On the difference between Poincaré and Lorentz gravity

Abstract

The Poincaré invariance of GR is usually interpreted as Lorentz invariance plus diffeomorphism invariance. In this paper, by introducing the local inertial coordinates (LIC), it is shown that a theory with Lorentz and diffeomorphism invariance is not necessarily Poincaré invariant. Actually, the energy-momentum conservation is violated there. On the other hand, with the help of the LIC, the Poincaré invariance is reinterpreted as an internal symmetry. In this formalism, the conservation law is derived, which has not been sufficiently explored before.

de Sitter-covariant Hamiltonian formalism of Einstein--Cartan gravity

“…Firstly, the diffeomorphism symmetry is a fundamental symmetry, which does not correspond to any conservation law directly. In fact, the energy-momentum conservation results from both the translation and diffeomorphism invariance, and likewise, the angular momentum conservation results from both the Lorentz and diffeomorphism invariance [8]. Secondly, the distributional dS constraint satisfies {H(Ω), H(Ω ′ )} = −H([Ω, Ω ′ ]), indicating that the localization of the dS group does not deform the dS algebra, including the translation algebra embedded in it.…”

“…Then the gauge transformations consist of the P/dS/AdS and diffeomorphism transformations, acting on the 5d connection and a 5d vector field ξ A . Actually, ξ A constitutes a system of local 5d Minkowski coordinates, named the local inertial coordinates (LIC) [5,8].…”