PHYSICAL REVIEW D VOLUME 39, NUMBER 6 Existence of global strings coupled to gravity 15 MARCH 1989 Gary W. Gibbons, Miguel E. Ortiz, and Fernando Ruiz Ruiz Department ofApplied Mathematics and Theoretical Physics, University of Cambridge, 3 Silver Street, Cambridge CB3 9EW, United Kingdom (Received 28 November 1988) We consider o.-model strings and U(1) global strings coupled to gravity and look for static solu- tions with whole-cylinder symmetry. We prove that in both cases there are no regular spatially compact solutions. The U(1) global string admits no asymptotically well-behaved solutions, whereas cr-model strings can be constructed under certain assumptions. Finally we present some singular solutions and show that one of them corresponds to a physical global string. I. INTRODUCTION Over the past few years cosmic strings have attracted much attention due to their important role in cosmolo- gy. ' These strings are expected to form after the break- ing of both local and global symmetries in the corre- sponding field theory. Much progress has been made in the understanding of local strings. Nielsen and Olesen considered the Abelian Higgs model and constructed a static cylindrically symmetric solution with boost invari- ance in the axial direction in flat spacetime. Garfinkle studied the exact gravitational field for such a solution and proved the existence of an asymptotically conical metric with deficit angle 5=8~6p, where p is the mass per unit length of the string. Laguna-Castillo and Matzner found a numerical solution to Garfinkle's equa- tions, and Gregory proved its stability. The situation for global strings is rather more compli- cated and requires clarification. Using a scaling argu- ment originally due to Derrick, one can easily prove that there exist no stable global strings of finite energy per unit length in a flat background. In brief, consider any scalar model with arbitrary non-negative potential. Any static straight string in 3+ 1 dimensions would give rise to a static soliton solution in 2+ 1 dimensions whose (2+ 1)-dimensional energy E is the energy per unit length of the string. This solution would be an extremum of E. Under a scale transformation x"~ax",p = 1,2, 3, the en- ergy goes to E = T+a V, where T and V are the kinetic and potential energies, both non-negative. Static finite- energy solutions are thus ruled out since there is no a&0 which extremizes E . In fact, it turns out that there do exist solutions representing global strings in flat space- time, but their energy density falls off as 1/r so that their energy per unit length diverges. In a cosmological context, one might argue that a spacetime containing a global cosmic string will not be asymptotically flat, and so that there is no necessity for considering only strings of finite energy per unit length. One would still like to know, however, whether or not it is sensible to talk about the gravitational field of an isolat- ed global cosmic string in the same way that one can talk about the asymptotic gravitational field of a local string. In the latter case an appropriate concept of total energy, including gravitational, is given by Thorne's C energy which should be minimized by any stable solution. One may attempt to apply the scaling argument to the C ener- gy but one must check that the initial-value constraint R =16~GTt' is invariant under the scaling x"~nx" or equivalently g„~o.g„, and this is only true if the po- tential term is identically zero. For global strings it does not seem to be easy to define a suitable notion of total en- ergy per unit length. Studies of the gravitational field of a U(1) global string have already encountered difficulties. While Harari and Sikivie have calculated the metric of a global string us- ing the weak-field approximation, Gregory has shown that, under certain assumptions about how the Higgs field approaches the potential minimum, there are no asymptotically well-behaved static cylindrically sym- metric solutions. In Sec. II we derive the field and Ein- stein equations and prove in agreement with Ref. 5 that they have no regular asymptotic solutions with the Higgs field tending to its vacuum value. These results suggested that the spacetime could compactify. However, in Sec. III we prove a no-go theorem for regular compact warped product solutions for a general class of nonlinear o. models, including static global strings as a special case, showing that they cannot compactify. The results of Sec. III also apply to o.