Imaginary-Frequency Interior Modes of Black Holes

Details Imaginary-Frequency Interior Modes of Black Holes Imaginary-Frequency Interior Modes of Black Holes

Nov 1980

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1980rspsa.373..223m&link_type=abstract

Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 373, Issue 1753, pp. 223-233

Physics

5

Scientific paper

Perturbations of black holes (Schwarzschild, Reissner-Nordstrom 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 (cosθ, θ real) for such functions is the black hole interior, rin(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-Nordstrom 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 rin(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 imaginary-frequency modes. Here l is the angular momentum eigen-number associated with the angular decomposition.

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