Simultaneity, radar 4-coordinates and the 3+1 point of view about accelerated observers in special relativity

TL;DR

This paper develops a 3+1 Hamiltonian framework for accelerated observers in special relativity, providing a global notion of simultaneity, analyzing rotating frames, and applying the concepts to practical systems like GPS and spacecraft synchronization.

Contribution

It introduces a new 3+1 splitting approach with non-orthogonal simultaneity surfaces, extending M"oller’s transformations, to better understand non-inertial frames and simultaneity in special relativity.

Findings

Explicit foliations for non-inertial observers are constructed.

A good notion of simultaneity is achieved for certain accelerated motions.

Application to GPS and spacecraft time transfer demonstrates practical relevance.

Abstract

After a review of the 1+3 point of view on non-inertial observers and of the problems of rotating reference frames, we underline that was is lacking in their treatment is a good global notion of simultaneity due to the restricted validity of the existing 4-coordinates associated to an accelerated observer (like the Fermi normal ones). We show that the relativistic Hamiltonian 3+1 point of view, based on a 3+1 splitting of Minkowski space-time with a foliation whose space-like leaves are both simultaneity and Cauchy surfaces, allows to find a solution to such problems, if we take into account M$\o$ller's definition of allowed 4-coordinate transformations extended to radar 4-coordinates. Rigidly rotating relativistic reference frames are shown not to exist. We give explicit foliations, with simultaneity surfaces (also space-like hyper-planes) non orthogonal to the non-inertial observer world-line, which correspond to a good notion of simultaneity for suitable (mutually balancing) translational and rotational accelerations. This allows to evaluate the simultaneity-dependent one-way speed of light and to give the 3+1 description of both the rotating disk and the Sagnac effect. It is also shown how a GPS system of spacecrafts may establish a grid of admissible radar 4-coordinates and how the ACES mission can employ a variant of this method adapted to Earth's rotation in the evaluation of the one-way time transfer to detect the deviation from Einstein convention on the synchronization of inertial clocks implied by such a notion of simultaneity. We show that in parametrized Minkowski theories all the admissible notions of simultaneity are gauge equivalent ({\it conventionality of simultaneity}) and, as an example, we describe Maxwell theory in non-inertial systems with any admissible notion of simultaneity.