Recovery of the Configuration of a Set of Space-time Events from Their Mutual Minkowski Distances
نویسندگان
چکیده
The recovery of a set of points in an N -dimensional vector space from their set of mutual distances has a long history. In 1938 Young and Householder gave necessary and sufficient conditions for a set of numbers to be the set of mutual distances between a collection of real points in Euclidean space and gave methods for determining the configuration of these points up to translation and rotation. In this paper we generalise these results to pseudo-Euclidean spaces with indefinite metrics, such as Minkowski space-time. We give necessary and sufficient conditions for a matrix to be the pseudo-Gram matrix of a set of vectors in a space with indefinite metric. We give necessary and sufficient conditions for a set of mutual squared distances to be consistent a set of points laying in a particular pseudo-Euclidean space and discuss the extent to which the configuration of a set of points can be recovered from their mutual distances. We use our methods to develop fast tests for the common causality of a set of time-of-arrival events from mutual Minkowski distances
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