Re: gravitational embedding diagrams

[Leigh Palmer <palmer@SFU.CA>]

> At 02:03 PM 2/21/01 -0800, Leigh Palmer wrote:
>
> > embedding diagram ....... made with the
> > depression transformed into a peak! Why didn't the light rays get
> > deflected *away* from the Sun in that case, the naive listener might
> > easily think.
>
> This is a most excellent point.
>
> > science museum exhibit in which ball bearings or coins are rolled to
> > simulate planetary orbits
>
> We agree that the exhibit is misleading.
>
> > Box 1.6 of MTW. It is, in fact, just plain RONG,
>
> Not so fast.  M, T, and W are pretty smart dudes, way too smart to get
> sucked in by something like that.

This comment refers to the error in their diagram. Making it go
downward is not an error. The error is in plane sight, and I will
reveal what it is when all have had a chance to look at it.

> The difference is that they never roll ball bearings on their embedding
> diagrams.  They show light rays and sometimes they show ants walking
> around, sensitive to distance but not to the pull of the embedding world's
> gravity.
>
> ================
>
> We can somewhat improve the museum exhibit as follows:
>    a) Use trained ants, or
>    b) Make a little "tricycle" as follows:  two drive wheels in front, one
> little castering freewheel in the back.  The drive wheels always turn at
> the same rate, and they don't slip, so they always travel the same distance.

Now that is a really neat idea! I don't think the castering wheel
is necessary or desirable, however. How about a bicycle (of sorts)
with two wheels affixed to the same axle and the drive mechanism
depending from the center of that axle. It should have o-ring tires
and should be as small as is practical. Gravity will keep the drive
mechanism hanging downward on any slope sufficiently shallow that
the o-rings don't slide sideways. To make it mysterious, one could
use a closed cylindrical can with o-rings on its edges and the
drive mechanism hidden inside.

> -- On a planar surface, the tricycle moves in a straight line.
>
> -- On a curved surface (curved up or down, it doesn't matter) the tricycle
> will be deflected toward the pit or peak.
>
> -- Unfortunately you can't demonstrate a closed orbit without using a pit
> or peak so extreme that it has cylindrical walls.  There would be technical
> problems due to slippage.

There is only one closed orbit for light anyway. The demonstration
(or analog) merely shows that the deflection of light is independent
of the orientation of the surface. It would work in zero gravity
with sticky o-rings.

> Maybe the demo would work in microgravity.  Even
> then there is only one closed orbit through a point, so you have to get the
> initial conditions just right, which is hard.  So basically the tricycle is
> behaving like a photon in this model.

It is moving on a "null geodesic", a trajectory which is locally a
straight line on flat space.

> -- Maybe somebody can figure out a way to model a nonrelativistic particle.

by which I assume you mean a free particle - one on which no forces
act. (The "force of gravity" does not exist in GR. Gravity is solely
a geometrical effect.) I'm just learning about this stuff now, John.
I'll keep that goal in mind. If I can figure out how to do that I
will have learned something valuable. I know the name of such a
trajectory. It is called a "timelike geodesic". I have no equivalent
geometrical feel for the meaning of such a trajectory in curved
space. I will also point out that the Kepler problem hasn't been
solved in GR. Has the restricted two-body problem (a planet of
negligible mass in orbit about a massive object) been solved for the
Schwarzschild geometry?

Leigh