- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Inextendible Schwarzschild black hole with a single exterior: How thermal is the Hawking radiation?

Jorma Louko*,†

Donald Marolf‡

  • Department of Physics, University of Maryland, College Park, Maryland 20742-4111
  • Max-Planck-Institut für Gravitationsphysik, Schlaatzweg 1, D-14473 Potsdam, Germany

  • Department of Physics, Syracuse University, Syracuse, New York 13244-1130

  • *Electronic address: louko@aei-potsdam.mpg.de
  • †Present address.
  • ‡Electronic address: marolf@suhep.phy.syr.edu
PDF

Phys. Rev. D 58, 024007 – Published 15 June, 1998

DOI: https://doi.org/10.1103/PhysRevD.58.024007

Abstract

Several approaches to Hawking radiation on Schwarzschild spacetime rely in some way or another on the fact that the Kruskal manifold has two causally disconnected exterior regions. To assess the physical input implied by the presence of the second exterior region, we investigate the Hawking(-Unruh) effect for a real scalar field on the RP3 geon: an inextendible, globally hyperbolic, space and time orientable eternal black hole spacetime that is locally isometric to Kruskal but contains only one exterior region. The Hartle-Hawking-like vacuum |0G〉, which can be characterized alternatively by the positive frequency properties along the horizons or by the complex analytic properties of the Feynman propagator, turns out to contain exterior region Boulware modes in correlated pairs, and any operator in the exterior that only couples to one member of each correlated Boulware pair has thermal expectation values in the usual Hawking temperature. Generic operators in the exterior do not have this special form; however, we use a Bogoliubov transformation, a particle detector analysis, and a particle emission-absorption analysis that invokes the analytic properties of the Feynman propagator, to argue that |0G〉 appears as a thermal bath with the standard Hawking temperature to any exterior observer at asymptotically early and late Schwarzschild times. A (naive) saddle-point estimate for the path-integral-approach partition function yields for the geon only half of the Bekenstein-Hawking entropy of a Schwarzschild black hole with the same ADM mass: possible implications of this result for the validity of path-integral methods or for the statistical interpretation of black-hole entropy are discussed. Analogous results hold for a Rindler observer in a flat spacetime whose global properties mimic those of the geon.

View figure in article

References (49)

  1. S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
  2. J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973).
  3. J. D. Bekenstein, Nuovo Cimento Lett. 4, 737 (1972).
  4. J. D. Bekenstein, Phys. Rev. D 9, 3292 (1974).
  5. W. G. Unruh, Phys. Rev. D 14, 870 (1976).
  6. J. B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188 (1976).
  7. W. Israel, Phys. Lett. 57A, 107 (1976).
  8. R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (The University of Chicago Press, Chicago, 1994).
  9. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
  10. S. W. Hawking, in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, England 1979).
  11. D. N. Page, in Black Hole Physics, edited by V. D. Sabbata and Z. Zhang (Kluwer Academic Publishers, Dordrecht, 1992).
  12. J. D. Brown and J. W. York, Phys. Rev. D 47, 1407 (1993).
  13. J. D. Brown and J. W. York, Phys. Rev. D 47, 1420 (1993).
  14. S. W. Hawking, G. T. Horowitz, and S. F. Ross, Phys. Rev. D 51, 4302 (1995).
  15. C. Teitelboim, Phys. Rev. D 51, 4315 (1995); ibid.52, 6201(E) (1995).
  16. J. I. Kapusta, Finite-Temperature Field Theory (Cambridge University Press, Cambridge, England, 1989).
  17. J. Polchinski, Rev. Mod. Phys. 68, 1245 (1996).
  18. G. T. Horowitz, in Black Holes and Relativistic Stars, edited by R. M. Wald (The University of Chicago Press, Chicago, 1998), gr-qc/9704072.
  19. J. L. Friedman, K. Schleich, and D. M. Witt, Phys. Rev. Lett. 71, 1486 (1993); ibid.75, 1872(E) (1995).
  20. G. W. Gibbons, Nucl. Phys. B271, 497 (1986).
  21. G. ’t Hooft, J. Geom. Phys. 1, 45 (1984).
  22. D. Boulware, Phys. Rev. D 11, 1404 (1975).
  23. S. A. Fulling, J. Phys. A 10, 917 (1977).
  24. B. S. DeWitt, in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1979).
  25. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England, 1982).
  26. S. Takagi, Prog. Theor. Phys. Suppl. 88, 1 (1986).
  27. G. W. Gibbons, in Proceedings of the 10th Sorak School of Theoretical Physics, edited by J. E. Kim (World Scientific, Singapore, 1992).
  28. A. Chamblin and G. W. Gibbons, Phys. Rev. D 55, 2177 (1996).
  29. R. Laflamme, Phys. Lett. B 196, 449 (1987).
  30. J. A. Wolf, Spaces of Constant Curvature (McGraw-Hill, New York, 1967).
  31. J. S. Dowker and R. Banach, J. Phys. A 11, 2255 (1978).
  32. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965).
  33. Ref. [25], Sec. 5.3.
  34. M. Holschneider, Wavelets: An Analysis Tool (Clarendon Press, Oxford, 1995).
  35. M. Ferraris and M. Francaviglia, Gen. Relativ. Gravit. 10, 283 (1979).
  36. G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970).
  37. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1991), Vol. I.
  38. N. Dunford and J. S. Schwartz, Linear Operators (Interscience, New York, 1963), Vol. II.
  39. S. M. Christensen and S. A. Fulling, Phys. Rev. D 15, 2088 (1977).
  40. J. P. Boersma, Phys. Rev. D 55, 2174 (1997).
  41. J. W. York, Phys. Rev. D 33, 2092 (1986).
  42. B. F. Whiting and J. W. York, Phys. Rev. Lett. 61, 1336 (1988).
  43. J. J. Halliwell and J. B. Hartle, Phys. Rev. D 41, 1815 (1990).
  44. J. J. Halliwell and J. Louko, Phys. Rev. D 42, 3997 (1990).
  45. B. L. Hu and A. Matacz, Phys. Rev. D 49, 6612 (1994).
  46. B. S. Kay and R. M. Wald, Phys. Rep. 207, 49 (1991).
  47. K. Fredenhagen and R. Haag, Commun. Math. Phys. 127, 273 (1990).
  48. R. Brandenberger and R. Kahn, Phys. Lett. 119B, 75 (1982).
  49. K. Schleich and D. M. Witt (private communication).

Outline

Information

Phys. Rev. D 58, 024007– Published 15 June, 1998
  • Received 2 March 1998

© 1998 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation