Black holes with massive graviton hair

No-hair theorems exclude the existence of nontrivial scalar and massive vector hair outside four-dimensional, static, asymptotically flat black-hole spacetimes. We show, by explicitly building nonlinear solutions, that black holes can support massive graviton hair in theories of massive gravity. These hairy solutions are, most likely, the generic end state of the recently discovered monopole instability of Schwarzschild black holes in massive graviton theories.

Figure 1

Spacetime mass $M$ (left panel), parameter $u_H$ (center panel) and inverse of the temperature $T\equiv T_g=T_f$ (right panel) as functions of the graviton mass $\mu M_S$. Quantities are normalized by the mass of the corresponding Schwarzschild BH, $M_S$. For graviton masses close to the critical threshold $\mu M_S\sim 0.438$ the solutions merge smoothly with a Schwarzschild BH, as they should since the latter are marginally stable at this point [11, 12].

Figure 2

Conjectured diagram of the parameter space for BHs with massive graviton hair in bimetric massive gravity. See main text for details.

Figure 3

The function $U^{\prime }(r)$ for different values of the mass $\mu M_S$. The behavior is similar for any value ${\alpha }_3$ and ${\alpha }_4$ near the threshold $\mu M_S\sim 0.438$ but for small $\mu M_S$ it can be very different depending on the specific values of the parameters. Top panel: ${\alpha }_3=2$, ${\alpha }_4=-2$. Middle panel: ${\alpha }_3=1$, ${\alpha }_4=-1$. Bottom panel: ${\alpha }_3=1$, ${\alpha }_4=0$.

Figure 4

Metric functions for small masses $\mu M_S$. Top panel: ${\alpha }_3=2$, ${\alpha }_4=-2$. Bottom panel: ${\alpha }_3=1$, ${\alpha }_4=-1$.