26
3 Scalar-Tensor Gravities
This is a very broad class of metrics including Schwarzschild, Reissner–Nordström
and many other metrics. As we already said, in this case the non-zero components of
the curvature tensor are R [ab][ab] , so, the consistency condition (3.17) is automatically
satisfied. For this metric, one has only non-zero components of the Ricci tensor
R
r
r =
A
r A 2 , R
θ
θ =
1
r 2 (1 −
1
A
) +
A
r A 2 . Then, we can consider the vacuum case T
μν
=
0, and choose the vector v μ = ∂ μ ϑ to be purely timelike, v μ = (
1
μ
, 0), with μ =
const, i.e. ϑ =
t
μ
. In this case, the components C
00 and C
0i
= C
i0 of the Cotton
tensor immediately vanish [39]. A bit more involved calculation (see details in [39])
allows to show that the C
i j components also vanish. As a result, we conclude that
the spherically symmetric static solutions of the usual Einstein equations solve the
modified Eq. (3.16) as well. It is clear that if one suggests the ϑ to be dynamical, the
Eq. (3.19) for ϑ will be satisfied if the potential is zero, and ϑ =
t
μ
. We note that this
choice for ϑ is a particular case of the expression ϑ = k μ x
μ used within studies of
the Lorentz symmetry breaking in CSMG which we will discuss further.
Moreover, it has been shown in [41] that all, even non-static ones, spherically
symmetric metrics given by
ds
2
= g μν (x
λ
)dx
μ dx
ν
+
2
(x
ρ
)d
2
,
(3.21)
where d
2 is the 2-sphere line element, so that the coordinates on the sphere are x
i ,
and x
μ are two remaining coordinates (one of them is necessarily timelike), solve
the modified Einstein equations (3.16) for
ϑ = F(x
μ
) + (x
μ
)G(x
i
),
(3.22)
where G(x
i
) and F(x
γ
) are the arbitrary functions of sphere coordinates and remaining coordinates respectively, and is defined in (3.21). The class of spherically
symmetric metrics (3.21) involves not only the static ones (3.20) but also many
other metrics, including the FRW cosmological metric (the cosmological aspects of
CSMG were also discussed in many papers, f.e. in [42]). Some types of metrics with
cylindrical symmetry were also shown in [41] to be consistent within the CSMG.
Now, let us discuss the consistency of the Gödel-type metric (2.26) in CSMG. We
consider the equations of motion (3.16) in the tetrad base, following [43].
In the non-dynamical case, with appropriate choice of units, the Eq. (3.16) imply
R AB + C AB = κ
T AB −
1
2
η AB T
+ η AB ;
(3.23)
C
AB
= −
1
2
[ε
C ADE
(∇ D R
B
E )∂ C ϑ +
∗ R
E AFB
∇ E ∇ F ϑ] + (A ↔ B).
The divergence of modified Einstein equations is
∇ A C
AB
=
1
8
∗ R R∂
B
ϑ.
(3.24)
3 Scalar-Tensor Gravities
This is a very broad class of metrics including Schwarzschild, Reissner–Nordström
and many other metrics. As we already said, in this case the non-zero components of
the curvature tensor are R [ab][ab] , so, the consistency condition (3.17) is automatically
satisfied. For this metric, one has only non-zero components of the Ricci tensor
R
r
r =
A
r A 2 , R
θ
θ =
1
r 2 (1 −
1
A
) +
A
r A 2 . Then, we can consider the vacuum case T
μν
=
0, and choose the vector v μ = ∂ μ ϑ to be purely timelike, v μ = (
1
μ
, 0), with μ =
const, i.e. ϑ =
t
μ
. In this case, the components C
00 and C
0i
= C
i0 of the Cotton
tensor immediately vanish [39]. A bit more involved calculation (see details in [39])
allows to show that the C
i j components also vanish. As a result, we conclude that
the spherically symmetric static solutions of the usual Einstein equations solve the
modified Eq. (3.16) as well. It is clear that if one suggests the ϑ to be dynamical, the
Eq. (3.19) for ϑ will be satisfied if the potential is zero, and ϑ =
t
μ
. We note that this
choice for ϑ is a particular case of the expression ϑ = k μ x
μ used within studies of
the Lorentz symmetry breaking in CSMG which we will discuss further.
Moreover, it has been shown in [41] that all, even non-static ones, spherically
symmetric metrics given by
ds
2
= g μν (x
λ
)dx
μ dx
ν
+
2
(x
ρ
)d
2
,
(3.21)
where d
2 is the 2-sphere line element, so that the coordinates on the sphere are x
i ,
and x
μ are two remaining coordinates (one of them is necessarily timelike), solve
the modified Einstein equations (3.16) for
ϑ = F(x
μ
) + (x
μ
)G(x
i
),
(3.22)
where G(x
i
) and F(x
γ
) are the arbitrary functions of sphere coordinates and remaining coordinates respectively, and is defined in (3.21). The class of spherically
symmetric metrics (3.21) involves not only the static ones (3.20) but also many
other metrics, including the FRW cosmological metric (the cosmological aspects of
CSMG were also discussed in many papers, f.e. in [42]). Some types of metrics with
cylindrical symmetry were also shown in [41] to be consistent within the CSMG.
Now, let us discuss the consistency of the Gödel-type metric (2.26) in CSMG. We
consider the equations of motion (3.16) in the tetrad base, following [43].
In the non-dynamical case, with appropriate choice of units, the Eq. (3.16) imply
R AB + C AB = κ
T AB −
1
2
η AB T
+ η AB ;
(3.23)
C
AB
= −
1
2
[ε
C ADE
(∇ D R
B
E )∂ C ϑ +
∗ R
E AFB
∇ E ∇ F ϑ] + (A ↔ B).
The divergence of modified Einstein equations is
∇ A C
AB
=
1
8
∗ R R∂
B
ϑ.
(3.24)