Stars and (furry) black holes in Lorentz breaking massive gravity

We study the exact spherically symmetric solutions in a class of Lorentz-breaking massive gravity theories, using the effective-theory approach where the graviton mass is generated by the interaction with a suitable set of Stückelberg fields. We find explicitly the exact black-hole solutions which generalizes the familiar Schwarzschild one, which shows a nonanalytic hair in the form of a powerlike term $r^{\gamma }$. For realistic self-gravitating bodies, we find interesting features, linked to the effective violation of the Gauss law: (i) the total gravitational mass appearing in the standard $1/r$ term gets a multiplicative renormalization proportional to the area of the body itself; (ii) the magnitude of the powerlike hairy correction is also linked to size of the body. The novel features can be ascribed to the presence of the Goldstones fluid turned on by matter inside the body; its equation of state approaching that of dark energy near the center. The Goldstones fluid also changes the matter equilibrium pressure, leading to an upper limit for the graviton mass, $m\lesssim {10}^{-28÷29}\mathrm{eV}$, derived from the largest stable gravitational bound states in the Universe.

Figure 1

Plot, in the linear regime, of $J-1$ and $p/{\rho }_0$ in the case of massive gravity (14, 17), both for ${\mu }^2$ positive and negative; also the standard GR case is shown for comparison. The numerical values used are ${\mu }^2{R}^2=\pm 3.06$, $R_s/R=8.6\times {10}^{-7}$, and $\gamma =-2.5$. The units are such that $G{M}_0/R=1$.

Figure 2

Excluded regions, in the (${\mu }^2{R}^2$-$\gamma$) plane, from positivity of the pressure [$p(0)$ solid line, $p^{\prime }(R)$ dashed line] and of the renormalized mass $M$ (dotted line). The central white region is allowed.

Figure 3

Numerical results for $\Delta M=M-M_0$ (left) and $S$ (right) for different values of the matter density and of the graviton mass scale ${\mu }^2{R}^2=0.1,\dots ,1$ (lower to upper). The dashed lines correspond to the perturbative values (18).

Figure 4

Numerical exclusion plot of the graviton mass [from $p(0)>0$] toward the strong field. Here $\gamma$ is fixed and the perturbative limit from (22) is ${\mu }^2{R}^2<0.338$ (dashed line).