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arXiv:0906.0554v3 [hep-th] 25 Jun 2009
The G¨ odel solution in the modified gravity
C. Furtado,1T. Mariz,2J. R. Nascimento,1A. Yu. Petrov,1and A. F. Santos1
1Departamento de F´ ısica, Universidade Federal da Para´ ıba
Caixa Postal 5008, 58051-970, Jo˜ ao Pessoa, Para´ ıba, Brazil∗
2Instituto de F´ ısica, Universidade Federal de Alagoas
CEP 57072-970, Macei´ o, Alagoas, Brazil†
Abstract
We consider the modified gravity whose action represents itself as a sum of the usual Einstein-
Hilbert action and the gravitational Chern-Simons term and show that the G¨ odel metric solves
the modified equations of motion, thus proving that the closed timelike curves whose presence is
characteristic for the G¨ odel solution are not forbidden in the case of the Chern-Simons modified
gravity as well.
∗Electronic address: furtado,jroberto,petrov,alesandro@fisica.ufpb.br
†Electronic address: tmariz@if.ufal.br
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The idea of modifications of the general relativity has a long history. The motivations for
such modifications arise, first, from the perturbative studies of quantum gravity which have
shown that the general relativity is non-renormalizable [1], second, from the astrophysical
observations which have showed the accelerated expansion of the Universe [2]. The most
popular modifications are based on introduction of additive higher-derivative terms which
are known to improve essentially renormalization properties of the theories [3]. At the same
time, actually large scientific attention is attracted by the Lorentz and/or CPT breaking
modifications of the gravity [4]. The most popular of such modifications, with no doubts,
is the additive gravitational Chern-Simons term [5] which represents itself as an example of
the higher-derivative term. In [5] it was shown that this term introduces parity violation,
i.e., each of two polarizations of gravitational waves travels with the speed of light, but with
different intensity, however without Lorentz violation. Also, in [5] it was shown that the
Schwarzschild solution is compatible with this modification of the gravity. Some interesting
results were obtained for this theory, for example, effects of this modification for bodies on
orbits around the Earth were described in [6], the post-Newtonian expansion was studied in
[7], and some cosmological effects were analyzed in [8, 9]. A conserved, symmetric energy-
momentum (pseudo)tensor for the Chern-Simons modified gravity was constructed in [10],
with in [11] the conserved charges in this theory are discussed in details, in [12] it was shown
that in this theory the Poincar´ e invariance holds, and in [13] the dynamical generation of
this term via perturbative corrections was carried out.
One of the most important issues related to the Chern-Simons modified gravity is the
search for the solutions of the equations of motion for modified gravity. A discussion on
this subject is presented in [14]. In general, inclusion of the gravitational Chern-Simons
term is characterized by the so-called external field θ, and it was shown in [14] that a
wide class of solutions of the usual general relativity involving, in particular, spherically
symmetric and axisymmetric solutions, persists to solve the modified gravity equations for
specific forms of the θ. However, some important metrics, such as Kerr metric, fail to solve
the modified gravity equations and require essential modification [15, 16], with the modified
metrics resolving the modified Einstein equations can be found only via the perturbative
approach.
One of very important solutions in general relativity is the G¨ odel metric [17], representing
itself as a first cosmological solution with rotating matter.This solution is stationary,
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spatially homogeneous, possessing cylindrical symmetry, and its highly nontrivial property
consists in breaking of causality implying in the possibility of the closed timelike curves
(CTCs) in the G¨ odel space, whereas, as it was conjectured by Hawking [18], presence of CTCs
is physically inconsistent. Furthermore, in [19] this metric was generalized in cylindrical
coordinates and the problem of causality was examined with more details, thus it turned
out to be that one can distinguish three different classes of solutions. These solutions are
characterized by the following possibilities: (i) there is no CTCs, (ii) there is an infinite
sequence of alternating causal and noncausal regions, and (iii) there is only one noncausal
region. In the paper [20] the quantities called superenergy and supermomentum which can
be used as criteria of possibility of existence the CTCs were introduced. In [21] the CTC
solutions in the G¨ odel space are discussed in the string context (for a review of different
aspects of CTCs see also f.e. [22]). Another reasons for interest to the G¨ odel solution
consists in the fact that the G¨ odel universe allows for nontrivially embedded black holes
[23]. Different aspects of the G¨ odel solutions are discussed also in [24]. In this paper we
are interested in verifying whether the G¨ odel solution holds in the Chern-Simons modified
gravity.
We start our study with introducing the G¨ odel metric which is written as [17]
ds2= a2?
dt2− dx2+1
2e2xdy2− dz2+ 2exdtdy
?
,
(1)
where a is a positive number. The non-zero Christoffel symbols corresponding to this metric
look like
Γ0
01= 1,
Γ0
12=1
2ex,
Γ1
02=1
2ex,
Γ1
22=1
2e2x,
Γ2
01= −e−x.
(2)
The non-zero components of the Riemann tensor are
R0101= −1
2a2, R0112=1
2a2ex, R0202= −1
4a2e2x, R1212= −3
4a2e2x.
(3)
The corresponding non-zero components of the Ricci tensor look like
R00= 1,R02= R20= ex,R22= e2x.
(4)
Finally, the Ricci scalar is
R =1
a2.
(5)
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It is easy to check that the G¨ odel metric solves the Einstein equations
Rµν−1
2gµνR = 8πGρuµuν+ Λgµν,
(6)
where u is a unit time-like vector whose explicit contravariant components look like uµ=
(1
a,0,0,0) and the corresponding covariant components are uµ= (a,0,aex,0). Indeed, let
us for example consider the (00) component of the Einstein equations, that is
R00−1
2g00R = 8πGρu0u0+ Λg00,
(7)
which leads to
1
2
= 8πGρa2+ Λa2.
