Proposal for the proper gravitational energy–momentum tensor

Abstract

We propose a gravitational energy–momentum (GEMT) tensor of the general relativity obtained using Noether’s theorem. It transforms as a tensor under general coordinate transformations. One of the two indices of the GEMT labels a local Lorentz frame that satisfies the energy–momentum conservation law. The energies for a gravitational wave, a Schwarzschild black hole and a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe are calculated as examples. The gravitational energy of the Schwarzschild black hole exists only outside the horizon, its value being the negative of the black hole mass.