Effective Dynamics of Spherically Symmetric Static Spacetime

TL;DR

This paper explores the effective dynamics of spherically symmetric static spacetimes in general relativity, incorporating matter fields and quantum corrections, leading to potential singularity resolution and new models of black hole interiors.

Contribution

It introduces a framework for deriving effective Hamiltonians for gravity coupled with matter and quantum fields, enabling analysis of non-trivial evolutions and singularity resolution in black hole models.

Findings

Effective Hamiltonians describe semi-classical and quantum-corrected dynamics.

Existence of singularity-free black-hole-like solutions stabilized by quantum matter.

Transition from constant Ricci scalar interior to Schwarzschild exterior achieved.

Abstract

In general relativity, the Einstein equations provide spherically symmetric static spacetimes with dynamics defined as an evolution along the radial coordinate $r$. The geometrical sector becomes a one-dimensional mechanical system, with the Misner-Sharp mass and lapse as canonically conjugate variables, and a vanishing Hamiltonian for pure gravity. Coupling classical or quantum matter fields, or introducing (quantum) corrections to general relativity, then generate a non-vanishing effective Hamiltonian, leading to non-trivial evolutions of the mass and lapse. We illustrate this mechanism through various examples of classical matter fields and identify Hamiltonians describing the effective dynamics of gravity coupled to perfect fluids with linear barotropic equation of state. Finally, we derive effective Hamiltonians that reproduce the gravitational semi-classical dynamics coupled to renormalized quantum matter fields and discuss the conditions for which the singularity at $r=0$ is resolved. In particular, we find a singularity-free black-hole-like solution, stabilized by quantum matter, smoothly transitioning from a bulk with constant negative Ricci scalar to the standard outside Schwarzschild metric. This opens new possibilities for the modeling of both semi-classical corrections and deep quantum effects on the interior structure of self-gravitating compact objects and black holes.