Re: Question about light clock and derivation of time dilation

Usually, this construction is used in proving the invariance of the
interval, as on pages 67-72 of Taylor, Wheeler 'Spacetime Physics' text. Yes
it does raise the question of why does the photon travel at just the right
angle? It certainly does not seem obvious to students. Especially since
Taylor and Wheeler, on page 115 Exercise L-9, finally get around to deriving
the 'headlight effect' that gives the angle. The case of 90 degrees is a
special 'degenerate' case so maybe the student should have seen it?

In any case, the invariance of the interval and time dilation can all be
derived geometrically in the x-t plane, without having to use a second
spatial dimension. This is done in two papers:

Dieter Brill & Ted Jacobson, Spacetime and Euclidean Geometry,
arXiv:gr-qc/0407022 v1 6 Jul 2004
http://arxiv.org/abs/gr-qc/0407022

N. David Mermin, From Einstein's 1905 Postulates to the Geometry of Flat
Space-Time, arXiv:gr-qc/0411069 v1 15 Nov 2004
http://arxiv.org/abs/gr-qc/0411069

I believe that the approach used in these papers is much easier to
understand than the standard introductions to special relativity. The key
point was to show that equal intervals always have equal areas (of the
rectangle whose diagonal is the interval and whose sides are light rays.)
The papers rather inspired me to write a Mathematica notebook that derives
all the basic relativity results with a lot of graphics and animation and a
minimum of algebra (which Mathematica easily handles in any case.) The
notebook can be obtained on the Mathematica page of my web site:

http://home.earthlink.net/~djmp/Mathematica.html

It also requires the DrawGraphics Mathematica package on the same page,
which is used to create the graphics and animations.

David Park
djmp@xxxxxxxxxxxxx
http://home.earthlink.net/~djmp/




<john_doe_ph_d@xxxxxxxxx> wrote in message
news:1120401164.004302.67760@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> Time dilation is typically the first SR effect derived in text books,
> and it is usually done with the example of a light clock as follows,
> where S is the source and M is a mirror. Since it appears to be so
> important, I'm trying to understand every aspect of this example.
>
> --- M
>
> ^
> |
> |
> |
> |
> |
> |
>
> | | S
>
> The well-known derivation is based on the fact that, to an observer B
> moving horizontally w.r.t. the clock, the light is taking a longer path
> from S to M as shown below:
>
> --- --- ---
> ^
> |
> ^
> |
>
> ^
> |
> | | | | | |
>
> But the speed of light is constant and dimensions perpendicular to the
> direction of motion don't change. Therefore, B concludes that the time
> for the light to go from S to M is longer than the time determined by
> an observer stationary w.r.t. the clock.
>
> Fine, but I am trying to reconcile B's observation of the light path
> with the fact that the light source is pointed in the vertical
> direction. Doesn't that tell B that the light is "really" just moving
> up and down? Wouldn't B have to see the light source tilted to make
> sense of his observation of the light path?
>
> To make this more concrete, let's suppose that the source S and mirror
> M are connected with a narrow fiber optic (with some scattering
> material included so that B can still observe the light). How can B
> reconcile the observation that the light path is tilted with his
> knowledge that the light is confined to move in the fiber? Doesn't he
> "really" know that the light is just moving straight up and down?
>


.

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