Re: Variable speed of light

[John Mallinckrodt <ajm@CSUPOMONA.EDU>]

> Just a small remark (later I hope to be able to write more). In textbooks, one
> finds essentially the following statement: "While  clocks in the rest frame
> tick 5 times, clocks on the (very fast) train tick only once". Doesn't this
> mean that t' < t? If it doesn't, textbooks appear to be fatally confused.

Not in general because different clocks are used and they may be out
of sync due to the relativity of simultaneity.

In your case, BOTH sets of observers will agree that the two clocks
being used in the STATION frame are in sync because their separation
is perpendicular to the relative motion.  Since there is no
complication from the relativity of simultaneity, the train
observer's observation that the "moving" clocks in the station "run
slow" is all we need to conclude that t < t'.

But what about the clocks in the TRAIN frame?

The train observers will, of course, maintain that those clocks are
in sync, so they obtain t' as the difference between the readings of
those clocks.

The station observers, however, find that the train clock used to
time the ENTRANCE of the beam is out of sync with and LAGS the clock
used to time the EXIT of the beam.  This is because of the relativity
of simultaneity and the fact that the ENTRANCE clock is "in front of"
the EXIT clock in the direction of their mutual motion (according,
once again, to the station observers.)  Thus, the station observers
have to contend with two effects:  1) The train clocks run slow,
which would tend to make t' < t, and 2) the ENTRANCE clock lags the
EXIT clock, which would tend to make t' > t.  If you work out the
details, you will find that the second effect is larger than the
first effect so the station observers also conclude, properly and
consistently, that t < t'.

--
John Mallinckrodt        mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm