Anisotropy-refined Borde–Sorkin conjecture for Morse spacetimes

Let $(\mathcal{M},h,f,\zeta)$ be a Morse geometry of dimension $n$, and let $(M,g)$ be the corresponding Morse spacetime. Assume that $f$ has a single critical point for each critical value. Let Condition $\mathrm{condrelax}$ denote the anisotropy bound specified in the paper at a critical point. Anisotropy-refined Borde–Sorkin conjecture. The spacetime $(M,g)$ is causally continuous if and only if both of the following hold: no critical point has index $1$ or $n-1$, and Condition $\mathrm{condrelax}$ is satisfied at every critical point whose index is different from $0$ and $n$. This refinement incorporates a bound on anisotropy because the paper presents a potential highly anisotropic counterexample to the original Borde–Sorkin conjecture, although the authors do not prove that the example is causally discontinuous. The refined statement is therefore proposed as an open conjecture.