Generalization of Einstein's gravitational field equations

TL;DR

This paper introduces a novel 4-index gravitational field equation that explicitly incorporates the Riemann and Weyl tensors, extending Einstein's original equations within the framework of general relativity.

Contribution

It proposes the first rigorous mathematical generalization of Einstein's equations using a 4-index tensor formulation that includes the Riemann and Weyl tensors.

Findings

The new equation contains the energy-momentum tensor for matter and gravity.

It is a natural extension of Einstein's equations within general relativity.

The formulation justifies the use of a fourth-order theory due to additional information from the Weyl tensor.

Abstract

The Riemann tensor is the cornerstone of general relativity, but as everyone knows it does not appear explicitly in Einstein's equation of gravitation. This suggests that the latter may not be the most general equation. We propose here for the first time, following a rigorous mathematical treatment based on the variational principle, that there exists a generalized 4-index gravitational field equation containing the Riemann curvature tensor linearly, and thus the Weyl tensor as well. We show that this equation, written in $n$ dimensions, contains the energy-momentum tensor for matter and also that of the gravitational field itself. This new 4-index equation remains completely within the framework of general relativity and emerges as a natural generalization of the familiar 2-index Einstein equation. Due to the presence of the Weyl tensor, we show that this equation contains much more information, which fully justifies the use of a fourth-order theory.