Timelike Hilbert geometry of the spherical simplex

Abstract

We first review the notion of timelike metric spaces. This is a metric theory developed by Herbert Busemann, as a geometrical setting for the theory of general relativity. We review in particular the notions of timelike Hilbert and timelike spherical Hilbert geometry. We prove the following result on the timelike spherical Hilbert geometry of simplices:

Let ${\Delta }_2$ be a simplex on the 2-sphere and ${\tilde{\Delta }}_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated with the pair ${\Delta }_2,{\tilde{\Delta }}_2$ is isometric to a union of six copies of vector spaces equipped with a timelike norm, isometrically and transitively acted upon by the group ${\mathbb{R}}_{>0}^2\times {\mathbb{Z}}_3\times {\mathbb{Z}}_2$.

This is a timelike spherical analogue of a well-known result (due to Busemann) stating that the Hilbert metric of a Euclidean simplex is isometric to a metric induced by a normed vector space. At the same time, this gives a new example of timelike space.