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Focusing of geodesic congruences in an accelerated expanding Universe

F.D. Albareti (a), J.A.R. Cembranos (a) and A. de la Cruz-Dombriz (b,c)

(a) Departamento de F́ısica Teórica I, Universidad Complutense de Madrid, E-28040 Madrid, Spain
(b) Astrophysics, Cosmology and Gravity Centre (ACGC),

University of Cape Town, 7701 Rondebosch, Cape Town, South Africa and
(c) Department of Mathematics and Applied Mathematics,

University of Cape Town, 7701 Rondebosch, Cape Town, South Africa.
(Dated: January 9, 2013)

We study the accelerated expansion of the Universe through its consequences on a congruence of
geodesics. We make use of the Raychaudhuri equation which describes the evolution of the expansion
rate for a congruence of timelike or null geodesics. In particular, we focus on the space-time geometry
contribution to this equation. By straightforward calculation from the metric of a Robertson-Walker
cosmological model, it follows that in an accelerated expanding Universe the space-time contribution
to the Raychaudhuri equation is positive for the fundamental congruence, favoring a non-focusing
of the congruence of geodesics. However, the accelerated expansion of the present Universe does not
imply a tendency of the fundamental congruence to diverge. It is shown that this is in fact the case
for certain congruences of timelike geodesics without vorticity. Therefore, the focusing of geodesics
remains feasible in an accelerated expanding Universe. Furthermore, a negative contribution to the
Raychaudhuri equation from space-time geometry which is usually interpreted as the manifestation
of the attractive character of gravity is restored in an accelerated expanding Robertson-Walker
space-time at high speeds.

PACS numbers: 04.50.-h, 98.80.-k, 04.20.-q

I. INTRODUCTION

General Relativity (GR) as a theory that describes the
geometrical structure of space-time and its connection
with the energy content of the Universe has been highly
successful in the last one hundred years. GR is not only
able to give explanation of some discrepancies that had
been arisen with Newtonian theory in the Solar System,
but also provides a satisfactory description of the cosmo-
logical evolution of the space-time.

However, GR is not able to account for the latest
cosmological and astrophysical observations with stan-
dard matter sources. Among these observations are the
anomalous rotational velocities of objects near the edge
of the galaxies, the dynamics of galaxies in clusters, for-
mation of large scale structures or gravitational lensing
of background objects by galaxy clusters, such as the
Bullet Cluster. For the explanation of these observa-
tions, more matter than the standard one is required,
the so-called dark matter (DM) which has an attractive
gravitational character that decelerates the expansion of
the Universe. Although there are many plausible origins
for this component [1], DM is usually assumed to be in
the form of thermal relics that naturally freeze-out with
the right abundance in many extensions of the standard
model of particles [2]. Future experiments will be able to
discriminate among the large number of candidates and
models, such as direct and indirect detection designed
explicitly for their search [3], or even at high energy col-
liders, where they could be produced [4]. Furthermore,
the authors of [5], recently awarded with the Nobel Prize
in physics, discovered the accelerated expansion of the

Universe. This result has been one of the main reasons
for the revision of the cosmological evolution as predicted
by GR. Neither the properties of usual matter and en-
ergy nor the believed properties of DM can account for
this acceleration. One way of circumventing this prob-
lem is to postulating a new kind of fluid, the so-called
dark energy, which has a non-standard equation of state
allowing a positive energy density while having a nega-
tive pressure [6, 7]. Another possibility to generate the
present accelerated expansion of the Universe is by the
modification of GR [8, 9]. Indeed, this type of modifi-
cations could produce an inflationary epoch in the very
early Universe. This epoch is postulated to evade the
problems with the initially homogeneous and isotropic
state of the Universe which arise from considering parti-
cle horizons in Robertson-Walker cosmological models.

In this work, we will analyze the accelerated expan-
sion of the Universe by its description through its effects
on a congruence of geodesics without assuming any spe-
cific gravitational theory. This description has the ad-
vantage of being coordinate-independent. In an acceler-
ated expansion it is expected that neighboring geodesics
with similar direction will increasingly recede from each
other. This evolution may be studied either through the
geodesic deviation equation [10] or alternatively by the
definition of a kinematic quantity, the expansion, which is
quite analogous to the spatial divergence of a vector field.
The expansion rate along the congruence will provide us
with the appropriate tool to analyze the consequences of
the cosmological accelerated expansion of the Universe.
This rate is given by the Raychaudhuri equation (RE),
which will be discussed and used throughout this work.

http://arxiv.org/abs/1208.4201v2


RE was first derived by Raychaudhuri in [11] for congru-
ences of timelike geodesics. Later, Sachs [12] introduced
the corresponding RE for null geodesics. Ehlers et al.

[13] provided a generalization of hydrodynamics to GR
where results analogous to well-known theorems of hy-
drodynamics using RE equation were obtained. Given
a congruence of timelike or null geodesics this equation
provides the contraction or expansion rate of the con-
gruence according to some kinematic properties of the
congruence and the curvature of the space-time where
the congruence is embedded. In GR, the RE together
with assumed energy conditions guarantee the attractive
character of gravity represented by a non-positive con-
tribution to this equation from the space-time geome-
try. Furthermore, the RE has a fundamental role in the
demonstration of important theorems due to Hawking
and Penrose. These theorems assert that singularities of
manifolds representing physical space-times are intrinsic
to GR and not a special property of particular solutions
of Einstein’s equations provided that some energy condi-
tions hold [14, 15].

