, though for our analysis only its positivity is relevant. Plugging the expansions for the deformation matrix and G(z), J(z) in the metric function (4.15) and (4.22), respectively, yields to C(z) = C̃1(zc)δ1 δ2(z − zc)2 +O(z − zc) −3/2 , (4.46) D(z) = − 3 ( 1 + 2τ2c ) (γ1/γc − 1) 2 zcγ2 (3 + 4 τ2c )ω 1/2 2 √ z − zc (4.47) + 3 ( γ2 − γ1z 2 c ) 2 γ2(2 + sinh 2h(zc)) +O(z − zc) 1/2 , 1 Ω1D(z) = 3 zc γ2 ω 1/2 2 cosh 2h(zc) 2(γ1/γc − 1)(3 + 2τc) √ z − zc +O(z − zc) , (4.48) where τc ≡ τ(zc) and recall zc the minimum value of the coordinate z. The first expression corresponds to f(R) whereas the last two are the time and radial metric components of EiBI gravity. Let us derive our attention first to the f(R) case, where the metric function C(z) diverges always at z = zc as a consequence of the poles present in the fR factor as well as in the G(z) function. Moreover, due to the positivity of C̃1(z), δ1 and δ2 in this expression, one finds that this divergence goes always to +∞. Together with the asymptotically flat character of the solutions, as given by (4.31), provides the structure of horizons for these solutions. Indeed, as depicted in Fig.4.3a, this structure resembles the one of the RN solution of GR, namely, two-horizon BHs, extreme BHs (with a single degenerated horizon) and naked configurations. However, a systematic classification of the values of {|α̃|, δ1, δ2} yielding any such configurations are hard to find, and require instead direct inspection case-by-case. We also see that the single horizon BHs of the EH electrodynamics in GR has been lost, due to the modifications on the geometry caused by the presence of the WH throat at a finite distance zc. Focusing now on the EiBI case, a glance at the expansion (4.47) shows that such a classification can be performed according to the ratio γ1/γc, since it controls the sign of the divergence of the metric function D(z) at z = zc. Thus, • γ1/γc < 1, then D(zc) → −∞, and the corresponding solutions are Schw-like BHs with a single horizon. • γ1/γc > 1, then D(zc) → +∞ and one finds configurations with the same structure as the one of the RN solution of GR: BHs with two horizons, extreme BHs, or naked configurations, depending on the value of the constant γ2. 47 CHAPTER 4. APPLICATIONS 1 2 3 4 5 z -1.0 -0.5 0.5 1.0 C(z) (a) The metric function C(z) in Eq.(4.15) with the choices of |α̃| = 5, δ2 = 1/2 and three values of δ1 = 50 (purple), δ1 ≈ 85 (blue) and δ1 = 120 (red). 1 2 3 4 5 z -1.0 -0.5 0.5 1.0 D(z) (b) The metric function D(z) in Eq.(4.22) for l2ϵ = −1 and the choices of {γ1 = 1, γ2 = 1/3} (green), {γ1 = 5, γ2 = 1/3} (violet), {γ1 = 5, γ2 ≈ 0.4675} (violet), {γ1 = 5, γ2 = 1} (red) and (γ1 = γc ≈ 3.260)(oranges) with (γ2 = 0.6) and (γ2 = 1.5). Figure 4.3: Numerically integrated metric functions for the negative brunch, α̃, l2ϵ < 0, for f(R) and EiBI gravity. All solutions are asymptotically flat, C(z), D(z) ≈ z→∞ 1. The vertical dotted line represents the throat of the object placed at z = zc. In both cases, the red, blue and violet curves correspond to RN-like configurations: BHs with two horizons, extreme BHs, and naked solutions, respectively. Additionally, the Born-Infeld case has a richer variety with Schw-like BHs with a single horizon (green) and some configurations (orange) where D(zc) has a finite value with one horizon or none. • γ1 = γc requires a bit more effort. First replace this constraint into the metric functions (4.21) as well as (4.22) and expand in series of zc. In this case, the first term in Eq.(4.47) goes away and only the finite contribution at z = zc remains. Consequently, the corresponding configurations are solutions with either a single non-degenerate horizon or none, depending on the value of γ2 and γ1z 2 c . It should be pointed out that the Schw/RN-like structure of horizons resemble the original one of the EH electrodynamics within GR. While in GR the ratio between the total mass of the spacetime, M , and the total (finite) energy stored in the electrostatic field is the one playing the role in classifying such a structure, here is the ratio γ1/γc instead. However, the remaining configurations (constant curvature) are a novel feature of these Palatini spacetimes, having no counterpart in the GR-EH system. 4.2.5 Properties of the solution: geodesic behaviour and regularity To gain deeper knowledge on the innermost geometry of these solutions let us study their geodesic structure. Making use of their derivation in Sec.3.3, both theories will keep the same geodesic equation structure as in Eq.(3.68) and more specifically are given by 1 f2R ( dx du )2 = E2 − C(x) ( −k + L2 x2 ) , (4.49) 1 Ω2 1 ( dx du )2 = E2 −D(x) ( −k + L2 x2 ) , (4.50) where the first equation is for f(R) solution (4.13) and the second for EiBI (4.20). It is convenient to rewrite the above equations in terms of the dimensionless radial function z(x) by using their relationship 48 4.2. RBGS COUPLED TO EULER-HEISENBERG ELECTRODYNAMICS 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 E u ˜ (a) Ingoing (blue) and outgoing (orange) null radial geodesics for α̃ = 0.5 in terms of the radial coordinate z. -1.5 -1.0 -0.5 0.5 1.0 1.5 x -0.5 0.5 1.0 1.5 E u ˜ (b) Null radial geodesics for |l2ϵ | = 0.01 (green), |l2ϵ | = 0.2 (orange) and |l2ϵ | = 1 (blue) in terms of the radial variable x. Figure 4.4: The affine parameter E ũ(x) versus the radial coordinate for null radial geodesics in f(R) (left panel) and EiBI gravity (right panel). The vertical dashed purple line corresponds to the WH throat, z = zc, while the black dashed lines correspond to null radial geodesics in GR. For f(R) gravity it is enough to consider just one side of the WH, as radial light rays will never reach it. Conversely, in EiBI gravity, they will get there in a finite affine parameter so one needs needs to plot the x coordinate since they can be smoothly extended across x = 0. Eq. (4.14) and (4.21). For radial null geodesics (k = 0 and L = 0), the above differential equations become E dũ dz = ± 1 + z fR,z 2 fR f 1/2 R , (4.51) E dũ dz = ±Ω 1/2 2 Ω1 ( 1 + zΩ2,z 2Ω2 ) , (4.52) where, due to the change of variables from x to z, a derivative of the deformation matrix appears. The affine time parameter is also redefined as ũ = u/rc , while the +(-) sign come from taking the square-root of the equation and are linked to ingoing (outgoing) geodesics, that is trajectories of particles leaving (getting to) the throat at z = zc. At large distances, z ≫ 1, fR,Ω1,2 → 1, so these equation can be integrated as Eũ ≃ ±z, which is the expected GR behaviour, in agreement with the fact that in this limit both solutions should boil down to the RN one. However, departures are expected as the throat is approached. Unfortunately, to find a closed expression for ũ(z(x)) everywhere is not possible, but we can resort to series expansions around the WH throat z = zc. Indeed, using the deformation matrices expansion in Eq. (4.36), (4.37) and (4.38), we find the integrated expansion of (4.52) at z = zc as Eũ(z) ≈ ∓ √ 8/3 zc f1(zc) 1√ z − zc , (4.53) E(ũ− ũ0) ≈ ±ω 1/2 2 zc ω1 √ z − zc . (4.54) From the f(R) expression (4.53), it is readily seen that the affine parameter ũ(z) diverges to ±∞ as the WH throat is approached (see Fig. 4.4a). This implies that null radial geodesics require an infinite 49 CHAPTER 4. APPLICATIONS affine time to get to (or to depart from) the WH throat which, consequently, lies on the future (or past) boundary of the spacetime1. This way, they are complete in the geometry explored in this section as opposed to the RN spacetime, where null radial geodesics get to r = 0 in finite affine time without any possibility to further extension beyond this point. We point out that this mechanism for the removal of geodesic incompleteness via the displacement of any potentially pathological region to the boundary of the spacetime has been discussed in detail in Refs.[54, 55] on very general grounds, and explicitly implemented in other settings within Palatini theories of gravity [56, 57]. On the contrary, the EiBI case, Eq. (4.54), tells us that geodesics will reach the WH throat at a finite affine parameter time. However, with the radial function z we can only analyze what happens in one side, thus let us translate this geodesic into the x variable defined in the whole real axis, E(ũ− ũ0) ≈ ± x ω1 +O(x2) , (4.55) where we have made use of the fact that the radial function can be expanded in series of x as z ≈ zc + x2 z2cω2 +O(x4) . (4.56) Then, nothing prevents the affine parameter in Eq.(4.54) to cross the WH throat and be indefinitely extended to the asymptotic infinity x = −∞. This is shown in Fig. 4.4b, where we numerically integrate the geodesic equation (4.52) in full range, showing that any such geodesic starting from a certain ũ0 at x = +∞ departs from the GR behaviour as the WH throat, x = 0, is approached, and continues its path to another asymptotically flat region of spacetime, x = −∞. Therefore, null radial geodesics are complete in this geometry. Moving on to the remaining geodesics, the simplest case to discuss is f(R) gravity. Null geodesics with L ̸= 0 and time-like geodesics are complete in a quite similar way as in their RN counterparts. Due to C(x) diverges to +∞ at z = zc, any such geodesics will not be able to get to the throat. The reason is the infinitely repulsive potential barrier they see that make them to bounce off at a certain radius. This radius is given by the vanishing of the right hand side of (4.49). The bottom line of this discussion is the null and time-like geodesic completeness of the full spectrum of solutions of quadratic f(R) gravity with EH electrodynamics in the α̃ < 0 branch. Regarding the behaviour of curvature at x = 0, a quick computation revels the existence of curvature divergences there of size K ≡ Kα βγδKα βγδ ∼ 1/(z − zc) 2, which are nonetheless much weaker than in their GR counterparts K ∼ 1/r8. Meanwhile, the argument used before does not exactly apply here, due to the complexity of the metric function D(z). Depending on the combination of the parameters we can be dealing with a Schw-like, RN- like or a curvature-constant solution. Thus, one needs to analyze the behavior of the effective potential according to the expansion of the metric function at z = zc (x = 0), as follows from Eq.(4.47). As a 1In some sense this means that we have half a WH, in that the region x > 0 (x < 0) is not accessible to observers living in the region x < 0 (x > 0). 50 4.2. RBGS COUPLED TO EULER-HEISENBERG ELECTRODYNAMICS result, the effective potential series expansion is Veff ≈ − a |x| − b+O(x) , (4.57) with constants a = 3(1 + 2τ2c )(γc − γ1) 2 γ2γc (3 + 4 τ2c ) ( −k + L2 r2cz 2 c ) , (4.58) b = 3 ( γ2 − γ1z 2 c ) 2 γ2(2 + sinh 2h(zc)) ( −k + L2 r2cz 2 c ) . (4.59) Indeed, likewise the structure of horizons, the fate of any such geodesic depends on the ratio γ1/γc. For Schw-like configurations, γ1 < γc, the potential (4.57) is infinitely attractive and, therefore, any such geodesic crossing the event horizon of these configurations will unavoidably get to the WH throat in finite affine time. At such a point the geodesic equation (4.50) behaves as dũ dx = |x|1/2 ω1a1/2 +O(x3/2) → ũ(x) = 2x|x|3/2 3ω1a1/2 +O(x5/2) . (4.60) Since the coordinate x extends over the whole real axis, it is clear that these geodesics can be naturally extended across x = 0 and will be therefore complete for any values of the parameters of the model within the constraint γ1 < γc. It should be stressed that, despite the geodesically complete character of these spacetimes, any extended observer crossing the WH throat will find curvature divergences of size K ∼ 1/(z − zc) 3 there, which are much weaker than their GR counterparts, K ∼ 1/r8. For RN-like configurations, γ1 > γc, the effective potential (4.57) flips sign and it is infinitely repulsive at z = zc. Therefore, any of these geodesics will bounce at some z > zc and will continue its path within the x > 0 (or x < 0) region, which is the same behaviour as the one found in the RN solution of GR. Finally, for constant curvature configurations, γ1 = γc, the expansions (4.47) and (4.48) are not valid. As before, one needs to replace first this value of γ1 to the metric components before making the expansion around z = zc, which yields the result gtt ≈ − 3(1 + z2cγc γ2 ) 2 ( cosh 2h(zc) 1 + 2 cosh 2h(zc) ) +O(z − zc) , (4.61) grr ≈ 3 2 ( 1 + z2cγc γ2 ) ( cosh 2h(zc) 1 + 2 cosh(2h(zc)) ) +O(z − zc) . (4.62) This implies that the effective potential takes the same form as (4.57) with a = 0, that is Veff ≈ −b + c(zc)x 2, where c(zc) > 0 is a constant with an involved dependence on zc. Therefore, any particle with energy E above the maximum of this potential will be able to get to the throat. At that point, its affine parameter will behave as λ̃(x) ≈ x√ b+ E2 +O(x3) , (4.63) and therefore will find no impediment to continue its trip to the x < 0 region. Moreover, as opposed to the Schw-like and RN-like configurations, in this case curvature scalars are all finite at the throat. 51 CHAPTER 4. APPLICATIONS In summary, we have shown that all null and time-like geodesics in these geometries (in the branch l2ϵ < 0) are complete, no matter the values of mass and charge of the solutions or the value of the EH scale. The mechanism is, though different from the f(R) case, in that now the WH throat is accessible to different sets of geodesics, but all of them can be smoothly extended across the region x = 0. Therefore, these geometries represent nonsingular spacetimes. 4.3 Mapping As one could already witness from the previous section, solving the field equations in modified theories of gravity by brute force might become a real headache due to the additional complexities appearing in them ( e.g., their fourth-order nature and/or their highly non-linear character). This is even worst when one looks for an exact analytical axisymmetric solutions, being a trendy topic in the field thanks to the current strong-limit observations such as gravitational waves or the direct observation of the supermassive object in the center of galaxies. Up to now, the rotating solutions have been found via two different approaches; on the one hand, one can introduce additional simplifications such as constant curvature solutions, slowly-rotating scenarios or reasonable ansatz found by some means, see e.g. [58–60]. On the other hand, one can also use advanced numerical codes to face this issue, but they are extremely connected to the structure of the GR field equations, thus one must adapt them to each proposal of modified gravity. In this section, we discuss a recently established correspondence or mapping between RBGs and GR that actually reduces significantly our efforts to solve the field equations. Indeed, this method takes advantage of the RBG’s particular structure allowing to map not only their field equations to the ones of GR and vice versa, but also the solutions of both theories. The proof of concept of this correspondence has been explicitly established for electromagnetic [49, 61] and scalar [62] fields, where the form of some known exact solutions was obtained. Therefore, instead of solving the highly non-linear field equations of the original RBG, one can translate the problem to the GR counterpart, find the solution and map it back to the original RBG via purely algebraic transformations. As a consequence of their properties, when we map the field equations of RBG coupled with a given matter source, the GR side will have the same matter source but with a modified Lagrangian and vice versa. This means that if, for example, we have a Maxwell electrodynamics, on the one side, the matter sector would be governed by a NED on the other. The best way to grasp the relation between both frameworks is by glancing at the field equation (3.13), which is evoking us to rewrite it in the standard Einsteinan form of General Relativity by defining a new stress-energy tensor as T̄µν(q) = 1 |Ω̂|1/2 [ Tµν − δµν ( LG + T 2 )] . (4.64) This equation relates the effective stress-energy tensor generated by a matter source coupled to the Ricci Based Gravity, Tµν(g), and the one of General Relativity, T̄µν(q). So, if one finds a T̄µν(q) whose dependence on the metric gµν can be totally replaced by qµν and matter fields, then a correspondence 52 4.3. MAPPING between the two frameworks is guaranteed. Additionally, this map is given at the level of field equations, thus, it is not restricted to matter sources or symmetries of the solution. Among the different kind of matter sources available in the market, this Thesis has been mainly focus on nonlinear electrodynamics but also fluids. In this way, let us work as general as possible and consider an anisotropic fluid that, as has been shown in section 3.2.3, NEDs fit under its umbrella. Therefore, let us consider such stress-energy tensor on the RBG side written as in (3.56) and, as mentioned before, since the same kind of matter fields is found in the other gravitational sector, the energy-momentum tensor on the GR side will be of the form, T̄µν = (ρq + pq⊥)v µvν + pq⊥δ µ ν + (pqr − pq⊥)ξ µξν , (4.65) with {ρq, pqr, p q ⊥} being the density, radial and tangent pressure of the fluid in the q-metric, and qµνv µvν = −1, qµνξ µξν = +1 the timelike and spacelike vectors, respectively. Then, from Eq. (4.64) is easy to check the relation between matter sectors is pq⊥ = 1 |Ω̂|1/2 [ ρ− pr 2 − LG ] , (4.66) ρq + pq⊥ = ρ+ p⊥ |Ω̂|1/2 , (4.67) pqr − pq⊥ = pr − p⊥ |Ω̂|1/2 . (4.68) These scalar relations must be supplemented with the relations uµuν = vµvν , (4.69) χµχν = ξµξν . (4.70) However, with this correspondence it is not enough to map the solutions, since we still need the relation between the space of solutions given by Ωµν in Eq.(3.11). This deformation matrix depends on the RBG Lagrangian, but also on the matter fields, so it will have the following structure Ωµν = αδµν + βuµuν + γχµχν , (4.71) where α, β, and γ are functions of the variables ρ, pr, p⊥ ( or in terms of the GR side with ρq, pqr, p q ⊥. The explicit form of all these functions depends on the specific RBG Lagrangian that one considers). Therefore, to illustrate the above discussion, we are going to preform the procedure with a particular RBG theory. 4.3.1 Mapping example: EiBI gravity + NED Before proceeding to a particular case, let us briefly synthesize what we want to obtain. The basic idea is to take advantage of the “simpler” form of the field equations in GR. This can e done in two different ways, 53 CHAPTER 4. APPLICATIONS 1. Using known solutions of GR to find its equivalent in the particular RBG that we have considered. 2. Given a particular coupling between RBG and matter, mapping it to the GR case with its corre- sponding matter solve it there and map the solution back again. Bearing this in mind, we will go ahead with the first case, taking GR+Maxwell electrodynamics as the seed solution which corresponds to the RN solution, and map it back to the EiBI framework. Even if we are going to work with electrodynamics to find the RBG solution, for the sake of generality, let us derive all the methodology with an anisotropic fluid following the above derivation. Regardless of the matter source we are going to choose, the Lagrangian densities on RBG and GR sides will be related non-linearly (see Eq.(4.64) for example). This would mean that the Maxwell electrodynamics will lead to non-linear electromagnetic fields on the other side. Finally, the remaining ingredient is the gravitational theory inside the RBGs, in this case we are only going to focus on the EiBI gravity. As we did in the previous section, we begin with the definition of the deformation matrix written in Eq. (3.29), which for the specific energy-momentum tensor (3.56), becomes |Ω̂|1/2(Ωµν)−1 = (λ− ϵκ2p⊥)δ µ ν − κ2ϵ [(ρ+ p⊥)u µuν + (pr − p⊥)χ µχν ] , (4.72) or equivalently written in terms of the Einstein frame variables using Eqs.(4.66), (4.67) and (4.68). Once we have found the form of the deformation matrix, we can take advantage of the fundamental relation (3.11) and find the spacetime metric in the RBG frame as gµν = ( λ− ϵκ2 2 [ρq − pqr] ) qµν − κ2ϵ [(ρq + pq⊥)vµvν + (pqr − pq⊥)ξµξν ] . (4.73) This last relation is extremely powerful as it provides a solution for the EiBI theory starting with any known solution in GR supported by an anisotropic fluid source. From here, we are going to specify the fluid function to reassemble the typical NED structure of the energy-momentum tensor. In particular, pr = −ρ and p⊥ = K(ρ). After these choices, the mapping equations (4.66), (4.67) and (4.68) for the EiBI gravity with Lagrangian (3.30) and the deformation matrix (4.72) become [49] ρ̃BI = λρ̃GR − (λ− 1) 1− ρ̃GR , (4.74) K̃BI = λK̃GR + (λ− 1) 1 + K̃GR , (4.75) here, tildes denote an implicit factor ϵκ2. If one keeps in mind the relation between the functions of an anisotropic fluid and the nonlinear electrodynamics written in Eqs. (3.60) and (3.61), these equations tells us again that in general any matter described by NED on the RBG side, K̃BI, will be mapped into a different NED on the GR side, K̃GR, and vice versa. To be more illustrative, let us consider the standard Maxwell electrodynamics on the GR side, φGR(X) = X, which satisfies K̃GR = ρ̃GR = X. Thus, Eqs. (4.74) and (4.75) allows to relate the EiBI matter functions as K̃BI = ρ̃BI 1 + 2ρ̃BI , (4.76) 54 4.3. MAPPING where we have restricted ourselves to asymptotically flat solutions, λ = 1. In order to get the NED associated to this fluid on the EiBI side, we write the relations (3.60) and (3.61) under the explicit form φ = ρBI 1 + 2ρ̃BI , (4.77) φ− 2XφX = −ρBI , (4.78) where φ is the Lagrangian of the electrodynamics. The solution to this system of equations is φ(X) = 1 2κ2ϵ ( 1− √ 1− 4κ2ϵX ) , (4.79) where an integration constant has been set to get Maxwell electrodynamics in the limit ϵ → 0, that is, limϵ→0 φ(X) ≈ X. It is worth noting the similarities between this NED and the well known BI theory of electrodynamics (3.44). By identifying β2 → 1/(2κ2ϵ) the correspondence is exact. One should note that the sign of ϵ is not necessarily positive, whereas β2 is typically seen as positive in NED scenarios. Let us further elaborate on the result above. To be precise, we have found a correspondence between SGR+Max = 1 2κ2 ∫ d4x √ −gR+ ∫ d4x √ −gX , (4.80) and SEiBI+BI = 1 κ2ϵ ∫ d4x [√ −|gµν + ϵRµν | − √ −g ] + ∫ d4x √ −gφBI(X) . (4.81) In some previous works on electrostatic fields within EiBI gravity, for example in the previous section 3.2.2, it was assumed ϵ < 0, which implies that the corresponding NED would have the same sign as compared to the standard BI electrodynamics. Now it is time to find the solution of (4.81) using the mapping procedure and the brute force, check if the result coincides and compare the effectiveness of both ways. 4.3.2 Finding the solution Old-school way Let us first derive the solution by brute force as we did in Sec. 4.1. We should start by taking (3.29) with the diagonal deformation matrix ansatz defined in (4.6) together with stress-energy tensor components (3.47) of the BI electrodynamics, leading to Ω1 = 1 2 ( 1 + r2√ r4 + 4sr4c ) , (4.82) Ω2 = 1 2 ( 1 + √ r4 + 4sr4c r2 ) , (4.83) 55 CHAPTER 4. APPLICATIONS where r4c = 2κ2|ϵ|Q2 has been redefined. As before, the radial coordinates defined in (4.1) and (4.5) are linked via (3.11) that for the above components of the deformation matrix leads to the explicit expressions r2 = x4 − sr4c x2 , (4.84) x2 = 1 2 ( r2 + √ r4 + 4sr4c ) . (4.85) Finally, if we introduce the standard ansatz (4.2) to the Rθθ component of the field equations (3.12) as we did in (4.7) leads to (after writting everything in terms of the variable r using (4.84)) M,r = κ2Ω 1/2 2 2Ω1 (LG + T2)r 2 [ 1 + r 2 Ω2,r Ω2 ] = κ2Q2 √ 2 r √ r2 + √ r4 + 4r4cs( r2 ( r2 + √ r4 + 4r4cs ) + 4r4cs ) . (4.86) The integration of this function is straightforward M(r) =M0 − κ2Q2 √ 2 1√ r2 + √ r4 + s4r4c , (4.87) where M0 is an integration constant that can be identified with the asymptotic Schw mass. It is worth noting that using (4.85) the above mass function turns into M(r(x)) =M0 − κ2Q2 2x , (4.88) which is nothing but the RN mass function in terms of the radial coordinate x. This fact makes sense since the solution of an electrostatic, spherically symmetric space-rime coupled to a Maxwell electrodynamics is indeed the RN solution. Finally, the line element of the solution is given by (4.8), where Ω1,2 functions and the metric component A(x) are given by (4.82), (4.83) and the standard ansatz (4.2), respectively and can be directly expressed in terms of the coordinate x as Ω1 = 1 1 + sr4c x4 , (4.89) Ω2 = 1 1− sr4c x4 , (4.90) A(x) = 1− 2M0 x − κ2Q2 x2 . (4.91) Note that setting rc → 0, we recover the RN solution as expected since the deformation matrix reduces to the identity. Via mapping We will now use the final mapping equation presented in (4.73) to derive the above solution. As we already showed, there is a correspondence between GR + Maxwell and EiBI+ BI electrodynamics. Therefore, we can use the former known solution to find ours, that is via the RN as the seed. On more practical 56 4.4. A ROTATING BLACK HOLE SOLUTION grounds, this is translated to have the Einstein-frame metric qµν functions in Eq.(4.1) as ψ = 0 ; A(x) = 1− rS x + r2q 2x2 , (4.92) where r2q = 2κ2Q the charge radius. For a Maxwell field, in the fluid view its components are given by ρq = 2Q2 x4 , (4.93) pq⊥ = ρq = −pqr , (4.94) vµ = (−A(x)1/2, 0, 0, 0) , (4.95) ξµ = (0, A(x)−1/2, 0, 0) . (4.96) With these definitions and the deformation matrix written in Eq.(4.72), Ω1,2 functions take the form Ω1 = 1 1 + ϵκ2ρq , (4.97) Ω2 = 1 1− ϵκ2ρq , (4.98) where we have also taken advantage of the 2x2 structure of the deformation matrix as in Eq. (4.6). One can readily check that these expressions coincide with those given in (4.89) and (4.90), when ρq is substituted using (4.93) and the definition of r4c = 2κ2|ϵ|Q2. It is thus immediate to verify that the metric functions in (4.5) read B(r(x)) = A(x)/Ω1, C(r(x)) = 1/(A(x)Ω1), and r(x)2 = x2/Ω2, in complete agreement with the line element found by brute force. This verifies that not only does Maxwell theory coupled to GR, as defined by action (4.80), maps into a BI-type NED coupled to EiBI gravity, as defined by action (4.81), but also are their solutions mapped into each other. In order to generate a rotating solution in the EiBI theory starting from a known solutions in GR, we need to upgrade the above case to axial symmetric solutions. Thus, following the same concept, the rotating solution of GR coupled with a Maxwell electrodynamics is Kerr-Newman. Again, a rotating Maxwell electric field is equivalent to a rotating anisotropic fluid with K̃GR = ρ̃GR. Mapping this solution into the EiBI theory implies coupling it to the BI-type NED of Eq.(4.79). 4.4 A rotating black hole solution Before progressing on the generalization of the above case to the axisymmetric solutions, we should first suitably adapt the machinery explained in the previous section to the new particular features. The process to achieve this goal goes as follows: 1. Write the GR metric as the Kerr-Newman solution and calculate the corresponding Einstein tensor. 