Metric convergence of sequences of static spacetimes with the null distance

Abstract

How should one define metric space notions of convergence for sequences of spacetimes? Since a Lorentzian manifold does not define a metric space directly, the uniform convergence, Gromov-Hausdorff (GH) convergence, and Sormani-Wenger Intrinsic Flat (SWIF) convergence does not extend automatically. One approach is to define a metric space structure, which is compatible with the Lorentzian structure, so that the usual notions of convergence apply. This approach was taken by C. Sormani and C. Vega [Sormani, C., Vega, C.: Null distance on a spacetime, Classical Quantum Gravity 33, no. 8, 085001, 29. (2016) MR3476515] when defining the null distance. In this paper, we study sequences of static spacetimes equipped with the null distance under uniform, GH, and SWIF convergence, as well as Hölder bounds. We use the results of the Volume Above Distance Below (VADB) theorem of the author, R. Perales, and C. Sormani [Allen, B., Perales, R., Sormani, C.: Volume above distance below. J. Differential Geom. 126(3), 837–874 (2024)] to prove an analog of the VADB theorem for sequences of static spacetimes with the null distance. We also give a conjecture of what the VADB theorem should be in the case of sequences of globally hyperbolic spacetimes with the null distance.