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When is the Busemann product a lattice? A relation between metric spaces and corresponding space-time models. (English) Zbl 1288.06017

Summary: 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 \(\geq t + \frac{d(x, y)}{c}\). 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 \(s \geq t + d(x, y)\). 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 \(X\). From the mathematical viewpoint, 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).

MSC:

06B99 Lattices
54E35 Metric spaces, metrizability
83A05 Special relativity
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