  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.
