Black hole interacting with matter as a simple dynamical system

TL;DR

This paper develops a variational framework for a spherically symmetric black hole and matter system, simplifying the phase space and deriving explicit solutions, with implications for understanding black hole-matter interactions.

Contribution

It introduces a new variable transformation that regularizes the phase space and simplifies the constraint equations for a black hole with matter, providing explicit solutions and analyzing symmetries.

Findings

Phase space boundary is complex but manageable with new variables.

Explicit solutions for conserved momenta are obtained.

Lower-dimensional models reveal additional properties.

Abstract

Recently, a variational principle has been derived from Einstein-Hilbert and a matter Lagrangian for the spherically symmetric system of a dust shell and a black hole. The so-called physical region of the phase space, which contains all physically meaningful states of the system defined by the variational principle, is specified; it has a complicated boundary. The principle is then transformed to new variables that remove some problems of the original formalism: the whole phase space is covered (in particular, the variables are regular at all horizons), the constraint has a polynomial form, and the constraint equation is uniquely solvable for two of the three conserved momenta. The solutions for the momenta are written down explicitly. The symmetry group of the system is studied. The equations of motion are derived from the transformed principle and are shown to be equivalent to the previous ones. Some lower-dimensional systems are constructed by exclusion of cyclic variables, and some of their properties are found.