Low regularity approach to Bartnik's conjecture

Didier A. Solis; J\'onatan Herrera; Jos\'e Luis Flores

REVIEW 3 major objections 5 minor 38 references

Low regularity approach to Bartnik's conjecture

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Bartnik's splitting conjecture is proven at low regularity for Lorentzian length spaces: compactness, nonnegative timelike curvature, and complete vertical fibres force a metric Lorentzian product.

desk verdict Plausible and interesting low-regularity Bartnik splitting, but the proof of Theorem 4.1 has a load-bearing gap in the angle-boundedness step. read the letter →

arxiv 2412.08967 v1 pith:BJMWJRTD submitted 2024-12-12 math.DG gr-qcmath-phmath.MP

José Luis Flores , Jónatan Herrera , Didier A. Solis This is my paper · ORCID

classification math.DGgr-qcmath-phmath.MP MSC 53C5053C2383C75

keywords BartniksplittingconjectureLorentzianlengthspacescausalboundarytimelikecurvatureboundstheoremglobalhyperbolicitylow-regularitygeometrymetricproduct

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a synthetic, low-regularity version of Bartnik's splitting conjecture in the setting of Lorentzian length spaces. It proves that a connected, regularly localisable, globally hyperbolic Lorentzian length space of the form X = Σ × R, with Σ compact, nonnegative timelike curvature bounds, and timelike-complete vertical fibres, must split as a metric Lorentzian product S × R. The proof works by showing that the future causal boundary of X is a single point, which rules out null lines and turns a complete causal line into a timelike line; the synthetic Lorentzian splitting theorem then applies. This matters because it extends a rigidity statement from smooth general relativity to causal spaces with low regularity, where no smooth metric or constant-mean-curvature hypersurface is available.

What carries the argument

The future causal boundary, built from indecomposable past sets, is the central object. Theorem 4.1 uses a comparison-angle argument: for a timelike triangle with vertices A1, Ai, Bi, comparison in 2-dimensional Minkowski spacetime and the zero lower curvature bound give α_i ≥ \bar{α}_i; combined with the law of cosines and the unboundedness of τ(A1, Bi), this produces a contradiction unless the two vertical lines have identical pasts. The other machinery is the causal limit curve lemma and the synthetic splitting theorem for Lorentzian length spaces.

What would settle it

A concrete way to falsify the claim is to find a connected, regularly localisable, globally hyperbolic Lorentzian length space X = Σ × R with compact Σ, nonnegative timelike curvature, timelike geodesic prolongation, and complete vertical fibres whose future causal boundary contains more than one point; Theorem 4.1 predicts such a space cannot exist, and the angle comparison would have to break in a way that can be checked in explicit warped-product models.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: under the stated hypotheses there is a (τ,≤)-preserving homeomorphism f : S × R → X with S a proper, strictly intrinsic metric space of Alexandrov curvature ≥ 0, so X is a metric Lorentzian product. The heart of the argument is Theorem 4.1, which shows that all vertical lines share the same past. If two vertical lines had different pasts, points can be chosen along them so that the Lorentzian comparison angles are bounded while the law of cosines in Minkowski space forces a divergence; the contradiction shows I⁻(δ) = I⁻(γ) for every pair of vertical lines. With Σ compact this makes the future causal boundary a singleton; Corollary 3.2 then excludes inextensible null lines, and the complete causal line produced by global hyperbolicity is upgraded to a timelike line, so the splitting theorem applies.

Load-bearing premise

The load-bearing premise is that every vertical fibre is a timelike-complete line with τ((x,t),(x,s)) > 0 for all t,s, together with the assumption that X is already the topological product Σ × R; if a fibre fails this, two vertical lines can have distinct pasts and the future causal boundary need not be a single point.

Editorial extensions

If this is right

If Theorem 1.1 is correct, every such space admits a global time function with a compact slice, and the Lorentzian distance factorises through the metric product S × R.
A singleton future causal boundary implies the absence of inextensible null lines, so any complete causal line in such a space is necessarily timelike.
The theorem provides a synthetic route to Bartnik rigidity that avoids constant-mean-curvature hypersurfaces, which are unavailable in low regularity.
For product spacetimes with compact Cauchy surface, the no-horizon condition emerges from curvature and completeness rather than being assumed.

