Two null gravitational cones in the theory of GPS-intersatellite communications between two moving satellites. I. Physical and mathematical theory of the space-time interval and the geodesic distance on intersecting null cones

TL;DR

This paper develops a relativistic theoretical framework for intersatellite communication between moving satellites, introducing null cones to accurately determine space-time intervals and geodesic distances without relying on traditional delay formulas.

Contribution

It introduces a novel null cone approach to model intersatellite signals in General Relativity, providing a rigorous mathematical basis for calculating distances and intervals in satellite navigation.

Findings

Space-time interval can be zero, negative, or positive.

Geodesic distance is greater than Euclidean distance, proven without Shapiro delay.

Compatibility condition is essential for accurate intersatellite measurements.

Abstract

Several space missions such as GRACE, GRAIL, ACES and others rely on intersatellite communications (ISC) between two satellites at a large distance one from another. The main goal of the theory is to formulate all the navigation observables within the General Relativity Theory. The same approach should be applied also to the intersatellite GPS-communications (in perspective also between the GPS, GLONASS and Galileo satellite constellations). In this paper a theoretical approach has been developed for ISC between two satellites moving on (one-plane) elliptical orbits, based on the introduction of two gravity null cones with origins at the emitting-signal and receiving-signal satellites. The two null cones account for the variable distance between the satellites during their uncorrelated motion.The intersection of the two null cones defines a distance, which can be found from a differential equation in full derivatives. This distance is the space-time interval in GRT. Applying some theorems from higher algebra, it was proved that this space-time distance can become zero, consequently it can be also negative and positive. But in order to represent the geodesic distance travelled by the signal, the space-time interval has to be "compatible" with the Euclidean distance. So this "compatibility condition", conditionally called "condition for ISC", is the most important consequence of the theory. The other important consequence is that the geodesic distance turns out to be the space-time interval, but with account also of the "condition for ISC". This interpretation enables the strict mathematical proof that the geodesic distance is greater than the Euclidean distance - a result, entirely based on the "two null cones approach" and moreover, without any use of the Shapiro delay formulae. Consistency between several other newly derived numerical parameters is noted.