In this investigation, we shall analyze the evolution of
the expansion of a congruence of geodesics. This is car-
ried out in a Robertson-Walker cosmological model from
a kinematical point of view since no gravitational theory
is assumed beyond the equivalence principle. This work
is organized as follows: First, in Section II, we present a
short review of the derivation of the RE for illustrative
purposes and stress some aspects usually disregarded in
the available literature. Then, the kinematics of a con-
gruence of geodesics in a Robertson-Walker space-time is
studied in Section III. We conclude the paper by giving
our conclusions in Section IV.

II. RAYCHAUDHURI EQUATION

DERIVATION

The RE provides the expansion rate of a congruence
of timelike or null geodesics. In the following, we will
summarize the main results for both cases separately ac-
cording to the notation in [15]. For different reviews and
approaches to the RE, we refer the reader to [16]. In
the following let us use abstract index notation (see [15]
for details). Thus, an expression like ξa represents the
vector itself and not a component in a certain basis. An
expression like gabξ

aφb must be understood as g (ξ,φ).
When explicit mention to the components of a tensor in
a certain basis is made, greek indices will be used, i.e.,
ξµ. The principal advantage of this notation is that it
distinguishes equations that hold between tensors and
equations that only hold for components in a certain ba-
sis. We use a metric signature (−,+,+,+) and the Rie-
mann and Ricci tensors and the Ricci scalar are defined
respectively as R d

abc ≡ ∂bΓ
d
ac− ∂aΓ

d
bc +Γe

acΓ
d
eb −Γe

bcΓ
d
ea ,

Rac ≡ R b
abc and R ≡ Ra

a.

Congruences of timelike geodesics

Let us consider a congruence of timelike geodesics and
its tangent vector field ξa. It is assumed that the tan-
gent vector field is normalized, i.e., ξaξa = −1, which is
always feasible by choosing an affine parameterization of
the congruence. Let us define the (0,2) tensor

Ξab ≡ ∇bξa . (1)

It is easy to verify that

ξaΞab = 0 ; ξbΞab = 0 . (2)

The first expression is a consequence of ξa being normal-
ized whereas the second one is the geodesic equation. As
usual the projector tensor hab is defined by

hab ≡ gab + ξaξb . (3)

This is a spatial metric provided that ξa is hypersurface
orthogonal. This property of the vector field means that
there exists a foliation of the manifold by hypersurfaces
orthogonal to ξa. The metric induced by gab on these
hypersurfaces is hab. For cases where an orthogonal hy-
persurface cannot be found, the projector tensor hab is
still well-defined as the tensor that projects tensors in
a subspace orthogonal to ξa at each point of the mani-
fold. It is worth noticing that Ξab has no components in
the direction of ξa according to expression (2). Conse-
quently, Ξab is rather defined in the subspace orthogonal
to ξa. This subspace is the hypersurfaces tangent space
foliating the manifold when ξa is hypersurface orthogo-
nal. Finally, let us consider the following quantities

θ ≡ Ξabhab , (4)

σab ≡ Ξ(ab) −
1

3
θhab , (5)

ωab ≡ Ξ[ab] ; (6)

which are the trace (with respect to the metric hab, but
the trace is the same respect to gab because of (2)), the
trace-free symmetric part and the antisymmetric part of
Ξab respectively. These quantities are known as the ex-
pansion θ, shear σab and twist ωab of the congruence.
Thus, Ξab can be decomposed as

Ξab =
1

3
θhab + σab + ωab , (7)

which is analogous to the decomposition of the deforma-
tion velocity tensor in hydrodynamics. The meaning of
θ, σab and ωab is clear from the analogy with hydrody-
namics. The expansion θ takes into account the diver-
gence or convergence of nearby geodesics. It may also
be interpreted as a relative change in volume of a fluid
particle. The shear σab provides information about the
deformation of the geodesic congruence without change

2



in volume. Finally, the twist ωab contains the vorticity
of the congruence. Equations for the rate of these kine-
matic quantities along each geodesic can be derived in
the following way. After some calculus, it can be proved

ξc∇cΞab = −Ξc
bΞac +R d

cba ξcξd , (8)

where the Leibniz’s rule and the geodesic equation satis-
fied by ξa have been used. Since we are interested in the
equation for the expansion θ, the trace of (8) becomes

dθ

dτ
= −

1

3
θ2 − σabσ

ab + ωabω
ab −Rabξ

aξb , (9)

where τ is an affine parameter of the geodesics, i.e., the
proper time of an observer moving along a geodesic of the
congruence. Equation (9) is known as the RE [11, 14, 15].
Let us recall that the expansion θ provides a natural
coordinate-independent way of measuring the focusing or
divergence of free-falling matter following a certain con-
gruence of geodesics in the gravitational field, i.e., θ > 0
means divergence and θ < 0 means convergence. The
quantities of the r.h.s. of (9) are scalars, thus any conve-
nient coordinate system can be used to calculate them.
However, it is important to stress that equation (9) pro-
vides the expansion rate as seen by an observer moving
along a geodesic of the congruence. The equivalent coun-
terparts for shear σab and the twist ωab are obtained from
equation (8) by taking the trace-free symmetric part and
the antisymmetric part respectively.

Let us now discuss the character of the contribution
of the different terms in expression (9). The quantities
σabσ

ab and ωabω
ab are non-negative quadratic scalars.

