Timelike Hilbert and Funk geometries

Athanase Papadopoulos (1) , Sumio Yamada (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Department of Mathematics [Gakushuin]

A timelike metric 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 containing the diagonal. Distances between triples of points, whenever they are defined, satisfy the so-called time inequality, which is a reversed triangle inequality. In the 1960s, Herbert Busemann developed an axiomatic theory of timelike metric spaces and of locally timelike metric 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 metric spaces introduced by Busemann are the timelike analogues of the Funk and Hilbert geometries. In this paper, we investigate these geome-tries. We shall find new interactions among the Euclidean, affine, projective and spherical timelike geometries. In particular, the de Sitter metric is described as a special case of a timelike spherical Hilbert metric.

Mots clés: Timelike space; timelike Hilbert geometry; timelike Funk geometry; time inequality; convexity; metric geometry; Busemann geometry; Lorentzian geometry; light cone; relativity.

Domaines: Topologie géométrique [math.GT]; Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]

Athanase Papadopoulos, Sumio Yamada. Timelike Hilbert and Funk geometries. 2016. ⟨hal-01276762v2⟩