[Phys-L] Re: "moving clock runs slower" (yes)

[John Denker <jsd@AV8N.COM>]

Bob Sciamanda wrote:
> | Why do you "DEFINE" length to behave that way with respect to boosts,
> | when you  "DEFINE" length to behave otherwise with respect to rotations?
>
> This is tantamount to the question: "Why do you adopt the kinematical model
> of Special Relativity?"

I don't agree.  Lots of people can do kinematics in Special Relativity
even though they see a close analogy between rotations and boosts.

> I'll let Einstein answer both questions. Read his 1905 paper.

1) Wow, a naked appeal to authority.  What fun.  I will not debase this
list by replying in kind.  I prefer logical arguments.

2) Einstein does not get the last word on such matters.  Things change.
Einstein wrote of a "mass" that changed with velocity.  Nowadays relativists
take the word "mass" to refer to something that is invariant, so that:
  -- "proper mass" is redundant    ==> just say "mass"
  -- "invariant mass" is redundant ==> just say "mass"
  -- "rest mass" is redundant      ==> just say "mass"

> Further, our
> measuring instruments of length and time do follow the Einstein definitions.
> No special "relativistically corrected" instruments are necessary.

Here is a nice picture of Saturn, observed with a fine modern instrument:
    http://astrogeology.usgs.gov/assets/wallpaper/saturn.jpg

  -- Are you saying we should not apply *any* corrections to what the
   instrument is telling us?
  -- Are you sure the rings are elliptical, with eccentricity greater
   than 2?  That's what the instrument is telling us.
  -- Are you sure that the eccentricity of the rings changes from
   month to month?  In a time-lapse sequence of such photons, that's
   what the instrument is telling us.

Maybe that's what you choose to believe, but as for me, I choose to
believe that the image is heavily foreshortened by the geometry of
the situation.  I'm not sure I know how to make a "geometrically
corrected" instrument, but certainly downstream of the instrument
I can make a "geometrically corrected" calculation of the orbital
elements of the rings.

And in case it wasn't obvious:  I choose to correct for the geometry
of spacetime, not just the geometry of space.  I correct for boosts
just as I correct for rotations.

*WHY* do I choose this?  Not because some authority told me to do so.

I choose this because it makes the physics simpler.  I suppose you
could work out the hypothetical "physics" of highly elliptical rings,
but it would be bizarrely complex.  A far simpler hypothesis is that
the rings have very little eccentricity, and that the eccentricity
does not change from month to month, and the "uncorrected" instrument
is giving us a foreshortened view.  This is not due to any easily-
rectifiable defect in the instrument;  it is simply due to the geometry
of the situation.

I make this choice for rulers foreshortened by rotations, and I make
the same choice for clocks foreshortened by boosts.  I make this
choice for the same reason, because it makes the physics simpler.

> Proper length is an idea which arises in the narrow
> consideration of a rigid body - a very special (and in principle impossible)
> case.

I categorically disagree, particularly as to the notion of proper
time ticked off along a world-line.  That is a very well defined
concept.  I cannot think of any theoretical or practical reason for
not using it.  This is precisely the operational definition of "clock"
that I was taught:  imagine a little light that blinks at a regular
well-calibrated interval.  Each blink is an _event_ in spacetime.
These events are arrayed on the clock's world line like a string
of pearls.  Being _events_, they have a fixed reality that transcends
any particular reference frame.  However, they are most simply
described by saying they are regularly spaced in proper time.

Rick Tarara wrote:
> Excuse me if I bring this back to the level of those of use who introduce
> special relativity to our HS, and Gen-Ed (and our other INTRO courses for
> that matter) where 4-vectors and different algebras (normal algebra for that
> matter) are not going to be useful.  I'm back to the twins reunited on
> earth--with one much younger than the other.  The only conclusion I can
> really latch onto here is that the twin who was moving, relative to earth,
> had a slow moving clock.