The fact that these scalars are non-negative is due to the
spatial character of the tensor Ξab reflected by (2). This
can be understood if one considers a framework with the
spatial components of hab, i.e., the orthogonal compo-
nents to ξa, taking the values δµν with µ, ν = 1, 2, 3 at
a certain point of the manifold, whereas the components
in the direction of ξa being zero. Then, taking into ac-
count the spatial character of Ξab the values of σµν and
σµν are the same in this special framework. Therefore,
σabσ

ab ≥ 0. The same reasoning is valid for ωab.
Thus, it is clear that both the shear σab and the ex-

pansion θ always tend the congruence to focus whereas
the twist ωab always tends to make it diverge. For the
contribution of space-time geometry, i.e., for the sign of
−Rabξ

aξb, there appear three possibilities:

−Rabξ
aξb > 0 Positive contribution, (10)

− Rabξ
aξb < 0 Negative contribution, (11)

−Rabξ
aξb = 0 Zero contribution. (12)

The implications of the last three inequalities will be the
keystone of this investigation. The effect that the two

first possibilities produce in the congruence may be ex-
pressed as a tendency to diverge and a tendency to focus
respectively. It can also be understood as accelerated ex-
pansion in the case of (10) and decelerated expansion in
the case of (11). On the other hand, the value of Rabξ

aξa

depends both on the point of the manifold where it is
calculated and on the direction of the vector field ξa at
that point. Thus, a positive contribution to the RE may
be obtained at some points of the manifold for certain
directions whereas may be negative at those points for
different directions.

Although it is rarely considered, the expression
Rabξ

aξb admits a geometrical interpretation [17]. Let
ξa(0) be a unit vector and ξa(α) with α = 1, 2, 3 an arbi-
trary set of unit vectors orthogonal to ξa(0). The Gaussian
curvature K(0α) of the geodesic surface generated by ξa(0)
and ξa(α) is

K(0α) = e(0) e(α) Rabcd ξ
a
(0) ξ

b
(α) ξ

c
(0) ξ

d
(α) , (13)

where e(0) and e(α) are the usual Eisenhart e’s which

take the values +1 or −1 according to gab ξ
a
(0) ξ

b
(0) = e(0)

and gab ξ
a
(α) ξ

b
(α) = e(α). As is widely known, the sign

convention for Riemann and Ricci tensors is arbitrary a
priori. However, let us recall that the sign of the Gaussian
curvature possesses an intrinsic meaning sign-convention
independent. Thus, when all these Gaussian curvatures
K(0α) are summed one gets

3
∑

α=0

K(0α) = e(0) Rac ξ
a
(0) ξ

c
(0), (14)

where we have considered that K(00) is zero and the spec-
tral decomposition of the metric

gab =
3

∑

α=0

e(α) ξ
a
(α) ξ

b
(α) (15)

has been used. Manifestly, the sum (14) does not depend
on the election of the set of orthogonal unit vectors ξa(α)
under consideration. Then, eRabξ

aξb is the sum of all the
Gaussian curvatures of the geodesics surfaces generated
by ξa and any set of orthogonal unit vectors to it. This
has been referred to as the mean curvature in the direc-
tion of ξa[17]. Thus, by following our sign convention
e = −1 and the mean curvature Mξa in the direction of
a timelike vector ξa becomes

Mξa ≡ −Rabξ
aξb . (16)

Hence, the contribution of the space-time geometry to the
RE (9) has a clear geometrical interpretation as the mean
curvatureMξa in the direction of the tangent vector field
ξa. Furthermore, the inequalities (10), (11) and (12) can
be rewritten in terms of the mean curvature Mξa in the
direction of ξa as follows

Mξa > 0 Positive contribution, (17)

3



Mξa < 0 Negative contribution, (18)

Mξa = 0 Zero contribution. (19)

Thus, if the mean curvature for a given timelike vector
ξa at one point of the manifold is positive (negative) the
space-time geometry would tend to make geodesics with
tangent vector ξa passing through this point diverge (fo-
cus). This formulation represents a clear advantage over
the previous one in terms of Rabξ

aξa since it does not
depend upon any sign convention nor upon the metric
signature. Let us recall that when energy conditions are
assumed in GR, one always obtains (18) and (19) but
not (17). The fact that in GR Mξa is always negative
or zero is interpreted as a manifestation of the attractive
character of gravity.

Let us return to our discussion about the RE. It is
clear that the sign of the expansion rate dθ/dτ will de-
pend upon all the contributions of the r.h.s of (9). From
their respectively definitions, it is obvious that all these
terms depend on the congruence of geodesics under con-
sideration. However, the terms θ2, σabσ

ab and ωabω
ab

depend on the behavior of the congruence in the neigh-
borhood of the point where they are calculated. Let us
recall that these terms come from ∇bξa, then the tangent
vector field ξa must be known locally in order to calculate
its covariant derivative. Nevertheless, the contribution of
space-time geometry Mξa depends only upon the direc-
tion of the congruence at a given point. Thus, all the
congruences pointing in the same direction at a given
point, will have the same contribution from space-time.
Moreover, there are only two terms that may have a pos-
itive contribution: the term ωabω

ab and the one given
by Mξa . Specifically, we shall restrict ourselves to con-
gruences with ωab = 0 as we shall explain at the end of
this section. Therefore, in this type of congruences the
only term that may have a positive contribution to (9) is
Mξa .

Congruences of null geodesics

An equivalent version of the RE (9) for null geodesics
can be obtained by considering the null tangent vector
field ka to a congruence of null geodesics [14–16]. As
expected the result is

dθ̂

dλ
= −

1

2
θ̂2 − σ̂abσ̂

ab + ω̂abω̂
ab −Rabk

akb , (20)

where the quantities σ̂ab, ω̂ab and θ̂ are defined with re-
spect to the congruence of null geodesics, the tensor ĥab

is the tensor analogous to hab for null geodesics and λ
is an affine parameter in the following sense. When null
geodesics are considered the tangent vector field has al-
ways zero norm and consequently an affine parameter

defined as the one which normalizes the tangent vec-
tor field does not hold. For this reason, an affine pa-
rameter can be defined in a more general way as the
one preserving the geodesic equation in its usual form.
This definition remains valid both for timelike and null
geodesics. The factor 1/2 in (20) instead of 1/3 appear-
ing in (9) is due to the fact that the involved subspace in
the case of null geodesics is two-dimensional rather than
three-dimensional. Thus, equation (20) is the RE for null
geodesics.

Finally, let us remind that the same type of space-
time geometry contributions provided by the inequali-
tites (10), (11) and (12) applies replacing Rabξ

aξb by
Rabk

akb. However, an analogous interpretation of the
term Rabk

akb as a mean curvature is not practicable in
this case [17].

Further remarks

At this stage, we estimate important to stress that the
Raychaudhuri equations (9) and (20) are identities, i.e.,
geometrical statements which do not require any under-
lying gravitational theory to be derived. They are quite
general and apply whenever a Riemannian1 geometry for
the space-time is assumed.

A more general equation for the evolution of the ex-
pansion of a congruence results if the assumption of being
a congruence of geodesics is discarded. Moreover, the RE
can be generalized to spaces with torsion [18]. For our
purposes (9) and (20) are sufficient.

As was briefly mentioned, there are analogous equa-
tions to (9) and (20) for the shear σab(σ̂ab) and twist
ωab(ω̂ab) that provide the rate of these quantities along
geodesics. These equations form a coupled system of non-
linear first order differential equations. They also receive
the generic name of Raychaudhuri equations. For in-
stance, the equation for the twist ωab is an homogeneous
differential equation in ωab and thus if ωab is zero initially
it will remain zero along the geodesics. This result does
not prove that the unique solution is ωab = 0 because of
the non-linearity of the equation. However, this conclu-
sion can be eventually proved more rigorously (see [13]).
This is in fact the analogous result to Kelvin’s theorem
in hydrodynamics showing the analogy between theorems
of hydrodynamics and theorems concerning congruences
of geodesics [13].

The initial value problem of Raychaudhuri equations is
still an open question. However, in the case of equation
(9) if congruences that satisfy ωab = 0 are considered,

1 Let us stress that we use the adjective Riemannian in an extended
sense, i.e., Lorentzian metrics are included.

4



there would exist hypersurfaces orthogonal to the vector
field that foliate the space-time. This is a consequence
of Frobenius’ theorem [15]. Therefore, for a normalized
vector field defined orthogonally to an arbitrary hyper-
surface – not necessarily in the whole manifold – its shear
and its expansion (with components only in that hyper-
surface (2)) can be determined. This way, the expansion
rate of the congruence of geodesics defined by this vector
field can also be determined.

Therefore, the rate of the kinematic quantities of the
congruence (θ, ωab and σab for timelike geodesics and θ̂,
ω̂ab and σ̂ab for null geodesics) is influenced by the geome-
try of space-time. In particular, in the case of expansion
θ (θ̂), the term −Rabξ

aξb (−Rabk
akb) is the contribu-

tion of space-time geometry for timelike geodesics (null
geodesics). Let us recall that the contribution of space-
time geometry has more general implications than the
other contributions since the former depends solely on
the direction of the tangent vector field at one point but
not on the local behavior of the tangent vector field as θ
(θ̂), σab (σ̂ab) and ωab (ω̂ab). Thus, the contribution of
space-time at one point of the manifold is the same for
every congruence of geodesics with the same value of the
tangent vector field at that point. However, the quanti-
ties θ, σab and ωab will be different if the congruences are
not locally the same.

Finally, let us remind here that the field equations de-
rived from the assumed gravitational theory determine
the geometry of the space-time. For instance, in the case
of GR, the usual Einstein’s field equations enable to re-
late the quantities Rabξ

aξb and Rabk
akb which appear in

(9) and (20) respectively with the energy-momentum ten-
sor Tab in a very simple way. Hence, the expansion rate
is straightforwardly related with the energy-momentum
content. As a result, general properties of the behavior of
the geodesic congruence can be established provided that
some energy conditions on the energy-momentum tensor
are assumed [14, 15]. More precisely, if the strong energy
condition and the null energy condition are assumed (al-
though the last one also follows by continuity if the strong
energy condition holds) one reaches the geodesic focusing

theorem, which assures the convergence until zero size of
a congruence of geodesics without vorticity which is ini-
tially focusing. This result is used in the demonstration
of the singularity theorems by Hawking and Penrose and
it has recently drawn wide attention due to its prominent
role in the holographic principle [19].

III. KINEMATIC CONSIDERATIONS IN

ROBERTSON-WALKER SPACE-TIMES

In this section, let us start by considering a spatially
flat Robertson-Walker metric [20],

ds2 = −dt2 + a2(t)
(

dx2 + dy2 + dz2
)

, (21)

that according to the ultimate measurements of cosmo-
logical densities of the Universe [21], represents correctly
the Universe at sufficiently large scales in agreement with
the hypotheses of homogeneity and isotropy stated in the
Cosmological principle. The connection between the in-

equality (10) – or equivalently (17) – with the accelerated
expansion of the Universe is not straightforward. The
accelerated expansion of the Universe in a Robertson-
Walker cosmological model is governed by the condition
ä > 0 where the derivatives are taken with respect to
cosmic time. However, provided that the inequality (17)
holds for all the timelike directions ξa in a region of
the manifold, the contribution of space-time geometry
then favors a tendency to diverge of all the congruence of
geodesics in that region. Therefore, the natural question
that arises is whether the condition ä > 0, when satisfied
in a region, implies Mξa > 0 for every timelike direc-
tion ξa. This question will be addressed throughout the
present section.

