GENERALIZED RADAR 4-COORDINATES AND EQUAL-TIME CAUCHY SURFACES FOR ARBITRARY ACCELERATED OBSERVERS

Abstract

All existing 4-coordinate systems centered on the world-line of an accelerated observer are only locally defined, as for Fermi coordinates both in special and general relativity. As a consequence, it is not known how non-inertial observers can build equal-time surfaces which (a) correspond to a conventional observer-dependent definition of synchronization of distant clocks, and (b) are good Cauchy surfaces for Maxwell equations. Another type of coordinate singularities generating the same problems are those connected to the relativistic rotating coordinate systems used in the treatment of the rotating disk and the Sagnac effect. We show that the use of Hamiltonian methods based on 3+1 splittings of space–time allows one 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 Møller (for instance, only the differentially rotating relativistic coordinate system, but not the rigidly rotating ones of non-relativistic physics, are allowed). All these conventional notions of an instantaneous 3-space for an arbitrary observer can be empirically defined by introducing generalizations of the Einstein ½ convention for clock synchronization in inertial frames. Each admissible 3+1 splitting has two naturally associated congruences of time-like observers: as a consequence every 3+1 splitting gives rise to non-rigid non-inertial frames centered on any one of these observers. Only for Eulerian observers are the simultaneity leaves orthogonal to the observer world-line. When there is a Lagrangian description of an isolated relativistic system, its reformulation as a parametrized Minkowski theory allows one to show that all the admissible synchronization conventions are gauge equivalent, as also happens in the canonical metric and tetrad gravity, where, however, the chrono-geometrical structure of space–time is dynamically determined. The framework developed in this paper is not only useful for a consistent description of the rotating disk, but is also needed for the interpretation of the future ACES experiment on the synchronization of laser-cooled atomic clocks and for the synchronization of the clocks on the three LISA spacecrafts.