Relativistic Localizing Processes Bespeak an Inevitable Projective Geometry of Spacetime

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.