Properties of the Null Distance and Spacetime Convergence

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.

Initial data and black holes for matter models

“…Since a (0) and a 0 are positive for all r > 0 by Step 1, and a (1) is negative for r > r * − δ by construction, we have that a v (v 0 , r) < 0, for all r > r * − δ.…”

“…In [9] we have shown that this solution is, although singular, still surprisingly well-behaved in a way that it satisfies the second Bianchi identity weakly. The stability of this solution may be studied using metric convergence, e.g., in the sense of Gromov-Hausdorff convergence or Sormani-Wenger intrinsic flat convergence [1,6,22,29,30].…”

“…Without loss of generality we may assume that δ ≤ min{ r * 2 , ∆}, hence δ 2 +3r 2 * r * −δ ≤ 13r * 2 , so that we obtain 0 < −a (1)…”

Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes

“…For results in Lorentzian length spaces, see [1,3,6,22,30]. Lastly, for results related to the null distance function and other notions of distance defined on a spacetime, see [2,23,39,40].…”

A Lorentzian analog for Hausdorff dimension and measure

“…Finally, there have been several approaches to a synthetic or axiomatic description of (parts of) Lorentzian geometry and causality in the past: The causal spaces of Kronheimer and Penrose [KP67], and the timelike spaces of Busemann [Bus67]. A closely related direction of research is the recent approach of Sormani and Vega [SV16] and its further development by Allen and Burtscher in [AB21] of defining a metric on a (smooth) spacetime that is compatible with the causal structure in case the spacetime admits a time function satisfying an anti-Lipschitz condition. Recently, this approach has been extended to the setting of Lorentzian length spaces in [KS21] and it was shown that these two approaches are in a strong sense compatible.…”