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3
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84931516461
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In D dimensions, a cross section of the horizon is a (D 2) dimensional space, and so the surface area A refers to the volume of such a slice.
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16
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84931516460
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V. Iyer and R.M. Wald, Chicago Report No. gr qc/9403028 (unpublished).
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19
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4243894352
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in Directions in General Relativity, edited by B.L. Hu, and C., (Cambridge University Press, Cambridge, England, 1993), Vol. 1. For a closely related work, see also J.
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(1991)
Phys. Rev. Lett.
, vol.66
, pp. 2281
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Wald, R.M.1
Ryan, M.2
Vishveshwara, V.3
Brown, D.4
Martinez, E.A.5
York, J.W.6
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28
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84931516459
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The identification of the black hole temperature as kappa/2π is a result of quantum field theory in curved space time, Ref. 1, and so is completely independent of which theory of gravity determines the space time geometry. Alternatively, this identification presumably follows from a semiclassical evaluation of the partition function or the density of states in quantum gravity with any action, as has been shown explicitly in the case of Einstein gravity see for example S.W. Hawking, in General Relativity, An Einstein Centenary Survey, edited by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1979);
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31
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84931516458
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In higher dimensions, one has the possibility of commuting rotations in totally orthogonal planes. Hence the black hole is characterized by the set of angular momenta δ J(α) in the maximal set of mutually orthogonal planes.
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34
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84931516457
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In the present context of asymptotically flat solutions for Einstein gravity plus higher derivative perturbations, a straightforward calculation shows that the asymptotic variations are dominated by contributions from the Einstein term, and do indeed yield the desired mass and angular momentum variations see, Ref. 6, for a more general discussion.
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38
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84931516456
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Here { cal L}ξ and iξ denote the Lie derivative and interior multiplication by the vector ξa, respectively. We use the identity { cal L}ξ=diξ+iξ d. We do not consider the possibility that an additional exact form appears in the variation of L, which would still be compatible with diffeomorphism invariance of the theory. In the case of spinor fields and/or gauge fields, one must choose a spin frame or gauge in which the field components have vanishing Lie derivatives in order to apply the method as given in, Ref. 6, The identity { cal L}ξ=diξ+iξ d will fix our definition of the Lie derivative of such gauge or frame dependent fields.
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39
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84931516500
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It should be emphasized that is only the pullback of Cab tilde chib to the horizon that vanishes by this argument. For example, the covariant Killing field tilde{ chi}a itself does not vanish on the horizon, although its pullback does vanish.
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41
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84931516501
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To compare Eq. ( ref{current}) with the results of, Ref. 7, one uses the identity hat{ epsilon}ab hat{ epsilon}cd=gad⊥gbc⊥ -gac⊥gbd⊥.
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42
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84931516504
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We are grateful to Andy Strominger for relating this observation to us.
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44
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84931516503
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Note that to order λ2, we are free to interchange gab and bar{g}ab in any first order terms.
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53
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0039176116
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edited by, K.S. Thorne, R.H. Price, D.A. MacDonald, Yale University Press, New Haven, CT
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(1986)
Black Holes: The Membrane Paradigm
, pp. 280
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Thorne, K.S.1
Zurek, W.H.2
Price, R.H.3
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54
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84931516508
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edited by, B. Bertotti, F. de Felice, A. Pascolini, Consiglio Nazionale delle Ricerche, Roma, Vol. 2;
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(1983)
General Relativity and Gravitation, Proceedings of the 10th International Conference, Padova, Italy, 1983
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Sorkin, R.D.1
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