Properties of the Null Distance and Spacetime Convergence. (6th January 2021)

• Record Type: Journal Article
• Authors: Allen, Brian; Burtscher, Annegret
• Is Part Of: International mathematics research notices. Volume 2022:Number 10(2022)
• Page Start: 7729
• Page End: 7808
• Publication Date: 2021-01-06
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa311
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov–Hausdorff, and Sormani–Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that nonuniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.