Relativistic Stereometric Coordinates from Relativistic Localizing Systems and the Projective Geometry of the Spacetime Manifold

TL;DR

This paper introduces relativistic stereometric coordinates derived from a five-satellite system, linking spacetime localization to projective geometry and an advanced parallax concept.

Contribution

It defines a new class of coordinates using five satellites and celestial angles, connecting spacetime localization with conformal and projective geometry.

Findings

Coordinates relate to spacetime's conformal structure

Spacetime has a local projective geometry modeled on real projective space

Localization uses an enhanced space-time parallax concept

Abstract

Relativistic stereometric coordinates supplied by relativistic auto-locating positioning systems made up of four satellites supplemented by a fifth one are defined in addition to the well-known emission and reception coordinates. Such a constellation of five satellites defines a so-called relativistic localizing system. The determination of such systems is motivated by the need to not only locate (within a grid) users utilizing receivers but, more generally, to localize any spacetime event. The angles measured on the celestial spheres of the five satellites enter into the definition. Therefore, there are, up to scalings, intrinsic physical coordinates related to the underlying conformal structure of spacetime. Moreover, they indicate that spacetime must be endowed everywhere with a local projective geometry characteristic of a so-called generalized Cartan space locally modeled on four-dimensional, real projective space. The particular process of localization providing the relativistic stereometric coordinates is based, in a way, on an enhanced notion of parallax in space and time generalizing the usual parallax restricted to space only.