From (Idealized) Exact Causality-Preserving Transformations to Practically Useful Approximately-Preserving Ones: A General Approach

It is known that every causality-preserving transformation of Minkowski space-time is a composition of Lorentz transformations, shifts, rotations, and dilations. In principle, this result means that by only knowing the causality relation, we can determine the coordinate and metric structure on the space-time. However, strictly speaking, the theorem only says that this reconstruction is possible if we know the exact causality relation. In practice, measurements are never 100% accurate. It is therefore desirable to prove that if a transformation approximately preserves causality, then it is approximately equal to an above-described composition. Such a result was indeed proven, but only for a very particular case of approximate preservation. In this paper, we prove that simple compactness-related ideas can lead to a transformation of the exact causality-preserving result into an approximately-preserving one. Causality-preserving mappings: formulation of the general problem. One of the fundamental notions of physics is the notion of causality, the description of which events can causally influence others. In particular, the Minkowski space-time of special relativity is an (n + 1)-dimensional space-time E = R, in which the causality relation a ≤ b between events a = (a0, a1, . . . , an) ∈ E and b = (b0, b1, . . . , bn) ∈ E is described by the formula a ≤ b ↔ a = b ∨ (b0 ≥ a0 & (b − a) 2 ≥ 0), where a def = a20 − a 2 1 − . . . − a 2 n. It is known that for every n ≥ 2, every bijection E → E which preserves the Minkowski causality relation is linear (moreover, it is a composition of Lorentz transformations, shirts, rotations, and dilations). This theorem was first proven by A. D. Alexandrov [1, 5]; see also [2, 3, 7, 8, 16, 22, 26, 27, 29, 30, 31, 32, 33, 35].