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Black hole boundaries

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Ivan Booth

General Relativity and Quantum Cosmology · Submitted 26 Aug 2005 · 54 references

Pith's verdict: A genuinely useful nomenclature-clearing review of quasilocal horizons, with one harmless factor-of-two slip in the topology argument and a real dependence on unpublished companion papers.

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arXiv IDgr-qc/0508107 (version 2)
Pith IDMQETRPOD, Pith's permanent record of this paper and its review
CitationIvan Booth. "Black hole boundaries." arXiv:gr-qc/0508107 (2005). https://pith.science/paper/MQETRPOD
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The paper

The paper claims

Quasilocal horizons, surfaces where outgoing light neither expands nor contracts, can stand in for the global event horizon as physical black hole boundaries, and this review unifies their many definitions under one nomenclature.

The reading

Physicists usually define a black hole as the region from which no signal can ever escape. That definition is exact but awkward: to know where the edge is, you have to know the entire future of the universe, because the edge is drawn by light rays that only fail to escape after infinite time. In a computer simulation of two black holes merging, nobody can wait that long. The alternative is to look at local geometry. A trapped surface is a closed two-surface where both families of light rays crossing it are converging, inward and outward. The boundary of the region containing trapped surfaces is a surface where the outward expansion is exactly zero. This review walks through the family of definitions built on that idea: apparent horizons, isolated horizons (equilibrium, no flux across), dynamical horizons (growing, spacelike), trapping horizons (classified by whether trapped surfaces lie just inside or just outside), and slowly evolving horizons (almost isolated, where thermodynamic laws apply to first order). The paper's contribution is a clean map. A single table lines up the acronyms and the conditions that separate them, and the text works through examples, including spherical dust collapse where horizons can appear, jump, and disappear. It also states plainly where the program is incomplete: the surfaces are not unique, they depend on how you slice spacetime into instants, and some spacetimes contain these surfaces without containing black holes.
Extended reading notes

Core claim

The review's load-bearing synthesis is that a black hole boundary can be identified quasilocally by a three-surface that is marginally trapped (outward null expansion zero, inward expansion negative) and that bounds a region of trapped surfaces. From section 5: 'we have argued that one can quasilocally identify a black hole boundary by finding a three-surface that is marginally trapped (with theta(l) = 0 and theta(n) < 0) and bounds a region in which all points lie on a trapped surface.' If correct, the many named objects (apparent, isolated, trapping, dynamical, slowly evolving) are one family distinguished by a small set of conditions, and black hole boundaries become measurable by local observers.

Load-bearing premise

The sufficiency of the marginal-trapping conditions. The paper itself supplies counterexamples: Senovilla's vanishing-scalar-invariant spacetimes (section 3.2, citing [31]) contain marginally trapped tubes and dynamical horizons but no black holes, so theta(l) = 0 plus theta(n) < 0 plus a trapped interior does not by itself pick out a black hole boundary. The review patches this with extra genericity and energy conditions (sigma sigma + 8 pi T(l)(l) != 0, null energy condition) in section 4.2, but those conditions are assumptions about the matter and shear, not consequences of the definition, and the paper does not prove they hold in generic collapse.

Original 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.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,143 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Classical black holes and event horizons are highly non-local objects, defined in relation to the causal past of future null infinity. background 1.00 0.05 0.05 asserted 1.00
s1 Alternative, quasilocal characterizations of black holes are often used in mathematical, quantum, and numerical relativity. background 1.00 0.04 0.03 asserted 0.99
s2 These include apparent, killing, trapping, isolated, dynamical, and slowly evolving horizons. background 1.00 0.04 0.03 asserted 1.00
s3 All of these are closely associated with two-surfaces of zero outward null expansion. background 1.00 0.07 0.04 asserted 0.99
s4 This paper reviews the traditional definition of black holes and provides an overview of some of the more recent work on alternative horizons. contribution 0.98 0.65 0.14 asserted 0.95

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
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 ±1. i ± is future/past timelike infinity (the destination/origin of timelike curves as “time” goes to ±∞), i o is the corresponding spacelike infinity, and I ± is future/past null infinity (the destination/origin origin of null curves as “time” goes to ±∞). Note that point A can send signals to I + while B, which is part of the black hole, cannot. explore the mathematical implications of these definitions,…
Figure 2. A two-dimensional schematic of the evolution of the event horizon (the heavy black line) as two concen￾tric, 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. said to…
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
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. generically spacelike. Howevever, this initial expectation is confounded by our second example — Oppenheimer-Snyder collapse [33]. These spacetimes are generated using spacetime surgery to piece together sections of Schwarzschild and…
Figure 5. Collapse of a dust ball to form a black hole Since our spacetime is spherically symmetric, we search for MOTS which share this symmetry. Focussing our attention of these surfaces one can show that, up to multiplication by positive functions of r and τ : θ(ℓ) = 2(R˙(τ, r) + p 1 − 2m(r)/r) R(τ, r) and θ(n) = R˙(τ, r) − p 1 − 2m(r)/r R(τ, r) . (13) Then, consulting (12) we then find that θ(ℓ) = 0 if and only if R(τ, r) = 2m(r), and on that surface we always have θ(n) < 0. Thus, all quasilocal horizons in this spacetime are marginally trapped tubes (or future trapping horizons whenever…
Figure 6. Large shell of constant density falls “into” a small black hole growth followed by a “jump” at τ ≈ 3000m as a new horizon forms at R ≈ 1202 around the old. It is fairly easy to show that the segment connecting the new horizon to the old is a timelike membrane. Alternatively, we can trace the evolution as a function of R. Then, it begins as an isolated horizon (before the first dust shells hit), becomes a dynamical horizon (during the initial expansion), transitions through a null (but not isolated) instant to become a timelike membrane (heading backwards in time), and then goes back…
Figure 7. Complicated dust distributions collapse to form black hole seems possible that the multiple θ(ℓ) = 0 surfaces sometimes seen inside a numerical “apparent” hori￾zon [36] may actually represent one (or possibly a few) MTTs weaving backwards and forwards in time. We will come back to this point in section 5. 4. Properties of quasilocal horizons We now review some properties that follow from the definitions set forth in the last section. 4.1. Existence and Uniqueness While it is clear from both numerical simulations and our examples that the quasilocal horizons discussed in the previous…
Figure 8. A schematic showing an MTS which transitions from isolated-dynamical-TLM-dynamical-isolated. The large dots indicate transitions points between these behaviours. Dashed lines are null surfaces and shading indicates regions where the uniqueness theorems forbid the existence of WTSs. the example of horizons generated by spacetime foliations, we then know that while different foliations will necessarily generate different horizons, those horizons will just as necessarily be constrained to repeatedly intersect each other. Finally, before moving on, we will briefly discuss trapping…

Figures reproduced from arXiv:gr-qc/0508107 by the authors.

References
How later papers used it

93 later papers cite it

Evidence: substantial. 86 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 11 of the 135 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 5 papers with this label
How the literature uses it

83 arXiv papers cite this in 123 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 62 sentences without a confident label

Uses its method or tool 3 papers
Cites it as background 45 papers

Discussion

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