Description of Black Hole Microstates by Means of a Free Affine-Scalar Field

In this article we will invetsigate the origin of central charges in the Poisson algebra of charges, which arise in dimensionally reduced theories describing black holes. In order to do this we will analyze the equations of motion arising from the dimensionally reduced EinsteinHilbert action with the ansatz of spherical symmetry. We will show that the equations of motion and the constraints, which involve two fields, namely the dilaton (i.e. the radial coordinate) and the conformal factor of the 2-metric can be integrated by means of a free field. The transformation properties of this field must be found. The solution of the free field, which gives the Rindler space-time, which is the near-horizon approximation of the Schwarzschild solution, is equivalent to the solution of the conformal factor of the 2-metric. The conformal factor is a piece of a metric and therefore under conformal transformations it transforms like an affine scalar. The free field being in this case equivalent to the conformal factor must therefore also behave like an affine scalar. The stress-energy tensor generating such an affine transformation is the improved stress-energy tensor. The second derivative term in this tensor gives origin to the central term in the charges algebra. It is therefore the affine transformation property of the field that gives origin to the central charge used to compute the black hole entropy via Cardy formula. ∗e-mail: [email protected]