We construct a self-consistent model which describes a black hole from formation to evaporation including the back reaction from the Hawking radiation. In the case where a null shell collapses, at the beginning the evaporation occurs, but it stops eventually, and a horizon and singularity appear. On the other hand, in the generic collapse process of a continuously distributed null matter, the black hole evaporates completely without forming a macroscopically large horizon nor singularity. We also find a stationary solution in the heat bath, which can be regarded as a normal thermodynamic object. †

We analyze time evolution of a spherically symmetric collapsing matter from a point of view that black holes evaporate by nature. We first consider a spherical thin shell that falls in the metric of an evaporating Schwarzschild black hole of which the radius a(t) decreases in time. The important point is that the shell can never reach a(t) but it approaches a(t) − a(t) da (t) dt . This situation holds at any radius because the motion of a shell in a spherically symmetric system is not affected by the outside. In this way, we find that the collapsing matter evaporates without forming a horizon. Nevertheless, a Hawking-like radiation is created in the metric, and the object looks the same as a conventional black hole from the outside. We then discuss how the information of the matter is recovered. We also consider a black hole that is adiabatically grown in the heat bath and obtain the interior metric. We show that it is the self-consistent solution of Gµν = 8πG Tµν and that the four-dimensional Weyl anomaly induces the radiation and a strong angular pressure. Finally, we analyze the internal structures of the charged and the slowly rotating black holes.

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. †

Abstract:We analyze the time evolution of a spherically-symmetric collapsing matter from the point of view that black holes evaporate by nature. We consider conformal matters and solve the semi-classical Einstein equation G µν = 8πG T µν by using the four-dimensional Weyl anomaly with a large c coefficient. Here, T µν contains the contribution from both the collapsing matter and Hawking radiation. The solution indicates that the collapsing matter forms a dense object and evaporates without horizon or singularity, and it has a surface, but looks like an ordinary black hole from the outside. Any object we recognize as a black hole should be such an object.

We study a classical many-particle system with an external control represented by a timedependent extensive parameter in a Lagrangian. We show that thermodynamic entropy of the system is uniquely characterized as the Noether invariant associated with a symmetry for an infinitesimal non-uniform time translation t → t + η β, where η is a small parameter, is the Planck constant, β is the inverse temperature that depends on the energy and control parameter, and trajectories in the phase space are restricted to those consistent with quasi-static processes in thermodynamics.