Black-hole normal modes: A WKB approach. II. Schwarzschild black holes

We employ a semianalytic technique, based on a modified WKB approach, to determine the complex normal-mode frequencies of Schwarzschild black holes. 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,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}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 0.13% in the real part of the frequency and 0.22% in the imaginary part. The agreement for both the real and imaginary parts of the low overtones is better than 0.5%. The relative agreement improves with increasing angular harmonic.