Horizon-penetrating form of parametrized metrics for static and stationary black holes

The Rezzolla-Zhidenko (RZ) and Konoplya-Rezzolla-Zhidenko (KRZ) frameworks provide an efficient approach to characterize agnostically spherically symmetric or stationary black-hole spacetimes in arbitrary metric theories. In their original construction, these metrics were defined only in the spacetime region outside of the event horizon, where they can reproduce any black-hole metric with percent precision and a few parameters only. At the same time, numerical simulations of accreting black holes often require metric functions that are regular across the horizon, so that the inner boundary of the computational domain can be placed in a region that is causally disconnected from the exterior. We present a novel formulation of the RZ/KRZ parametrized metrics in coordinate systems that are regular at the horizon and defined everywhere in the interior. We compare the horizon-penetrating form of the KRZ and RZ metrics with the corresponding forms of the Kerr metric in Kerr-Schild coordinates and of the Schwarzschild metric in Eddington-Finkelstein coordinates, remarking the similarities and differences. We expect the horizon-penetrating formulations of the RZ/KRZ metrics to represent new tools to study via simulations the physical processes that occur near the horizon of an arbitrary black hole.