Relativistic Localizing Processes Bespeak an Inevitable Projective Geometry of Spacetime

TL;DR

This paper introduces relativistic localizing systems that reveal a four-dimensional projective structure of spacetime, enhancing localization accuracy and offering new insights into the geometry of spacetime in relativity.

Contribution

It proposes a novel relativistic localizing framework that uncovers an inherent four-dimensional projective geometry of spacetime beyond the traditional Riemannian view.

Findings

Spacetime possesses an unexpected local four-dimensional projective structure.

The proposed systems enable precise localization of events and users in spacetime.

Spacetime can be modeled as a generalized Cartan space with both projective and Riemannian structures.

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 to locate users endowed with receivers and, in addition, to localize any spacetime event. These localizing systems are made up of relativistic auto-locating positioning sub-systems 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, i.e., a spacetime with both a local four dimensional projective structure and a compatible (pseudo-)Riemannian structure. Localization protocols are presented in details, while possible applications to astrophysics are also considered.