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 µν = 8πG T µν . 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. †

A Model of Black Hole Evaporation and 4D Weyl Anomaly

“…A Planck-like distribution can be obtained even if there is no horizon [3,5,12]. 4 We keep using the term "black hole" even though the system is different from the conventional black hole that has a horizon. 5 See also [13][14][15][16][17].…”

“…Finally, we take the continuum limit in the multi-shell model and construct the candidate metric [3][4][5]. Especially, we focus on a configuration in which each shell has already come close to R(a i ):…”

“…We note that the dominant energy condition [21] is violated, − T T T T θ θ , and that the interior is not a fluid in the sense T r r T θ θ [3][4][5]. We can check the validity of the classical gravity in (5).…”

“…Here, a = 2GM, and l p ≡ √h G. However, these two effects become comparable near the black hole. Recently, it has been discussed that the inclusion of the back reaction plays a crucial role in determining the time evolution of a collapsing matter [3][4][5][6][7][8][9].…”

On the near-horizon geometry of an evaporating black hole

“…However, even the time-dependent Schwarzschild metric can not be a good approximation for the geometry close to the Schwarzschild radius. It was shown [14][15][16][17][18] that the timedependent Schwarzschild metric does not allow any infalling causal trajectory to cross the Schwarzschild radius, assuming that the Schwarzschild radius monotonically shrinks to zero within finite time, regardless of how slowly the Schwarzschild radius shrinks. As previous works on the near-horizon geometry is unreliable, we shall thus focus on the geometry around the Schwarzschild radius, and study the time-dependence of the wormhole-like geometry.…”