Black hole boundaries

TL;DR

Booth surveys local and non-local notions of black-hole boundaries, contrasting event horizons with quasilocal constructions based on marginally trapped surfaces. It introduces MOTS/MOTT, isolated, trapping, and dynamical horizons, and develops their existence, uniqueness, flux laws, topology, and slow-evolving limits within a Hamiltonian framework. The paper emphasizes that quasilocal horizons offer physically meaningful, locally identifiable proxies for black-hole boundaries in dynamical spacetimes and numerical relativity, while acknowledging non-uniqueness and potential connections to the event horizon. Through analytic examples (e.g., Vaidya, Tolman-Bondi), it illustrates rich horizon dynamics, including horizon jumps and transitions between regimes. The work points toward future developments in understanding horizon thermodynamics, near-field interactions, and the refinement of trapping boundaries.

Abstract

Classical black holes and event horizons are highly non-local objects, defined in relation to the causal past of future null infinity. Alternative, quasilocal characterizations of black holes are often used in mathematical, quantum, and numerical relativity. These include apparent, killing, trapping, isolated, dynamical, and slowly evolving horizons. All of these are closely associated with two-surfaces of zero outward null expansion. This paper reviews the traditional definition of black holes and provides an overview of some of the more recent work on alternative horizons.

Figures (8)

• Figure 1: A Penrose-Carter diagram showing a spacetime in which a matter distribution collapses to form a black hole. As usual in these diagrams, null curves are drawn with slopes of $\pm 1$. $i^\pm$ is future/past timelike infinity (the destination/origin of timelike curves as "time" goes to $\pm \infty$), $i^o$ is the corresponding spacelike infinity, and $\mathscr{I}^{\pm}$ is future/past null infinity (the destination/origin origin of null curves as "time" goes to $\pm \infty$). Note that point $A$ can send signals to $\mathscr{I}^+$ while $B$, which is part of the black hole, cannot.
• Figure 2: A two-dimensional schematic of the evolution of the event horizon (the heavy black line) as two concentric, spherically symmetric shells of matter (the sets of parallel arrows) collapse to form a black hole (the grey region). As noted in the text, the evolution is highly non-causal: the horizon forms in vacuum and then expands continuously with infalling matter decreasing the rate of expansion. The expansion only stops when the last shell crosses the the horizon. In this diagram the horizon is non-expanding (and so null) when it is vertical. Otherwise it is expanding.
• Figure 3: A schematic of a Vaidya spacetime in which two concentric shell of null dust collapse onto a pre-existing isolated horizon. Isolated horizon sections are labelled IH while dynamic horizon (and so spacelike) regions are labelled DH. Note the difference between the behaviour of the MTT versus the event horizon shown in Fig. \ref{['EHex']}.
• Figure 4: An Oppenheimer-Snyder spacetime in which three consecutive shells of constant density dust fall into a pre-existing black hole. When no dust crosses the MTT, it is an isolated horizon while when it is dynamic it is a timelike membrane. Note that the trapped region may be either inside or outside of the MTT depending on whether it is dynamic or isolated.
• Figure 5: Collapse of a dust ball to form a black hole