But that conclusion is not logical.  I discussed this before.  Suppose Moe
and Joe travel from point A to point B in separate cars.  Each car has an
odometer and a clock.  If Moe's odometer shows more elapsed distance than
Joe's, would you say that "the only conclusion" is that Moe's odometer is
faulty?  I hope not.  A far simpler explanation is that Moe took the scenic
route, while Joe took the direct route.

The analogy to spacetime seems blindingly obvious:  If Moe's clock shows
less elapsed time than Joe's, it is by no means "the only conclusion" that
Moe's clock is faulty!  A far simpler explanation is that Moe took the scenic
route, while Joe took the direct route.

Yes, Moe returns younger.  No, that does not imply his clock ran slower.
He just took a different path through spacetime.

The geometry of spacetime is such that the scenic route always racks up
more distance and less time than the direct route.  (In some sense this
defines what we mean by "direct" route.)

> Here is another scenario.  I, after studying physics, set off for Alpha
> Seti-6, a mere 30 light years away.  I know that the speed of light is the
> galactic speed limit and I am pretty damn sure that the universe does not
> contract and expand (despite appearances) when I move, especially because it
> does not do so when I sit still and others move.  I arrive at AS-6 with an
> hour having passed on my trusty Dick Tracy wrist watch.  To be sure, AS-6,
> looked to be awfully close to the earth during my flight, BUT I know it is
> 30 light years distance.  How can I come to any conclusion other than my
> watch (and my biological clocks) ran very slowly during the trip?

Here's how:  Realize that the first conclusion that comes to mind is not the
only possible conclusion.  Realize that the conclusion you found in some
"authoritative" book is not the only possible conclusion.

The far simpler conclusion is that
  -- Elapsed time is path-dependent, for the same reason that path-length is
   path dependent.
  -- There is nothing wrong with our clocks or rulers.
  -- We should use the proper length of our rulers to define length, and use
   the proper time of our clocks to define time.
  -- Commonly when projecting one reference frame onto another, lengths and
   times get foreshortened because of the rotations and/or boosts.  This is
   a property of the projective geometry;  it is not a property of the rulers
   or clocks.

I repeat:  Yes, Moe returns younger.  No, that does not imply his clock ran
slower.  He just took a different path through spacetime.

> 4-vectors and different algebras (normal algebra for that
> matter) are not going to be useful.

You can teach this using spacetime diagrams, and simple statements like
"Elapsed time is path-dependent for the same reason that path-length is
path-dependent."   This requires *less* algebra than the other approach.
This has *less* severe conceptual barriers than the other approach.

I abhor the use of "paradoxes" in teaching introductory relativity.
It just teaches the kids to be afraid of the subject.  It emphasizes
how weird the subject is.  I take diametrically the opposite approach,
emphasizing how *little* the geometry of spacetime differs from the
familiar geometry of space.  A boost mixes x and t in much the same way
that a rotation mixes x and y.   A vector that used to be just t
becomes mostly t and a little bit of x.  Piece of cake.

You can explain this to 12-year-olds;  it's just not that tricky.  I'll
concede there's not much textbook support for this approach, but IMHO
that's why the teacher gets the big bucks:  to figure out the right
thing and teach it, no matter what the book says.  Also:  if you want to
do it without algebra, it will require a lot of diagrams.  It is a bit
time-consuming to freehand the required diagrams with sufficient accuracy,
especially if you don't do it every day ... so find a way to draw them
in advance.  One good option is to draw them on acetate foils and use an
overhead projector.  If you stack a blank foil on top of the pre-drawn
foil, you can draw on it extemporaneously without spoiling the original.

Also note that drawing Joe's space/time grid on one foil and Moe's grid
on another foil -- in a contrasting color -- allows you to introduce them
separately and then stack them to make the comparison.  (Of course if you
are using a computer to make your drawings, there are other ways to skin
this cat.)
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