Black-hole normal modes: A WKB approach. I. Foundations and application of a higher-order WKB analysis of potential-barrier scattering

We present a semianalytic technique for determining the complex normal-mode frequencies of black holes. The method makes use of the WKB approximation, carried to third order beyond the eikonal approximation. Mathematically, the problem is similar to studying one-dimensional quantum-mechanical scattering near the peak of a potential barrier, and determining the scattering resonances. Under such conditions, a modification of the usual WKB approach must be used. We obtain the connection formulas that relate the amplitudes of incident, reflected, and transmitted waves, to the third WKB order. By imposing the normal-mode (resonance) boundary condition of a zero incident amplitude with nonzero transmitted and reflected amplitudes, we find a simple formula that determines the real and imaginary parts of the normal-mode frequency of perturbation (or of the quantum-mechanical energy of the resonance) in terms of the derivatives (up to and including sixth order) of the barrier function evaluated at the peak, and in terms of the quantity (n+(1/2)), where n is an integer and labels the fundamental mode (resonance), first overtone, and so on. This higher-order approach may find uses in barrier-tunneling problems in atomic and nuclear physics.