3.2 Chern–Simons Modified Gravity
27
In tetrad base, the components of Ricci tensor for Gödel-type metric are constant,
which is an essential advantage of this base. Actually, one has
R 00 = 2ω
2
, R 11 = R 22 = 2ω
2
− m
2
, R = 2(ω
2
− m
2
).
(3.25)
Following the methodology described in [44], we consider three cases of H and D
consistent with the conditions of space-time homogeneity of the metric (2.27):
(i) hyperbolic, H =
2ω
m 2 [cosh mr − 1], D =
1
m
sinh mr;
(ii) trigonometric, H =
2ω
μ 2 [1 − cos μr ], D =
1
m
sin μr ; μ
2
= −m
2 ;
(iii) linear, H = ωr
2 , D = r .
Repeating the argumentation from [44], one immediately sees that for 0 < m
2
< 4ω
2 ,
there is a noncausal region with r > r c , where sinh
2 mr c
2
= (
4ω
2
m 2 − 1).
So, at m
2
≥ 4ω
2 there is no problems with causality.
Now let us choose the matter. We have three most important its examples [43,
44]:
(i) Fluid, T AB = (ρ + p)u A u B + pη AB , u
A
= (1, 0, 0, 0), T 00 = ρ, T 11,22,33 = p.
(ii) Scalar, ψ = s(z − z 0 ), T 00,33 =
s
2
2
, T 11,22 = −
s
2
2
.
(iii) Electromagnetism,
F 03 = −F 30 = e sin[2(z − z 0 )],
F 12 = −F 21 = −
E cos(2(z − z 0 )), T 00,11,22 =
e
2
2
, T 33 = −
e
2
2
.
The matter can be presented by composition of these three types. Then, the nonzero components of the Cotton tensor in this base look like
C 00 = 2
∂ϑ
∂z
ω(4ω
2
− m
2
); C 11 = C 22 =
1
2
C 00 ;
C 01 = −
1
2
∂
2
ϑ
∂z∂t
H
D
(4ω
2
− m
2
);
C 02 = −
1
2
∂
2
ϑ
∂z∂r
(4ω
2
− m
2
);
C 03 = −
1
2
∂ϑ
∂t
ω(4ω
2
− m
2
);
C 13 = −
1
2
∂
2
ϑ
∂t 2
H
D
(4ω
2
− m
2
);
C 23 =
1
2
∂
2
ϑ
∂r ∂t
(4ω
2
− m
2
).
(3.26)
It is clear that the Cotton tensor is traceless, C
A
A = 0. To cancel the off-diagonal
components of C AB we choose ϑ(z) = b(z − z 0 ) which matches the suggestion done
above that the vector v M = ∂ M ϑ is constant, which will be further used to study the
Lorentz symmetry breaking. We introduce also k = bω, and require 4ω
2
= m
2 .
The system of the modified Einstein equations (for 00, 11=22, 33 components
respectively) looks like: