Properties of radiation near the black-hole horizon and the second law of thermodynamics

By considering a gedanken experiment of adiabatically lowering a box containing matter with rest energy E and entropy S into a black hole, Bekenstein claimed that the necessary condition for the validity of the generalized second law of thermodynamics is S/E≤2πR, where R is the effective radius of the box. Unruh and Wald claimed that this condition is not necessary but the acceleration radiation can guarantee the generalized second law. In this paper, we point out that the Unruh-Wald conclusion does not hold because Hawking radiation near the horizon is not thermal. Bekenstein’s conclusion does not hold because the thin box approximation is not correct near the horizon. Neither Hawking radiation (or acceleration radiation) nor S/E≤2πR can guarantee the second law. We have sufficient reasons to conjecture that gravitation can influence the matter equation of state. For radiation, the usual equation of state ρ=α${\mathit{T}}^4$ and S=4/3α${\mathit{T}}^3$ does not hold in the strong gravitation field, e.g., near the black-hole horizon. We derive the equation of state for radiation near the horizon and find that it is very different from the equation in flat spacetime. The second law of thermodynamics can be satisfied if we impose some restrictions on one parameter of the equation of state. As a corollary, we get an upper bound on S/E which resembles Bekenstein’s result.