-model strings, whose existence is dis- cussed in Sec. IV. Section V is dedicated to explaining the relationship between Harari and Sikivie s weak-field approximation and singular solutions and we show that as the radial distance increases, the Higgs field will inevit- ably blow up. We also give a second asymptotically singular solution to the full equations, but this appears to have no relevance to cosmic strings. II. BASIC EQUATIONS AND NONEXISTENCE OF EVERYWHERE-WELL-DEFINED GLOBAL U(1) VORTICES Let us consider the model for a U(1) global string, that is, a complex scalar field P in a "Mexican hat" potential. The corresponding Lagrangian density is (2.1) 39 1546 1989 The American Physical Society 39 EXISTENCE OF GLOBAL STRINGS COUPLED TO GRAVITY 1547 where A, and g are positive. The field and Einstein equa- tions are (2.2) (2.3) assumption about the way in which f approaches 1 at infinity. From Eqs. (2.7)—(2.10) it follows that 8,(82——48')=8rrGH T„" with the energy-momentum tensor given by T„=B„QB,Q —g„+ . (2.4) g 2=2rrG A rl H f ' ~—q + (f 1)— ds =e "[—dt +(dx ) ]+dp +p e dg y=rlf (p)e", (2.5) (2.6) where A, B, and f are functions of p only. Such a solu- tion is usually called a vortex. Notice that in Eq. (2.6) we have taken the winding number of the string to be 1. Substituting Eqs. (2.5) and (2.6) into Eqs. (2.2) and 2.3) we get f"+ 2'+ + —f' +f (1——f )=0, 8' 1, f ~2e 8 (2.7) A 'B' 2 "+2' + + = —4m. G (f 1)— 2 T (2.8) t2 pB + 2B 2 p' 4mGrI [4f' +—(f 1) ], — Br2 IB"+ + 3 'B'+ + 2 I" T (2.9) 4@2= —4m.Gg " + (f 1)— 2 8 (2.10) We are interested in finding a static cylindrically sym- metric string along the x axis with boost invariance in the x direction. This amounts to finding a solution of Eqs. (2.2) and (2.3) of the form (2.12) where H =re ~+~, 0, =HA', 02 ——H'. Using analytical arguments one can prove that the right-hand side of this equation approaches zero as r ~~ (see the Appendix), and thus the only two possibilities for the asymptotic be- havior of the metric are 0&~0 or 02 — —,'0, ~0. The first case gives an asymptotically conical metric but this can- not correspond to the physical solution. This can be seen by evaluating the deficit angle 5. The Gauss-Bonnet theorem yields 21677 6 p 4 J r re 0 which diverges logarithmically. The second case, 02 — —,'0, ~0, is a Kasner metric with equivalent asymp- totics to those of Melvin's magnetic universe. However, this does not describe a physical solution either since for this metric the metric coefficient r e goes to zero as r ~~ and then the energy density T,' diverges at infinity, contradicting our previous assumptions. These results confirm that there are no well-behaved asymptotic solutions to Eqs. (2.7)—(2.10). Both the solu- tions we have ruled out have the common feature that they correspond in some sense to a closing up of space- time. However, in the next section we further prove that there are no regular static spatially compact solutions. f(0)=0, f(r)~l as r~oo . (2.1 1) It is easy to show, using the Bianchi identities and the regularity of the metric at the axis, that Eq. (2.9) follows from Eqs. (2.7), (2.8), and (2.10). We further require one more condition, that the energy density Tt — 9 f&2+ f +1(f2 1)2 vanishes as r —+ ~, which from a physical point of view is reasonable. As we have already mentioned, assuming that f~1 faster than some negative power of r, there are no asymp- totically well-behaved static U(1) global vortices. Here we briefly present a difFerent proof, along the lines of that used by Garfinkle for local strings, without making any the prime denoting difFerentiation with respect to r =—&Zap. We impose standard boundary conditions:3 the metric is locally Hat at the z axis, i.e., A (0)=0, e —1 as r~ co, and III. A NO-GO THEOREM FOR STATIC COMPACT GLOBAL STRINGS where V(P) is any non-negative potential. Notice that the U(1) global