Imaginary-frequency interior modes of black holes

Abstract

Perturbations of black holes (Schwarzschild, Reissner-Nordstrøm and Kerr) can be treated by simple radial wave equations. It is shown that the massless scalar radial equation is a form of the spin-weighted spheroidal wave equation. The region in r corresponding to the usual angular argument (cos 0, 0 real) for such functions is the black hole interior, r E (r - r + ) where r - , r + are the inner and outer horizon radii respectively. We restrict ourselves to axisymmetric scalar waves. (Because of the spherical symmetry this is no restriction in the Schwarzschild and Reissner-Nordstrøm backgrounds, but it is a physical restriction in the Kerr background.) In these cases the spin-weighted spheroidal harmonics correspond to imaginary-frequency waves, i.e. to exponentially growing or decaying waves that fall inward across the outer horizon r+ and are converted to waves moving in the opposite direction as they cross the r _-horizon (r is a timelike coordinate when r e( r _ , r + )). These modes are exactly analogous to the external quasi-normal modes of the black hole. There is always one zero-frequency mode, and l non-zero imaginaryfrequency modes. Here l is the angular momentum eigen-number associated with the angular decomposition.