ADDENDUM: Low regularity approach to Bartnik’s conjecture (2025 Class. Quantum Grav. 42 215020)

Abstract

This note serves as an addendum to our previous work (Flores et al 2025 Class. Quantum Grav.42 215020), where the Bartnik Splitting Conjecture (BSC) was first established for globally hyperbolic Lorentzian length spaces. Here, we strengthen and generalize that result by removing the assumption of a global topological product structure, which is not intrinsic in the setting of Lorentzian length spaces. Instead, we only require the existence of a compact Cauchy set. Consequently, there is no hypothesis on the asymptotic behaviour of vertical curves—whose existence is now obtained a posteriori—but rather the more natural timelike geodesic completeness condition is assumed. This refinement (theorem 1.2) yields a stronger and more flexible version of the BSC, extending its applicability and bringing it closer to its smooth counterpart.