string is a particular case. In Eq. (3.1) g„ and G~~ are the metrics on the spacetime manifold Md and the target space, respectively. The former is assumed to have the form ds = 8'(x')5,bdy'dy +h „(x')dx dx", IMd g, I —[&d I &ab ~J && [I'D h—m. ] (3.2) p, v=0, 1, . . . , d —1, a, b =0, 1, . . . , d —D —1, m, n =d —D, . . . , d —1, The question of compactification is of interest in a variety of contexts, and so we derive a no-go theorem for a general class of nonlinear o. models in d-dimensional Euclidean spacetime. The static global string will appear as a special case after we have used the time indepen- dence of the equations to perform a Wick rotation. The model we consider has the Lagrangian (3.1) 1548 GIBBONS, ORTIZ, AND RUIZ 39 where the metric g„ is of Euclidean signature. It has d —D hypersurface-orthogonal Killing vectors B/By '. The energy-momentum tensor for this model is Ti =G~a(4')B„4"BA"+gi,A which together with the Einstein equation (2.3) gives (3.3) R„„=S~G G„(y)B„y'B„y'+ g„.V(y) 2 (3.4) co = — —,'(VW) + —, ' W(V'„K, )(Vi'K ) . On the other hand, the Bianchi identities give Clw=2(V' K )(V'"K ) 2R„K~—K' . From these two equations we get V' VR' =6 —2 (3.5) (3.6) Under these assumptions the following theorem holds: if the Ricci tensor is non-negative, W&0, and (PD, h „) is regular and compact, then 8' is constant and R„K"E =0 for K" any of the above Killing vectors. To prove this we introduce the twist three-form co„&=K(„V+i ) and note that, for any of the d DKil-— ling vectors, 8'=K„K". After some simple algebra we obtain =eh „, where ~ is the Gauss curvature. Integrating we obtain J d~xV h i~= f d x&hh "G„(P)B P"B„P ~0. The left-hand side is equal to 4~(1 —g), where g is the genus of P2, and we deduce that g =0 or g =1. The latter does not correspond to a compact solution since it requires ~ to vanish everywhere. We conclude that the only possible topology for P2 is that of a two-sphere. Furthermore, if the target space is fiat then P"=const (see Sec. IV below). Let us see how these results apply to the global string of Eq. (2.1). The Lagrangian density is of the form (3.1), with V(P) =(k/4)( ~P~ —i) ) . The target space has dimension 2, metric Gzz =5~&, and coordinates P'= —,'(P+P), P =(i/2)(P —P). We perform a Wick ro- tation, t = —i ~, and work with Euclidean signature since the field equations (2.7)—(2.10) are time independent. The spacetime metric is then ds =e "(dr +dz )+dr +r e dB which is of the type (3.2) and has a non-negative Ricci tensor. According to the above result the existence of an (r, B) compact solution requires V(P) =0 and A and P to be constants, and so the spacetime does not contain a glo- bal string. Since the K" are hypersurface orthogonal we have that ~„&=0. Integrating now over Md —the y independence of the metric reduces this to an integration over the corn- pact manifold PD —the total derivative term vanishes and we are left with d x~hW~ =R K—~K.=O.pv This implies that R„K"K =0 throughout Md since R„)0. To see that 8' is a constant we again use Eqs. (3.5) and (3.6) to obtain IV. o -MODEL STRINGS In general, nonlinear o. models also admit stringlike solutions. To see this let us consider the Lagrangian den- sity of Eq. (3.1) with a zero potential: ,' g'G„(y)g——B„y'B,y', where we have introduced a coupling constant g with di- mensions of (length) ' since we are taking the fields P to be dimensionless. We are interested in stringlike solu- tions, that is, static solutions with an axial Killing vector so that W'R„K"K —3' + —,'V( WV'W) —(V'W') =0 . ds =e [ dt +(dx ) ]+—e dz dz, (4.1) Od -D R, R „)0. mn (3.7) Equating Eqs. (3.4) and (3.7) we have that the potential must be identically zero, V(P)=0. Having shown that V(P) =0, the fields P are now harmonic maps from Md into the target space. Finding the possible P" is then a problem of harmonic maps. ' In the case of four spacetime dimensions, Eqs. (3.4) and (3.7) also imply that P" is independent of y' provided that Gz& is Riemannian. We are thus left with R „=Gzs($)B P "B„P . For the case D =2, R We now integrate and use co =O=R„K"K to get f d x&h W~" '"(V W)'=-O from which it follows that 8 is constant. This result gives the following form for the Ricci ten- sor: where (z,z) is a complex local coordinate patch and X and 4' are functions of z and z only. Spatially compact solutions of the type (4.1) are governed by the results of the previous section. Here we shall discuss the existence of o.