Shortcuts to high symmetry solutions in gravitational theories

TL;DR

The paper addresses the challenge of efficiently obtaining exact, highly symmetric solutions in gravity theories across dimensions. It employs the Weyl method, justified by Palais' symmetric criticality theorems, to insert highly symmetric ansatze into the action and vary a reduced one-dimensional radial action, yielding first-order, integrable field equations. The authors derive and classify static solutions across models such as GR with $oxed{\Lambda}$ and Maxwell terms, Einstein–Gauss–Bonnet, conformal $C^2$ and cubic $C^3$ theories, pure $R^2$ gravity, and a traceless Ricci model, uncovering results like the exclusion of Schwarzschild in certain higher-curvature theories and a Birkhoff-type theorem in tensor-only sectors; they also obtain BTZ-like geometries in $D=3$. The work provides a streamlined, cross-model toolkit for generating exact, highly symmetric geometries and suggests extensions to brane metrics and exploration of novel spacetimes via controlled symmetry constraints.

Abstract

We apply the Weyl method, as sanctioned by Palais' symmetric criticality theorems, to obtain those -highly symmetric -geometries amenable to explicit solution, in generic gravitational models and dimension. The technique consists of judiciously violating the rules of variational principles by inserting highly symmetric, and seemingly gauge fixed, metrics into the action, then varying it directly to arrive at a small number of transparent, indexless, field equations. Illustrations include spherically and axially symmetric solutions in a wide range of models beyond D=4 Einstein theory; already at D=4, novel results emerge such as exclusion of Schwarzschild solutions in cubic curvature models and restrictions on ``independent'' integration parameters in quadratic ones. Another application of Weyl's method is an easy derivation of Birkhoff's theorem in systems with only tensor modes. Other uses are also suggested.