Scattering of massless scalar waves by a Schwarzschild black hole: A phase-integral study.

Abstract

Plane scalar waves scattered off a Schwarzschild black hole are studied in the partial-wave picture. A new approximate formula for the relevant phase shifts is derived using the phase-integral method. This formula is, in principle, uniformly valid for all frequencies and agrees with well-known approximations for frequencies well above and below the top of the curvature potential barrier. The reliability of the phase-integral formula is assessed in two different ways. First we use the fact that higher orders of approximation are easily implemented in the phase-integral method. The accuracy of each phase shift can be estimated by the contribution to it by the following order of approximation. Second, we use the approximate phase shifts to construct physically meaningful quantities, such as the deflection function and cross section, for several scattering frequencies. The features of these quantities, especially those associated with the prominent black-hole glory in the backward direction, are in excellent agreement with results of previous studies of the problem. The new phase-shift formula is thus shown to be reliable and provides a useful and efficient tool, especially for intermediate and high frequencies.