Globally hyperbolic spacetimes can be defined without the 'causal' condition

E. Minguzzi; R. A. Hounnonkpe

Globally hyperbolic spacetimes can be defined without the 'causal' condition

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For non-compact spacetimes of dimension at least three, compact causal diamonds alone imply global hyperbolicity, so the causality condition can be dropped from the definition.

desk verdict A clean, genuinely new simplification of global hyperbolicity for non-compact spacetimes of dimension at least three; causality drops out, the proof holds up, and it deserves a serious referee. read the letter →

arxiv 1908.11701 v2 pith:PJFHRMER submitted 2019-08-30 gr-qc math-phmath.MP

R. A. Hounnonkpe , E. Minguzzi This is my paper · ORCID

classification gr-qcmath-phmath.MP MSC 53C50

keywords globalhyperbolicitycausalsimplicitydiamondscausallyconvexhullclosedconestructuresLorentziancausalitynon-compactspacetimesMorse-Sardtheorem

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

This paper proves that 'reasonable' spacetimes—non-compact and of dimension at least three—are globally hyperbolic exactly when their causal diamonds are compact. In that setting the causality condition, usually included as a separate requirement, follows automatically, so it can be removed from the definition. The same simplification is obtained for causal simplicity when the spacetime is not totally vicious, using closed rather than compact causal diamonds. For rougher cone structures, where a smooth metric may not be available, the paper proves a parallel characterization that keeps causality: global hyperbolicity is equivalent to causality together with compactness of the causally convex hull of every compact set, and it gives an example showing causality cannot be dropped there. The conceptual payoff is that the strongest term in the causal hierarchy is fixed solely by compactness of the sets of events that a causal curve can connect.

What carries the argument

The central object is the causal diamond, $J^+(p)\cap J^-(q)$, and its set-level version, the causally convex hull $J^+(K)\cap J^-(K)$ of a compact set. Proposition 2.3 equates compactness (or closedness) of all causal diamonds with compactness (or closedness) of all causally convex hulls of compact sets. The proof that causality is redundant then works by showing that closed diamonds make the causal relation closed, that a reflecting non-totally vicious spacetime is chronological, and that any failure of causality would appear as a closed achronal lightlike geodesic; the Morse-Sard theorem is used to pick a point of the achronal boundary off that geodesic, and a corner in a piecewise lightlike curve yields the contradiction. In the closed cone structure setting, the causally convex hull property is combined with the limit curve theorem to prove closedness of the causal relation, and Example 2.12 shows that without causality compact hulls need not give global hyperbolicity.

What would settle it

A direct falsifier would be a non-compact three-dimensional $C^2$ Lorentzian spacetime whose causal diamonds are all compact but that contains a closed causal curve; Theorem 2.8 says none exists, so any explicit such example would settle the claim negatively. A more technical check would be to test whether the Morse-Sard step can be bypassed in dimension two, since the proof's point off the lightlike geodesic relies on the achronal boundary having dimension at least two.

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Extended reading notes

Core claim

Let $(M,g)$ be a connected time-oriented Lorentzian manifold of dimension $n+1\ge 3$ with a $C^2$ (or $C^{1,1}$) metric. The paper's central result is Theorem 2.8: if $M$ is non-compact, then $M$ is globally hyperbolic if and only if every causal diamond $J^+(p)\cap J^-(q)$ is compact. The causality condition—no closed causal curves—is a consequence, not an assumption. Theorem 2.7 gives the companion statement for causal simplicity: with dimension $n+1\ge 3$ and non-total viciousness, causality can be dropped so that closed causal diamonds (equivalently, closed causally convex hulls of compact sets) define causal simplicity, and compact versions give global hyperbolicity. For upper semi-continuous closed cone structures, the paper proves that global hyperbolicity is equivalent to causality plus compactness of the causally convex hull of every compact set (Corollary 2.11), and it exhibits a non-causal closed cone structure with compact causally convex hulls, showing the causality assumption is essential there.

