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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.