2. Interpret the solution as if it was generated by the stress-energy tensor of an anisotropic fluid. A direct comparison of the Kerr-Newman Einstein tensor with this stress-energy tensor provides concrete expressions for the fluid density/pressure functions and the fluid’s angular velocity ω. 57 CHAPTER 4. APPLICATIONS 3. Use those expressions to generate the corresponding solution in the EiBI theory using the relations (4.73). Once the path is shown, it is simpler to proceed. Let us start by the first step and write the Kerr-Newman solution of GR using (t, x, θ, ϕ) coordinates in Boyer-Lindquist form as [63] qµν =  −(1− f) 0 0 −af sin2 θ 0 Σ ∆ 0 0 0 0 Σ 0 −af sin2 θ 0 0 ( x2 + a2 + fa2 sin2 θ ) sin2 θ  , (4.99) where 0 ≤ a ≤ 1 is the spin parameter and f = rSx− r2q/2 Σ = x2 + a2 −∆ Σ , Σ = x2 + a2 cos2 θ , (4.100) ∆ = x2 − rSx+ a2 + r2q/2 , remember that r2q = 2κ2Q2 is the charge radius. The corresponding Einstein tensor for this line element can be conveniently written as Gµν(q) =  −r2q(r 2+a2[1+sin2 θ]) 2Σ3 0 0 ar2q(a 2+x2) sin2 θ Σ3 0 −r2q 2Σ2 0 0 0 0 r2q 2Σ2 0 −ar2q Σ3 0 0 r2q(x 2+a2[1+sin2 θ]) 2Σ3  . (4.101) Moving on to the second step we first note that now the fluid four-velocity has an additional spatial component as compared to the static case (4.95) and can be written(assuming motion only in the ϕ direction) as vµ = (vt, 0, 0, vϕ) , (4.102) with vϕ = ωvt, and ω being the angular velocity of observers comoving with a fluid, i.e., ω ≡ dϕ/dt. The normalization of the vector vµvνqµν = −1 implies that vt = 1√ −(qtt + ωqtϕ + ω2qϕϕ) , (4.103) but the form of ω is still not known. Conversely, the normalization of the spatial vector ξµ, qµνξ µξν = 1, is enough to fully constraint its only spatial component, ξµ = (0, ξx, 0, 0), giving ξx = 1/ √ qxx. According to Eq.(4.65), the fluid stress-energy tensor becomes T̄µν =  − ρq F1 0 0 F2 0 −ρq 0 0 0 0 ρq 0 F3 0 0 ρq F1  , (4.104) 58 4.4. A ROTATING BLACK HOLE SOLUTION where remember pq⊥ = ρq = −pqr, and F1, F2, and F3 are complicated expressions of the fluid properties as well as metric variables. In particular, without substituting the metric components, the F’s are F1 = qtt + ω qtϕ + ω2qϕϕ qtt − ω2qϕϕ , (4.105) F2 = ρq qtϕ + 2ω qϕϕ qtt + ω qtϕ + ω2qϕϕ , F3 = ρq 2ω qtt + ω2qtϕ qtt + ω qtϕ + ω2qϕϕ . Bearing in mind the usual GR field equations (3.15) it is trivially seen that the radial components lead to the relation κ2ρq = r2q 2Σ2 , (4.106) which can be interpreted as the energy density of an axially symmetric electromagnetic field described by Aµ = (At, 0, 0, Aϕ) = Qr Σ (1, 0, 0,−a sin2 θ) , (4.107) from which the components of the field strength tensor are immediately obtained in the usual way. Moreover, a glance at the expression for F1 in Eq.(4.105) evokes to solve a quadratic algebraic equation to find ω as ω = a a2 + x2 . (4.108) From the comparison between the Einstein and the stress-energy tensors, we have been able to find all the key ingredients to map the Kerr-Newman solution of GR into a rotating solution in the EiBI gravity coupled to BI electrodynamics, as given by the action (4.81). To this end, we follow the same approach for the static case presented in Section 4.3.2 and apply it to the rotating case. Starting by the Kerr-Newman metric (4.99) with its corresponding definitions (4.100) and the expressions found up to now ρq = r2q 2κ2Σ2 = pq⊥ = −pqr , (4.109) vµ = ( − √ ∆ Σ , 0, 0, a sin2 θ √ ∆ Σ ) , (4.110) ξµ = ( 0, √ Σ ∆ , 0, 0 ) , (4.111) we can find the EiBI solution by using the relation between metrics presented in (4.73), with λ = 1 for asymptotically flat solutions. Indeed, it is possible to write the spacetime metric gµν as the Einsteinian one plus a EiBi gravity correction as gµν = qµν + ϵκ2ρqhµν , (4.112) 59 CHAPTER 4. APPLICATIONS where the additional term suppressed by the gravitational parameter reads hµν =  − (∆+a2 sin2 θ) Σ 0 0 a(∆+x2+a2) Σ sin2 θ 0 Σ ∆ 0 0 0 0 −Σ 0 a(∆+x2+a2) Σ sin2 θ 0 0 − [ (x2+a2)2+a2∆sin2 θ Σ ] sin2 θ  . (4.113) Thus, putting all together, the final solution for a rotating BH of EiBI gravity coupled to BI electrody- namics is obtained, in Boyer-Lindquist coordinates, as ds2 = − ( 1− f + ϵκ2ρq (∆ + a2 sin2 θ) Σ ) dt2 (4.114) − 2a ( f − ϵκ2ρq (∆ + x2 + a2) Σ ) sin2 θdtdϕ+ (1 + ϵκ2ρq)Σ ∆ dx2 + (1− ϵκ2ρq)Σdθ2 + [ ( x2 + a2 + fa2 sin2 θ ) − ϵκ2ρq (x2 + a2)2 + a2∆sin2 θ Σ ] sin2 θdϕ2 . The additional pieces coming from the modified theory are apparent. A basic aspect to note is the fact that the vacuum solution, rq = 0 =⇒ ρq = 0, coincides with the Kerr solution of GR as it was expected from the general behaviour of RBGs in absence of matter; only when the density function ρq in Eq.(4.109) is not zero we do have modifications as compared to the GR solution. Indeed, in the weak field limit, r → ∞, the line element above boils down to ds2 = − [ 1− 2M r +O(r−2) ] dt2 + [ 4aM sin2 θ r +O(r−2) ] dtdϕ+ [ 1 +O(r−1) ] (dr2 + r2dΩ2) , (4.115) which is nothing but the asymptotic limit of an axisymmetric rotating body in Boyer-Lindquist coordi- nates, and implies the consistence of these objects with the observations of orbital motions around any such object. The changes induced by the RBG theory on GR solutions will be stronger in those regions where the energy density reaches its highest values, which in the present case is the innermost region. On the other hand, for slowly-rotating BHs, a≪ 1, the expression (4.114) boils down to ds2 = [ − (x4 + sr4c )∆0 x6 + (cos2 θ∆0 x4 + sr4c ( − 1 + sin2 θ x6 + 3 cos2 θ∆0 x8 )) a2 +O(a4) ] dt2 + [ 2a sin2 θ ( r2q(x 4 + sr4c )− 2x(−2sr4cx+ rS(x 4 + sr4c )) 2x6 ) +O(a3) ] dtdϕ + [ x4 + sr4c x2∆0 + −x2(x4 + sr4c ) + cos2 θ(x4 − sr4c )∆0 x2∆2 0 a2 +O(a4) ] dx2 (4.116) + [( x2 − sr4c x4 ) + cos2 θ ( 1 + sr4c x4 ) a2 +O(a4) ] dθ2 + [( x2 − sr4c x2 ) + x2(sr4c (−2 + 3 cos2 θ) + x4)− sin2 θ(r2q(x 4 + sr4c )/2 + x(sr4cx− rS(x 4 + sr4c )) x6 +a2O(a4) ] dϕ2 , where ∆0 ≡ ∆(a = 0) = x2 − rSx + r2q/2. Upon fully neglecting the spin, a = 0, the above expression 60 4.4. A ROTATING BLACK HOLE SOLUTION recovers the spherically symmetric solutions described in Sec. 4.2 and given by the line element (4.5). Before concluding this section, for completeness, we write the line element (4.114) in Eddington- Finkelstein coordinates given by the transformations dv = dt+ (x2+a2) ∆ dx and dφ = dϕ+ adx/∆: ds2 = − ( 1− f + ϵκ2ρq (∆ + a2 sin2 θ) Σ ) dv2 + 2 ( 1 + ϵκ2ρq ) dvdx (4.117) − 2a sin2 θ ( f − ϵκ2ρq (∆ + x2 + a2) Σ ) dvdφ− 2a sin2 θ ( 1 + ϵκ2ρq ) dxdφ + (1− ϵκ2ρq)Σdθ2 + [ ( x2 + a2 + fa2 sin2 θ ) − ϵκ2ρq (x2 + a2)2 + a2∆sin2 θ Σ ] sin2 θdφ2 . It is now time to study the properties of this solution. 4.4.1 Horizons and ergoregions As we did for the static case, let us begin by analyzing the structure of horizons and, since we are in the rotating case, also the ergoregions. Starting by the horizons, this time as compared to what we did in Sec. 4.2 we will get them analytically. In order to do it so, we define the normal vectors to the x =constant hypersurfaces, nµ ≡ gµν∂νx, in this rotating solution obtained in Eq. (4.114) have a norm given by nµnµ = Σ(x− xKN+ )(x− xKN+ ) Σ2 + sr4c , (4.118) where remember r4c = |ϵ|r2q and xKN± are the roots of ∆ which correspond to xKN± = rS ± √ r2S − 4(a2 + r2q/2) 2 . (4.119) From this expression, it is evident that the hypersurfaces x = xKN± have vanishing norm, whereas there is a divergence if the denominator vanishes when s = −1. Since the hypersurface (Σ− sr4c/Σ) = constant has vanishing norm at Σ = r2c when s = −1, it can be shown that at such hypersurface the metric is singular. On the other hand, the two hypersurfaces x = xKN± turn out to be Killing horizons of the combination ξ = χ+ a x2 + a2 m , (4.120) where χ = ∂t and m = ∂ϕ are the Killing vectors associated to time translations and rotations around the symmetry axis, respectively. A glance at the norm of this vector ξ puts forward that Σ = r2c is also a Killing horizon when s = −1: ξµξµ = (x− xKN+ )(x− xKN+ )(Σ2 + sr4c ) (x2 + a2)Σ2 . (4.121) Let us now consider the conditions for the existence of ergoregions. To this end, let us derive the attention to the norm of the time-like Killing vector χ given by χµχµ = (x− xe−)(x− xe+) (x2 + a2 cos2 θ) + sr4c (x− ye−)(x− ye+) (x2 + a2 cos2 θ)3 , (4.122) 61 CHAPTER 4. APPLICATIONS where we have introduced the definitions xe± = rS ± √ r2S − 2r2q − 4a2 cos2 θ 2 , (4.123) ye± = rS ± √ r2S − 2r2q − 4a2(sin2 θ + 1) 2 . (4.124) The norm of this Killing vector will vanish when (x− xe−)(x− xe+) (x2 + a2 cos2 θ) = −sr4c (x− ye−)(x− ye+) (x2 + a2 cos2 θ)3 , (4.125) where, in general, the solutions of such condition are difficult to obtain analytically. However, some useful information can be obtained on simple grounds. Firstly, note that xe− ≤ ye− ≤ ye+ ≤ xe+ , (4.126) and that the sign of the lhs of (4.125) must be the same as that on the right-hand side since the norm of the time-like Killing vector vanishes in the ergoregion. According to this, if s = 0 (the GR case) then the only solutions are x = xe±, which define the usual ergoregions of the Kerr-Newman solution. If s = +1 then the radius of the ergoregion must satisfy xe− ≤ x ≤ ye− and ye+ ≤ x ≤ xe+ , (4.127) while if s = −1 then the solutions satisfy instead x ≤ xe− ye− ≤ x ≤ ye+ and x ≥ xe+ . (4.128) Note that on the rotation axis (θ = 0, π), xe± = ye±, which indicates that the new ergoregions are smooth deformations of those present in the GR solutions. Away from the rotation axis, a perturbative approach can be used to estimate the leading order corrections in some cases of interest. Given that in the rc → 0 limit one recovers the GR solutions xe±, approximated solutions can be obtained in the astrophysical limit, in which rc is much smaller than xe− (assumed to be real). In fact, taking x = xe± + δ±, using Eq.(4.125) it is easy to verify that δ± = ∓ sr4c ((xe±) 2 + a2 cos2 θ)2 (xe± − ye−)(x e ± − ye+) (xe+ − xe−) +O(r8c ) , (4.129) In the slow-rotation limit, this quantity boils down to δ± ≈ ∓s 8a2 sin2 θr4c√ r2S − 2r2q ( r2q + rS ( −rS + √ r2S − 2r2q ))2 +O(a4) , (4.130) This indicates that rotating BHs in extensions of GR can indeed exhibit an external ergoregion different from that expected in GR. Bounds on potential deviations could be obtained by the observation of 62 4.4. A ROTATING BLACK HOLE SOLUTION accretion disks [64], shadows [65], and other means [66], to be explored in future works. In this sense, we point out that by turning back to the exact expression (4.122), one can verify that on the hypersurface Σ = r2c it boils down to χµχµ = 2a2 sin2 θ/r2c , which vanishes on the equatorial plane θ = π/2. 4.4.2 Metric and curvature divergences We have already seen that the hypersurface Σ = r2c leads to metric singularities when s = −1, as it induces the vanishing of the cross-term dvdx in the representation (4.117). Additional problems can be guessed by looking at the zeros of the gθθ component. From Eq.(4.114) there are, in principle, two such cases, Σ ≡ x2 + a2 cos2 θ = 0 ; Σ2 − sr4c = 0 , (4.131) The first one corresponds to the region where the electromagnetic field energy density in GR diverges, see (4.106), that is x = 0 ; θ = π/2 . (4.132) As for the second, it only admits a real solution in the branch s = +1, where it becomes Σ = r2c ↔ x2 + a2 cos2 θ = r2c . (4.133) In the spherically symmetric limit a → 0 leading to the result shown in Sec. 4.2, the s = +1 case that we are considering here was characterized by the vanishing of the area of the two-spheres at x = rc, preventing x from taking smaller values (since, otherwise, the metric signature would change). Now we see that a non-vanishing angular momentum has an important impact on this relation. As is evident, on the equatorial plane (θ = π/2) this condition occurs exactly at x = rc, like in the spherical case. Considering smaller values of x on the plane θ = π/2 would imply a change of signature in the metric and suggests that we should discard the first case in (4.131), Σ = 0, as unphysical for this value of s. On the contrary, for s = −1, there is no signature change and Σ = 0 still represents a physically acceptable region (a non-spherical boundary of infinite area). Further useful information on the relevance of the conditions (4.131) can be extracted by looking at curvature invariants. The Ricci scalar, for instance, has a structure of the form R = Ps(x, θ; a, rc 4, rS , r 2 q) Σ (Σ− r2c ) 3 , (4.134) where Ps(x, θ; a, r 4 c , rS , r 2 q) is a polynomial that vanishes if rc → 0. The denominator of this quantity shows the critical cases identified in (4.131) and puts forward that the Ricci scalar has problems at Σ = r2c regardless of the sign of ϵ. Something similar happens to RµνR µν and RαβµνRα βµν but they have a much more complicated rational structure that does not illuminate the discussion. We thus see that on the hypersurface Σ2 = r4c curvature invariants diverge. This happens, in particular, when x = rc on the equatorial plane θ = π/2. As one moves back from θ = π/2 towards θ → 0 (the rotation axis), the condition Σ = r2c extends the range of x below the spherical limit x = rc. If a 2 > r2c , we may reach x = 0 63 CHAPTER 4. APPLICATIONS at a critical angle θc such that a2 cos2 θc = r2c . (4.135) For angles within 0 ≤ θ < θc and π/2 + θc ≤ θ < π, the condition Σ = r2c cannot be satisfied. In the limit r2c/a 2 ≪ 1, it is clear that the divergence will be located very near the equatorial plane, being completely confined on the plane in the GR limit rc → 0 (yielding the well-known ring singularity). If a2 ≤ r2c then the condition can be satisfied for some 0 < x < rc and any azimuthal angle 0 ≤ θ ≤ π. A deeper analysis in terms (possibly of some extension) of Cartesian Kerr-Schild coordinates could help better understand the geometry of this singular region. However, together with its maximal analytical extension, this is a non-trivial aspect to be explored elsewhere because the general structure of the metric shown in (4.73) indicates that gµν cannot be written in standard Kerr-Schild form even if qµν admits such a decomposition. The conformal factor in front of qµν is just one of the reasons against that possibility. Before concluding this section, it should be noted that the energy density at Σ = r2c is finite in the wormhole case (s = −1) but divergent in the other case (s = +1), as it follows from (4.74), because ρ̃GR ≡ ϵκ2ρq = sr4c/Σ 2 = s when Σ = r2c . The behavior of the angular pressures is just the opposite, as shown by (4.75), being divergent in the wormhole case and finite in the other. On the other hand, a glance at the induced geometry on the surfaces of t and x constant, shows that at Σ = r2c the equatorial and polar circumferences (with θ = π/2 and ϕ = constant, respectively) in the wormhole case have proper length given by leq = 2π √ 2 (r2c + a2) rc and lpol = 2 √ 2πrc , (4.136) which illustrates the lack of spherical symmetry due to the non-vanishing angular momentum. 4.5 Conclusion Let us recap the main results of this chapter. We have started solving a general solution for a static, spherically symmetric spacetime in a RBG. Later on, we have used it to find the solutions of Palatini f(R) and EiBI gravities coupled to the Euler-Heisenberg electrodynamics. The reason to choose them is because their new gravitational dynamics are fed differently by the matter fields: in the f(R) case the new effects in the gravitational sector are oblivious to anything but to the trace of the stress-energy tensor, while in the EiBI case they are sensible to its full content. Both of them were found starting from the Einstein-like representation of the field equations and they suggested that only a particular combination of the gravity and matter sign parameters may yield to non-singular solutions. Therefore, we focused on the characterization of such branches according to the behaviour of the metric functions on the innermost region and their asymptotic limit, the horizon structure, and on the completeness of geodesics. The main conclusions of this analysis is that both settings boil down to the expected RN solution of GR at large distances, whereas the differences arise when the innermost region is reached. Additionally, these solutions do yield to null and time-like geodesically complete spacetimes as long as the aforementioned constraint on the signs of the parameters is met. Indeed, such completeness is a consequence of a wormhole structure that can be identified thanks to the radial behaviour; however, the singularity-regularization is 64 4.5. CONCLUSION achieved via two different mechanisms: • In the f(R) gravity case, which has the same structure of horizons as in the RN solution of GR, the focusing point is pushed to an infinite affine distance, so that null radial geodesics would take an infinite time to get there, while for time-like geodesics or null non-radial geodesics the presence of an infinitely repulsive potential near the throat prevents them to get near it. Thus, only half of the wormhole (which may be covered by two horizons, a single extreme one, or be naked) is available for travel within the x > 0 and x < 0 regions. • In the EiBI gravity, the throat can be reached in finite affine time by some sets of observers, depending on the ratio γ1/γc, which classifies the corresponding configurations. Indeed, they are split as Schw-like, RN-like, or constant curvature spacetime with one or no horizons. For the first case, the wormhole is a one-way structure and any observer departed from one asymptotic region that crosses the event horizon will reach the wormhole throat in an finite affine time. For RN-like configurations, any time-like observer could only get as close to the throat as its energy permits (given the existence of the infinite potential barrier), while null radial geodesics would only require a finite time to get to the throat and cross it. Finally, the constant curvature configurations have a finite maximum of its effective potential, thus allowing any observer whose energy is larger than it to cross the wormhole throat. At the same time, curvature divergences generally appear at the throat (except for this latter case, where curvature scalars are well behaved everywhere). The fact that they are much weaker than their GR counterparts, ∼ (z − zc) −3, together with the lessons from previous research in the topic showing that extended observers are not necessarily destroyed in the transit through such regions [67], raises questions on their true meaning when both the matter fields and the trajectories of idealized observers are well behaved. These two basic mechanisms for such a singularity avoidance are shared by several other theories, and in agreement with these last results of model-independent analysis in spherically symmetric spacetimes [54, 55]. However, a challenge may arise with such results concerning their sustainability when moving to axially symmetric solutions. Additionally, the difficulties that one usually faces when solving the field equations for rotating scenarios in modified gravity is translated to a lack of exact solutions. Therefore, the investigation on such a field in modified theories of gravity demands the development of new strategies and ideas to address the specific challenges that one may find during the process. Particularly, for Ricci-based theories of gravity, a recently discovered correspondence that allows to map the field equations formulated in metric-affine spaces and GR. Via this correspondence any RBG coupled to some matter source can be mapped into GR coupled to the same matter type but described by a different Lagrangian density, in such a way that the solutions of the former can be obtained from the solutions of the latter using purely algebraic transformations. While in previous works the aim was to prove the validation of this method by re-deriving previously known spherically symmetric solutions, in this Thesis we used it to find rotating solutions. We have begun in Sec. 4.3.1 with the static case considering the GR sector coupled to Maxwell electrodynamics (whose solution is the well-known RN) and showed that can be mapped into EiBI gravity coupled to a BI-type electrodynamics. Thus, the last 65 CHAPTER 4. APPLICATIONS step in this section was to derive the same solution by solving directly the field equations as well as with this novel procedure, and then checked that they yield exactly the same result (verifying once again its effectiveness) in a simpler and more straightforward way. After considering the static, spherically symmetric solution, the main highlight of this chapter was to use this framework in order to find the rotating counterpart of the Kerr-Newman solution of GR in the EiBI theory. Doing this required a bit more care as compared with the previous case but, after all, we managed to obtain this solution under a closed exact form, where the new gravitational corrections to the GR solution appear as new terms supressed by ϵ. We have furthermore characterized some basic properties of the resulting rotating BH solution, such as its event horizons, ergoregions, and curvature divergences. This solution has a number of distinctive properties, for example different structure of horizons and ergoregions as compared to that of the Kerr-Newman one and a nontrivial surface with curvature divergences and a non-spherical (exact) wormhole structure. 66 Part II Phenomenology 67 Chapter 5 Optical appearance of compact objects The optical appearance of a BH has recently been a hot topic in the field motivated by the first observation of such an object in the center of the galaxy M87 in 10 April 2019 by the EHT collaboration [4]. In the corresponding image and all the next two results published lately by the same collaboration, we can observe two distinct regions: a dark circular center called the shadow and a light ring involving it produced by the extremely hot accretion disk surrounding the astrophysical object. A BH may be illuminated by two different cases, • A distant source, for example a distant star which is loosely bounded to the compact object. • An accretion disk emitting around it, such a possibility might be given in a tighter binary system or in a dense center of a galaxy. As we did before, we are going to build up the complexity of our framework, where in the first part we consider a distant luminous source which highly simplifies our analysis as we expect the light rays to be concentrated on the critical curves. However, the brightness of the corresponding light rings may be dimmer than those coming from a surrounding accretion disk, but such a case involves a much hard machinery to model and study the corresponding light rings and shadow. 5.1 Asymmetric thin-shell wormholes with two critical curves This is our very first approach to comprehend the optical appearance of compact objects. Indeed, the analysis carried out through this work is rather simple, consisting in a naked WH where the observed luminous rings are originated from photons emitted by external sources and passing close by to the object. Despite its simplicity, it allowed to understand the basic tools to build up more realistic scenarios in later projects such as adding an accretion disk. Thin-shell wormholes whcih are usually constructed assuming the same spacetime patch on both sides of its throat. However, nothing prevents to consider reflection-asymmetric wormholes, namely, different 69 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS solutions on each side [68, 69]. This section is motivated byt he results obteined in Ref. [70], where the authors construct reflection-asymmetric wormholes from two patches of RN spacetimes of different masses and charges. They show that, in addition to the standard shadow on each side, on the side with the lower peak in the effective potential another shadow is present, which is originated from those photons whose impact factor allows them to overcome the maximum of the effective potential and bounce back across the wormhole throat after hitting the potential slope on the other side (for an stability analysis of these solutions see [71]). This phenomenon, known as double shadows, could hint at the existence of new physics beyond the one of GR. The construction of [70], despite being gravity-model-independent, still has the potential problem of the violation of the energy conditions at the shell. However, this difficulty can go away if we frame wormholes within the context of modified gravity, where the existence of additional gravitational corrections, understood as a kind of matter fluid, are able to restore geodesic completeness without unavoidably violating the energy conditions as we showed in Sec. 3.3.3. The aim of this section is to look for qualitative new phenomena that would allow us to identify the exotic object as well as our gravitational theory. In particular, we would like to obtain stable (under radial perturbations), traversable WHs held by positive energy densities that have two photon rings (or critical curves) on each side of the WH throat. One way to achieve such aim is making use of the thin-shell formalism for which two manifolds are surgically cut and joined in a thin-shell. 5.1.1 Thin-shell formalism in Palatini f(R) gravity Let us start by introducing the thin-shell formalism which consists on matching two smooth manifolds M± (with metrics g±µν), bounded by some hypersurfaces Σ± to form a single manifold M = M+ ∩M−. The time-like hypersurface Σ = Σ± is where both manifolds are joined on their respective boundaries. Since there may be discontinuities on several geometric and matter quantities across the hypersurface, one should upgrade the concept of tensorial functions to that of tensorial distributions. The corresponding conditions that the geometry and the matter fields need to satisfy at the matching hypersurface and across it are called junction conditions [72, 73]. The most basic condition is that the spacetime metric components gµν have to be continuous across the hypersurface, that is g+µν |Σ = g−µν |Σ . (5.1) The remaining junction conditions are strongly influenced by the gravitational theory chosen. Therefore, we need to define our specific framework. In this case, we perform our analysis within f(R) gravity defined in Sec. 3.1.2. For the sake of this section, the fact that the field equations of Palatini f(R) gravity are second-order introduces significant changes in the shape of its junction conditions not only with respect to their metric cousins [74], but surprisingly with GR itself. After setting the gravitational theory, it is time to introduce the matter content on M± and on the hypersurface Σ given by some energy-momentum tensors T± µν and Sµν -with S ≡ Sµµ denoting its trace-, respectively. The distributional version of the field equations of Palatini f(R) gravity straightforwardly 70 5.1. ASYMMETRIC THIN-SHELL WORMHOLES WITH TWO CRITICAL CURVES leads to the conditions [75] [T ] = 0 and S = 0 , (5.2) where brackets denote the discontinuity of the inside quantity across Σ, which as an example the above case translates to [T ] = T+|Σ − T−|Σ . Comparing such conditions to the ones in GR one can see that the second one is absent [72, 73]. To proceed with the junction conditions, we shall define the pullback of the first fundamental form on the hypersurface as hµν = gµν − nµnν , where nµ is the unit vector normal to Σ, and the pullback of the second fundamental form (the extrinsic curvature) as K± µν ≡ hρµh σ ν∇± ρ nσ . (5.3) Plugging this result into the singular part of the Palatini field equations on the shell and applying the junction conditions found in Eq.(5.2) one gets − [Kµν ] + 1 3 hµν [K ρ ρ] = κ2 Sµν fRΣ , (5.4) where the subscript Σ denotes evaluated in the shell. Note that this equation departs from its GR counterpart, −[Kµν ] + hµν [K ρ ρ] = κ2Sµν , since the Lagrangian density of GR with a cosmological constant, f(R) = R− 2Λ, is a singular case in this formalism -which justifies why the second condition of (5.2) does not hold in GR, being replaced by [Kρ ρ] = S/2. The last two junction conditions are obtained from the Bianchi identities on the shell. These can be expressed as energy conservation equations DρSρν = −nρhσν [Tρσ] , (5.5) (K+ ρσ +K− ρσ)S ρσ = 2nρnσ[Tρσ]− 3R2 T f 2 RR fR [b2] , (5.6) where RT ≡ dR/dT and b ≡ nµ[∇µT ]. With all these junction conditions we can already built a WH from two surgically joined spacetimes, in particular we are going to focus on (electro-)static, spherically symmetric solution. 