Vacuum States in Spacetimes with Killing Horizons

  • Robert M. Wald
Part of the NATO ASI Series book series (NSSB, volume 230)


Soon after the discovery by Hawking in 1974 that a black hole formed by gravitational collapse will radiate a thermal distribution of field quanta, a number of authors showed that, in particular spacetimes, when a quantum field is in a certain “natural vacuum state,” then appropriate observers “see” a thermal distribution of particles. For the ordinary vacuum state of Minkowski spacetime, Unruh (1976) obtained such a result for accelerating observers. In extended Schwarzschild spacetime, Hartle and Hawking (1976) and Israel (1976) defined a natural vacuum state (the “Hartle-Hawking vacuum”), which is a thermal state for a static observer. Similarly, for de Sitter spacetime, Gibbons and Hawking (1977) defined the “Euclidean vacuum state” and showed that it has thermal properties for any inertial (geodesic) observer.


Black Hole Curve Spacetime Weyl Algebra Cauchy Surface Hyperbolic Spacetime 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ashtekar, A., and Magnon, A., 1975, Quantum fields in curved space-times, Proc. Roy. Soc. Lond., A346:375.MathSciNetADSGoogle Scholar
  2. Choquet-Bruhat, Y., 1968, Hyperbolic partial differential equations on a manifold, in: “Battelle Rencontres,” C. M. DeWitt and J. A. Wheeler, eds., Benjamin, New York.Google Scholar
  3. Fulling, S. A., Narcowich, F. J. and Wald, R. M., 1981, Singularity structure of the two-point function in quantum field theory in curved spacetime. II, Ann. Phys., 136:243.MathSciNetGoogle Scholar
  4. Geroch, R. P., 1970, Domain of dependence, J. Math. Phys., 11:437.MathSciNetADSMATHCrossRefGoogle Scholar
  5. Gibbons, G.W., and Hawking, S. W., 1977, Cosmological event horizons, thermodynamics and particle creation, Phys. Rev., D15:2738.MathSciNetADSGoogle Scholar
  6. Hartle, J. B., and Hawking, S. W., 1976, Path-integral derivation of black hole radiance, Phys. Rev. D13:2188.ADSGoogle Scholar
  7. Hawking, S. W., and Ellis, G. F. R., 1973, “The Large Scale Structure of Spacetime,” Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  8. Israel, W., 1976, Thermo-field dynamics of black holes, Phys. Lett., 57A:107.MathSciNetADSGoogle Scholar
  9. Kay, B. S., 1978, Linear spin-zero quantum fields in external gravitational and scalar fields, Commun. Math. Phys., 62:55.MathSciNetADSCrossRefGoogle Scholar
  10. Kay, B. S., 1985, The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes, Commun. Math, Phys., 100:57.MathSciNetADSMATHCrossRefGoogle Scholar
  11. Kay, B. S., and Wald, R. M., 1989, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasi-free states on spacetimes with a bifurcate Killing horizon, to be published.Google Scholar
  12. Reed, M., and Simon, B., 1972, “Methods of Modern Mathematical Physics, Vol. I: Functional Analysis,” Academic Press, New York.Google Scholar
  13. Simon, B., 1972, Topics in functional analysis, in “Mathematics of Contemporary Physics,” R. F. Streater, ed., Academic Press, New York.Google Scholar
  14. Treves, F., 1975, “Basic Linear Partial Differential Equations,” Academic Press, New York.MATHGoogle Scholar
  15. Unruh, W. G., 1976, Notes on black hole evaporation, Phys. Rev., D14: 870.ADSGoogle Scholar
  16. Unruh, W. G., and Wald, R. M., 1984, What happens when an accelerating observer detects a Rindler particle, Phys. Rev., D29:1047.ADSGoogle Scholar
  17. Wald, R. M., 1977, The back reaction effect in particle creation in curved spacetime, Commun. Math. Phys., 54:1.MathSciNetADSMATHCrossRefGoogle Scholar
  18. Wald, R. M., 1978, Trace anomaly of a conformally invariant quantum field in curved spacetime, Phys. Rev., D17:1477.MathSciNetADSGoogle Scholar
  19. Wald, R. M., 1979, Existence of the S-matrix in quantum field theory in curved spacetime, Commun. Math, Phys., 70:221.MathSciNetADSCrossRefGoogle Scholar
  20. Wald, R. M., 1974, “General Relativity,” University of Chicago Press, Chicago.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Robert M. Wald
    • 1
  1. 1.Enrico Fermi Institute and Department of PhysicsUniversity of ChicagoChicagoUSA

Personalised recommendations