Interior Metric of Slowly Formed Black Holes in a Heat Bath

We study a spherical black hole formed slowly in a heat bath in the context of ordinary field theory, which we expect to have the typical properties of black holes. We assume that the matter field is conformal and that the metric satisfies the semi-classical Einstein equation $G_{\mu\nu}=8\pi G \langle \psi| T_{\mu\nu}|\psi \rangle$, where $|\psi \rangle$ is the wave function of the matter field. Then as a necessary condition, its trace part must be satisfied, $G^\mu{}_\mu=8\pi G \langle \psi| T^\mu{}_\mu|\psi \rangle$, whose right-hand side is independent of $|\psi \rangle$ and is determined only by the metric through the 4-dimensional Weyl anomaly. With some physically reasonable assumptions, this equation restricts the interior metric to a certain class. Such metrics are approximately warped products of $AdS_2$ and $S^2$ with almost Planckian curvature. Among them, we find one that is consistent with Hawking radiation and is smoothly connected to the exterior Schwarzschild metric slightly outside the Schwarzschild radius. This leads to a picture that the black hole is a dense object with a surface (not a horizon), which evaporates due to Hawking-like radiation when taken out of the bath. Other solutions represent objects that are larger in size than a black hole, which may be useful for analyzing ultra-dense stars.