Black hole normal modes - A semianalytic approach

Abstract

A new semianalytic technique for determining the complex normal mode frequencies of black holes is presented. The method is based on the WKB approximation. It yields a simple analytic formula that gives the real and imaginary parts of the frequency in terms of the parameters of the black hole and of the field whose perturbation is under study, and in terms of the quantity (n + 1/2), where n = 0, 1, 2,... and labels the fundamental mode, first overtone mode, and so on. In the case of the fundamental gravitational normal modes of the Schwarzschild black hole, the WKB estimates agree with numerical results to better than 7 percent in the real part of the frequency and 0.7 percent in the imaginary part, with the relative agreement improving with increasing angular harmonic. Carried to higher order the method may provide an accurate and systematic means to study black hole normal modes.