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

TL;DR

The paper answers a long-standing question about spacetime maximality and closed timelike curves by constructing a concrete 2D spacetime $\mathbf{M}^-$ obtained from time-rolled Minkowski space via a fractal barrier $B$. The barrier is a Cantor-like fractal of trapezoids arranged in $[0,1]\times\mathbb{R}$, ensuring no causal curve can cross this region without meeting $B$, and enabling a rolled-up spacetime with no $CTCs$ while remaining $ ext{U}$-extendible. The key result is that every proper extension of $\mathbf{M}^-$ must introduce $CTCs$, proving the existence of a $CTC$-free spacetime whose extensions inherently acquire $CTCs$; the authors then generalize the construction to higher dimensions using $B\times\mathbb{R}^{d-2}$. This work resolves a question posed by Geroch and provides a tangible, fractal-based method for analyzing spacetime maximality and its interaction with causality, with implications for the notion of singular points via extensions.

Abstract

By removing a fractal from time-rolled Minkowski spacetime, we construct an extendible spacetime without closed timelike curves whose every extension contains closed timelike curves. This settles a question posed by Geroch.

Figures (7)

• Figure 1: Construction of the barrier $B$ by an infinite iteration.
• Figure 2: The figure illustrates the main construction step of barrier $B$ together with the variables of equation \ref{['eq-selfsim']} that is used to derive the condition $2<\lambda<3$ of self-similarity. The light gray part is the set that we remove from the trapezoid, and the three dark gray trapezoids are the remaining closed trapezoids which are similar to the original one. Analogously to the construction of the Cantor set, we repeat this removal step ad infinitum.
• Figure 3: No causal curve can cross region $[0,1]\times\mathbb{R}$ without intersecting $B$, cf. Proposition \ref{['prop-blocking']} below.
• Figure 4: Inside the smallest light rhombus containing a trapezoid used in the construction of $B$, the vertical centerline intersects $B$ only in one point, which is an eventually middle point.
• Figure 5: Through every eventually middle point $e$ there are timelike curves $\tau$ and $\tau'$ respectively connecting $e$ to points $(-\ell,x)$ and $(\ell,x)$ for some $x\in\mathbb{R}$ such that these curves contain no other point of $B$ apart from $e$, cf. Lemma \ref{['lem-em-intersection']}.

Theorems & Definitions (14)

• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof