40
4 Vector-Tensor Gravities and Problem of Lorentz …
in these power series, up to 1/r
3 terms in large r limit have been explicitly found in
certain cases.
For example, treating the black holes solutions, one can show [83] that the metric
ds
2
=
1 −
2M
r
+
2β M
2
r 2
dt
2
−
1 −
2γ M
r
(dr
2
+ r
2 d
2
).
(4.6)
is consistent in this theory, with γ = 1 (the usual value characteristic for
Schwarzschild metric) and β expressed in terms of coefficients c 1 , c 2 , c 3 . Actually
this solution is the Schwarzschild metric modified by the additive term.
Similarly, much more solutions for the Einstein-aether gravity can be obtained,
in particular, the cosmological ones. In this context, the detailed study of various
cosmological aspects of this theory has been performed in [84] where the model
involving two scalar fields coupled to Einstein-aether gravity was considered, and it
has been explicitly demonstrated on the base of the numerical analysis of solutions
that the consistent potential for these fields is the exponential one, and the de Sitterlike solutions can arise both in the past (inflationary Universe) and in the future (de
Sitter attractor). Earlier the idea of using the Einstein-aether model in order to explain
the cosmic acceleration has been claimed in [85]. All this allows to conclude that the
Einstein-aether gravity can be considered as an acceptable solution of the dark energy
problem. Besides of this, a detailed discussion of various aspects of Einstein-aether
gravity, including discussion of plane wave solutions and observational constraints on
parameters of the theory, can be found in [86]. Also, we note that the Einstein-aether
gravity also displays some similarity to the Einstein–Maxwell theory, see [82].
However, it is clear that the Einstein-aether model is problematic from the quantum
viewpoint. Indeed, its action involves a constraint. As it is well known (see f.e.
[87]), a theory with constraints, being considered at the perturbative level, requires
special methodologies like 1/N expansion which clearly cannot be applied to the
Einstein-aether gravity since it involves only four fields u
μ . Moreover, in principle
such a theory, when treated in an improper manner, can display various instabilities.
Therefore, the natural idea consists in introducing the spontaneous Lorentz symmetry
breaking not through constraints but through introducing some potential of the B μ
field displaying a set of minima. This idea gave origin to the bumblebee gravity [78,
79] which we begin to discuss now.
4.3 Bumblebee Gravity
So, let us start with considering the bumblebee gravity. Our initial point will be the
action (4.1). The key features of this action, in comparison with the Einstein-aether
theory, are the following ones.
First, this action is characterized by a generic potential, instead of the constraint,
which makes it better for quantum studies since the usual perturbative methodology
can be applied. Second, the kinetic term is Maxwell-like which is essential to avoid
4 Vector-Tensor Gravities and Problem of Lorentz …
in these power series, up to 1/r
3 terms in large r limit have been explicitly found in
certain cases.
For example, treating the black holes solutions, one can show [83] that the metric
ds
2
=
1 −
2M
r
+
2β M
2
r 2
dt
2
−
1 −
2γ M
r
(dr
2
+ r
2 d
2
).
(4.6)
is consistent in this theory, with γ = 1 (the usual value characteristic for
Schwarzschild metric) and β expressed in terms of coefficients c 1 , c 2 , c 3 . Actually
this solution is the Schwarzschild metric modified by the additive term.
Similarly, much more solutions for the Einstein-aether gravity can be obtained,
in particular, the cosmological ones. In this context, the detailed study of various
cosmological aspects of this theory has been performed in [84] where the model
involving two scalar fields coupled to Einstein-aether gravity was considered, and it
has been explicitly demonstrated on the base of the numerical analysis of solutions
that the consistent potential for these fields is the exponential one, and the de Sitterlike solutions can arise both in the past (inflationary Universe) and in the future (de
Sitter attractor). Earlier the idea of using the Einstein-aether model in order to explain
the cosmic acceleration has been claimed in [85]. All this allows to conclude that the
Einstein-aether gravity can be considered as an acceptable solution of the dark energy
problem. Besides of this, a detailed discussion of various aspects of Einstein-aether
gravity, including discussion of plane wave solutions and observational constraints on
parameters of the theory, can be found in [86]. Also, we note that the Einstein-aether
gravity also displays some similarity to the Einstein–Maxwell theory, see [82].
However, it is clear that the Einstein-aether model is problematic from the quantum
viewpoint. Indeed, its action involves a constraint. As it is well known (see f.e.
[87]), a theory with constraints, being considered at the perturbative level, requires
special methodologies like 1/N expansion which clearly cannot be applied to the
Einstein-aether gravity since it involves only four fields u
μ . Moreover, in principle
such a theory, when treated in an improper manner, can display various instabilities.
Therefore, the natural idea consists in introducing the spontaneous Lorentz symmetry
breaking not through constraints but through introducing some potential of the B μ
field displaying a set of minima. This idea gave origin to the bumblebee gravity [78,
79] which we begin to discuss now.
4.3 Bumblebee Gravity
So, let us start with considering the bumblebee gravity. Our initial point will be the
action (4.1). The key features of this action, in comparison with the Einstein-aether
theory, are the following ones.
First, this action is characterized by a generic potential, instead of the constraint,
which makes it better for quantum studies since the usual perturbative methodology
can be applied. Second, the kinetic term is Maxwell-like which is essential to avoid