Introduction to Modified Gravity by Albert Petrov

Author:Albert Petrov
Language: eng
Format: epub
ISBN: 9783030528621
Publisher: Springer International Publishing

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

A. PetrovIntroduction to Modified GravitySpringerBriefs in Physicshttps://doi.org/10.1007/978-3-030-52862-1_4

4. Vector-Tensor Gravities and Problem of Lorentz Symmetry Breaking in Gravity

Albert Petrov1

(1)Department of Physics, Federal University of Paraíba, Joao Pessoa, Brazil

Albert Petrov

Email: [email protected]

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 is defined to satisfy the condition , this condition cannot be applied in a curved space since it breaks the general covariance. A possible “covariant extension” of this condition like 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 ). 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 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 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”

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