Stars and black holes in massive gravity

Generically, massive gravity gives a nonunique 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 subfamily of these massive gravity theories with nonsingular at horizon black holes.

Figure 1

Semilogarithmic 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 (15). Einstein theory is recovered at small radii since $-\lambda /\nu$ approaches the Schwarzschild 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 Schwarzschild, 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 $r$.