# Quantum effects for near-extremal black holes ## Two puzzles The first puzzle about extremal or near-extremal black holes is that, above extremality, $\delta E=E-M_0=T^2/M_\text{gap}$, implying that the black hole does not have enough energy to emit even a single quantum of average energy $T$ at small $T<M_\text{gap}$ (because $\delta E/T<1$). So, either semiclassical physics breaks down below this temperature, or there is a literal gap between extremal black holes and the lightest near-extremal black hole in the spectrum of black holes, which is weird for non-SUSY black holes like RN. The second puzzle that the large degeneracy of states, as suggested by the non-zero entropy, is incompatible with our usual observations, which is that a quantum system should not have a large ground state degeneracy unless protected by symmetry. ## Resolution Both puzzles are resolved by considering the quantum effects near extremality. For non-SUSY black holes, there is no mass gap, but the quantum corrected relation becomes$E-M_{0} \sim \frac{3}{2} T>T,$which is enough energy for a Hawking quantum to get emitted, resolving the first puzzle. Correctly accounting for quantum effects for near-extremal black holes gives$Z_{\mathrm{grav}}\sim T^{\#}e^{S_0+T S_1}$so that the degeneracy goes to zero as $T\to 0$, resolving the second puzzle. For SUSY black holes, there is a delta function density of states at exact extremality and a gap between the ground state and excited states, resolving both puzzles. ## Refs - review - [[2023#Turiaci (Newsletter)]] - original puzzles - [[1991#Preskill, Schwarz, Shapere, Trivedi, Wilczek]]: raises the first puzzle (about) - [[1998#Maldacena, Michelson, Strominger]] - [[2000#Page]]: ground state degeneracy - resolution of the puzzles - [[2020#Iliesiu, Turiaci]] - full higher-dimensional perspective - [[2024#Kolanowski, Marolf, Rakic, Rangamani, Turiaci]] - $T^{3/2}$ scaling from [[0631 DHS formula|DHS formula]] - [[2024#Kapec, Law, Toldo]] - Kerr - Reissner-Nordström - strange metal - [[2024#Liu, Nian, Zayas]]