One may thus wonder about the effects of the cos-
mological acceleration in the kinematical behavior of a
congruence of timelike geodesics. In particular, it is of
special interest to study whether a congruence of time-
like geodesics tends to focus – or to diverge – and under
which conditions this happens. The kinematic quantity
of a congruence known as expansion θ introduced in Sec-
tion II provides us with a useful tool to study the focusing
of timelike geodesics. In this sense, the inequality

ξc∇cθ =
dθ

dτ
> 0 , (22)

states the tendency of nearby geodesics to diverge mea-
sured by a comoving observer. The proper time τ is the
time of the comoving observer. Since the l.h.s. of the pre-
vious expression (22) can be replaced by the r.h.s. of the
RE (9), a vector field tangent to a congruence of time-
like geodesics must be chosen in order to determine the
expansion θ, shear σab and twist ωab.

Fundamental congruence

Since we are dealing with a Robertson-Walker metric
(21), the most natural choice for a vector field would be

ξaRW =
∂

∂t
, (23)

which is the preferred direction of a Robertson-Walker
space-time because of the local isotropy in hypersur-
faces orthogonal to ξaRW. The vector field (23) gives
rise to a congruence of timelike geodesics of the form
xµ = (t− t0, ~x0) in comoving coordinates with t0 and ~x0

constants 2. These geodesics are followed by the funda-

2 Therefore, being steady is a geodesic motion in Robertson-Walker
metric.

5



mental observers, those seeing homogeneity and isotropy
in constant-time hypersurfaces. Thus, a fundamental
congruence refers to the congruence of timelike geodesics
generated by ξaRW. As it is easy to verify by straightfor-
ward calculation – or by using symmetry arguments –,
the kinematic quantities of the fundamental congruence
are all zero except the expansion

θRW = 3
ȧ

a
. (24)

Thus, by inspection of (9) it follows that the rate of the
expansion seen by a fundamental observer will be entirely
determined by the value of the expansion θRW and the
contribution of space-time geometry

MξaRW
= −Rabξ

a
RWξbRW = 3

ä

a
. (25)

Hence

dθRW
dt

= 3

[

−

(

ȧ

a

)2

+
ä

a

]

, (26)

where the proper time of the fundamental observer is t,
i.e., the time coordinate in (21). A fundamental observer
will then see nearby geodesics with a tendency to diverge
if and only if dθRW/dt > 0. By imposing this condition
on (26) we get

ä >
ȧ2

a
, (27)

which can be expressed in the following way

q ≡ −
1

H2

ä

a
< −1 ⇔

dθRW

dt
> 0 , (28)

where q is the deceleration parameter and H ≡ ȧ/a is the
Hubble parameter. Hence, a fundamental observer will
perceive nearby geodesics with a tendency to diverge if
and only if q < −1. In other words, getting a decel-
eration parameter smaller than zero is not sufficient for
fundamental timelike geodesics to be perceived by a fun-
damental observer as diverging in an increasingly fashion.

In the standard cosmological model ΛCDM, the value
of q today can be determined by using the values of the
present relative abundances of matter (including dark
matter) and cosmological constant, ΩM and ΩΛ respec-
tively by the expression

q =
1

2
ΩM − ΩΛ , (29)

where flatness in spatial sections was again assumed.
Thus, using the approximate values ΩM ≃ 0.27 and
ΩΛ ≃ 0.73 given in [21], the present value of the decelera-
tion parameter q0 results q0 ≃ −0.6, with an uncertainty
of the order of 0.01. This value of q0 does not imply a ten-
dency of nearby geodesics to diverge since q0 > −1. One
may argue that we have assumed GR to calculate the
deceleration parameter q0, and therefore the discussion

is not independent of a gravitational theory. However,
there are straightforward estimations of the deceleration
parameter using Supernovae Ia [22] without appealing to
a gravitational theory. The preferred values obtained for
q0 in these analyses are always larger than −1, although
with uncertainties of the order 0.1.

Before continuing with the study of the kinematics of
a congruence of timelike geodesics, let us mention that
in general, the determination of other geodesics is not
possible in a closed form since an expression for a(t) is
usually missing. Therefore, it will not be possible in gen-
eral to obtain an expression for the tangent vector field
ξa of a congruence of geodesics. However, if our interest
is focused on calculating the space-time geometry contri-
bution to the RE for timelike geodesics, the knowledge of
the vector field ξa expression is not necessary. The reason
is that the value of Mξa only depends on the direction
of the vector field at a given point. Then, given a time-
like vector, the value of Mξa can be determined by using
the expressions for the components of the Ricci tensor.
Moreover, the value of Mξa will be constant in each hy-
persurface orthogonal to ξaRW because of homogeneity in
these hypersurfaces.

