The UV divergences of the gravitational action are computed by the heat kernel method using the small
expansion (69
). For a minimal massless field (
in the scalar field equation) one finds
The tree-level entropy can be obtained by means of the same replica trick, considered in
Sections 3.8, 3.9 and 3.10, upon introduction of the conical singularity with a small angle deficit
,
. The conical singularity at the horizon
manifests
itself in that a part of the Riemann tensor for such a manifold
behaves as a distribution
having support on the surface
. Using formulas (56
) – (59
) one finds for the tree-level entropy
The UV divergent part of the entanglement entropy of a black hole has already been calculated, see
Eq. (82). For a minimal massless scalar, one has
It should be noted that the proof of the renormalization statement is based on a nice property of the
heat kernel coefficients (68
) on space with conical singularity. Namely, up to
terms the
exact coefficient
on conical space
is equal to the regular volume coefficient
expressed in terms of the complete curvature, regular part plus a delta-like contribution, using
relations (55
)
That the leading divergence in the entropy can be handled by the standard renormalization of
Newton’s constant
has been suggested by Susskind and Uglum [213
] and by Jacobson [141
].
That one also has to renormalize the higher curvature couplings in the gravitational action in
order to remove all divergences in the entropy of the Schwarzschild black hole was suggested by
Solodukhin [196
]. For a generic static black hole the renormalization statement was proven
by Fursaev and Solodukhin in [112
]. In a different approach based on ’t Hooft’s “brick-wall
model” the renormalization was verified for the Reissner–Nordström black hole by Demers,
Lafrance and Myers [62
]. For the rotating black hole described by the Kerr–Newman metric
the renormalization of the entropy was demonstrated by Mann and Solodukhin [170]. The
non-equilibrium aspect (as defining the rate in a semiclassical decay of hot flat space by black hole
nucleation) of the black hole entropy and the renormalization was discussed by Barbon and
Emparan [12].
http://www.livingreviews.org/lrr-2011-8 |
Living Rev. Relativity 14, (2011), 8
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