Towards a Covariant Theory of Gravitation

TL;DR

This paper presents a covariant reformulation of General Relativity from multiple perspectives, emphasizing the role of nonmetricity and reference frames to develop a fully covariant theory of gravitation.

Contribution

It introduces a unified covariant approach to gravity using geometrodynamics, Lagrangian, and gauge theories, highlighting nonmetricity's role in Einstein's equations.

Findings

Decomposition of Einstein tensor into affine and nonmetric parts.

Reformulation of Einstein's equations invariant under reference frame changes.

Development of a covariant energy-momentum tensor for gravity.

Abstract

A covariant reformulation of General Relativity is briefly considered from three points of view: geometrodynamics, Lagrange-Euler field theory, and gauge field theory. From a geometrodynamics perspective, a definition of the reference frame as a differential manifold with an affine connection results in separation of the respective contributions of inertial and gravitational fields represented by the affine connection and the tensor of nonmetricity within the Levi-Civita connection of GR. Resulting decomposition of the Einstein curvature tensor into affine and nonmetric parts allows to recast Einstein's field equations in a form invariant with respect to the choice of a reference frame wherein the gravity is described by nonmetricity of space-time . A covariant Lagrangian is proposed leading to the same field equation. All three approaches ultimately lead to the same fully covariant theory of gravitation with a covariant tensor of energy-momentum of the gravitational field and differential and integral conservation laws. The role of the frames of reference, as distinguished from coordinate systems, is discussed.