Global Spacetime Similarity

TL;DR

This paper introduces a new topology for Lorentz metrics on manifolds that better captures the concept of global spacetime similarity, addressing limitations of existing topologies.

Contribution

The paper proposes a novel topology on the space of Lorentz metrics that overcomes the shortcomings of compact-open and Whitney topologies in representing global spacetime similarity.

Findings

The new topology better reflects physical and mathematical notions of spacetime similarity.

It addresses the limitations of existing topologies in capturing global properties.

The topology has desirable mathematical and physical properties.

Abstract

There are two classes of topologies most often placed on the space of Lorentz metrics on a fixed manifold. As I interpret a complaint of R. Geroch [Relativity, 259 (1970); Gen. Rel. Grav., 2, 61 (1971)], however, neither of these standard classes correctly captures a notion of global spacetime similarity. In particular, Geroch presents examples to illustrate that one, the compact-open topologies, in general seems to be too coarse, while another, the open (Whitney) topologies, in general seems to be too fine. After elaborating further the mathematical and physical reasons for these failures, I then construct a topology that succeeds in capturing a notion of global spacetime similarity and investigate some of its mathematical and physical properties.