Black Hole Entropy from Entropy of Hawking Radiation

Abstract

We provide a simple way for calculating the entropy of a Schwarzschild black hole from the entropy of its Hawking radiation. To this end, we show that if a thermodynamic system loses its energy only through the black body radiation, its loss of entropy is always 3/4 of the entropy of the emitted radiation. This proposition enables us to relate the entropy of an evaporating black hole to the entropy of its Hawking radiation. Explicitly, by calculating the entropy of the Hawking radiation emitted in the full period of evaporation of the black hole, we find the Bekenstein-Hawking entropy of the initial black hole. 1

Conserved charges and first law of thermodynamics for Kerr–de Sitter black holes

“…Specifically for the Einstein-Hilbert (EH) gravity, the entropy (which is called Bekenstein-Hawking entropy) was found to be S = A H 4G , in which A H is the area of the event horizon [2,3]. One could also find the same result for the entropy of a black hole, by studying directly the entropy of the Hawking radiation emitted in the whole process of the evaporation [4,5]. Nonetheless, a robust classical, semi-classical or quantum description for the microstates corresponding to the origin of this entropy is still an open question, although some interesting proposals have been suggested.…”

Black hole thermodynamics from decoherence

“…In the current case, this can be translated via (6) (cf. also [15]) into the information-theoretic context as S n S BH (10) which, by Schumacher's noiseless channel coding theorem for quantum information [16], means that the von Neumann entropy S BH of Hawking radiation measures the minimal physical resources to restore the information encoded in Hawking radiation. Indeed, the decoherence performs a projection from the original state to the diagonal density matrix, which can be envisioned as a data compression process provided that the whole system including the state and the observer is closed.…”

“…For a Schwarzshcild black hole, it is simply δE BH = T BH δS BH with E BH = M. In the current situation, this geometric relation can be derived from the relative entropy. Indeed, it has been shown in [23] that, since the relative entropy is a smooth nondegenrate function of states, if S(ρ(λ)||$ e (ρ)) depends on the perturbation parameter λ and ρ(λ) = ρ(0) + λρ ′ , $ e (ρ) = ρ(0), the first derivative of S(ρ(λ)||$ e (ρ)) vanish at λ = 0, which entails ∆S = ∆ H (15) for the first order variation of the entanglememt entropy S and the expectation of the modular Hamiltonian H = − log(ρ). Now suppose two nearby equilibrium states are in the Hartle-Hawking states, $ d (ρ) ∼ exp[−E/T BH ] 2 , with different energies.…”

“…Now suppose two nearby equilibrium states are in the Hartle-Hawking states, $ d (ρ) ∼ exp[−E/T BH ] 2 , with different energies. Then (15) gives…”