Reading between the lines

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

A likely next step is to relax the topological-product assumption: the proof uses only that a compact base allows the shared-past condition to propagate along curves, so a more general fibration over a compact Alexandrov space might satisfy the same argument.
The causal-boundary-singleton strategy is transferable: in any synthetic spacetime where a complete causal line exists and the future causal boundary is a single point, the same upgrade to a timelike line and splitting should go through.
Dropping the vertical completeness condition can produce multiple future boundary points, which would pinpoint that hypothesis as the one responsible for forcing the singleton boundary in this argument.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper has a genuinely new idea: prove a synthetic Bartnik splitting by first showing the future causal boundary is a singleton, then invoking the existing splitting theorem. The main new statement, Theorem 4.1, says that under a strong vertical completeness assumption and global non-negative timelike curvature, the past of every vertical line is the whole space. That is a real geometric result, and the strategy is the right one for Lorentzian length spaces. The paper is well-written, carefully positioned, and the reliance on [11] and [16] is legitimate use of published background.

The soft spot is in the proof of Theorem 4.1. The step 'by continuity of angles, α_i → ∡_{A1}(δ, λ)' is not justified. The vertical curve δ is not a geodesic, and the limit curve λ from the Limit Curve Lemma is not the limit of the geodesics δ_i; their limit is some timelike geodesic δ_hat from A_1 to the limit point of the A_i, which need not lie on δ. Even after replacing δ by δ_hat, finiteness of the limiting angle is not automatic. A sequence of unit timelike geodesics can have initial directions approaching the null cone, so the hyperbolic angles can be unbounded. The cited continuity result does not supply a uniform upper bound. If the α_i are unbounded, the right-hand side of (10) need not diverge, and the contradiction collapses. This is the step where the curvature bound must do the work, and it is currently just asserted. The finite-subcover extension from nearby base points to all of Σ is also sketched rather than proven; it looks patchable, but a rigorous ε-δ argument is missing.

So the overall strategy is coherent and the theorem is plausible, but the proof as written has a load-bearing gap. This is not a desk reject. Send it to a referee who knows hyperbolic angles in Lorentzian length spaces and the c-boundary, and expect a major revision before publication. The paper is for people working in synthetic Lorentzian geometry and mathematical relativity; if the gap is fixed, it would be a significant step.

Referee Report

3 major / 5 minor

Summary. The paper establishes a synthetic version of Bartnik's splitting conjecture for globally hyperbolic Lorentzian length spaces. Under the assumptions that X is a topological product Σ×R with Σ compact, that the vertical lines are timelike complete, and that X has nonnegative timelike curvature bounds, the authors prove that the future causal boundary is a singleton (Theorem 4.1). This allows them to upgrade a limit causal line to a timelike line and then apply the Lorentzian splitting theorem of Beran–Ohanyan–Rott–Solis to obtain a (τ,≤)-preserving homeomorphism to S×R with S an Alexandrov space of nonnegative curvature (Theorem 1.1). The proof combines causal boundary techniques from [16] with the angle comparison framework of [12].

Significance. If completed, the result is a valuable low-regularity analogue of Bartnik's splitting conjecture, extending modern synthetic Lorentzian geometry to a rigidity problem of independent interest. The paper builds in a natural way on substantial recent machinery: the Lorentzian splitting theorem [11], hyperbolic angle comparisons [12], and the c-completion of Lorentzian metric spaces [16]. The core idea, showing that all vertical lines share a common past by a comparison-angle contradiction, is attractive and promising. However, as written, the proof has two load-bearing gaps, one in the angle-bound argument of Theorem 4.1 and one in Proposition 3.1, so the main theorem is not yet fully established.

major comments (3)

