Apparent horizon tracking in supercritical solutions of the Einstein-scalar field equations in spherical symmetry in affine-null coordinates

Choptuik's critical phenomena in general relativity is revisited in the affine-null metric formulation of Einstein's equations for a massless scalar field in spherical symmetry. Numerical solutions are obtained by evolution of initial data using pseudo-spectral methods. The underlying system consists of differential equations along the outgoing null rays which can be solved in sequential form. A new two-parameter family of initial data is presented for which these equations can be integrated analytically. Specific choices of the initial data parameters correspond to either an asymptotically flat null cone, a black hole event horizon or the singular interior of a black hole. Our main focus is on the interior features of a black hole, for which the affine-null system is especially well adapted. We present both analytic and numerical results describing the geometric properties of the apparent horizon and final singularity. Using a re-gridding technique for the affine parameter, numerical evolution of initially asymptotically flat supercritical data can be continued inside the event horizon and track the apparent horizon up to the formation of the final singularity.