Birkhoff's theorem for stable torsion theories

Abstract

We present a novel approach to establish the Birkhoff's theorem validity in the so-called quadratic Poincar\'e Gauge theories of gravity. By obtaining the field equations via the Palatini formalism, we find paradigmatic scenarios where the theorem applies neatly. For more general and physically relevant situations, a suitable decomposition of the torsion tensor also allows us to establish the validity of the theorem. Our analysis shows rigorously how for all stable cases under consideration, the only solution of the vacuum field equations is a torsionless Schwarzschild spacetime, although it is possible to find non-Schwarzschild metrics in the realm of unstable Lagrangians. Finally, we study the weakened formulation of the Birkhoff's theorem where an asymptotically flat metric is assumed, showing that the theorem also holds.