Quantum gravitational corrections to black hole geometries

We calculate perturbative quantum gravity corrections to generic eternal two-dimensional dilaton gravity spacetimes. We estimate the leading corrections to the AdS2 black hole entropy and determine the quantum modification of N -dimensional Schwarzschild spacetime. ∗ [email protected] † [email protected] In recent years, two-dimensional dilaton gravity has earned a great deal of attention because of its relevance to the study of classical and quantum properties of (higher-dimensional) black holes, branes, string theory, and gravitational collapse. At the quantum level, attention has been mainly focussed on semiclassical theory (see for instance [1] and references therein). Matter fields are quantized whereas gravity is classical in this formulation. Here, we take a different approach and quantize the gravitational sector of the theory. We apply the perturbative quantization algorithm described in Ref. [2] to generic two-dimensional dilaton gravity spacetimes in vacuo and evaluate pure quantum gravity corrections to the geometry of eternal dilatonic black holes. Quantum corrections to the classical solutions are obtained by a perturbative expansion on powers of the curvature. Thus quantum gravitational effects vanish in the limit of large ADM mass and in the asymptotic region far away from the black hole horizon where the black hole behaves classically. Quantum effects become instead significant at finite distances from the black hole horizon. This leads to a modification of the physical quantities which are associated to the classical geometry. As two illustrative examples, we will estimate the leading quantum gravity corrections to the Bekenstein-Hawking entropy of the AdS2 black hole and to N -dimensional spherically symmetric gravity. The two-dimensional dilaton gravity action is SG = ∫ dx √−g [φR+ V (φ)] , (1) where φ is the dilaton field, R is the two-dimensional Ricci scalar constructed from the metric g (2) ab , and V (φ) is the dilaton potential. In the following we will consider the power-law dilaton potential V (φ) = bφ , (2) where b is a positive real number. In the gauge φ = x the general classical black hole solution of the theory is [3] ds(2) ≡ g (2) ab (x)dx dx = [N(φ)−M ] dt − [N(φ)−M ] dφ , (3) where N(φ) = ∫ φ dφV (φ) = φ . (4) The constant parameter M is proportional to the ADM mass. The geometry is singular at φ = 0 and asymptotically flat as φ → ∞. The black hole