Re: [Phys-l] how to prove relativity

[John Denker <jsd@av8n.com>]

On 06/03/2010 02:41 PM, Stefan Jeglinski wrote:

> Thanks for a great discussion, JD! 

:-)

> But when I think about 
> communicating relativity to a layperson, I need help still on one 
> sticking point (down below, after this summary):
>
> distance != path

Yes, they are indeed not the same.

> easy to visualize with the ruler/odometer pairing

Yes.

> clock is not to ruler; clock is to odometer; clock ticks are to ruler

Yes, yes, yes.  That is true and very important.

To say the same thing in other words, words that may ring a bell
with some students:  the ruler measures the _chord_, whereas 
the odometer measures the _arc length_.  See figure 4 in
  http://www.av8n.com/physics/odometer.pdf

  Again: beware that the first part of that document is unsuitable 
  for an introductory pedagogical situation, because it starts by 
  mentioning things that ought not be mentioned until much later, 
  if at all.  I really need to rewrite that document.

> Although when we ended the trip and compared rulers, they were still 
> identical, and when we compared clock ticks at the end of the trip, 
> they were still identical too, even while the odometers and times 
> were different, there remains, in the layperson's eyes, this one 
> amazing asymmetry: one person is actually older (without, for 
> example, also being shorter).

That so-called asymmetry is bogus.  As Indiana Jones famously
said about his own worn-out beat-up body:  "It's not the years, 
its the miles".

There are some parts of a person -- or a car -- that wear
out with age, but there are other parts that wear out in
proportion to usage aka mileage.  Certainly used-car buyers
are sensitive to the odometer reading as well as to the age
of the car.  This is perhaps less obvious for human bodies,
but it is simply unfair to overemphasize the distinction 
between age and mileage.

I can sympathize with those who focus attention in this
general area, because the aging issue is a big shocker to
newcomers to special relativity.  Still, I would make the 
point that there is no real asymmetry.  We suffer from a
bogus notion of asymmetry, and also from a false assumption 
that elapsed time "should be" independent of path.  Once 
you accept (even hypothetically) the idea of four dimensions,
and accept the fact that elapsed odometer distance is clearly
path-dependent, then you should at least be willing to accept 
the /possibility/ that elapsed clock-time is path-dependent.

We can sympathize with the false assumption, because in
ordinary life the magnitude of the clock's path-dependence
is small.  So this is a teachable moment.  The dependence
is small, but nonzero.

By way of analogy, for _small angles_ we don't need to
worry about the difference between the chord and the arc
length;  all small-angle paths from A to B have very nearly 
the same length.  Similarly for small speeds, we don't need 
to worry about the path-dependence of the elapsed time;  all
low-speed paths from A to B take very nearly the same amount 
of time.  It helps to draw pictures of this; see below.

Remember, a rotation mixes the x and y coordinates.  If
we start out with a vector purely in the x direction and
then rotate it, then for small angles the slope is proportional 
to the angle, the change in the y-projection is proportional 
to the angle, and the x-projection is unchanged to first order.
A picture of this can be found in figure 1 of
  http://www.av8n.com/physics/spacetime-trig.pdf

So it is for boosts.  A boost mixes the x and t coordinates.
Suppose we start out with a particle at rest, just sitting
there, moving purely in the t direction at the rate of 60
minutes per hour.  Then we give it a small-angle boost.  The
velocity is proportional to the angle, and the change in hte
time-projection is zero to first order.  Let t denote the 
/projection/ of the proper time (τ) onto the lab-frame 
coordinate system.  To repeat, all low-speed paths age at 
very nearly the same rate, so the distinction between t 
and τ is zero to first order.  See figure 3 of
  http://www.av8n.com/physics/spacetime-trig.pdf

In contrast ... for a large-angle xy rotation, the projection 
of the rotated vector onto the lab frame coordinates will no 
longer be anywhere near the same.  For a large-angle xt rotation, 
i.e. for a big boost in the x direction, the projection of 
the new vector on the lab frame coordinates will no longer be 
anywhere near the same.  That is, the rate of aging will be 
very very different.

Special relativity is not weird or paradoxical.  It is just 
the geometry and trigonometry of spacetime.

> The OP was about a layperson who would not accept relativity. I'd bet 
> that the arguments laid out here (even without data) would be 
> extremely persuasive, except for this one last part (the aging). And 
> what do we then have left:
>
> 1. Talk about esoteric evidence (eg muon lifetimes or accelerator 
> operation) that they will resist?

By itself, that will never be convincing ... and indeed, by 
itself, no single piece of evidence *should* be convincing ... 
especially if it is at all esoteric.

The better approach is to marshal a mountain of evidence
in support of one simple idea.

We get to combine the muon data with the somewhat less
esoteric clocks-on-plane data and GPS data, and also with 
a dozen other lines of evidence;  see below.

> 2. Try to convince them that our aging must also be dictated by some 
> biological clock that simply cannot work differently (proof by 
> blatant assertion)?

All sufficiently accurate clocks are affected the same way.
Aging is just the passage of proper time.  There's nothing 
special about it.  Used-car buyers care about the mileage 
of the car /and/ about the age.  There is no mystery about 
this.

If you went on a trip and found that your clock aged at one
rate, your car aged at another rate, and your body aged at
yet a different rate, THAT would be weird.  Special relativity
is not weird;  it affects all timing the same way.

> 3. Merely quote Minkowski, that we must abandon space and time and 
> combine them instead into spacetime?

If we leave off the "merely quote", then yes, I would use 
Minkowski's idea as a starting point, because the idea is 
stunningly simple and can be supported with multiple diverse 
lines of evidence.
 -- It unifies space and time
 -- It unifies momentum and energy
 -- It unifies electricity and magnetism
 -- It explains rest energy: m c^2
 --    as observed in positron (etc.) annihilation
 --    which is connected to low-speed kinetic energy: .5 p v
 --    and also connected to photon energy: 1.0 p c
 -- It unifies rotations and boosts
 -- It explains odometers in motion
 -- It explains clocks in motion
 --     including muons in motion
 -- It (plus curvature) explains stationary clocks in a g field
 -- It (plus curvature) explains moving particles in a g field

Any one of these things would be interesting.  Any two of them
would be an amazing coincidence.  The whole package is far beyond
amazing.
 
> OK fine, fair enough, perhaps that *is* all we got. 

Seems like a lot to me.  The ratio of what we get out, compared
to what we put in, is stunning.

> More than enough 
> for us, but in every discussion I can imagine with a layperson, I'll 
> lose them at the very end of what is otherwise a brilliant 
> presentation.

Well, if they are smart they /shouldn't/ be easily convinced.
Keep in mind that most relativistic effects are unobservably 
small in their everyday life, so every bit of the evidence
will seem somewhat esoteric.  Then again, when you're talking
to a non-physicist, even things we consider non-esoteric (such
as magnetism) will seem esoteric to them.

That's why you need to persist.  The hit-and-run approach is
not going to work.  That is, you can't just say a few words 
about muons (or whatever) and expect it to stick.  Besides,
the important thing isn't the muons.  The important thing is
spacetime itself.