Shortcuts to high symmetry solutions in gravitational theories

TL;DR

This paper demonstrates how the Weyl method simplifies finding highly symmetric solutions in various gravitational theories, revealing new results and restrictions in models beyond standard Einstein gravity.

Contribution

It introduces a systematic application of the Weyl method to derive symmetric solutions and uncover novel constraints in diverse gravitational models.

Findings

Exclusion of Schwarzschild solutions in cubic curvature models

Restrictions on integration parameters in quadratic models

Simplified derivation of Birkhoff's theorem

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.