5.1.2 Electrovacuum spherically symmetric spacetimes Even though we have defined before the line element for static, spherically symmetric spacetimes, let us write it again in order to differentiate between M±, ds2 = A±(r)dt 2 −B−1 ± (r)dr2 − r2dΩ2 , (5.7) where A±(r), B±(r) are the metric functions on each side and dΩ2 = dθ2 + sin2 θdφ2 is the unit volume element on the two-spheres. On the other hand, the induced metric on the shell is written as ds2Σ = −dτ2 +R 2 (τ)dΩ2 , (5.8) 71 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS and is parameterized in terms of the proper time τ of a comoving observer in Σ, with R(τ) repre- senting its areal radius. Thus, the non-vanishing components of the second fundamental form Ki j = diag (Kτ τ , K θ θ, K θ θ) in Eq.(5.3) are given by the expressions [76] Kτ τ± = ± R 2 τ ( AR±B± −A±BR± ) + 2A±B±Rττ +AR±B 2 ± 2A±B± √ B± +R 2 τ , Kθ θ± = ± √ B± +R 2 τ R , where AR ≡ dA/dR. For the matter content of our shell we assume a perfect fluid of the form Sµν = diag(−σ,P,P), where σ and P are the energy density and pressure, respectively. Remarkably, the second of the junction conditions (5.2) tells us that P = σ/2, which means that the energy content of the shell is entirely determined by its energy density, and therefore no equation of state is required to close the system. The junction condition (5.4) with this matter content yields the equation [Kτ τ ]− [ Kθ θ ] = 3κ2 2fRΣ σ . (5.9) We now call upon the energy conservation equation (5.5), which for the present spherically symmetric case reads −DρS ρ ν = −σ̇ + 2Ṙ R (σ + P) = −nρhσν [Tρσ] , (5.10) where dots mean derivatives with respect to the proper time of the thin shell and on the second equality we have made use of Eq. (5.5). The above equation relates quantities of the fluid defined on the shell with the discontinuity in the energy-momentum tensor across the normal direction to it. Recall that for any electrostatic, spherically symmetric field described by a given NED (defined by the two field invariants of the electromagnetic field), the corresponding energy-momentum tensor can be written as Eq.(3.42). In such a case, one can show that the quantities in the bracket of the rhsright-hand side of Eq.(5.10) vanish on both sides of Σ± [76] and, therefore, also does its discontinuity across Σ. Under such conditions, and recalling that P = σ/2, then Eq.(5.10) integrates as σ = C R 3 , (5.11) where C is an integration constant and nicely relates the energy density of the matter fields with the radius of the shell. This completes our formalism for the analysis of reflection-asymmetric thin-shell WH solutions. In the next two sections we shall seek explicit examples of this construction using electrovacuum spacetimes. 72 5.1. ASYMMETRIC THIN-SHELL WORMHOLES WITH TWO CRITICAL CURVES 5.1.3 Traversable wormholes from surgically joined Reissner-Norström space- times Our target configuration here are a thin-shell WHs supported by matter sources with a positive energy density and stable against small (radial)perturbations. We first note that for any f(R) gravity theory, the fact that the dependence on the curvature goes via the trace of energy-momentum tensor, R = R(T ) (as shown by Eq. (3.22)) implies that in vacuum or traceless tensors (Maxwell), one gets κ2/fRΣ = κ̃2 =constant on the rhs of Eq. (5.4), regardless of the form of the f(R) Lagrangian. Moreover, thanks to this property the first junction condition of (5.2) and (5.5) automatically hold. If we turn now our attention to Eq.(5.9), recalling the property above, and combining with (5.11), one gets (Kτ+ τ −Kθ+ θ )− (Kτ− τ −Kθ− θ ) = 3κ̃2 2R 3C . (5.12) Let us now assume two RN spacetimes, described by their respective masses, (M+,M−), and charges, (Q+, Q−), that is ds2+ = − ( 1− 2M+ r + Q2 + r2 ) dt2 + dr2 1− 2M+ r + Q2 + r2 + r2dΩ2 , ds2− = − ( 1− 2M− r + Q2 − r2 ) dt2 + dr2 1− 2M− r + Q2 − r2 + r2dΩ2 . This turns Eq.(5.12) into γ R 3 = 3M+ + 2Q2 + +R 2 ( RRττ −R 2 τ − 1 ) R 3 √ 1− 2M+ R + Q2 + R 2 +R 2 τ + 3M−R− 2Q2 − +R 2 ( RRττ −R 2 τ − 1 ) R 3 √ 1− 2M− R + Q2 − R 2 +R 2 τ , (5.13) with γ = 3κ̃C 2 , (5.14) and remember that C is an integration constant whereas R refers to the shell radius where the two spacetimes are matched at. By having a first glance at Eq.(5.13), one can note that there is a more appropriated form to study the stability of such equation. In particular, if one assumes that there is an equilibrium configuration when the shell radius is R = R0, then Rτ = 0 there. Therefore, once can investigate the linear stability by isolating Rττ from the above equation and expand around this equilibrium radius to first order, Rττ ≈ C1(R0) + C2(R0)(R−R0) +O(R−R0) 2, (5.15) where C1 and C2 are some complicated functions of the shell radius R0, the parameter γ, the masses and the charge of each side. In order to have an equilibrium configuration, the zeroth order in this expression must be zero, i. e. C1 = 0, while the second must be positive to be stable. This first condition 73 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS allows us to find the expression for the parameter γ as γ = − R 2 0 − 3M−R0 + 2Q2 −√ R 2 0−2M−R0+Q2 − R 2 0 − R 2 0 − 3M+R0 + 2Q2 +√ R 2 0−2M+R0+Q2 + R 2 0 . (5.16) Remember that this parameter is proportional to the energy density as shows Eq.(5.14), therefore the above expression links the energy density to the masses and charges of the solutions as well as to the radius of the shell. As you might have already notice, there are many parameters involved that complicate the lecture of our parameter spaces. For this reason, let us get rid of one mass and normalize all other parameters as follows R = xM− , (5.17) M+ = ξ M− , (5.18) Q2 + = η Q2 − , (5.19) Q2 − = yM2 − , (5.20) Additionally, we can eliminate the dependence with another parameter by making use of the continuity condition of the metric across the shell Eq.(5.1), ξ = 1− y 2x0 (1− η) , (5.21) thus, if one puts all together in Eq. (5.16) such an equation reduces to γ̃ = −x0 4(x0 − 3)x0 + (η + 7)y 2 √ (x0 − 2)x0 + y , (5.22) where γ̃ = γ/M2 − has also suffered a normalization. Since we are interested in WHs supported by positive energy sources, γ̃ > 0, the above expression allows us to classify two regions fulfilling this condition, 2/3 < x0 < 2 and 0 < η < 3x0 − 2 2− x0 , (5.23) x0 ≥ 2 and η > 0 , (5.24) for both regions, the dimensionless charge y has to fulfill the additional condition x0(2− x0) < y < 4x0(3− x0) η + 7 . (5.25) Let us now deal with the stability of these solutions. Introducing the expression of γ̃ into the expansion of Rττ (5.15) leads to the compact differential equation δtt + ω2δ(t) = 0, (5.26) where δ ≡ R − R0 encapsulate the dependence with the shell radius R, τ = tM− is the dimensionless 74 5.1. ASYMMETRIC THIN-SHELL WORMHOLES WITH TWO CRITICAL CURVES γ ˜ >0 y=9/7 y=1 ω>0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x00.0 0.2 0.4 0.6 0.8 1.0 1.2 y (a) Schw-RN solution. (b) RN-RN solution. Figure 5.1: Parameter space of γ̃ and of ω2 > 0 in Eq.(5.22) and (5.28), respectively, as a function of the dimensionless radius of the shell, x0, the charge-to-mass ratio y = Q2 −/M 2 − and also the charge ratio, η (only for the RN-RN). The blue curve corresponds to y = 4x0(3−x0)/(η+7) which is the upper limit of the (blue-shaded) region where the energy density is positive (check Eq.(5.25)). On this boundary curve, we have γ̃ = 0. On the other hand, the orange curve is given by y = x0(2− x0). The portion in contact with the orange shaded region, from 2/3 < x0 < 2, represents the lower limit of γ̃ > 0 and the upper limit of the region ω2 > 0. Thus, it is not possible to have both properties at the same time for the Schw -RN case. Conversely, the right plot has two additional surfaces depicted in green and purple which delimit configurations stable under perturbations and are given by Eq.(5.29). Remarkably, there is a overlapping region between the green and blue surfaces, described by Eq.(5.32), thus leading to stable solutions with positive energy density. variable and the parameter ω2 = − 8 ( 2x20 − 8x0 + 9 ) x20 + 4x0y(η − (η − 7)x0 − 17)− ( (η − 1)2 + 16 ) y2 8x40 ((x0 − 2)x0 + y) . (5.27) The perturbations will have a bounded amplitude if ω2 > 0. This only happens when y < x0(2− x0) , (5.28) a− 2 √ b (η − 1)2 + 16 < y < a+ 2 √ b (η − 1)2 + 16 , (5.29) where we have introduced the following constants a = 2x0 (η(x0 − 1) + 17− 7x0) , (5.30) b = −3η2x40 − 6ηx40 − 19x40 + 14η2x30 + 16ηx30 +34x30 − 17η2x20 + 2ηx20 − 17x0 . (5.31) Our next goal is to inspect if there is an overlap of the regions between the positivity of energy, γ̃ > 0, and the stability of the corresponding solutions, ω2 > 0. To this end, let us start with a simpler non- trivial case which corresponds to Schw-RN WH. This implies that one of the sides will be chargeless, i.e. η = 0, whereas the other one has a charge Q. Once this simplification is made, we have reduced 75 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS γ ˜ >0, ω2>0 x0=1.264 x0=1.680 x0=1.755 x0=0.912 η=0.268 η=3.732 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 η 1.0 1.2 1.4 1.6 1.8 x0 Figure 5.2: Parameter space for the region contained within (5.32) in terms of the charge ratio, η, and of the dimensionless shell radius, x0. All those values falling inside the spaceship-shaped orange region will lead to stable, positive energy configurations. The blue dashed lines will be useful later when illustrating the double shadows. the dimensions of the parameter space from three to two -ratio of masses and one charge-, simplifying our efforts to understand the configurations supported by positive energy sources and stable under radial perturbations. The result is depicted in Fig.5.1a, from which can be easily check that there is no overlap between the two regions of the parameters positiveness, since the black curve y = x0(2− x0) bounds the regions in which each condition is satisfied. Thus, both conditions cannot be met by matching a Schw and RN spacetime for any values of the masses and charge of the solutions. Going back to two matched RN solutions, the parameter space is represented in Fig. 5.1b where the limiting surfaces for each condition is given in terms of y, x0 and η. The orange surface is the same as the orange line depicted in Fig.5.1a but adding the η-dimension and exactly the same happens for the blue blue surface/line being the upper limit of γ̃ > 0. However, the green and purple surfaces only appearing in Fig.5.1b limit a second region where ω > 0 coming from the second condition (5.29). Now, the existence of such an overlapping region can be clearly seen, being the region between the blue and green surface. Moreover, we can reduce one dimension of the parameter space by checking that the dimensionless charge on the overlaping region fulfills a− 2 √ b η2 − 2η + 17 < y < 4x0(3− x0) η + 7 . (5.32) We shall use the above condition to find limits on the parameter space of x0 and η for which stable, positive-energy configurations are found, indeed, these boundaries are plotted in Fig. 5.2, where the values that we choose have to fell inside the orange-shaded region. Afterwards, the limiting values of the charge-to-mass ratio y can be found by substituting these values in the above relations. In order to make the above explanation more transparent, let us cut different x0 − y planes at several η of Fig. 5.1b, some of these cuts are depicted in Fig. 5.3. In this last figure, we cn see the evolution of the different regions with η are shown. In particular, the blue region corresponds to ω2 > 0 while the orange one to γ̃ > 0. Now it is a piece of cake to see that there is a overlap between them (shaded in red) thanks to the new blue domain above the parabola y = x0(2−x0) (compare with Fig. 5.1a) that emerges when η ̸= 0 is not too big. Precisely, in the first figure 5.3a corresponding to η = 0.2, despite that has 76 5.1. ASYMMETRIC THIN-SHELL WORMHOLES WITH TWO CRITICAL CURVES 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x0 y (a) η = 0.2. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x0 y (b) η = 0.6. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x0 y (c) η = 3.2. Figure 5.3: Representation of the domains with ω2 > 0 (shaded light blue regions) and with γ̃ > 0 (shaded orange) for different values of η. Note the emergence of a new region with ω2 > 0 above the parabola y = x0(2−x0). Note that now there is a non-empty intersection with the positive energy region (shaded red) which represents the parameter space with stable positive energy solutions. the new domain above it is not big enough to intersect the orange region. Indeed, as η grows the upper blue patch changes in size and shape and slightly moves from left to right. Moreover, the orange region also changes with η according to (5.25) and, therefore, the overlapping region sweeps as η grows. 5.1.4 Structure of the asymmetric wormhole: horizons and photon spheres We shall now look if it is possible to hold γ̃ > 0, ω2 > 0 while having the shell radius properly placed in order to have traversable WHs with a distinct photon sphere on both sides of Σ. Regarding the horizon radius, since we are dealing with two RN spacetimes, their well-known locations (on M±) are rh± =M± + (M2 ± −Q2 ±) 1/2 , (5.33) where we have only kept the positive sign in front of the square-root because we are just interested on the outermost (event) horizon. This is so, since it will be the one to pop up first when the shell radius is small enough to ruin our aim to find traversable WHs. It is convenient to write such radius in terms of the same dimensionless functions introduced in Eq.(5.17)-(5.20) and (5.19), which yields x−h = 1 + √ 1− y , (5.34) x+h = ξ (1 + √ 1− η y/ξ2) , (5.35) in M− and M+, respectively. Therefore, event horizons are absent in M− when y > 1, and in M+ when y > ξ2/η. Conversely, the photon ring radius is found by dVeff/dr = 0 written in Eq. (3.74) which in our two RN spacetimes, we find r2γ − 3M±rγ + 2Q± = 0 . (5.36) 77 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS Using again our set of normalized variables, these two equations can be solved as x−γ = 3 + √ 9− 8 y 2 , (5.37) x+γ = ξ ( 3 + √ 9− 8η y/ξ2 2 ) . (5.38) Once we have found the positions of all necessary radius, it is the best moment to recall this subsection aim; to look for traversable (horizonless) WHs with distinct photon sphere on each side. With this in mind, our next task is to classify all possible combinations between the locations of the horizons, photon spheres on each side and the shell radius. Let us first study the manifold M−, and combine the horizon equation (5.34) with the photon sphere one (5.37) based on the position of the WH throat. One then finds seven different possibilities: Condition x0 Condition y Horizon Photon sphere 0 < x0 ≤ 1 0 < y ≤ 1 ✓ ✓ 0 < x0 ≤ 1 1 < y ≤ 9 8 x ✓ 0 < x0 < 3 2 y > 1 x ✓ 3 2 < x0 < 2 0 < y < x0(3− x0) 2 x ✓ 1 < x0 < 2 x0(2− x0) < y < 1 x ✓ 2 ≤ x0 < 3 0 < y ≤ x0(3− x0) 2 x ✓ x0 > 3 - x x Table 5.1: Conditions of the shell radius, x0, the charge, y, in order to have (or not) horizon and photon sphere at the at the manifold M−. Note that other combinations of throat radius and the charge-mass ratio will not be part of our target configuration. At this stage, we need to add to the above discussion the previously obtained parameter space for the existence of stable and positive-energy solutions, namely, Eq.(5.32). This region is depicted in Fig. 5.2 in the plane (η, x0). As it can be seen, there is a minimum and maximum absolute values for x0, given by xmin0 ≈ 0.911 and xmax0 ≈ 1.75497. In view of this constraint, from the above list it is clear that only the full case 3, and some parts of cases 2, 4, and 5 have the potential to represent traversable thin-shell WHs having a photon sphere. We now focus our attention upon the side M+ of the WH. In this case, one might expect a similar discussion as in the M− case save by the replacement y → yη/ξ2 and a global factor ξ in the scaling of the corresponding horizon and photon sphere radius (recall Eqs.(5.35) and (5.38)). However, enforcing the constraint (5.21), which allows to eliminate one of these variables in terms of the others and of the shell radius x0, and by requiring that x+h ≤ x0 ≤ x+γ , the complexity of classifying the different regions becomes far more noticeable than before as shows Tab.5.2. Note that for some cases there is an upper bound for the charge ratio namely yL = 2 η x20 + (η − 1)x0 − √ η2 x40 + 2 (η − 1) η x30 (η − 1)2 , (5.39) 78 5.1. ASYMMETRIC THIN-SHELL WORMHOLES WITH TWO CRITICAL CURVES in other words, it is the maximum allowed charge in order to have a horizon in M+. It shall be also noticed that in some cases we may need to surpass this limit so we can find stable configurations hold by non-exotic matter sources. Condition x0 Condition η Condition y 0 < x0 ≤ 1 0 < η ≤ x0 8− 3x0 2x0 − x20 ≤ y ≤ 2x0(3− x0) η + 3 0 < x0 ≤ 1 x0 8− 3x0 < η < − x0 x0 − 2 2x0 − x20 ≤ y ≤ yL 1 < x0 < 8 5 0 < η ≤ x0 8− 3x0 2x0 − x20 ≤ y ≤ 2x0(3− x0) η + 3 1 < x0 < 8 5 x0 8− 3x0 < η < 1 2x0 − x20 ≤ y ≤ yL 1 < x0 < 8 5 1 < η < − x0 x0 − 2 2x0 − x20 ≤ y ≤ yL 8 5 ≤ x0 < 2 0 < η ≤ x0 8− 3x0 2x0 − x20 ≤ y ≤ 2x0(3− x0) η + 3 8 5 ≤ x0 < 2 0 < η ≤ x0 8− 3x0 2x0 − x20 ≤ y ≤ 2x0(3− x0) η + 3 8 5 ≤ x0 < 2 0 < η ≤ x0 8− 3x0 2x0 − x20 ≤ y ≤ 2x0(3− x0) η + 3 x0 = 2 0 < η < 1 0 < y ≤ 4 η + 3 x0 = 2 1 < η < 4 0 < y ≤ 4(η + 1− 2η1/2) (η − 1)2 2 < x0 < 32 13 0 < η ≤ x0 8− 3x0 0 < y ≤ 2x0(3− x0) η + 3 2 < x0 < 32 13 x0 8− 3x0 < η < 4 0 < y ≤ yL 32 13 ≤ x0 < 3 0 < η < 4 0 < y ≤ 2x0(3− x0) η + 3 Table 5.2: Conditions of the shell radius, x0, the charge ratio, η and the charge, y, in order to do not have horizon but photon sphere at the at the manifold M+. Additionally, we have used the definition of Eq. 5.39. For a more visual discussion of the different regions in parameter space that lead to i) stable solutions, ii) stable and positive energy solutions, iii) stable solutions with one or two photon spheres, and iv) stable and positive energy solutions with one or two photon spheres check Fig.5.4. 5.1.5 Double photon sphere Once we have identified the different regions that one may encounter in this reflection-asymmetric WH spacetime, we can now focus on those stable, positive-energy configurations having a shell radius which lies above the event horizon (when present) but below the photon sphere (on each side) in order to characterize their shadows. For concreteness of our analysis, and given the large complexity of the spectrum of solutions depending on model parameters, we shall illustrate our results by a list of the steps followed in this section, 1. Set the charge-to-mass ratio to η = 2 (see Fig. 5.5). 2. For stable solutions hold with positive energy density within this choice, the corresponding dimen- sionless shell radius will have the lower and upper limits given by 1.264 < x0 < 1.680 (see Fig. 79 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x0 y (a) η = 0.6. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x0 y (b) η = 1.0. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x0 y (c) η = 3.2. Figure 5.4: Representation of the parameter space regions that have a photon sphere in a traversable wormhole geometry on M− (shaded green) and an analogous scenario on M+ (shaded gray) for different values of η. Note that for η = 0.6 the shaded gray region has a large overlap with the green one and completely contains the orange region that represents γ̃ > 0. Whenever gray and green overlap, we find a traversable wormhole with two photon spheres. When gray and green overlap on a blue region, those configurations are stable. If they meet on the red region, then they are stable and the thin shell has positive energy density. When η = 1.0, the gray and green shaded regions exactly coincide. Note how the upper blue patch representing positive frequencies changes its shape and location with η, reducing its overlap with the orange region as η grows. Though the green shaded region remains always the same, the gray one has significant changes as η grows. Remarkably, there exist large regions with two photon spheres, and one can always find stable regions with positive energy within this range of η. x0= 1.680x0 = 1.264 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x 0 y Figure 5.5: Same representation as in Fig. 5.4 but for η = 2.0. The relevant points used in the discussion of the double shadow of Sec. 5.1.5 are highlighted. 5.2). 3. Choose a suitable value, in particular we picked x0 = 1.5. 4. Plugging these values in Eq.(5.32), one finds the limits for the charge ratio 0.92 < y < 1. 5. We choose then for convenience the value y = 0.95, which is clearly (see Fig. 5.5) within the desired (red) region of stability and positive-energy density. 6. Finally, the mass ratio for this case is ξ ≈ 1.316 (from Eq.(5.21)). These choices correspond to a RN solution on M− that would have an event horizon, but with the shell radius above it. On the other hand, if we compare this result with the upper limit of the charge given 80 5.1. ASYMMETRIC THIN-SHELL WORMHOLES WITH TWO CRITICAL CURVES Figure 5.6: Euclidean embedding of the reflection-asymmetric RN-RN wormhole of Sec. 5.1.5 on M− (top orange) and M+ (bottom blue), corresponding to the parameters η = 2, x0 = 1.5 and y = 0.95, which via the constraint (5.21) imply ξ ≈ 1.316. ℳ- 246810 0.00 0.02 0.04 0.06 0.08 VeffM- 2 ℳ+ x0 1/bc- 2 1/bc+ 2 2 4 6 8 10 x Figure 5.7: The effective potential Veff = f(r)/r2 on each side of the wormhole (blue curve on M+ and orange on M−) as a function of the dimensionless radius x for the parameters η = 2, x0 = 1.5, y = 0.95 and ξ ≈ 1.316. The vertical black dashed line is the wormhole throat (the location of the shell), x = x0, while the green and purple dotted vertical lines are the photon sphere radius of M− and M+, as given by (5.40) and (5.41), respectively. The horizontal black dashed lines are the maxima of the effective potential of each side and correspond to the critical impact parameters 1/b2c±, as given by Eq.(5.42) after its evaluation for the chosen values of this problem, yielding b+c = 4.969 and b−c = 4.095. in Eq.(5.39), we find yL− = 0.803848. This implies that x+h is imaginary, which means that there are no horizons on M+ regardless on where we place the matching hypersurface. The reflection-asymmetric WH built this way is depicted in Fig. 5.6. Regarding the location of the photon sphere radius, from Eqs.(5.37) and (5.38) we can compute their values for the choices above, which yields x−γ = 2.09161 , (5.40) x+γ = 2.29221 , (5.41) they are larger than the shell radius x0 on both sides, as expected. Recall that the effective potential Veff (r) attains a maximum at these two radii, as shown in Fig. 5.7. With this setup we can analyze the critical parameter bc leading to circular photon orbits of radius xγ using Eq.(3.73). Thus, rewriting this 81 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS Figure 5.8: Pictorial illustration (the colors are exaggerated) of the double shadow as seen from the M+ side using celestial coordinates (α, β). From the M− region one would only see the inner ring. The dark region between the two rings represents the region of impact parameters contained between the critical values of M+ and M− (i.e. b̃c− < b̃c < b̃c+), whereas the dark region within the inner ring corresponds to impact parameters smaller than b̃c−. We have used a gradient of colors to better illustrate the different intensities one would expect in the different regions. equation in terms of our dimensionless variables and evaluating at xγ of each side, one finds b̃c± = xγ±√ f±(xγ) , (5.42) where b̃ ≡ b/M−. As it can be seen from Fig. 5.7, a light ray travelling from M+ to M− will not be able to overcome the potential barrier and cross the throat, if the impact parameter is larger than b̃c+, being deflected at a radius x > x+γ . However, when the impact parameter satisfies b̃c− < b̃c < b̃c+, we are in the region above the maximum of V + eff but below the one of V − eff , then the light rays will pass through the WH throat and will bounce back to M+ due to the larger potential barrier of M−. Therefore, observers on the manifold M+ will be able to see a double photon ring (since there is no event horizon preventing the information to travel back), corresponding to the radii associated to the critical impact parameter on each side, while those observers on M− will see the single shadow of its side, as usual. Finally, for the case b̃c > b̃c+, light rays will not be reflected since they are above the maximum of both potentials, thus the shadow of the compact object in each side will not be totally black due to the presence of photons of the other side. Finally, by substituting the different values of the parameters into Eq.(5.42), the corresponding photon rings can be plotted and an illustration of the effect is shown in Fig. 5.8, where the yellowish thick rings represent the regions with accumulation of photons that an observer from M+ would see. The radius of these light rings are given by the critical impact parameter since when we observe the light rings of black hole, we can assume that the object is far away from us, thus, light rays will reach us almost parallel among themselves; in other words, the photons will be received with a certain impact parameter and, therefore, the light ring radius is given by this parameter. 