Scattering of Massless Scalar Waves by a Schwarzschild ``Singularity''

This paper investigates the scattering and absorption of scalar waves satisfying the equation ${\phi }_{\mathrm{;\mu }}^{\mathrm{;\mu }}\text{=0}$ in the Schwarzschild metric. This problem has been previously considered by Hildreth. We find, for a Schwarzschild mass m, the following cross sections in the zero‐frequency limit for s‐waves: σ(absorption) = 0, dσ/dΩ ≃ [c + ⅓(2m) ln (2mω)]², where c is a constant of order m. These results disagree with the previous calculation. We exhibit a method of solution for the equation. Its limiting (Newtonian) form, with suitable identification of the coefficients, is the problem of Coulomb scattering in non‐relativistic quantum mechanics. By demanding coordinate conditions which for large l allow the usual Coulomb results in a partial‐wave expansion, we are able to define a partial‐wave cross section. The (summed) differential cross section for small frequencies inherits the logarithmic behavior of the s‐wave part, which is the only contribution explicitly calculated. (The l ≠ 0 contributions and the behavior of the cross sections for ω ≠ 0 are qualitatively indicated.) Cosmological considerations are given which cut off this divergence.