Small Kerr–anti-de Sitter black holes are unstable

Superradiance in black hole spacetimes can trigger instabilities. Here we show that, due to superradiance, small Kerr–anti-de Sitter black holes are unstable. Our demonstration uses a matching procedure, in a long wavelength approximation.

Figure 1

Range of black hole parameters for which one has stable and unstable modes. Regularity condition implies that $a/\ell <1$, and for small Kerr-AdS black holes we have $r_{+}/\ell <1$. Region I represents a stable mode zone, while regions II and III represent black holes that can have unstable modes. To be accurate, in the approximations we used, we can only guarantee the presence of an instability in region II. There is however no reason to doubt that the instability also exists in region III. The frontier between regions I and II is the parabola $a/\ell =r_{+}^2/{\ell }^2$. To ascertain the complete instability zone, numerical work is needed.