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

This paper investigates the scattering and absorption of scalar waves satisfying the equation φ;μ;μ=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.