Re: A mixture of time dilations and constrictions

[Pentcho Valev <pvalev@BAS.BG>]

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