Numerical investigation of the interior geometry of semiclassical evaporating spherical charged black holes
Gil Arad
TL;DR
This study numerically evolves the semiclassical Einstein equations for a spherically symmetric, evaporating charged black hole to probe the interior geometry near the inner horizon. By implementing a double-null grid and testing an analytic near-IH approximation developed by Ori and Zilberman, the authors validate that the region-3 metric derivatives $S_{,v}$, $S_{,u}$, $R_{,v}$, and $R_{,u}$ closely follow their predicted behaviors, with deviations scaling predictably with the semiclassical parameter $oldsymbol{ ilde{oldsymbol{ au}}}_{0}$. The simulations also demonstrate robustness against details of the semiclassical source terms, and under certain flux configurations they reveal the possible formation of a spacelike $R=0$ singularity inside the evaporating black hole. Overall, the work supports the analytic IH-backreaction picture and sets the stage for iterative, self-consistent semiclassical treatments. Key methodological contributions include a second-order finite-difference scheme in double-null coordinates, careful initialization to avoid EH irregularities via the $ ilde{g}_{1}$ and $g_{1}$ constructions, and a framework to test analytic approximations under varied horizon flux inputs. The results strengthen confidence in semiclassical interior dynamics for charged BHs and offer a path toward fully self-consistent RSET computations in evolving spacetimes.
Abstract
We developed a numerical code which evolves the semiclassical Einstein's equation (with the quantum stress-energy contribution added as a source term) for the spherically symmetric metric inside an evaporating semiclassical charged black hole. An analytical approximation for the evolving semiclassical metric was recently developed by Ori and Zilberman (and will be briefly overviewed here). We seek to numerically check the validity of this analytical approximation. The Einstein equations in this case are partial differential equations for the two unknown metric functions which fully describe the spherically symmetric metric. We begin our numerical simulation close to the event horizon with regular initial data specified by a variant of the charged Vaidya metric. We then evolve the metric functions deep into the neighborhood of the inner horizon. We explore the results of running this numerical code in several representative cases. Our numerical simulations confirm the validity of the above mentioned analytical approximation in all these cases.