Interior of Black Holes and Information Recovery

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 $a(t)-a(t)\frac{d a(t)}{d t}$. 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 $G_{\mu\nu}=8\pi G \langle T_{\mu\nu} \rangle$ 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.Comment: Appear in Physical Review D. Typos fixed. References, clarifications and new appendixes adde

Higher derivative three-form gauge theories and their supersymmetric extension

We investigate three-form gauge theories with higher derivative interactions and their supersymmetric extensions in four space-time dimensions. For the bosonic three-form gauge theories, we show that derivatives on the field strength of the 3-form gauge field yield a tachyon as far as the Lagrangian contains a quadratic kinetic term, while such the term with opposite sign gives rise to a ghost. We confirm that there is neither a tachyon nor a ghost when all higher derivative terms are given by functions of the field strength. For this ghost/tachyon-free Lagrangian, we determine the boundary term necessary for the consistency between the equation of motion and energy-momentum tensor. For supersymmetric extensions, we present ghost/tachyon-free higher derivative interactions of arbitrary order of the field strength and corresponding boundary terms as well.Comment: 46 pages; v2: references added, published versio

Black Hole as a Quantum Field Configuration

We describe 4D evaporating black holes as quantum field configurations by solving the semi-classical Einstein equation $G_{\mu\nu}=8\pi G \langle \psi|T_{\mu\nu}|\psi \rangle$ and quantum matter fields in a self-consistent manner. As the matter fields we consider $N$ massless free scalar fields ($N$ is large). We find a spherically symmetric self-consistent solution of the metric $g_{\mu\nu}$ and state $|\psi\rangle$. Here, $g_{\mu\nu}$ is locally $AdS_2\times S^2$ geometry, and $|\psi\rangle$ provides $\langle \psi|T_{\mu\nu}|\psi \rangle=\langle0|T_{\mu\nu}|0 \rangle+T_{\mu\nu}^{(\psi)}$, where $|0\rangle$ is the ground state of the matter fields in the metric and $T_{\mu\nu}^{(\psi)}$ consists of the excitation of s-waves that describe the collapsing matter and Hawking radiation with the ingoing negative energy flow. This object is supported by a large tangential pressure $\langle0|T^\theta{}_\theta|0 \rangle$ due to the vacuum fluctuation of the bound modes with large angular momenta. This describes the interior of the black hole when the back reaction of the evaporation is considered. The black hole is a compact object with a surface (instead of horizon) that looks like a conventional black hole from the outside and eventually evaporates without a singularity. If we count the number of self-consistent configurations $\{|\psi\rangle\}$, we reproduce the area law of the entropy. This tells that the information is carried by the s-waves inside the black hole. $|\psi\rangle$ also describes the process that the negative ingoing energy flow created with Hawking radiation is superposed on the collapsing matter to decrease the total energy while the total energy density remains positive. As a special case, we consider conformal matter fields and show that the interior metric is determined by the matter content of the theory, which leads to a new constraint to the matter content.Comment: ver4: We added a new paragraph to Sec.2.1. and made Appendix

Phenomenological Description of the Interior of the Schwarzschild Black Hole

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