However, if we wish to know the total value of dθ/dτ
from the RE, an expression for the tangent vector field
is needed. It means that we must know the vector field
locally at each point of the manifold. Nevertheless, we
can use the property stated in equations (2) which as-
serts the spatial character of Ξab = ∇bξa. The procedure
consists on defining a smooth vector field on a spacelike
hypersurface. The only conditions on the definition of
this vector field are that it be normalized ξaξa = −1
and that it be orthogonal to the hypersurface where it is
defined. This slice of a vector field will give rise to a con-
gruence of timelike geodesics passing by the hypersurface
with the direction of the vector field in each point. This
congruence will have ωab = 0 because being zero in the
initial hypersurface it will remain so. However, the shear
σab and the expansion θ do not need to be zero. Then,
instead of considering a vector field tangent to a con-
gruence of geodesics defined in all the manifold, a vector
field is instead defined at a spacelike hypersurface. Let us
remark that this procedure restricts us to consider con-
gruences of timelike geodesics without vorticity. By this
way Ξab at the initial hypersurface can be determined.
Thus, the expansion rate dθ/dτ of the congruence eval-
uated in the hypersurface will be known from a slice of
the geodesic tangent vector field.

One-parameter dependent congruences of timelike

geodesics

The congruences under consideration from now on are
obtained from a one-parameter dependent slice of a vec-
tor field. First of all, we must choose a spacelike hy-

6



persurface where the vector field will be defined. The
simplest one-parameter dependent ǫ family of spacelike
hypersurfaces defined by a function f = 0 in terms of the
coordinates (21) is

f = t− ~ǫ · ~x = 0 , (30)

where 0 ≤ |~ǫ |/a < 1 in order the hypersurfaces to be
spacelike. This constraint is easily understood if one con-
siders the normal vector to the hypersurfaces (30) and
this vector is then constrained to be timelike. This fam-
ily of hypersurfaces depends in fact on three parameters,
namely the three components of ~ǫ. However, thanks to
the spatial isotropy, the direction of ~ǫ does not affect the
results. Thus, the relevant parameter is |~ǫ | = ǫ. More-
over, because of spatial homogeneity and the arbitrari-
ness in the election of the time origin we have eliminated
the constant that may appear in (30), i.e., the hypersur-
faces pass through the origin of coordinates. The above
restriction on the value of ǫ means that the hypersurface
is defined only in the manifold region a > ǫ. Otherwise,
the hypersurfaces are not spacelike. Let us now define
the following quantity

β ≡
ǫ

a2(t)
. (31)

The slice of the timelike vector field which is normalized
and orthogonal to the hypersurface given by (30) then
becomes

ξa(β) = γ̃

(

∂

∂t
+ β

∂

∂x

)

, (32)

where

γ̃ ≡
1

√

1− a2β2
. (33)

The tangent vector field that gives rise to the fundamen-
tal congruence is obtained in the specific case ǫ = 0 or
equivalently β = 0. In order to guarantee ξa(β) to be time-
like, one imposes 0 ≤ aβ < 1. Then, it follows the re-
striction on the value of ǫ commented above. From (32),
β can be interpreted as the spatial coordinate velocity of
a particle passing through the hypersurfaces defined by
(30). The dependence of β on the scale factor (31) means
that this velocity is not the same in the whole hypersur-
face defined by a certain value of the parameter ǫ. Note
that in expression (32) we have chosen the x-axis in the
direction of the spatial coordinate velocity β without loss
of generality thanks to the spatial isotropy. The γ̃ factor
just ensures the normalization gabξ

a
(β)ξ

b
(β) = −1.

One realizes of the similarity of the former derivation
with the usual definitions of β and γ in Special Rela-
tivity. Consequently, the slice of the timelike vector field
orthogonal to the hypersurfaces (30) in Special Relativity
becomes

ua = γ

(

∂

∂t
+ β

∂

∂x

)

(34)

with 0 ≤ β < 1 in order to guarantee ηabu
aub < 0, with

ηab the Minkowskian metric. Hence, expressions (32) and
(33) are completely valid both in Special Relativity (set-
ting a = 1) and in Robertson-Walker space-times. How-
ever, when we are considering a Robertson-Walker met-
ric, the maximum coordinate velocity βmax that a phys-
ical entity can reach, in the sense that it moves along
null geodesics, depends on the value of scale factor a(t)
at each point. In order to guarantee that the vector field
as given by (32) is timelike, the required condition reads
0 ≤ aβ < 1. Therefore the maximum coordinate velocity
is expressed as βmax = 1/a(t). Nevertheless, let us recall
that the physical spatial velocity which is measured is
aβ. One may interpret a physical velocity as the ratio
between a spatial and time distances. These distances
hold for the ones induced by the timelike geodesic one
follows. Therefore, for a particle with four-velocity ξa(β),
its physical velocity yields

Vphysical = lim
∆t→0

∫ t+∆t

t

√

hRW
ab ξa(β)ξ

b
(β) dt

′

∫ t+∆t

t
−ξRWa ξa(β) dt

′

= aβ (35)

where ξRWa is given by (24) and hRW
ab = gab + ξRWa ξRWb

is the spatial metric induced by the flow of time in the
direction of ξaRW. Alternatively, one may also build a unit
vector orthogonal to ξaRW being the spatial projection of

ξa(β), namely ha ≡ hRW
ab ξb(β)/

√

gachRW
ab ξb(β)h

RW
cd ξd(β), and

then calculate the ratio between the projection of ξa(β) in
the direction of ha and in the direction of ξaRW, finally
obtaining the same result as in (35).

Therefore, after considering the slice of the timelike
vector field ξa(β) (32) defined in the hypersurface (30),
the tensor Ξab = ∇bξa is calculated in order to obtain
both the expansion θ and the shear σab (the twist ωab is
zero by construction). By following this procedure, the
value of dθ/dτ in the hypersurfaces f = t− ~ǫ · ~x will be
determined. This will provide us with information about
the focusing or divergence of the congruence generated
by the vector field slice in terms of the scale factor a(t),
its derivatives and the spatial coordinate velocity β.

