Timelike Hilbert and Funk geometries

Abstract

A timelike space is a Hausdorff topological space equipped with a partial order relation $<$ and a distance function $\rho$ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the product of the space with itself that contains the diagonal, namely, $\rho(x,y)$ is defined if and only if $x<y$ or $x=y$. Distances between pairs of distinct points in a triple $x,y,z$, whenever these distances are defined, satisfy the so-called \emph{time inequality}, which is a reverse triangle inequality $\rho(x,y)+\rho(y,z)\leq \rho(z,y)$. In the 1960s, Herbert Busemann developed an axiomatic theory of timelike spaces and of locally timelike spaces. His motivation comes from the geometry underlying the theory of relativity, and he tried to adapt to this setting his geometric theory of metric spaces, namely, his theory of $G$-spaces (geodesic spaces). The classical example he considers is the $n$-dimensional Lorentzian space. Two other interesting classes of examples of timelike spaces he introduced are the timelike analogues of the Funk and Hilbert geometries. In this paper, we investigate these two geometries, and in doing this, we introduce variants of them, in particular the timelike relative Funk and Hilbert geometries, in the Euclidean and spherical settings. We describe the Finsler infinitesimal structure of each of these geometries (with an appropriate notion of Finsler structure) and we display the interactions among the Euclidean and timelike spherical geometries. In particular, we characterize the de Sitter geometry as a special case of a timelike spherical Hilbert geometry. The final version of this paper will appear in Differential Geometry and its Applications.