[Section 4, Theorem 4.1, proof after defining δ_i and λ_i] The claim that 'by continuity of angles ... α_i → ∡_{A1}(δ, λ)' is not justified. The curve δ is not assumed to be a geodesic, angles are defined between geodesics, and the sequence δ_i, which consists of geodesics from A1 to Ai, is not shown to converge to δ; the Limit Curve Lemma was applied only to λ_i. The boundedness of α_i is essential for the contradiction derived from Eq. (10), because if α_i is unbounded then the term -2τ(A1,A_i)cosh(\bar α_i) need not be bounded and the right-hand side of (10) need not diverge. To repair this step, the authors should prove that δ_i subconverges to a timelike geodesic \bar δ (using properness and global hyperbolicity) and then apply the appropriate upper-semicontinuity property of angles to obtain limsup_i α_i ≤ ∡_{A1}(\bar δ, λ) < ∞.
[Section 3, Proposition 3.1, proof of 'Consequently, I^-(ς) should be a PIP'] The inference that I^-(ς) must be a proper indecomposable past set (PIP) is not a consequence of the preceding facts. A nonconvergent future causal chain whose past is a proper IP can define a terminal IP (TIP), and a singleton future causal boundary does not by itself rule out I^-(ς) being that unique TIP; this would require proving that the unique TIP is the whole space X, which is not established in the proposition. Corollary 3.2 relies on this proposition to rule out inextensible null lines. In the application to Theorem 1.1, the stronger conclusion of Theorem 4.1 (that every TIP contains the common past of all vertical lines, and hence equals X) may supply the missing fact, but the proposition and corollary as stated need either an added hypothesis or a corrected proof.
[Section 4, Theorem 4.1, hypotheses versus tools used] The proof invokes the Limit Curve Lemma [11, Thm 2.23] and compactness/continuity arguments, but Theorem 4.1 is stated only for a strongly causal, causally closed, timelike geodesically connected Lorentzian length space with continuous τ. The cited limit curve theorem typically requires global hyperbolicity and properness, and those hypotheses are not listed or verified in Theorem 4.1. Since Theorem 1.1 does assume global hyperbolicity and a proper metric, adding these hypotheses to Theorem 4.1 would align the statement with the proof and with the application in the main theorem.

minor comments (5)

[Section 4, Theorem 4.1, notation] The symbol α_i is used first for the angle ∡_{A1}(δ_i, λ_i) and then for the comparison angle \bar∡_{A1}(A_i, B_i); the second occurrence should be labelled \bar α_i to avoid the notational clash.
[Theorem 1.1, statement] The hypothesis 'τ((x,t),(x,s)) > 0 for any t,s ∈ R' should read 'for any t,s ∈ R with t < s', as is correctly stated in Theorem 4.1; otherwise the condition conflicts with the fact that τ(p,q)>0 is equivalent to p ≪ q.
[Proposition 3.1, proof] The text says 'Since X is a causally continuous, thus strongly causal, spacetime', but causal continuity is not among the assumptions; the intended phrase is likely 'causally simple' (which already includes strong causality).
[Section 4, Theorem 4.1, finite cover argument] In the final step for arbitrary a,b, the existence of points \bar t_i with r(\bar t_i) ∈ U_{t_i} ∩ U_{t_{i+1}} is not guaranteed for an arbitrary finite subcover; the authors should use the connectedness of the curve r([0,1]) and the fact that the intersection graph of the subcover is connected.
[Section 4, Theorem 4.1, proof of lim τ = ∞] The statement 'Since lim_{s→∞} τ((x,t),(x,s)) = ∞' is not explicitly among the hypotheses; it follows from timelike completeness of the vertical lines via the reverse triangle inequality, but this derivation should be indicated for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new Theorem 4.1 is an independent geometric argument, and the cited splitting theorem is external published support, not an input equivalent to the conclusion.

