Birkhoff's Theorem in Higher Derivative Theories of Gravity

TL;DR

This paper identifies a class of higher derivative gravity theories where Birkhoff's theorem holds, showing that under symmetry conditions, the complex field equations simplify to second order similar to Lovelock theories.

Contribution

It demonstrates that certain higher derivative gravity theories preserve Birkhoff's theorem by reducing to second order equations under symmetry, extending the class of theories with this property.

Findings

Field equations reduce to second order under symmetry

Theories resemble Lovelock gravity in structure

Birkhoff's theorem applies in these higher derivative theories

Abstract

In this paper we present a class of higher derivative theories of gravity which admit Birkhoff's theorem. In particular, we explicitly show that in this class of theories, although generically the field equations are of fourth order, under spherical (plane or hyperbolic) symmetry, all the field equations reduce to second order and have exactly the same or similar structure to those of Lovelock theories, depending on the spacetime dimensions and the order of the Lagrangian.