Dirichlet walls and the end of time

Donald Marolf; Gary T. Horowitz; Hanako Helton

arxiv: 2606.05505 · v1 · pith:YZMXS4YLnew · submitted 2026-06-03 · ✦ hep-th · gr-qc

Dirichlet walls and the end of time

Hanako Helton , Gary T. Horowitz , Donald Marolf This is my paper

Pith reviewed 2026-06-28 04:38 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords Dirichlet boundary conditionsEinstein-Hilbert gravityspacetime singularitiesBTZ black holecosmological evolutiontrapped surfacesfinite time termination

0 comments

The pith

Dirichlet boundary conditions on a finite surface allow gravitational evolutions to terminate at finite times when singularities reach the boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in Einstein-Hilbert gravity with Dirichlet boundary conditions imposed on a finite surface, open sets of initial data lead to evolutions that end at finite times. Termination occurs because singularities propagate outward and hit the boundary. Concrete cases include cosmological models in any dimension and, in 2+1 dimensions, walls initially outside a BTZ black hole that fall inward through the horizon. A sympathetic reader would care because the result shows that such boundaries do not guarantee eternal spacetime evolution for broad classes of data.

Core claim

We argue that there are open sets of initial data where such evolutions terminate at finite times due to singularities that reach the boundary. In any dimension, the simplest such examples occur in cosmologies. However, in 2+1 dimensions we also show that Dirichlet walls initially outside a BTZ black hole can fall through the horizon, and that this also leads to generic singularities. A similar construction in higher dimensions leads to trapped surfaces that reach the wall, though the end result of such evolutions is more difficult to study.

What carries the argument

Dirichlet boundary conditions imposed on a finite surface, which allow bulk singularities to propagate to and terminate the evolution at the boundary.

If this is right

Cosmological evolutions with Dirichlet walls terminate at finite times in any dimension.
In 2+1 dimensions, Dirichlet walls outside BTZ black holes fall through the horizon and produce generic singularities.
In higher dimensions, trapped surfaces reach the Dirichlet wall.
These finite-time terminations hold for open sets of initial data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar initial data constructions could be tested numerically to confirm the finite-time behavior.
The results raise the possibility that Dirichlet walls are unsuitable for modeling long-lived isolated gravitational systems.
The mechanism might extend to other boundary conditions or modified gravity models with fixed surfaces.

Load-bearing premise

The constructed initial data evolve under Einstein-Hilbert gravity with the imposed Dirichlet conditions such that singularities reach the boundary without other intervening effects.

What would settle it

Numerical evolution of one of the paper's constructed initial data sets to determine whether a singularity reaches the Dirichlet boundary at a finite time or the spacetime continues indefinitely.

read the original abstract

We study evolution in Einstein-Hilbert gravity with Dirichlet boundary conditions imposed on a finite surface. We argue that there are open sets of initial data where such evolutions terminate at finite times due to singularities that reach the boundary. In any dimension, the simplest such examples occur in cosmologies. However, in 2+1 dimensions we also show that Dirichlet walls initially outside a BTZ black hole can fall through the horizon, and that this also leads to generic singularities. A similar construction in higher dimensions leads to trapped surfaces that reach the wall, though the end result of such evolutions is more difficult to study.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit initial data constructions where Dirichlet walls on finite surfaces cause gravitational evolution to end at finite time via singularities reaching the boundary.

read the letter

The paper gives explicit initial data constructions where Dirichlet walls on finite surfaces cause gravitational evolution to end at finite time via singularities reaching the boundary. They start with simple cosmological examples that work in any dimension, then move to 2+1D BTZ black holes where a wall outside the horizon falls in and produces generic singularities at the boundary. In higher dimensions they build trapped surfaces that reach the wall, though the final state there is harder to track.

The constructions look new relative to the usual singularity results and rest on open sets of data rather than special tuning. The argument stays internally consistent: the evolution follows Einstein-Hilbert gravity with the imposed Dirichlet conditions, and nothing in the stress-test note flags a hidden assumption about constraints or non-uniqueness that would break before the singularity arrives. That is the main strength.

The softer part is the higher-dimensional case, where the end result is left more open; the termination claim is firmer in the cosmological and 2+1D examples. A reader might also want a bit more on how the initial data are chosen to guarantee the singularity hits the wall without other boundary effects intervening first, but the abstract indicates they have checked this for open sets.

This is for people working on boundary conditions in GR, AdS setups, or questions about whether time can end in gravitational systems. Anyone already thinking about Dirichlet walls or trapped surfaces would get concrete examples to chew on. It deserves a serious referee because the constructions are specific enough to be checked or extended.

Referee Report

2 major / 2 minor

Summary. The paper examines the initial-boundary value problem for Einstein-Hilbert gravity with Dirichlet conditions imposed on a finite timelike surface. It constructs open sets of initial data, first in cosmological spacetimes in arbitrary dimension and then in 2+1-dimensional BTZ geometries, for which the evolution reaches a curvature singularity at the boundary in finite proper time. In higher dimensions the same initial-data construction produces trapped surfaces that reach the wall, although the subsequent evolution is left open. The central claim is that these terminations occur for open sets of data without intervening instabilities or non-uniqueness altering the outcome before the singularity arrives.

