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arxiv: 2505.02210 · v4 · submitted 2025-05-04 · 🌀 gr-qc · math-ph· math.GT· math.MP

Wormhole Nucleation via Topological Surgery in Lorentzian Geometry

Alessandro Pisana , Barak Shoshany , Stathis Antoniou , Louis H. Kauffman , Sofia Lambropoulou This is my paper

Pith reviewed 2026-05-22 16:57 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.GTmath.MP
keywords wormhole nucleationtopological surgeryLorentzian geometryMorse theoryclosed timelike curvesgeneral relativityCP2energy conditions
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The pith

A wormhole can be created without singularities in classical general relativity by using 0-surgery and replacing the critical point with closed timelike curves via connected sum with CP².

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

The paper models wormhole nucleation as a 0-surgery process inside a compact region of Lorentzian spacetime that connects two spacelike slices with different topologies. This surgery initially produces a singular cobordism at the Morse critical point. The authors avoid the singularity by taking a connected sum with the closed 4-manifold CP², which replaces the naked singularity with a region of closed timelike curves while keeping the metric everywhere nondegenerate. The resulting spacetime is nonsingular but violates all standard energy conditions. A sympathetic reader cares because the construction gives an explicit classical mechanism for wormhole formation that never breaks down at a singularity.

Core claim

Using techniques from topological surgery and Morse theory, a 0-surgery process is applied to describe the neighborhood of the nucleation point, yielding a singular Lorentzian cobordism. The Misner trick of taking a connected sum with CP² produces an everywhere nondegenerate Lorentzian metric that replaces the naked singularity at the Morse critical point with a region containing closed timelike curves. The obtained spacetime is nonsingular but violates all the standard energy conditions, showing that a wormhole can be created without singularities in classical general relativity.

What carries the argument

The 0-surgery on a 3-manifold embedded in 4-dimensional Lorentzian spacetime, resolved by the Misner connected-sum construction with CP² to eliminate the Morse critical-point degeneracy.

If this is right

  • A wormhole connecting regions of different topology can form inside a compact region of spacetime without any singularity.
  • The spacetime must violate all standard energy conditions.
  • Closed timelike curves appear in the finite region that replaces the former Morse critical point.
  • Topological surgery can be carried out on Lorentzian cobordisms while preserving nondegeneracy of the metric.
  • Wormhole nucleation occurs through a process that changes the topology of spatial slices.

Where Pith is reading between the lines

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

  • If the same replacement trick works for other Morse critical points, many additional topological transitions in spacetime might be realizable without singularities.
  • The unavoidable appearance of closed timelike curves near the nucleation site suggests that causality violation is a built-in feature of this classical construction.
  • Similar surgery-plus-connected-sum methods could be applied to other 4-manifolds or to non-compact spacetimes to generate further exotic geometries.
  • The construction raises the possibility that wormhole formation in classical gravity is always accompanied by regions where time travel becomes locally possible.

Load-bearing premise

The connected sum with CP² produces an everywhere nondegenerate Lorentzian metric that replaces the naked singularity without introducing new singularities or other pathologies.

What would settle it

An explicit local computation of the metric after the connected sum that finds even one point where the metric becomes degenerate or curvature becomes unbounded would falsify the nondegeneracy claim.

Figures

Figures reproduced from arXiv: 2505.02210 by Alessandro Pisana, Barak Shoshany, Louis H. Kauffman, Sofia Lambropoulou, Stathis Antoniou.

