Skip to main content
SpringerLink
Log in

Fluctuations in a Thermal Field and Dissipation of a Black Hole Spacetime: Far-Field Limit

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

We study the backreaction of a thermal field ina weak gravitational background depicting the far-fieldlimit of a black hole enclosed in a box by the closedtime path (CTP) effective action and the influence functional method. We derive the noise anddissipation kernels of this system in terms ofquantities in quasiequilibrium, and formally prove theexistence of a fluctuation-dissipation relation (FDR) at all temperatures between the quantumfluctuations of the thermal radiance and the dissipationof the gravitational field. This dynamical self-consistent interplay between the quantum field and theclassical spacetime is, we believe, the correct way totreat backreaction problems. To emphasize this point wederive an Einstein–Langevin equation whichdescribes the nonequilibrium dynamics of thegravitational perturbations under the influence of thethermal field. We show the connection between our methodand the linear response theory (LRT), and indicate howthe functional method can provide more accurate results than prior derivations of FDRs via LRT in thetest-field, static conditions. This method is inprinciple useful for treating fully nonequilibrium casessuch as backreaction in black hole collapse.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. B. L. Hu, A. Raval, and S. Sinha, In Vishveshw ara Festschrift, B. Bhawal and B. Iyer, eds., Kluwer, Amsterdam, (1998).

    Google Scholar 

  2. B. L. Hu, Physica A 158, 399 (1979); Quantum statistical fields in gravitation and cosmology, in Proceedings Third International Workshop on Thermal Field Theory and Applications, R. Kobes and G. Kunstatter, eds., World Scientific, Singapore (1994) [grgc/ 9403061]; E. Calzetta and B. L. Hu, Phys. Rev. D 49, 6636 (1994); B. L. Hu and A. Matacz, Phys. Rev. D 51, 1577 (1995); B. L. Hu and S. Sinha, Phys. Rev. D 51, 1587 (1995); A. Campos and E. Verdaguer, Phys. Rev. D 53, 1927 (1996); F. C. Lombardo and F. D. Mazzitelli, Phys. Rev. D 55, 3889 (1997); E. Calzetta, A. Campos, and E. Verdaguer, Phys. Rev. D 56, 2163 (1997); A. Campos and E. Verdaguer, Int. J. Theor. Phys. 36, 2525 (1997); J. J. Halliwell, Phys. Rev. D 57, 2337 (1998); R. Martin and E. Verdaguer, in preparation.

    Google Scholar 

  3. J. Schwinger, J. Math. Phys. 2, 407 (1961); P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4, 1, 12 (1963); L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Engl. trans. Sov. Phys. JEPT 20, 1018 (1965)]; K. Chou, Z. Su, and B. Hao, Phys. Rev. B 22, 3385 (1980); G. Zhou, Z. Su, B. Hao, and L. Yu, Phys. Rep. 118, 1 (1985); Z. Su, L. Y. Chen, X. Yu, and K. Chou, Phys. Rev. B 37, 9810 (1988); B. S. DeWitt, in Quantum Concepts in Space and Time, R. Penrose and C. J. Isham, eds., Clarendon Press, Oxford (1986); R. D. Jordan, Phys. Rev. D 33, 444 (1986); E. Calzetta and B. L. Hu, Phys. Rev. D 35, 495 (1987); 37, 2878 (1988).

    Google Scholar 

  4. R. Feynman and F. Vernon, Ann. Phys. (NY) 24, 118 (1963); R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965); A. O. Caldeira and A. J. Leggett, Physica 121A, 587 (1983); H. Grabert, P. Schramm, and G. L. Ingold, Phys. Rep. 168, 115 (1988); B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992); 47, 1576 (1993).

    Google Scholar 

  5. A. Einstein, Ann. Physik 17, 549 (1905); 19, 371 (1906); H. Nyguist, Phys. Rev. 32, 110 (1928); H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951); H. B. Callen and R. F. Greene, Phys. Rev. 86, 702 (1952) J. Weber, Phys. Rev. 101, 1620 (1956); R. Kubo, J. Phys. Soc. Japan 12, 570 (1957).

    Google Scholar 

  6. P. Candelas and D. W. Sciama, Phys. Rev. Lett. 38, 1372 (1977); D. W. Sciama, In Relativity, Quanta and Cosmology— Centenario di Einstein, F. DeFinis, ed., Editrici Giunti Barbera Universitaria, Florence (1979); D. Sciama, P. Candelas, and D. Deutsch, Adv. Phys. 30, 327 (1981).