-model strings for noncom- pact spacetimes. We will restrict ourselves to the case of two-dimensional target spaces, for which the metric can always be written as Gz~ =Q 6~~ for suitable B. We seek solutions of the field and Einstein equations with metric of the form (4.1) and for which the field P de- pends only on z and z. The P field equation is BBP+(B&)(BP)+(B&)(BP) +2 (By)(By) +(By)(By) =0, By where B and B stand for B/Bz and B/Bz, respectively. Now the energy density is minimized by fields satisfy- 39 EXISTENCE OF GLOBAL STRINGS COUPLED TO GRAVITY 1549 ing Bogomol'nyi-type equations, which are solved by any holomorphic function P (Ref. 11). Thus configurations of minimal energy satisfy a/=0, and for these the field equation simplifies to (ar)(ay) =o, which has two solutions. The first is a/=0, which, to- gether with a/=0, is equivalent to the statement that /=const. The other is aX=0, from which it follows that 2=const since X is real. For /=const the solutions of type (4.1) to the Einstein equations are no more than vac- uum solutions with one Killing vector and have been widely studied. We shall concentrate on the more in- teresting possibility X=const, which is equivalent to X=0 since this amounts to a rescaling of t and x in (4.1). The physical meaning of solutions with X=O is clear: the problem of stringlike solutions of type (4.1) has been reduced to that of finding static solitons in 2+ 1 dimen- sions, or equivalently instantons in two dimensions. The Bogomol'nyi equations ensure that any solution will cor- respond to a stable soliton, subject to the assumption that the holomorphic function tends to a fixed map at infinity. For X=0 the only nontrivial Einstein equation is ave= —2~Gg'n'(ay)(a y) . (4.2) In Ref. 11 some solutions to this equation are given for the case in which the target space is CP' with metric 0=2/( I+PP). For P=P/z, with P a constant, 4 takes the form 'I'= —8~ Gg'»( I zl' +Iftl') . (4.3) Thus the metric is asymptotically conical with deficit an- gle 32m Gg . Independently of whether one is looking for compact or noncompact solutions some restrictions can be ob- tained on the geometry of the target space when looking for stringlike solutions. In Ref. 10 the following identity is derived: For large values of z one can make the change (1—a)p =Izl" ', with a—= 8vrGg, which gives the following asymptotic form for the metric (4.1), (4.3): ds — dt +—(dx ) +d8 +a p dp f&—g d x X.„'"=f& —g d x P„„P""' f. &—g —d x(R„~ cDQ""P„P 'P —R„P""P"„), (4.4) (4.5) R y APyBycvyD () (4.6) and where P „„=a„P„and the spacetime metric is assumed to be regular. Typically, a harmonic map corresponding to a soliton in 2+ 1 dimensions will tend to a fixed map at least as 1/z as the proper length goes to infinity. For such behavior, the boundary term containing the La- grangian density will vanish. If the Ricci tensor of the spacetime manifold is non- negative, we can exclude target spaces with negative- semidefinite curvature R~~&D as follows. In order to satisfy Eq. (4.4) the following identities must hold every- where: q'= 8~G g'»( lzl' — I pl'), for which the scalar curvature is R =4a/( IPI—Izl ) "+ ', where a=8irGg . R takes arbitrarily large values as Izl~lPI and so the transverse spatial sections become more and more curved as z ~ IPI. Notice that such sections are topologically noncompact but they have finite volume since I@I"~+2 1+2' If we visualize one of these sections by embedding it isometrically in a three-dimensional Euclidean space we obtain a teardrop of the kind derived by Gell-Mann and Zwiebach' in the context of higher-dimensional super- gravity theories. Q Appv o (4.7) V. SINGULAR SOLUTIONS AND THEIR PHYSICAL SIGNIFICANCE Now, since the target space is two dimensional, Eq. (4.7) reduces to (ay)(ay) =o, from which it follows that / =const, which is again the trivial case. Naively one could think of repeating the construction of stringlike solutions for