Figure 1
Figure 1. Figure 1: The 3-dimensional 0-surgery removes an embedding [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The transition occurs within a compact region [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) The trivial 3-dimensional cobordism W is rep￾resented by a tube in 3D Minkowski space. (b) To smooth out the corners, the solid cylinder D 1 × D 2 is deformed to D 3 , and the vector field is transformed accordingly. Because the kink number vanishes on purely spacelike and timelike boundary components, the only contributions to the integral (4) come from the two circles {0} × S 1 and {1} × S 1 , where … view at source ↗
Figure 4
Figure 4. Figure 4: (a): The boundary ∂W consists of three components—Σi, Σf , and T—whose interiors are spacelike, spacelike, and timelike, respectively. On each of these interiors the kink number is zero, independently of whether Σi or Σf is multiply connected. (b): The simplest cobordism for the 3-dimensional 0-surgery is obtained by attaching a (D 1 × D 3 )- handle to Σi × D 1 along their common boundary S 0 × D 3 at the … view at source ↗
Figure 6
Figure 6. Figure 6: Embedding diagrams for the wormhole metric [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Numerical solutions of Eqs. (15a) and (15b) for timelike and lightlike geodesics passing through the wormhole. The integration is performed over the affine interval λ ∈ [0, 10] with t(0) = 1 and ρ(0) = 5. Blue curve: Timelike geodesic with t˙0 = 1, ˙ρ0 = −1. Orange curve: Lightlike geodesic with t˙0 = 1/2, ρ˙0 = − p 1 + 2√ 26/2. Taking variational derivatives of L and employing the constraint L = ϵ, one ob… view at source ↗
Figure 8
Figure 8. Figure 8: The region in which the expansion scalars [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The stress–energy tensor changes type in different [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Two-dimensional streamline plots of the vector field [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Numerical solution of the geodesic equations for the [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Two-dimensional projection (with x = t = 0) of the region in which the weak energy condition is violated by the vector field w µ in the U1 chart. closed manifold CP2 possesses a unique singular point, every maximal integral curve originates and terminates at this singularity8 . These integral curves determine the future orientation of the light-cones for the metric (22), which explains why the geodesics a… view at source ↗
Figure 13
Figure 13. Figure 13: The two annuli are glued, respectively, to [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

We construct a model for the nucleation of a wormhole within a Lorentzian spacetime by employing techniques from topological surgery and Morse theory. In our framework, a 0-surgery process describes the neighborhood of the nucleation point inside a compact region of spacetime, yielding a singular Lorentzian cobordism that connects two spacelike regions with different topologies. To avoid the singularity at the critical point of the Morse function, we employ the Misner trick of taking a connected sum with a closed 4-manifold -- namely $\mathbb{CP}^{2}$ -- to obtain an everywhere nondegenerate Lorentzian metric. This connected sum replaces the naked singularity with a region containing closed timelike curves. The obtained spacetime is nonsingular, but violates all the standard energy conditions. Our construction, thus, shows that a wormhole can be "created" without singularities in classical general relativity.

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

1 major / 2 minor

Summary. The manuscript constructs a model for wormhole nucleation in Lorentzian geometry by applying 0-surgery to create a singular cobordism between spacelike regions of different topologies, then uses a connected sum with CP² via the Misner trick to eliminate the Morse critical point singularity, replacing it with a region of closed timelike curves while violating energy conditions. The central claim is that this yields a nonsingular spacetime in which a wormhole is created within classical general relativity.

Significance. If the metric construction is rigorously verified, this work offers an interesting topological approach to modeling wormhole formation without curvature singularities, highlighting the trade-off with causality violations and energy condition breaches. It builds on established tools like Morse theory and connected sums but applies them in a Lorentzian setting, potentially stimulating further research on exotic spacetimes. The direct construction from topological surgery is a strength, though the lack of explicit verification of the resulting metric reduces immediate applicability.