82 5.2. IMAGING MODEL IMPROVEMENT 5.2 Imaging model improvement The imaging of a BH can be possible due to two different types of illumination, from a distant star loosely bounded, like it was in the previous case, or from an accretion disk surrounding the astrophysical object. The later scenario is more involved, since not only is modelling the accretion disk by itself a tough nut to crack as a consequence of the many branches of physics implicated such as GR and magnetohydrody- namics, but also one needs to precisely know the trajectories of the light rays in the particular spacetime. In this section we aim to fully explain both issues and the assumptions/simplifications that we use. 5.2.1 Ray tracing The first step is to have under control the trajectories followed by photons passing near the compact object, bearing in mind that they will be deflected as a consequence of the gravitational field. Before getting into the mathematics, let us begin by explaining the main concept behind the ray tracing. The fundamental idea is to classify the light rays depending on the number of turns around the object. This information is necessary since, at the end of the day, when an accretion disk is added, the light rays are going to intersect it at most the number of times it turns around the object. This could potentially be additional luminous “boosts”, of course this will depend on the features of the accretion disk. Additionally, as we mentioned before, the optical appearance of a compact object is tightly related with the impact parameter, as the photon rings radius are constrained by the impact parameter. Using this fact, we trace back the trajectories of the light rays arriving to our screen with a certain impact parameter b using the geodesic equation (3.75). With this, we know the total number of turns around the compact object. This procedure is known as ray-tracing. In order to define how we count the number of turns we shall explain the setup of the object-observer system. Let us locate the observer at the asymptotic infinity, so a photon not being deflected at all by the compact object has “turned” around the gravitational source a number n = 1/2 times. Indeed, the total number of orbits made by a single light ray is the (normalized) change in the azimuthal angle, that is, n(b) ≡ ϕ 2π . As a consequence, the number of intersections with the equatorial plane of a particular right ray is [2n]. Finally, recall that as we get close to the critical impact parameter, b ≳ bc, a light ray will have a longer tour around the vicinity of a BH until formally being there forever at the critical value (or until a small perturbation makes it either fall into the object or depart from it). The number of orbits will obviously depend on how close the impact parameter is to the critical one. Under the conditions above, the typical relevant contributions to the total luminosity on the observer’s screen will be provided by three types of trajectories indexed by an integer number m that counts the number of intersections of a particular light ray with the vertical axis, i.e.: m 4 − 1 4 ≤ n < m 2 + 1 4 , (5.43) except for the first case, m = 1, for which the lower limit corresponds to n = 1/2 and remember that n is the normalized change on the azimuthal angle. We shall use this number in order to classify the different 83 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS contributions to the optical appearance1: • Direct emission: it represents light rays that intersect the accretion disk m = 1 times (1/2 ≤ n < 3/4). This is the dominant contribution to the optical appearance of the object, in terms of luminosity and width of the associated ring of radiation. However, it essentially reproduces the accretion disk features rather than those of the background geometry and its critical curve. • Lensed emission: corresponding to light rays intersecting the equatorial plane twice (on its front and its back, respectively), and defined by 3/4 < n ≤ 5/4 (m = 2), being the sub-dominant contribution to the luminosity. • Photon ring emission: corresponding to light rays intersecting the equatorial plane at least thrice, and defined by n > 5/4 (m = 3). • Higher-order emissions: m > 3: typically their addition to the luminosity is utterly negligible (see [78, 79] for a general discussion), as a consequence of the shrinking of their impact parameter range. For this reason, they are usually integrated in the photon ring emission. These modes are much more sensitive to the features of the background geometry than the other emissions. Despite we have said that higher-order emissions are usually omitted, the shape of the effective potential plays a main role on the contribution of these lower-order trajectories being exponentially diminished images as they approach the critical curve [80] in such a way that beyond n ≥ 7/4 all further images add negligibly to the luminosity of the object, so that they all can be accumulated in the mode m = 3 (what the authors of [79] dub as the “photon ring”). However, in Sec. 5.4, the geometry studied is wealthier as compared with the Schw case allowing the higher-order emissions to contribute meaningfully to the total luminosity. In the Schw geometry these lower-order trajectories contribute to exponentially diminished images approaching the critical curve [80] in such a way that beyond n ≥ 7/4 all further images contribute negligibly to the luminosity of the object, so that they all can be accumulated in the mode m = 3. Last but not least, note that for impact parameters b < bc the light ray will also perform a number of half-orbits. These light rays would be emitted close by the central region of the object inside the photon sphere. However, when calculating their trajectories with the ray tracing we will see two different cases: if the object corresponds to a BH, the backtrack of such trajectories intersects the event horizon, while for traversable WH, such trajectories continue their path all the way down to the throat given the absence of event horizon. Though, there could be some light rays that do not intersect the equatorial plane. This is so because such light rays intersect the event horizon of the BH without finding the accretion disk on their path, and therefore the inner shadow [14], b = bis, defines the brightness depression of a BH no matter the emission properties of the geometrically thin accretion disk. However, such an inner shadow may be missing for a horizonless compact object. Examples of this are traversable WHs (i.e. horizonless), 1This notation is slightly different from the (perhaps) more canonical in the community, in which m = 0 is reserved for the direct emission and m = 1, 2, . . . for photon ring images, see e.g. [77]. Given the degeneracy between m numbers and photon ring images in our case, we find it clearer to use m for the number of intersections with the disk, which means that our m is always one unit greater than the usual convention. 84 5.2. IMAGING MODEL IMPROVEMENT in which the inner shadow limit may be defined by the size of its throat (though for b < bis light rays traveling from the other side of the throat could still reach the observer on our side [81]), or by objects without a critical curve such as gravastar/boson star/Proca stars [82, 83], where no multi-ring structure contributing to the direct image is present. In the Appendix A.1, we have explained the part of the Mathematica code used to study the ray tracing of different objects for a Schw BH. 5.2.2 Accretion disk model Up to now, we have analysed the optical appearance only considering the background geometry, however a realistic astrophysical image is also fuelled by the accretion disk around the compact object. Nonetheless, as the reader should already thought, treating the surrounding disk can become a nightmare easily, as its modeling requires the use of general relativistic magneto-hydrodynamic (GRMHD) simulations. Nonetheless, significant progress can also be made on the theoretical front by using analytical models of static accretion disks with a localized emission on a given geodesic starting from a finite-size region of the disk. Even though this last assumption simplifies the problem noticeably, we will add the following further assumptions, • Place the accretion disk on the equatorial plane: the image seen from the observer will be face-on. • Optically thin: on each intersection with the equatorial plane the light ray will pick up additional brightness in a way that largely depends on the particular assumed emission modelling of the disk. • Geometrically thin: produces a infinite sequence of concentric rings from photons that have com- pleted n half-orbits in their approach to the critical curve. • The specific luminosity only depend on the radial coordinate, r. • Neglecting absorption effects. • Emit isotropically , Iemν = I(r), with I being the intensity and ν the emission frequency in the rest frame and monochromatically, jν ∼ ν2. The next step in our analysis is to set an emission profile for I(r) in the effective region of the (infinitely thin) disk. The last two assumptions above are mainly motivated to find an analytic solution of emission of a finite-size disk (neglecting scattering) given by the radiative transfer equation [84] d dλ ( dIν dν3 ) = ( jν ν2 ) − (ναν) ( Iν ν3 ) , (5.44) where Iν is the intensity for a given frequency ν, jν is the emissivity, αν the absorptivity, and quantities inside parenthesis are frame-independent. The resolution of the above equation requires feeding it with precise knowledge of the plasma fluid (e.g. number density, angular momentum, emissivity and absorp- tivity) making up the disk, to be implemented in GRMHD simulations. For the purpose of simulating different stages in the temporal evolution of such an accretion disk, we are modelling such a profile by 85 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS truncating the inner edge of the disk, rie, at different radius. Also, we assume that there the intensity actually takes its maximum value, and smoothly falls off outwards until asymptotic infinity (so that the outer edge of the disk is assumed to be infinitely far away) with a given radial decay. To simplify the analysis of this aspect, typically in the literature different decay profiles for the emission are taken ad hoc depending on how close to the innermost region of the geometry the inner edge of the disk is. Specifically, such models are defined as follows: • Model I: The emission starts at the ISCO for time-like observers, while vanishing in the region internal to it and falling off asymptotically to zero beyond of it, IIem =  1 (r−(risco−1))2 if x ≥ risco 0 if r < risco (5.45) • Model II: The emission has a sharp peak at the critical curve also known as photon ring, (5.52), having a qualitatively similar central and asymptotic behaviour as Model I. This is described by IIIem =  1 (r−(rpr−1))3 if r ≥ rpr 0 if r < rpr (5.46) • Model III: The emission starts right off the event horizon (in the BH case2) or to the throat (in the WH case) and falls off more smoothly to zero than in the previous two cases, IIIIem =  π/2− arctan[r − 5] π/2− arctan[rhor − 5] if r ≥ rhor 0 if r < rhor (5.47) Once the emission profile of the accretion disk is set, we shall proceed to turn our attention to the observed intensity received by the observer. If we assume that the photons emitted reach us directly after being deflected by the compact object, then the observed intensity is the emitted but altered due to the gravitational redshift and the optical properties of the accretion disk. We already know that the former phenomena will affect the emitted frequency; in particular, if the frequency of the photon in the rest frame of the plasma in the disk is given by νe with associated intensity Iνe , then the photon frequency measured by the distant observer will be νo with intensity Iob. To relate both intensities we use the assumption of a geometrically (infinitesimally) thin accretion disk, for which Eq.(5.44) implies that Iν/ν 3 is conserved along a photon’s trajectory. Thus in the spherically symmetric geometry considered in this work Iobν′ = A3/2(r)I(r). On the other hand, the implications of an optically thin disk are less known. The raw idea is that each additional intersections of the trajectories with the accretion disk will contribute to pick up additional luminosities according to the emission profile of the disk. Therefore, the total observed intensity will be the integration over the whole range of received frequencies as Iob = ∫ Iν′dν′ resulting 2From the point of view of the GRMHD simulations relevant for the EHT observations this is the most suitable scenario [85]. 86 5.3. BLACK BOUNCE SOLUTIONS in Iob(b) = ∑ m A2I|r=rm(b) , (5.48) where the so-called “transfer function” rm(b) contains the information about the radius of the disk where a given light ray with impact parameter b will have its mth-intersection with the disk (in the coordinate r). This is the last improvement to our previous model. It is time now to use this machinery in some specific backgrounds by making use of the Mathematica code explained in the Appendix A.2. 5.3 Black Bounce solutions Let us start by considering a static, spherically symmetric solution of the form (4.5). Even though we know BHs are rotating objects, this assumption turns out to be a good approximation since the size and shape of the shadow, as seen by an asymptotic observer, depends very weekly on the spin of the BH in combination with the inclination with respect to the line of sight, with deviations from circularity lying within ∼ 7% for ultra-fast spinning BHs [86]. The first spacetime we are going to consider is the so-called BB which is a uniparametric family of solutions based on the Schw one with a modified radial sector [15]. Despite its simple mathematical structure, it has several intriguing properties: • Smoothly interpolates between the Schw space-time, a family of regular BH solutions and of traversable WH solutions. • Has the same critical parameter as in the Schw solution. • Removes the presence of space-time singularities. • Does not have Cauchy horizons, thus avoiding their associated instability issues [87]. • Can be taken as parameterized deviations from the Schw solution in a theory-agnostic way (for an example where solutions of this type arise as solutions to modified gravity equations, see [57, 88]). Since BHs and traversable WHs are conceptually and operationally two different types of objects, the BB geometry allows to study the light rings and shadows for each object and compare them to that of the Schw solution. The line element of such solution is ds2 = − ( 1− 2M r(x) ) dt2 + dx2( 1− 2M r(x) ) + r2dΩ2 ; r2(x) = x2 + a21 , (5.49) where the radial coordinate x span the entire real line, x ∈ (−∞,+∞), dΩ2 = dθ2 + sin2 θdϕ2 is the line element on the two spheres and a1 is the model parameter. Soon after, a generalized version of BB-type geometry was introduced in [16], which this time interpolates between the Schw solution, a family of two horizon BHs (with an extreme) and a family of traversable WHs with one or two photon spheres. Nonetheless, the critical curves will not coincide anymore with the one of the Schw allowing to 87 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS differentiate them more easily. The line element for this novel BB is given by ds2 = − ( 1− 2Mx2 (x2 + a22) 3/2 ) dt2 + dx2( 1− 2Mx2 (x2+a22) 3/2 ) + r2dΩ2 ; r2(x) = x2 + a22 , (5.50) where this time a2 is its model parameter. In both cases, the behaviour of the radial function guarantees the extensibility of geodesics beyond x = 0. The areal radius is measured by S = 4πr2(x) and, in bouncing geometries such as in WH, the radial function r(x) is bounded by r ≥ rth in a model-dependent way [53]. In these geometries, one has the WH throat located at r2th = a21 and a22, respectively. The most noticeable feature of such geometries is the bounce (hence its name) in the radial function. It is a simple implementation of a WH geometry extending the Schw solution via the replacement x→ r(x), such that in the limit a1, a2 → 0 one has r2(x) ≈ x2. Additionally, in the asymptotic limit, x → ±∞, we also recover the Schw solution for both cases. While for the generalized version, the limit x → 0 tends to a de Sitter core-type behaviour A(x) ≈ 1− 2M a32 x2, a usual mechanism invoked in the literature to prevent the divergence of curvature scalars [89, 90]. In this sense, it is worth pointing out that (5.50) is actually an extension of the well known Bardeen solution [91] by allowing a non-trivial dependence in the radial function r2(x). For the purpose of classifying the different configurations within these families, one needs to calculate first the location of the horizons, since depending on the position of the horizon with respect to the throat, the bounce will be hidden behind an event horizon or not. For the original BB, the horizons are placed at x±h = ± √ 4M2 − a21 , (5.51) where the ± signs refer to the location of the horizon on both sides of the throat. From this equation, it can be easily seen that the bounce will be hidden by an event horizon if a1 < 2M , finding a regular BH geometry, while if a1 > 2M the bounce lies above the would-be horizon and the geometry represents instead a traversable WH solution with its throat located at xth = 0. The case a1 = 2M was argued in [15] to correspond to a non-traversable WH and we shall use it as a limiting case in the transition BH/WH. On the contrary, the horizon expression for the generalized BB case is analytical, but quite cumber- some, and for our purposes we do not need to explicitly write it here. Instead, we just need to know that two horizons arise on each side of the throat (x = 0) as far as the condition 0 < a2 < 4 √ 3 9 M is hold. These two horizons merge into a single one (extreme BH) when the model parameter satisfies a2 = 4 √ 3 9 M , whereas for larger values no horizon arises, i.e., the WH throat uncloaks and one finds a family of traversable configuration. Apart from the position of horizons, there is another main radial distance which plays a role when analyzing the optical appearance of a compact object, the critical curve (for which we shall also reserve the word “photon sphere”). For the first BB, it corresponds to xps = √ 9M2 − a21 → rps = 3M . (5.52) 88 5.3. BLACK BOUNCE SOLUTIONS -6 -4 -2 2 4 6 x 0.01 0.02 0.03 0.04 V(x) (a) a1 = 0 (dashed black), a1 = 3/2 (orange), a1 = 2 (blue), a1 = 5/2 (red) and a1 = 3 (gray). -4 -2 2 4 x 0.5 1.0 1.5 2.0 2.5 V(x) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.04 -0.02 0.00 0.02 0.04 0.06 (b) a2 = 0 (dashed black), a2 = 2/3 (orange), a2 = 4 √ 3 9 (blue), a2 = 6/7 (red), a2 = 2 √ 5 5 (green) and a2 = 1 (purple). Figure 5.9: The effective potential V (x) for both cases with M = 1 as a function of x and different a1,2. Each color depicts an example of all the classified configurations, starting from the Schw solution, BHs with one, two horizons or one degenerated, non-traversable and traversable WHs with one or two photon spheres. Note that only when a1,2 > 0 both sides of this figure are physically connected, since in the a = 0 case, r2 ≈ x2 and because r > 0 then the two regions x ∈ (−∞, 0), x ∈ (0,+∞) are causally disconnected. A remarkable property of the BB family of solutions is that, when (5.52) is introduced in the impact parameter (3.72), it yields the critical impact parameter bc = 3 √ 3M ≈ 5.19615M . This is shown in Fig,5.9a, where all the curves corresponding to different values of a1 have the same maximal value defined by 1/b2c . Therefore, all BB solutions have the same critical impact parameter as the Schw one, no matter if it is a BH or a traversable WH. Note that the condition (5.52) implies that such innermost circular orbits will exist if a1 < 3M . On the other hand, due to the more complicated form of the line element for the extended version of the BB (5.50), let us discuss the presence of photon spheres graphically. In particular, Fig. 5.9b depicts the effective potential for several relevant values of the parameter a2. As it can be seen there, any non-vanishing value of a2 induces strong changes in the shape of the effective potential as compared with the one of the Schw solution. Indeed, the most salient feature of this family of solutions is that the potential attains its absolute maximum value at x = 0, which effectively acts as a photon sphere at some (model-dependent) impact parameter bc 1 = a2. This photon sphere, however, is hidden behind an event horizon if a2 < 4 √ 3 9 M , preventing any light ray with origin below the event horizon to reach an asymptotic observer. On the other hand, a second impact parameter bc 2, whose expression is quite tedious, is present associated to a (local) maximum in this potential provided that a2 ≤ 2 √ 5 5 M , which is always accessible since the event horizon (when present) lies below. In addition, the fact that the maximum values change with a2, means that each BB solution of this type will have its own critical parameters depicted in Fig. 5.10 to clarify this discussion. To summarize, the configurations relevant for shadows (i.e, having a photon sphere) are naturally classified into BHs and WHs depending on their model parameter as: We can now proceed to study the optical appearance of the different families of BB solutions. We have split them depending on the type of object we are dealing with. Starting with the Schw/BHs and, 89 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS Type of object Original BB Novel BB Schwarzschild a1 = 0 a2 = 0 Black hole 0 < a1 < 2M 0 < a2 ≤ 4 √ 3 9 M Wormhole 1 photon sphere 2M < a1 < 3M a2 ≤ 2 √ 5 5 M 2 photon sphere - 4 √ 3 9 M < a2 < 2 √ 5 5 M Table 5.3: Classification of the type of object depending on the model parameters a1 and a2 for the original and generalized BB spacetimes, respectively. Bear in mind that for the BH solutions, the first model has only one horizon, whereas the second one has two or one degenerated (when a2 = 4 √ 3/9M). bc 2 bc 1 Black hole Traversable wormhole 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 a/M b c /M Figure 5.10: The critical impact parameter bc for the second BB model as a function of a2 for the outer curve (bc 2, solid blue) and the inner one (bc 1, dashed) within the range a2 ∈ [0, 1]. In the latter we have depicted in red the case in which both critical curves exist for the same configuration, and in green when only bc 1 is present. afterwords, WHs. Independently of the object, we are going to follow the same procedure, first do the ray tracing in order to classify the photons depending on the number of half-turns with their corresponding impact parameter and, then, add the three different accretion disk models. 5.3.1 Black hole To heat up, let us consider first the BH cases where there is only a photon sphere accessible above the event horizon for both cases. From the above classification, we take the model parameters’ value to be well inside the range of BH (see table 5.3), for example a1 = 3/2 and a2 = 2/3. The next step in our analysis is to integrate the geodesic equation for a bunch of light rays spanning the whole region of impact parameter values. The corresponding trajectories can be therefore classified according to the number of (half-)orbits around the solution as follows: Schwarzschild BB Generalised BB Direct b > 6.15 b > 6.17 b > 6.02 Lensed b ∈ (5.23, 6.15) b ∈ (5.22, 6.17) b ∈ (4.95, 6.02) Photon Ring b ∈ (5.19, 5.23) b ∈ (5.19, 5.22) b ∈ (4.87, 4.95) Lensed b ∈ (5.02, 5.19) b ∈ (5.02, 5.18) b ∈ (4.59, 4.87) Direct b ∈ (2.85, 5.02) b ∈ (2.57, 5.02) b ∈ (2.39, 4.59) Inner Shadow b < 2.85 b < 2.57 b < 2.39 Table 5.4: Impact parameter range for direct, lensed and photon ring emissions for BHs. 90 5.3. BLACK BOUNCE SOLUTIONS -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 Figure 5.11: Ray-tracing of the BH configurations (in units ofM = 1) with a1,2 = 0 (Schwarzschild, left), a1 = 3/2 (one-horizon BH, middle) and a2 = 2/3 (two-horizons BH, right) for a range of relevant values of the impact factor, making use of the radial function r. The observer’s screen is located in the far right side of this plot and the type of emission is defined with respect to the number of intersections with the equatorial plane (vertical line): for b > bc we have direct (green), lensed (orange) and photon ring (red) emissions reaching to a minimum distance from the photon sphere (dashed yellow circumference) before running away, while for b < bc we also have direct (cyan), lensed (purple) and photon ring (blue) emissions. The latter three trajectories intersect the BH horizon (black central circle) after crossing the photon sphere. The bunch of black curves do not intersect the equatorial plane and therefore no emission can come out on them no matter the accretion disk model, therefore corresponding to the inner shadow of the solutions. where we have ordered them from the outermost to the innermost emission. There are several aspects to be underlined in the modifications of the light rays’ impact parameters as compared to the Schw solution. For the first solution the impact parameter range for each trajectory slightly increases, whereas the second one has an overall smaller value of the impact parameter defining the beginning/ending of a given type of emission than the Schw, being more obvious for the lensed and photon ring. This result should not surprise us, since the critical curve of the original BB geometry coincides with the one of the Schw, while the critical impact parameter of extended version changes with the model parameter as showed Fig.5.10. To illustrate this general discussion, the trajectories of a bunch of photons are depicted in Fig.5.11 for b ∈ (0, 10). We point out that the observer’s screen is located at the far right side of this plot in all these cases. In these figures one can see the direct (green), lensed (orange) and photon ring (red) trajectories outside the photon sphere (dashed yellow). In addition, we have plotted the photon ring (blue), lensed (purple), and direct (cyan) emission originated from inside the photon sphere, b < bc, while the black trajectories correspond to the inner shadow, those light rays that do not cross at any time the vertical axis. As we saw on the above classification, the differences are minimal between the first BB and the Schw solution. Alternatively, the second solution (right panel) enlarges a little bit more the lensed emission and, as we discussed, the limits of each type of emission are getting closer to the center. For these reasons, let us omit the original case and only focus on the Schw and the BH of the generalized BB when a2 = 2/3. At this point, we already know which is the impact parameter range for each type of photon trajectory. Despite with Fig. 5.11 we can visually reason why we only classify the null geodesics into these three groups, as we can barely see the photon ring emissions (red and blue), there is a better way to understand it graphically. This is Fig. 5.12 which depict the transfer functions, xm, for the BH (right) as compared to 91 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS 2 4 bc 6 8 10 b/M0 2 4 6 8 10 xm/M (a) Schwarzschild solution. 2 4 bc 6 8 10 b/M0 2 4 6 8 10 xm/M (b) Black hole with a2 = 2/3. Figure 5.12: The first three transfer functions for the direct (blue), lensed (orange) and photon ring (green) emissions. bc denotes the location of the corresponding photon sphere, with the BH cases having the standard single one. The slope of each curve is interpreted as the demagnification factor of the corresponding emission. the Schw solution of GR (left). Remember that the transfer functions xm account for the location of the m-th intersection between the light ray and the vertical axis (i.e., the disk). Therefore, the information one can subtract about this plot is how demagnified the light ring will be by the slope of the transfer function; the steeper it is, the lesser the contribution will be due to the ring width will be continuously shrinking as its thikness depends on the impact parameter range. Bearing this in mind, the direct emission is the largest contribution to the total luminosity by far, and the lensed and photon ring are highly diminished as we expected from the previous section. Finally, it is now time to add the accretion disks. The first model is extended up to the ISCO for time-like observers. Its emitted intensity profile written in Eq. (5.45) is depicted in Fig. 5.13 (left) together with the observed (middle) and the optical appearances with its intensity legend (right) for the BH configuration (bottom) as compared to the one of the Schw solution (top). The fact that the emission starts at the ISCO allows to clearly identify the impact parameter regions on the observed intensity corresponding to (from larger to smaller b’s) a small reproduction of the emission profile and two spikes representing the direct, lensing and photon ring emissions, respectively. This is translated into a clean view of the three kinds of light rings in the optical appearances image (after zooming in a little bit). The direct emission is largely dominating the total luminosity with a broad ring very bright at the inner edge and smoothly fading out for larger impact parameters. This ring encloses a thinner and dimmer ring (the lensed emission) and inside this latter an even thinner photon ring which is barely visible at naked eye. The modifications introduced by the BB solution moderately increase the width and luminosity of the lensed and photon ring emissions, thanks to the enhanced impact parameter region previously discussed. This is more noticeable on the observed intensity profile, where not only the lensed “spike” is thicker, but brighter as compared with the Schw case and among its spike companions. This is transferred to a more intense ring as compared to the Schw. Therefore, the main contribution to the total luminosity in the optical appearance is provided by the direct emission yielding a wide rim, while inner to it we find a brighter lensing ring and in the innermost region the barely visible photon ring. In Model II, depicted in the middle panel of Fig. 5.13, the direct, lensed, and photon ring types are 92 5.3. BLACK BOUNCE SOLUTIONS 2 4 6 8 10 12 14 b 0.2 0.4 0.1 0.3 Iob I (b) 2 4 6 8 10 b 0.05 0.10 0.15 0.20 0.25 Iob II (b) 2 4 6 8 10 12 14 b 0.3 0.1 0.2 0.4 0.5 0.6 Iob III (b) (a) Swcharzschild. 