the CP' model above for spaces of negative curvature. Although one is able to construct such solutions, they correspond to singular spacetime geometries. Consider, for example, the unit disc with Poincare metric, 0=2/(I —PP), and take P=z/P with Izl & IPI. The solution to Eq. (4.2) is, then, The system of Eqs. (2.7)—(2.10) does at least admit a solution in the weak-field approximation. By studying Eq. (2.7) we shall show that for the physical solution picked out by this approximation the Higgs field blows up. The corresponding configuration turns out to be singular at large radius (the spacetime will close up at a curvature singularity' ), and this result is thus compati- ble with those of Secs. II and III, as it does not satisfy condition (2.11). In addition we present a second singular solution, for which exact asymptotics are known, but ar- gue that it does not correspond to a string configuratio. Harari and Sikivie, using the weak-field approximation and assuming that f—:1, found the following form for the 1550 GIBBONS, ORTIZ, AND RUIZ 39 metric 1 1 p, — — exp 8nGil. is much larger than the present horizon scale. Consider now Eq. (2.7), glf"+ 2'+ +—f' — +f(1—f )=0. We may regard this as the equation for a particle in an inverted Mexican hat potential, —V(P), under a friction- al force (since 3'+ ,'B'+I/—r)0) and a force f/r e When f is close to 1, in the sense of the weak-field ap- proximation, the frictional force is negligible, and the constancy off implies that =2e where f =1+e .1 2 B The existence of a family of flat-space solutions [i.e., solu- tions to (2.7) when e =1 and e"=1] shows that if the force is 1/r, the "particle" will just reach the peak. However, the weak-field approximation predicts that e ) 1 near f =1 and that 1/r e will begin to increase as r approaches r, . This qualitative behavior will remain valid if we relax the weak-field approximation, demand only that e be small, and solve the resulting equations in 3 and B. We see from such an analysis that the force indeed exceeds 1/r and that the friction is less than the flat-space value. Thus as we approach r„e will reach zero. From this point on, the behavior of the system is evident: f will rapidly grow, and such growth will drive e and e even more rapidly toward zero. The result is a curvature singularity at the point of nonregular compactification. A study of the cylindrically symmetric vacuum Ein- stein equations with definite cosmological constant yields a Kasner metric: e ~=cos 3A 4 1/2 p 2B 4 . 2 p e = sin 3A 3A 4 ' 1/2 p cos 3A 4 ' 1/2 p This metric is singular at po=ir/i/3A and corresponds to a solution to Eqs. (2.7)—(2.10) with f—=0, and A=2m. GA, g . This is also, however, an asymptotic solu- tion to the equations for f~ 1 and f ' ~0 as p~ po. While this solution would appear to have some physical significance, since the Higgs field approaches an energy minimum, a numerical analysis has shown that if we evolve f back from po it will grow rather than decrease. eB=1—2p lnr+c, e "=1—p lnr, where p=4+Gg . This approximation turns out to be valid out to p «p, where This demonstrates that we cannot match this solution with one satisfying our imposed boundary conditions at the origin, since if f ) 1 then f cannot have a turning point. VI. CONCLUSIONS The purpose of this paper was to see whether one can talk meaningfully about the gravitational field of a static isolated global string. There are just two boost-invariant vacuum metrics with whole-cylinder symmetry. One cor- responds to an asymptotically flat conical metric, the second to an asymptotically Melvin-type metric. In the first case the transverse (p, 9) space opens out to form a flat cone with finite deficit angle 5 &2m. . In the second it closes up but never quite "compactifies" —the (p, 8) sur- face resembles an infinitely long onion. The conical metric corresponds to the gravitational field of a local string. The onion metric corresponds to the gravitational field of a Melvin flux tube. Before un- dertaking this investigation we had thought that the onion metric might be relevant for the gravitational field of a global string. In fact this is not so. We have shown that there is no universal