Dynamical theory for spherical black holes in modified gravity

TL;DR

This paper develops a general algorithm to construct Hamiltonians for spherically symmetric static metrics, enabling the study of regular black holes and their dynamical properties within modified gravity theories.

Contribution

It introduces a method to derive Hamiltonians for static spherical metrics in modified gravity, facilitating dynamical analysis of black hole models without higher-derivative terms.

Findings

Explicit Hamiltonians for deformed Schwarzschild geometries

Framework for coupling matter and analyzing backreaction

Application to regular black hole models

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.