Abstract
In case of spacetimes with single horizon, there exist several well- established procedures for relating the surface gravity of the horizon to a thermodynamic temperature. Such procedures, however, cannot be extended in a straightforward manner when a spacetime has multiple horizons. In particular, it is not clear whether there exists a notion of global temperature characterizing the multi-horizon spacetimes. We examine the conditions under which a global temperature can exist for a spacetime with two horizons using the example of Schwarzschild–De Sitter (SDS) spacetime. We systematically extend different procedures (like the expectation value of stress tensor, response of particle detectors, periodicity in the Euclidean time etc.) for identifying a temperature in the case of spacetimes with single horizon to the SDS spacetime. This analysis is facilitated by using a global coordinate chart which covers the entire SDS manifold. We find that all the procedures lead to a consistent picture characterized by the following features: (a) In general, SDS spacetime behaves like a non-equilibrium system characterized by two temperatures. (b) It is not possible to associate a global temperature with SDS spacetime except when the ratio of the two surface gravities is rational. (c) Even when the ratio of the two surface gravities is rational, the thermal nature depends on the coordinate chart used. There exists a global coordinate chart in which there is global equilibrium temperature while there exist other charts in which SDS behaves as though it has two different temperatures. The coordinate dependence of the thermal nature is reminiscent of the flat spacetime in Minkowski and Rindler coordinate charts. The implications are discussed.
Similar content being viewed by others
References
Padmanabhan T. (2002). Class. Quant. Grav. 9: 5387
Padmanabhan T. (2002). Mod. Phys. Lett. A17: 923 gr-qc/0202078
Padmanabhan T. (2002). Class. Quant. Grav. 19: 3551 gr-qc/0110046
Padmanabhan T. (2005). Phys. Rep. 406: 49 gr-qc/0311036
Padmanabhan, T.: Braz. J. Phys. (special issue) 35, 362 (2005), gr-qc/0412068
Gibbons G.W. and Hawking S.W. (1977). Phys. Rev. D15: 2738
Shankaranarayanan S. (2003). Phys. Rev. D67: 084026 gr-qc/0301090
Srinivasan K. and Padmanabhan T. (1999). Phys. Rev. D60: 024007 gr-qc/9812028
Shankaranarayanan S., Srinivasan K. and Padmanabhan T. (2001). Mod. Phys. Lett. A16: 571 gr-qc/ 0007022
Vagenas E.C. (2002). Nuovo Cim. 117B: 899 hep-th/0111047
Tadaki S.-I. and Takagi S. (1990). Prog. Theor. Phys. 83: 941
Tadaki S. and Takagi S. (1990). Prog. Theor. Phys. 83: 1126
Markovic D. and Unruh W.G. (1991). Phys. Rev. D43: 332
Bousso R. and Hawking S.W. (1998). Phys. Rev. D57: 2436 hep-th/9709224
Nojiri S. and Odintsov S.D. (1999). Phys. Rev. D59: 044026 hep-th/9804033
Nojiri S. and Odintsov S.D. (2000). Int. J. Mod. Phys. A15: 989 hep-th/9905089
Wu Z.C. (2000). Gen. Relat. Gravity 32: 1823 gr-qc/9911078
Wu Y.-Q., Zhang L.-C. and Zhao R. (2001). Int. J. Theor. Phys. 40: 1001
Zhao R., Zhang J.-F. and Zhang L.-C. (2001). Mod. Phys. Lett. A16: 719
Hiscock W.A. (1989). Phys. Rev. D39: 1067
Deser S. and Levin O. (1997). Class. Quant. Gravity 14: L163 gr-qc/9706018
Zhao R., Zhang L.-C. and Li Z.-G. (1998). Nuovo Cim. B 113: 291
Deser S. and Levin O. (1999). Phys. Rev. D59: 064004 hep-th/9809159
Myung Y.S. (2001). Mod. Phys. Lett. A16: 2353 hep-th/0110123
Wu S.Q. and Cai X. (2001). Nuovo Cim. 116B: 907 hep-th/0108033
Garattini R. (2001). Class. Quant. Grav. 18: 571 gr-qc/0012078
