A dynamical formulation of ghost-free massive gravity

TL;DR

This paper develops a non-linear, ghost-free massive gravity formulation suitable for numerical simulations, explores higher-order operators for well-posed dynamics, and investigates gravitational collapse outcomes in a simplified model.

Contribution

It introduces a fully non-linear, numerically implementable formulation of ghost-free massive gravity with higher-order operators, and studies collapse behavior in a minimal model.

Findings

Weak initial data leads to matter dispersal.

Strong initial data causes breakdown of classical evolution.

No apparent horizon formation in the minimal model.

Abstract

We present a formulation of ghost-free massive gravity with flat reference metric that exhibits the full non-linear constraint algebraically, in a way that can be directly implemented for numerical simulations. Motivated by the presence of higher order operators in the low-energy effective description of massive gravity, we show how the inclusion of higher-order gradient (dissipative) terms leads to a well-posed formulation of its dynamics. While the formulation is presented for a generic combination of the minimal and quadratic mass terms on any background, for concreteness, we then focus on the numerical evolution of the minimal model for spherically symmetric gravitational collapse of scalar field matter. This minimal model does not carry the relevant interactions to switch on an active Vainshtein mechanism, at least in spherical symmetry, thus we do not expect to recover usual GR behaviour even for small graviton mass. Nonetheless we may ask what the outcome of matter collapse is for this gravitational theory. Starting with small initial data far away from the centre, we follow the matter through a non-linear regime as it falls towards the origin. For sufficiently weak data the matter disperses. However for larger data we generally find that the classical evolution breaks down due to the theory becoming infinitely strongly coupled without the presence of an apparent horizon shielding this behaviour from an asymptotic observer.