On the difference between Poincar\'e and Lorentz gravity

Abstract

The Poincar\'e 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 Poincar\'e invariant. Actually, the energy-momentum conservation is violated there. On the other hand, with the help of the LIC, the Poincar\'e invariance is reinterpreted as an internal symmetry. In this formalism, the conservation law is derived, which has not been sufficiently explored before.