Phenomenological Description of the Interior of the Schwarzschild Black Hole

Abstract

We discuss a sufficiently large 4-dimensional Schwarzschild black hole which is in equilibrium with a heat bath. In other words, we consider a black hole which has grown up from a small one in the heat bath adiabatically. We express the metric of the interior of the black hole in terms of two functions: One is the intensity of the Hawking radiation, and the other is the ratio between the radiation energy and the pressure in the radial direction. Especially in the case of conformal matters we check that it is a self-consistent solution of the semi-classical Einstein equation, $G_{\mu \nu }=8\pi G⟨T_{\mu \nu }⟩$. It is shown that the strength of the Hawking radiation is proportional to the c-coefficient, that is, the coefficient of the square of the Weyl tensor in the 4-dimensional Weyl anomaly.

Comments: 10 pages. Detail discussions and references added. Accepted Int. J. Mod. Phys. A