When is the Busemann Product a Lattice? A Relation between Metric Spaces and Corresponding Space-Time Models
Hans-Peter A. Künzi, Francisco Zapata, Владик Крейнович · scholarworks - UTEP (The University of Texas at El Paso) · 2011
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).