No abstract

Ghosh and Mitra Reply: The Comment [1] criticizes its own Eq. (1). This equation was neither written nor used in [2].The technical observation made in [1] is that configurations with a 2 ϳ ͑r 2 r 1 ͒ 21 near the horizon ͑r ϳ r 1 ͒ and with extremal topology, i.e.,

We present a statistical mechanical calculation of the thermodynamical properties of (nonrotating) isolated horizons. The introduction of the Planck scale allows for the definition of a universal horizon temperature (independent of the mass of the black hole) and a well-defined notion of energy (as measured by suitable local observers) proportional to the horizon area in Planck units. The microcanonical and canonical ensembles associated with the system are introduced. Black hole entropy and other thermodynamical quantities can be consistently computed in both ensembles and results are in agreement with Hawking's semiclassical analysis for all values of the Immirzi parameter.

A proper counting of states for black holes in the quantum geometry approach shows that the dominant configuration for spins are distributions that include spins exceeding one-half at the punctures. This raises the value of the Immirzi parameter and the black hole entropy. However, the coefficient of the logarithmic correction remains -1/2 as before. * amitg@theory.saha.ernet.in † mitra@theory.saha.ernet.in

Various approaches to black hole entropy yield the area law with logarithmic corrections, many involving a coefficient 1/2, and some involving 3/2. It is pointed out here that the standard quantum geometry formalism is not consistent with 3/2 and favours 1/2.There has long been an association of the area of the horizon of a black hole with an 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 naïve 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. This divergence may be thought of as a contribution to the bare or classical gravitational constant G, which is to be renormalized to a finite G R 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 quantum geometry [6]. Even though a complete and universally accepted quantum theory of gravity is not quite at hand, both of these approaches can accommodate 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. The area of the horizon of an extremal dilatonic black hole vanishes, and in this case the matter field approach was seen to lead to a logarithm of the mass of the black hole [7] in the expression for the entropy. For black holes with non-vanishing area, the logarithm of the area appears as a sub-leading term after the dominant term proportional to the area. The coefficient of the logarithm depends on the black hole and is 1/90 in the Schwarzschild case. These coefficients are expected to be renormalized, as indicated above. Logarithmic corrections to the gravitational entropy, with coefficients which are negative, appeared later in many models. One approach [8] was related to the quantum geometry formulation but eventually mapped the counting problem to conformal blocks, leading to a negative coefficient of magnitude 3/2. Another [9] started from ideas about conformal symmetry in the near-horizon degrees of freedom and considered corrections to the Cardy formula, reaching the same coefficient. There were variations on thes...