tabletop geodesics, general relativity, embedding diagrams

["John S. Denker" <jsd@MONMOUTH.COM>]

At 02:03 PM 2/21/01 -0800, Leigh Palmer brought up the topic of embedding
diagrams, including:

> science museum exhibit in which ball bearings or coins are rolled to
> simulate planetary orbits

The ball bearing is affected by the embedding world's gravity more than it
is affected by the curvature of the "space" in which it rolls.  This is a
fine demonstration of motion in a peculiar classical potential, but is not
a good demonstration of the effects of curvature.

We agree that any alleged connection between this exhibit and general
relativity is essentially 100% wrong.

=========

Here is a convenient way to _correctly_ demonstrate the effect of curved
space on geodesics.

Lay a piece of dark-colored construction paper on the table.  On top of
that, put a large dark-colored bowl, upside down.  I have a large salad
bowl that works beautifully.

To make the geodesics, use masking tape!!!!!!

Masking tape has the wonderful property that it very un-stretchy.  (If your
masking tape is stretchy, replace it with some that isn't.)  Therefore it
will follow straight lines in the D=2 world.  Meanwhile, it is so thin that
it can bend as necessary in the extra dimension, the embedding dimension.

1) The most basic thing is to lay out some tape on the construction
paper.  Put down a few inches of tape to get things started, and then lay
down the rest bit by bit, just by pushing down the next bit of
tape.  Observe how straight the result is.  Hold the supply of tape
slightly slack;  do not try to "force" the result to be straight.  The
point is to build your confidence that the already-stuck tape can
effectively guide the addition of the next bit of tape.

As long as you don't allow "crumpling" or "air pockets" under the tape, it
should guide itself quite well.

2) Lay out another line of tape that is initially parallel to the
first.  Extend it, letting the tape itself do the guiding, and observe that
the two lines remain parallel for a long ways.

3) Make a hump under the construction paper, in such a way that it doesn't
stretch the paper.  Observe that this has *no* consequences for the
geodesics;  things that start out parallel remain parallel, et
cetera.  This is important because it demonstrates extrinsic curvature,
which is much less interesting than the intrinsic curvature to be
demonstrated in the next items:

4) Lay out a geodesic that refracts off the bowl.

5) Lay out a geodesic that starts out parallel to the previous one.  It
will hit the bowl with a different impact parameter, and refract differently.

*) Et cetera.  You get the idea.


Note:  I suggested dark construction paper and dark bowls, to contrast with
the typical blonde color of the tape.  If you can get multiple colors of
non-stretchy tape, things get even more interesting.

Another suggestion:  You can pile smaller bowls onto the back of the larger
bowl, to change the shape of the potential.

Remark:  The embedding world's gravity has no effect on the tape.

Related remark:  The tape does not care whether the curvature is a bump or
a pit.  You could put the bowl under the paper, right-side-up.  Then cut a
hole so the tape can drop down and follow the curvature of the inside of
the bowl.  In all cases the geodesic will be bent toward the region of high
intrinsic curvature.

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

One thing that my tape cannot model very nicely is planetary orbits.  You
can create some closed orbits by putting a cylindrical glass on the
landscape.  But that is not a particularly good model of planetary orbits,
because all model orbits in that vicinity have the same radius.

The physics behind this is that the masking tape is essentially modeling a
photon.  There are no closed orbits for photons, except at the horizon of a
black hole, and in that case all orbits in the vicinity have the same radius.

To model Keplerian orbits and other things that nonrelativistic particles
do requires a universe that is curved across the _time_ dimension as well
as the space dimensions;  see MTW page 33, figure C.   I haven't figured
out to model this properly, but I haven't given up.  I'm open to suggestions!