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Black Holes and Wormholes in 2+1 Dimensions

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Stefan Aminneborg (Norra Reals gymnasium, Stockholm), Ingemar Bengtsson (Fysikum, Stockholm University), Dieter Brill (Department of Physics, University of Maryland), Soren Holst, Peter Peldan (Fysikum

General Relativity and Quantum Cosmology · Submitted 15 Jul 1997 · 17 references

Pith's verdict: A clean, exactly solvable family of non-eternal 2+1 black holes with wormhole interiors; the §V horizon identification is asserted rather than proved, but it is a short lemma away from being solid.

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arXiv IDgr-qc/9707036 (version 1)
CitationDieter Brill (Department of Physics, Ingemar Bengtsson (Fysikum, Peter Peldan (Fysikum, Soren Holst, Stefan Aminneborg (Norra Reals gymnasium, Stockholm), Stockholm University), University of Maryland). "Black Holes and Wormholes in 2+1 Dimensions." arXiv:gr-qc/9707036 (1997). https://pith.science/paper/gr-qc/9707036
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The paper

The paper claims

Any interior topology with at least one handle can host a black hole in 2+1 dimensional anti-de Sitter space, and these black holes are born late rather than eternal.

The reading

The authors ask which black-hole spacetimes can be built by taking 2+1 dimensional anti-de Sitter space and identifying points under a discrete group of isometries, subject to two restrictions: the identification must preserve a moment of time symmetry, and the resulting spacetime must have exactly one asymptotic region. Under those restrictions they claim a complete answer, and the answer is a family of non-eternal black holes whose spatial slices have wormhole topology, with genus one (a torus with one asymptotic region) as the simplest case. The interest is that in 2+1 dimensions there are no gravitational waves to obscure the causal structure, so the event horizon, its birth, its caustics, and the trapped curves inside can all be worked out exactly rather than numerically. If the construction is right, it gives a clean laboratory model of the kind of non-eternal black hole that in four dimensions is usually studied only by simulation.

What carries the argument

The load-bearing device is the fundamental region of the discrete isometry group drawn inside anti-de Sitter space, most usefully in sausage coordinates where time is static and the whole region fits in one picture. The event horizon is then read off geometrically: it is the boundary of the part of the fundamental region that cannot be reached by past-directed null geodesics from the conformal boundary. On the time-symmetric slice $U = 0$ the same group acts as Möbius transformations of the Poincaré disk, so the horizon at that moment is the unique geodesic flow line of the identifying transvection, and the interior topology is fixed by choosing a geodesic polygon whose sides are identified in pairs. This dictionary between group elements, polygons, and horizons is what makes the whole classification tractable.

Extended reading notes

Core claim

The paper's central claim, stated in the abstract as [abstract:s2], is that the spacetimes obtained from anti-de Sitter space by a discrete isometry group, restricted to time-symmetric initial data and a single asymptotic region, are non-eternal black holes with collapsing wormhole topologies. The concrete instance is worked out in Section V: the group is generated by two transvections $a = e^{2J_{TX}}$ and $b = e^{2J_{TY}}$ with a common parameter, the fundamental region is the 'tent' bounded by $X/T = pm alpha$ and $Y/T = pm alpha$ with $1/sqrt{2} < alpha < 1$, and the event horizon is the past light cone of the last point $P$ on null infinity, born at sausage time $t_{birth} = t_P - pi/2$ with $tan t_P = sqrt{2alpha^2 - 1}$. Because the horizon is born after the universe itself, the black hole is not eternal, and its horizon grows outward from a pair of spacelike caustics with kinks that move faster than light. The authors also show that the domain of outer communication is isometric to the exterior of the BTZ black hole even though the spacetime as a whole admits no global Killing vector, that active topological censorship holds while passive topological censorship fails, and that the interior of the initial data slice is densely filled with closed marginally trapped geodesics.

Load-bearing premise

The paper locates the event horizon by asserting that the chosen identifications can never cause a causal curve to turn back in sausage time, so that the horizon is exactly the backward light cone of the last point on null infinity; that monotonicity claim is stated rather than proved, and if a word in the group did turn a curve back, the birth time and the caustic picture would need rederiving.