well-behaved form of the metric to which all static global strings tend. We have also proved that there are no static solutions in which the transverse (p, O) directions are compactified. In fact the gravitational field of a global string is singular at infinity and our results show that the Higgs field also blows up at infinity. From a physical point of view this singularity is prob- ably unimportant. The distance away from the string at which its effects would be felt is of the order of 1 1 p, — — exp SirGi) For realistic values of X and g this distance is enormous. It is far larger than the horizon size at present. At such large distances it is unreasonable to consider only the field of the string alone. The other matter in the Universe would completely modify the metric. However, a consequence of the impossibility of finding a well- defined asymptotic field of a global string is that, just as in Oat spacetime, one cannot talk about the total energy per unit length of a global string. This is quite different from the local case, where the deficit angle 5 or equivalently Thorne's energy play this role. The situation for nonlinear o.-model strings is some- what different. As in the global case, there are no static solutions with compact transverse (r, 8) sections. But now there are regular everywhere-well-defined metrics with conical asymptotics. Finally, one also has solutions in which the spacetime closes up at a singularity.¹teadded in proof. The critical radius p, is an ex- tremely sensitive function of q. As has recently been pointed out in a paper we received after the completion of this work (Cohen and Kaplan' ), if i1=2X10' CreV (which is rather high for grand unified theory strings) p, = 10 cm, the horizon radius. 39 EXISTENCE OF GLOBAL STRINGS COUPLED TO GRAVITY 1551 ACKNOWLEDGMENTS We would like to thank Tanmay Vachaspati and Ruth Gregory for valuable discussions. F.R.R. is indebted to the Department of Applied Mathematics and Theoretical Physics for their kind hospitality in Cambridge and to the Spanish Ministry of Education and Science for sup- port. APPENDIX Here we prove that the right-hand side of Eq. (2.12), 0, (82 ——38, ) = SIGHT„.", tends to zero at large distances. In the case of local strings, certain assumptions about the way in which f~ 1 are used to prove that lim„„H T„'=0. We present a proof for the global string case where we assume only that the metric is well behaved and that f~ l. From Eqs. (2.7)—(2.10) we get that H" =4rrH — —12(f' —1)4 r'e~ H'= 1 — 4' 0 + 12(f —1) dr' . r&2eB and so, using the appropriate boundary conditions on the axis, Because H remains positive, H' is bounded below, and so we have 4 2 H +12 —1 dr'& ~,~2 8 which implies that Hf Ir e and H(f 1)—fall off as r', where a & —1. Now H =r — dr' 4' +12 ~ —1 dr" 0 0 B and so, since H is positive and hence less than r, asymp- totically H f /r e and H (f 1) wil—l behave as r, where b &0. Hence, H2 2 hm 2 ~ =0= lim H (f 1)2 . — p' —+ oo r r~ oo It remains to show that lim„H f' =0. Now Eq. (2.7) tells us that if at some finite radius f =1, then f blows up as r —+ac. Hence f ( I for all r and, since we cannot have oscillatory behavior, there will be some ro after which f will tend monotonically to 1, so that f' ~0 faster than (f 1) ~0. — On leave of absence from Departamento de Fisica Teorica, Universidad Complutense, Madrid, Spain. ~A. Vilenkin, Phys. Rep. 121, 263 (1985). H. B.Nielsen and P. Olesen, Nucl. Phys. B61,45 (1973). D. Garfinkle, Phys. Rev. D 32, 1323 (1985). 4P. Laguna-Castillo and R. Matzner, Phys. Rev. D 36, 3663 (1987). 5R. Gregory, Phys. Rev. Lett. 59, 740 (1987). G. H. Derrick, J. Math. Phys. 5, 1252 (1964). 7K. Thorne, Phys. Rev. 138, B251 (1965). 8D. Harari and P. Sikivie, Phys. Rev. D 37, 3448 (1988). 9D. Kramer, H. Stephani, M. MacCallum, and E. Herlt, Exact Solutions of Einstein's Field Equations (VEB Deutscher Ver- lag der Wissenschaften, Berlin, 1980). C. W. Misner, Phys. Rev. D 18, 4510 (1978). A. Comtet and G. W. Gibbons, Nucl. Phys. B299, 719 (1988). M. Gell-Mann and B.Zwiebach, Phys. Lett. 147B, 111 (1984). R. Gregory, Phys. Lett. B 215, 663 (1988). ~4A. D. Cohen and D. B.Kaplan, Phys. Lett. B 215, 67 (1988).