Synchronous Lagrangian variational principles in General Relativity

TL;DR

This paper develops synchronous Lagrangian variational principles for General Relativity, restoring gauge symmetry and manifest covariance, and applies it to Einstein-Maxwell equations with Vlasov matter sources.

Contribution

It introduces an alternative synchronous variational formulation for Einstein-Hilbert and Palatini principles, enhancing gauge symmetry and covariance in GR.

Findings

Synchronous variational principles restore gauge symmetry.

Manifest covariance is maintained at all levels.

A joint variational principle for Einstein-Maxwell-Vlasov systems is formulated.

Abstract

The problem of formulating synchronous variational principles in the context of General Relativity is discussed. Based on the analogy with classical relativistic particle dynamics, the existence of variational principles is pointed out in relativistic classical field theory which are either asynchronous or synchronous. The historical Einstein-Hilbert and Palatini variational formulations are found to belong to the first category. Nevertheless, it is shown that an alternative route exists which permits one to cast these principles in terms of equivalent synchronous Lagrangian variational formulations. The advantage is twofold. First, synchronous approaches allow one to overcome the lack of gauge symmetry of the asynchronous principles. Second, the property of manifest covariance of the theory is also restored at all levels, including the symbolic Euler-Lagrange equations, with the variational Lagrangian density being now identified with a $4-$scalar. As an application, a joint synchronous variational principle holding both for the non-vacuum Einstein and Maxwell equations is displayed, with the matter source being described by means of a Vlasov kinetic treatment.