Load-bearing premise

The load-bearing premise is that the spacetime is non-compact and at least three-dimensional with a metric smooth enough for the Morse-Sard argument to rule out closed causal curves; in rougher cone structures the paper's own example shows causality cannot be dropped.

Editorial extensions

If this is right

In every non-compact $C^2$ spacetime of dimension $n+1\ge 3$, checking global hyperbolicity reduces to checking that $J^+(p)\cap J^-(q)$ is compact for all $p,q$; no separate search for closed causal curves is needed.
For non-totally vicious spacetimes of dimension at least three, causal simplicity can be defined by closedness of causal diamonds (or of causally convex hulls of compact sets), with causality following automatically.
For closed cone structures, global hyperbolicity is exactly causality plus compactness of the causally convex hull of every compact set; this improves the known formulations for low-regularity theories.
The dimension at least three is not a technical decoration: the argument uses $\dim\partial I^+(p)=n\ge 2$ to find a point outside the lightlike geodesic, so the causality-free statement cannot be expected in two spacetime dimensions by this proof.
Compactness of causally convex hulls of compact sets is singled out as the operative property that characterizes global hyperbolicity across both regular spacetimes and general cone structures.

Reading between the lines

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

If Theorem 2.8 is right, numerical or computational checks of global hyperbolicity in smooth non-compact spacetimes can be reduced to boundedness and compactness checks on causal diamonds, which are more local and easier to verify than absence of closed causal curves.
The non-causal cone structure of Example 2.12 has compact causally convex hulls but is built from integral curves asymptotic to two compact slabs; this suggests that the obstruction to dropping causality in rough settings is tied to branching or asymptotic behavior of cone curves, and that a modest regularity condition beyond upper semi-continuity might recover the causality-free statement.
A testable extension would be to identify the exact regularity threshold: since the proof's Morse-Sard step fails only in low regularity, one could determine whether $C^1$ metrics with rough connections admit compact-diamond non-causal examples, which would mark the precise boundary between Theorem 2.8 and Example 2.12.

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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

Dear colleague,

The short version: the main result is real. Hounnonkpe and Minguzzi prove that for a non-compact spacetime of dimension n+1 ≥ 3, global hyperbolicity is equivalent to compactness of causal diamonds, with no causality condition imposed. For that class, causality follows from compactness of causal diamonds plus non-compactness of the manifold. The analogous simplification holds for causal simplicity, and they show by example that for general closed cone structures the causality assumption cannot be dropped. This settles a genuine gap left after Bernal–Sánchez weakened strong causality to causality.

What is new: the removal of causality entirely for the physically relevant class. The argument is clever and mostly self-contained. The key step, Theorem 2.7, uses a dimension-based Sard/Hausdorff-dimension argument to find a point on ∂I+(p) off a closed achronal lightlike geodesic γ, then a corner-smoothing argument to get a contradiction. The reduction from non-compact to non-totally vicious is a one-line argument but it is sound. The counterexample for closed cone structures (Example 2.12) is a nice touch and shows the boundary of the result.

Where are the soft spots? They are minor. The paper leans heavily on Minguzzi's own recent review and closed-cone-structure papers. That is not circular—the cited results are published independently—but a referee should check that the quoted theorems match the usage. The C^{1,1} claim is asserted rather than proved in detail; the main theorem only requires C^2, so that is not load-bearing. There is also a small overstatement in the abstract: the title says defined "without the causal condition", but Theorem 2.8 uses non-compactness and dimension ≥ 3, not all reasonable spacetimes. The paper itself is honest about this and about the open question for proper cone structures, which speaks well of it.

I see no load-bearing flaw. The proofs are checkable, the new characterization is clean, and the literature is handled fairly. This is a paper for causality theorists and anyone using global hyperbolicity in mathematical relativity; it will likely become the standard citation for the definition. I would send it to a competent referee; it does not deserve a desk reject.

Best,

Referee Report

0 major / 4 minor

Summary. The paper simplifies the definitions of global hyperbolicity and causal simplicity in Lorentzian causality theory. Its main result, Theorem 2.8, states that a non-compact spacetime of dimension n+1 ≥ 3 is globally hyperbolic if and only if all its causal diamonds are compact, so that the causality condition becomes redundant in this physically natural setting. The key new ingredient is Theorem 2.7, which shows that for a non-totally vicious spacetime of dimension n+1 ≥ 3, the causality assumption can be dropped from both definitions: closedness of the causal relation (or of causal diamonds) forces chronology, and a failure of causality would produce a closed achronal lightlike geodesic γ that must lie in the achronal boundary ∂I^+(p); the dimension assumption supplies a point q in ∂I^+(p) off γ, and closedness of J then gives a lightlike geodesic p→q whose corner with γ gives q∈I^+(p), a contradiction. The paper also treats low-regularity settings: for closed cone structures it proves that global hyperbolicity is equivalent to causality plus compactness of causally convex hulls of compact sets, and that causality cannot be dropped there (Example 2.12). An honest limitation is stated in §2.2.1: the authors do not know whether Theorem 2.7 passes to proper cone structures.

Significance. If the main theorem is correct, it is a clean and useful simplification: in dimension ≥ 4 (or n+1≥3) and for non-compact spacetimes, global hyperbolicity is checked solely by compactness of causal diamonds, with no separate causality condition. This is a foundational result that should interest the relativity community. The proof is detailed and self-contained apart from standard tools, and the role of the dimension and regularity assumptions is made explicit: the dimension enters through the Hausdorff-dimension argument in Theorem 2.7, and Example 2.12 shows the causality condition cannot be dropped for general closed cone structures. The paper also gives a new characterization of causal simplicity via closed causally convex hulls. The authors are appropriately cautious about the limits of the argument for proper cone structures. The work is honest about its reliance on prior results, including the second author's review and closed-cone-structure paper, which are published and independent.

minor comments (4)

[§1, Definition of spacetime] The opening asserts that the metric is C^2 and that C^{1,1} would be enough, but the proof of Theorem 2.7 uses the fact that a closed achronal lightlike geodesic is C^3 because the connection is C^1, which requires a C^2 metric. Since the authors later indicate in §2.2.1 that the C^{1,1} case does hold via Lorentz-Finsler theory, they should either state Theorem 2.7 explicitly for C^2 metrics and mention the C^{1,1} extension as a separate assertion, or provide the needed argument in the proof.
[§2.1, Theorem 2.7 proof] The Morse-Sard argument and its Hausdorff-dimension alternative assume that the C^0 achronal hypersurface ∂I^+(p) has Hausdorff dimension n. Since ∂I^+(p) is a priori only a topological hypersurface, the authors should justify that it is locally Lipschitz (as is standard for achronal boundaries) so that its Hausdorff dimension is indeed n. This is a presentation issue, not a correctness gap, but a short clarifying sentence or reference would make the proof fully rigorous.
[§2.1, Theorem 2.7 statement] The theorem says that the causality conditions '(a) and (α)' in Definitions 1.1 and 1.2 can be dropped, but condition (a) is strong causality and condition (α) is distinguishing. The proof uses the Bernal-Sánchez weakening of these to ordinary causality; this reduction should be recalled in the statement or just before the proof to avoid confusion.
[§2, after Definition 2.4] There is a typographical error in the sentence beginning 'In fact troug h this version...'; it should read 'through this version'. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is proved from prior independent causality-theory results and does not assume its conclusion.

full rationale

The paper's central result, Theorem 2.8, is a mathematical implication: compact causal diamonds in a non-compact spacetime of dimension n+1 ≥ 3 force causality, and hence global hyperbolicity. The proof does not define compactness of causal diamonds in terms of global hyperbolicity, and it does not define global hyperbolicity in terms of compactness; the causality condition is genuinely shown to follow rather than being assumed. No data are fitted, no fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to rule out alternatives. The repeated citations to [31] and [32] supply standard and externally published background facts, such as the existence of a closed achronal lightlike geodesic in a non-causal spacetime and the achronal-boundary hypersurface theorem; these results do not contain the target conclusion and are not made circular by the fact that one of the present authors also wrote them. The authors even state explicitly where their proof does not extend, as in §2.2.1 for proper cone structures, which further confirms that the derivation is not forced by a self-citation chain. The Clarke-Joshi chronology result, the Morse-Sard/Hausdorff-dimension argument, and the corner-smoothing contradiction all contribute independent content. No specific circular step can be quoted or exhibited, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's results rest on standard theorems of Lorentzian causality and differential topology. No free parameters are fitted and no new physical entities are postulated. The main non-standard inputs are self-cited results from the coauthor's earlier work, which are published and whose proofs do not assume the present theorems.

