Dynamical theory for spherical black holes in modified gravity

TL;DR

The paper develops a covariant Hamiltonian framework for static, spherically symmetric spacetimes to provide a dynamical theory for regular black holes. It introduces a reconstruction algorithm that, given a target line element, yields a Hamiltonian constraint $\mathcal{H}_{(h_1,h_2,h_3)}$ built from deformations of general relativity, while preserving the hypersurface-deformation algebra. For Schwarzschild-like geometries, the authors show how GR is recovered (with additive corrections possible when deforming shape functions such as $h_1,h_3$) and demonstrate the method by deriving explicit Hamiltonians for a wide class of deformed spacetimes, including Bardeen, Hayward, Simpson-Visser, and loop-quantum-gravity-inspired metrics. The framework accommodates covariant matter coupling, enabling backreaction analyses and assessments of whether proposed regular black-hole geometries can emerge from dynamical gravitational collapse. This covariant formulation thus provides a unified route to connect static regular geometries with their possible dynamical origins and experimental implications.

Abstract

We provide a general algorithm to construct a Hamiltonian, such that its dynamical flow covariantly defines any given spherically symmetric and static metric. This Hamiltonian is defined as a linear combination of the standard (general relativistic) radial diffeomorphism constraint plus a Hamiltonian constraint that is appropriately deformed as compared to its corresponding form in general relativity though it does not include higher-derivative terms. Therefore, given a static model of spherical gravity, it is possible to obtain its Hamiltonian, and, thus, its canonical (second-order) equations of motion. A particularly relevant application of this construction is the study of regular black holes, where proposed geometries often lack an underlying dynamical theory. The present method provides such a theory. In particular, for a wide class of deformations of the Schwarzschild geometry, we explicitly obtain their corresponding Hamiltonian. This construction can be further used to covariantly couple matter. In this way, one can analyze the backreaction of matter fields on the geometry of interest, and, specifically, whether a particular black-hole model may emerge as the end state of a dynamical collapse.