Re: tabletop geodesics, general relativity, embedding diagra

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding John Denker's comments:
> ...
> 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!

As ingenious as John's idea is for using masking tape to model geodesics
on a curved surface, I should point out that if the curved surface model
is supposed to represent a curved spatial section of spacetime, (i.e.
curved space), then these geodesics so modelled do *not* model the paths
of photons, nor planets nor anything else that is realizable (as far as
we know).  What the geodesics *do* model are the paths of zero energy
tachyons since these are the only conceivable particles that can cover
all of the space of their world lines at the same instant of time.  The
worldline geodesics of massive particles or of light (i.e. timelike or
null geodesics) do not go through space per se, but through spacetime
instead.

If we project these geodesic world lines of such realizable particles
down onto a static spatial section of spacetime, the resulting orbit is
*not* a geodesic of that curved spatial section.  The slower the particle
moves the greater the error made by making the projection.  Projections
of geodesics *onto* a submanifold do not usually result in geodesics *of*
that submanifold.

In the case of the light rays the result of the masking tape model is
qualitatively OK (as long as the overall curvatures of space and
spacetime are relatively weak as in our solar system).  In such
circumstances the deflections obtained by the tape method have their
asymptotes deflect in the correct direction, but the amount of that
deflection is 1/2 of what it is supposed to be for real photon orbits in
space.  The reason for this is that the amount of the curvature of
spacetime across time is about the same as the amount of curvature across
space, and the null geodesics of the light's world lines tilt at 45
degrees from the time axis toward the spatial axes.  When these geodesics
are projected onto the spatial 3-space half of their deflection is due to
the internal curvature of space itself, and half is due to the
projection-suppressed curvature w.r.t. time.

In the case of an orbiting planet the timelike geodesic in spacetime is a
very shallow pitched helix oriented along the time axis where the time
axis is the symmetry axis for the helix (assuming of course, that the
spatial origin is taken to be at the center of the Sun, so that the Sun's
world line coincides with the time axis).  If the planet is the Earth the
radius of the helix is 1AU, but the pitch period is 1 ly.  This makes the
pitch period some 6.3 x 10^4 times longer than the radius of the helix in
spacetime.  Suppose we follow the world line of the Earth from one vernal
equinox to the next one (and neglect any precession effects).  This
single turn of the helix, being a timelike geodesic, has a longer proper
time along it than the path which stays fixed in space relative to the
Sun at the position of the Vernal Equinox, and just moves in time
parallel to the time axis to the next equinox when the Earth return's
there.  Thus, the curvature of spacetime makes the shallow pitched
helical path *straighter* than the ostensibly straight path of a
motionless object hovering above the Sun at the position of the Vernal
Equinox.

Remember, for timelike paths the geodesic straightest possible paths are
those that *maximize* the proper time rather than minimize the proper
length (as it the case for spacelike paths).  The reason the helix path
*looks* curved to us is that we are using a flat spacetime model for
the actual curved spacetime when we plot the spacetime events in
R^4.  This is just like how a typical projection of the Earth's
curved surface onto a flat map representation can result in the
geodesic Great Circle paths appear to be curved paths on such a flat
map.

When this helical worldline is projected down onto the spatial subspace
perpendicular to the time axis, this projection is a closed figure
(ellipse with an eccentricity of 0.017).  The reason this path is so
strongly curved in space is that the cumulative effects of a very slight
curvature of spacetime along the time axis have been telescoped down onto
the spatial orbital plane.  (If you draw a slightly curved path on a
piece of paper, and then look at it nearly edge-on down along the curve,
it looks much more curved.)

Even though the degree of curvature across space is similar to the amount
of curvature across time in spacetime, the fact that nonrelativistic
timelike paths in spacetime stay so close to being nearly parallel to the
time axis causes the curvature of their projection onto a spatial section
to be completely dominated by the spacetime curvature across time along
the axis of collapse.  The tiny spatial curvature effects across space
itself are quite negligible in most circumstances, and only show up at
all in the planet's orbit in space as a very tiny GR correction to the
Newtonian orbit.

BTW, all the effects of Newtonian gravitation come from the GR-induced
curvature of spacetime across *time* since Newtonian physics only treats
particles which move nearly parallel to the time axis, because they have
speeds slow compared to c.  The GR-induced curvature effects of space
itself do not show up at all in the Newtonian limit.

David Bowman
David_Bowman@georgetowncollege.edu