We consider line elements of the form $-2Gdu(dx+Bdu)+R^2(\dots )$, where (...) does not contain $dx$. Surfaces of constant $u$ are then null surfaces, and their affinely parametrized generators have tangent vector $G^{-1}{\partial }_x$. Considering $u$ as the time coordinate, we can evolve either $R$ or $G$, with the other one found by solving the Raychaudhuri equation along the null generators, or we can evolve both. This choice of formulation is independent from the remaining gauge choice $x\to x^{\prime }(u,x,\dots )$ in the line element above, which is fixed incrementally by the choice of $B$. For example, we can evolve $G$, in order to be able to evolve through an event horizon, and use $B$ to adapt the coordinates to type-II critical collapse. As a demonstration of these ideas, we consider a charged scalar field in spherical symmetry. We consider two settings: a domain where the outgoing null cones emanate from a regular center $R=0$, and a domain where they emanate from an ingoing-null boundary. In both settings, we demonstrate convergence with resolution, within each formulation and between the three formulations. As test beds, we compute the critical exponents and periodic fine structures of the black hole charge and mass scaling laws in a one-parameter family of charged regular initial data, and examples of perturbed extremal Reissner-Nordström solutions.

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• General relativity
• Numerical relativity