Re: GR1916, available online.

On Nov 29, 1:27 pm, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:
Alen wrote:
On Nov 23, 5:14 am, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:
Let's imagine the earth is in the region of the model, and is scaled
down to 1 mm in radius; the moon is much smaller, and orbits the earth
about 35 mm away from the earth; ignore all other objects. Let's arrange
the mapping of world => model so that the earth follows a straight line
right along the time axis of the model (because it is timelike, and we
can choose the mapping to do this). Now the moon follows a helix,
orbiting the earth but also moving along the time axis (it, too, is
timelike). Indeed, as the speed of the moon relative to earth is
something like 10^-6 c, for every 35 mm traveled along the time axis, it
will make something like 10^6/2pi orbits around the earth -- a very
tight "corkscrew".

Yes, I understand this visualisation. The problem
I have with it is that the earth is a line,

No. The earth is not a line, its TRAJECTORY is a line.

so that its
gravitational field at a distance d, say, from the line,
is a constant along the time direction, because it, too
is formed as a line in the time direction.

The gravitational field is not a scalar or vector, it is a rank-2 tensor
and has more structure than that.

Thus the
gravitational manifold of the earth in this picture is
a constant_field cylinder of radius d.

No. The "gravitational field" is not such a cylinder, but a contour line
of the field MAGNITUDE is. The field itself is, at each point of that
cylinder, such that a test particle there will be accelerated INWARD
(i.e. not along the time direction, but rather along a vector pointed
toward the earth at the center of the cylinder).

This would suggest that, at different velocities, the helix
of the moon's orbit would be stretched or compressed in
the time direction, but remain at the same orbital distance,
d, i.e., confined to the cylindrical manifold, making the orbit
independent of velocity, which it obviously is not.

The structure of the field is more complex than your visualization.

Tom Roberts

Yes - you are right, and that is my point. It seems
impossible to create an adequate visualisation of the
manifold to take account of all possible trajectories at
the same time. That leads me to suspect something like
the following:

The GR equations appear to be extremely generalised,
so that they have to be severely curtailed in order to
create a particular solution like, for example, the
Schwarzschild solution. It therefore occurs to me that
this process may not only limit the equations to
describing a specified trajectory, but also limit them
to specifying a particular manifold as well. That is
to say: perhaps the generalised equations do not really
specify a single manifold only, but an entire array of possible
manifolds, from which one is automatically selected when
a particular solution is selected. In such a case, GR
would no longer be the fundamental insight it is supposed
to be, but really only a more complicated way of examining
a trajectory, including one in a simple force field in flat space.

Take a smooth curved 4-d trajectory. At any point, P, you
can get a radius of curvature, in whatever direction, and
describe a small segment of a spherical surface about
the point P. The small segment of the trajectory, through P,
will then be a geodesic of the spherical surface. Move a
small distance, ds, along the trajectory, and repeat the
process, and do the same all along the trajectory, and
smoothly join together all the small spherical segments,
and you will have a manifold of which the trajectory is a
geodesic. If the trajectory is a solution to the GR equations,
will this manifold not be precisely that to which the GR
equations must refer, even if it is relevant only to a flat
space, forcefield interpretation? In other words, might
GR not be merely a more complicated, geometrical
way of describing any kind of trajectory?

This would certainly get over the apparent impossibility
of visualising some global 4-d manifold, which is supposed
to allow for all possible trajectories at the same time.

Alen
.

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