full rationale

The paper does not derive its conclusion from an assumption that already contains it. Theorem 1.1 assumes X = Σ × R with compact Σ, vertical timelike completeness, and the chronology condition τ((x,t),(x,s)) > 0; these are explicit hypotheses, not the splitting conclusion. The proof proceeds by constructing a causal line via the limit curve theorem and then using Theorem 4.1 to show the future causal boundary is a single point, which upgrades the causal line to a timelike line; Theorem 2.1 then supplies the splitting. Theorem 4.1 is a new argument: given two vertical lines, it constructs sequences with τ(A_i,B_i) → 0 and τ(B_i,B_{i+1}) > 1, and derives a contradiction from the law of cosines and the curvature bound. The conclusion that the past of every vertical line is X is not assumed anywhere in the proof. The appeal to [11, Theorem 1.4] is a normal use of a prior published splitting theorem; although one author of the present paper is also an author of [11], that cited theorem has its own proof and its assumptions do not include the Bartnik conclusion, so it constitutes independent support rather than a self-citation chain. No parameter is fitted and no quantity is renamed as a prediction; there is no equation that reduces by construction to an earlier assumption. A possible technical gap exists in the continuity-of-angles step (the geodesics δ_i need not converge to the vertical curve δ), but a gap in justification is not circularity. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's central claim rests on several background theorems from the authors' own recent work, particularly the synthetic splitting theorem [11] (coauthored by Solis) and the c-completion framework [16] (coauthored by Flores and Herrera). These are published results used as black boxes. The additional assumptions of product structure and vertical timelike completeness are explicit hypotheses that carry much of the burden. There are no free parameters or invented entities.

assumptions (7)

domain assumption Theorem 2.1, the synthetic Lorentzian splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature ([11, Theorem 1.4])
The main proof invokes this theorem to obtain the splitting once a complete timelike line is produced; it is a published result but not proved in this paper.
domain assumption Proposition 2.20, angle comparison from lower timelike curvature bounds ([12, Corollary 3.8 and Theorem 4.13])
Used in the proof of Theorem 4.1 to relate angles in X to comparison angles in Minkowski spacetime, i.e., α_i ≥ ar{α}_i.
domain assumption Limit Curve Lemma ([11, Theorem 2.23])
Used twice: to extract limit geodesics from the sequences δ_i and λ_i in Theorem 4.1, and to construct a complete causal line in the proof of Theorem 1.1.
domain assumption Properties of the c-completion and causal boundary ([16, Section 5.2])
The definitions and structural properties of the c-completion, including embeddedness, density, and chronological completeness, are imported from [16].
domain assumption Continuity of the time separation function τ in globally hyperbolic Lorentzian length spaces
Theorem 4.1 explicitly assumes τ continuous, and the proof of Theorem 1.1 must ensure this hypothesis holds under the stated assumptions; this is a standard result in the literature but is not proved or explicitly cited here.
standard math Law of cosines in 2-dimensional Minkowski spacetime for timelike triangles
Used in equation (10) of the proof of Theorem 4.1 to derive the contradiction from the boundedness of the comparison angles.
domain assumption Existence of comparison triangles in Minkowski spacetime for timelike triangles in a Lorentzian pre-length space with the stated curvature bounds
The curvature bound definitions assume such comparison triangles exist; this is part of the framework from [9,12].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Low regularity approach to Bartnik's conjecture." pith.science (2026). https://pith.science/paper/BJMWJRTD

@misc{pith2026241208967,
  author       = {Pith},
  title        = {Pith review of: Low regularity approach to Bartnik's conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJMWJRTD}},
  note         = {Machine review of arXiv:2412.08967}
}

abstract

In this work we establish a version of the Bartnik Splitting Conjecture in the context of Lorentzian length spaces. In precise terms, we show that under an appropriate timelike completeness condition, a globally hyperbolic Lorentzian length space of the form $\Sigma\times \mathbb{R}$ with $\Sigma$ compact splits as a metric Lorentzian product, provided it has non negative timelike curvature bounds. This is achieved by showing that the causal boundary of that Lorentzian length space consists on a single point.

Figures

Figures reproduced from arXiv: 2412.08967 by the authors.

**Figure 2.** The picture on the left shows the curves [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages

[11]

Beran, A

T. Beran, A. Ohanyan, F. Rott, and D. A. Solis, The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature. Lett. Math. Phys. 113(2) 48, 2023

work page 2023
[16]

Burgos, J.L

S. Burgos, J.L. Flores, and J. Herrera. The c-completion of Lorentzian metric spaces. Class. Quantum Grav. 40 20501, 2024

work page 2024
[12]

Beran, and C

T. Beran, and C. S¨ amann. Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds. J. London Math. Soc. 107(5) pp. 1823-1880, 2023

work page 2023
[1]

