Re: GR1916, available online.

Alen wrote:
The GR equations appear to be extremely generalised,

Hmmm. The field equation of GR is quite general -- it must hold for any valid manifold modeling any physical world, in GR.


so that they have to be severely curtailed in order to
create a particular solution like, for example, the
Schwarzschild solution.

This is not "curtailed", but rather APPLIED. Just like the equations of Newtonian mechanics are quite general and must be APPLIED to any specific physical situation.

In the case of the Schwarzschild solution, the physical situation is that the world being modeled is both static and spherically symmetric, with vacuum everywhere but at a single point in space. This is the simplest solution that could describe a planet or star (outside its surface).


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.

Yes, if by "manifold" you really mean "manifold with metric" (which is the convention in physics, but not in mathematics). A particular manifold (with metric) is a model of a specific world.

The field equation of GR does not describe trajectories, it describes the geometry of the manifold (via the metric). The metric determines geodesics, and is an important component in determining trajectories in general (which need not be geodesics).


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.

Again yes -- that is what one means by a differential equation (which is what the field equation of GR is), and by finding a particular solution to such an equation. For GR, the "solution" is the metric on the manifold, so it is the geometry itself which is determined by the field equation.


< 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.

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.

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).


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.

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
.

Relevant Pages

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