Re: A mixture of time dilations and constrictions

[Ken Caviness <caviness@SOUTHERN.EDU>]

It may be misleading in this thought-experiment to refer to the rotating arm
as an "extended clock".  A common device in elementary SR treatments is the
"light-clock", a beam of light bouncing back and forth between two closely
placed mirrors.  One can think of the reflections (bounces) as ticks of the
clock.  This gives a very simple visualization of the "flow of time".  It's
immediately clear to that in a frame where the mirrors are stationary, the
interval between the ticks is constant and can be used to measure time
(provide a standard for comparison of the duration of other processes or the
separation between events in that reference frame).

We might also use the orbit of the tip of a rotating clock hand as a
standard of comparison.  For instance, a "tick" of the clock could be
thought of as occuring when the hand is straight up -- in the reference
frame where the axle of the clock has no translational motion.  It is not
hard to show that these "ticks" are also subject to normal SR time
dilation:  any observer in relative motion (in ANY direction) with respect
to the clock will observe the period between the ticks to take longer
(according to _her_ clock) than does the observer in the clock's rest frame.

Pentcho, in your example you are effectively treating _all_ positions of the
clock's arm as "ticks", relying on the idea that in the extended clock's own
rest frame, the clock's arm's rotational speed is constant, and thus the
tangential speed of the tip of the arm is constant.  But clearly this speed
is not constant in any reference frame in uniform motion with respect to the
clock's axle, the motion of the tip will be cycloidal.  To be very careful
we might let a "tick" be when the clock's arm passes a certain set of
positions, such as multiples of 1/60 of a rotation (6 degrees), or even
1/360 of a rotation (.1 degrees).  It would indeed be rather complicated to
consider the velocity of the tip at each position according to different
observers, and we might become confused in treating the situation, but we
really don't need to do this.  We can imagine a smaller, faster clock arm,
that makes 60 or 360 rotations in the time our original clock arm made one!
Minute and second hands, of course.  ;-)  Clearly normal time dilation
effects will be observed in regard to the "ticks" of these new hands, and
this resolves the confusion:  although our original clock hand tip may
instantaneously be moving forward or backward (or standing still) with
respect to some observer, there will be no "time constriction".  Please keep
in mind that the direction of the relative motion is inconsequential:
observers in relative motion both see the other as experiencing time
dilation, neither experiences "time constriction".  There ain't any such
animal.

Enjoy,

Ken


Pentcho Valev wrote:
> Stephen Speicher wrote:
>
> > I do not know what you have been told on other lists, but special
> > relativity is a geometric theory with the notion of a point-like
> > event as a fundamental concept.  Clocks are idealized to be
> > present at any given event, not as an extended object but as a
> > point-like particle. One can deal with a clock as an extended
> > object in relativity, but such techniques are _vastly_ more
> > complex than standard analysis.
>
> Still let us try. If the extended-clock analysis gives results different
> from point-like-clock analysis, I hope you would agree the problem is
> serious.
>
> Consider the traditional SR setup: two inertial frames, S and S', with
> relative speed v and an EXTENDED clock situated at the beginning of the
> primed system. I am going to prove that, according to an observer in S,
> the movement of the tip of the clock's hand can be characterized by time
> CONSTRICTION in some positions of the hand but by time DILATION in
> others.
>      Let the hand rotate in the x'-y'-plane, and the linear velocity of
> the tip of the hand be greater than v. Accordingly, either in its higest
> or its lowest position, with respect to the y'-axis, the tip moves in the
> negative direction of the x-axis in the UNPRIMED system. Let this happen
> as the tip is at its lowest position (with respect to the y'-axis), and
> let this lowest position of the tip be EVENT 1 which coincides with the
> origin (x=x'=t=t'=0).  Then some close subsequent position of the tip is
> EVENT 2, characterized by (x<0, x'<0, t>0, t'>0).  By substituting x<0
> and t>0 in the right side of Lorentz second equation
>
> t' - t = gamma[t(1 - 1/gamma) - x(v/c^2)]
>
> we obtain
>
> t' > t
>
> which means that this movement of the tip is characterized by time
> CONSTRICTION, according to the observer in the unprimed system.
>      Now let the position of the tip have the highest value of x'. This
> means that, at this moment, the movement of the tip is perpendicular to
> the x'-axis, in the primed system. This position of the tip will be the
> new EVENT 1 which coincides with the origin (x=x'=t=t'=0). Then some
> close subsequent position (EVENT 2) is characterized by (x=vt, x'=0, t>0,
> t'>0).  Substituting x=vt in Lorentz second equation yields
>
> t' = (1/gamma)t < t
>
> which means that this movement of the tip is characterized by time
> DILATION, according to the observer in the unprimed system.
>
> I would be grateful if you could stop hinting at my ignorance and analyse
> the case. If I am wrong, please give your own solution. For the two
> analysed positions of the hand, how will the observer in the unprimed
> system find the movement of the tip -  characterized by time dilation or
> constriction?
>
> Pentcho