Admittedly, the above calculation is restricted to the leading correction to the semiclassical approximation. It has
been done for a ﬁxed large A
by taking the spins on all the punctures to be 1/ 2 so that we have the largest number
of punctures. But it is not diﬃcult to argue that the coeﬃcient of the lnA
term is robust in that inclusion of spin
values higher than 1/2 do not aﬀect it, although the c onstant term a nd the coeﬃcients of sub-leading corrections
with powers of O(A
) might get aﬀected. The same appears to be true for values of the level k away from the
asymptotic value which we have ass igned it above: the coeﬃcient of the lnA
is once again unaﬀected. Thus, the
leading logarithmic correction with co e ﬃcient -3/2 that we have discerned for the black hole entropy is in this sens e
universal. Moreover, although we have set β = β
ln2 in the above formulae, the coeﬃcient of the lnA
independent of β, a feature not shared by the s e miclassical area law.
It is therefore clear that the leading correction (and maybe also the subleading ones) to the BH entropy is negative.
One way to understand this could be the information-theoretic approach of Bekenstein : black hole entropy
represents lack of information about quantum states which arise in the various ways of gravitational collapse that
lead to formation of black holes with the same mass, charge and angular momentum. Thus, the BH entropy is the
‘maximal’ entropy that a black hole can have; incorporation of leading quantum eﬀects reduces the entropy. The
logarithmic nature of the leading correction points to a possible existence of w hat might be called a ‘non-perturbative
ﬁxed point’. That this happens in the physical world of four dimensions is perhaps not without interest.
Recently, the zeroth and ﬁrst law of black hole mechanics have been derived for situations with radiation present
in the vicinity of the horizon, using the notion of the isolated ho rizon . Our conclusions ab ove for the case of
non-rotating black holes hold for such generalizations [1 5 ] as well. Note however that while, the foregoing analysis
involves SU (2)
Chern Simons theory, for large k this reduces to a sp e c iﬁc U(1) theory presuma bly related to the
‘gauge ﬁxed’ clas sical theory discussed in . The charge spectrum of this U (1) theor y is discrete and bounded from
above by k. The SU (2) origin of the theory thus provides a natural ‘regularization’ for calculation of the number of
Note Added: After the ﬁrst version of this paper appeared in the Archives, it has been brought to our attention
that correc tions to the area law in the form of logarithm of horizon area have been obta ined earlier  for extremal
Reissner-Nordstrom and dilatonic black holes. These corrections are due to quantum scalar ﬁelds pro pagating in ﬁxed
classical backgrounds appropriate to these black holes. The coeﬃcient of the lnA
term that appears in ref.  is
diﬀerent from ours. This is only expected, since in contras t to ref. , our corrections originate from non-perturbative
quantum ﬂuctuations of spacetime ge ometry (for generic non-rotating black holes), in the absence of matter ﬁelds.
Thus, this correction is ﬁnite and independent of any arbitrary ‘renormalization sc ale’ asso c iated with divergences
due to quantum matter ﬂuctuations in a ﬁxed classical background.
We thank Prof. A. Ashtekar for many illuminating discussions and Prof. R. Mann for useful correspondence.
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