    Google Scholar 

  7. S. W. Hawking, Nature 248, 30 (1974); Commun. Math. Phys. 43, 199 (1975).

    Google Scholar 

  8. A. Campos and B. L. Hu, Phys. Rev. D58, 125021 (1998).

    Google Scholar 

  9. D. N. Page, Phys. Rev. D 25, 1499 (1982).

    Google Scholar 

  10. A. Campos, B. L. Hu, and A. Raval, Fluctuation-di ssipation relation for a quantum black hole in quasi-equilibrium with its Hawking radiation, in preparation.

  11. B. L. Hu, N. Phillips, and A. Raval, Fluctuations of the energy-momentum tensor of a quantum field in a black hole spacetime, in preparation.

  12. P. R. Anderson, W. A. Hiscock, and D. A. Samuel, Phys. Rev. Lett. 70, 1739 (1993); Phys. Rev. D 51, 4337 (1995); W. A. Hiscock, S. L. Larson, and P. A. Anderson, Phys. Rev. D 56, 3571 (1997).

    Google Scholar 

  13. B. P. Jenson, J. G. McLaughlin, and A. C. Ottewill, Phys. Rev. D 51, 5676 (1995).

    Google Scholar 

  14. P. Hajicek and W. Israel, Phys. Lett. 80A, 9 (1980); J. M. Bardeen, Phys. Rev. Lett. 46, 382 (1981).

    Google Scholar 

  15. J. W. York, Jr., Phys. Rev. D 28, 2929 (1983); 31, 775 (1985); 33, 2092 (1986).

    Google Scholar 

  16. R. Parentani and T. Piran, Phys. Rev. Lett. 73, 2805 (1994); S. Massar, Phys. Rev. D 52, 5857 (1995).

    Google Scholar 

  17. J. B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188 (1976).

    Google Scholar 

  18. G. Gibbons and M. J. Perry, Proc. R. Soc. Lond. A 358, 467 (1978).

    Google Scholar 

  19. E. Mottola, Phys. Rev. D 33, 2136 (1986).

    Google Scholar 

  20. W. Bernard and H. B. Callen, Rev. Mod. Phys. 31, 1017 (1959); R. Kubo, Rep. Prog. Phys. 29, 255 (1966); L. Landau, E. Lifshitz, and L. Pitaevsky, Statistical Physics, Pergamon, London (1980), Vol. 1; R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II, Springer-Verlag, Berlin (1985).

    Google Scholar 

  21. D. J. Gross, M. J. Perry, and L. G. Yaffe, Phys. Rev. D 25, 330 (1982).

    Google Scholar 

  22. A. Rebhan, Nucl. Phys. B 351, 706 (1991); 369, 479 (1992).

    Google Scholar 

  23. A. P. de Almeida, F. T. Brandt, and J. Frenkel, Phys. Rev. D 49, 4196 (1994), and references therein; F. T. Brandt and J. Frenkel, Phys. Rev. D58, 085012 (1998).

    Google Scholar 

  24. S. W. Hawking and D. Page, Comm. Math. Phys. 87, 577 (1983).

    Google Scholar 

  25. H. A. Weldon, Phys. Rev. D 26, 1394 (1982).

    Google Scholar 

  26. B. L. Hu, Phys. Lett. 108B, 19 (1982); 123B, 189 (1983).

    Google Scholar 

  27. N. D. Birrell and P. C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge (1982).

    Google Scholar 

  28. C.-I. Kuo and L. H. Ford, Phys. Rev. D 47, 4510 (1993); E. Flanagan and R. M. Wald, Phys. Rev. D 54, 6233 (1996); N. G. Phillips and B. L. Hu, Phys. Rev. D 55, 6123 (1997).

    Google Scholar 

  29. P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959); L. P. Kadanoff and P. C. Martin, Ann. Phys. (NY) 24, 419 (1963).

    Google Scholar 

  30. D. S. Jones, The Theory of Generalized Functions, Cambridge University Press, Cambridge (1982).

    Google Scholar 

  31. R. Mills, Propagators for Many Particle Systems, Gordon and Breach, New York (1969).

    Google Scholar 

  32. M. Le Bellac, Thermal Field Theory, Cambridge University Press, Cambridge (1996).

    Google Scholar 

  33. B. L. Hu, Alpan Raval, and S. Sinha, Fluctuation-dissipation relation and back reaction for a radiating quantum black hole, in preparation.

  34. B. L. Hu, J. Louko, H. Phillips, and J. Simon, In preparation.

Download references

Authors
  1. Antonio Campos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. L. Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Campos, A., Hu, B.L. Fluctuations in a Thermal Field and Dissipation of a Black Hole Spacetime: Far-Field Limit. International Journal of Theoretical Physics 38, 1253–1271 (1999). https://doi.org/10.1023/A:1026670816596

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026670816596

Keywords

Search

Navigation