New Equations for Gravitation with Riemann Tensor and 4-Index Energy-Momentum Tensors for Gravitation and Matter

TL;DR

This paper derives a generalized Einstein equation in 4-index form involving the Riemann tensor and constructs corresponding energy-momentum tensors for gravity and matter, providing new insights into gravitational energy distribution.

Contribution

It introduces a novel 4-index formulation of Einstein's equations and constructs associated energy-momentum tensors for gravity and matter.

Findings

Gravitational energy-momentum density tensor is expressed via the Weyl tensor.

A 4-index energy-momentum tensor for matter is constructed.

The generalized equations offer new perspectives on gravitational energy localization.

Abstract

A generalized version of the Einstein equations in the 4-index form, containing the Riemann tensor linearly, is derived. It is shown, that the gravitational energy-momentum density tensor outside a source is represented across the Weyl tensor vanishing at the 2-index contraction. The 4-index energy-momentum density tensor for matter also is constructed.