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

David Alba, Luca Lusanna|ArXiv.org|Nov 18, 2003

Geophysics and Sensor Technology133 references20 citations

TL;DR

This paper proposes a 3+1 Hamiltonian framework in special relativity that establishes a global, physically consistent notion of simultaneity for accelerated observers using radar 4-coordinates. By extending Møller’s allowed 4-coordinate transformations to radar-based coordinates, it resolves the limitations of traditional Fermi normal coordinates and shows that rigid rotating reference frames do not exist, while enabling empirical synchronization via GPS-like systems and modeling relativistic effects like the Sagnac and Shapiro delays.

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$ø$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.

Motivation & Objective

• To address the lack of a global, consistent notion of simultaneity for accelerated observers in special relativity, particularly in rotating frames.
• To overcome the coordinate singularities and limited validity of existing 4-coordinate systems like Fermi normal coordinates.
• To provide a 3+1 splitting of Minkowski spacetime with simultaneity and Cauchy surfaces, enabling a physically meaningful foliation for non-inertial observers.
• To demonstrate that rigid rotating relativistic reference frames are incompatible with special relativity under the proposed framework.
• To enable empirical simultaneity through radar 4-coordinates, as realized in GPS-like systems, and to model post-Newtonian effects like Shapiro delay.

Proposed method

• Adopt a 3+1 Hamiltonian point of view with a foliation of Minkowski spacetime into spacelike hypersurfaces that are both simultaneity and Cauchy surfaces.
• Use radar 4-coordinates derived from light travel time to define a global coordinate system adapted to an arbitrary accelerated worldline.
• Extend Møller’s definition of allowed 4-coordinate transformations to radar coordinates, ensuring compatibility with the relativistic Hamiltonian formalism.
• Construct explicit foliations with simultaneity surfaces non-orthogonal to the worldline, valid for combined translational and rotational accelerations.
• Derive the one-way speed of light and spatial distance as functions of the chosen simultaneity convention, showing their dependence on the observer’s motion.
• Apply the formalism to model the rotating disk 3-geometry and the Sagnac effect, and to compute time delays (including Shapiro delay) between Earth stations and satellites.

Experimental results

Research questions

• RQ1 Can a global, physically consistent notion of simultaneity be defined for accelerated observers in special relativity, avoiding the singularities of traditional coordinate systems?
• RQ2 How does the choice of simultaneity convention affect the 3-geometry of a rotating disk and the one-way speed of light?
• RQ3 To what extent can radar 4-coordinates provide an empirical, operational definition of simultaneity in non-inertial frames?
• RQ4 What is the impact of non-Einstein simultaneity conventions on time delay measurements in Earth-satellite systems, such as those in the ACES mission?
• RQ5 Can the framework be extended to canonical metric gravity, where simultaneity is dynamically determined by the ADM equations?

Key findings

• Rigid rotating relativistic reference frames do not exist in special relativity under the proposed 3+1 Hamiltonian framework.
• The 3+1 foliation with simultaneity and Cauchy surfaces allows a global, non-singular definition of radar 4-coordinates for arbitrary accelerated observers.
• The one-way speed of light and spatial distance are no longer invariant but depend on the chosen simultaneity convention, which is physically meaningful and empirically measurable.
• Explicit foliations are constructed for observers with combined translational and rotational accelerations, showing that simultaneity surfaces are not orthogonal to the worldline.
• The Sagnac effect and the rotating disk’s 3-geometry are consistently described as dependent on the choice of simultaneity, resolving long-standing ambiguities.
• The framework enables the empirical determination of time delays (including Shapiro delay) between Earth and satellite systems without assuming Einstein’s convention, offering a testable alternative for missions like ACES.