Almost isometries of non-reversible metrics with applications to stationary spacetimes

TL;DR

This paper develops a theory of almost isometries for quasi-metric spaces, focusing on non-reversible Finsler metrics, with applications to stationary spacetimes in General Relativity, exploring conformal diffeomorphisms and their structures.

Contribution

It introduces a framework for almost isometries in quasi-metric spaces, especially for non-reversible Finsler metrics, and applies it to analyze conformal maps in stationary spacetimes.

Findings

Topology of conformal maps is characterized.

Lie group structure of conformal diffeomorphisms is established.

Connections between almost isometries and spacetime symmetries are demonstrated.

Abstract

We develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. The case of non-reversible Finsler (more specifically, Randers) metrics is of particular interest, and it is studied in more detail. The main motivation arises from General Relativity, and more specifically in spacetimes endowed with a timelike conformal field K, in which case \emph{conformal diffeomorphisms} correspond to almost isometries of the Fermat metrics defined in the spatial part. A series of results on the topology and the Lie group structure of conformal maps are discussed.