assumptions (6)

domain assumption Standard Lorentzian causality theory for C^2 spacetimes: closed causal diamonds imply J closed, hence reflecting; Clarke-Joshi theorem (reflecting and non-totally vicious implies chronological).
Invoked in Section 2.1 to go from compactness or closedness of causal diamonds to chronology.
domain assumption Closed cone structure causal ladder and limit curve theorem from Minguzzi [31].
Used in Section 2.2.2 in the proof of Theorem 2.10 and Corollary 2.11.
standard math Morse-Sard theorem for C1 maps between manifolds, and the fact that locally Lipschitz maps do not increase Hausdorff dimension.
Used in Theorem 2.7 to show a closed achronal lightlike geodesic does not cover the whole achronal boundary of dimension at least 2.
domain assumption Achronal boundaries are C0 achronal hypersurfaces of dimension n (Minguzzi [32, Thm. 2.87]).
Used in Theorem 2.7 to treat N = ∂I+(p) as a manifold of dimension n = dim M - 1.
domain assumption Existence of local chronological diamonds around each point of a smooth spacetime.
Used in Proposition 2.3 to cover a compact set K by finitely many sets I+(q_i) ∩ I-(r_i) with q_i, r_i in a larger compact set.
standard math Spacetimes are Hausdorff, second countable, paracompact, connected, time-oriented Lorentzian manifolds.
Stated in the introduction as the standing definition of spacetime.

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Pith. "Pith review of Globally hyperbolic spacetimes can be defined without the 'causal' condition." pith.science (2026). https://pith.science/paper/PJFHRMER

@misc{pith2026190811701,
  author       = {Pith},
  title        = {Pith review of: Globally hyperbolic spacetimes can be defined without the 'causal' condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJFHRMER}},
  note         = {Machine review of arXiv:1908.11701}
}

abstract

Reasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non $C^{1,1}$ metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

A. N. Bernal and M. S´ anchez. On smooth Cauchy hypersurfac es and Ge- roch’s splitting theorem. Commun. Math. Phys. , 243:461–470, 2003

work page 2003

[2] [2]

A. N. Bernal and M. S´ anchez. Globally hyperbolic spacetimes can be de- ﬁned as ‘causal’ instead of ‘strongly causal’.Class. Quantum Grav. , 24:745– 749, 2007

work page 2007

[3] [3]

Bernard and S

P. Bernard and S. Suhr. Lyapounov functions of closed cone ﬁe lds: from Conley theory to time functions. Commun. Math. Phys. , 359:467–498, 2018

work page 2018

[4] [4]

Caponio, M

E. Caponio, M. Javaloyes, and M. S´ anchez. On the interplay bet ween Lorentzian causality and Finsler metrics of Randers type. Rev. Mat. Iberoamericana, 27:919–952, 2011

work page 2011

[5] [5]

V. Chernov. Conjectures on the relations of linking and causality in causally simple spacetimes. Class. Quantum Grav. , 35:105010, 2018

work page 2018

[6] [6]

Choquet-Bruhat and R

Y. Choquet-Bruhat and R. Geroch. Global aspects of the Cauc hy problem in general relativity. Commun. Math. Phys. , 14:329–335, 1969

work page 1969

[7] [7]

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

work page 2012

[8] [8]

P. T. Chru´ sciel, J. D. E. Grant, and E. Minguzzi. On diﬀerentiabil- ity of volume time functions. Ann. Henri Poincar´ e, 17:2801–2824, 2016. arXiv:1301.2909

work page Pith review arXiv 2016