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Path-integral derivation of black-hole radiance

J. B. Hartle

S. W. Hawking*

  • Department of Physics, University of California, Santa Barbara, California 93106 and California Institute of Technology, Pasadena, California 91125

  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England and California Institute of Technology, Pasadena, California 91125

  • *Sherman Fairchild Distinguished Scholar at the California Institute of Technology.
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Phys. Rev. D 13, 2188 – Published 15 April, 1976

DOI: https://doi.org/10.1103/PhysRevD.13.2188

Abstract

The Feynman path-integral method is applied to the quantum mechanics of a scalar particle moving in the background geometry of a Schwarzschild black hole. The amplitude for the black hole to emit a scalar particle in a particular mode is expressed as a sum over paths connecting the future singularity and infinity. By analytic continuation in the complexified Schwarzschild space this amplitude is related to that for a particle to propagate from the past singularity to infinity and hence by time reversal to the amplitude for the black hole to absorb a particle in the same mode. The form of the connection between the emission and absorption probabilities shows that a Schwarzschild black hole will emit scalar particles with a thermal spectrum characterized by a temperature which is related to its mass, M, by T=ℏc38πGMk. Thereby a conceptually simple derivation of black-hole radiance is obtained. The extension of this result to other spin fields and other black-hole geometries is discussed.

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The Work of Stephen Hawking in Physical Review

To mark the passing of Stephen Hawking, we gathered together his 55 papers in Physical Review D and Physical Review Letters. They probe the edges of space and time, from "Black holes and thermodynamics” to "Wave function of the Universe."

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Outline

Information

Phys. Rev. D 13, 2188– Published 15 April, 1976
  • Received 17 November 1975

© 1976 American Physical Society

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