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3
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84927903122
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J. D. Brown, Ph.D. thesis, University of Texas at Austin, 1985; published as a book, Lower Dimensional Gravity (World Scientific, Singapore, 1988).
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5
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84927903121
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The Euclidean continuation may be obtained by setting t = - i tau in (2). This makes the metric complex when J != 0. The Euclidean action is given by iI=-IEuc. Although correct in the case at hand, making imaginary a particular coordinate in a generally covariant theory is disturbing. A more satisfactory way to define the Euclidean continuation is the following. One solves the Euclidean field equations but introduces complex constants of integration in the solutions, thus generating complex solutions to real equations. In the present example, one sets JEuc= - i J, and, when there is electric charge, QEuc= - i Q. Here J and Q are the physical (Minkowskian signature) values of the angular momentum and electric charge. This point of view makes it clear that the procedure is coordinate invariant.
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9
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84927903110
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Already S. Hawking [in General Relativity, An Einstein Centenary Survey, edited by S. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1979), pp. 779,780] had pointed out that the difference between the action and the mass multiplied by the time was due to a contribution at the horizon. However, his somewhat ad hoc regularization procedure—involving a subtraction at infinity—obscured the fact that the entropy itself is a surface term at the horizon. Expressing the entropy as a surface integral on the horizon plays a key role in an ongoing program by F. Wilczek (Princeton lecture notes, unpublished) to elucidate the physical origin of the black hole entropy. We thank him for inspiring discussions on black hole physics.
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10
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84927903109
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The constant r0 is an arbitrary reference point for the energy, which cannot be taken to be at infinity as in the 3+1 case. If r0 is replaced by r0sprime, M is replaced by M + 1/2 Q2ln ( r0sprime/ r0). However, the variation of the free energy keeping the temperature and the chemical potentials fixed is independent of r0, and so are T, S, and Nφ( r+).
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84927903108
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We thank E. Witten for pointing this out to us.
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