Re: GR1916, available online.

On Dec 9, 11:57 am, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:
Alen wrote:
[...]
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.

Huh? The fundamental insight of GR is that the geometry of the manifold
is determined by the contents of the manifold, and not by any God-given
fiat (as it was previous to GR -- Euclidean geometry was universally
assumed).

Once the manifold with metric is determined, then the set of all
geodesic paths on it are also determined.

Yes, that is supposed to be the case, but we appear
to have the problem of an unvisualisable manifold.

Note that in GR, if one "adds" a "force field" to a previously found
manifold, one destroys the validity of the solution. One must start over
and re-solve the field equation, with the "force field" included. That
is, any such "force field" will itself alter the geometry of spacetime
(in addition to altering the trajectories of objects subject to it).

I did not intend to suggest the 'addition' of a force field but,
rather, a geometrical interpretation of the force field picture,
as a possible equivalent alternative to GR. An applied force
causes an equal and opposite reaction, via acceleration,
which produces a trajectory. Therefore, along the trajectory,
there is no nett force acting, so that you can examine the
trajectory from a purely geometrical perspective, if you
create a manifold, of which it is the geodesic.

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.

This is just wrong -- a manifold is not constructed that way, at least
in physics (in math there are lots of ways to construct a manifold).

Even mathematically your procedure will not create a
suitable manifold, because the "spherical segments" you
constructed cannot possibly fill a 4-d spacetime manifold.

No, but that doesn't necessarily matter. A trajectory, or
geodesic, is fundamentally a one-dimensional reality, which
can therefore have a resultant curvature in one direction only,
at any particular point along it. This will automatically make
it a geodesic of a 2-d curved surface, which must be the
resultant of the 4-d manifold at that point. In other words, for
a 1-d geodesic, in a 4-d manifold, it will always be possible,
at any particular point on the geodesic, to find 4-d coordinate
reference frames, having two of the curvilinear coordinate axes
lying along the 2-d spherical-surface segment, with the others
off it, such that there will be two g_uu values less than unity,
with the other two equal to unity, and all g_uv values zero (u != v)
This is a consequence solely and immediately based
on the fact that a geodesic has an intrinsically 1-d nature.

I think that it should be possible to elevate this to the quality
of a theorem in n dimensions. That is, a point on a 1-d curved
geodesic of an n-dimensional manifold will always have reference
frames in which only two g_uu values are less than unity, with
the others being unity, and all g_uv values 0, where u != v. This
will mean that a 1-d geodesic in an n-dimensional manifold is
always accompanied by a local 2-d curved surface, into which
the manifold is resolved, as it were, which twists and turns its
way through the n-dimensional manifold space.

In physics, trajectories come logically after the manifold. That is, one
must have spacetime before one has trajectories through spacetime.

If the trajectory is a solution to the GR equations,

The field equation of GR determines the metric of the manifold, not any
trajectories. See above -- that's why the manifold (with metric) is
logically prior to trajectories.

> [...]

Tom Roberts

Yes, the manifold should theoretically be prior to and
independent of the trajectories, but this appears to be
impossible to visualise. We have a massive object,
in 3 dimensions, scanned into 4 dimensions along the
time direction, thus degrading the conceivable complexity
of a stationary 4-d manifold, but which is nevertheless
supposed to create a gravitational manifold prior to, and
able to support, all possible trajectories.

But even consider only trajectories that move directly
towards the centre of gravity of the massive body. An
object approaching from outer space, at near the velocity
of light, will have a 4-d trajectory that is almost a straight
line. Similar objects, at progressively lesser velocities,
will have trajectories progressively more curved, in 4-d.
Stationary objects, released at different heights in the
gravitational field (zero initial approach velocity) will have
curved trajectories, in 4-d, that will be dependent on the
height of their initial release. How can we have a single,
pre-existing, 4-d manifold that takes account of all these
possible trajectories at the same time, not to mention
countless others. This proposition just isn't convincing,
to me at least. The illustration of a heavy ball on a
stretched rubber ***, to represent such a gravitational
manifold, doesn't even remotely approach a representation
of such a complex array of possible trajectories.
Do you not see a difficulty?

I, therefore, am not inclined to trust that the GR picture
of a comprehensive, pre-existing manifold is not merely
an illusion. Perhaps a manifold is actually created for
each trajectory, by the interaction of the object with the
field, while the purely geometrical analysis of it makes
the GR picture only look as if it is right, because it gives
the correct numerical results?

I am not able to comprehensively argue such an
alternative, but I think, nevertheless, that the severe
problems involved in any attempt to visualise the
pre-existing gravitational manifold should be taken
seriously, in case we are really living under an illusory
'profound GR insight' into the nature of gravitation.

Alen

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