Ghezelbash A.M. and Mann R.B. (2002). JHEP 01: 005 hep-th/0111217
Cvetic M., Nojiri S. and Odintsov S.D. (2002). Nucl. Phys. B628: 295 hep-th/0112045
Wu S.Q. and Cai X. (2002). Int. J. Theor. Phys. 41: 559 gr-qc/0111045
Danielsson U.H. (2002). JHEP 03: 020 hep-th/0110265
Nojiri S., Odintsov S.D. and Ogushi S. (2003). Int. J. Mod. Phys. A18: 3395 hep-th/0212047
Guido D. and Longo R. (2003). Annales Henri Poincare 4: 1169 gr-qc/0212025
Gomberoff A. and Teitelboim C. (2003). Phys. Rev. D67: 104024
Corichi A. and Gomberoff A. (2004). Phys. Rev. D69: 064016 hep-th/0311030
Davies P.C.W. and Davis T.M. (2002). Founds. Phys. 32(12): 1877 astro-ph/0310522
Davis T.M., Davies P.C.W. and Lineweaver C.H. (2003). Class. Quant. Grav. 20: 2753 astro-ph/0305121
Teitelboim, C.: (2002) hep-th/0203258
Kim, Y.-b., Oh, C.Y., Park, N. (2002) hep-th/0212326
Cai R.-G. and Guo Q. (2004). Phys. Rev. D69: 104025 hep-th/0311020
Cai R.-G. (2002). Nucl. Phys. B628: 375 hep-th/0112253
Cai R.-G. (2002). Phys. Lett. B525: 331 hep-th/0111093
Cai R.-G., Ji J.-Y. and Soh K.-S. (1998). Class. Quant. Grav. 15: 2783 gr-qc/9708062
Klemm D. (2002). Nucl. Phys. B625: 295 hep-th/0106247
Chao W.-Z. (1997). Int. J. Mod. Phys. D6: 199 gr-qc/9801020
Maeda K., Koike T., Narita M. and Ishibashi A. (1998). Phys. Rev. D 57: 3503 gr-qc/9712029
Lin F.-L. and Soo C. (1999). Class. Quant. Gravity 16: 551 gr-qc/9708049
Perlmutter S., Aldering G., Goldhaber G., Knop R.A., Nugent P., Castro P.G., Deustua S., Fabbro S., Goobar A. and Groom D.E., (1999). Astrophys. J. 517: 565
Padmanabhan T. (2003). Phys. Rept. 380: 235
Padmanabhan T. (2005). Curr. Sci. 88: 1057 astro-ph/0411044
Sahni V. and Starobinsky A.A. (2000). Int. J. Mod. Phys. D 9: 373 astro-ph/9904398
Peebles P.J. and Ratra B. (2003). Rev. Mod. Phys. 75: 559
Padmanabhan T. and Choudhury T.R. (2003). Mon. Not. R. Astron. Soc. 344: 823
Choudhury T.R. and Padmanabhan T. (2005). Astron. Astrophys. 429: 807 astro-ph/0311622
Padmanabhan T. and Choudhury T.R. (2002). Phys. Rev. D66: 081301 hep-th/0205055
Bagla J.S., Jassal H.K. and Padmanabhan T. (2003). Phys. Rev. D67: 063504
Padmanabhan T. (2002). Phys. Rev. D 66: 021301 hep-th/0204150
Medved A.J.M. (2002). Phys. Rev. D66: 124009 hep-th/0207247
Birrell N.D. and Davies P.C.W. (1982). Quantum Fields in Curved Space. Cambridge University Press, Cambridge
Sriramkumar L. and Padmanabhan T. (2002). Int. J. Mod. Phys. D11: 1 gr-qc/9903054
Boulware D.G. (1975). Phys. Rev. D11: 1404
Hartle J.B. and Hawking S.W. (1976). Phys. Rev. D13: 2188
Unruh W.G. (1976). Phys. Rev. D14: 870
Christensen S.M. and Fulling S.A. (1977). Phys. Rev. D15: 2088
Wald R.M. (1984). General relativity. University of Chicago Press, Chicago
Bousso R. and Hawking S.W. (1995). Phys. Rev. D52: 5659
Ginsparg P. and Perry M.J. (1983). Nucl. Phys. B 222: 245
Gibbons G.W. and Hawking S.W. (1979). Commun. Math. Phys. 66: 291
Rovelli C. (1998). Living Rev. Relat. 1: 1
Padmanabhan T. (2004). Class. Quant. Gravity 21: L1 gr-qc/0310027
Choudhury T.R. and Padmanabhan T. (2004). Phys. Rev. D69: 064033 gr-qc/0311064
Cardoso V., Natario J. and Schiappa R. (2004). J. Math. Phys. 45: 4698 hep-th/0403132
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Choudhury, T.R., Padmanabhan, T. Concept of temperature in multi-horizon spacetimes: analysis of Schwarzschild–De Sitter metric. Gen Relativ Gravit 39, 1789–1811 (2007). https://doi.org/10.1007/s10714-007-0489-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10714-007-0489-0