An extendible spacetime without closed timelike curves whose every extension contains closed timelike curves

This paper resolves a question posed by Geroch by constructing an extendible spacetime devoid of closed timelike curves, formed by removing a fractal from time-rolled Minkowski spacetime, such that every possible extension of it inevitably contains closed timelike curves.

The Gist

The Big Question: Can You "Fix" a Broken Universe?

Imagine you have a map of a city. In this city, there are no loops where you can drive around and end up back in your driveway before you left (these loops are called Closed Timelike Curves, or CTCs, in physics—they represent time travel).

Now, imagine this city is "broken" or incomplete. There are some holes in the pavement. The paper asks a very specific question: If we fill in those holes to make the city complete, will we accidentally create a time-travel loop?

Usually, in physics, if you have a universe without time travel, you expect that if you expand it or "fix" it, it should still be a safe, time-travel-free universe. The authors of this paper say: "Not always."

They constructed a specific, weird universe that has no time travel, but the moment you try to "fix" it or extend it to make it complete, time travel becomes inevitable.

The Analogy: The Infinite Wall of Fractal Bricks

To build this universe, the authors used a clever trick involving Minkowski spacetime. Think of this as a flat, infinite sheet of paper representing space and time.

1. Rolling the Paper: First, they "rolled" this paper up like a carpet. If you walk forward in time on this rolled paper, you eventually circle around and bump into your past self. This creates a time loop (a CTC).
2. The Problem: We don't want time loops. So, we need to cut a hole in the carpet so you can't complete the circle.
3. The Solution (The Barrier): Instead of cutting a simple hole, they built a fractal wall (a shape that repeats itself infinitely) across the path.
   • Imagine a wall made of bricks.
   • But these aren't normal bricks. They are made of a "Cantor set" pattern (a shape where you keep removing the middle third, leaving gaps, then removing the middle of the remaining pieces, and so on forever).
   • They built this wall using light-speed bricks. The edges of these bricks are angled exactly at the speed of light.

Why This Wall is Special

This fractal wall acts like a perfect barrier.

• No Time Travel in the Original: Because the wall is so intricate and covers every possible path, you cannot walk from one side to the other and loop back without hitting the wall. So, in this "punctured" universe, time travel is impossible.
• The Catch: The wall is made of "dust" and "gaps." It's not a solid, continuous solid wall; it's a mathematical fractal. It has holes, but they are so tiny and structured that you can't pass through them without hitting the "dust."

The "Extension" Trap

Here is the magic trick of the paper.

In physics, a "maximal" universe is one that is complete and cannot be made any bigger. If our fractal wall universe is "extendible," it means there is a bigger universe that contains our wall universe as a subset.

The authors proved that any way you try to fill in the gaps of this fractal wall to make a complete universe, you must create a time loop.

The Metaphor: The "Ghost" Door
Imagine the fractal wall is a fence made of infinitely many tiny, shifting pickets.

• In our current universe, the pickets are arranged so perfectly that you can't squeeze through. No time travel.
• But, the fence is "incomplete." There are mathematical gaps between the pickets.
• If you try to "complete" the fence (the extension), you have to fill in those gaps.
• The authors show that the only way to fill those gaps without breaking the laws of physics is to connect the pickets in a way that creates a door.
• Once that door exists, you can walk through it, go around the world, and come back to your starting point before you left.

The "Time Machine" Misconception

The authors are careful to say this isn't a "Time Machine" in the sci-fi sense.

• In the original universe, time travel is impossible.
• In the extended universe, time travel exists, but it is trapped in the past.
• Imagine the time loop is a secret tunnel that only exists in the "basement" of the universe, behind the wall you built. You can't access it from the "living room" (the original universe). You can only access it if you expand the house to include the basement.

Why Does This Matter?

This paper solves a puzzle posed by the famous physicist Robert Geroch over 50 years ago.

