Chapter 4
Vector-Tensor Gravities and Problem of
Lorentz Symmetry Breaking in Gravity
4.1 Introduction and Motivations
The interest to vector-tensor gravity models strongly increased in recent years. One
of the main motivations to studying these models arises from the idea of the Lorentz
symmetry breaking. Indeed, as it is well known, in the flat space the explicit Lorentz
symmetry breaking is implemented through introduction of a constant vector (tensor)
generating a space-time anisotropy (see f.e. [73, 74]). As we already noted in the
previous chapter, this methodology allowed to define, for example, the Carroll–
Field–Jackiw term (3.7) as well as many other terms discussed in [73]. However,
in the curved space the explicit Lorentz symmetry breaking faces serious problems.
First of all, the definition of the constant vector (tensor) itself in this case becomes
highly controversial: for example, while in the flat space the constant vector k
μ
is defined to satisfy the condition ∂ ν k
μ
= 0, this condition cannot be applied in a
curved space since it breaks the general covariance. A possible “covariant extension”
of this condition like ∇ ν k
μ
= 0 would imply in extra restrictions for the space-time
geometry (and, moreover, nobody could guarantee these restrictions to be satisfied
for a general choice of the vector k μ ). In principle, one can also deal with derivative
expansions of corresponding effective actions, where various orders of derivatives
of “constant” tensors can be obtained (see f.e. [75]), however, it is clear that in this
case the definition of a constant vector (or tensor) simply loses its sense, and such
a vector becomes an extra field. Moreover, in many cases such possible new terms
are not gauge invariant which means that together with the Lorentz symmetry, the
general covariance for such terms is broken as well (the problem of breaking the
general covariance in modified gravity is discussed in details in [76]; in principle,
it should be noted that breaking of general covariance occurs for the term u
μ u
ν R μν
proposed in [77] as a possible example of a CPT-even Lorentz-breaking term for
gravity, as well as for the one-derivative linearized term discussed in [47]).
Therefore, the most appropriate method for implementing the Lorentz symmetry
breaking into a curved space-time turns out to be based on the spontaneous symmetry
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_4
37
Vector-Tensor Gravities and Problem of
Lorentz Symmetry Breaking in Gravity
4.1 Introduction and Motivations
The interest to vector-tensor gravity models strongly increased in recent years. One
of the main motivations to studying these models arises from the idea of the Lorentz
symmetry breaking. Indeed, as it is well known, in the flat space the explicit Lorentz
symmetry breaking is implemented through introduction of a constant vector (tensor)
generating a space-time anisotropy (see f.e. [73, 74]). As we already noted in the
previous chapter, this methodology allowed to define, for example, the Carroll–
Field–Jackiw term (3.7) as well as many other terms discussed in [73]. However,
in the curved space the explicit Lorentz symmetry breaking faces serious problems.
First of all, the definition of the constant vector (tensor) itself in this case becomes
highly controversial: for example, while in the flat space the constant vector k
μ
is defined to satisfy the condition ∂ ν k
μ
= 0, this condition cannot be applied in a
curved space since it breaks the general covariance. A possible “covariant extension”
of this condition like ∇ ν k
μ
= 0 would imply in extra restrictions for the space-time
geometry (and, moreover, nobody could guarantee these restrictions to be satisfied
for a general choice of the vector k μ ). In principle, one can also deal with derivative
expansions of corresponding effective actions, where various orders of derivatives
of “constant” tensors can be obtained (see f.e. [75]), however, it is clear that in this
case the definition of a constant vector (or tensor) simply loses its sense, and such
a vector becomes an extra field. Moreover, in many cases such possible new terms
are not gauge invariant which means that together with the Lorentz symmetry, the
general covariance for such terms is broken as well (the problem of breaking the
general covariance in modified gravity is discussed in details in [76]; in principle,
it should be noted that breaking of general covariance occurs for the term u
μ u
ν R μν
proposed in [77] as a possible example of a CPT-even Lorentz-breaking term for
gravity, as well as for the one-derivative linearized term discussed in [47]).
Therefore, the most appropriate method for implementing the Lorentz symmetry
breaking into a curved space-time turns out to be based on the spontaneous symmetry
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_4
37