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Re: tabletop geodesics, general relativity, embedding diagrams



At 10:57 AM 3/2/01 -0500, I wrote about how to find geodesics in a D=2
curved space on your tabletop, using only masking tape and other items
available to almost everybody.

Here are some recent refinements to the idea. Let's model a piece of
spacetime that has D=1+1, i.e. one spacelike dimension and one timelike
dimension. We will illustrate the gravitational field of a planet that
exists for a long time (a long ways along the time axis). The
gravitational field decreases as we move away from the planet in the +-X
directions.

This can be modeled using _darts_. The darts are arranged as shown here:

A2 B2
----====#### ####====----

----====#### ####====----

----====#### ####====----

----====#### ####====----

----====#### ####====----

----====#### ####====----
A1 B1

... where the time axis runs vertically along the page, and the X-axis runs
horizontally across the page. The 6 _ribs_ are made of two darts each, for
a total of 12 darts. Each dart is about 4" long in the long direction, and
has an equilateral triangular cross-section everywhere. At the base of the
dart (near the center of the diagram) each side of the triangle is 1/2".

Many of the following fabrication ideas are due to Paul Fuoss....

Maple is an excellent material for making darts. (Any hard, fine-grained
wood will do.) The grain should run down the long axis of the dart; this
makes fabrication harder but makes the final result nicer. We found that
Paul's _compound miter saw_ was the smart way to make them. We tried
making them with a table saw but the compound miter saw was a much better
choice.

Attach the hardwood piece (from which you are cutting the darts) to a much
larger carrier piece, using wood screws, so you can hold it very securely
without endangering your fingers. (Security is vastly more important for
miter cuts than for regular cuts, where slight vertical motions are harmless.)

My 16-year-old neice Jessica pointed out that you can make darts that are
just as functional (but perhaps not as beautiful) out of clay. If you
don't have access to a compound miter saw, this may be your best option.

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

The stores around here sell masking tape that is 3/4" or 1" wide. Narrower
tape is better for this demo. So I divided the tape in half lengthwise. I
did this while it was still on the roll, using a very sharp knife to cut
through many layers.

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

If you attach the darts to a piece of acetate foil, you can set the whole
thing on an overhead projector so everybody can see it. But letting people
play with it hands-on is the best.

My favorite thing is to bring in a trajectory at point A1, in the purely
timelike direction, and let it "orbit" the planet. In D=1+1, the orbit
looks like a sine-wave when viewed from above the page. An orbit that
starts at B1 (on a timelike heading) will cross the orbit that starts at A1.

Also, you can ask people "what is the shortest path from point A1 to point
A2". The path in the presence of the darts is very different from what it
would be in the absence of the darts. Actually in the presence of the
darts (spacetime curvature) there are a couple of different geodesics that
connect A1 to A2.

You can find many other lessons that can be demonstrated with this setup.