Timelike Hilbert and Funk geometries
Résumé
A timelike space is a Hausdorff topological space equipped with a partial order relation and a distance function satisfying a set of axioms including certain compatibility conditions between these two objects. The distance function is defined only on a certain subset (whose definition uses the partial order) of the product of the space with itself that contains the diagonal. Distances between triples of points, whenever they are defined, satisfy the so-called \emph{time inequality}, which is a reversed triangle inequality. 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 the classical example he gives is the $n$-dimensional Lorentzian spaces. Two other interesting classes of examples of timelike spaces introduced by Busemann 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, we call timelike relative Hilbert geometries, in the Euclidean and spherical settings. We display new interactions among the Euclidean and spherical timelike geometries. In particular, we characterize the de Sitter geometry as a special case of a timelike spherical Hilbert geometry.
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