Dynamical Horizons and their Properties

TL;DR

This paper develops a quasi-local framework for growing black holes by introducing dynamical horizons—spacelike 3-surfaces foliated by marginally trapped 2-surfaces. It derives local flux expressions for matter and gravitational energy across these horizons, proves an area balance law, and constructs an integral first law incorporating angular momentum, leading to a notion of horizon mass that reduces to the Hawking or Kerr mass in appropriate limits. It further analyzes angular momentum flux, the transition to equilibrium with weakly isolated horizons, and the matching of physical quantities across the dynamical-to-equilibrium transition. The framework provides foundational tools for numerical relativity, mathematical proofs (e.g., Penrose-type inequalities), and potential quantum-gravity applications by linking strong-field dynamics to horizon geometry. Overall, it offers a comprehensive, local description of black-hole growth and relaxation toward steady states in full general relativity.

Abstract

A detailed description of how black holes grow in full, non-linear general relativity is presented. The starting point is the notion of dynamical horizons. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local and the energy flux is positive. Change in the horizon area is related to these fluxes. A notion of angular momentum and energy is associated with cross-sections of the horizon and balance equations, analogous to those obtained by Bondi and Sachs at null infinity, are derived. These in turn lead to generalizations of the first and second laws of black hole mechanics. The relation between dynamical horizons and their asymptotic states --the isolated horizons-- is discussed briefly. The framework has potential applications to numerical, mathematical, astrophysical and quantum general relativity.

Figures (2)

• Figure 1: $H$ is a dynamical horizon, foliated by marginally trapped surfaces $S$. ${\widehat{\tau}}^a$ is the unit time-like normal to $H$ and ${\widehat{r}}^{\,a}$ the unit space-like normal within $H$ to the foliations. Although $H$ is space-like, motions along ${\widehat{r}}^{\,a}$ can be regarded as time evolution with respect to observers at infinity. In this respect, one can think of $H$ as a hyperboloid in Minkowski space and $S$ as the intersection of the hyperboloid with space-like planes. $H$ joins on to a weakly isolated horizon $\Delta$ with null normal $\bar{\ell}^a$, at a cross-section $S_o$.
• Figure 2: A plot of the Kerr surface gravity $\kappa_o$ (from eq. (\ref{['kerrkappa']})) as a function of $R$ (with $JG$ set equal to $1$ for definiteness). The part $\kappa_o<0$ is the Kerr forbidden side while $\kappa_o>0$ is the Kerr allowed regime. Since $R$ increases monotonically with time, this graph shows that the dynamical horizon always evolves toward the Kerr allowed region under time evolution.