A new topology on the space of Lorentzian metrics on a fixed manifold

TL;DR

The paper introduces a covariant topology on the space of Lorentzian metrics on a manifold, based on functions measuring volume form differences and causal structure, creating a Hausdorff topology for strongly causal metrics.

Contribution

It defines a new covariant topology on Lorentzian metrics using functions that compare volume forms and causal structures, ensuring Hausdorff separation for strongly causal metrics.

Findings

The topology is generated by a family of functions measuring geometric differences.

The topology is Hausdorff on the space of strongly causal Lorentzian metrics.

The functions depend on parameters like volume scale, length scale, and submanifold labels.

Abstract

We give a covariant definition of closeness between (time oriented) Lorentzian metrics on a manifold M, using a family of functions which measure the difference in volume form on one hand and the difference in causal structure relative to a volume scale on the other hand. These functions will distinguish two geometric properties of the Alexandrov sets $ A(p,q), \tilde{A} (p,q) $ relative to two space time points $q$ and $p$ and metrics $g$ and $ \tilde{g} $. It will be shown that this family generates uniformities and consequently a topology on the space of Lorentzian metrics which is Hausdorff when restricted to strongly causal metrics. This family of functions will depend on parameters for a volume scale, a length scale (relative to the volume scale) and an index which labels a submanifold with compact closure of the given manifold M.