Birkhoff's Theorem and Uniqueness: A Peek Beyond General Relativity

TL;DR

This paper investigates the validity of Birkhoff's theorem in various modified gravity theories, showing that Schwarzschild remains unique outside spherically symmetric configurations, with implications for observational tests of gravity models beyond General Relativity.

Contribution

It extends Birkhoff's theorem to a broad class of modified gravity theories, revealing conditions under which Schwarzschild solutions remain unique and highlighting observational implications.

Findings

Schwarzschild spacetime remains the unique Einstein branch solution in these theories.

Breakdown of junction conditions prevents spherically symmetric, horizonless stars from matching Einstein branch metrics.

Provides a potential observational test for theories beyond General Relativity.

Abstract

In General Relativity, Birkhoff's theorem asserts that any spherically symmetric vacuum solution must be static and asymptotically flat. In this paper, we study the validity of Birkhoff's theorem for a broad class of modified gravity theories in four spacetime dimensions, including quadratic and higher-order gravity models. We demonstrate that the Schwarzschild spacetime remains the unique Einstein branch solution outside any spherically symmetric configuration of these theories. Consequently, unlike black holes, the breakdown of junction conditions at the surface of the star further implies that the actual spacetime metric outside a horizonless star in these modified theories cannot simultaneously be spherically symmetric and remain within the Einstein branch. This insight offers a unique observational probe for theories beyond General Relativity.