Riemann's problems as per TVF
As MTW comment on p. 32, Riemann spent his entire life in an unsuccessful search for a curved space theory.
In general, the primary reasons for failure of such theories are:
(a) Only a single curvature can be attached to a single point in space at a single instant of time. Yet bodies of different speeds passing through that point all curve by different amounts. And a taut rope passing through it is Euclidean straight.
(b) A stationary body cannot be induced to commence motion by space curvature alone. An external force must act.
(c) Changes of momentum require a source for the new momentum which must itself possess that momentum (i.e., must move) before it can deliver it to a target body. Curved space postulates new momentum ex nihilo, which is a miracle, forbidden in physics.
How does your theory deal with these allegedly fatal objections that stopped all attempts by the minds that preceded you?
- Tom Van Flandern
Let's start here ...
We postulate that particles have internal properties that cause them to perform actions in place. That is, a particle jumps around within a confined volume, rotates on some axis or even disappears and reappears within that volume ... in accordance with the uncertainty principle.
If it exists in a general Euclidean manifold, the locus of such actions will remain in place having no particular reason to go in any preferred direction. But if we now composite a distant spherical reference frame with that Euclidean frame, the locus of action will move toward the center of the spherical reference frame, i.e. the vertical component of the action vector remains straight while the lateral component is bent closer to the center of the spherical frame than it would otherwise have been ... or ... the probability of any action occurring will be similarly skewed. This accelerates the particle in that direction.
The particle itself possesses its own spherical reference frame. Due to the necessarily finite rate at which information is transmitted in any field, its frame is "left behind" until notification of movement of the particle arrives at any point in that frame. Hence, the accelerated frame is distorted into an ellipsoidal shape, the curvature of which opposes the direction of motion (inertia).
Let us then imagine a small cube of this composite, curved reference frame placed in a flat Euclidean frame. An object moving along a straight line in the Euclidean frame encounters the curved cube and upon entering it, has its trajectory altered by some amount depending on the amount of time it spends inside the cube. Thus, the trajectory of a faster moving object is "bent" less by the same cube.
And a taut rope in such a composite curved field will approximate a catenary curve no matter how tight it is pulled. It cannot assume an opposite curvature to conform to that reference frame because the accelerations forcing it into a catenary curve swamps any other miniscule effect.
Changes in Momentum
I agree that a stationary body cannot commence movement in response to a curved space alone. It requires the aforementioned "internal actions" to proceed appropriately.
Physics does not forbid the appearance of momentum so long as an equal and opposite momentum is produced ... which is the case here. Objects are mutually pulled toward one another with exactly cancelling momentums. If we say that they must invariably be "pushed", we have the counter-example of magnetic and electric phenomena which give no appearance of such everyday pushes (which underlie our expectation of finding them at all levels of physical inquiry). Electromagnetism seemingly produces equal and opposite momentums out of nothing but no one regards this as magic.