Restoring Lorentz symmetries in parity-even Einstein–Cartan domain walls conformal to Bianchi I metric embedded in weak gravity Weitzenböck vacuum manifold

Abstract It is well known that a weak gravity limit can be associated with a restoration of Lorentz covariance. In this paper we present a physical example of a parity-even Einstein–Cartan domain wall (ECDW) embedded in a Weitzenböck zero connection vacuum manifold in the weak torsion gravity limit. Off the wall, the manifold is almost Minkowskian, and the teleparallel equivalent of general relativity is restored (TEGR). ECDW non-dynamical torsion gravity implies that the regions off the wall would naturally be Minkowskian. Thus, our generalization here is used for a parity-even ECDW in Weitzenböck geometry to allow for coupling the wall to the weak gravitational waves in teleparallel gravity. Recently, Li et al. reported giving up Weitzenböck geometry, and they also argued that local Lorentz invariance (LLI) does not affect experiments. But here we argue that in these lab experiments, weak Weitzenböck gravity is used and the specific ECDW embedded in the Weitzenböck manifold vacuum can be considered equivalent to GR (TEGR). Since conformal Bianchi type I metrics are in general not Bianchi I, we do not address here the use of the proper Weitzenböck frame instead of the orthonormal frame which introduces the local covariance violation.