Isolated and dynamical horizons and their applications

TL;DR

The paper presents a quasi-local program for black holes based on isolated and dynamical horizons, unifying equilibrium and dynamical regimes across quantum gravity, numerical relativity, and gravitational phenomenology. It develops rigorous definitions, area increase laws, and first-law-like relations for horizons, then demonstrates practical applications in simulations, and extends the framework to non-Einstein theories and quantum gravity. A central achievement is the horizon-based derivation of a mass and entropy consistent with Kerr and Hawking results, using a boundary Chern–Simons theory and loop-quantum-gravity techniques. The work offers a versatile toolkit for extracting physically meaningful horizon data from simulations, clarifying black-hole thermodynamics in non-equilibrium settings, and guiding future research on horizon dynamics and quantum aspects of gravity.

Abstract

Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modeled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity; suggested a phenomenological model for hairy black holes; provided novel techniques to extract physics from numerical simulations; and led to new laws governing the dynamics of black holes in exact general relativity.

Figures (10)

• Figure 3: Set-up of the general characteristic initial value formulation. The Weyl tensor component $\Psi_0$ on the null surface $\Delta$ is part of the free data which vanishes if $\Delta$ is an IH.
• Figure 4: Penrose diagrams of Schwarzschild--Vaidya metrics for which the mass function $M(v)$ vanishes for $v\le 0$kuroda. The space-time metric is flat in the past of $v=0$ (i.e., in the shaded region). In the left panel, as $v$ tends to infinity, $\dot{M}$ vanishes and $M$ tends to a constant value $M_0$. The space-like dynamical horizon $H$, the null event horizon $E$, and the time-like surface $r = 2M_0$ (represented by the dashed line) all meet tangentially at $i^+$. In the right panel, for $v\ge v_0$ we have $\dot{M} =0$. Space-time in the future of $v=v_0$ is isometric with a portion of the Schwarzschild space-time. The dynamical horizon $H$ and the event horizon $E$ meet tangentially at $v=v_0$. In both figures, the event horizon originates in the shaded flat region, while the dynamical horizon exists only in the curved region.
• Figure 5: $H$ 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. In the figure, $H$ joins on to a weakly isolated horizon $\Delta$ with null normal $\bar{\ell}^a$ at a cross-section $S_0$.
• Figure 6: The region of space-time $\mathcal{M}$ under consideration has an internal boundary $\Delta$ and is bounded by two Cauchy surfaces $M_1$ and $M_2$ and the time-like cylinder $\tau_\infty$ at infinity. $M$ is a Cauchy surface in $\mathcal{M}$ whose intersection with $\Delta$ is a spherical cross-section $S$ and the intersection with $\tau_\infty$ is $S_\infty$, the sphere at infinity.
• Figure 7: The world tube of apparent horizons and a Cauchy surface $M$ intersect in a 2-sphere $S$. $T^a$ is the unit time-like normal to $M$ and $R^a$ is the unit space-like normal to $S$ within $M$.