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We numerically investigate the interior of a four-dimensional, asymptoticallyflat, spherically symmetric charged black hole perturbed by a scalar field$\Phi$. Previous study by Marolf and Ori indicated that late infallingobservers will encounter an effective shock wave as they approach the leftportion of the inner horizon. This shock manifests itself as a sudden change inthe values of various fields, within a tremendously short interval of propertime $\tau$ of the infalling observers. We confirm this prediction numericallyfor both test and self-gravitating scalar field perturbations. In both cases wedemonstrate the effective shock in the scalar field by exploring $\Phi(\tau)$along a family of infalling timelike geodesics. In the self-gravitating case wealso demonstrate the shock in the area coordinate $r$ by exploring $r(\tau)$.We confirm the theoretical prediction concerning the shock sharpening rate,which is exponential in the time of infall into the black hole. In addition wenumerically probe the early stages of shock formation. We also employ a familyof null (rather than timelike) ingoing geodesics to probe the shock in $r$. Weuse a finite-difference numerical code with double-null coordinates combinedwith a recently developed adaptive gauge method in order to solve the (Einstein+ scalar) field equations and to evolve the spacetime (and scalar field) $ - $from the region outside the black hole down to the vicinity of the Cauchyhorizon and the spacelike $r=0$ singularity.

Numerical study of the gravitational shock wave inside a spherical charged black hole