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Re: tabletop geodesics, general relativity, embedding diagrams



OK, folks, I figured out how to build an embedding diagram that exhibits
Keplerian orbits. The trick is to make a model that contains a time
dimension as well as a space dimension (in contrast to my earlier model
that had two space dimensions and no time dimension).

David Bowman's note
http://mailgate.nau.edu/cgi-bin/wa?A2=ind0103&L=phys-l&F=&S=&P=7442
helped clarify my thinking about this.

Imagine a gravitating planet, such as the earth, but assume there is no
atmosphere to get in our way. We drill a hole from the north pole through
to the south pole, and we drop in a particle, from ground level or from
some other height. It will oscillate. It will move along a line, so we
only need 1 spatial dimension and 1 time dimension to describe its motion.

An embedding diagram with a distribution of curvature that approximately
models this situation can be found at:
http://www.monmouth.com/~jsd/physics/gr-orbit.gif
and if anybody cares, the scilab program that produced it can be found at:
http://www.monmouth.com/~jsd/physics/gr-orbit.sci

You could carve a physical model of this contour, and find the geodesics
using masking tape as described in my previous note:
http://mailgate.nau.edu/cgi-bin/wa?A2=ind0103&L=phys-l&F=&S=&P=3541

The center of the planet is at X=0. The planet has radius=1. Far away,
the space is essentially flat. Nearer to X=0, the ripples get bigger. It
is possible to have ripples with no intrinsic curvature (just uninteresting
extrinsic curvature) but these ripples do have some intrinsic
curvature. In particular, it is obvious that at the very bottom of the
pits there is substantial intrinsic curvature.

The key idea is that the length from (X=0, t=0) to (X=0, t=24) has to be
longer than the length from (X=10, t=0) to (X=10, t=24). The model has
this property.

This model is, however, imperfect. In the real world, the curvature
produced by the gravitating planet does not vary in time, but alas I don't
think there is any way to create a D=2 contour that has curvature that
depends on position but not on time. So the next best thing is to create
something where, averaged over some modest time, the average curvature is
correct. You can imagine building a sequence of models like this, with
using ripples with shorter and shorter pitch (also less and less amplitude)
so that the averaging is more thorough.

Decreasing the amplitude is a good thing in itself. In the interests of
clarity, the diagram shows ripples and pits that are deeper than
desirable. Making them 1/10th as deep would be a lot better. You want
each pit to "attract" the masking tape geodesic only a little, so that the
orbit is produced by the average effect of many pits.

Remember, the key idea is that the length from (X=0, t=0) to (X=0, t=24)
has to be longer than the length from (X=10, t=0) to (X=10, t=24). If you
can think of a better way to "crumple up" the extra length along the X=0
line, please let me know.