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Re: [Phys-l] Relativity Question



There is a really fun essay by David Mermin about something he calls "The Amazing Relativity Engine". I read it in his collection: "Boojums All the Way Through". It is about a simple way to illustrate that time dilation and length contraction follow automatically from the relativity of simultaneity.

In brief: he sets up two rocket trains, moving in opposite directions, each numbered car with a clock on it. The scientists on each train have been told that their clocks are synchronized though we can see in a our spectator's frame that "really" they are not. We collect photographs showing pairs of opposite train cars with their clocks. By analyzing well-chosen pairs of such photos, each set of scientists can prove that the other (moving) team's clocks are running slow, are not synchronized and that the moving train is shorter. And this is all because they "erroneously" believe that their own clocks are synchronized.

I am not doing this justice, but I highly recommend the essay and the approach, especially for less algebraically skilled students (Mermin's originally intended audience). Also, I have a windows compatible program that implements this idea, written by some former students. Contact me if you'd like a copy.

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Thursday, May 14, 2009 10:04 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Relativity Question

On 05/14/2009 05:00 AM, chuck britton wrote:

I've always used the Lorentz dilation/contraction
approach and
realized early on that it is the impossibility of simultaneity that
is often overlooked.

Agreed: all too often overlooked.

There's no nice 'equation' to sum it up as there
is for time and space intervals.

As much as I hate to say anything nice about the dilation/contraction
approach, there *is* a perfectly nice equation for the breakdown of
simultaneity at a distance. It's the "other" term in the Lorentz
transformation for t' in terms of (t,x). Specifically, -γvx/c^2.
See Feynman volume I equation 15.3.

Psychologically speaking, it is easy for people to blow off this term
because the corresponding term is entirely absent from ye olde
non-relativistic equations of motion. Compare equation 15.3 to 15.2.
In each case compare the x' row to the t' row; see how similar they
are in 15.3 and how different they are in 15.2.

Or you could take the modern (post-1908) approach:
-- Do away with dilation/contraction.
-- Use the invariant length and proper time.
-- Draw the spacetime diagram.
-- Exploit the profound analogy between 3-vectors and 4-vectors.
-- Exploit the profound analogy between old-fashioned rotations
and boosts.

The spacetime diagram makes the qualitative relationships unforgettable
(including simultaneity or lack thereof). The (t,x) rotation matrix
makes the quantitative relationships unforgettable.

Using 4-vectors starts by exploiting what students already know about
3-vectors, but it also reinforces and greatly deepens their
understanding
about what vectors are and why they are important.

Minkowski said in 1908:

Space of itself and time of itself will sink into mere shadows,
and only a kind of union of them will survive.

And then he explained in vast depth and breadth what he meant by that.
Einstein was shocked.

Space of itself and time of itself will sink into mere shadows,
and only a kind of union of them will survive.

That must be one of the greatest sentences in this history of the
world.

Two events that are simultaneous in one frame CANNOT be simultaneous
in ANY OTHER frame.
(unless of course they are at the same location as well
as the same
time -
in which case I'll claim that they are only ONE event -
not two.)

That's exactly right if there is only one spacelike dimension.

OTOH if there are more dimensions, we need to consider the possibility
that the separation vector (between events) is perpendicular to the
velocity vector (between frames). In that case, there is a whole
family
of frames that involve no dilation, no contraction, and no breakdown of
simultaneity for transversely-separated events.

That is to say, any (t,x) rotation matrix is independent of y and z.

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