Orders on sets of conformal classes applied to Bartnik's conjecture

TL;DR

This work investigates whether an order-theoretic organization of conformal classes can illuminate Bartnik's splitting conjecture, introducing the order $\le_2$ on conformal classes of compact Cauchy slabs with fixed past boundary and a hierarchy of orders $\le_0$, $\le_1$, and $\le_3$ to study their properties. It shows $\le_0$ is only a preorder, while $\le_1$ becomes an actual order on relevant classes (e.g., long spacetimes form a $\le_1$-future subset) and that $\le_3$ can be described via order-preserving correspondences; these results are grounded by a catcher-theorem and Malament-type separation of diffeomorphism classes. The first part indicates long Cauchy slabs form a $\le_2$-future set, suggesting a path to attack the conjecture from the $\le_2$-past using conformal conditions. The second main finding shows that replacing the strong energy condition $SEC$ with the null energy condition $NEC$ in dimension $\ge 3$ destroys Bartnik's conjecture: every globally hyperbolic spatially compact conformal class admits future complete metrics satisfying $NEC$, with analogous statements for spatially noncompact cases, demonstrated via a constructive conformal deformation and flatzoomer techniques. Overall, the paper clarifies both the promise and the limits of order-theoretic, conformal approaches to rigidity questions in Lorentzian geometry.

Abstract

In the first part, after showing that the most natural approach to define an order on sets of conformal classes fails, we define a nontrivial order $\leq_2$ on the set of conformal classes of compact Cauchy slabs with fixed past boundary that could help structuring approaches to the Bartnik splitting conjecture via conformal conditions. In the second part we show that if we replace the strong energy condition in Bartnik's splitting conjecture with the null energy condition, then in any dimension greater or equal to $3$ the conclusion of the conjecture would be wrong, more precisely: On a manifold of dimension $\geq 3$, {\em every} globally hyperbolic spatially compact conformal class contains future complete metrics satisfying the null energy condition. In the spatially noncompact case, the same is true in the future of any Cauchy surface.

Theorems & Definitions (11)

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• Theorem 9
• Lemma 1