Re: A mixture of time dilations and constrictions

[pvalev <pvalev@BAS.BG>]

--- Chuck Britton <britton@NCSSM.EDU> wrote:

> Two points, from a total non-'expert' in the field. But I DO
remember
> some brief discussions with our Freshman physics professor back in
> the old days.
>
> 1)      An example of a 'real' clock that is straightforward to
> analyze, is presented in many intro texts. Instead of a 'pointlike'
> clock, it is a 'one-dimensional' apparatus that is carefully aligned
> to be perpendicular to the axis of relative motion. If we accept the
> result that dilation only occurs along this axis (a separate
question
> - but easily addressed) the extent of the clock perpendicular to
this
> axis should introduce problems.
>         This is the clock that has a photon bouncing between two
> mirrors. The 'clock holding frame' sees the 'beam' of light bouncing
> a given distance in a given time that is unaffected by relative
> motions.
>         Does this satisfy the nature of the clock requirement?


Let us see what happens when the light clock is again situated at the
beginning of the primed system but is PARALLEL to the axis of
relative motion (the photon bounces along the x'-axis). I am going to
show that, when v is small, this clock is characterized by time
CONSTRICTION, according to Lorentz transform.
     Let us first consider the forward movement of the photon, in the
positive direction. For simplicity, the movement starts from the
origin (x=x'=t=t'=0). The time it takes will be t' in the primed
system and t1 in the unprimed system. In accordance with Einstein
postulate, at the end of the movement,

x = c*t1; x' = ct'            /1/

Substituting these into Lorentz first equation gives

t' = gamma(1 - v/c)*t1          /2/

(the same result would be obtained by substituting x=c*t1 into
Lorentz second equation).
     Now let us consider the backward movement of the photon, in the
negative direction of the x'-axis. Again, for simplicity, the
movement starts from the origin. The time it takes will be t' in the
primed system and t2 in the unprimed system. In this case Einstein's
postulate gives

x = -c*t2; x' = -ct'            /3/

Substituting these into Lorentz first equation gives

t' = gamma(1 + v/c)*t2             /4/

Now the time it takes the photon to move forward and backward is 2t'
in the primed system and t1+t2 in the unprimed system. In accordance
with /2/ and /4/, these times are related by

2t' = gamma(t1 + t2) - gamma(v/c)(t1 - t2)     /5/

Clearly, when v is small, the last term in /5/ can be neglected and
we obtain

2t' = gamma(t1 + t2)                    /6/

This means that, for a small v, 2t'>(t1+t2). Unlike the "vertical"
clock, this "horizontal" clock is characterised by time CONSTRICTION
in the primed system.
     I may be mistaken but I hope many on the list would agree that
constantly testing the (taught) theory is better than avoiding any
testing. I would be grateful if someone could find my mistake.

Pentcho