- Access by Xinjiang University
Gravitational Field of a Particle Falling in a Schwarzschild Geometry Analyzed in Tensor Harmonics
Phys. Rev. D 2, 2141 – Published 15 November, 1970
DOI: https://doi.org/10.1103/PhysRevD.2.2141
Abstract
We are concerned with the pulse of gravitational radiation given off when a star falls into a "black hole" near the center of our galaxy. We look at the problem of a small particle falling in a Schwarzschild background ("black hole") and examine its spectrum in the high-frequency limit. In formulating the problem it is essential to pose the correct boundary condition: gravitational radiation not only escaping to infinity but also disappearing down the hole. We have examined the problem in the approximation of linear perturbations from a Schwarzschild background geometry, utilizing the decomposition into the tensor spherical harmonics given by Regge and Wheeler (1957) and by Mathews (1962). The falling particle contributes a -function source term (geodesic motion in the background Schwarzschild geometry) which is also decomposed into tensor harmonics, each of which "drives" the corresponding perturbation harmonic. The power spectrum radiated in infinity is given in the high-frequency approximation in terms of the traceless transverse tensor harmonics called "electric" and "magnetic" by Mathews.
Comments & Replies
Radial Motion of a Spinning Test Body in the Field of a Black Hole
References (30)
- F. Dyson (private communication)
- Ya. Zel'dovich and I. Novikov, Dokl. Akad. Nauk SSSR 155, 1033 (1964) [Soviet Phys. Doklady 9, 246 (1964)]
- L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley, Reading, Mass., 1962)
- R. Isaacson, Phys. Rev. 166, 1263 (1968) ibid.166, 1272 (1968)
- A. Trautman, Lectures on General Relativity, lecture notes, Kings College, London, 1958 (unpublished)
- T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957)
- J. Mathews, J. Soc. Ind. Appl. Math. 10, 768 (1962)
- F. Zerilli, J. Math. Phys. 11, 2203 (1970)
- J. Stachel, Nature 220, 779 (1968)
- F. Zerilli, Ph.D. thesis, Princeton University, 1969 (unpublished)
- K. Thorne and A. Campolattaro, Astrophys. J. 149, 591 (1967)
- J. A. Wheeler, "Superspace and the Nature of Quantum Geometrodynamics," in Battelle Rencontres (Benjamin, New York, 1968) A. E. Fischer, Ph.D. thesis, Princeton University, 1969 (unpublished)
- C. W. Misner, K. S. Thorne, and J. A. Wheeler in "An Open Letter to Relativity Theorists," 1968 (unpublished)
- C. Lanczos, Z. Physik 31, 112 (1925) E. Lifshitz, J. Phys. USSR 10, 116 (1946) P. C. Peters, Phys. Rev. 146, 938 (1966)
- J. P. Vajk, Ph.D. thesis, Princeton University, 1968 (unpublished)
- M. D. Kruskal, Phys. Rev. 119, 1743 (1960) R. Fuller and J. A. Wheeler, ibid. 128, 919 (1962)
- R. Geroch, J. Math. Phys. 9, 450 (1968)
- C. V. Vishveshwara, Ph.D. thesis, University of Maryland, 1968 (unpublished)
- F. Zerilli, Phys. Rev. Letters 24, 737 (1970)
- R. Ruffini and J. A. Wheeler, "Cosmology from Space Platforms," European Space Research Organization report (unpublished)
- L. Edelstein, Ph.D. thesis, University of Maryland, 1969 (unpublished)
- R. Price and K. Thorne, Astrophys. J. 155, 163 (1969)
- B. L. van der Waerden, Appl. Sci. Res. B2, 33 (1960) H. A. Lauwerier, Asymptotic Expansions (Mathematisch Centrum, Amsterdam, 1966)
- E. T. Copson, Asymptotic Expansions (Cambridge U. P., New York, 1965), p. 21
- C. W. Misner (private communication)
- A. Doroshkevich, Ya. Zel'dovich, and I. Novikov, Zh. Eksperim. i Teor. Fiz. 49, 170 (1965) [Soviet Phys. JETP 22, 122 (1966)]
- K. Thorne (private communication)
- F. Pirani, Phys. Rev. 105, 1089 (1957)
- F. PiraniHandbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1965), Chap. 15, p. 562
- F. PiraniHigher Trancendental Functions, edited by A. Erdélyi (McGraw-Hill, New York, 1953), Vol. I, Chap. 2