The Principle of Symmetric Criticality in General Relativity

TL;DR

This paper examines the applicability of Palais' Principle of Symmetric Criticality to gravitational theories, identifying specific conditions under which symmetry reduction of Lagrangian equations is valid in general relativity.

Contribution

It establishes two independent conditions necessary for PSC validity in metric-based gravitational theories, generalizing previous results and providing criteria based solely on group actions.

Findings

Two conditions determine PSC validity in gravitational theories.

One condition generalizes the unimodularity condition in cosmology.

Conditions are based on pointwise properties of the symmetry group.

Abstract

We consider a version of Palais' Principle of Symmetric Criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. PSC asserts that, given a group action, for any group-invariant Lagrangian the equations obtained by restriction of Euler-Lagrange equations to group-invariant fields are equivalent to the Euler-Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric. It is shown that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in the context of transverse symmetry group actions, provides a generalization of the well-known unimodularity condition that arises in spatially homogeneous cosmological models. The other condition seems to be new. The conditions that determine the validity of PSC are equivalent to pointwise conditions on the group action alone. These results are illustrated with a variety of examples from general relativity. It is straightforward to generalize all of our results to any relativistic field theory.