Ala˜ na, and J.L

V. Ala˜ na, and J.L. Flores.The causal boundary of product spacetimes. Gen. Rel. Grav. 39 pp. 1697–1718, 2007

work page 2007
[2]

Alexander, and R

S. Alexander, and R. Bishop. Lorentz and semi-Riemannian spaces with Alexandrov curvat ure bounds. Commun. Anal. Geom. 16(2) pp. 251-282, 2008

work page 2008
[3]

Ake Hau, S

L. Ake Hau, S. Burgos, and D.A. Solis. Causal completions as Lorentzian pre-length spaces. Gen. Rel. Grav. 54 108, 2022

work page 2022
[4]

Ake Hau, A

L. Ake Hau, A. Cabrera Pacheco, and D. A. Solis. On the causal hierarchy of Lorentzian length spaces. Class. Quantum Grav. 37(21), 2020

work page 2020
[5]

S. B. Alexander, V. Kapovitch, and A. Petrunin. Alexandrov Geometry: Foundations. (Preprint) https://arxiv.org/abs/1903.08539, 2023

arXiv 1903

Show all 38 references

[6]

Barrera, L

W. Barrera, L. Montes de Oca, and D.A. Solis. On the space of compact diamonds of Lorentzian length spaces. Class. Quantum Grav. 41 065012, 2024

2024
[7]

Barrera, L

W. Barrera, L. Montes de Oca, and D. A. Solis. Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds. Gen. Rel. Grav. 54(9) 107, 2022

2022
[8]

R. Bartnik. Remarks on cosmological spacetimes and constant mean curva ture surfaces. Comm. Math. Phys. 117 pp. 615-624, 1988

1988
[9]

Beran, M

T. Beran, M. Kunzinger, and F. Rott. On curvature bounds in Lorentzian length spaces. J. London Math. Soc. 110(2), 2024

2024
[10]

Beran, L

T. Beran, L. Napper, and F. Rott. Alexandrov’s Patchwork and the Bonnet-Myers Theorem for Lo ren- tzian length spaces. (Preprint) arXiv:2302.11615, 2023 10

2023 arXiv
[13]

M. R. Bridson and A. Haeﬂiger. Metric spaces of non-positive curvature . Fundamental Principles of Mathematical Sciences, vol. 319, Springer, Berlin Heidelberg, 1999

1999
[14]

Burago, Y

D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry. Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001

2001
[15]

Burago, M

Y. Burago, M. Gromov, and G. Perelman. Alexandrov spaces with curvature bounded below. Russian Math. Surveys 47(2) pp. 1-58, 1992

1992
[17]

Burtscher and L

A. Burtscher and L. Garc ´ ıa-Heveling. Time functions on Lorentzian length spaces. 2024. https://doi.org/10.1007/s00023-024-01461

2024 doi
[18]

First M87 event horizon telescope results

Collaboration, T.E.H.T. First M87 event horizon telescope results. I. The shadow of t he supermassive black hole. Astrophys. J. Lett. 875, pp 1-17, 2019

2019
[19]

Observation of gravitational waves from a binary black hole merger

Collaboration, L.S., Collaboration, V. Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116 061102, 2016

2016
[20]

Chru´ sciel and J.D.E

P.T. Chru´ sciel and J.D.E. Grant. On Lorentzian causality with continuous metrics Class. Quantum Grav. 29 145001, 2012

2012
[21]

Costa e Silva, J.L

I.P. Costa e Silva, J.L. Flores and J. Herrera. Omniscient foliations and the geometry of cosmological spacetimes, Gen. Rel. Grav. 54 147, 2022

2022
[22]

Eschenburg

J.H. Eschenburg. The splitting theorem for space-times with strong energy co ndition. J. Diﬀ. Geom. 27 pp. 477-491, 1988

1988
[23]

Galloway

G.J. Galloway. The Lorentzian splitting theorem without the completeness assumption. J. Diﬀ. Geom. 29 pp. 373-387, 1989

1989
[24]

Galloway

G.J. Galloway. Splitting theorems for spatially closed spacetimes. Comm. Math. Phys. 96 pp. 423-429, 1984

1984
[25]

