On the Energy Problem in General Relativity

TL;DR

This paper examines the longstanding energy problem in general relativity, analyzing its local and global aspects, and proposes a covariant framework to better define gravitational energy-momentum.

Contribution

It introduces a covariant definition of gravitational energy-momentum and a rigorous Lagrange Field Structure on manifolds, addressing inconsistencies in GR's energy concepts.

Findings

Decomposition of Levi-Civita connection into affine and nonmetric parts

Covariant definition of gravitational energy-momentum tensor

Generalization of Noether's theorem for LFS

Abstract

The Energy Problem (EP) in General Relativity (GR) is analyzed in the context of GR's axiomatic inconsistencies. EP is classified according to its local and global aspects. The local aspects of the EP include noncovariance of the energy-momentum pseudotensor (EMPT) of the gravitational field, non-uniqueness of the EMPT, asymmetry of EMPT, and vanishing metric energy-momentum tensor. The global aspect of the EP relates to the lack of integral conservation laws due to the general difficulties in defining invariant integrals of tensors in non-Euclidean space. These difficulties are related to the lack of precise definition of a reference frame in the GR. A reference frame is defined here as a differential manifold with an affine connection. The resulting unique decomposition of the Levi-Civita connection into its affine and nonmetric parts allows for a covariant definition of the gravitational energy-momentum tensor. It is pointed out that the invariance of the Lagrangian (or action functional) is a necessary but not sufficient condition to secure the covariance of the Lagrange-Euler field theory. A rigorous definition of the Lagrange Field Structure (LFS) on differential manifolds is proposed. A covariant generalization of the first Noether theorem for LFS is obtained. Different approaches to the EP are discussed.