However, in the following we shall focus on the mean
curvature Mξa . This is due to the fact that although
σab and θ may not be zero, their contribution to the RE
is always non-positive. Moreover, as stressed in Section
II, the implications of the mean curvature Mξa are more
general than those of the other terms. Furthermore, Mξa

is the unique term that may have a positive contribution
to the RE (9) since for the congruence given rise by the
vector field slice (32) we have ωabω

ab = 0. Hence, let us
proceed with the calculation of Mξa

(β)
in order to extract

information about the focusing of these one-parameter
dependent congruences.

7



Evaluation of the condition Mξa
(β)

> 0

In the following we will obtain a condition on a(t)
and the spatial coordinate velocity β that guarantees
Mξa

(β)
> 0 , i.e., a positive contribution to the RE (17).

The motivation of choosing a positive contribution lies
in the fact that the fundamental congruence satisfies
MξaRW

> 0 in the present epoch as follows from (25).

Then, by using the Ricci tensor components derived
from the metric (21) and the expression for the timelike
vector field slice given in (32), we get

Rabξ
a
(β)ξ

b
(β) = γ̃2

[

−3
ä

a
+ β2

(

aä+ 2ȧ2
)

]

. (36)

Thus, imposing Mξa
(β)

= −Rabξ
a
(β)ξ

b
(β) > 0, yields

− 3
ä

a
+ β2

(

aä+ 2ȧ2
)

< 0 ⇒ ä >
2aβ2ȧ2

3− a2β2
, (37)

where a2β2 − 3 < 0 (because of aβ < 1) have been used.
This last inequality tells us that the condition ä > 0 is
not sufficient for a timelike vector of the form (32) to get
a positive contribution to the RE from the space-time
geometry Mξa

(β)
. Consequently, ä must be larger than a

certain positive value. Therefore, a positive contribution
to the RE from space-time geometry (17) for all timelike
congruences is not equivalent to ä > 0. The inequality
(37) can be recast as

q <
−2a2β2

3− a2β2
, (38)

which depends only on the physical velocity aβ. In Figure
1, the condition (38) has been depicted. In that figure,
the values (aβ, q) ensuring a positive contribution to the
RE from the mean curvature are plotted.
Moreover, it is worthwhile to stress that the require-

ment Mξa
(β)

> 0 implies lower values of the deceleration

parameter q for timelike vectors approaching the null
cone. Then, in the limit aβ → 1, i.e., timelike vectors
near the null cone, we get q < −1. Therefore, provided
that q < −1 all the congruences of timelike geodesics will
have a positive contribution to the RE from space-time
geometry which means Mξa

(β)
> 0 for all timelike vectors

ξa(β). The implication of this condition must not be con-

fused with (28) : the condition (28) implied dθRW/dt > 0
for the fundamental congruence whereas in this case the
condition q < −1 implies a positive contribution from
the space-time geometry, namely Mξa

(β)
> 0, for every

congruence of timelike geodesics. Since the present value
of the deceleration parameter is not smaller than −1, it
remains possible to get Mξa

(β)
< 0 for timelike congru-

ences whose tangent vector field is sufficiently inclined
with respect to ξaRW = ∂/∂t. Therefore, using (38), the
following possibilities for the physical velocity aβ may be

0.0 0.2 0.4 0.6 0.8 1.0
-1.0

-0.8

-0.6

-0.4

-0.2

0.0

aΒ

q

Figure 1. Mean curvature sign for congruence of geodesics gener-
ated by (32): The line represents the limiting case q = −2a2β2/(3−
a2β2). For a given spatial coordinate velocity β and scale factor
a(t) if the deceleration parameter q is below the line (region in
blue) a congruence of geodesics with a physical velocity aβ ac-
quires a positive contribution to the RE from the mean curvature
Mξa or equivalently from the term −Rabξ

aξb. In the region above
the line (white color) we depict the opposite case, i.e., negative
contribution.

fulfilled

aβ <

√

−3q

2− q
Positive contribution, (39)

aβ >

√

−3q

2− q
Negative contribution, (40)

aβ =

√

−3q

2− q
Zero contribution; (41)

to the RE from the space-time geometry.
If β = 0 in (37) we obtain ä > 0 which means that

the fundamental congruence will experience a positive
contribution from the space-time if ä > 0 and therefore
in agreement with (25).

In Figure 2, a future light cone is represented. Using
the value for the present deceleration parameter (q0 ≃
−0.6), the limiting value obtained from (41), aβ ≃ 0.8
separates the regions of different contribution to the RE.
The requirement aβ > 0.8 constitutes a necessary con-
dition to ensure Mξa < 0. However, due to the non-
positive character of all the other terms of the RE for
a congruence without vorticity, dθ/dτ might be negative
despite a positive contribution Mξa > 0.

8



-1.0 -0.5 0.0 0.5 1.0
0.0

0.2

0.4

0.6

0.8

1.0

ax

t

Figure 2. Modifed future light cone in Robertson-Walker metric:
The line t = ax corresponds to the usual light-cone. A positive
contribution to the RE (i.e., Mξa > 0), for timelike vectors ξa lies
in the inner (red) area and expression (39) is there satisfied. For
vectors in the outer (blue) zone Mξa < 0 and condition (40) is now
obeyed. The value of the slope of the straight line that separates the
two regions corresponds to 1/(aβ) = 1/0.8. This value is obtained
using q0 ≃ −0.6 in (41) according to WMAP7 [21]. For smaller
values of q, i.e. closer to −1, the inner region (red) would increase
its area and the outer region (blue) would become smaller. For the
value q = −1, the blue region vanishes and there will be only red
zone which corresponds, according to the condition q < −1 to all
timelike directions having a positive contribution to the RE from
the space-time geometry.

