Massive scalar field instability in Kerr spacetime

We study the Klein-Gordon equation for a massive scalar field in Kerr spacetime in the time-domain. We demonstrate that under conditions of super-radiance, the scalar field becomes unstable and its amplitude grows without bound. We also estimate the growth rate of this instability.

Figure 1

Black hole bomb: Massless scalar field sampled at $r_{*}=-20M$ in Kerr spacetime with $a/M=0.9999$ surrounded by perfect mirrors. The field grows without bound upon successive reflections. All quantities in units of black hole mass, $M$.

Figure 2

Black hole bomb: Electromagnetic field sampled at $r_{*}=-20M$ in Kerr spacetime with $a/M=0.9999$ surrounded by perfect mirrors. The field grows without bound upon successive reflections. All quantities in units of black hole mass, $M$.

Figure 3

Instability: Massive scalar field ($\ell =m=1$) sampled at $r_{*}=20M$ in Kerr spacetime with $a/M=0.9999$. The field grows without bound, demonstrating the existence of instability. All quantities in units of black hole mass, $M$.

Figure 4

Massive scalar field ($\ell =m=1$) sampled at $r_{*}=20M$ in Kerr spacetime with $a/M=0.9999$. Here the super-radiance condition is not satisfied. The field decays, demonstrating the absence of an instability. All quantities in units of black hole mass, $M$.

Figure 5

Instability: Massive scalar field ($\ell =m=2$) sampled at $r_{*}=20M$ in Kerr spacetime with $a/M=0.9999$. The field grows without bound, demonstrating the existence of instability. All quantities in units of black hole mass, $M$.

Figure 6

Dependence of the growth rate of the instability on the mass of the scalar field $\mu$. All quantities in units of black hole mass, $M$.