Birkhoff's Theorem in Lovelock Gravity for General Base Manifolds

TL;DR

This paper generalizes Birkhoff's theorem within Lovelock gravity, demonstrating that solutions with a warped product structure are static and impose specific constraints on the base manifold and transverse metric.

Contribution

It extends Birkhoff's theorem to arbitrary base manifolds in Lovelock gravity using an elementary approach, revealing conditions on the base manifold and metric functions.

Findings

Solutions are static for warped product metrics in Lovelock gravity.

Base manifold's intrinsic Lovelock tensors are constant and arbitrary.

Transverse metric function satisfies an algebraic equation involving constants.

Abstract

We extend the Birkhoff's theorem in Lovelock gravity for arbitrary base manifolds using an elementary method. In particular, it is shown that any solution of the form of a warped product of a two-dimensional transverse space and an arbitrary base manifold must be static. Moreover, the field equations restrict the base manifold such that all the non-trivial intrinsic Lovelock tensors of the base manifold are constants, which can be chosen arbitrarily, and the metric in the transverse space is determined by a single function of a spacelike coordinate which satisfies an algebraic equation involving the constants characterizing the base manifold along with the coupling constants.