Phenomenological Description of the Interior of the Schwarzschild Black Hole

TL;DR

This paper models the interior of a large Schwarzschild black hole in equilibrium with a heat bath, expressing the metric via Hawking radiation intensity and radiation-pressure ratio, and links Hawking radiation strength to Weyl anomaly coefficients.

Contribution

It provides a phenomenological description of the black hole interior using semi-classical Einstein equations and relates Hawking radiation to Weyl anomaly in conformal matter.

Findings

Hawking radiation intensity is proportional to the Weyl anomaly coefficient.

The interior metric is expressed through two key functions: radiation intensity and energy-pressure ratio.

The solution is self-consistent for conformal matter in equilibrium with a heat bath.

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, $G_{\mu\nu}=8\pi G \langle T_{\mu\nu}\rangle$. 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.