Symmetry reduction of gravitational Lagrangians

TL;DR

This paper systematically classifies all symmetry reductions of gravitational Lagrangians in four dimensions, providing a comprehensive list of invariant metrics and analyzing the implications of symmetries on the equations of gravity.

Contribution

It offers a complete classification of symmetry reductions for 4D gravitational Lagrangians using advanced Lie algebra techniques and provides accompanying computational tools.

Findings

Complete list of symmetry group actions for gravitational Lagrangians.

Identification of invariant metrics and their relations.

A Mathematica package for symmetry reduction of Lagrangians.

Abstract

We analyze all possible symmetry reductions of Lagrangians that yield fully equivalent field equations for any 4-dimensional metric theory of gravity. Specifically, we present a complete list of infinitesimal group actions obeying the principle of symmetric criticality, identify the corresponding invariant metrics (and $l$-chains), discuss relations among them, and analyze simplifications allowed by the residual diffeomorphism group and Noether identities before and after variation of the reduced Lagrangian. We employ rigorous treatment of the symmetry reduction by Fels and Torre and use the Hicks classification of infinitesimal group actions by means of Lie algebras pairs of isometries and their isotropy subalgebras. The classification is recast in the coordinates in which the well-known symmetries and geometries are easily recognizable and relatable to each other. The paper is accompanied by a code for the symmetry reduction of Lagrangians that we developed in \texttt{xAct} package of \textsc{Mathematica}.