On Newtonian frames

TL;DR

This paper classifies Newtonian frames and coordinate systems into four causal classes, contrasting them with the 199 Lorentzian classes, and explores their geometric and physical interpretations, including positioning systems without synchronization.

Contribution

It extends the classification of frames to coordinate systems in Newtonian space-time and compares these with relativistic classes, highlighting new geometric and physical insights.

Findings

Four causal classes of frames in Newtonian space-time identified

Coordinate systems can be classified into four causal classes

Positioning systems can locate events without synchronization

Abstract

In Newtonian space-time there exist four, and only four, causal classes of frames. Natural frames allow to extend this result to coordinate systems, so that coordinate systems may be also locally classified in four causal classes. These causal classes admit simple geometric descriptions and physical interpretations. For example, one can generate representatives of the four causal classes by means of the {\em linear synchronization group}. Of particular interest is the {\em local Solar time synchronization}, which reveals the limits of the frequent use of the concept of `causally oriented oordinate', such as that of `time-like coordinate'. Classical {\em positioning systems}, based in sound or light signals, are, by themselves, interesting examples of location systems, i.e. of physically constructible coordinate systems. They show that one can locate events in Newtonian space-time {\em without} any use of the concept of synchronization. In fact, the coordinate systems associated to positioning systems, belong to all the classes but the standard one, i.e. the one based in the simultaneity synchronization. The relativistic analogs of these examples, emphasize the contrast between the four Newtonian and the one hundred and ninety nine Lorentzian causal classes of frames of classical and relativistic space-times, respectively.