LAUSR

We calculate quantum gravitational corrections to the entropy of black holes using the Wald entropy formula within an effective field theory approach to quantum gravity. The corrections to the entropy are calculated to second order in curvature and we calculate a subset of those at third order. We show that, at third order in curvature, interesting issues appear that had not been considered previously in the literature. The fact that the Schwarzschild metric receives corrections at this order in the curvature expansion has important implications for the entropy calculation. Indeed, the horizon radius and the temperature receive corrections. These corrections need to be carefully considered when calculating the Wald entropy. E-mail: [email protected] E-mail: [email protected] Black holes are fascinating objects for many different reasons. Hawking’s groundbreaking intuition that black holes are not black but have a radiation spectrum that is very similar to that of a black body makes black holes an ideal laboratory to investigate the interplay between quantum mechanics, gravity and thermodynamics. This has led to the notion of Bekenstein-Hawking entropy or black hole entropy which has attracted much attention over the last almost 50 years. The calculation of quantum corrections to this entropy has been the subject of many publications, see e.g. [1, 2] for reviews. In this work we revisit the calculation of the entropy of a Schwarzschild black hole in quantum gravity and identify new important subtleties that have been overlooked in previous calculations. To be very specific, we use effective field theoretical methods to calculate quantum gravitational corrections to the entropy of this black hole using the Wald entropy formula [3]. We highlight new intriguing relations between the quantum corrections to the entropy, the Euler characteristic and quantum corrections to the metric of the Schwarzschild black hole. Previous calculations within the effective theory approach to quantum gravity [4– 6] have used the Euclidean path integral formulation of the entropy. We present a systematic approach that can easily be extended to any order in perturbation theory or to any black hole metric. The Wald approach to the calculation of a black hole entropy is very elegant and does not involve the Wick rotation to Euclidean time which is known to be tricky in quantum gravity. The Wald entropy formula reads [3] SWald = −2π ∫ dΣ ǫμνǫρσ ∂L ∂Rμνρσ ∣