Formation of extremal regular black holes

Static and slowly evolving regular black holes with a non-extremal inner horizon are generally expected to suffer from mass inflation. Here we consider the scenario of asymptotic gravitational collapse into extremal regular black holes. To that end, we first construct geometric models of static spherically symmetric single-horizon extremal and double-horizon inner-extremal regular black holes for generic black hole mass as the only dimensionful scale. These spacetimes may be interpreted as non-fine-tuned vacuum solutions of modified gravitational theories defined implicitly by the requirement that their spherical reduction yields an integrable two-dimensional Horndeski theory. As such, these so-called general quasi-topological gravities admit exact Vaidya solutions in which the mass becomes a time-dependent function. We use this effectively two-dimensional second-order dynamical framework to model the asymptotic formation of extremal regular black holes without a violation of energy conditions. The latter is illustrated explicitly by the equivalence between the strong energy condition for generalised Vaidya solutions of general relativity, and the kinematic timelike convergence condition reformulated dynamically as an onshell condition on the theory-dependent functions characterising the spherical reduction and correspondingly Vaidya solutions of a general quasi-topological gravity. The defocusing of geodesics necessary for regular black holes is thus triggered by the modified gravitational dynamics rather than by the addition of exotic matter degrees of freedom.