Analytic study of the null singularity inside spherical charged black holes

TL;DR

This paper analytically examines the null singularity inside charged black holes, deriving exact expressions for divergence rates of key quantities and confirming them with numerical verification.

Contribution

It provides the first exact analytic expressions for the divergence rates of the scalar field and area coordinate derivatives at the Cauchy horizon inside charged black holes.

Findings

Derivatives grow as 1/r along the Cauchy horizon.

Exact expressions for divergence rates are derived.

Results are verified numerically across the horizon.

Abstract

We study analytically the features of the Cauchy horizon (CH) singularity inside a spherically-symmetric charged black hole, nonlinearly perturbed by a self-gravitating massless scalar field. We derive exact expressions for the divergence rate of the blue-shift factors, namely the derivatives in the outgoing direction of the scalar field $\Phi$ and the area coordinate $r$. Both derivatives are found to grow along the contracting CH exactly like $1/r$. Our results are valid everywhere along the CH singularity, up to the point of full focusing. These exact analytic expressions are verified numerically.