Generalized Radar 4-Coordinates and Equal-Time Cauchy Surfaces for Arbitrary Accelerated Observers

TL;DR

This paper develops a Hamiltonian framework to define globally valid radar coordinate systems and equal-time surfaces for arbitrary accelerated observers, extending the notion of clock synchronization beyond local approximations in relativity.

Contribution

It introduces a method to construct globally defined, observer-dependent 4-coordinate systems using 3+1 splittings, generalizing Einstein synchronization and linking it to gauge equivalence in relativistic theories.

Findings

Hamiltonian methods enable global radar coordinate systems.

Generalized synchronization conventions are gauge equivalent.

Admissible 3+1 splittings correspond to non-inertial frames.

Abstract

All existing 4-coordinate systems centered on the world-line of an accelerated observer are only locally defined like it happens for Fermi coordinates both in special and general relativity. As a consequence, it is not known how non-inertial observers can build {\it equal-time surfaces} which a) correspond to a conventional observer-dependent definition of synchronization of distant clocks; b) are good Cauchy surfaces for Maxwell equations. Another type of coordinate singularities are those connected to the relativistic rotating coordinate systems (the rotating disk).We show that the use of Hamiltonian methods based on 3+1 splittings of space-time allows to define as many observer-dependent globally defined radar 4-coordinate systems as nice foliations of space-time with space-like hyper-surfaces admissible according to M$\o$ller (for instance only differentially rotating relativistic coordinate system are allowed). All these conventional notions of an {\it instantaneous 3-space} for an arbitrary observer can be empirically defined by introducing generalizations of Einstein ${1\over 2}$ convention for clock synchronization in inertial frames. Each admissible 3+1 splitting corresponds to a non-rigid non-inertial frame centered on the observer. When there is a Lagrangian description of an isolated relativistic system, its reformulation as a parametrized Minkowski theory allows to show that all the admissible synchronization conventions are {\it gauge equivalent}, as it also happens in canonical metric and tetrad gravity, where, however, the chrono-geometrical structure of space-time is dynamically determined.