Einstein's
clock synchronization method is based on the principle that,
in every inertial frame, the
oneway speed of light in vacuum is the same as its average roundtrip speed.^{(a)}
In synchronization terms, it means the
following. Let clocks A and B be at rest in an inertial frame; measure the coordinate distance d between two clocks; send a timestamped light pulse from A to B; at the instant of reception, add the propagation delay (d/c) to the time stamp and set clock B accordingly. Clocks A and B are now 'Einsteinsynchronized' in that inertial frame.
The
relativistic principle and hence Einstein's method of clock synchronization
is a convention, not an absolute truth.^{(b)} This may sound like relativistic heresy, but there are other
valid ways of synchronizing clocks  it is just that none works as well as
Einstein's.^{(c)} This Blog article attempts to show why by means of
the Sagnac effect.
The
figure (right)^{(d)} is largely selfexplanatory. The two red arrows
represent two signals moving in opposite directions at speed k relative
to the blue ring. If the ring is nonrotating, each signal will circumnavigate the ring in time t = 2Πr/k, as measured by a laboratory clock.
In Newtonian mechanics, this is also true if the ring is rotating, because in
the lab frame, the difference in the
distances that the signals have to travel in the two directions exactly cancels
the apparent faster or slower speed of the signals (u = v ± k), depending on signal direction. Signals traveling in opposite
directions arrive at the same time and remain in phase; hence there should be
no Sagnac effect. This Newtonian result is in conflict
with observations of the Sagnac effect.
In
Einstein dynamics, the principles are the same, accept that we cannot
simply add the speed of the signal to the speed of the ring, i.e. u = v ± k does not work. We must use the relativistic speed addition equation u = (v±k)/(1±kv/c^{2}). This gives a definite difference in the time for signals circumnavigating the rotating ring in opposite directions: Δt = 4Πrν/(c^{2}v^{2}), as measured in the inertial laboratory frame.^{(d,e)}
This
time difference is independent of the speed of the signals (k) and just depends on the
radius and the rotation rate of the ring. Hence, in contrast to
classical Galileo/Newton theory, Einstein's theory does predict the Sagnac
effect and it agrees with all experiments performed so far. The Sagnac effect is a very simple proof that the Einstein clock synchronization method works. But, why is it the best scheme?
The answer is simple: by convention, it forces light to propagate at the same speed in both directions around the ring. All other clock synchronization schemes imply that light moves at different speeds in the two directions. As we have seen, in Newton mechanics this results in zero Sagnac effect. In others, having time dilation and Lorent contraction, like Lorentz ether theory (LET), it results in horribly complex equations for the observed Sagnac effect.^{(f)}
Einstein clocks are cool...
J
PS: see reply #29 below for a summary of the topic.
Notes:
(a)
More precisely stated: the observed oneway speed of light
in vacuum is constant and isotropic in every inertial frame.
(b) A very good
discussion of the conventionality of relativity can be found in this Wiki and
also in this
Blog entry.
(c) One alternative clock synchronization is due to Selleri, which has
preferred inertial frame, but with time dilation and Lorentz contraction
in any other frame. It is discussed in "The relativistic Sagnac Effect: two derivations", section 3.5. The paper contains a complete mathematical treatment.
(d)
I borrowed the graphics from a physicsinsights.org article, which is excellent for a introductory discussion of the Sagnac effect and shows the calculations involved. It is a lot more accessible than "The relativistic Sagnac Effect: two derivations" in (b) above.
(e) Because v^{2 }« c^{2} in the usual Sagnac interferometers, v^{2 }is normally ignored and just Δt = 4Πrν/c^{2} is used. This is
precisely double the Einstein clock synchronization offset between the
transmitter and the receiver for each direction.
(f) For an example, see eq. (23) of "The relativistic Sagnac Effect", referenced in endnote (c).
Note that the equations given there are for time as measured by a ring clock and
not a lab clock, as used above, but there is only a factor γ difference.

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