Geroch was worried about how we define "singularities" (like the center of a black hole or the Big Bang). Physicists often define a singularity as a place where spacetime "breaks" and can't be extended.

• If you have a universe without time travel, and you extend it, you should expect it to still be free of time travel.
• If it suddenly gains time travel just by being extended, it suggests our definition of "completeness" or "singularity" is flawed.

The Conclusion:
The authors built a universe that is "safe" (no time travel) but "fragile." The moment you try to make it "complete," it becomes "unsafe" (time travel appears). This proves that the property of "having no time travel" is not stable under extension.

Summary in One Sentence

The authors built a universe with a fractal, light-speed wall that prevents time travel, but proved that the only way to fill in the holes of that wall to make the universe complete is to accidentally create a time loop, solving a 50-year-old mystery about the nature of spacetime.

Technical Summary

1. Problem Statement and Context

The paper addresses a fundamental question in General Relativity regarding spacetime maximality and the definition of singularities.

• The Core Question: Does the property of "having no Closed Timelike Curves" (CTCs) satisfy a specific condition regarding maximality?
  • Let $\mathcal{U}$ be the collection of all spacetimes. Let $P \subseteq \mathcal{U}$ be a collection of spacetimes with a specific property (e.g., "no CTCs").
  • A spacetime is $P$-maximal if it cannot be isometrically embedded as a proper subset of another spacetime in $P$.
  • A spacetime is $P$-extendible if it is not $P$-maximal.
  • Condition (*): "Every $P$-maximal spacetime is $\mathcal{U}$-maximal."
• Geroch's Inquiry: Robert Geroch questioned whether condition (*) holds for the collection $P$ of all spacetimes without CTCs.
  • If (*) holds, one could refine the notion of a "singular point" by extending spacetimes until they hit a CTC barrier.
  • If (*) fails, there exists a spacetime without CTCs that can be extended, but every possible extension introduces CTCs.
• The Goal: Construct a counterexample to condition (*) for the "no CTCs" property. Specifically, find a spacetime $(M, g)$ that:
  1. Contains no CTCs.
  2. Is extendible (not maximal in $\mathcal{U}$).
  3. Every proper extension $(M', g')$ of $(M, g)$ contains at least one CTC.

2. Methodology and Construction

The authors construct a specific 2-dimensional spacetime (later generalized to $d$-dimensions) by modifying time-rolled Minkowski spacetime.

A. The Base Spacetime

• Start with 2D Minkowski spacetime $(\mathbb{R}^2, \eta)$.
• "Roll" it up along the time direction by identifying points $(-\ell, x)$ and $(\ell, x)$ for a sufficiently large $\ell$. This creates a cylinder with CTCs.
• The goal is to remove a set of points (a "barrier") from this cylinder such that all CTCs are broken, but the remaining spacetime is "fragile" enough that any attempt to fill in the missing points restores CTCs.

B. The Fractal Barrier ($B$)

The authors construct a closed set $B \subset [0, 1] \times \mathbb{R}$ using a fractal iteration similar to the Cantor set, but with lightlike legs.

• Construction Step: Start with a closed trapezoid $B^0_0$ with horizontal bases and lightlike (null) legs.
• Iteration: Inside this trapezoid, remove a central region such that three smaller, self-similar closed trapezoids remain.
  • The similarity ratio is $\lambda$ (where $2 < \lambda < 3$).
  • The removed regions are chosen such that the legs of the new trapezoids remain lightlike.
• Limit Set: The barrier $B$ is the intersection of the unions of these trapezoids over infinite iterations.
  • $B$ is a generalized Cantor set.
  • The complement $\mathbb{R}^2 \setminus B$ is connected.
• The Spacetime $M_-$: The authors define the spacetime as the time-rolled Minkowski space with the set $B$ removed (punctured).
  • $M_- = (\text{Rolled } \mathbb{R}^2 \setminus B, \eta)$.

