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 Psi 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, down to the point of full focusing. These exact analytic expressions are verified numerically.