Let us recall that a value of the deceleration parameter
q equals to −1 is obtained for a(t) ∝ eH0t which is the
solution in Robertson-Walker cosmological model of the
Einstein’s equations with cosmological constant Λ = 3H2

0

in vacuum. This solution is usually referred as the de
Sitter expanding universe. This means that a value of
q lower than −1 implies an expansion rate higher than
the rate of expansion of a de Sitter solution. Therefore,
it seems very unlikely to have q < −1 in any epoch of
the evolution of the Universe. Let us recall that even in
the inflation phase the rate of expansion is expected to
be exponential and therefore qinflation = −1. Thus, let
us conclude this section by remarking that for timelike
vectors sufficiently close to the light cone one obtains
Mξa < 0 in any epoch of the evolution of the Universe.

IV. CONCLUSIONS

In this investigation we have studied the RE and its
implications on the evolution of geodesics. We addressed

the kinematics of a congruence of timelike geodesics with-
out vorticity in a Robertson-Walker cosmological model
with no underlying gravitational theory assumed a pri-
ori. From a general point of view, we have established
the difference between the implications of a cosmic accel-
eration ä > 0 and a positive contribution to the RE from
the space-time geometry, i.e. Mξa = −Rabξ

aξb > 0.

We have then considered the fundamental congruence
generated by the vector field ξaRW. We proved that the
tendency to focus or diverge for the fundamental con-
gruence depends of the fact that q is either bigger or
smaller than −1 respectively. Given the fact that an
approximate value for the deceleration parameter today
q0 ≃ −0.6 can be found either by assuming GR or just
by using Supernovae Ia data in theory-independent cal-
culations, one concludes that although the fundamental
congruence is expanding because of θRW = 3ȧ/a > 0, it
has a tendency to focus since dθRW/dt < 0.

Moreover, we have examined the congruence of time-
like geodesics generated by a vector field orthogonal to a
one-parameter ǫ dependent spacelike hypersurface. The
spatial coordinate velocity β = ǫ/a2(t) represented –
at each point of the hypersurface – the inclination of
the vector field with respect to the preferred direction
of Robertson-Walker models ξaRW. In the case of the
slice of a vector field to be normalized and orthogonal
to the hypersurface, it is possible to determine the con-
dition for the deceleration parameter q which ensures
the contribution of the space-time geometry to the RE,
Mξa = −Rabξ

aξb to be positive. The aforementioned
condition depends both upon the spatial coordinate ve-
locity β and the scale factor a(t) but only through the
physical spatial velocity aβ. In congruences without vor-
ticity, if Mξa > 0 does not hold all the contributions to
the RE are non-positive since ωab = 0 and the remain-
ing terms are non-positive as we discussed. We found
that this condition is not satisfied in the present Uni-
verse provided that aβ > 0.8. This means that con-
gruences of geodesics without vorticity with its tangent
vector fulfilling aβ > 0.8 will not tend to diverge. In the
Robertson-Walker preferred framework this situation oc-
curs for geodesics moving near the light cones. These
congruences will tend to focus provided that the other
terms of the RE are negative. Besides, a negative contri-
bution from space-time geometry to the RE would mean
that the attractive character of gravity is restored at high
speeds for −1 < q < 0. The term high speeds must be
understood with respect to the preferred framework of
a Robertson-Walker cosmological model which is usually
identified with the CMB because of its high isotropy.

In the previous sense, systems may decouple from the
cosmological accelerated expansion. This result can help
to understand the structure formation of system moving
fast, that is favored with respect to slow systems. Even
taking into account the present value of the deceleration

9



parameter q0, aβ > 0.8 gives a negative contribution to
the RE from space-time geometry. This means that in
order to get a negative contribution to RE from the space-
time geometry, it is necessary to move with a velocity
bigger than 0.8 c with respect to the preferred framework
of Robertson-Walker space-times. Although this value is
quite large, the deceleration parameter q was smaller in
the past and consequently the velocity respect the CMB
necessary to accomplish a negative contribution to the
RE from space-time geometry was also smaller.

To conclude, we would like to highlight the fact that
the results obtained in this investigation provide us with
a mechanism for a gravitational system to decouple from
the accelerated expansion of the Universe. This mech-
anism does not need to appeal to the gravitational in-
teraction of the system itself. Although the decoupling
of astrophysical systems from the cosmic acceleration is
usually carried out by the stronger effect of their own
gravitational forces, our investigation may help to un-
derstand how geodesic motion in an expanding Universe
influences this decoupling.

Acknowledgments. We would like to thank prof.
G. F. R. Ellis for drawing our attention to reference
[10]. This work has been supported by MICINN (Spain)
projects numbers FIS2011-23000, FPA2011-27853-C02-
01 and Consolider-Ingenio MULTIDARK CSD2009-
00064. FDA would like to thank Theoretical Physics I
department, Complutense University of Madrid for pro-
viding all the required academic resources. AdlCD ac-
knowledges financial support from NRF and URC re-
search fellowships (South Africa). JARC acknowledges
the kind hospitality of ACGC/University of Cape Town
while elaborating the manuscript.

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11

http://arxiv.org/abs/1109.4519
http://arxiv.org/abs/gr-qc/9709060
http://arxiv.org/abs/gr-qc/0511123
http://arxiv.org/abs/1202.4689