[Phys-L] Re: "moving clock runs slower" (not)

[John Denker <jsd@AV8N.COM>]

Fayngold, Moses wrote:
> I find the kind of logics used by John here, rather unusual.

It's not unusual ... although it may be unfamilar to some people.

> It boils=
>  down to this:
>
>   A moving car does not move. Why? Because the true properties of an =
> object can be observed only in the object's rest frame, and any drive=
> r knows that in the rest frame of a car the car does not move. Welcom=
> e back to Zeno!

1) First of all, in spacetime, a car at rest *does* move.  It moves
toward the future at the rate of 60 minutes per hour.

That is, its four-vector velocity (u) is not zero.  The components of (u) are
[1, 0, 0, 0].

This may be unfamiliar to some people, but it is useful as the starting point
of many a calculation.

2) While we are discussing analogies, I find it odd that people who defend the
notion that "moving clocks run slow" don't seem willing to defend the notion
that a ruler is shorter when viewed nearly end-on.

If you want to be logically consistent, you can't have one without the other.

I'm not talking about Lorentz contraction here;  I'm talking about plain old
rotation.  Note that the rotation group is a subgroup of the Lorentz group.

Most people consider it obvious that the proper length of a ruler is independent
of the viewing angle.

The rotational analog of the twins "paradox" makes the "paradoxical" claim that
an ordinary one-foot ruler is less than an inch long, allegedly because I can
arrange two rulers end-to-end in a narrow V shape, with the starting point of
the first ruler only one inch from the ending point of the second ruler.  This
V-shape is closely analogous to the travelling twin's world line, as diagrammed
at e.g.
   http://www.av8n.com/physics/twins.htm#fig-twins-joe
I put "paradox" in scare quotes, because it's not really a paradox;  it's just
silly.  The rulers are being used improperly.

There is a profound analogy between rotations and boosts.  Viewing a ruler
end-on doesn't change what the ruler _is_.  Similarly viewing a clock from
some boosted reference frame doesn't change what the clock _is_ or _does_.

The projection of the clock or ruler onto your field of view will depend
on your viewpoint, but that is a property of the projection, not a property
of the clock or ruler.

=====================

Several people have asked me about the fact that elementary particles (e.g.
muons) are observed to decay more slowly when they are in motion relative
to the lab frame.  Doesn't that mean that the muon's on-board clock is
running slow?  I say no.

Consider another spatial analogy.  Our heros, the twins Moe and Joe, drive
from point A to point B in separate cars.  Each car has an odometer and a
clock.
  -- Joe's odometer shows that he travelled a distance of 100 miles.
  -- Moe's odometer shows that he travelled a distance of 200 miles.

Do we conclude from this that the odometer on Moe's care is somehow broken,
and measures some sort of weird "contracted" miles?  Of course not.  A far
simpler explanation is that Moe took the scenic route, while Joe took the
direct route.

In the non-relativistic case, we find that even though the odometers give a
path-dependent notion of elapsed distance, the clocks give a path-independent
notion of elapsed time.

The only halfway-tricky thing about relativity is that if the twins travel
at high speeds, their clock readings become just as path-dependent as their
odometer readings.

Do we conclude from this that Moe's clock is somehow broken, and measures
some sort of weird "slow" seconds?  Of course not.  A far simpler explanation
is that Moe took the scenic route, while Joe took the direct route.

Because of the hyperbolic geometry of Minkowski space, a scenic route will
always rack up _more_ distance and _less_ time than a direct route.

This is a property of the routes!!!  It is not a property of the clocks or
odometers.

I repeat:  It is the ordinary Muggle experience that the elapsed time from
A to B is independent of path.  This is not true in general!  Get used to
it.  All the "proofs" that Moe's clock runs slow are based on the implicit
assumption that time "should" be path-independent.  However, there is another
interpretation that is far simpler and far more consistent with the structure
of the Lorentz group, namely that the clocks are not broken or distorted, and
that the notion of elapsed time is path-dependent, just as path-length is
path-dependent.

=====================

Ludwik Kowalski wrote:

> > The two approaches to SR (special relativity) described by JohnD
> > reminded me of two approaches of introducing QM (quantum mechanics).
> > The first starts with Bohr's model and imposes nonclassical assumptions
> > (orbiting electrons do not radiate and only certain orbits are
> > allowed). The second starts with Schroedinger's equations and solves it
> > for different potential energies (different systems, such an H atom,
> > CO2 molecule or U-235 nucleus) and with different boundary conditions.
> > The second approach is probably preferable for those who are very
> > comfortable with advanced calculus, partial differential equations,
> > etc.

That analogy is colorful, but I don't think it is apt.  Quite the
reverse.  IMHO the spacetime approach is *less* mathematical, *less*
abstract, *more* intuitive, *more* readily visualized, and *more*
suitable for an introductory-level course.

My approach #2 -- i.e. drawing spacetime diagrams and taking the dot
product of four-vectors -- is not analogous to solving a notoriously
intractable partial differential equation.  It's just multiplication
and addition, and not even very much of that.  It is in fact the same
amount of arithmetic and *less* of a conceptual burden than approach
#1 (which requires weird stuff like clocks that run slow and rulers
that contract, and even then is not a complete description of what is
happening). The four-vector approach is easier to teach, easier to learn,
and easier to remember, because it is so closely parallel to familiar,
visualizable three-vector concepts.

Just because a tool works better doesn't mean it is harder to use.
Cooking your soup on a modern stovetop is both easier and safer than
cooking it over a campfire.  If somebody gives you a modern, well-engineered
apparatus, why not use it?

================

I don't think it is helpful to argue that things "are" as they appear to be,
or that things "are" whatever we measure them to be.  Appearances can be
deceiving, and some measurements are incomplete and/or non-intrinsic.  A
ruler subtends a certain angle in my field of view.  I can measure this
angle, but it may not tell me anything about the proper length of the ruler,
if I don't account for other factors such as orientation.

A ruler is a ruler.  Its length is its proper length.  A clock is a clock.
Its time is its proper time.  The projection of a ruler onto the wall of
the cave is not a ruler;  it is just a projection.

   http://www.constitution.org/pla/repub_07.htm
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