Spherically Symmetric Solutions in Ghost-Free Massive Gravity

TL;DR

The paper develops spherically symmetric solutions in a ghost-free massive gravity theory realized as bigravity with two dynamical metrics and a ghost-free potential. It identifies two flat backgrounds (LI with $c=1$ and LB with $c\neq1$) and two solution branches: Type I (analytic, D $\neq0$) yields Schwarzschild-like geometries with no Yukawa-like modification, while Type II (D = 0) is more complex, often requiring perturbative treatment and exhibiting vDVZ-type behavior in the weak-field limit. In the LB phase, a massive tensor sector with two degrees of freedom propagates while the static potential remains Newtonian for Type I; Type II introduces more intricate interplays between massless and massive modes and can challenge the Vainshtein mechanism in a frozen-auxiliary-metric limit. The work demonstrates that the bigravity formulation is essential to the theory’s structure and phenomenology, and it provides a careful comparison to the Stückelberg approach, highlighting both consistencies and crucial differences in various limits. These results have potential implications for astrophysical and cosmological applications of massive gravity and for understanding how to realize consistent infrared modifications of gravity.

Abstract

Recently, a class of theories of massive gravity has been shown to be ghost-free. We study the spherically symmetric solutions in the bigravity formulation of such theories. In general, the solutions admit both a Lorentz invariant and a Lorentz breaking asymptotically flat behaviour and also fall in two branches. In the first branch, all solutions can be found analitycally and are Schwarzschild-like, with no modification as is found for other classes of theories. In the second branch, exact solutions are hard to find, and relying on perturbation theory, Yukawa-like modifications of the static potential are found. The general structure of the solutions suggests that the bigravity formulation of massive gravity is crucial and more than a tool.