38
4 Vector-Tensor Gravities and Problem of Lorentz …
breaking. Its essence is as follows. One considers the action of the metric tensor
coupled to the vector field (again, similarly to the previous chapter, this vector field
is treated as an ingredient of gravity model itself but not a matter, thus, we have
the vector-tensor gravity) so that the purely metric sector is presented by the usual
Einstein–Hilbert action, and the dynamics of the vector field is described by the
Maxwell-like term, plus a potential whose minimum yields a vector implementing
the Lorentz symmetry breaking, and maybe also some extra terms responsible for
a vector-gravity coupling. The paradigmatic example is the bumblebee action [78]
(the name “bumblebee” itself was introduced in [79]), looking like
S =
d 4 x
|g|
1
16πG
(R + ξ B μ B ν R μν ) −
1
4
B μν B μν − V (B μ B μ ± b 2 )
. (4.1)
Here ξ is a dimensionless constant, B μν = ∂ μ B ν − ∂ ν B μ is the stress tensor for the
bumblebee field B μ , and V is the potential possessing an infinite set of minima B 0μ
satisfying the condition B
μ
0 B 0μ = ±b
2 (the difference of signs reflects that the vector
B 0μ can be either time-like or space-like, while b
2
> 0). So, actually choosing of one
of the vacua B 0μ allows to introduce the privileged direction. The potential is usually
chosen to be quartic in the field B μ by renormalizability reasons. Alternatively, one
can deal with Einstein-aether theory where, instead of this, the minima arise due to a
constraint multiplied by a Lagrange multiplier σ, so that one has V = σ(B
μ B μ ± b
2
),
but the kinetic term is not Maxwell-like being a more generic quadratic function of
covariant derivatives of the vector B μ . In principle, one can consider the vector-tensor
gravity models without any potential [80], however, in this case the spontaneous
Lorentz symmetry breaking cannot occur. Such theories are considered mostly within
the cosmological context (see f.e. [80]).
Within this chapter, we discuss some interesting classical results for the Einsteinaether gravity and for the bumblebee gravity. At the end of the chapter, we also
will review some terms proposed in [73, 74] as possible extensions of the Einstein
gravity allowing to break the Lorentz symmetry explicitly. As for the Horava-Lifshitz
gravity, although it represents itself as an example of non-Lorentz-invariant gravity
model, it is described in terms of the essentially distinct methodology and will be
discussed in the next chapter.
4.2 Einstein-aether Gravity
So, let us implement the spontaneous Lorentz symmetry breaking in a curved spacetime. To justify importance of this approach, one can remind that namely the spontaneous breaking mechanism has been initially proposed to explain the origin of
the Lorentz symmetry breaking in the low-energy limit of the string theory [81].
Following this concept, one considers a vector field B μ with a constant square, i.e.
B
μ B μ = ±b
2 , which is implemented via introducing the constraint with use of the
Lagrange multiplier σ, adding to the Lagrangian the potential V = σ(B
μ B μ ± b
2
).
4 Vector-Tensor Gravities and Problem of Lorentz …
breaking. Its essence is as follows. One considers the action of the metric tensor
coupled to the vector field (again, similarly to the previous chapter, this vector field
is treated as an ingredient of gravity model itself but not a matter, thus, we have
the vector-tensor gravity) so that the purely metric sector is presented by the usual
Einstein–Hilbert action, and the dynamics of the vector field is described by the
Maxwell-like term, plus a potential whose minimum yields a vector implementing
the Lorentz symmetry breaking, and maybe also some extra terms responsible for
a vector-gravity coupling. The paradigmatic example is the bumblebee action [78]
(the name “bumblebee” itself was introduced in [79]), looking like
S =
d 4 x
|g|
1
16πG
(R + ξ B μ B ν R μν ) −
1
4
B μν B μν − V (B μ B μ ± b 2 )
. (4.1)
Here ξ is a dimensionless constant, B μν = ∂ μ B ν − ∂ ν B μ is the stress tensor for the
bumblebee field B μ , and V is the potential possessing an infinite set of minima B 0μ
satisfying the condition B
μ
0 B 0μ = ±b
2 (the difference of signs reflects that the vector
B 0μ can be either time-like or space-like, while b
2
> 0). So, actually choosing of one
of the vacua B 0μ allows to introduce the privileged direction. The potential is usually
chosen to be quartic in the field B μ by renormalizability reasons. Alternatively, one
can deal with Einstein-aether theory where, instead of this, the minima arise due to a
constraint multiplied by a Lagrange multiplier σ, so that one has V = σ(B
μ B μ ± b
2
),
but the kinetic term is not Maxwell-like being a more generic quadratic function of
covariant derivatives of the vector B μ . In principle, one can consider the vector-tensor
gravity models without any potential [80], however, in this case the spontaneous
Lorentz symmetry breaking cannot occur. Such theories are considered mostly within
the cosmological context (see f.e. [80]).
Within this chapter, we discuss some interesting classical results for the Einsteinaether gravity and for the bumblebee gravity. At the end of the chapter, we also
will review some terms proposed in [73, 74] as possible extensions of the Einstein
gravity allowing to break the Lorentz symmetry explicitly. As for the Horava-Lifshitz
gravity, although it represents itself as an example of non-Lorentz-invariant gravity
model, it is described in terms of the essentially distinct methodology and will be
discussed in the next chapter.
4.2 Einstein-aether Gravity
So, let us implement the spontaneous Lorentz symmetry breaking in a curved spacetime. To justify importance of this approach, one can remind that namely the spontaneous breaking mechanism has been initially proposed to explain the origin of
the Lorentz symmetry breaking in the low-energy limit of the string theory [81].
Following this concept, one considers a vector field B μ with a constant square, i.e.
B
μ B μ = ±b
2 , which is implemented via introducing the constraint with use of the
Lagrange multiplier σ, adding to the Lagrangian the potential V = σ(B
μ B μ ± b
2
).