The Fully Covariant Energy Momentum Stress Tensor For Gravity and the Einstein Equation in General Relativity

TL;DR

This paper introduces a fully covariant energy-momentum stress tensor for gravity, providing a general derivation of Einstein's equations without restrictive assumptions, and explores implications for spacetime dimensions and curvature.

Contribution

It presents a new covariant energy-momentum tensor for gravity and derives Einstein's equations under minimal assumptions, extending understanding of gravitational energy in general relativity.

Findings

Energy-momentum tensor for gravity is physically motivated and covariant.

Derivation of Einstein's equations without assuming properties of matter fields.

Implication that divergence-free matter fields lead to 4D spacetime or constant scalar curvature.

Abstract

We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the surce matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter and fields must be divergence free, when spacetime is 4 dimensional. Moreoverr, if the total source matter fields are assumed to be divergence free, then either spacetime is dimension 4 or the spacetime has constant scalar curvature.