2 4 6 8 10 12 14 b 0.1 0.2 0.3 0.4 Iob I (b) 2 4 6 8 10 b 0.05 0.10 0.15 0.20 Iob II (b) 2 4 6 8 10 12 b 0.1 0.2 0.3 0.4 0.5 0.6 Iob III (b) (b) BB. Figure 5.13: The observed luminosity (top) and the optical appearance (bottom) for the Schw BH and BHs with a2 = 2/3 with an accretion disk based on the Model I written in Eq.(5.45) (left), Model II (5.46) (middle) and Model III (5.47) (right panel), viewed from a face-on orientation. In the emission profiles we have made use of the radial coordinate x, related to the radial function as r2 = x2 + a2 [recall Eq.(5.49)], which for the Schw case reads simply as r2 ≈ x2. The observed profiles and the optical appearance are plotted as functions of the impact parameter. overlapped in the observed intensity as a consequence of the inner edge location of the accretion disk being on the critical curve itself, which enables the direct emission via the gravitational redshift correction to pierce well inside the critical impact factor region and become the dominant contribution there, while for larger impact parameter values the combined lensed and photon ring emissions occurring roughly at the same location produce a large but narrow spike in the observed emission. Indeed, if we zoom in, we see a split between the photon ring (being fainter and closer to the direct peak) and the lensed spikes. After this luminosity boosts, the direct emission dominates again in a fainter way. The net result is that in the optical appearance the lensing and photon rings are superimposed with the direct emission. The lensing ring contribution can be appreciated in this figure, though the one of the photon ring is highly diluted and barely visible. As a comparison with the Schw solution, the overall ring is less brighten but 93 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS thicker. Lastly, Model III is depicted in right panel of Fig. 5.13. Since the inner edge of the disk extends all the way down to the event horizon this translates into a much wider region of luminosity in the observed emission, thanks to the stretching of the direct emission to a larger distance. As a consequence, the photon ring and lensed emissions appear now as two separated but superimposed spikes with the direct emission. Another meaningful feature is the enlargement of the range of lensed emission. At the same time, this discussion is reflected in the optical appearance which shows a much wider region of luminosity with the contributions of the direct, lensed and photon ring emission. However, the second type of light rays encloses a wide ring right on the middle of it, whereas another (dimmer) one right on the inner boundary comes from the photon ring emission. 5.3.2 Traversable wormholes Things get far more interesting in the traversable WH solutions, in particular, when having two photon spheres. As before, we choose values of the model parameters inside the range of our target objects. In this case, they will be a1 = 5/2, a2 = 6/7. The ray tracing procedure allows us to classify the different types of emissions with respect to the impact parameter range as follows: Schw BB Generalised BB Direct b > 6.15 b > 6.64 b > 5.93 Lensed b ∈ (5.23, 6.15) b ∈ (5.27, 6.46) b ∈ (4.70, 5.93) Photon Ring b ∈ (5.19, 5.23) b ∈ (5.19, 5.27) b ∈ (4.59, 4.70) In-outer photon Ring - - b ∈ (3.10, 4.59) In-outer lensed - - b ∈ (1.59, 3.10) In-outer direct - - b ∈ (0.88, 1.59) In-outer lensed - - b ∈ (0.8572, 0.886) Inner photon Ring b ∈ (5.04, 5.19) b ∈ (0.8571, 0.8572) Inner lensed b ∈ (5.02, 5.19) b ∈ (4.59, 5.04) b ∈ (0.857139, 0.857143) Inner direct b ∈ (2.32, 5.02) b < 2.32 b ∈ (0.72, 0.857139) Inner Shadow b < 2.85 b < 2.60 b < 0.72 Table 5.5: Impact parameter range of direct, lensed and photon ring emissions for WHs where we have followed the same strategy as in the BH case. Remember that the novel BB solution has two photon spheres, thus we need to add an additional classification for those light rays laying between the inner and outer critical curves, called inner-outer photon ring/lensed/direct. If we compare the above classification with the BH cases, we see a moderate change. On the one hand, the solution with model parameter a1 = 5/2 still keeps the same critical impact parameter but the range of the photon ring and lensed emission have gained ground to the direct. Conversely, the addition of a new critical curve has completely changed the limits for each type of emission. Again, to illustrate this classification, we have depicted the trajectories obtained by the ray tracing in Fig. 5.14. The left panel corresponds to the first WH solution whereas the other two trajectories above the outer critical impact parameter (middle panel) and below it (right panel) for the second one. Thus, the first panel follows the same pattern explained for the BH, while for the second case, those light rays with b > bc 2 follow a similar pattern as those of the BH case discussed above. Those in the region 94 5.3. BLACK BOUNCE SOLUTIONS -10 -5 0 5 10 -10 -5 0 5 10 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -4 -2 0 2 4 -4 -2 0 2 4 Figure 5.14: Ray-tracing of the traversable WH configurations having one photon sphere, a1 = 5/2, (left) and with two, a2 = 6/7, above (middle) and below (right plot) the outer critical one bc 2 (dashed yellow circumference). On the left and middle plot, the direct (green), lensed (orange) and photon ring (red) trajectories of the outermost part follow the same logic as in the BH case of Fig. 5.11. On the left and right plot, we find for the innermost photon ring (cyan), lensed (purple), and direct (green) trajectories as well as the inner (blank rather than black) shadow (black), which is only potentially accessible to those light rays travelling from the other side of the WH. For the right plot, we also find the inner-outer direct (blue), lensed (orange) and photon ring (red) trajectories. 2 4 bc 6 8 10 b/M0 2 4 6 8 10 xm/M (a) Schwarzschild solution. 2 4 bc 6 8 10 b/M0 5 10 15 20 xm/M (b) Wormhole with a1 = 5/2. bc 1 2 4 bc 2 6 8 10 b/M0 2 4 6 8 10 xm/M (c) Wormhole with a2 = 6/7. Figure 5.15: The three transfer functions for the direct (blue), lensed (orange) and photon ring (green) emissions. bc denotes the location of the corresponding photon sphere. between the two photon spheres, 0.85714 ≈ bc 1 < b < bc 2 ≈ 4.5888, have a frenzied behaviour. This is particularly true for those light rays which hover just slightly below the critical impact parameter b ≲ bc 2, having quite a chaotic pattern of trajectories between the two photon spheres before being finally able to exit the outer one. Let us discuss the differences appearing in the ray tracing; both spacetimes show an enhancement of the range for the outer lensed/photon ring emissions as compared to the BH. This is even more patent for the novel BB case due to the complex interplay between the two photon spheres. On the contrary, for the original BB the inner photon ring and lensed emissions are magnified, whereas for the new BB solution, the lensing/photon ring emissions driven by the inner critical curve are extremely narrow: due to this fact, for the sake of the emission from accretion disks of Sec. 5.2.2 we shall extend the inner region to the photon ring trajectories above to include the full region bis < b < bc 1. The inner shadow is named here as blank since light rays coming from the other side might flow through the WH throat and reach eventually the observer, colouring the central darkness region of the BH case (and also reaching significantly smaller impact factor values), though the detailed analysis of such a scenario goes beyond the scope of this work. 95 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS Before adding the accretion disk, we shall discuss Fig. 5.15, where the transfer function is plotted in terms of the impact parameter. Due to photons can be emitted from the very center of the traversable WH solution, hence the transfer function reaches xm = 0. Apart from this fact, there are more obvious differences. For a1 = 5/2 represented in Fig. 5.15b, it is clearly seen how the lensed emission is largely enhanced but still keeping the same structure as the Schw (Fig.5.15a), meaning that at most we will only see 3 light rings in the optical appearance. Contrasting with these two cases, a2 = 6/7, the transfer functions have a totally different shape. The presence of the second (inner) photon sphere not only largely enhances the window of impact parameters for the lensed and photon ring emissions, but also significantly reduces the slope of both curves in the intermediate region between the two critical curves, as shown in Fig.5.15c. This will supposedly have a larger impact in the total share of the observed luminosity budget of these contributions as compared to the direct one, as we expect to deal at most with 6 different luminous rings (one for each peak), 1 for the direct, 2 for the lensed and 3 for the photon ring emissions. The last step is to add the accretion disk. Fig 5.16 depicts the observed intensity profile (top) and the optical appearances (bottom panels) for Model I (left), Model II (middle) and Model III (right panel) of the accretion disk for the first BB and novel BB WHs. We begin the discussion of the first model where, again, the fact that the inner edge of the disk only extends up to the ISCO for time-like observers allows one to see the expected peaks isolated. For a1 = 5/2, the structure of rings is similar to the one of BHs, but with the lensed ring a little bit thicker, whereas the direct emission has a maximum intensity smaller. Additionally, the photon ring is brighter allowing us to see it without zooming in. For a2 = 6/7, we can clearly see the three additional spikes in the inner part of the observed luminosity, leading to the expected total of 6 spikes. These additional peaks are associated to (starting from the innermost one) the photon ring and lensing emission of the inner photon sphere, and to the photon ring emission of the inner part of the outer photon sphere. Additionally, from Fig. 5.15c, the almost vertical slope of the inner photon sphere implies that the luminosity of the innermost additional light ring is very dimmed (roughly one part in 106) making it almost invisible at naked eye. The net result on the optical appearance is the structure of five clear luminous concentric rings, that even the outer photon ring is visible here, despite the total observed intensity is smaller than the other emissions. The presence of these additional light rings and their associated luminosities are trademarks of this kind of compact object having two critical curves, and could potentially be able to act as smoking guns of the existence of any of such objects in the cosmic zoo. In Model II, second column, we have a similar result to the BH case for the original BB, where the three contributions are overlapped. However, this time the spikes of photon ring and lensed emissions are a bit more separated, even though in the optical appearance are untangled. Indeed, the overall result is a thicker ring starting from a inner region and with a larger decay of the luminosity as compared to the Schw case. Nevertheless, the outer thin ring associated to the photon and lensed emissions is much more brighten now. Conversely, the novel BB solution has again six separated peaks of observed intensity, though now we have some overlapping of different emissions. From the innermost to the outermost, the spike of the new photon ring and the new lensing emissions are clearly seen isolated, followed by 96 5.3. BLACK BOUNCE SOLUTIONS 2 4 6 8 10 12 14 b 0.1 0.2 0.3 0.4 Iob I (b) 2 4 6 8 10 b 0.05 0.10 0.15 0.20 Iob II (b) 2 4 6 8 10 12 b 0.1 0.2 0.3 0.4 0.5 0.6 Iob III (b) (a) BB. 2 4 6 8 10 12 b 0.1 0.2 0.3 Iob I (b) 0 2 4 6 8 10 b0 0.05 0.10 Iob II (b) 2 4 6 8 10 b 0.5 1. 1.5 Iob III (b) (b) Extended BB. Figure 5.16: The observed intensity profile (top) and optical appearance (bottom) of the wormholes a1 = 5/2 as well as a2 = 6/7 (right) for Model I (left), Model II (middle) and Model III (right panel) of the accretion disk. The observed profiles are plotted in terms of the impact parameter, whereas the optical appearance (in celestial coordinates) represents what we would see in a face-on orientation. the dominance and smooth fall off of the direct emission. Next another spike by the new photon ring emissions is superimposed with the direct one before quickly going off, and finally the lensed and photon ring emissions associated to the outer critical curve appear superimposed (at almost the same impact parameter) with the direct emission. In the corresponding optical appearance plot, the innermost photon ring is again suppressed and therefore not visible, while next to it a lensing ring is apparent. Going to larger impact parameters, the direct emission (plus the additional photon ring contribution) yields a wide ring of radiation, and in the outermost part another thin ring made up of the lensing and photon ring emissions of the outer critical curve appears. Note that in this Model II the contribution of the direct emission to the total luminosity is significantly decreased (compared this with the Schw case). 97 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS Lastly, in Model III (Fig. 5.16, right panel), the first issue we should point out is that now the emitted profile is extended up to the WH throat and this will have a clear impact mainly on the case with two critical curves. Still, let us focus on the middle panels (a1 = 5/2), where the total impact parameter range primarily related to the direct emission is enlarged, allowing a broader ring to appear on the predicted observation. Following a comparable behaviour, the photon ring and lensed contributions are enhanced, translating to a wider luminous ring well inside the underlying contribution of the direct emission. On the contrary, things get very messy for the right panels (a2 = 6/7) given the fact that the combination of direct, lensed, and photon ring emissions driven by the two critical curves has a complicated pattern of peaks and intermediate valleys in the observed intensity, which never goes completely to zero. Despite this, one can still identify six different peaks associated to the six contributions. The two innermost peaks are a combination of the direct/lensed/photon ring emissions attaining the maximum of the luminosity, which revels itself in the optical appearance plot as a bright ring of radiation extending all the way down to the inner blank shadow of the WH, bis. As we move away to larger impact factors, the chain of peaks manifests as additional blurred rings in the optical appearance, with intermediate dark-brown regions rather than totally black associated to the non-vanishing valleys of observed intensity. Moreover, since the peaks have a relatively low height except at the innermost region, this manifests as a large contrast between this region (containing up to a ∼ 85% of the total luminosity) and the rest of the optical appearance of the object. Despite of being able to find new fingerprints with a Schw-like spacetimes, only the WH with two photon spheres could clearly be differentiated from the Schw. This fact opens up a question if new BH/WH spacetimes with different geometry in their innermost part could lead to a unique optical feature that make it undoubtedly distinguished. 5.4 The eye of the storm The spacetime analyzed in this section was proposed by Simpson and Visser [17] which were inspired by the regular BH with asymptotically Minkowski core [92]. In the non-rotating limit, this family of configurations is given by the line element non-rotating limit of the family of configurations [17] as ds2 = − ( 1− 2Me−l/r r ) dt2 + 1 1− 2Me−l/r r dr2 + r2dΩ2 , (5.53) where l > 0 is a new scale parameterizing the deviations with respect to the Schw solution and, as expected, these configurations reduce to such spacetime in the asymptotic limit, r → ∞. On the contrary, the deviations do manifest as we approach the center of the solutions, where A(r → 0) → 1 as long as l > 0, which in turn removes the point-like central curvature singularity, replacing it by a Minkowski-like core. This model can be taken in a theory-agnostic approach as a proxy for the features induced by the new behaviour of the metric function3. Horizons are present in this model at the locations rh as far as 3From the inferred size of the shadow’s size of Sgr A* by the EHT Collaboration [93], the authors of [94] argue on the compatibility of this kind of models with such observations. 98 5.4. THE EYE OF THE STORM 1 2 3 4 5 r -0.15 -0.10 -0.05 0.05 0.10 0.15 0.20 V(r) Figure 5.17: The effective potential (in units of M = 1) for l = 0 (Schwarzschild, dashed black), l = 0.7 (blue), l ≈ 0.73576 (red), l = 0.75 (orange), l = 0.8 (purple) and l = 1 (green). the following equation has positive and real roots: rhe l/rh = 2M , (5.54) which holds provided that l/M ≲ 0.73576, corresponding to BHs without curvature singularities. Other- wise in absence of horizons the line element (5.53) represents a regular naked compact objects. Concerning the null geodesics equation, the effective potential defined in (3.71) is depicted in Fig. 5.17, where one can see that it strongly departs from the Schw at the center of the solutions. Indeed, V (r → 0) → +∞ (instead of going to −∞). Moreover, a local maximum and minimum is present provided that l < 0.8. Combining the parameter space for horizon and critical curves, one finds a particularly interesting region within the range 0.73576 ≲ l/M < 0.8 in which we have a kind of regular naked object with an accessible critical curve as well as an anti-photon sphere4, supplemented with a infinite reflective potential barrier at its center. For the sake of our computations, we shall take l = 3/4 (using M = 1, which we shall assume in the rest of the paper). The photon sphere is reached when the impact parameter is (3.73). For the model and parameter’s choice considered in this work, the critical curve/photon sphere are characterized by bc ≈ 3.4263 and rph ≈ 1.5529, which are significantly smaller than their corresponding values in the Schw case (bc = 3 √ 3 ≈ 5.1962 and rph = 3). Due to we are only dealing with the configuration l = 3/4 for this solution, let us go directly to the ray-tracing procedure. This time, as a consequence of the peculiar effective potential, only considering the three usual emissions it is not enough, so we cut our integrations at the m = 7 emission. The corresponding results are depicted in Fig. 5.18, where bunches of curves with the same color belong to the same emission type. We first notice that approaching to the critical curve (depicted as a black dashed circumference in this plot) by above b > bc and by below b < bc, has large differences in the contribution of the corresponding modes to the luminosity of the object. Indeed, for b > bc we neatly 4While the presence of anti-photon spheres can be able to trigger a non-perturbative instability [95–99], the time scales at which this may occur are model-dependent, thus not being a completely unsurmountable argument against the viability of solutions holding them. 99 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS -4 -2 0 2 4 -4 -2 0 2 4 Figure 5.18: Ray-tracing of the direct emission m = 1 (green, cyan and yellow), the lower-order modes m = 2 (orange and blue) and m = 3 (red and brown), and the higher-order modes m = 4 (gray), m = 5 (magenta), m = 6 (purple), m ≥ 7 (pink). The black dashed circumference corresponds to the critical curve, which is the major force driving these trajectories. Beyond the direct emission (m = 1), the modes with m = 2, 3 are relevant for the total luminosity both above and below the critical curve, while those with m = 4, 5, 6 are non-negligible only below of it. Modes with m ≥ 7 are too demagnified and can be dismissed from our subsequent analysis. identify the direct (m = 1) emission in green, followed by the lower-order trajectories m = 2 (orange) and m = 3 (red), while contributions to the total luminosity from higher-order modes with m > 3 will be so demagnified that they can be removed from our analysis. As for b < bc we identify the relevant contributions by m = 1 (yellow, cyan), m = 2 (blue), m = 3 (brown), m = 4 (gray), m = 5 (magenta), m = 6 (purple) and m = 7 (pink). In order to better understand the above discussion, we shall study the transfer function, rm in terms of the impact parameter, depicted in Fig. 5.19 up to the mode m = 7. We find that only the higher-order emissions m = 4, 5, 6 will contribute non-negligibly to it, while those modes with m ≥ 7 can be safely neglected. Intuitively, their enhance on the contribution is a direct consequence of the presence of the potential well inside the critical curve. Clearly, the direct emission, with its almost constant (unit) slope, dominates the image at every impact parameter value, being insensitive to the existence of the critical curve, which is in agreement with the fact that it reflects source-dependent features rather than probing the background geometry. The sequence of lower-order (lensed and photon ring) and higher-order emissions contributes non- negligibly to the luminosity within a (decreasing with higher m) range of impact parameter values whose outer edge is roughly located near the critical curve b = bc. As compared to the usual structure of the Schw solution (see Fig. 5.15a), the domain of existence of the transfer function for lower-order emissions are greatly enhanced in the inner region of the critical curve due to the presence of the effective potential barrier discussed before, giving more weight to the lower-orders and a non-negligible role to the higher- order ones. Indeed, in the Schw case every transfer function associated to both a lower-order as well as 100 5.4. THE EYE OF THE STORM 2 4bc b/M0 2 4 rm/M Figure 5.19: The transfer function rm as a function of b for the trajectories appearing in the ray-tracing plot (5.18): m = 1 (direct emission, blue), m = 2 (green), m = 3 (red), m = 4 (orange), m = 5 (purple), m = 6 (cyan), and m = 7 (pink). In this figure, bc ≈ 3.4263 denotes the location of the critical curve, which is much more reduced than in its Schwarzschild counterpart, bc = 3 √ 3 ≈ 5.1962. a higher-order mode meets its end shortly after crossing the b ≤ bc region due to the effective potential becoming negative, while in the present case the divergence of the potential at the center allows to extend the bottom end of every mode much deeper inside the critical curve. From Fig. 5.19, we can also subtract that some type of emission may lead to two different luminous rings thanks to this new U shape that we already saw in the previous case (see Fig. 5.15c). However, not all the emissions will be responsible of two separated light rings, instead the lensed (green) may be in charge of two additional luminous rings, whereas m > 2 are a different story. In fact, look again at the same figure these later emissions have a distinct separated slope well inside bc; conversely, approaching this critical value, the curves start to get more and more closer, being unable to distinguish m > 3 (orange, purple, cyan and pink), so they will all contribute to a single ”optical” ring. Therefore, one can expect at most 9 luminous rings on the optical appearance, being 1 direct (blue), 2 lensed (green), 2 photon ring (red), 1 for 4 ≤ m ≤ 6 (3 in total) and 1 for all higher-order emissions together. For the objects considered here, their inner shadow will be mainly determined by the position of the inner edge of the disk, since the emission profiles chosen in this work will be truncated (and take their maximum values) at such a surface, smoothly decreasing onwards. This means that when such an edge is allowed to extend close enough to the center of the object, the latter can become shadowless, as shall be verified later on. There is yet another interesting feature worth of comment: those light rays with a small enough impact parameter, b ≲ 0.2 (yellow trajectories in Fig. 5.18), are slightly scattered as if the object lens effect (rather than convergent). The closest to the center, the weaker the deviation is, whereas the maximum divergent lens effect occurs around b ≈ 0.2, and is immediately followed by the standard convergent behaviour for larger impact parameters. This effect can be traced back to the effective energy sources that (within the context of GR) generate this geometry. Indeed, one can verify that the stress energy tensor of this line element can be mimicked by a nonlinear theory of electrodynamics whose (positive) energy density is peaked at rmax = l/4 = 0.1875, and has negative pressures with minima at rrad = l/4 = 0.1875 (radial component) and at rang = 0.13485 (angular components). Typically, in WH 101 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS 2 4 6 8 10 b 0.2 0.4 0.1 0.3 Iob I (b) Figure 5.20: The observed intensity (5.48) as function of the impact parameter b (left) and its associated multi-ring structure (right panel). We have taken for convenience the emission model (5.46) with rie = 5 in order to clearly see the peaks associated to the (eight) narrow rings, and where the outermost curve corresponds to the direct emission, which is much more spread than those of the multi-rings. [53] and BB space-times [100] negative pressures are required in order to prevent the focusing of geodesics via a local repulsive gravity effect it produces, and we similarly interpret this shell of negative pressure as responsible for the divergent lens effect. Since the center has vanishing energy and pressures, the lowest impact factors experience little or no deflection at all. The next step in our analysis is to set the emission profile for I(r) in the effective region of the (infinitely thin) disk. Here, we are interested in analyzing how the lower-order and higher-order ring structures get modified as we move the inner edge of the disk closer to the inner regions of the geometry while keeping the decay of the emission profile fixed. To this end, we consider the three different decay laws and move the inner edge of the disk for each of them in discrete steps, starting from the ISCO for time-like observers, downward until getting to the very center of the object, r = 0, since the absence of an event horizon does not prevent the accreting material to keep falling all the way down (though the infinite slope of the potential will do it so). The goal of such an approach is to compare the evolution in size and luminosity of the lower and higher-order photon rings (created by its respective emissions), for a fixed emission model while also allowing us to compare the optical appearance of the object at fixed inner edge but different emission profiles. We next plug the emission profiles into Eq.(5.48) to feed the light trajectories within our background geometries according to the data of the ray-tracing collected. A first glimpse on the new features of the image, we first use a sample of an observed intensity shown in the left panel of Fig. 5.20 to depict a suitable cut of the associated multi-ring structure in the right panel (see Sec. 5.4.1 below for an interpretation and discussion of the observed structure). Next, we build the sequence of optical appearances of Figs. 5.21a, 5.21b and 5.21c, corresponding to the three emission models (5.45), (5.46), and (5.47), respectively. In these figures we start the simulations by locating the inner edge of the disk, rie, at the ISCO radius, left figure of each of these plots. This choice of the initial surface is motivated, besides its obvious physical relevance, on the grounds that it maximizes the chances of neatly seeing at naked eye the multi-ring structure associated to the 2 ≤ m ≤ 6 modes, as follows from Fig. 5.20. Moving the value of rie from the ISCO downward to the photon sphere, 102 5.4. THE EYE OF THE STORM rph ≈ 1.5528, (middle) and the very center of the solution, r = 0, (bottom right) to produce a total of three figures per emission model. From these plots one can observe the great influence of the choices of both the location of the inner edge of the disk (by comparing the different panels within the same figure) and the decay of the emission profile (by comparing the different figures) in the distribution of luminosity among the different rings. 