The problem of motion in gauge theories of gravity

TL;DR

This paper explores how gauge symmetries influence the motion of matter in gravity theories, showing that uncharged particles follow geodesics and gauge-charged particles experience generalized Lorentz forces, within a broad gauge-invariant framework.

Contribution

It demonstrates the gauge invariance of particle motion and field equations in gravity theories without relying on specific Lagrangians, extending the understanding of gauge symmetries in gravitational contexts.

Findings

Uncharged particles move along Riemannian geodesics due to gauge translational invariance.

Gauge-charged particles experience a generalized Lorentz force from gauge symmetry.

Derived identities linking gravitational and gauge field equations from matter field equations.

Abstract

In this article we consider the problem to what extent the motion of gauge-charged matter that generates the gravitational field can be arbitrary, as well as what equations are superimposed on the gauge field due to conditions of compatibility of gravitational field equations. Considered problem is analyzed from the point of view symmetry of the theory with respect to the generalized gauge deformed groups without specification of Lagrangians. In particular it is shown, that the motion of uncharged particles along geodesics of Riemannian space is inherent in an extremely wide range of theories of gravity and is a consequence of the gauge translational invariance of these theories under the condition of fulfilling equations of gravitational field. In the cause of gauge-charged particles, the Lorentz force, generalized for gauge-charged matter, appears in equations of motion as a consequence of the gauge symmetry of the theory under the condition of fulfilling equations of gravitational and gauge fields. In addition, we found relationships of equations for some fields that follow from the assumption about fulfilling of equations for other fields, for example, relationships of equations of the gravitational field and the gauge field of internal symmetry which follow from the assumption about fulfilling of equations of matter fields. In particular, we obtained the identity that generalizes in the case of arbitrary gauge field (and in the presence of gauge-charged matter) the identity found by Hilbert for the electromagnetic field. At the end of the article there is an Appendix, which briefly describes the main provisions and facts from the theory of generalized gauge deformed groups and presents the main ideas of a single group-theoretical interpretation of gauge fields of both external (space-time) and internal symmetry, which is an alternative to their geometric interpretation.