Dynamics of dRGT ghost-free massive gravity in spherical symmetry

A bstract 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.