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Spectrum of Hawking radiation and the Huygens principle
Phys. Rev. D 33, 3573 – Published 15 June, 1986
DOI: https://doi.org/10.1103/PhysRevD.33.3573
Abstract
It has been established that the Minkowski vacuum observed by a uniformly accelerating detector can be equivalently described by a canonical ensemble of states. However, recent calculations have revealed that in a spacetime of odd dimensions free massless scalar particles are detected with the Fermi-Dirac distribution instead of the Planck one. The purpose of this paper is to solve this apparent puzzle. It is concluded that the cause of this phenomenon is the lack of the Huygens principle in odd dimensions: the expectation value of the commutator of massless fields does not vanish in the timelike region. The case of a spinor field is treated equally. Other spacetimes, de Sitter space and Schwarzschild space, are also discussed.
References (11)
- S. W. Hawking, Nature (London) 248, 30 (1974).
- For reviews, see D. W. Sciama, P. Candelas and D. Deutsch, Adv. Phys. 30, 327 (1981); N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England, 1982); S. A. Fulling and S. N. M. Ruijsenaars, Phys. Rep. (to be published), and references therein.
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- C. R. Stephens, University of Maryland report, 1985 (unpublished).
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- See, for example, S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, Cambridge, England, 1973).
- It would be worthwhile to note here that, in the original paper of the black-hole radiation, Hawking has made use of the optical approximation to derive the Planck spectrum of the emitted particle. This approximation, however, is not applicable in odd dimensions because of the lack of the Huygens principle.
- The author is informed by S. Takagi that he has also found this phenomenon in de Sitter space. See S. Takagi, Prog. Theor. Phys. 47, 501 (1985); ibid. 47, 1219 (1985).