A Proposal of Positive-Definite Local Gravitational Energy Density in General Relativity

TL;DR

This paper introduces a new local gravitational energy density in general relativity using a (2,2)-splitting and Kaluza-Klein approach, which is positive-definite and consistent with known global energy measures.

Contribution

It proposes a novel positive-definite local gravitational energy density derived from a (2,2)-splitting and Kaluza-Klein formalism, aligning with Bondi and ADM energy concepts.

Findings

The local energy density reproduces Bondi and ADM integrals at infinity.

The formalism yields a positive-definite local gravitational energy density.

The approach derives the Bondi mass-loss formula as a flux integral.

Abstract

We propose a 4-dimensional Kaluza-Klein approach to general relativity in the (2,2)-splitting of space-time using the double null gauge. The associated Lagrangian is equivalent to the Einstein-Hilbert Lagrangian, since it yields the same field equations as the E-H Lagrangian does. It is describable as a (1+1)-dimensional Yang-Mills type gauge theory coupled to (1+1)-dimensional matter fields, where the minimal coupling associated with the diffeomorphism group of the 2-dimensional spacelike fibre space automatically appears. Written in the first-order formalism, our Lagrangian density directly yields a non-zero local Hamiltonian density, where the associated time function is the retarded time. From this Hamiltonian density, we obtain a positive-definite local gravitational energy density. In the asymptotically flat space-times, the volume integrals of the proposed local gravitational energy density over suitable 3-dimensional hypersurfaces correctly reproduce the Bondi and the ADM surface integral, at null and spatial infinity, respectively, supporting our proposal. We also obtain the Bondi mass-loss formula as a negative-definite flux integral of a bilinear in the gravitational currents at null infinity.