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Re: Variable speed of light (was: Relativity conundrum)



Ken,

The essential problem is Einstein's premise

x' = 0 <-> x = vt

and its implications.

Pentcho

Ken Caviness wrote:

I haven't had much time for this forum recently (making up and grading
finals), but I'm taking a half-hour breather now, so here goes!

Ken Caviness wrote:

1. For events located at the same point in the train frame (ticks of a
train clock, for instance), the time interval delta t' (as measured by
the train observer) is less than the time interval delta t (the temporal
separation of the same events measured in the station frame).

Pentcho Valev responded:

Perhaps "events located at the same point in the train frame (ticks of a
train clock, for instance)" is not a well defined concept. There are processes in the
clock developing in different directions and Lorentz equations predict different time
changes for them. The problem can be resolved by assuming, in accordance with alledged
experimental evidence, that ALL processes in the train frame obey t' = (1 / gamma)*t.

No, I can't accept that generalization. There is automatically a
distinction that must be made between events that occur at the same
place (for some observer) and events that are separated in space
(according to the same observer) since in the Lorentz transformation
both the x- and t-coordinates depend on both the x'- and t'-coordinates
(and vice versa). Simply stated, an observer for whom the events are
coincident will measure the minimum temporal separation between the
events, others measure longer times. Another way to say this is that
the formula delta-t' = (1 / gamma) * delta-t can only be correctly
applied to pairs of events where delta-x' = 0. On the other hand, if
delta-x = 0, we have the delta-t = (1 / gamma) * delta-t'.

Surely we can agree on (1) whether two events occur at the same position
(x' coordinate) according to the train observer, (2) whether two events
occur at the same position (x coordinate) according to the station
platform observer. (Since the train is moving with respect to the
station platform, the answers to the above questions cannot both be
"yes".) I mentioned ticks of a clock (which is stationary with respect
to some reference frame) as an easy example of a process consisting of
events which one observer will see as all occuring at the same spatial
location, while other observers (moving with respect to the first
observer) will not.

A look at the Lorentz transformations shows that the disagreement
(between relatively moving observers) on lengths and time intervals is
because the spatial and temporal intervals aren't invariant, but are
merely components of the 4-d spacetime interval, the length of which
_is_ invariant. Thus if a clock is stationary with respect to the
train, according to the train observer the interval between two ticks is
entirely a time, there was no change in spatial position. But according
to the platform observer the events occurred at two different
positions. A facile explanation is that a portion of the time was
converted into space. Actually this doesn't work like the trade-off of
spatial components under a rotation (more x means less y, using a
rotation matrix to find the new components) but as a rotation through an
imaginary angle (or using hyperbolics instead of trig functions in the
rotation matrix). So the station observer sees the events as being
separated by a non-zero distance AND a longer time interval: she sees
the moving clock as running slow. The inverse situation applies to a
clock stationary on the station platform: the moving observer sees it
as running slow.

This is why I thought it important to be clear in the proposed thought
experiment about which observer sees the beam of light entering and
leaving the train at the same x-coordinate. If I now understand
correctly, this is the platform observer. She sees the light beam
travel a short distance (the width of the train, w) in a short time
(delta-t = w/c). The observer on the train sees the light enter the
train at an angle, cover a greater distance ( sqrt(w^2 + (v delta-t')^2)
) in a longer time (delta-t' = gamma delta-t, as explained above), still
travelling at the speed of light, c. For easy consideration, suppose
that according to the station clocks the light enters the train on one
tick and exits the train on the next. [Either a very wide train, or our
clocks are ticking in nanoseconds or deca-nanoseconds! ;-) ] Now we
simply consider those events as viewed in the two reference frames,
either by plotting them on a spacetime diagram or using the Lorentz
transformation to find the various components of the interval.
Results: all observers see the light beam as travelling at speed c, but
disagree on the distance covered and the time needed, the relationships
given by the familiar time dilation formula. The temporal separation
between the events is _smallest_ in the frame in which the events are
separated by the smallest spatial separation (here, same x-coordinate).

The fun thing about special relativity is that it's all quite consistent
(although still counter-intuitive) on the important things. It just
turns out that spatial and temporal separations aren't (individually)
important things. ;-)

Back to grading now, :-(

Ken