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arxiv: 2402.09312 · v3 · submitted 2024-02-14 · 🌀 gr-qc · math-ph· math.DG· math.MP

Coverings and Non-Hausdorff Extensions of Misner Spacetime

N. E. Rieger This is my paper

Pith reviewed 2026-05-24 04:07 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.DGmath.MP
keywords Misner spacetimenon-Hausdorff extensionsboost actioncovering spacesHawking-Ellis extensionisocausalitygeneral relativityclosed timelike curves
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The pith

Misner spacetime admits a family of non-Hausdorff extensions from boost-compatible coverings of the punctured plane.

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

The paper separates the notions of covering and extension for Misner spacetime. It classifies the connected coverings of the punctured Minkowski plane that are compatible with the discrete boost action. From these, it constructs induced quotient spacetimes and shows explicit embeddings of Misner spacetime into each. This produces a natural family that includes the Hawking-Ellis extension, its universal-cover analogue, and intermediate finite cyclic coverings. The work proves a precise statement about non-Hausdorffness and identifies a causal adjacency invariant that distinguishes the finite-sheeted cases from the universal cover. It further compares these spacetimes to two-dimensional Schwarzschild-type metrics using the notion of isocausality.

Core claim

Misner spacetime is obtained by quotienting a timelike wedge of two-dimensional Minkowski spacetime by a discrete boost. The connected coverings of the punctured model that are compatible with the boost action induce a family of quotient spacetimes into which Misner spacetime embeds explicitly. This family consists of the Hawking-Ellis extension, its universal-cover analogue, and the intermediate finite cyclic coverings. A classification theorem holds within the covering-compatible class, along with a precise non-Hausdorffness statement for the punctured quotient and a causal adjacency invariant distinguishing finite-sheeted and universal-cover cases.

What carries the argument

The boost-compatible connected coverings of the punctured Minkowski plane, which induce the quotient spacetimes and allow explicit embeddings of Misner spacetime.

If this is right

  • Explicit embeddings of Misner spacetime exist in each member of the family.
  • The family is classified within the covering-compatible class.
  • A causal adjacency invariant distinguishes the finite-sheeted cases from the universal-cover case.
  • The spacetimes admit comparison to two-dimensional Schwarzschild-type metrics via isocausality.

Where Pith is reading between the lines

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

  • The covering technique could extend to constructing extensions of other quotient spacetimes with discrete isometries.
  • The causal adjacency invariant might offer a general tool for classifying non-Hausdorff spacetimes beyond this model.
  • These extensions could affect the analysis of initial-value problems in spacetimes containing closed timelike curves.

Load-bearing premise

The connected coverings of the punctured model are compatible with the boost action.

What would settle it

A boost-compatible covering that produces an extension of Misner spacetime outside the described family of Hawking-Ellis, finite cyclic, and universal-cover cases would falsify the classification theorem.

Figures

Figures reproduced from arXiv: 2402.09312 by N. E. Rieger.

Figure 1
Figure 1. Figure 1: Misner space depicted as a ”room” in flat 2-dimensional spa [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Minkowski spacetime with some events related by the boos [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A helicoid, or more precisely a Riemann surface constructe [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Construction of a helicoid universal cover starting with a s [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Curve homotopy on the multiply connected covering space [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The covering space S∞ consists of an infinite number of sheets. Two observers, On and O′ n , that start their journey at the same event pn in region In of sheet n won’t meet anymore once they have passed the singularity Q on opposite sides. • Observer O′ k goes leftward (and downward), then we have the following situation in S1: O1 and O′ 1 meet in region IV1 in S2: O1 and O′ 2 meet in region IV2 O2 and O′… view at source ↗
read the original abstract

Misner spacetime is obtained by quotienting a timelike wedge of two-dimensional Minkowski spacetime by a discrete boost. The familiar Hausdorff extensions and the Hawking--Ellis non-Hausdorff extension are classical, but the passage from covering constructions of the punctured Minkowski plane to genuine extensions of Misner spacetime is subtler than is often stated. In this article we separate systematically the notions of covering and extension, classify the connected coverings of the punctured model that are compatible with the boost action, construct the induced quotient spacetimes, and exhibit explicit embeddings of Misner spacetime into each of them. This yields a natural family consisting of the Hawking--Ellis extension, its universal-cover analogue, and the intermediate finite cyclic coverings. We prove a precise non-Hausdorffness statement for the punctured quotient, formulate and prove a classification theorem for the resulting family within the covering-compatible class, and identify a causal adjacency invariant distinguishing the finite-sheeted and universal-cover cases. Finally, we compare these spacetimes with two-dimensional Schwarzschild-type metrics from the viewpoint of isocausality.

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

0 major / 2 minor

Summary. The paper separates the notions of covering and extension for Misner spacetime (obtained by quotienting a timelike wedge of 2D Minkowski space by a discrete boost). It classifies the connected coverings of the punctured model that are compatible with the boost action, constructs the induced quotient spacetimes, exhibits explicit embeddings of Misner spacetime into each, proves a precise non-Hausdorffness statement for the punctured quotient, formulates and proves a classification theorem within the covering-compatible class, identifies a causal adjacency invariant distinguishing finite-sheeted and universal-cover cases, and compares the resulting family (including the Hawking–Ellis extension and its variants) with two-dimensional Schwarzschild-type metrics from the viewpoint of isocausality.

Significance. If the results hold, the work supplies a systematic, explicit classification of boost-compatible non-Hausdorff extensions of Misner spacetime together with a causal invariant that distinguishes the finite and universal cases. The explicit embeddings and quotient constructions, the separation of covering from extension, and the isocausality comparison constitute concrete strengths that clarify the structure of these spacetimes and may serve as a reference for similar constructions in Lorentzian geometry.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'precise non-Hausdorffness statement' is used without a one-sentence indication of its content; adding a brief parenthetical description would improve immediate readability.
  2. The notation for the boost generator and the punctured model is introduced gradually; a short preliminary subsection collecting the basic objects and the compatibility condition would aid readers who wish to follow the classification theorem directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and positive summary of our work, the assessment of its significance, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on standard covering-space constructions in differential geometry, explicitly scoped to boost-compatible coverings of the punctured model, followed by induced quotients and embeddings of Misner spacetime. The classification theorem, non-Hausdorffness statement, and causal adjacency invariant are obtained directly from these constructions without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The abstract and structure indicate a self-contained mathematical argument independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities are identifiable; the work relies on standard notions from differential geometry and Lorentzian geometry.

pith-pipeline@v0.9.0 · 5723 in / 1162 out tokens · 45223 ms · 2026-05-24T04:07:21.062699+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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