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

TL;DR

The paper investigates the validity of Birkhoff's theorem in 4D modified gravity, focusing on the Einstein branch of quadratic gravity and related theories. Using a perturbative expansion in the higher-curvature couplings and a spherically symmetric ansatz, it demonstrates that the exterior spacetime must be static and Schwarzschild, up to a renormalized mass, within the Einstein branch. The result extends to a broader class of vacuum theories with G_{mu nu} + ε K_{mu nu} = 0 when K_{mu nu}[g^{Sch}] = 0, via a unified perturbative approach. A key observational implication is that the exterior spacetime of horizonless stars cannot reside entirely in the Einstein branch, due to junction-condition failures, offering a novel probe of beyond-GR physics.

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.