Remnant group of local Lorentz transformations in f(T) theories

TL;DR

This paper investigates the residual local Lorentz invariance in f(T) gravity theories, showing that the set of tetrads preserving the spacetime structure is large and includes Abelian subgroups, with implications for symmetry and frame choice.

Contribution

It demonstrates the existence of a remnant group of local Lorentz transformations in f(T) theories, expanding understanding of their symmetry properties and frame dependence.

Findings

The remnant group includes one and two-dimensional Abelian subgroups.

Minkowski spacetime's symmetry reflects the absence of gravity's influence.

The set of tetrads for spacetime parallelization is quite extensive.

Abstract

It is shown that the extended teleparallel gravitational theories, known as f(T) theories, inherit some on shell local Lorentz invariance associated with the tetrad field defining the spacetime structure. We discuss some enlightening examples, such as Minkowski spacetime and cosmological (Friedmann-Robertson-Walker and Bianchi type I) manifolds. In the first case, we show that the absence of gravity reveals itself as an incapability in the selection of a preferred parallelization at a local level, due to the fact that the infinitesimal local Lorentz subgroup acts as a symmetry group of the frame characterizing Minkowski spacetime. Finite transformations are also discussed in these examples and, contrary to the common lore on the subject, we conclude that the set of tetrads responsible for the parallelization of these manifolds is quite vast and that the remnant group of local Lorentz transformations includes one and two dimensional Abelian subgroups of the Lorentz group.