A bound on the log correction to the black hole area law

There has recently been a controversy regarding logarithmic corrections to the area law of the black hole entropy: different approaches give different negative corrections, some with a coefficient 3/2, some with 1/2. It is pointed out here that the standard quantum geometry formalism is consistent with 1/2 but not with 3/2. It has long been argued that the area of the horizon of a black hole can be interpreted as its entropy [1]. This was not initially understood according to the Boltzmann definition of entropy as a measure of the number of quantum states of a black hole, because of the absence of a proper quantum theory of gravity. As a first step, however, considering gravity to be a statistical system, the näıve Lagrangian path integral was seen quite early to lead to a partition function from which the area law of entropy was obtained [2] in the leading semiclassical approximation ignoring all quantum fluctuations. Subsequent support was obtained from considerations of quantum fields in black hole backgrounds [3, 4]. The entropy calculated for the fields may be regarded as an additional contribution to the entropy of the black hole matter system, and the gravitational entropy of the black hole itself may be imagined to get modified in this way. In these field theory calculations the leading term has a divergent multiplicative factor with the area of the horizon. The divergence has been interpreted as unphysical, to be understood as a contribution to the bare or classical gravitational constant G, which is to be renormalized to a finite GR in the presence of quantized matter fields. Recently some statistical derivations of the area law have appeared in more elaborate models of quantum gravity – in string theory [5] as well as in what is called quantum geometry [6]. Even though a universally accepted quantum theory of gravity is not quite at hand, both of these approaches yield the expected number of quantum micro-states of a black hole. With the area law so well established for the entropy of large black holes, it is not surprising that even corrections to the area formula have been studied. [email protected] [email protected]