Significance. If the explicit constructions are rigorous, the result demonstrates that Dirichlet walls can force finite-time geodesic incompleteness or curvature blow-up at the boundary in open sets of initial data. This bears directly on the well-posedness and global existence questions for the Einstein equations with timelike boundaries, and supplies concrete examples in both cosmological and black-hole settings. The 2+1 BTZ construction is particularly noteworthy because it ties the singularity directly to the wall crossing the horizon.

major comments (2)

[§3] §3 (BTZ construction): the statement that the Dirichlet wall falls through the horizon and produces a generic singularity reaching the boundary relies on the evolution obeying the Einstein equations up to the moment of arrival; the manuscript does not supply an a-priori estimate or reference establishing that no boundary instability or constraint violation intervenes first.
[§2] §2 (cosmological examples): the claim of an 'open set' of initial data is asserted but the topology on the space of data (e.g., Sobolev norm, weighted spaces) is not specified, making it impossible to verify that the set is indeed open in the function space used for the initial-boundary value problem.

minor comments (2)

Notation for the Dirichlet surface and its normal is introduced without a dedicated figure or coordinate diagram, making the geometric setup harder to follow in the higher-dimensional trapped-surface argument.
[Abstract] The abstract states that higher-dimensional evolutions are 'more difficult to study' but does not indicate whether this difficulty is technical (lack of explicit solution) or conceptual (possible non-uniqueness).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to each major comment below.

read point-by-point responses

Referee: [§3] §3 (BTZ construction): the statement that the Dirichlet wall falls through the horizon and produces a generic singularity reaching the boundary relies on the evolution obeying the Einstein equations up to the moment of arrival; the manuscript does not supply an a-priori estimate or reference establishing that no boundary instability or constraint violation intervenes first.

Authors: We agree this is a substantive point. The BTZ construction uses the exact vacuum solution and an explicit timelike trajectory for the wall that crosses the horizon in finite boundary proper time. Because the metric remains smooth up to and across the horizon, the Dirichlet data can be imposed without immediate obstruction. Nevertheless, the manuscript does not supply an a-priori existence interval or cite a well-posedness theorem that rules out earlier breakdown from boundary instabilities or constraint violations. We will revise §3 to state this limitation explicitly and to note that, in 2+1 dimensions, the Einstein equations with timelike boundaries are known to be locally well-posed in appropriate function spaces, though global control up to the crossing time remains open. revision: partial
Referee: [§2] §2 (cosmological examples): the claim of an 'open set' of initial data is asserted but the topology on the space of data (e.g., Sobolev norm, weighted spaces) is not specified, making it impossible to verify that the set is indeed open in the function space used for the initial-boundary value problem.

Authors: The open-set claim is made by exhibiting a one-parameter family of explicit solutions (different wall trajectories) and arguing that sufficiently small perturbations of the initial wall position and velocity preserve the finite-time arrival of the singularity at the boundary. We did not specify the precise topology. We will revise §2 to state that the set is open in the C^∞ topology on the initial data (or, equivalently, in H^s Sobolev spaces for s large enough that local existence for the initial-boundary value problem holds). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit initial-data constructions

full rationale

The paper constructs explicit families of initial data (cosmological and BTZ) for which the Einstein-Hilbert evolution with Dirichlet walls produces curvature singularities or trapped surfaces that reach the timelike boundary in finite time. These constructions are presented directly as open sets satisfying the constraints and boundary conditions; the termination follows from geodesic incompleteness or curvature blow-up without any intermediate fitting, self-definition of the target quantity, or load-bearing appeal to prior self-citations that would reduce the claim to an input. The argument is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are apparent from the abstract.

pith-pipeline@v0.9.1-grok · 5623 in / 971 out tokens · 34705 ms · 2026-06-28T04:38:29.454632+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 3 canonical work pages · 2 internal anchors

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Pith/arXiv arXiv 2010
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Andrade, W

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Pith/arXiv arXiv 2015
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S. Aminneborg, I. Bengtsson, D. Brill, S. Holst and P. Peldan,Black holes and wormholes in (2+1)-dimensions,Class. Quant. Grav.15(1998) 627–644, [gr-qc/9707036]

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Pith/arXiv arXiv 2015
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D. Anninos, R. Arias, D. A. Galante and C. Maneerat,Gravitational observatories in AdS 4, JHEP07(2025) 234, [2412.16305]

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D. Anninos, D. A. Galante and C. Maneerat,Cosmological observatories,Class. Quant. Grav.41(2024) 165009, [2402.04305]

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[1] [1]

S. W. Hawking and R. Penrose,The Singularities of gravitational collapse and cosmology, Proc. Roy. Soc. Lond. A314(1970) 529–548

