42
4 Vector-Tensor Gravities and Problem of Lorentz …
discussed in [79], as a result, the Schwarzschild-like solution will carry extra factor
e
±2K i r
α , where α is a constant, the sign + is for the temporal component, and the sign
− for the radial one, with the values of K i are different for these two components.
Therefore, we conclude that the Lorentz symmetry breaking generates the black hole
solutions.
Another important example is the cosmological FRW metric. Here we review
its description within the bumblebee context presented in [89]. Explicitly, as a
first attempt, we suggest the vector B μ to be directed along the time axis, B μ =
(B(t), 0, 0, 0). Evidently, in this case the stress tensor for the bumblebee field vanishes, and the only nontrivial component of the equations of motion for the B μ is
V
−
3
2κ 2
¨
a
a
B = 0.
(4.12)
Thus, the bumblebee field either vanishes or, at ξ = 0, stays at one of the minima
of the potential. In this case, it is possible to show numerically that one has the de
Sitter-like expansion of the Universe.
More generic solutions can be obtained for B μν = 0. However, in this case the
numerical analysis is necessary. Explicit studies carried out in [89] show that in
this case, de Sitter-like solutions arise for many values of parameters of the theory
confirming this a possibility to have a cosmic acceleration due to the bumblebee
field, therefore, one can conclude that the spontaneous Lorentz symmetry breaking
can explain the dark energy problem.
Finally, we consider also the Gödel solution (1.8). Within the bumblebee context
it has been considered in [90]. In this case, the energy-momentum tensor is suggested
to be a sum of that one for the relativistic fluid (we note that namely this form has
been employed in [3]):
T
M
μν = ρv μ v ν + g μν ,
(4.13)
and that one for the bumblebee:
T
B
μν = B μα B
α
ν −
1
4
g μν B λρ B
λρ
− V g μν + 2V
B μ B ν ,
(4.14)
where V
is a derivative of the potential with respect to its argument. Therefore, the
modified Einstein equation (in an appropriate system of units where κ = 1) looks
like
G μν = T
M
μν + T
B
μν .
(4.15)
The Einstein tensor G μν and the matter energy-momentum tensor T
M
μν (4.13) in the
bumblebee gravity are the same as in the usual Einstein gravity with the cosmological
term. Therefore, the Gödel metric continues to be solution in our theory if and only
if the energy-momentum tensor of the bumblebee field will vanish. To achieve this
4 Vector-Tensor Gravities and Problem of Lorentz …
discussed in [79], as a result, the Schwarzschild-like solution will carry extra factor
e
±2K i r
α , where α is a constant, the sign + is for the temporal component, and the sign
− for the radial one, with the values of K i are different for these two components.
Therefore, we conclude that the Lorentz symmetry breaking generates the black hole
solutions.
Another important example is the cosmological FRW metric. Here we review
its description within the bumblebee context presented in [89]. Explicitly, as a
first attempt, we suggest the vector B μ to be directed along the time axis, B μ =
(B(t), 0, 0, 0). Evidently, in this case the stress tensor for the bumblebee field vanishes, and the only nontrivial component of the equations of motion for the B μ is
V
−
3
2κ 2
¨
a
a
B = 0.
(4.12)
Thus, the bumblebee field either vanishes or, at ξ = 0, stays at one of the minima
of the potential. In this case, it is possible to show numerically that one has the de
Sitter-like expansion of the Universe.
More generic solutions can be obtained for B μν = 0. However, in this case the
numerical analysis is necessary. Explicit studies carried out in [89] show that in
this case, de Sitter-like solutions arise for many values of parameters of the theory
confirming this a possibility to have a cosmic acceleration due to the bumblebee
field, therefore, one can conclude that the spontaneous Lorentz symmetry breaking
can explain the dark energy problem.
Finally, we consider also the Gödel solution (1.8). Within the bumblebee context
it has been considered in [90]. In this case, the energy-momentum tensor is suggested
to be a sum of that one for the relativistic fluid (we note that namely this form has
been employed in [3]):
T
M
μν = ρv μ v ν + g μν ,
(4.13)
and that one for the bumblebee:
T
B
μν = B μα B
α
ν −
1
4
g μν B λρ B
λρ
− V g μν + 2V
B μ B ν ,
(4.14)
where V
is a derivative of the potential with respect to its argument. Therefore, the
modified Einstein equation (in an appropriate system of units where κ = 1) looks
like
G μν = T
M
μν + T
B
μν .
(4.15)
The Einstein tensor G μν and the matter energy-momentum tensor T
M
μν (4.13) in the
bumblebee gravity are the same as in the usual Einstein gravity with the cosmological
term. Therefore, the Gödel metric continues to be solution in our theory if and only
if the energy-momentum tensor of the bumblebee field will vanish. To achieve this