Quasi-Local Black Hole Horizons: Recent Advances

TL;DR

This review argues for quasi-local horizons as robust, teleology-free boundaries for black holes, replacing event horizons in dynamical GR and numerical relativity. It develops a cohesive framework around non-expanding, weakly isolated, isolated, and dynamical horizon segments, with a rigorous second-law-like area growth for dynamical horizons tied to local fluxes of energy and gravitational radiation. The work highlights invariant horizon multipoles and their time evolution, revealing deep links to null infinity observables and BMS structure, and demonstrates how horizon dynamics can be integrated with gravitational-wave signals in a process called gravitational-wave tomography. These insights, supported by analytic results and high-precision NR simulations, advance our understanding of BH mergers, the approach to equilibrium of remnants, and the interplay between strong-field horizon physics and asymptotic gravitational radiation, with implications for GW data interpretation and tests of GR.

Abstract

While the early literature on black holes focused on event horizons, subsequently it was realized that their teleological nature makes them unsuitable for many physical applications both in classical and quantum gravity. Therefore, over the past two decades, event horizons have been steadily replaced by quasi-local horizons which do not suffer from teleology. In numerical simulations event horizons can be located as an `after thought' only after the entire space-time has been constructed. By contrast, quasi-local horizons naturally emerge in the course of these simulations, providing powerful gauge-invariant tools to extract physics from the numerical outputs. They also lead to interesting results in mathematical GR, providing unforeseen insights. For example, for event horizons we only have a qualitative result that their area cannot decrease, while for quasi-local horizons the increase in the area during a dynamical phase is quantitatively related to local physical processes at the horizon. In binary black hole mergers, there are interesting correlations between observables associated with quasi-local horizons and those defined at future null infinity. Finally, the quantum Hawking process is naturally described as formation and evaporation of a quasi-local horizon. This review focuses on the dynamical aspects of quasi-local horizons in classical general relativity, emphasizing recent results and ongoing research.

Figures (11)

• Figure 1: Left Panel: Penrose diagram of an Oppenheimer-Snyder (OS) stellar collapse. The event Horizon (EH) $\mathcal{E}$ is the future boundary of the part of space-time from which signals can reach future null infinity $\mathcal{I}^{+}$. The spherical dynamical horizon (DH) $\mathcal{H}$ has zero radius at formation and grows in the past direction to join the EH. Middle Panel: Collapse of a null fluid. The fluid (shaded region) is incident from $\mathcal{I}^{-}$. To the past of the null fluid, space-time is flat. The EH forms and grows already in this flat region although there is nothing happening there. DH forms in the fluid region and grows in area and joins the EH once all of the null fluid has fallen in. Right Panel: OS stellar collapse followed by a null fluid collapse, say, a billion years later. 'New EH' is the actual EH of this space-time after the stellar and the null fluid collapse while 'Old EH' denotes the would be EH, had there been no null fluid collapse in the distant future. DH grows in area only when there is flux of matter passing through it --first in the stellar region and then in the null fluid region. In the intervening billion years, it becomes an isolated horizon. It has been shown Booth_2010 that the this isolated horizon segment is extremely close to the actual EH, a feature that is difficult to see because distances are not faithfully represented in Penrose diagrams.
• Figure 2: Left Panel: Disappearance of a WH via emissions of a null fluid. This is a time reversal of the middle panel of Fig. 1. The spherical DHS is again space-like but now a AT-DHS. Middle Panel: Disappearance of a while hole by emission of an OS star. This is a time reversal of the left panel of Fig. 1. DHS is again time-like but now a AT-DHS. Right Panel: Two roles of infalling null fluids. An infalling null fluid with $\mathcal{E} >0$ leads to a BH of mass $M_1$. Here we have a space-like T-DHS. A second in-falling null fluid but now with $\mathcal{E} <0$ reduces its mass to $M_2 < M_1$. Here we have a time-like T-DHS. This process mimics the formation of a BH by Vaidya collapse and subsequent (partial) evaporation due to infalling negative energy flux as in the Hawking process Sawayama:2005mwHayward_2006. Note that now there are MTSs outside the EH $\mathcal{E}$; this is possible because the energy condition is violated by the second infalling null fluid. Finally, all these examples, QLHs are spherically symmetric. More general examples of MTSs and DHSs can be found in Podolsky:2009an
• Figure 3: Space-like DHS:$\Delta \mathcal{H}$ is a portion of the DHS $\mathcal{H}$ bounded by two cross-sections $S_1$ and $S_2$ and an intermediate cross-section $S_t$ at time $t$.
• Figure 4: (Figure from Pook-Kolb:2018igu) Sequence for MTS for Brill-Lindquist data: In each of the plots the masses are fixed at $m_1 = 0.2$, $m_2 = 0.8$, and the separation $d$ becomes successively smaller. Each panel shows four MTSs: the two associated with the individual progenitor BHs (larger one in red, and smaller one in magenta), the outer MTS associated with the remnant BH (in blue and referred to as the apparent horizon (AH)), and the inner MTS (in green).
• Figure 5: (Figure from PhysRevLett.123.171102Snapshots of the MTS structure of a simulation of a Brill-Lindquist initial datum: Taken at different simulation times, they show the progression of the marginally trapped surfaces as the two BHs merge to form a final remnant BH. The first panel, shortly after the start of the simulation, shows the two marginal surfaces for the two individual BHs labeled $S_1$ and $S_2$. The common horizon has bifurcated into an inner and outer branch labeled $S_{\rm inner}$ and $S_{\rm outer}$ respectively. The middle panel shows the situation just before $S_1$ and $S_2$ touch, i.e. at the time $T_{\rm touch} \approx 5.5378\,\mathcal{M}$ where $\,\mathcal{M}$ is the ADM mass. Here $S_{\rm inner}$ is on the verge of forming a cusp, while on the other hand $S_{\rm outer}$ has lost most of its distortions. Finally, the right panel shows the surfaces after $T_{\rm touch}$. Here $S_{\rm outer}$ is very close to its final spherically symmetric state, while $S_{\rm inner}$ has developed a self-intersection, and $S_1$ and $S_2$ have penetrated each other.