(8)
This equation is satisfied if
Λ = −1
2a2,
8πGρ =1
a2,
(9)
or, Λ = −4πGρ. For the other components of the Einstein equations we find the same
condition, i.e., the G¨ odel metric solves the Einstein equations if and only if this condition is
satisfied. This result was found in [17].
Now, let us modify the gravity action by adding the gravitational Chern-Simons term [5].
The resulting Chern Simons modified gravity action looks like
S =
1
16πG
?
d4x
?√−gR +1
4θ∗RR
?
+Smat,
(10)
where R is the scalar curvature, Smatis the matter action and∗RR is the topological invariant
called the Pontryagin term whose explicit definition is
∗RR ≡∗Ra
bcdRbacd,
(11)
where Rbacdis the Riemann tensor and∗Rabcdis the dual Riemann tensor given by
∗Ra
bcd=1
2ǫcdefRabef.
(12)
The function θ is an external scalar field. Alternatively, θ can be interpreted as a dynamical
variable [14, 15], however, in this case the equations of motion acquire additional terms
which makes the analysis more complicated [15], therefore we consider θ as an external field
henceforth. After integration by parts, introducing vµ= ∂µθ, we get
1
4
?
d4xθ∗RR = −
?
d4xvµǫµαβγ(1
2Γσ
ατ∂βΓτ
γσ+1
3Γσ
ατΓτ
βηΓη
γσ),
(13)
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that is, the usual gravitational Chern-Simons term. Varying the action (10) with respect to
the metric, we obtain the modified Einstein equations
Rµν−1
2gµνR + Cµν= 8πGρuµuν,
(14)
where Cµν is the Cotton tensor arising due to the varying of the additive Chern-Simons
term, i.e. (cf. [5]).
δ1
4
?
d4xθ∗RR =
?√−gCµνδgµν.
(15)
The explicit form of the Cotton tensor is [25]
Cµν= −
1
2√−g
?
vσǫσµαβDαRν
β+1
2vστǫσναβRτµ
αβ
?
+(µ ←→ ν),
(16)
with vσ≡ ∂σθ, vστ≡ Dσvτ. Taking the covariant divergence of this equation we find
DµCµν=
1
8√−gvν∗RR.
(17)
Using the Bianchi identity, DµGµν= 0, and suggesting that the mattter terms are diffeo-
morphism invariant, i.e. DµTµν= 0, with Tµν= ρuµuν, one finds that the solution of the
equation (14) requires a consistency condition,
∗RR = 0.
(18)
This consistency condition implies that the diffeomorphism symmetry breaking is suppressed
on-shell (for more details see [5, 12]).
As the G¨ odel metric solves the usual Einstein equations, it can solve the modified ones
if and only if the Cotton tensor vanishes for such a metric. Let us verify whether it is the
case.
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The nontrivial components of the Cotton tensor can be explicitly found and look like
C00=
1
√−g
2√−g
2√−g
1
2√−g
1
√−g
2√−g
1
√−g
1
2√−g
ex
a2
?
2v3+ v31
?
,
C01=
11
a2
?
1
a2
v32− exv30
?
?
?
?
,
C02= −
1
2v3+ v31
,
C03=
1
a2
ex
2a2v3,
1
v2− 2exv0− exv01
?
,
C11=
C13= −
1
2a2
?
exv1+ v20− 2exv00+ v02
?
,
C22=
e−x
a2v3,
1
2a2
C23=
?
−2e−xv2+ 2v0+ v10+ v01
?
.
(19)
The remaining components, that is, C12and C33, are identically equal to zero. Using the
definition of vσand writing vστ= ∂σ∂τθ − Γλ
of the Cotton tensor as
στvλ, we can express the non-zero components
C00=
1
√−g
2√−g
2√−g
1
2√−g
1
√−g
2√−g
1
√−g
1
2√−g
ex
a2
?
2∂3θ + ∂3∂1θ
?
,
C01=
1
1
a2
?
1
a2
∂3∂2θ − ex∂3∂0θ
?
?
?
,
C02= −
1
2∂3θ + ∂3∂1θ
?
,
C03=
1
a2
ex
2a2∂3θ,
1
2a2
e−x
a2∂3θ,
1
2a2
−ex∂0θ − ex∂0∂1θ
?
,
C11=
C13= −
1
?
−2ex∂0∂0θ + ∂0∂2θ + ∂2∂0θ
?
,
C22=
C23=
?
∂1∂0θ + ∂0∂1θ
?
.
(20)
Now, analysing these results, we note that the G¨ odel metric can solve the modified gravity
equation only for the specific form of the external field θ, that is,
θ = F(x,y),
(21)
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which, in particular case, can be rewritten as θ = F(x) + xG(y), i.e., as a reminiscence of
the structure of the θ used in [14] for the spherically symmetric cases. Indeed, in this case
all components of the Cotton tensor vanish, thus making the G¨ odel metric to be a solution
of the modified Einstein equations. Therefore, for this specific choice of the θ function, we
find that the G¨ odel metric is compatible with the Chern Simons modified gravity. As a
consequence, the highly nontrivial property of the G¨ odel metric, that is, the situation with
the possibility for existence of the CTCs in the Chern-Simons modified gravity in the case
of the external field θ in the form (21) does not differ from the case of the usual Einstein
gravity, and the discussion of [19, 20] is applicable in the modified gravity case as well as in
the usual case.
Acknowledgements.
The work has been supported by Conselho Nacional
de Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq),by Coordena¸ c˜ ao de Aper-
fei¸ coamento de Pessoal de N´ ıvel Superior (CAPES: AUX-PE-PROCAD 579/2008) and by
CNPq/PRONEX/FAPESQ.
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