The Chrono-geometrical Structure of General Relativity and Clock Synchronization
Luca, L.
INFN

While Newton physics has absolute non-dynamical notions of time and of instantaneous Euclidean 3-space (with the associated spatial distance), special relativity has only the absolute notion of Minkowski space-time with Lorentz signature. A clock synchronization convention has to be stipulated by a time-like observer to define instantaneous (in general Riemannian) 3-spaces. This requires a (M$\o$ller admissible) 3+1 splitting of Minkowski space-time and the definition of an either inertial or non-inertial frame centered on the observer. Einstein's midpoint synchronization convention identifies the inertial frames centered on inertial observer: only in this case the 1-way velocity of light coincides with the 2-way one appearing in the light postulates. In this way the either Fermi or rotating coordinate singularities of the 1+3 point of view are avoided and it can be shown that relativistic rigid rotations are not allowed. The natural parametrization of non-inertial frames is obtained by using radar 4-coordinates.

By means of parametrized Minkowski theories isolated systems can be described in such a way that a change of synchronization convention is a gauge transformation due to the presence of a special relativistic form of general covariance. To fix the gauge means to identify the description of the system in the associated non-inertial frame with the associated pattern of relativistic inertial forces. The gauge freedom in the choice of the instantaneous 3-space and of their 3-coordinates induces the spatio-temporal appearances of the phenomena in non-inertial frames, i.e. in extended physical laboratories whose metrology is compatible with the chosen notion of simultaneity (think to the description of the Earth given by GPS: it is a metrological standard of space-time around the Earth ).

The inertial frame having the simultaneity hyper-planes (named Wigner hyper-planes) orthogonal to the conserved 4-momentum of the isolated system allows to get the Wigner-covariant rest-frame instant form of dynamics and consistent theory of relativistic orbits. Instead non-inertial rest frames have the associated simultaneity 3-surfaces tending to Wigner hyper-planes at spatial infinity.

In general relativity the full chrono-geometrical structure of the space-time is dynamical. Global inertial frames are not admitted due to the equivalence principle, which replaces the relativity principle. Einstein's geometrical view of the gravitational field implies that the 4-metric of space-time, the dynamical field mediating the gravitational interaction, also determines the line element of the space-time, teaching relativistic causality to all other fields. In the ADM canonical formulation of gravity the allowed space-times must be globally hyperbolic, the gauge equivalence of the allowed non-inertial frames (i.e. of the clock synchronization conventions) is implied by general covariance and asymptotically flatness is needed to have the asymptotic ADM Poincare' generators and a real temporal evolution (no frozen picture) governed by the ADM energy (rest-frame instant form of metric gravity. Dirac's theory of constraints allows to separate the two pairs of canonical variables describing the {\it tidal effects} (the gravitational waves of the linearized theory) from the gauge variables associated with the {\it relativistic generalized inertial effects} off-shell, i.e. {\it before} solving Hamilton equations. Only after a complete gauge fixing, namely after the choice of a kinematical non-inertial frame with its pattern of relativistic inertial forces, the Hamilton equations for the tidal degrees of freedom and matter (if present) become deterministic. Their solution with suitable Cauchy data on a Cauchy surface (a leaf of the non-inertial frame) identifies a unique Einstein space-time with its family of dynamical (and on-shell gauge equivalent) non-inertial frames. As a consequence, both the admissible clock synchronization conventions, i.e. the admissible instantaneous 3-spaces, emerge dynamically.

The general relativistic inertial effects (the gauge variables) are associated with the lapse (local unit of proper time) and shift (gravito-magnetism) functions, with the choice of 3-coordinates on the simultaneity surfaces (the analogue of the special relativistic inertial effects) and with the choice of the synchronization convention (the momentum conjugate to the conformal factor of the 3-metric is the gauge variable determining the instantaneous 3-space, i.e. the shape of the simultaneity surface).

A review is given of the application of this framework to the PN metrics of the IAU conventions and to the ESA ACES mission for the synchronization of high-precision laser-cooled atomic clock: new metrological protocols are needed for time dissemination at the 1/c3 level of precision.