5.4.1 Multi-ring structure In the Schw solution of GR, the sequence of higher-order photon rings is on a one-to-one correspondence with the modes they are produced from, converging to the critical curve (in spherical models [101]) or to the inner shadow (in both thin [14] and thick models [77]). The luminosity of each photon ring is exponentially suppressed so rings beyond m = 3 can be safely dismissed in their contributions to the optical appearance of the BH. From the plots 5.21, one can notice several new features in the optical appearance of this object as compared to the Schw BH. A general comment looking at the different panels from the figures is that a given emission m may have associated more than one photon ring as we discussed before, and moreover they can be overlapped with each other. The net effect is that, in the images of the three emission profiles above, a maximum of up to eight rings can be separately seen, though this is only possible when the starting surface of the emission (given by the inner edge of the disk) is pushed to sufficiently far away distances (as in Fig. 5.20 (right panel) where rie = 5, or for rie = rISCO represented in all left panels of Fig. 5.21), with a sub-dominant influence on the choice of the decay of the emission (i.e., on the three models above). As rie gets closer to the center of the object, the direct emission can (and actually does) superimpose with one or more of such rings5 to yield fewer but wider rings. Moreover, while their location is almost unchanging with respect of the location of rie, their luminosity can be significantly boosted depending on the degree of superimposition with the direct emission. Indeed, the choice of the emission profile plays the role of redistributing the luminosity of the direct contribution over those of the lower and higher-order rings in such a way that the smoother the decay is, the greater effect on this redistribution. One can easily appreciate it when going from Fig. 5.21a (cubic decay of Eq.(5.45)) to Fig. 5.21b (quadratic decay of model (5.46)), where the luminosity of the lower and higher-order rings of the latter is much more pronounced than in the former due to the larger spread of the luminosity of the direct contribution. This effect is strongly exaggerated in the model III (5.47) depicted in Fig. 5.21c, where the larger spread of the emission profile infuses all the higher-order rings with additional luminosity at all choices of rie. In view of this analysis, the shadow, understood as the dark central area of the image, is bounded by the innermost of the higher-order rings as long as the location of the inner edge of the disk is not located deep enough into the region inside the critical curve (left and middle figures). When this is the case (bottom figures), the shadow will be bounded instead by the (gravitationally redshifted) location 5Note that one can always split these contributions by viewing the distribution of the observed intensity rather than the full image, since in the former one clearly differentiates the peaks associated to every ring, see Fig. 5.20 for an example of this. Furthermore, adding more modes to the ray-tracing would result in additional peaks appearing at random locations due to the gravitational redshift, but their contributions to the total luminosity of the object can be utterly neglected due to their very reduced height. 103 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS (a) Model I. (b) Model II. (c) Model III. Figure 5.21: The optical appearance of the regular naked compact object considered in this section in the emission model I (5.45), II (5.46) and III (5.47) with the location of its inner edge at (from left to right and top to bottom): r = {rISCO, rph, 0}. The structure of lower-order and higher-order photon rings is mostly apparent in the first figure of the sequence, while its superimposition with the direct emission as we move through the sequence produces more complex and hard-to-split contributions of each ring to the image. of the inner edge of the disk, at which the emission profile is truncated. Thus, below some value of rie, as it gets closer to the center of the object, r = 0, the shadow’s size gets diminished. In the limit r → 0 one would arrive to a shadowless naked core (right figure in all these plots) formed by those light rays that have acquired the divergent lens effect besides the little deflection at low enough values of b, though one could however argue that the infinite potential barrier of the effective potential (for both null and time-like trajectories) would actually prevent the particles forming the disk from penetrating down near r = 0. Nonetheless, if we keep pressing upon this possibility, we find that the objects lying at the bottom of these plots contain a central luminous core with a tiny shadow inside and surrounded by a series of diluted photon rings, whose visibility depends strongly on the emission profile, from the almost non-visible rings of Figs. 5.21a and 5.21b to the net sequence of Fig. 5.21c. Let us stress that the main ingredient behind the generation of higher-order ring images is the presence 104 5.5. CONCLUSIONS of both a photon sphere and an infinite potential at the center in the chosen solutions, together with the lack of an event horizon, which we recall it corresponds to the range 0.73576 ≲ l < 0.8. If we had chosen the branch of solutions below the lower limit a horizon would be present, and one would have found the usual set of photon rings of the Schw solution (though modified in their locations and luminosities) converging to its inner shadow, while above the upper limit the photon sphere disappears and one find just the infinite potential deflecting all light rays having no photon rings. 5.5 Conclusions We aimed to find here qualitative new features of compact objects described by modified gravity (or at least in a theory agnostic way). Our first attempt has been to build a reflection asymmetric WH supported by non-violating energy condition sources and stable under (radial) perturbations. These type of WH are constructed using the junction condition formalism. Indeed, such formalism involves restrictions on the geometrical quantities and matter fields at the shell and across it. The main feature for electrovacuum, spherically symmetric spacetimes is that the number of effective degrees on the shell is reduced to just one, characterized solely by its energy density. Using these conditions we have built a thin-shell WHs from surgically joined RN spacetimes on each side of the shell with different masses and charges leading to the asymmetric WH. In the case in which one of the sides has a vanishing charge (surgically joined Schw-RN spacetimes), it is not possible to have (linearly) stable solutions supported by positive energy density matter sources at the shell. However, in the full RN-RN case a non-null overlapping region with stable and positive-energy solutions is found, via some restrictions involving the radius of its throat with the mass and charge of each side. Indeed, after finding the parameter space of such stable, positive-energy solutions, we have carefully analyzed the conditions upon which the radius of the thin-shell is above the event horizon (when present) but below the photon sphere radius of each manifold M±. In particular, this second set of constrains is aimed to find the traversable WH configuration. This property constrains even more the previously found parameter space (of stable under linear perturbations and positive energy sources holding the shell). Despite this reduction, we still do get a region where all these conditions can be all met at the same time. This proves that within the Palatini f(R) framework it is possible to find families of well behaved thin- shell WH geometries having two different photon spheres on each side, allowing for the generation of two luminous ring on one side. These results explicitly implement with a particular framework and specific solutions the detailed gravity-model-independent analysis carried out in Ref.[70] on double shadows from reflection-asymmetric WHs, and supports the feasibility of this approach with well behaved theories and solutions. However, typically the compact object is illuminated by a surrounding accretion disk instead of the fainter stars around (which was the above case). In this way, Sec. 5.2 is entirely devoted to improve the basic ground of the actual optical appearance analysis. Starting with the ray-tracing procedure which allows to classify the different light trajectories according to the number of orbits performed around the solution using the geodesic equation. This classification is usually split into three main contributions 105 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS according to the number of intersections with the equatorial plane: direct (one), lensed (two) and photon ring (three). Typically, it is assumed that other emissions intersecting the equatorial plane more that three times contribute negligibly to the total luminosity. After this analysis, one can proceed with the accretion disk. It is defined as a geometrically thin and with an emitted luminosity described by three different analytical canonical toy models whose inner edge extends up to three relevant surfaces: the ISCO for time-like observers, the critical curve itself, and the event horizon (in the BH case) or the throat (in the WH case). The observed emission corresponds to a gravitational redshift of the emitted one, bearing in mind the different intersections with the disk in the direct/lensed/photon ring trajectories and the additional intensities they pick up depending on the emission profile of the disk. These three models are chosen on the grounds that they simulate different physical scenarios and yield qualitative different observed emissions and their respective optical appearances for a given solution. In Sec. 5.3, we have considered two uniparametric families of (spherically symmetric) extensions of the Schw solution surrounded by an optically thin (i.e. transparent to its own radiation) accretion disk. These so-called BB geometries interpolate between the Schw solution, regular BHs and traversable WHs depending on the parameter’s value. Therefore, it allows to compare the shadows cast by conceptually different objects on an equal footing. This is why we decided to study the different category of compact object for both solutions in order to discuss the differences among them. From the ray-tracing procedure, we can already find some differences between both models. However, for the BH case, such differences are not big enough to potentially distinguish both solutions, since despite that there is a small enlargement of the impact parameter range, it was barely noticeable for the original BB (this is why we omitted adding the accretion disk for this case later on) and for the second family of solutions the main feature was the reduction of the critical impact parameter value allowing to enhance the range of the different type of emissions. Conversely, the WH case was far more intriguing thanks to the possibility of having two critical curves in the generalized solution. This allows to have additional contributions of these lower order emissions as compared with the previous cases that leading to a richer structure of light rays. On the other hand, the original version has a greater enlargement on the impact parameter range for the lensed and photon ring emissions as compared to the BH case. Once the impact parameter regions were neatly identified, we added the accretion disk as the main source of illumination of the BB solutions using the three standard toy models of its emitted intensity. We thus found the corresponding optical appearances, which show significant differences depending on the emission profile. Starting again by the BH case, remember that we only kept the second family of solution, whose critical impact parameter changes with the model parameter are greater. This allows the inner edge of the shadow to get closer to the center. Nevertheless, the main difference as compared with the Schw solution is the moderate enhancement of the width and luminosity of the light rings. On the contrary, the WH case is another story. Whereas the first BB solution keeps exactly the same pattern as before consisting on a sufficient increase in the contributions of the lensed and photon ring emissions to be perceived at naked eye in some of the optical appearances plots, the second solution with the presence of two photon spheres allows the existence of new lensed/photon ring contributions associated to the inner critical curve. This yields three additional peaks in the observed luminosity in such a way that 106 5.5. CONCLUSIONS the modelling of the disk has a much more dramatic impact on the corresponding optical appearances. Indeed, the three images of Fig. 5.16b barely resemble each other. This fact further supports how strongly does the optical appearance of these objects depend on the details of the accretion disk, and how once an accretion model is chosen, the corresponding images deviate dramatically from the BH (Schw) counterparts. The results found here are in agreement with the running discussion on the community regarding the difficulty for testing hints of new Physics given the many elements involved in this analysis, though pointing to some small differences in the shape of the light rings and the shadows of the BB solutions as compared to the Schw one. In view of the changes in the sizes and locations as well as on the contributions to the total luminosity of the different light rings induced by the traversable WH configurations, they are hardly a viable candidate to represent the supermassive object found at the center of the M87 galaxy by the EHT Collaboration, despite the low-resolution of the images available so far. The two-horizon BHs, on the contrary, yield mild modifications to the Schw predictions, which however would difficult their detectability. In Sec. 5.4, we have studied the images generated by a proposal to extend the Kerr BH by a new type of (analytically tractable) family of compact objects. In the non-rotating limit, it degenerates into a modification of the effective potential of the Schw case by introducing an infinite slope at the center driven by a new parameter. We follow the same strategy as before using the ray-tracing procedure for this particular geometry. The first main difference is that higher-order emissions, i.e. those light ray that intersects the vertical axis more than three times, contribute non-negligibly as a consequence of the particular shape of the effective potential. When an optically thin accretion disk is considered to surround the mimicker, we found up to six higher-order rings (besides the two lower-order ones of the Schw solution) superimposed on top of the direct emission main ring. The size and luminosity of each ring is strongly influenced by the choice of the background geometry parameter, the location of the inner edge of the disk, and the assumed decay of the emission intensity profile with distance. However, the above model is too contrived and modifies quantitatively far too much the basic features of the Schw images to represent a viable alternative to canonical BH candidates, particularly under the light of recent observations like those of the EHT (which actually infer a geometrically thick rather than thin accretion disk), but it supports the dire need to build a sort of shadowgraphy, namely, a thorough characterization of the qualitative possible shapes of the effective potential created by different background geometries. The results found here are in agreement with the running discussion on the community regarding the difficulty for testing hints of new Physics given the many elements involved in this analysis and the difficulty to disentangle the contributions from the background geometry and the astrophysics of the disk challenging the testing hints of new Physics. 107 CHAPTER 5. OPTICAL APPEARANCE OF COMPACT OBJECTS 108 Chapter 6 Pre-main Sequence evolution of low-mass stars Stars are less massive astrophysical objects but more usual objects in our neighbourhood. To be more specific, we will turn our attention to the early evolution of LMS, studying from their contracting stage to the start of (stable) thermonuclear reactions at their center. But before that, let us set the main concepts we are going to use throughout this chapter. First of all, the Hertzsprung–Russell (HR) diagram depicts the luminosity, L, in terms of the temperature, T . Since both quantities are large for these type of objects a logarithm scale is used and also one can find typically the horizontal axis reversed, in the sense that cold objects are placed on the right hand side of the diagram, opposed to the usual configuration. The most important feature of such diagram is that depending on the position of the stellar object we can know its evolution state. For example, a star will be located most of its lifetime on the so-called Main Sequence, which is the almost diagonal area (from hot-luminous to cold-darker objects) where all the stars are burning hydrogen. When the newborn baby star approaches the main sequence, it is a luminous but otherwise cold stellar object, which means that they are placed at the rhsright-hand side part of the Main Sequence band and a little bit above it similarly as for later phases (for example, red giants and supergiant stars). The evolutionary path that it follows is called a Hayashi Track (HT) [28], described by a relation between the effective temperature, luminosity, mass, and metallicity (where the last one is responsible for the shape of the curve). This track is depicted as an almost vertical line in the HR diagram, meaning that the proto-star suffers a severe decrease of it luminosity, keeping its exterior temperature almost the same. However, since we are going to deal with a rather simple toy-model description to understand the new features brought by the gravitational corrections of the EiBI gravity, that aspect will not be apparent in our subsequent analysis. A stellar object will end its HT when any of the following processes happens: • Radiative core development: Since the luminosity decreases as the baby star follows the HT but the effective (exterior) temperature remains almost constant; this means that, from the Stefan- 109 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS Boltzmann law, the star is contracting, as its surface shrinks. Therefore, it may happen that the star’s interior becomes radiative, as a consequence of increasing its interior temperature. In such a situation, it will reach a minimum and follow an almost horizontal line before getting to the main sequence, moving to higher effective temperatures, but keeping roughly the same luminosity. This stage of the early evolution is called a Henyey track [102–104], and it will not be studied here; however, in Sec. 6.4 we shall discuss in detail the onset of the radiative core development as a boundary condition of the fully convective star on the main sequence. • Hydrogen ignition: When the central temperature and pressure increase, the conditions present in the stellar core can become sufficiently high to ignite hydrogen and stop further gravitational contraction. If the process is stable, namely, when the energy radiated away through the photosphere is balanced by energy produced by the hydrogen burning in the core, the star has evolved to the next stage of the stellar evolution: the main sequence phase, which we analyze in Sec. 6.3. • Contraction stops at the onset of electronic degeneracy: This process will happen when none of the above ones takes place, that is, the interior of such an object is too cold to start hydrogen burning. Apart from the light elements burnt in the initial phase, those objects do not possess any source of energy production in their cores and, therefore, they will cool down with time when electron degeneracy pressure balances the gravitational contraction [105]. Such objects are called brown dwarfs. 6.1 Non-relativistic stellar structure equations Let us thus head to the non-relativistic limit of the field equations (3.27) above. To this end, we set the following ansatz for the time-independent metric: ds2 = −(1 + 2Φ)dt2 + (1− 2ψ)dx⃗dx⃗ , (6.1) with Φ and ψ depending only on x⃗. As for the stress-energy tensor, since the pressure is generally negligible for non-relativistic stars, we take a relativistic pressureless fluid, Tµν = ρ uµuν , where ρ is the energy density and gµνuµuν = −1 a normalized time-like vector. From Eq.(3.29), we can easily find the components of the corresponding deformation matrix and apply them to Eq.(3.11) to get the metric components of qµν ds2q = − (1 + 2Φ)√ 1 + κ2ϵρ dt2 + (1− 2ψ) √ 1 + κ2ϵρdx⃗dx⃗ . (6.2) From this expression it is clear that the time component depends on the functions x⃗. If one now expands the above metric up to linear order in Φ, ψ, ρ and their derivatives as well as the time component of the Ricci tensor and plugs them into Eq.(3.26), for a static, spherically symmetric spacetime, one finds 1 r2 d dr ( r2 dΦ dr ) = κ2 2 ρ+ κ2ϵ 4 r2 d dr ( r2 dρ dr ) . (6.3) 110 6.1. NON-RELATIVISTIC STELLAR STRUCTURE EQUATIONS This is the modified Poisson equation [19, 106] which can be rewritten in a more well-known way as ∇2Φ = κ2 2 ρ+ κ2ϵ 4 ∇2ρ , (6.4) where Laplacian operators retain their usual meanings in terms of the spacetime metric g, while the second term corresponds to the EiBI correction. Similarly, in this non-relativistic limit, all the RBG theories share a common set of extra pieces to the Poisson equation with theory-dependent coefficients [106]. Using the hydrostatic equation dΦ dr = −ρ−1 dP dr to the above equation and integrating it over the radial coordinate r, dP dr = −κ 2M(r)ρ 8πr2 − κ2ϵ ρ 4 dρ dr , (6.5) where the mass function M(r) is defined as M(r) = ∫ r 0 4πx2ρdx . (6.6) Since EiBI gravity has no additional fields contributing to the asymptotic mass of the spacetime, one can safely define the mass function above in a similar fashion as in GR itself, and the modifications to the shape of such a function will pop up via the corrections to the local energy density by EiBI-effects. The solutions of the hydrostatic equilibrium equations (6.5) and (6.6), equipped with an equation of state [given in Sec. 6.1.1], provide the main ingredient from the gravitational sector for the internal and external features of a LMS (LMS) on its early evolutionary stages. As shall be discussed later, they also contribute to the description of the boundary region between the star’s interior and its photosphere, and have an impact on the photospheric quantities. 6.1.1 Equation of state Throughout this chapter we will work with a rather simplified model of low-mass stars which assumes a fully convective interior, from the center up to the photosphere. Such objects are typically well described by a polytropic equation of state since they can be understood to be held by a mixture of electron degenerated matter and ideal gas, which both of them and their combination can be written as p = Kρ n+1 n , (6.7) where n is the polytropic index, while K is the degenerate parameter that carries the information about the microscopic features of the fluid, such as electron degeneracy, Coulomb force, and ionization. Let us introduce the following dimensionless variables ρ = ρcθ n, P = pcθ n+1, r = rcξ , (6.8) 111 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS where ρc and pc are the star’s central density and pressure, respectively, while rc is defined for convenience as r2c = 2(n+ 1)pc κ2ρ2c = 2(n+ 1)Kρ 1/n−1 c κ2 , (6.9) in order to rewrite the Poisson equation (6.3) in a more appropriate form, the modified Lane-Emden equation reads d dξ { ξ2 dθ dξ [ 1 + αθn−1 ]} = −ξ2θn . (6.10) In this equation the EiBI corrections are encapsulated into the only parameter α = ϵ n 2 r2c , (6.11) which is dimensionless and does not only depend on the polytropic parameters (as happens in GR) but also on the star’s central energy density ρc, as given by Eq.(6.9). This dependence on the local density of the matter sources is a general feature of Palatini theories of gravity (at least for the RBG family), caused by the particular way the matter fields source the new gravitational dynamics [107], and strongly departs from what happens in other theories of gravity, including GR itself. This implies that astrophysical constraints on EiBI gravity (and on the whole RBG family) parameter requires further information on the star’s central density, as we shall see later. Additionally, this parameter of Eq.(6.11) is the one that indeed appears in all the modified equations and, as a consequence, the constrains that one could set to the EiBI parameter by early-track stellar evolution will be unavoidably biased by the guess/assumption on the central density of the object under study. This implies a limitation on the comparison of bounds found for other compact (stellar or not) objects within this theory [19–26], unless more reliable methods are found in the future to fix the star’s central density. The generalized Lane-Emden equation (6.10) can be used to rewrite the main variables involved on the description of a star. Starting with the mass function (6.6) as M = ∫ R 0 4π ρ r2dr = 4πρcr 3 cωn, (6.12) where R⋆ is the total radius and ωn ≡ [ −ξ2 dθ dξ ( 1 + αθn−1 )] ξ=ξR , (6.13) with ξR being the value of ξ at the surface of the polytrope star. Now it is time to find an expression for the star radius, R. To do so, let us start with the definition of rc, R = rc ξR = √ 2K(n+ 1) κ2 ( Mξ3R 4πωnR3 ) 1−n 2n ξR , (6.14) In the last equality we have replaced the expression of ρc coming from Eq. (6.12). Isolating R again from the above equation, one finds, R = γn ( 2K κ2 ) n 3−n M 1−n 3−n , (6.15) 112 6.1. NON-RELATIVISTIC STELLAR STRUCTURE EQUATIONS with γn ≡ (n+ 1) n 3−n (4πωn) n−1 3−n ξR . (6.16) Another fundamental quantity is the central density which is found by introducing the mean density ρ̂ = 3M/(4πR3); after substituting M in Eq. (6.12), one gets the following relation ρ̂ ρc = 3 ξ−3 R [ −ξ2 θn−1 dθ dξ ( θ1−n − α )] ξ=ξR , (6.17) where ξ3R comes from the relation rc = R/ξR. Therefore, the central density is, ρc = δn ( 3M 4πR3 ) , (6.18) with δn = − ξR[ θn−1 dθ dξ (θ1−n − α) ] ξ=ξR . (6.19) Finally, the temperature is T = Kµ NAkB ρ1/nc θ1/n , (6.20) where kB is Boltzmann’s constant, NA the Avogadro number, µ the mean molecular weight and the central temperature is defined as Tc = Kµ kB ρ 1/n c . In general, analytic solutions to the generalized Lane-Emden equation (6.10) are not possible, so one has to resort to a numerical resolution procedure. Since the convective interior of a LMS is suitably described by the polytropic index n = 3/2, let us take such a value from now on. In this case, one can approximate the central behaviour of the solution of the Lane-Emden equation (6.10) by θ(ξ ≈ 0) = 1− ξ2 6(1 + α) ∼ exp ( − ξ2 6(1 + α) ) , (6.21) where the initial conditions θ(0) = 1 and θ′(0) = 0 have been applied. This result will prove its usefulness later on. 6.1.2 Simple photospheric model As already mentioned, the model described above holds up to the photosphere, which is the outer, luminous layer that delimits the star. It is formally defined as the radius for which the so-called optical depth equals the value 2/3 (see e.g. [108]), that is1 τ(r) = ∫ ∞ rph κop ρ dr = 2 3 , (6.22) 1It should be stressed that when one attempts, using the standard tensorial approach, to match an internal fluid with a polytropic equations of state of the form (6.7) to an external vacuum (Schw) solution, the particular way the matter sources the new gravitational corrections in Palatini theories of gravity of both f(R) [109] and EiBI types [19], curvature divergences may potentially appear at such a surface, thus questioning the viability of these theories. However, a re-assessment of this problem using the formalism of tensorial distributions within f(R) gravity [75] shows that the problematic range of polytropic indices get shifted beyond the region of physical interest. Even though a similar analysis for EiBI gravity has not been carried out yet, we expect similar conclusions to be reached there. 