Stars and Black Holes in Massive Gravity

TL;DR

The paper investigates nonlinear massive gravity, focusing on FP2 as a ghost-free two-parameter sub-family. It shows FP2 yields a unique static spherically symmetric exterior for a given source and that stellar solutions can reproduce Einstein gravity in the inner region while approaching linear massive gravity at large distances. Black holes in FP2 generally exhibit a near-horizon throat with diverging curvature, but a special FP1 sub-family can produce non-singular horizons. These results provide qualitative guidance for building viable massive gravity theories and clarify horizon-scale phenomenology relevant to testing gravity.

Abstract

Generically, massive gravity gives a non-unique gravitational field around a star. For a special family of massive gravity theories, we show that the stellar gravitational field is unique and observationally acceptable, that is close to Einsteinian. The black hole solutions in this family of theories are also studied and shown to be peculiar. Black holes have a near-horizon throat and the curvature diverging at the horizon. We show that there exists a sub-family of these massive gravity theories with non-singular at horizon black holes.

Figures (3)

• Figure 1: Semi-logarithmic plot of $-\lambda/\nu$ and redshift $z=e^{-\nu/2}-1$ as a function of $r$. $-\lambda/\nu$ linearly grows with $r$ at large values of $r$ in agreement with the linearized massive gravity solution \ref{['linear']}. Einstein theory is recovered at small radii since $-\lambda/\nu$ approaches the Schwarzchild metric value (i.e. $\lambda/\nu=-1$) outside the star, and $\lambda\to 0$ at the center. The surface of the star is located at $r_s=0.01$ and its mass is determined implicitly by $\nu=-0.005e^{-mr}/r$ at large $r$. The gravity parameters are $m=1$, $c_2=0$, $c_3=2$.
• Figure 2: $(1+z)dr/d\rho$ as a function of $(1+z)^{-1}$. Here $z$ is the redshift, $r$ is the radius as measured by the circumference of a constant-redshift sphere, $\rho$ is the proper radial distance. For the Schwarzschild metric, $(1+z)dr/d\rho =1$, which is indeed seen at intermediate redshifts. The negative values of $(1+z)dr/d\rho$ near the horizon ($z=\infty$) correspond to a throat. The gravity parameters are $m=1$, $c_2=0$, $c_3=2$. The mass of the black hole is given implicitly by $\nu=-0.06e^{-mr}/r$ at large $r$.
• Figure 3: $\lambda _2$, $w(1+z)-1$, indicating closeness to Schwrazschild, and $\sqrt{R_{\mu \nu}R^{\mu \nu}}/4$ are shown vs $(1+z)^{-1}$. Thin: $c_2=0.9$, thick: $c_2=0.999$. Other gravity parameters: $m=1$, $c_3=1$. The mass of the black hole is given implicitly by $\nu =-0.04e^{-mr}/r$ at large $r$.