Black hole in de Sitter space

If cosmological constant is positive, a black hole is naturally described by the Schwarzschild-de Sitter solution with two horizons. We use the global method to extract the topological information and the selection rule for the Gibbons-Hawking temperature for the thermal vacua. These are related to the Euler number of the Euclidean section whose topology is more complicated than expected. We also point out the failure of the usual local method of conical singularity approach in dealing with multi-horizon scenarios. VPI-IPPAP-98-4 July, 1998 Talk presented at the Sixth International Symposium on Particles, Strings and Cosmology (PASCOS-98), Northeastern University, March 22-29, 1998. b The author would like to thank Dr. Chopin Soo for useful comments on the manuscript. c email address: [email protected] Black hole in de Sitter space Feng-Li Lin Department of Physics, and Institute for Particle Physics and Astrophysics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, U.S.A. E-mail: [email protected] If cosmological constant is positive, a black hole is naturally described by the Schwarzschild-de Sitter solution with two horizons. We use the global method to extract the topological information and the selection rule for the Gibbons-Hawking temperature for the thermal vacua. These are related to the Euler number of the Euclidean section whose topology is more complicated than expected. We also point out the failure of the usual local method of conical singularity approach in dealing with multi-horizon scenarios. The evidence of the positive cosmological constant lead us to wonder how the usual black hole thermodynamics will be modified in this non-asymptotically flat de Sitter background. In this regard, we have to account for both the cosmological as well as the black hole horizons. It seems that no global thermal temperature could be defined unless these two horizons are coincident, however, then the Euclidean section in between the horizons shrink to zero, and not physically interesting. We may then ask if it is possible to define a global and unique Gibbons-Hawking temperature and the associated thermal vacuum in the generic static multi-horizon scenario. The answer is surprisingly in the affirmative. For the Schwarzschild black hole, the Hawking temperature is defined by requiring the removal of the conical singularity on the horizon. So unlike the quantum theory on flat R × S Euclidean background in which the thermal temperature is just a parameter and could be any values, in the curved background with horizon, the thermal temperature and thus the vacuum are prescribed by the geometry and not arbitrary. The conical method can not be applied to the multi-horizon spacetimes, such as the Schwarzschild-de Sitter spacetime, because one can not find a temperature to remove the conical singularities on both horizons simultaneously. If so, the one-loop quantum theory will blow up on the conical singularities at horizons, and be not well-defined. The more deep reason is that the local geometry information from conical singularity around each horizon is not enough to determine the global feature such as temperature in the multi-horizon scenario. This reminds us the similar situation about the singular string of the Dirac monopole, and implies a global method with the patching conditions is needed in dealing with the nontrivial