113 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS where κop is dubbed as the opacity, a phenomenological quantity playing a key role in the characterization of the star. Later on, we shall use various opacity models, depending on the physical features of the material filling the star. Actually, the photosphere is so close to the surface of the star that its radius, rph, can be well approximated by the star’s radius, R. In order to find a suitable relation between the photospheric quantities, we shall conveniently modify the hydrostatic equilibrium equation (6.5) in the following way p′ = −ρ ( g + κ2ϵρ′ 4 ) , (6.23) where primes indicate radial derivatives, while g is the surface gravity defined as g ≡ κ2M(r) 8πr2 ∼ κ2M 8πR2 = const . (6.24) Taking up to two derivatives in the equation above and using the definition (6.6), one can combine the resulting expressions to find ρ′ = −M(r) 2πr4 , (6.25) so that Eq.(6.23) evaluated in the photospheric region becomes p′ph = −ρg ( 1− ϵ R2 ) , (6.26) where in this equation we have set units κ2 = 8πG. This equation can be integrated with the help of (6.22), providing the photospheric pressure as pph = 2g ( 1− ϵ R2 ) 3κop . (6.27) It is clear now that the most relevant element of the photosphere’s modelling is its opacity. Depending on the physical conditions, mainly contained within the pressure and temperature regimes of the considered stages of the stellar evolution, we shall use different analytical expressions, which approximately reflect how opaque matter is to the electromagnetic radiation. Another main quantity of the photosphere is its temperature, often called “effective temperature”, which is tightly related to the luminosity of such outer layer via the Stefan-Boltzmann equation L = 4πσR2T 4 eff , (6.28) where σ is the Stefan-Boltzmann constant and L the luminosity. Therefore, we shall assume the star to radiate its energy as a black body with a temperature Teff. 114 6.2. HAYASHI TRACKS 6.1.3 Convective instability - modified Schwarzschild criterion Another crucial information in the description of stellar interiors is how the energy is transported through different regions of a star. Since our LMS is modelled by a fully convective sphere enveloped by a radiative photosphere, one needs a formal criterion involving the physical conditions responsible for any of those energy transports. This is given by the so-called Schwarzschild criterion, turning out to be dependent on the underlying theory of gravity as shown in [110]. Therefore, the heat is transported via radiative processes when the temperature gradient is smaller than the adiabatic one ∇rad < ∇ad . (6.29) For the rest of our setup we shall model the photosphere’s matter as an ideal, monoatomic gas, for which it can be shown that the adiabatic gradient has a constant value, ∇ad = 0.4 [108]. On the other hand, the radiative gradient is defined as ∇rad := ( d lnT d lnP ) rad , (6.30) where these operators corresponds to gradients of temperature computed with respect to coordinates of the metric g. To find its form and dependence on EiBI gravity parameter, we need to analyze the radiative heat transport equation, which is given by ∂T ∂m = − 3 64π2a κrcl r4T 3 , (6.31) with κrc being the radiative or/and conductive opacity, l the local luminosity, while a = 7.57×10−15 erg cm3K4 represents the radiation density constant. Combining the above expression with the modified hydrostatic equilibrium equation (6.5) differentiated with respect to the mass, one gets ∂T ∂P = 3κrc l 16πr2aT 3 ( Gm r2 + κ2ϵρ′ 4 )−1 . (6.32) Using this result into the definition (6.30) provides the radiative temperature gradient for EiBI gravity as ∇rad = 3κrc l p 16πr2aT 3 ( 1− ϵ r2 )−1 , (6.33) and therefore, depending on the value of the EiBI parameter ϵ, this modification has a (des-)stabilizing effect, altering a radiative region development. 6.2 Hayashi tracks From this section until the end of the chapter, we restore the speed of light units to ones of the CGS. In what follows, we will now focus on a simple description of the HT in EiBI gravity. Remember that before being settled down on the main sequence, or, in the case of objects with masses below the minimum threshold required to stably ignite sufficiently hydrogen, the proto-star is contracting, thus it is a luminous but rather cold stellar object. At this stage of the evolution they are all fully convective, and we shall 115 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS also assume that their interiors are made of a fully ionized monatomic gas with temperature T and mean molecular weight µ. In such a situation, the equation of state can still be formally recast as polytropic (6.7), p = K̃ T 1+n with K̃ = ( NAkB µ )−(n+1) K−n , (6.34) where we have used the ideal gas law given by ρ = µ p NAkBT . (6.35) It is worth stressing that K appering in Eq. (6.34) depends on the theory of gravity, since it can be expressed with respect to the solution of the modified Lane-Emden equation (6.10) via K = [ 4π ξn+1 R (−θ′n(ξR))n−1 ] 1 n G n+ 1 M1− 1 nR 3 n−1 . (6.36) Remember that the equation of state (6.34) is valid up to the photosphere, above which the energy transport is ruled by radiative processes instead. In that region we shall use a simplified relation for the absorption law, given by the Kramer formula κabs = κ0p iT j . (6.37) For the case of cold stars, whose surface temperatures lie between 3000 ≲ T ≲ 6000K, the surface layer is dominated by H− opacity [108]. When the hydrogen mass fraction is X ≈ 0.7, such opacity is given by κH− = κ0ρ 1 2 T 9 cm2g−1, (6.38) where κ0 ≈ 2.5× 10−31 ( Z 0.02 ) , where the metal mass fraction Z (or metallicity) is an important element in the stellar modelling. Its value is typically taken within the range 0.001 ≲ Z ≲ 0.03 [111]; as an example, the solar metallicity is Z = 0.02. Since the photosphere can be also modelled as an ideal gas, the opacity (6.38) can be expressed as κH− = κgp 1 2 T 8.5 cm2g−1 , (6.39) where we have redefined κg = κ0 ( µ NAkB ) 1 2 ≈ 1.371 × 10−33Zµ 1 2 . If one makes use of the photosphere definition (6.22), the photospheric pressure becomes pph = 2g ( 1− ϵ R2 ) 3κH− . (6.40) 116 6.2. HAYASHI TRACKS Applying the solution of the modified Lane-Emden (6.10) for n = 3/2, the Stefan-Boltzmann law (6.28) as well as the opacity expression (6.39), the photospheric pressure above takes the form pph = 8.11279× 1014 [ M β √ µLT 4.5Z ]2/3 , (6.41) where we have redefined the brackets appearing in Eq.(6.40) as β = 1− 2α 35/3δ2/3ω2/3 , (6.42) using the definition of ρc and α written in Eq. (6.18) and (6.11) respectively. In order to make the lecture more pleasant, we have removed from now on the sub-indices n appearing in the (6.42) and we will understand them as their values for n = 3/2. The obtained photospheric pressure must be matched to the one of ideal gas given by the relation (6.34) evaluated on the photosphere. The latter yields, after using the Stafan-Boltzmann law (6.28), the effective temperature as Teff = 9.1960× 10−6 ( µ5L3/2Mp2ph −θ′ξ5R )1/11 , (6.43) which after using the derived photospheric pressure (6.41) finally results Tph = 2482.10µ 13 51 ( L L⊙ ) 1 102 ( M M⊙ ) 7 51 β 4 51 Z 4 51 ( √ −θ′ξ5R) 1 17 , (6.44) where we have re-scaled both mass and luminosity to their solar values, {M⊙, L⊙}. Let us notice that the numerical value in the above expression is too low; it should be almost a twofold larger. The reason of this reduced value lies in the simplifications we have made, mainly related to the photospheric modelling. Notwithstanding, this analytical formula allows us to track down the modifications introduced by EiBI gravity. Therefore, for a given star with mass M , uniform mean molecular weight µ, and metallicity Z, the above expression gives the corresponding HT. These almost vertical lines are evolutionary tracks of infant stars with masses supposedly below ∼ 0.5M⊙, though such a limiting mass also depends on the theory of gravity. Its shape and position on the HR diagram does not only depend on the metallicity, but also on the theory of gravity, which in the present case is encapsulated in the parameter β appearing in Eq.(6.42) and through the solutions of the extended Lane-Emden equation (6.10). Note that, in addition to the phenomenological parameters above, and as discussed in the previous section, in EiBI gravity one has to face the fundamental feature of this theory regarding that the new gravitational dynamics includes a contribution from the energy density (here encapsulated in the star’s central density), as Eq.(6.11) tells us. Therefore, the usefulness of the crude modelling employed here, rather than providing a robust and reliable model of the early evolution of these stars, is to seek the expected modifications of these gravitational theories to GR predictions on an equal-footing. Besides the necessary amendments and upgrades to this model, more reliable predictions on the physics of these stars would require to go to full numerical simulations (using e.g. MESA [112]). Therefore, for the sake of our computations, we shall take in what remains of the paper a reference 117 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS Figure 6.1: The piece of the HR diagram representing shifted Hayashi tracks by the modifications intro- duced by EiBI gravity model in logarithmic scale. The curves are given by the equation (6.44) taking M/M⊙ = 1/2, for some chosen values of the parameter α defined in Eq.(6.11) as compared to the GR/Newtonian curve, α = 0. central density of ρc ≈ 103gr/cm3 . (6.45) According to the discussion above, bounds on α (or, for the sake of the argument, on β via (6.42) would be translated on bounds on ϵ using Eq.(6.11) (after choosing a polytropic index n and constant K). Such uncertainties on phenomenological quantities render such bounds on ϵ as much more unreliable than those obtained, for instance, in particle physics experiments [61]. Consequently, we shall refrain ourselves from getting to any deep conclusion on the practical consequences from observations of the analysis below. Having said this, as presented in Fig. 6.1, EiBI gravity shifts the tracks in opposite ways depending on the sign of the theory’s parameter ϵ, either in the direction of the forbidden zone (for ϵ > 0), which lies in the region of lower temperatures, or against it (for ϵ < 0), the size of such corrections encapsulated on the size of α in this plot. Let us also point out that in this plot we have chosen a value of α = 0.1, since this is the upper bound compatible with the minimum main sequence mass observations, which we describe in the next section. 6.3 Minimum main sequence mass The Minimum Main Sequence Mass (MMSM) is the minimal mass required by a star to ignite sufficiently stable thermonuclear reactions in its interior to compensate photospheric energy losses. In other words, when a protostar is contracting, its core temperature is increasing, thus this process can be halted by the pressure of electron degeneracy or by thermonuclear reactions. Depending on its mass, ignition will happen before the degeneracy or viceversa deciding the fate of the stellar object. If the central temperature is high enough to ignite thermonuclear reactions before the onset of degeneracy stops the contraction, the proto stellar object becomes a star. 118 6.3. MINIMUM MAIN SEQUENCE MASS For the case of low-mass stars, even though the central temperature is sufficient to start the p-p chain, it is not enough to complete it. The thermonuclear rates depend mainly on the temperature and density of the star, in such a way that the energy generation rate can be well approximated by power laws of the form (see [105] for details) ϵ̇pp = ϵ̇c ( T Tc )s( ρ ρc )u−1 and ϵ̇c = ϵ̇0 T s c ρ u−1 c ergs g−1s−1 , (6.46) where the two exponents can be phenomenologically fitted as s ≈ 6.31 and u ≈ 2.28 at the transition mass of the core, while ϵ̇0 is obtained phenomenologically from the sun reactions. Therefore, the corresponding luminosity of the hydrogen burning is found as Lpp = ∫ ϵ̇pp dM = 4πϵ̇cr 3 cρc ∫ ξR 0 θn(u+ 2 3 s) ξ2dξ , (6.47) where we have used the fact that (T/Tc) = (ρ/ρc) 2/3 along the adiabatic core. The last integral can be easily computed by using the approximation (6.21) and the definition (6.12), which yields the result Lpp = 6 √ 3π(α+ 1)3 ω3/2(2s+ 3u)3/2 ϵ̇cM . (6.48) In this approximation, we have considered that most of the hydrogen burning will be produced in a region near the star’s core. Additionally, we assume that the fraction of hydrogen in a high-mass brown dwarf is of 75%, and that the number of barions per electron can be approximated to µe = 1.143, besides setting the following degenerate polytropic constant K = (3π2)2/3ℏ 5mem 5/3 H µ 5/3 e ( 1 + αd η ) , (6.49) where ℏ is the reduced Plank constant, me is the electron mass and mH is the proton mass. Then, the luminosity (6.48) can be recast as Lpp = 1.54× 107L⊙ δ5.49 (1 + α)3/2 γ16.46ω M11.97 −1 η10.15 (αd + η) 16.46 , (6.50) where we have defined here, by convenience, M−1 = M/(0.1M⊙). In order to find the MMSM, we recall that it is the minimal mass a star can have to start the p-p chain, which means that the radiation pressure counters the gravitational collapse. If one translates the latter argument into luminosities instead of pressures, the hydrogen burning luminosity has to be the same to the photospheric. For the purpose of computing the latter, we take Eq.(6.40) as well as (6.42) and assume again that the components of the stellar atmosphere behave as ideal gas, that is ρphkBTph µmH = 2g β 3κop . (6.51) This equation will allow us to get a relation between ρph and M , where the dependency on the mass is encapsulated in the surface gravity g defined in Eq.(6.24). However, it does not only depend on the 119 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS desired quantity but also on the radius. For this reason, making use of definition of R in Eq. (6.15), the surface gravity becomes, g = G3M5/3 γ2K2 . (6.52) Moreover, the photospheric temperature can be found from the matching of the specific entropies of the gas/metallic phases there, which yields [105] Tph = 1.8× 106ρ0.42ph η1.545 . (6.53) Replacing the above two equations into (6.51), we find ρph = 2.957× 10−5 η 1.09G2.11M1.17 (γK) 1.41 ( µmHβ kBκop )0.70 . (6.54) Since, we are looking for the phtosopheric luminosity which mainly depends on the effective temperature (from the Stefan-Boltzmann law), let us insert the result above back to Eq.(6.53), so that Tph = 2.254× 104 G0.89M0.49 η1.09 (γK) 0.59 ( µmHβ kBκop )0.30 . (6.55) Therefore, the photospheric luminosity explicitly given by Lph = 4πR2σT 4 ph, can be expressed in terms of the star mass, M , as follows Lph = 0.534L⊙ M1.31 −1 β1.18 η3.99γ0.37(αd + η)0.37κ1.18−2 , (6.56) where we have defined the quantity κ−2 = κop/(10 −2cm2g−1). Finally, equalling the hydrogen burning luminosity (6.50) with the photospheric one (6.56), we find the following expression MMMSM −1 = 0.227 γ1.51ω0.09(αd + η)1.51β0.11 (α+ 1)0.14δ0.51η1.33κ0.11−2 , (6.57) where the EiBI dependences enter in this expression both via the coefficient α in Eqs.(6.11) and Eq.(6.42) and via the parameters {ω, γ, δ} obtained from the resolution of the modified Lane-Emden equation (6.10). This is the MMSM for EiBI gravity under the assumptions and simplifications above. In order to compute it for different values of the EiBI parameter, the main obstacle here is the fact that α depends on the central density of the star, as we already saw in the previous section. Therefore, for the sake of our calculations we shall take the same value for the central density (6.45), which allow us to compute the MMSM and thus set bounds on the size of α as coming from observational constraints. In Table 6.1 we actually compute the set of {γ, ω, δ} values for several choices of the parameter α in order to find the corresponding MMSM. For α = 0 (GR case) we get MMMSM ≈ 0.084M⊙, which is somewhat halfway between other analytical calculations [105] and the results of numerical simulations [113]. For non-vanishing values of α, this table shows that for positive (negative) α the MMSM is larger (smaller). Thus, the positive branch of α is the most interesting one for our purposes, since it allows us to constrain its size via comparison with the observations of the less massive main-sequence stars ever observed, 120 6.3. MINIMUM MAIN SEQUENCE MASS 0.02 0.04 0.06 0.08 0.10 α 0.082 0.084 0.086 0.088 0.090 0.092 MMSM/M Figure 6.2: The evolution of the MMSM (in units of solar masses) with the parameter α. The range in this plot goes from α ∈ (−0.003, 0.10), with the lower bound given by a well-defined solution of the extended Lane-Emden equation (6.10). which corresponds to the 0.0930 ± 0.0008M⊙ of the M-dwarf star G1 866C [114]. This way, in Fig. 6.2 we numerically depict the evolution of the MMSM with α > 0. Our results within our simplified model points that near values of the parameter α ≳ 0.1 the model is likely to run into conflict with observations. Therefore, setting a bound to the combination of the EiBI parameter and the star’s central density, the latter to be estimated by other means. On the other hand, in the negative branch we run into a problem related to the fact that when the parameter reaches α ≲ −0.003 there are non-physical solutions, since below that value the sign would change in the bracket of the extended Lane-Emden equation (6.10). Let us however note that, due to the same reasons stated above, this feature does depend on the star’s central density, hence for a less dense or a denser core we would deal with different singular values of the parameter and, thus, we do not extract any conclusion on the limit of validity of this branch within the formalism presented here. α γ3/2 ω3/2 δ3/2 MMSM/M⊙ 0.100 2.49 3.01 5.74 0.0933 0.010 2.37 2.74 5.96 0.0852 0.001 2.36 2.72 5.99 0.0845 0 (GR) 2.36 2.71 5.99 0.0844 -0.001 2.36 2.71 5.99 0.0843 -0.003 2.35 2.69 6.03 0.0837 Table 6.1: The MMSM (in units of solar masses) computed with Eq.(6.57) for several values of the parameter α defined in (6.11), including the intermediate values of the parameters {γ, ω, δ} obtained from the resolution of the extended Lane-Emden equation (6.10). 121 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS 6.4 Fully convective stars on the main sequence We will now focus on the final stage of the HT. Remember that during this evolutionary phase the proto-star is fully convective, so it might happen that the inner temperature increases enough to satisfy the conditions for radiative core development and, therefore, the object can have much more complex structure than the one we have considered. Since the star can either enter the Henyey evolutionary phase represented by the almost horizontal line to the main sequence lines, or it can stop contracting on the onset of nuclear processes in the core balancing the gravitational attraction. In the last situation, which is our concern now, the star begins its main sequence evolutionary phase, being still fully convective. Using the previous results found in this paper, we can obtain the Maximum Fully Convective Mass (MFCM) of a star on the main sequence. For the purpose of determining when the radiative processes take over in the core, one needs to analyze the Schwarzschild criterion, which tells us that the radiative processes start when ∇ad = ∇rad. In our simplified modelling above we have assumed that the star is made of an ideal, monoatomic gas, providing ∇ad = 0.4, while the radiative temperature gradient was already derived in Eq.(6.33). Applying the homology law, together with Eq.(6.34), we can express the latter as ∇rad = 5.21177× 1069 Lξ5(−θ′)κo µ5M2R3T 3.5β , (6.58) where L is the local luminosity (here evaluated at the core). Substituting the central temperature from (6.20) and subsequently the Stefan-Boltzmann law, the above expression yields ∇rad = 8.99× 10−13 ( L L⊙ )1.25 ξ10.83 (−θ′)2.17 κ0 δ2.33µ8.5M5.5 −1T . (6.59) Equaling this result with ∇ad = 0.4 one finds the maximum luminosity for a fully convective star on the onset of the radiative core development to be L = 2.0827× 109L⊙ β0.8δ1.87µ6.8T 0.8 ξ8.67 (−θ′)1.73 κ0.80 M4.4 −1 . (6.60) Now, by equaling this luminosity to the one of the hydrogen burning given by Eq.(6.56) yields the MFCM M−1 = 1.91 β0.11γ2.17µ0.90T 0.11ω0.13(αd + η)2.17 (α+ 1) 0.20 δ0.48η1.34ξ1.14 (−θ′)0.23 κ0.110 , (6.61) where similar comments as on the sources of EiBI corrections of the MMSM above apply here. To make quantitative estimates of this mass, let us first assume the usual values for LMS as αd = 4.82, η = 9.4, µ = 0.618 and Teff = 4000K. In addition, we have to set the opacity, keeping the Kramers’ form written in Eq.(6.37) with i = 1 and j = −4.5; this choice contains the total bound-free and the free-free opacities (see e.g. [108] for details) κbf0 ≈ 4× 1025µ Z(1 +X) NAkB cm2g−1, (6.62) 122 6.5. CONCLUSIONS Figure 6.3: The dependence of the Maximum Fully Convective Mass for both opacity models (6.62) and (6.63), on the parameter α ∈ (−0.003, 0.10). κff0 ≈ 4× 1022µ (X + Y )(1 +X) NAkB cm2g−1. (6.63) Once everything is settled, in Table 6.2 we calculate the MFCM for several values of the parameter α for both opacity models. Similarly as with the MMSM above, the MFCM increases (decreases) with positive (negative) gravitational parameter (note that the parameters {ω, γ, δ} are those appearing in Table 6.1). α Mbf/M⊙ Mff/M⊙ 0.100 0.108 0.225 0.010 0.0994 0.207 0.001 0.0985 0.205 0.000 0.0984 0.205 -0.001 0.0983 0.204 -0.003 0.0976 0.203 Table 6.2: Numerical values for the MFCM (in solar mass units), using the total bound-free and the free-free opacities, defined in Eq. (6.62) and (6.63), respectively, for different values of the composite EiBI parameter α appearing in Eq.(6.11). In addition, in Fig. 6.3 we depict the evolution with α of the two MFCM masses, corresponding to each opacity. In both table and plot it is clearly seen that the choice of the opacity model significantly affects (roughly a factor two) the value of the MFCM. As for the negative branch, we find the same feature as with the MMSM, namely, the fact that for α ≲ −0.003 the extended Lane-Emden equation (6.10) fails to provide a non-singular solution and, as such, those values are disregarded in our analysis of the MFCM. 6.5 Conclusions We have discussed several aspects of the early evolutionary phases of low-mass stars within the EiBI gravity. Its modifications on the stellar structure equations of non-relativistic stars are manifested as an extra piece to the Poisson equation transferring it to the Lane-Emden equation (its polytropic dimen- sionless version). As a consequence of their involvement on other fundamental equations and features of the stars such as the photospheric modeling and the criterion for convective instability, they will be 123 CHAPTER 6. PRE-MAIN SEQUENCE EVOLUTION OF LOW-MASS STARS also deviated from the standard results of GR. Bearing this in mind, we have investigated three different features on the early evolution of a LMS. The first feature deals with the effective temperature-luminosity relations in the evolutionary path of a proto-star, the so-called Hayashi tracks. We have shown that positive (negative) values of the EiBI parameter shift the corresponding Hayashi track in the sense of larger (smaller) effective temperature for a fixed luminosity, this might help us to constrain the gravitational parameter as no Hayashi track can be inside the so-called Hayashi forbidden zone. The second feature is the minimum required mass for a star to stably burn enough hydrogen to compensate photospheric losses, allowing it to belong to the main sequence. In this case, positive (negative) values of the EiBI parameter yield larger (smaller) minimum main-sequence masses, the former allowing to place constraints on the parameter α appearing in Eq.(6.11) via comparison with the lowest-mass main-sequence stars every observed. This poses a difficulty for this theory, since such constraints act upon a combination of the EiBI parameter and the star’s central density. This dependence of the stellar features not only on global quantities (such as the total mass) but also on local ones is a common feature of the RBG family. The immediate consequence is the difficulty to set strong bounds on the underlying parameter of this theory - ϵ - since it is unavoidably entangled with the parameters of the astrophysical modelling, thus requiring an upgrade of such a modelling and, beyond that, a full account of this problem via numerical simulations. The third feature deals with the development of a radiative core at the end of the Hayashi track, entering the main-sequence phase while still being fully convective. We found the maximum value of the mass for this to happen, again observing an increase (decrease) of this mass with positive (negative) EiBI parameter. Note, however, that for all these three features the main astrophysical ingredient determining their absolute values is the opacity, whose modelling is always a delicate issue. Its influence is obvious in the last feature (the MFCM), where two different models of opacities (bound-free and free-free ones) result in up to a factor two in the absolute value of this quantity. The above results highlight the viability of using Palatini gravities of the RBG type to study modifi- cations to the stellar model predictions of GR, in particular, within the non-relativistic regime. This is so because in such a regime, RBG modifications to the usual Poisson equation typically occur via a single additional parameter [106], allowing to study the phenomenology of several types of stars, particularly low-mass stars, without ruining the consistence of the theory with weak-field limit observations. 