Chambers, Robert (UK) wrote:
(The "clocks hypothesis" has also been
experimentally
verified, by the way.)
Slight glitch in the previous email - sorry!
To continue...
I found this on the clock hypothesis:
http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#5.%20Twin%20paradox
"The clock hypothesis states that the tick rate of a clock when measured in an
inertial frame depends only upon its velocity relative to that frame, and is independent
of its acceleration or higher derivatives."
Sounds right to me.
Are we not dealing with an accelerated frame, rather than an inertial one, in
the Sagnac effect?
Not necessarily. And here's the point of confusion, I think.
A "frame of reference" corresponds to a particular choice of coordinate
systems. In special relativity, we're constrained to use only "flat"
coordinates: orthonormal coordinate systems in which the metric is the
usual one, where lengths are measured as we expect, the "proper length"
of an object which is stationary relative to those coordinates is just
what we get using a ruler, time is measured by stationary clocks, all
stationary clocks can be synchronized, and the speed of light in vacuum
is exactly c.
However, we can examine objects which are moving or even accelerating
relative to the coordinate system we have chosen. That's perfectly
possible in special relativity. The Sagnac effect can be predicted
perfectly well from the "stationary" frame of reference of the
laboratory, in which we examine the fiber optic ring, which is an
accelerating object. Since we know the velocity of each point on the
ring at each moment, we can determine gamma for that location, we can
find dt/d(tau), and we can predict how fast time will pass for someone
riding on the ring. And we can use the composition of velocities
formula to determine how fast a signal will travel in the lab frame, if
we know how fast an observer on the ring sees it go as it passes through
an infinitesimal neighborhood around that observer.
If you want to run the analysis in the frame of reference of the
rotating disk, however, then you have a problem. There is no
"Lorentzian" coordinate system in in which the disk is stationary; the
disk's frame of reference is not inertial. So, you can't do that in
special relativity; you need to go the general route.
However, what we _can_ do is look at an inertial frame which happens to
be "momentarily co-moving" with an observer who is riding on the ring.
That frame of reference is Lorentzian, and it can be treated with the
techniques of SR. And for one instant in time, in the neighborhood of
the observer on the ring, that frame of reference will behave
identically with what the observer on the ring sees (except for the
acceleration, which doesn't affect clocks or small distances). Using
that approach, we can determine things like how fast proper time elapses
for the signal as it traverses the ring, without the necessity of
integrating d(tau)/dt in the laboratory frame.
In conclusion, one can answer many interesting questions about
accelerating bodies without ever using anything except "flat-space"
coordinates. What you can't do is produce a single complete solution in
the frame of reference of an accelerating observer, because that
requires non-flat coordinates. You also can't handle any problem that
involves gravity this way, because in the presence of gravity there is
no global Lorentz frame of reference.
As a general rule, if a problem can be handled with flat coordinates,
it's easier to understand the result if one does it that way. Curved
coordinates invariably introduce a lot of extra hair. One of the
primary rules of general relativity is "All coordinate systems are
equally valid", but a second rule is "Some coordinate systems are a
whole lot easier to work with than others."