C. Key Geometric Properties

1. No CTCs in $M_-$: By Proposition 2, any causal curve attempting to cross the "wall" $[0, 1] \times \mathbb{R}$ must intersect $B$. Since $B$ is removed, no causal curve can loop around the cylinder to form a CTC.
2. Extendibility: $M_-$ is clearly extendible because the full rolled Minkowski space (with $B$ present) is a valid extension.
3. The "Fragility" of $B$: The construction relies on "eventually middle" points in $B$. These are points in the fractal where the choice sequence of sub-trapezoids eventually becomes "Middle" forever.
   • Lemma 2: For an eventually middle point $e$, there is a "light rhombus" around it where the vertical centerline intersects $B$ only at $e$.
   • Lemma 3: Through any eventually middle point $e$, there exist timelike curves $\tau$ and $\tau'$ connecting $e$ to the "left" and "right" boundaries of the spacetime (which are identified in the rolled model). Crucially, these curves intersect $B$ only at $e$.

3. Key Results and Proofs

Proposition 3: Every Proper Extension Contains CTCs

The main theorem proves that if $(S, g)$ is any proper extension of $M_-$, then $S$ must contain a CTC.

Proof Logic:

1. Existence of a Limit Point: Since $M_-$ is a proper subset of $S$, and the original rolled Minkowski space is geodesically complete, there must be a geodesic $\gamma$ in $M_-$ that converges to a point $p \in S \setminus M_-$.
2. Convergence to the Barrier: The point $p$ must correspond to a point in the removed set $B$ (specifically, a point in the closure of the removed set).
3. Approximation by Eventually Middle Points: By Lemma 1, "eventually middle" points are dense in $B$. Therefore, arbitrarily close to $p$, there exists an eventually middle point $e \in B$.
4. Filling the Gap: Using Lemma 4 (a technical lemma on extensions and coordinate charts), the authors show that if the extension $S$ fills in the point $p$, it must also "fill in" the nearby point $e$ because the curves approaching them are arbitrarily close and short.
5. Formation of the CTC:
   • Once $e$ is included in the extension $S$, the timelike curves $\tau$ and $\tau'$ (from Lemma 3) which previously ended at $e$ (missing point) now connect through $e$.
   • Since the spacetime is rolled, the start of $\tau$ and the start of $\tau'$ are identified.
   • The path $\tau \to e \to \tau'$ (reversed) forms a Closed Timelike Curve entirely within $S$.

4. Generalization to Higher Dimensions

• The construction is extended to $d$-dimensions by taking the Cartesian product $B \times \mathbb{R}^{d-2}$.
• The barrier becomes a set of "bars" with 2D trapezoidal cross-sections.
• The logic holds: any causal curve crossing the barrier must intersect the set, and any extension filling in a point of the barrier allows for the reconstruction of a CTC via the 2D cross-sections.

5. Significance and Conclusion

• Resolution of Geroch's Question: The paper definitively answers Geroch's query in the negative. The property "no CTCs" does not satisfy condition (*).
• Implications for Singularity Theory: This result suggests that the program of defining singularities via "maximal extensions" is problematic for causality conditions. One cannot simply say a spacetime is singular because it cannot be extended without creating CTCs; the example $M_-$ is extendible, but all its extensions are causally pathological.
• Technical Innovation: The paper introduces a novel use of fractal geometry (specifically a Cantor-like set with lightlike boundaries) to control causal structure. It demonstrates that "time machines" (spacetimes with CTCs) can be created by filling in a set of measure zero (the fractal barrier) in a specific way.
• Clarification on "Time Machines": The authors note that $M_-$ itself is not a "time machine" in the traditional sense because all CTCs in its extensions are confined to the timelike past of the original spacetime $M_-$ within the extension.

In summary, the authors constructed a "punctured" spacetime that is causally safe but structurally unstable: any attempt to complete the spacetime (remove the punctures) inevitably reintroduces time travel, proving that the absence of CTCs is not a robust property under extension.