124 Chapter 7 Conclusion The problems of GR, such as the unexplained accelerated expansion of the Universe and the rotation curves of galaxies without adding the undetected dark energy/matter, or the singularities predicted in the central region of BHs and in early-cosmological models, opened the door to extend the theory. In this way, there are different paths one can take; in this Thesis we have enlarged the freedom granted to metric and affine connection, the so-called Palatini formalism. Within this formalism, we use the so-called RBG, where the action is given as a function of the Ricci tensor and the metric. If we assume the symmetric part of the Ricci (the torsionless case) the equivalence principle is fulfilled and there are only two propagating dof as in GR. This Thesis was intended to address two different aims within the RBGs: the first one was to find regular solutions of compact objects and on the second one to look for qualitative new phenomenology and observational tests for such theories. After setting down the theories and the main concepts repeated through out the Thesis, we have directly applied them in Sec. 4.2, where we found by the brute force the static, spherically symmetric solutions of Palatini f(R) and EiBI gravities coupled to the EH electrodynamics. Both theories can lead to regular solutions depending on the sign of the model parameters and their particular values lead to different patterns of horizons that resemble to other well known solutions. However, the regularization mechanisms used by each theory is different. On the one hand, f(R) gravity only has its focusing point is pushed to the infinite affine parameter. Therefore, any type of geodesic can get or depart from there. On the other hand, such a point is accessible to some types of geodesics, but since the solution is indeed a wormhole, those geodesics getting to the throat can cross to the other asymptotical region allowing them to be complete. Therefore, in this chapter we have also check that one does not need to consider exotic matter to find regular solutions when a modified gravity is considered. As one could already witnessed from Sec. 4.2, even solving the field equation for the static case can be difficult and one needs to use approximations. Thus, as one can imagine, rotating scenarios are yet a harder nut to crack. This fact causes a lack of exact solutions in modify gravity to compare with the current observations. This is where the mapping procedure presented in Sec. 4.3 enters to rescue us. This recently developed tool uses the structure of the RBG field equations and the tight relation with GR to map not only both theories but also their solutions. In particular, we have started by the static case using 125 CHAPTER 7. CONCLUSION the RN solution as the seed in the GR side to find the corresponding coupling of EiBI gravity. Indeed, applying the mapping equations, such modified theory of gravity is coupled to the BI electrodynamics. The last thing is to derive the solution by brute force with the mapping and compare the effectively of the latter. Finally, we have progressed to the rotating case, where in this case the seed solution is the Kerr-Newman. The resulting spacetime in the EiBI side has an analytic expression found in Eq. (4.114) and the analysis of the different limits and the main surfaces characterizing the problem. The bottom line of this discussion is the reliability of the mapping method to find exact analytical solutions of interest for metric-affine gravity of the kind considered here. The implementation of such solutions and the analysis of their features is extremely convenient given the open opportunities available for testing modified gravity in astrophysical settings using the pool of available data: accretion disks, strong gravitational lensing and shadows, generation of gravitational waves in binary mergers, and so on. Following this direction, the second part is intended to address its second main aim, which consists in finding qualitative new phenomenology that may allow us to identify black hole mimickers (such as WHs, boson stars, etc.) or new tests to constrain our model parameters. We have begun by studying the optical appearances of several compact objects in order to find new phenomenology and improve our technique. Indeed, in Sec. 5.1 we have started with a simple case where we have studied reflection-asymmetric thin-shell WHs within Palatini f(R) theories of gravity using a junction conditions formalism suitably adapted to the peculiarities of these theories. The aim is to find those traversable, stable, thin-shell wormholes hold by non-exotic matter that have a different photon ring on each side of its throat. This way, we have characterized the parameter space depending on the conditions that were fulfilled. After this process, we have been able to find the expected configuration and we have depicted a pictorial image Fig. 5.8 of what would see an observer in the side that has access to both critical curves. Despite the remarkable result obtained above, the width and luminosity of the luminous rings are unrealistic since they where set ad hoc. Instead, we had to improve our model and consider an accretion disk surrounding the object, which is going to be the one that distribute its luminosity. In order to do it so, we have to control first the geodesics in the background geometry, this is what is called the ray tracing. Once we know all the paths and we have classified the light rays depending on their half turns, we can move on to the modelling of the accretion disc. However, this is not an easy task, since it involves the magnetohydrodynamics of the plasma forming the disc. Thus, we have assumed several assumptions (some of them stronger than others) to drastically simplify the scenario. The main properties are that we consider a optically and geometrically thin disk. By optically thin we mean that the light is not reabsorbed when intersects the disk, whereas geometrically thin consists to a infinitesimally thin disk (which is a quite hard assumption). Both new upgrades have come with a Mathematica code described in the corresponding section, which is also one of the most important results of this Thesis. The next move is to consider a background geometry. In particular, we have studied three different cases, which can be doubled as six: BB, its generalization and the eye of the storm. The first two allowed to smoothly interpolate from Schw, to BH and traversable WH changing the model parameters. At the same time, each solution has added complexity on the analysis of light ring pattern. To be more precise, 126 the BB solution had the same photon sphere and horizon radius for all three types of objects disabling to see new phenomenology. On the contrary, the extended version not only does the the radius of the photon sphere change, but also has another one which is only accessible for some wormhole cases. Lastly, the eye of the storm has a subset of solution classified by the value of the model parameter that can change the contribution of higher-order emissions. Recall that typically, after three intersection with the disk, the impact parameter range of corresponding light ray is so diminished that do not contribute to the total luminosity. But this is not the case with this geometry as seventh-order emission still contributes leading to a pattern of nine light rings. Nonetheless, these two last solutions might lead to large changes as compared to the GR case that can risk the viability of it. Finally, moving on to the last part, we have discussed in Chap. 6 some possible tests of the EiBI gravity using the early evolutionary phases of low-mass stars. One of them is the HT, since it can be shifted changing the value of the gravitational parameter, entering to the so-called Hayashi forbidden zone, where any star can be placed there. The second one is the Minimum Main Sequence Mass (MMSM) that is the minimum mass a star burning hydrogen stably can have. Finally, the Maximum Fully Convective Mass (MFCM) is the maximum mass a star can have before developing a radiative core. Future prospects In the recent years, the observations of the strong gravitational limit have opened a window to test the theories of gravity since they explore the most extreme conditions of matter and spacetime. Following this idea, the first aspect that could be worth analyzing in the near future is the study of the Gravitational Wave (GW) signal within the RBGs. In particular, those produced by the merger of two compact stars solution of the RBGs like those in Ch.4. This is of particular interest thanks to the expected upgrades on the sensibility of future GW detectors and the breath-taking amount of data one can extract from a multi-messenger observation. Concerning the optical appearance of the simple models considered here, some of them illustrates the existence of qualitatively new features in some black hole mimickers having two critical curves or, in the Eye of the storm, the sharpness of higher-order ring images can be used either as discriminators of the existence of new Physics, or to rule them out as viable alternatives to the Schw (Kerr) solution. For example, future projects of very-long baseline interferometry could be able to search for the existence of additional light rings by resolving out diffuse but sharp fluxes in an image [115–118], and therefore the detection of such features. Investigating this kind of observational discriminators among compact objects that asymptote to the same spacetime is of timely interest, should the Kerr solution happen not to describe every single (ultra)-compact object in the universe. Thus, the search for clean and clear of these type of distinct observational properties can be done either on a case-by-case basis by finding new such compact objects from a well defined theory of gravity and matter action, or via parametric deviations of the Kerr black hole shadow in a theory-agnostic way, see e.g. [119–121]. It could also be interesting to check how some of the solutions found in the first part would be optically seen, however, the geodesic equation is modified in the presence of an electric field since they see an effective metric [43, 122]. In this sense, the analysis of the geodesic completeness in Sec.4.2, however this upgrade is not expected to spoil the completeness of the geodesics. 127 CHAPTER 7. CONCLUSION Additionally, one would need to include the rotating scenarios in order to study the deviations from the Kerr black hole expectations, as well as in the circularity of the shadow when getting close to extremely rotation scenarios. Though this is easier said than done in particular with the asymmetric WH, due to the intrinsic complexity of the field equations of modified gravity, that largely prevents the finding of axially symmetric solutions of physical interest. Additionally, more realistic rotating spacetimes would face additional difficulties, since in such a case the integrability of the geodesic equations cannot be taken for granted, and similar radial and angular potentials as in the photon shell of Kerr solution do not necessarily exist in every possible rotating solution. Should one manage to build theoretically well supported alternatives to the Kerr picture that, in particular, might agree with the inferred features of the shadow caster of the M87 and Sgr A* observations while qualitatively departing from the shape of the Schw potential. This could be addressed with the mapping procedure, where the exact rotating solutions analyzed here are very useful from the point of view of enriching the theoretical discussions regarding potentially observable deviations in the structure of such solutions as compared to GR ones. Thus, exploring further families of nonlinear electrodynamics and/or scalar fields paralleling our analysis, and extensions of these results to other RBGs, could be useful to build up a catalogue of new rotating solutions in more physically appealing models. Moreover, the description of the accretion disks could be improved from the optically and geometrically thin modelling to a geometrically thick one, and the face-on orientation should be upgraded to consider modest inclinations of the disks and their effects in the optical appearances of the cases considered here which, together with the addition of rotation, may significantly modify the total luminosity [123] and, therefore, the optical appearance. Finally, the presence of WH structures yields interesting new possibilities, such as shadows from objects without accretion disks due to contribution of those disks on the other side of the WH and flowing through its throat. To conclude, whether the combination of all the above elements could be able to lead to further enhances in the brightness of the photon ring region, now including also additional contributions of the bands with m > 3, is yet to be seen. Given the promises of the observational teams working on achieving better resolution for the optical appearance of black hole candidates in order to test GR to better precision, combined with some new ideas to test the photon sphere using interferometer [124] or via correlated intensity fluctuations [125], this field is ripe for the existing zoo of non-canonical compact objects to extract observational discriminators with respect to GR predictions. Last but not least, regarding the tests with low-mass stars, the main bottleneck in order to place observational constraints upon any such theories is the determination of the central density, which up to now we have been only able to fix by taking its assumed values within GR, though more reliable theoretical procedures to deal with this issue are being investigated. Additionally, to simplify our analysis we disregarded the deuterium burning process which happens during the pre-main sequence phase and in massive brown dwarfs, since the energy generated by this process is significantly smaller than the one of the hydrogen ignition. Another simplification of our analysis lies on the fact that in order to properly incorporate lithium burning in the low-mass stars or cooling process of a brown dwarf object, one needs to use a more realistic model of the electron degeneracy than the one used here, while the choice of the 128 opacities is always subject to discussion. 129 CHAPTER 7. 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[125] Shahar Hadar et al. “Photon Ring Autocorrelations”. In: Phys. Rev. D 103.10 (2021). 138 Appendix A Mathematica® Code A.1 Ray tracing Let us explain now explain how the ray tracing explained in 5.2.1 was implemented on Mathematica®. In order to make things as simple as possible we are going to consider the Schw solution. The first definitions that we find are the following: In[1]:= A=-(1-2/r); B=(1-2/r)^(-1); c=r^2; hor=2; Geo= b r^2 1 Sqrt[1-(A*b^2)/c] ; where the three first lines correspond to the time, radial and angular metric components (with M = 1). We also add the horizon radius (or the WH throat when applies) as well as the geodesic equation written in Eq. (3.75). These are the main ingredients needed to calculate the geodesic equation. We shall now proceed to define the initial and final values of the variables necessary for the numerical integration, In[2]:= r0 = 1000; rcent = hor; rend = rcent; b0 = Sqrt[27] (1 + 10^-6); Xp = 0; Yp = 0; Following the order of appearance, the first radius is its initial value, whereas the second one is the absolute closest radius to the center where light rays can still reach us (for BHs is the horizon and for WH its throat). Then, the third one is a definition that will change depending on the light ray as it will be the particular radius of maximum approach for a given impact parameter in this case defined in b0. Finally, Xp is used to identify which trajectories reach rcent and which not, on the contrary, Yp gives us 139 APPENDIX A. MATHEMATICA® CODE information on how many times the light rays cross the vertical axis. With this in mind, we can proceed to integrate numerically the geodesic equation (3.75). In[3]:= Sol0 = NDSolve[{{{ϕϕϕ’[r] == -Geo /. b →→→ b0, ϕϕϕ[r0] == 0, WhenEvent[ (Abs[1 - b^2*A/R^2] /. b →→→ b0) < 10^-15, {{{Xp = 1, rend = r, "StopIntegration"}}}, "LocationMethod"→→→"LinearInterpolation"], WhenEvent[ϕϕϕ[r] == πππ/2, {{{rb1 = r, Yp = 1}}}]; WhenEvent[ϕϕϕ[r] == 3 πππ/2, {{{rb2 = r, Yp = 2}}}]; WhenEvent[ϕϕϕ[r] == 5 πππ/2, {{{rb3 = r, Yp = 3}}}; "StopIntegration"]}}}, ϕϕϕ[r], {{{r, rcent, r0}}}, WorkingPrecision →→→ 30, PrecisionGoal →→→ 15, AccuracyGoal →→→ 15, MaxSteps →→→ 10000] // Quiet; P0 = ParametricPlot[{{{{{{r Cos[ϕϕϕ[r]], r Sin[ϕϕϕ[r]]}}} /. Sol0}}}, {{{r, rend, 30}}}, PlotStyle →→→ {{{Red}}}, PlotRange →→→ All, PlotPoints →→→ 200]; Sol0 is the numerical integration of the geodesic equation from r0 to rcent, given the initial value of ϕ, which we set it equal zero. The integration will end for two different cases: • The geodesic equation can be integrated until rcent, being completely defined. • The geodesic equation becomes 0 (or in this numerical approach smaller than 10−15), this means that we are at the turning point and, thus, we stop the integration, redefine rend to such a value of the radius and set Xp to 1. For the first case, the integration is finished, but for the second one, we have only calculated half of the geodesic, the ingoing part. Even though the next step will be different for each one, both of them will cross at least one time the vertical axis and this is calculated from line 4 to 6, where for each intersection we “save” the particular radius. Additionally, we are going to plot the trajectory in P0. Finally, we calculate the outgoing trajectory for the second case, when Xp=1, therefore, In[4]:= If[Xp > 0, {{{ rip = rend (1 + 10^-14); ϕϕϕrip = Evaluate[ϕϕϕ[r] /. Sol0[[1]]] /. r →→→ rip; Sol1 = NDSolve[{{{ϕϕϕ’[r] == Geo /. b →→→ b0, ϕϕϕ[rip] == ϕϕϕrip, WhenEvent[ϕϕϕ[r] == πππ/2, {{{rb1 = r, Yp = 1}}}], WhenEvent[ϕϕϕ[r] == 3 πππ/2, {{{rb2 = r, Yp = 2}}}], WhenEvent[ϕϕϕ[r] == 5 πππ/2, {{{rb3 = r, Yp = 3}}}; "StopIntegration"]}}}, ϕϕϕ[r], {{{r, rip, r0}}}, WorkingPrecision →→→ 30, PrecisionGoal →→→ 15, AccuracyGoal →→→ 15, MaxSteps →→→ 10000] // Quiet; P1 = ParametricPlot[{{{{{{r Cos[ϕϕϕ[r]], r Sin[ϕϕϕ[r]]}}} /. Sol1}}}, {{{r, rip, 30}}}, PlotStyle →→→ {{{Red}}}, PlotRange →→→ All, PlotPoints →→→ 200];}}}, False]; If[Yp > 0, Lista1br = {{{{{{b0, rb1}}}}}}, Lista1br = {{{{{{0, 0}}}}}}]; If[Yp > 1, Lista2br = {{{{{{b0, rb2}}}}}}, Lista2br = {{{{{{0, 0}}}}}}]; 140 A.1. RAY TRACING If[Yp > 2, Lista3br = {{{{{{b0, rb3}}}}}}, Lista3br = {{{{{{0, 0}}}}}}]; Figs = {{{If[Xp > 0, Show[P0, P1], Show[P0]]}}}; Here, we have followed the same strategy as before, but this time we change the sign of the equation. Moreover, the second and third lines are new definitions such as a slightly larger radius than the rend in order to make sure that the program will run correctly and the value of ϕ(rip). Now the numerical integration is from rip to r0 with the initial value ϕrip. We have also defined 3 lists with the corre- sponding impact parameter and the radius of each intersection between the trajectory and the vertical axis, respectively. Finally, the array Figs saves the plots P0 and P1 for the second case, or just P0 for the first case. Up to now, we have only calculated one trajectory, but as we have said before we want to calculate a bunch of trajectories. To this end, we code a Do loop of 200 iterations, starting from the initial impact parameter set before b0 to the last impact parameter bend with the same structure seen before. In[5]:= Nsteps = 200; bend=0; ϵϵϵ = (b0 - bend)/Nsteps; Do[ rcent = hor; rend = rcent; Xp = 0; Yp = 0; b0 = b0 - ϵϵϵ; Sol0 = NDSolve[{{{ϕϕϕ’[r] == -Geo /. b →→→ b0, ϕϕϕ[r0] == 0, WhenEvent[ (Abs[1 - b^2*A/R^2] /. b →→→ b0) < 10^-15, {{{Xp = 1, rend = r, "StopIntegration"}}}, "LocationMethod"→→→"LinearInterpolation"], WhenEvent[ϕϕϕ[r] == πππ/2, {{{rb1 = r, Yp = 1}}}], WhenEvent[ϕϕϕ[r] == 3 πππ/2, {{{rb2 = r, Yp = 2}}}], WhenEvent[ϕϕϕ[r] == 5 πππ/2, {{{rb3 = r, Yp = 3}}}; "StopIntegration"]}}}, ϕϕϕ[r], {{{r, rcent, r0}}}, WorkingPrecision →→→ 30, PrecisionGoal →→→ 15, AccuracyGoal →→→ 15, MaxSteps →→→ 10000] // Quiet; P0 = ParametricPlot[{{{{{{r Cos[ϕϕϕ[r]], r Sin[ϕϕϕ[r]]}}} /. Sol0}}}, {{{r, rend, 30}}}, PlotStyle →→→ {{{Red}}}, PlotRange →→→ All, PlotPoints →→→ 200]; If[Xp > 0, {{{ rip = rend (1 + 10^-14); ϕϕϕrip = Evaluate[ϕϕϕ[r] /. Sol0[[1]]] /. r →→→ rip; Sol1 = NDSolve[{{{ϕϕϕ’[r] == Geo /. b →→→ b0, ϕϕϕ[rip] == ϕϕϕrip, WhenEvent[ϕϕϕ[r] == πππ/2, {{{rb1 = r, Yp = 1}}}], WhenEvent[ϕϕϕ[r] == 3 πππ/2, {{{rb2 = r, Yp = 2}}}], WhenEvent[ϕϕϕ[r] == 5 πππ/2, {{{rb3 = r, Yp = 3}}}; "StopIntegration"]}}}, ϕϕϕ[r], {{{r, rip, r0}}}, WorkingPrecision →→→ 30, 141 APPENDIX A. MATHEMATICA® CODE PrecisionGoal →→→ 15, AccuracyGoal →→→ 15, MaxSteps →→→ 10000] // Quiet; P1 = ParametricPlot[{{{{{{r Cos[ϕϕϕ[r]], r Sin[ϕϕϕ[r]]}}} /. Sol1}}},{{{r, rip, 30}}}, PlotStyle →→→ {{{Red}}}, PlotRange →→→ All, PlotPoints →→→ 200];}}}, False]; If[Yp > 0, AppendTo[Lista1br,{{{b0, rb1}}}], False]; If[Yp > 1, AppendTo[Lista2br,{{{b0, rb2}}}], False]; If[Yp > 2, AppendTo[Lista3br,{{{b0, rb3}}}], False]; Figs = AppendTo[Figs,If[Xp > 0, Show[P0, P1], Show[P0]]]; ,{{{i,1,Nsteps}}} The only part that changes of the core code inside the loop is the last piece, where instead of just keeping the last pair of b-r, we add them into the list and the same happens with the figures. In[6]:= BR0 = ListPlot[{{{Lista1brL, Lista2brL, Lista3brL}}}, Joined →→→ True, PlotRange →→→ {{{{{{0, 10}}}, {{{0, 10}}}}}}, AxesLabel →→→ {{{"b/M", "x_m/M"}}}, LabelStyle →→→ Directive[FontSize →→→ 12]] Out[6]= 2 4 bc 6 8 10 b/M0 5 10 15 20 xm/M A.2 Accretion disk Now, it is the turn to implemented the accretion disk properties discussed in Sec.5.2.2 and the final optical appearance of a Schw BH. Indeed, here we are only giving the basic idea, with just focusing on the first emission profile of the disk whose inner edge starts at the ISCO and is described in Eq. (5.45) (these definitions are written in the firsts two lines). In[7]:= xisco = Sqrt[36]; Idisk[x_] = If[x > xisco, 1/(x - (xisco - 1))^2, 0]; IE1GR = Plot[Idisk[x], {{{x, 0, 15}}}, PlotPoints →→→ 300, AxesLabel →→→ {{{"x", "I_em^I(x)"}}}, LabelStyle →→→ Directive[FontSize →→→ 12], PlotRange →→→ {{{{{{0, 13}}}, {{{0, 1}}}}}}] Out[7]= xISCO 0 2 4 6 8 10 12 x0 0.2 0.4 0.6 0.8 1.0 Iem I (x) 142 A.2. ACCRETION DISK The next step is to calculate the observed intensity profile, to do so we will need to make use of Eq. (5.48), where we need the time-time component of the metric and the emission intensity profile. In[8]:= gg[x_] := 1 - 2/x; DirectIm = Table[{{{Lista1brL[[i, 1]], gg[Lista1brL[[i, 2]]]^2 Idisk[Lista1brL[[i, 2]]]}}}, {{{i, 1, Length[Lista1brL], 1}}}]//Sort; LensedIm = Table[{{{Lista2brL[[i, 1]], gg[Lista2brL[[i, 2]]]^2 Idisk[Lista2brL[[i, 2]]]}}}, {{{i, 1, Length[Lista2brL], 1}}}]//Sort; LightRIm = Table[{{{Lista3brL[[i, 1]], gg[Lista3brL[[i, 2]]]^2 Idisk[Lista3brL[[i, 2]]]}}}, {{{i, 1, Length[Lista3brL], 1}}}]//Sort; For the direct, lensed and photon ring images, we make a table where the first row keeps the impact parameter and on the second one the observed intensity of Eq. (5.48).From the above list, we can also take the impact parameters range, where the first element of the list is the initial value and the last one is the final, that is In[9]:= ain = DirectIm[[1, 1]] // N; afin = DirectIm[[Length[DirectIm], 1]] // N; bin = LensedIm[[1, 1]] // N; bfin = LensedIm[[Length[LensedIm], 1]] // N; cin = LightRIm[[1, 1]] // N; cfin = LightRIm[[Length[LightRIm], 1]] // N; Additionally, since we want a continuous function instead of a collection of data, we will use he interpo- lation function as follows, In[10]:= Ifdirect = Interpolation[DirectIm, InterpolationOrder →→→ 1]; Iflensed = Interpolation[LensedIm, InterpolationOrder →→→ 1]; Iflight = Interpolation[LightRIm, InterpolationOrder →→→ 1]; With the above functions and the definitions of the initial and final impact parameter range of each emission, the observed intensity will be a combination of different types of emissions as In[11]:= Pl1 = Plot[Ifdirect[x], {{{x, ain, bin}}}, PlotRange →→→ All, PlotPoints →→→ 500]; Pl2 = Plot[Ifdirect[x] + Iflensed[x], {{{x, bin, cin}}}, PlotRange →→→ All, PlotPoints →→→ 300]; Pl3 = Plot[Ifdirect[x] + Iflensed[x] + Iflight[x], {{{x, cin, cfin}}}, PlotRange →→→ All, PlotPoints →→→ 300]; Pl4 = Plot[Ifdirect[x] + Iflensed[x], {{{x, cfin (1 + 10^-4), bfin}}}, PlotRange →→→ All, PlotPoints →→→ 300]; Pl5 = Plot[Ifdirect[x], {{{x, bfin, afin}}}, PlotRange →→→ All, PlotPoints →→→ 300]; Show[Pl1, Pl2, Pl3, Pl4, Pl5, PlotRange →→→ {{{{{{0, 15}}}, {{{0, 0.4}}}}}}, 143 APPENDIX A. MATHEMATICA® CODE AxesLabel →→→ {{{"b", "I_ob^I (b)"}}}, LabelStyle →→→ Directive[FontSize →→→ 12], AxesOrigin →→→ Automatic] Out[11]= 2 4 6 8 10 12 14 b 0.2 0.4 0.1 0.3 Iob I (b) In order to know the quantitative contribution to the total luminosity of each type of emission, we integrate out each observed intensity function and calculate the ratio: In[12]:= IntDirect = NIntegrate[Ifdirect[x], {{{x, ain, afin}}}]; IntLensed = NIntegrate[Iflensed[x], {{{x, bin, bfin}}}]; IntLight = NIntegrate[Iflight[x], {{{x, cin, cfin}}}]; Weightd= IntDirect/(IntDirect + IntLensed + IntLight) Weightl= IntLensed/(IntDirect + IntLensed + IntLight) Weightpr= IntLight/(IntDirect + IntLensed + IntLight) Out[12]= 0.948853 0.049418 0.0017285 Finally, let us make a table of the impact parameter and the observed intensity there, however the length of each one is going to be different depending on the precision one needs to that particular region. In[13]:= DatosPl1 = Table[{{{x, Ifdirect[x]}}}, {{{x, ain, bin, (bin - ain)/51}}}]; DatosPl2 = Table[{{{x, Ifdirect[x] + Iflensed[x]}}}, {{{x, bin, cin, (cin - bin)/151}}}]; DatosPl3 = Table[{{{x, Ifdirect[x] + Iflensed[x] + Iflight[x]}}}, {{{x, cin, cfin, (cfin - cin)/151}}}]; DatosPl4 = Table[{{{x, Ifdirect[x] + Iflensed[x]}}}, {{{x, cfin (1 + 10^-4), bfin, (bfin - cfin)/151}}}]; DatosPl5 = Table[{{{x, Ifdirect[x]}}}, {{{x, bfin, afin, (afin - bfin)/101}}}]; Datos = Union[DatosPl1, DatosPl2, DatosPl3, DatosPl4, DatosPl5]; iDatos = Interpolation[Datos, InterpolationOrder →→→ 1]; Nnorm = NIntegrate[iDatos[x], x, ain, afin] where the last two inputs are requested to later on normalize our result. Moreover, with the above generated data we can already depict the optical appearance as follows 144 A.2. ACCRETION DISK In[14]:= LDatos = Table[{{{Datos[[1, 1]] Cos[t], Datos[[1, 1]] Sin[t], Datos[[1, 2]]/Nnorm}}}, {{{t, 0, 2 πππ, 2 πππ/100}}}]; Do[New = Table[{{{Datos[[i, 1]] Cos[t], Datos[[i, 1]] Sin[t], Datos[[i, 2]]/Nnorm}}}, {{{t, 0, 2 πππ, 2 πππ/100}}}]; LDatos = Union[LDatos, New];, {{{i, 2, Length[Datos], 1}}}]; ListDensityPlot[{{{LDatos}}}, ColorFunction →→→(ColorData["VisibleSpectrum"][Rescale[#,{{{1,0}}}, {{{565,750}}}]]&), ColorFunctionScaling →→→ True, PlotRange →→→ {{{{{{-10, 10}}}, {{{-10, 10}}}, {{{-10, 10}}}}}}, PlotLegends →→→ Automatic, LabelStyle →→→ Directive[FontSize →→→ 12]] Out[14]= As we have said at the begining of this subsection this is only for the first emission profile model, but it can be done for the other two or can be enlarged with more realistic models. 145 Title page Summary Abbreviations Contents Chapter 1. Preface Chapter 2. Introduction I Theory Chapter 3. Theoretical framework Ricci Based Gravities General Relativity Palatini f(R) gravity Eddington-inspired Born-Infeld gravity Stress-energy tensor Maxwell Electrodynamics Non-linear Electrodynamics Fluids Energy conditions Geodesics in symmetric RBGs Null geodesics in general static, spherically symmetric spacetime Timelike geodesics Geodesic completeness and energy conditions Chapter 4. Applications Derivation of a RBG spherically symmetric solution RBGs coupled to Euler-Heisenberg electrodynamics Solution Properties of the solution: asymptotic behaviour Properties of the solution: radial function Properties of the solution: inner behaviour and horizons Properties of the solution: geodesic behaviour and regularity Mapping Mapping example: EiBI gravity + NED Finding the solution A rotating black hole solution Horizons and ergoregions Metric and curvature divergences Conclusion II Phenomenology Chapter 5. Optical appearance of compact objects Pre-main Sequence evolution of low-mass stars Non-relativistic stellar structure equations Equation of state Simple photospheric model Convective instability - modified Schwarzschild criterion Hayashi tracks Minimum main sequence mass Fully convective stars on the main sequence Conclusions Asymmetric thin-shell wormholes with two critical curves Thin-shell formalism in Palatini f(R) gravity Electrovacuum spherically symmetric spacetimes Traversable wormholes from surgically joined Reissner-Norström spacetimes Structure of the asymmetric wormhole: horizons and photon spheres Double photon sphere Imaging model improvement Ray tracing Accretion disk model Black Bounce solutions Black hole Traversable wormholes The eye of the storm Multi-ring structure Conclusions Chapter 6. Pre-main Sequence evolution of low-mass stars Chapter 7. Conclusion Bibliography Mathematica® Code Ray tracing Accretion disk