Spherically symmetric analysis on open FLRW solution in non-linear massive gravity

TL;DR

This paper investigates non-linear massive gravity in spherically symmetric settings, revealing deviations from general relativity such as the breakdown of Birkhoff's theorem and potential superluminal energy emission due to the helicity-0 mode.

Contribution

It derives the effective energy-momentum tensor in non-linear massive gravity for spherically symmetric cases and shows how it differs from GR, especially regarding Birkhoff's theorem.

Findings

Effective energy-momentum tensor matches cosmological constant in specific cases

Birkhoff's theorem does not hold in non-linear massive gravity

Solutions exhibit arbitrary time-dependent parameters, indicating potential superluminal effects

Abstract

We study non-linear massive gravity in the spherically symmetric context. Our main motivation is to investigate the effect of helicity-0 mode which remains elusive after analysis of cosmological perturbation around an open Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. The non-linear form of the effective energy-momentum tensor stemming from the mass term is derived for the spherically symmetric case. Only in the special case where the area of the two sphere is not deviated away from the FLRW universe, the effective energy momentum tensor becomes completely the same as that of cosmological constant. This opens a window for discriminating the non-linear massive gravity from general relativity (GR). Indeed, by further solving these spherically symmetric gravitational equations of motion in vacuum to the linear order, we obtain a solution which has an arbitrary time-dependent parameter. In GR, this parameter is a constant and corresponds to the mass of a star. Our result means that Birkhoff's theorem no longer holds in the non-linear massive gravity and suggests that energy can probably be emitted superluminously (with infinite speed) on the self-accelerating background by the helicity-0 mode, which could be a potential plague of this theory.