### Abstract

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 + 1/3(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.

Original language | English (US) |
---|---|

Pages (from-to) | 163-170 |

Number of pages | 8 |

Journal | Journal of Mathematical Physics |

Volume | 9 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1968 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*9*(1), 163-170. https://doi.org/10.1063/1.1664470

**Scattering of massless scalar waves by a Schwarzschild " singularity".** / Matzner, Richard A.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 9, no. 1, pp. 163-170. https://doi.org/10.1063/1.1664470

}

TY - JOUR

T1 - Scattering of massless scalar waves by a Schwarzschild " singularity"

AU - Matzner, Richard A.

PY - 1968/1/1

Y1 - 1968/1/1

N2 - 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 + 1/3(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.

AB - 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 + 1/3(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.

UR - http://www.scopus.com/inward/record.url?scp=33748436884&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748436884&partnerID=8YFLogxK

U2 - 10.1063/1.1664470

DO - 10.1063/1.1664470

M3 - Article

AN - SCOPUS:33748436884

VL - 9

SP - 163

EP - 170

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 1

ER -