Angular momentum and conservation laws for dynamical black holes

TL;DR

This paper develops a coherent framework for defining and analyzing mass, angular momentum, and related laws for dynamical black holes using trapping horizons, extending classical black hole laws to non-stationary scenarios.

Contribution

It introduces a unique angular momentum measure satisfying a conservation law and formulates dynamical versions of the black hole laws using trapping horizons.

Findings

Defines a unique angular momentum from the Komar integral.

Derives dynamical energy and angular momentum conservation laws.

Formulates a dynamical first law and generalizes the zeroth law for black holes.

Abstract

Black holes can be practically located (e.g. in numerical simulations) by trapping horizons, hypersurfaces foliated by marginal surfaces, and one desires physically sound measures of their mass and angular momentum. A generically unique angular momentum can be obtained from the Komar integral by demanding that it satisfy a simple conservation law. With the irreducible (Hawking) mass as the measure of energy, the conservation laws of energy and angular momentum take a similar form, expressing the rate of change of mass and angular momentum of a black hole in terms of fluxes of energy and angular momentum, obtained from the matter energy tensor and an effective energy tensor for gravitational radiation. Adding charge conservation for generality, one can use Kerr-Newman formulas to define combined energy, surface gravity, angular speed and electric potential, and derive a dynamical version of the so-called "first law" for black holes. A generalization of the "zeroth law" to local equilibrium follows. Combined with an existing version of the "second law", all the key quantities and laws of the classical paradigm for black holes (in terms of Killing or event horizons) have now been formulated coherently in a general dynamical paradigm in terms of trapping horizons.