Analytic study of the null singularity inside spherical charged black holes

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 Φ and the area coordinate $r.\mathrm{}$ Both derivatives are found to grow along the contracting CH exactly like $1/r.$ Our results are valid everywhere along the CH singularity, down to the point of full focusing. These exact analytic expressions are verified numerically.