Black Holes as Frozen Stars: Regular Interior Geometry

Abstract

The authors have proposed a model geometry for the interior of a regular black hole (BH) mimicker, the frozen star, whose most startling feature is that each spherical shell in its interior is a surface of infinite redshift. The geometry is a solution of the Einstein equations which is sourced by an exotic matter with maximally negative radial pressure. The frozen star geometry is previously presented in singular coordinates for which $-g_{tt}$ and $g^{rr}$ vanish in the bulk and connect smoothly to the Schwarzschild exterior. Additionally, the geometry is mildly singular in the center of the star. Here, the authors present regular coordinates for the entirety of the frozen star. Each zero in the metric is replaced with a small, dimensionless parameter ε; in both $-g_{tt}$ and $g^{rr}$ thus maintaining maximally negative radial pressure. The authors also regularize the geometry, energy density and pressure in the center of the star in a smooth way. The frozen star solution presented here is a completely regular solution of Einstein's equations whose compactness is arbitrarily close to that of a Schwarzschild BH. It obeys the null energy condition, the universally agreed-upon energy condition, and it is free of any known pathology. As far as it is known, this is the first solution that obeys all of these constraints and, in addition, as will be shown in a future publication, can mimic all of the thermodynamic properties of a standard BH. Our initial analysis uses Schwarzschild-like coordinates and applies the Killing equations to show that an infalling, point-like object will move very slowly, effectively sticking to the surface of the star and never coming out. If one nevertheless follows the trajectory of the object into the interior of the star, it moves along an almost-radial trajectory until it comes within a small distance from the star's center. Once there, if the object has any amount of angular momentum at all, it will be reflected outwards by a potential barrier onto a different almost-radial trajectory. Finally, using Kruskal-like coordinates, the causal structure of the regularized frozen star and discuss its $\epsilon \to 0$ limit, for which the geometry degenerates and becomes effectively two dimensional is considered.