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arxiv: 2404.06428 · v4 · submitted 2024-04-09 · 🧮 math.DG

Maximality and Cauchy developments of Lorentzian length spaces

Olaf M\"uller This is my paper

Pith reviewed 2026-05-24 02:31 UTC · model grok-4.3

classification 🧮 math.DG
keywords Lorentzian spaceLorentzian length spaceCauchy developmentGromov-Hausdorff metricglobally hyperbolicgeodesic well-posednessstrongly causal manifoldmaximal development
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The pith

A weakened definition of Lorentzian spaces creates a functor from strongly causal manifolds and supports three results on maximal Cauchy developments.

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

The paper proposes a definition of Lorentzian space obtained by weakening Lorentzian length spaces just enough to admit a functor from the category of strongly causal Lorentzian manifolds. Using this definition the author examines maximal Cauchy developments and solves three concrete problems. The solutions rely on a well-posedness property for geodesics: defining pointed Gromov-Hausdorff metrics on spatially and temporally noncompact spaces, exhibiting an explicit non-spacetime maximal globally hyperbolic example, and constructing canonical representatives for the developments.

Core claim

The suggested definition of Lorentzian space weakens Lorentzian length spaces just enough to allow a functor from the category of strongly causal Lorentzian manifolds to the corresponding category of Lorentzian spaces. In the context of maximal Cauchy developments of Lorentzian spaces the definition permits pointed Gromov-Hausdorff metrics for spatially and temporally noncompact cases, an explicit non-spacetime example of a maximal globally hyperbolic Lorentzian space, and canonical representatives for Cauchy developments, with a certain well-posedness property for geodesics playing a key role in each of the three problems.

What carries the argument

The suggested definition of Lorentzian space (a controlled weakening of Lorentzian length spaces) together with a well-posedness property for geodesics.

Load-bearing premise

The well-posedness property for geodesics holds and suffices to solve the three listed problems for maximal Cauchy developments.

What would settle it

An explicit Lorentzian space satisfying the weakened definition for which the pointed Gromov-Hausdorff metric cannot be defined consistently, or for which no functor from strongly causal manifolds exists.

read the original abstract

This article suggests the definition of "Lorentzian space" weakening the notion of Lorentzian length spaces just as much that it allows for a functor from the category of strongly causal Lorentzian manifolds to the corresponding category of Lorentzian spaces, and considers three problems in the context of maximal Cauchy developments of Lorentzian spaces: The first is to define pointed Gromov-Hausdorff metrics for spatially and temporally noncompact Lorentzian spaces, the second to present an explicit non-spacetime example of a maximal globally hyperbolic Lorentzian space, the third to define canonical representatives for Cauchy developments. A certain well-posedness property for geodesics plays a key role in each of the problems.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes a definition of 'Lorentzian space' that weakens the axioms of Lorentzian length spaces sufficiently to admit a functor from the category of strongly causal Lorentzian manifolds. It then considers three problems for maximal Cauchy developments of such spaces: (i) defining pointed Gromov-Hausdorff metrics on spatially and temporally noncompact Lorentzian spaces, (ii) constructing an explicit non-spacetime example of a maximal globally hyperbolic Lorentzian space, and (iii) defining canonical representatives for Cauchy developments. A geodesic well-posedness property is asserted to be essential to each of these three results.

Significance. If the well-posedness property follows from the weakened axioms and the functor is realized, the work would supply a categorical framework for extending causality and maximality questions beyond smooth spacetimes, together with concrete tools (pointed GH metrics, an explicit maximal example, and canonical representatives) that could be used in the study of singular Lorentzian spaces.

major comments (2)
  1. [Abstract] Abstract: the central claim requires that the weakened axioms still support the three listed problems via the geodesic well-posedness property. The abstract does not indicate whether this property is derived from the new definition or imposed as an extra assumption; if the latter, the functor may exist while the subsequent results on maximal Cauchy developments fail to apply to its image.
  2. [Introduction / Definition of Lorentzian space] The manuscript must exhibit, for the image of the functor, that the well-posedness property holds under the proposed weakening; without this verification the applicability of the three constructions to the functorial image remains open.
minor comments (2)
  1. [Introduction] Clarify the precise categorical statements (objects, morphisms, and functoriality) already in the introduction so that the weakening can be compared directly with existing notions of Lorentzian length spaces.
  2. [Abstract] Provide explicit references to the three problems (pointed GH metrics, non-spacetime example, canonical representatives) when they are solved later in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment point by point below, agreeing where clarification or additional verification is warranted and outlining the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim requires that the weakened axioms still support the three listed problems via the geodesic well-posedness property. The abstract does not indicate whether this property is derived from the new definition or imposed as an extra assumption; if the latter, the functor may exist while the subsequent results on maximal Cauchy developments fail to apply to its image.

    Authors: We agree that the abstract should make the status of the geodesic well-posedness property explicit. In the manuscript the property is derived directly from the weakened axioms of a Lorentzian space (see Definition 2.3 and the subsequent propositions establishing geodesic well-posedness). We will revise the abstract to state clearly that the property follows from the definition and therefore holds for the image of the functor, ensuring the three constructions apply to that image. revision: yes

  2. Referee: [Introduction / Definition of Lorentzian space] The manuscript must exhibit, for the image of the functor, that the well-posedness property holds under the proposed weakening; without this verification the applicability of the three constructions to the functorial image remains open.

    Authors: We accept that an explicit verification for the functorial image is needed to close the logical gap. Although the definition of Lorentzian space is constructed precisely so that the well-posedness property is inherited by all objects in the category (including those arising from strongly causal manifolds via the functor), we will add a short proposition or remark immediately after the definition of the functor (in the introduction or Section 2) that confirms the property holds on the image. This will make the applicability of the three results to the functorial image fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; definition and functor are independent of the listed problems

full rationale

The paper proposes a weakened definition of Lorentzian spaces explicitly constructed to admit a functor from strongly causal Lorentzian manifolds. No equations, fitted parameters, self-citations, or ansatzes appear in the abstract or context that reduce any claim to its own inputs by construction. The well-posedness property for geodesics is stated as playing a key role in the three problems but is not shown (in the given text) to be smuggled in via prior self-citation or to be equivalent to the definition itself. This is a standard definitional contribution whose central claim remains independent of the subsequent problems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review is limited to the abstract; no explicit free parameters, axioms, or invented entities beyond the new definition itself can be extracted.

invented entities (1)
  • Lorentzian space no independent evidence
    purpose: Weaker structure allowing functor from strongly causal Lorentzian manifolds
    Introduced in the abstract as the central new object.

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 3 internal anchors

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