Editorial extensions

If this is right

  • Every spatial topology of genus $g geq 1$ with one asymptotic region admits such a black hole, with $6g - 3$ parameters (Fenchel-Nielsen lengths and twists) characterizing the initial geometry.
  • The event horizon is not eternal: it is born at a finite sausage time after the birth of the spacetime, and its early growth proceeds from spacelike caustics rather than from a smooth surface.
  • The domain of outer communication is exactly the BTZ exterior, so the wormhole is locally indistinguishable from an ordinary BTZ black hole at infinity even though the full spacetime has no global timelike Killing vector.
  • Active topological censorship holds, but passive topological censorship fails: light rays from the central line of the wormhole can reach null infinity through two different openings, so the topology is visible from far away even though it cannot be traversed.
  • The apparent horizon coincides with the event horizon, and the interior of the black hole is densely covered by closed marginally trapped curves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The construction suggests a general principle: in 2+1 dimensions the question 'which black holes exist' becomes a question about which discrete subgroups of $SO(2,2)$ act properly discontinuously while keeping fixed points behind a horizon, which is a purely group-theoretic classification problem rather than an analytic one.
  • The failure of passive topological censorship here is a useful counterexample to any conjecture that a simply connected domain of outer communication is the generic case; the anti-de Sitter boundary being a cylinder rather than a sphere is what makes room for it.
  • The result invites a check of whether the same 'tent' picture survives when the time-symmetry restriction is dropped and spin is added; the authors themselves flag the spinning case as open, and the obstruction they name is that the fixed-point singularities are replaced by hidden regions of closed timelike curves.
  • Because the horizon's caustics move outward faster than light, the horizon's spatial topology depends on the slicing, which is a concrete illustration that horizon topology statements are slicing-dependent rather than invariant.
Original abstract

A large variety of spacetimes---including the BTZ black holes---can be obtained by identifying points in 2+1 dimensional anti-de Sitter space by means of a discrete group of isometries. We consider all such spacetimes that can be obtained under a restriction to time symmetric initial data and one asymptotic region only. The resulting spacetimes are non-eternal black holes with collapsing wormhole topologies. Our approach is geometrical, and we discuss in detail: The allowed topologies, the shape of the event horizons, topological censorship and trapped curves.

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, 3,607 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

A large variety of spacetimes---including the BTZ black holes---can be obtained by identifying points in 2+1 dimensional anti-de Sitter space by means of a discrete group of isometries. We consider all such spacetimes that can be obtained under a restriction to time symmetric initial data and one asymptotic region only. The resulting spacetimes are non-eternal black holes with collapsing wormhole topologies. Our approach is geometrical, and we discuss in detail: The allowed topologies, the shape of the event horizons, topological censorship and trapped curves.

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 A large variety of spacetimes---including the BTZ black holes---can be obtained by identifying points in 2+1 dimensional anti-de Sitter space by means of a discrete group of isometries. background 1.00 0.11 0.06 asserted 1.00
s1 We consider all such spacetimes that can be obtained under a restriction to time symmetric initial data and one asymptotic region only. contribution 0.62 0.83 0.07 asserted 0.87
s2 The resulting spacetimes are non-eternal black holes with collapsing wormhole topologies. result 1.00 0.84 0.71 asserted 1.00
s3 Our approach is geometrical, and we discuss in detail: The allowed topologies, the shape of the event horizons, topological censorship and trapped curves. contribution 0.77 0.87 0.07 asserted 0.91

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • 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. The fundamental region of the BTZ black hole within ad
Figure 2. Isometries (a-c) and fundamental regions (d,e) of th
Figure 3. Initial data represented by fundamental regions and
Figure 4. The fundamental region of the new black hole. The poin
Figure 5. Different but equivalent ways to construct initial da
Figure 6. Construction of the class of spaces considered here b
Figure 7. Making it manifest that our initial data leads to one a
Figure 8. We follow the event horizon backwards in time. The firs
Figure 9. The event horizon (dotted curves) is born as two inter
Figure 10. A vertical slice through the stereographic image of
Figure 11. Trapped curves in the BTZ black holes can be obtained

Figures reproduced from arXiv:gr-qc/9707036 by the authors.

References
How the literature uses it

40 arXiv papers cite this in 61 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 · 33 sentences without a confident label

Builds on it 2 papers
Uses its method or tool 2 papers
  • Since in comoving coordinates the boundaries of the fundamental domain do not change under time evolution, the desired quotient of AdS$_3$ is obtained by applying the same such compactification to all surfaces of constant $τ$ [ref] .

    Dirichlet walls and the end of time · Constructing the solutions · 0.86

  • In this section, we will first review how to construct multi-boundary black holes by quotienting empty AdS$_3$ with isometries, following an algebraic approach [ref] [For construction of these geometries using explicit forms of the Killing vectors, see [ref] .].

    Traversability of Multi-Boundary Wormholes · Multi-boundary black holes in AdS$_3$ · 0.83

Compares with it 1 paper
Cites it as background 22 papers

Discussion

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