Emergence of an effective two-dimensional quantum description from the study of critical phenomena in black holes

We study the occurrence of critical phenomena in four-dimensional, rotating, and charged black holes, derive the critical exponents, and show that they satisfy the scaling laws. Correlation function critical exponents and renormalization group considerations assign an effective (spatial) dimension, d=2, to the system. The two-dimensional Gaussian approximation of the order parameter is shown to reproduce all the black hole's critical exponents. Higher order corrections (which are always relevant) are discussed. Identifying the two-dimensional surface with the event horizon and noting that a generalization of scaling leads to conformal invariance and then to string theory, we arrive at 't Hooft's string interpretation of black holes. From this, an effective model for dealing with a coarse-grained black hole quantization is proposed. We also give simple arguments that lead to a (first) quantization of the black hole mass in units of the Planck mass, i.e., M\ensuremath{\simeq}1/ \ensuremath{\surd}2${\mathit{M}}_{\mathrm{Pl}}$ \ensuremath{\surd}l with l a positive integer, and then, from this result, to the proportionality between quantum entropy and area.

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