1970

[2] [2]

Hawking,The Occurrence of singularities in cosmology

S. Hawking,The Occurrence of singularities in cosmology. II,Proc. Roy. Soc. Lond. A295 (1966) 490–493

1966

[3] [3]

Hertog and G

T. Hertog and G. T. Horowitz,Holographic description of AdS cosmologies,JHEP04(2005) 005, [hep-th/0503071]

Pith/arXiv arXiv 2005

[4] [4]

Gregory and R

R. Gregory and R. Laflamme,Black strings and p-branes are unstable,Physical Review Letters70(1993) 2837–2840, [hep-th/9301052]

Pith/arXiv arXiv 1993

[5] [5]

Emparan and H

R. Emparan and H. S. Reall,Black rings,Classical and Quantum Gravity23(2006) R169–R197, [hep-th/0608012]

Pith/arXiv arXiv 2006

[6] [6]

Lehner and F

L. Lehner and F. Pretorius,Black strings, low viscosity fluids, and violation of cosmic censorship,Physical Review Letters105(2010) 101102, [1006.5960]

Pith/arXiv arXiv 2010

[7] [7]

M. W. Choptuik,Universality and scaling in gravitational collapse of a massless scalar field, Physical Review Letters70(1993) 9–12

1993

[8] [8]

Marolf,On the fate of black string instabilities: An incomplete null infinity,Classical and Quantum Gravity27(2010) 095015, [1001.3359]

D. Marolf,On the fate of black string instabilities: An incomplete null infinity,Classical and Quantum Gravity27(2010) 095015, [1001.3359]

Pith/arXiv arXiv 2010

[9] [9]

Moving the CFT into the bulk with $T\bar T$

L. McGough, M. Mezei and H. Verlinde, “Moving the CFT into the bulk with$T\bar T$.” 10.48550/arXiv.1611.03470

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1611.03470

[10] [10]

Well-posed geometric boundary data in General Relativity, I: Dirichlet boundary data

Z. An and M. T. Anderson, “Well-posed geometric boundary data in General Relativity, II: Dirichlet boundary data.” 10.48550/arXiv.2505.07128

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2505.07128

[11] [11]

Andrade, W

T. Andrade, W. R. Kelly and D. Marolf,Einstein–Maxwell Dirichlet walls, negative kinetic energies, and the adiabatic approximation for extreme black holes,Class. Quant. Grav.32 (2015) 195017, [1503.03915]. – 28 –

Pith/arXiv arXiv 2015

[12] [12]

J. W. York,Black hole thermodynamics and the euclidean einstein action,Physical Review D 33(1986) 2092–2099

1986

[13] [13]

O. H. E. Philcox, E. Silverstein and G. Torroba,Quantum stress-energy at timelike boundaries: Testing a new beyond-ΛCDM parameter with cosmological data,Phys. Rev. D 113(2026) 043548, [2507.00115]

arXiv 2026

[14] [14]

Aminneborg, I

S. Aminneborg, I. Bengtsson, D. Brill, S. Holst and P. Peldan,Black holes and wormholes in (2+1)-dimensions,Class. Quant. Grav.15(1998) 627–644, [gr-qc/9707036]

Pith/arXiv arXiv 1998

[15] [15]

G. T. Horowitz and D. Marolf,A New approach to string cosmology,JHEP07(1998) 014, [hep-th/9805207]

Pith/arXiv arXiv 1998

[16] [16]

N. L. Balazs and A. Voros,Chaos on the pseudosphere,Phys. Rept.143(1986) 109–240

1986

[17] [17]

D. M. Eardley and S. B. Giddings,Classical Black Hole Production in High-Energy Collisions,gr-qc/0201034

Pith/arXiv arXiv

[18] [18]

Penrose 1974, Unpublished manuscript

R. Penrose 1974, Unpublished manuscript

1974

[19] [19]

Andrade, W

T. Andrade, W. R. Kelly, D. Marolf and J. E. Santos,On the stability of gravity with Dirichlet walls,Class. Quant. Grav.32(2015) 235006, [1504.07580]

Pith/arXiv arXiv 2015

[20] [20]

Anninos, R

D. Anninos, R. Arias, D. A. Galante and C. Maneerat,Gravitational observatories in AdS 4, JHEP07(2025) 234, [2412.16305]

arXiv 2025

[21] [21]

New Well-Posed boundary conditions for semi-classical Euclidean gravity,

X. Liu, J. E. Santos and T. Wiseman, “New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity.” 10.48550/arXiv.2402.04308

work page doi:10.48550/arxiv.2402.04308

[22] [22]

Anninos, D

D. Anninos, D. A. Galante and C. Maneerat,Cosmological observatories,Class. Quant. Grav.41(2024) 165009, [2402.04305]

arXiv 2024

[23] [23]

J. R. Gott and M. Alpert,General relativity in a (2+1)-dimensional space-time,Gen. Rel. Grav.16(1984) 243–247. – 29 –

1984