Publication Date
11-1-2021
Abstract
In the physical space, we define distance between the two points as the length of the shortest path connecting these points. Similarly, in space-time, for every pair of events for which the event a can causally effect the event b, we can define the longest proper time t(a,b) over all causal trajectories leading from a to b. The resulting function is known as kinematic metric. In practice, our information about all physical quantities -- including time -- comes from measurement, and measurements are never absolutely precise: the measurement result V is, in general, different from the actual (unknown) value v of the corresponding quantity. In many cases, the only information that we have about each measurement error dv = V -- v is the upper bound D on its absolute value. In such cases, once we get the measurement result V, the only information we gain about the actual value v is that v belongs to the interval [V -- D, V + D]. In particular, we get intervals [L(a,b), U(a,b)] containing the actual values of the kinematic metric. Sometimes, we underestimate the measurement errors; in this case, we may not have a kinematic metric contained in the corresponding narrowed intervals -- and this will be an indication of such an underestimation. Thus, it is important to analyze when there exists a kinematic metric contained in all the given intervals. In this paper, we provide a necessary and sufficient condition for the existence of such a kinematic metric. For cases when such a kinematic metric exists, we also provide bounds on its values.
Comments
Technical Report: UTEP-CS-21-100