Galloway and E

G.J. Galloway and E. Ling. Existence of CMC Cauchy surfaces from a spacetime condition . Gen. Rel. Grav. 50 108, 2018

2018
[26]

R. Geroch. Domain on dependence. J. Math. Phys. 11, pp. 437-449, 1970

1970
[27]

Geroch, E.H

R. Geroch, E.H. Kronheimer, and R. Penrose. Ideal points in space-time. Proc. Roy. Soc. Lond. A 327(1571) pp 545-567, 1972

1972
[28]

M. Graf. Singularity theorems for C1-Lorentzian metrics. Comm. Math. Phys. 378 pp. 1417-1450, 2020

2020
[29]

J. Hedicke. Lorentzian distance functions in contact geometry. J. Topol. Anal. pp. 1-21, 2022

2022
[30]

Kunzinger, A

M. Kunzinger, A. Ohanyan, B. Schinnerl and R. Steinbauer. The Hawking-Penrose singularity theorem for C1-Lorentzian metrics. Commun. Math. Phys. 391 1143-1179, 2022

2022
[31]

Kunzinger and C

M. Kunzinger and C. S¨ amann. Lorentzian length spaces Ann. Global Anal. Geom. 54 pp. 399-447, 2018

2018
[32]

Minguzzi and S

E. Minguzzi and S. Suhr. Lorentzian metric spaces and their Gromov-Hausdorﬀ conver gence. 114 73, 2024

2024
[33]

Minguzzi

E. Minguzzi. Causality theory for closed cone structures with applicati ons. Rev. Math. Phys. 31 1930001, 2019

2019
[34]

M¨ uller.Functors in Lorentzian geometry: three variations on a them e

O. M¨ uller.Functors in Lorentzian geometry: three variations on a them e. Gen. Rel. Grav. 55 39, 2023 11

2023
[35]

M¨ uller

O. M¨ uller. Dimensions of ordered spaces and Lorentzian length spaces (Preprint) arXiv:2303.11237, 2023

2023 arXiv
[36]

Podolsk´ y and R

J. Podolsk´ y and R. Steinbauer. Penrose junction conditions with Λ : geometric insights into low- regularity metrics for impulsive gravitational waves. Gen. Rel. Grav. 54 96, 2022

2022
[37]

F. Rott. Gluing of Lorentzian length spaces and the causal ladder. Class. Quantum Grav. 40(17), 2023

2023
[38]

S. Surya. The causal set approach to quantum gravity. Living Rev. Relativ. 22 5, 2019 12

2019

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [11]

Beran, A

T. Beran, A. Ohanyan, F. Rott, and D. A. Solis, The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature. Lett. Math. Phys. 113(2) 48, 2023

work page 2023

[2] [16]

Burgos, J.L

S. Burgos, J.L. Flores, and J. Herrera. The c-completion of Lorentzian metric spaces. Class. Quantum Grav. 40 20501, 2024

work page 2024

[3] [12]

Beran, and C

T. Beran, and C. S¨ amann. Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds. J. London Math. Soc. 107(5) pp. 1823-1880, 2023

work page 2023

[4] [1]

Ala˜ na, and J.L

V. Ala˜ na, and J.L. Flores.The causal boundary of product spacetimes. Gen. Rel. Grav. 39 pp. 1697–1718, 2007

work page 2007

[5] [2]

Alexander, and R

S. Alexander, and R. Bishop. Lorentz and semi-Riemannian spaces with Alexandrov curvat ure bounds. Commun. Anal. Geom. 16(2) pp. 251-282, 2008

work page 2008

[6] [3]

Ake Hau, S

L. Ake Hau, S. Burgos, and D.A. Solis. Causal completions as Lorentzian pre-length spaces. Gen. Rel. Grav. 54 108, 2022

work page 2022

[7] [4]

Ake Hau, A

L. Ake Hau, A. Cabrera Pacheco, and D. A. Solis. On the causal hierarchy of Lorentzian length spaces. Class. Quantum Grav. 37(21), 2020

work page 2020

[8] [5]

S. B. Alexander, V. Kapovitch, and A. Petrunin. Alexandrov Geometry: Foundations. (Preprint) https://arxiv.org/abs/1903.08539, 2023

arXiv 1903