Classical Liouville Theory and the Microscopic Interpretation of Black Hole Entropy

In this paper we will compute the entropy of a Schwarzschild black hole by finding a classical central charge of the Virasoro algebra of a Liouville theory and using the Cardy formula. This is done by performing a dimensional reduction of the Einstein–Hilbert action with the Ansatz of spherical symmetry and writing the metric in conformally flat form. We obtain two coupled field equations using the near horizon approximation the field equation for the conformal factor decouples becoming a Liouville equation. The generators of conformal transformations of a Liouville theory form a Virasoro algebra with a classical central charge. It is then possible to compute the black hole entropy via Cardy formula and so to count the microstates responsible for this entropy. This computation is independent from a specific quantum theory of gravity model.