  The phase shift due to the Sagnac Effect, for relativistic matter and
electromagnetic beams, counter-propagating in a rotating interferometer, is
deduced using two different approaches. From one hand, we show that the
relativistic law of velocity addition leads to the well known Sagnac time
difference, which is the same independently of the physical nature of the
interfering beams, evidencing in this way the universality of the effect.
Another derivation is based on a formal analogy with the phase shift induced by
the magnetic potential for charged particles travelling in a region where a
constant vector potential is present: this is the so called Aharonov-Bohm
effect. Both derivations are carried out in a fully relativistic context, using
a suitable 1+3 splitting that allows us to recognize and define the space where
electromagnetic and matter waves propagate: this is an extended 3-space, which
we call "relative space". It is recognized as the only space having an actual
physical meaning from an operational point of view, and it is identified as the
'physical space of the rotating platform': the geometry of this space turns out
to be non Euclidean, according to Einstein's early intuition.
