Dynamics of dRGT ghost-free massive gravity in spherical symmetry

TL;DR

This paper investigates the behavior of dRGT massive gravity in spherical symmetry, revealing limitations in regular solutions and implications for phenomenology, especially regarding matter pressure and singularities.

Contribution

It analyzes the dynamics of minimal and next-to-minimal dRGT models in spherical symmetry, highlighting conditions for regular solutions and phenomenological constraints.

Findings

No regular time-dependent solutions for non-relativistic matter in minimal model.

Next-to-minimal model requires positive pressure at the origin for regularity.

Realistic matter phenomenology may need resolution of singularities or beyond spherical symmetry.

Abstract

We focus on dRGT massive gravity in spherical symmetry in the limit of small graviton mass. Firstly we examine the minimal model. This does not exhibit a Vainshtein mechanism in spherical symmetry, but one may still ask what happens for spherical dynamics. We show that there are no regular time-dependent spherically symmetric solutions unless the matter has sufficiently large pressure. For matter that does not satisfy this, such as non-relativistic matter, any Cauchy slice of such a solution must necessarily have a point where the metric becomes singular. Only a weak assumption on the asymptotics is made. We then consider the next-to-minimal model. This has been argued to have a good Vainshtein mechanism in spherical symmetry, and hence be phenomenologically viable, provided the relative sign of the minimal and next-to-minimal mass terms is the same, and we restrict attention to this case. We find that regular behaviour requires the matter at the origin of symmetry to have positive pressure -- in particular a massive scalar field fails to satisfy this condition. Furthermore it restricts non-relativistic matter so that the pressure is bounded from below in terms of the density and graviton mass in a manner that is at odds with a reasonable phenomenology. This suggests that realistic phenomenology will either require a resolution of singularities, or will require dynamics beyond the non-generic setting of spherical symmetry.