Relativistic Localizing Processes Bespeak an Inevitable Projective Geometry of Spacetime

Jacques L Rubin

doi:10.1155/2017/9672417

Abstract

Surprisingly, the issue of events localization in spacetime is poorly understood and a fortiori realized even in the context of Einstein’s relativity. Accordingly, a comparison between observational data and theoretical expectations might then be strongly compromised. In the present paper, we give the principles of relativistic localizing systems so as to bypass this issue. Such systems will allow locating users endowed with receivers and, in addition, localizing any spacetime event. These localizing systems are made up of relativistic autolocating positioning subsystems supplemented by an extra satellite. They indicate that spacetime must be supplied everywhere with an unexpected local four-dimensional projective structure besides the well-known three-dimensional relativistic projective one. As a result, the spacetime manifold can be seen as a generalized Cartan space modeled on a four-dimensional real projective space, that is, a spacetime with both a local four-dimensional projective structure and a compatible (pseudo-)Riemannian structure. Localization protocols are presented in detail, while possible applications to astrophysics are also considered.

Highlights

The general principles of the relativistic localizing systems have been defined in a previous paper [1] with just a few details on the projective underlying structure provided by these localizing systems
The four emitters broadcast “emission coordinates” which are no more and no less than time stamps generated by onboard clocks and encoded within EM signals propagating in spacetime
In relativistic positioning systems, the emitters broadcast their own time stamps, and the time stamps they receive from the others

Summary

Introduction

The general principles of the relativistic localizing systems have been defined in a previous paper [1] with just a few details on the projective underlying structure provided by these localizing systems The latter are based on the so-called relativistic positioning systems [2,3,4,5,6,7,8,9]. In relativistic positioning systems, the emitters broadcast their own time stamps, and the time stamps they receive from the others This process of echoes undergone by the time stamps enables users to construct the fourdimensional emission grid because they can deduce the spacetime positions of the four emitters. Beforehand, to proceed in the difficult and delicate description of the relativistic localizing systems, we first need to define as clearly as possible the terminology and the different conventions and notations

Notations and Conventions

Consistency between the Positioning and Localizing Protocols: Identification

The Consistency between the Positioning and Localization Protocols

The Local Projective Structure

Conclusion