Numerical modeling of a Global Navigation Satellite System in a general relativistic framework

TL;DR

This paper models a GNSS within a Schwarzschild space-time framework, deriving satellite and signal trajectories analytically, and proposes a relativistic approach that could improve positioning accuracy without traditional corrections.

Contribution

It introduces an analytical relativistic model of GNSS in Schwarzschild space-time, enabling fast computation and potential improvements over existing post-Newtonian methods.

Findings

Analytical solutions for satellite and signal geodesics in Schwarzschild space-time.

Fast algorithm for computing GNSS user coordinates based on relativistic modeling.

Potential for enhanced stability and accuracy through inter-satellite links.

Abstract

In this article we model a Global Navigation Satellite System (GNSS) in a Schwarzschild space-time, as a first approximation of the relativistic geometry around the Earth. The closed time-like and scattering light-like geodesics are obtained analytically, describing respectively trajectories of satellites and electromagnetic signals. We implement an algorithm to calculate Schwarzschild coordinates of a GNSS user who receives proper times sent by four satellites, knowing their orbital parameters; the inverse procedure is implemented to check for consistency. The constellation of satellites therefore realizes a geocentric inertial reference system with no \emph{a priori} realization of a terrestrial reference frame. We show that the calculation is very fast and could be implemented in a real GNSS, as an alternative to usual post-Newtonian corrections. Effects of non-gravitational perturbations on positioning errors are assessed, and methods to reduce them are sketched. In particular, inter-links between satellites could greatly enhance stability and accuracy of the positioning system.