When Is Busemann Product a Lattice? A Relation Between Metric Spaces and Corresponding Space-Time Models

The causality relation of special relativity is based on the assumption that the speed of all physical processes is limited by the speed of light. As a result, an event (t, x) occurring at moment t at location x can influence an event (y, s) if and only if s ≥ t+ d(x, y) c . We can simplify this formula if we use units of time and distance in which c = 1 (e.g., by using a light second as a unit of distance). In this case, the above causality relation takes the form s ≥ t+d(x, y). Since the actual space can be non-Euclidean, H. Busemann generalized this ordering relation to the case when points x, y, etc. are taken from an arbitrary metric space X. A natural question is: when is the resulting ordered space – called a Busemann product – a lattice? In this paper, we provide a necessary and sufficient condition for it being a lattice: it is a lattice if and only if X is a real tree, i.e., a metric space in which every two points are connected by exactly one arc, and this arc is geodesic (i.e., metrically isomorphic to an interval on a real line). 1 Formulation of the Problem Special relativity: brief reminder. To uniquely describe an event, we need to describe the moment of time t at which it occurs and its spatial location x. In other words, an event can be characterized by a pair (t, x), where t ∈ IR is