major comments (1)
  1. [Abstract (resolution step via connected sum with CP²)] The central claim that the Misner connected-sum operation with CP² produces an everywhere nondegenerate Lorentzian metric (replacing the Morse critical-point degeneracy with CTCs) is load-bearing but unsupported by explicit local analysis. No coordinate chart or metric form is supplied near the S³ gluing locus to verify preservation of the (1,3) signature, consistent extension of the time-orientation, or global nondegeneracy after excising a 4-ball from CP² (whose Euler characteristic precludes a Lorentzian metric). This must be addressed to substantiate the nonsingularity assertion.
minor comments (2)
  1. [Construction section] The description of the 0-surgery process would benefit from an explicit diagram or local coordinate patches illustrating the transition between the two spacelike regions.
  2. [Introduction] Additional references to prior applications of the Misner trick or topological surgery in Lorentzian GR would help situate the novelty of the approach.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive report. The central concern regarding explicit verification of the Lorentzian metric after the connected-sum operation is well-taken, and we address it directly below while indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract (resolution step via connected sum with CP²)] The central claim that the Misner connected-sum operation with CP² produces an everywhere nondegenerate Lorentzian metric (replacing the Morse critical-point degeneracy with CTCs) is load-bearing but unsupported by explicit local analysis. No coordinate chart or metric form is supplied near the S³ gluing locus to verify preservation of the (1,3) signature, consistent extension of the time-orientation, or global nondegeneracy after excising a 4-ball from CP² (whose Euler characteristic precludes a Lorentzian metric). This must be addressed to substantiate the nonsingularity assertion.

    Authors: We agree that the absence of an explicit local coordinate description near the gluing locus leaves the nondegeneracy claim insufficiently substantiated in the current draft. The Misner construction proceeds by excising a 4-ball from the singular cobordism and from CP², then identifying the resulting S³ boundaries. In a neighborhood of this S³ we adopt Fermi normal coordinates along the gluing hypersurface, on which the metric takes the form ds² = −dt² + dr² + r² dΩ₂² plus a small perturbation that introduces the CTCs contributed by the CP² summand; the (1,3) signature is manifest and the time-orientation extends continuously across the interface because the normal to the S³ is spacelike on both sides. Because the final object is a cobordism (a manifold with two spacelike boundary components) rather than a closed 4-manifold, the Euler-characteristic obstruction that precludes a Lorentzian metric on closed CP² does not apply; a nowhere-vanishing timelike vector field can be chosen on the interior that matches the standard time-orientation on the original cobordism and remains non-vanishing after the sum. We will add a new subsection containing these local coordinates, the explicit metric ansatz, and a verification that the resulting tensor is nondegenerate and time-orientable everywhere. This constitutes a major but straightforward revision. revision: yes

Circularity Check

0 steps flagged

Direct construction from established topological surgery and Misner trick; no reduction to self-inputs

full rationale

The paper advances a explicit construction: a 0-surgery on a Lorentzian cobordism produces a singular Morse critical point, which is then replaced by a connected sum with CP² (after excising a 4-ball) to yield a nondegenerate metric containing CTCs. This step is asserted via the known Misner trick and standard results in 4-manifold topology; it does not define the final metric or the absence of singularities in terms of any quantity fitted or derived inside the paper itself. No equations equate a 'prediction' to an input parameter, and no load-bearing premise rests solely on an unverified self-citation chain. The derivation remains self-contained against external topological benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on standard background results from differential topology and general relativity; the only added element is the specific application of these tools to the nucleation problem. No numerical parameters are fitted to data.

axioms (2)
  • domain assumption Morse theory can be applied to a Lorentzian metric to locate and classify the critical point of the nucleation process.
    Invoked to model the singular point inside the compact region of spacetime.
  • domain assumption 0-surgery on a spacelike hypersurface produces a valid cobordism between regions of differing topology in Lorentzian geometry.
    Used to describe the neighborhood of the nucleation point.
invented entities (1)
  • Closed-timelike-curve region created by connected sum with CP² no independent evidence
    purpose: Replaces the naked singularity while keeping the metric nondegenerate everywhere.
    Introduced via the Misner trick; no independent physical evidence is supplied beyond the mathematical construction.

pith-pipeline@v0.9.0 · 5698 in / 1541 out tokens · 78356 ms · 2026-05-22T16:57:31.707735+00:00 · methodology

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Lean theorems connected to this paper

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

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    To avoid the singularity at the critical point of the Morse function, we employ the Misner trick of taking a connected sum with a closed 4-manifold—namely CP²—to obtain an everywhere nondegenerate Lorentzian metric. This connected sum replaces the naked singularity with a region containing closed timelike curves.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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