Curvature scaling and linear area response in charged black-hole interiors

Abstract

We identify the mechanisms and range of validity of two scaling laws reported in charged black-hole interiors. In spherical evolutions with a neutral massless scalar, mass inflation produces a large internal mass $m$, followed by a plateau. Small scalar and electric corrections give $K\simeq48m^2/r^6$, where $r$ is the areal radius and $K$ the Kretschmann scalar. The local exponent is near six when the mass and relative curvature correction also vary slowly with radius. The exact bound $K\geq28q^4/(5r^8)$, with $q$ the charge, excludes a finite-coefficient $r^{-6}$ asymptote, while a large plateau mass delays electric corrections to $r\sim q^2/m$. For sufficiently large plateau mass, this classical regime can reach Planckian curvature before electric corrections become significant. At fixed coordinates and initial gauge, a linear response of $ρ=r^2$ to the initial scalar amplitude $A$ gives the half-power radius law near a simple zero. Where both mechanisms apply, $K/m^2\propto(A_*-A)^{-3}$, with $A_*$ the area zero. Full evolutions confirm the electric correction, support the half-power law at several observation times, and test the joint inverse-cube law.