On the radial wave equation in Schwarzschild's space-time

Abstract

The radial factor of a separable solution of the wave equation in Schwarzschild's space‐time satisfies a second‐order linear differential equation. This equation is studied in detail. The behavior of the solutions near the singular points (the origin, the horizon, and infinity) of the equation is analyzed. By an appropriate transformation two simpler differential equations are obtained corresponding to retarded and advanced solutions with characteristic asymptotic expansions. Their properties permit the expression of the general solution of the radial equation in terms of a single contour integral. Finally, through a ``matching'' technique, the behavior of a solution at the singular points is determined from its behavior at a single singular point.