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Notes on angular momentum and rest frames in curved spaces
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Terry Bollinger
Terry Bollinger
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Addendum 2012-05-30.20:220 EST

Angular momentum

I skipped over angular momentum in defining the set of objects used to define the minimum energy SR frame. Unlike linear momentum, residual angular momentum in a set of object of course cannot be resolved simply by resetting the frame. Always interesting, angular momentum, since if it exists it necessarily "points" to additional matter that exists outside the current set, at least if you believe in absolute universal conservation of it (which I certainly do). Its energy can be quantified locally, though.

"Center of mass" in curved spaces

@LubošMotl and others brought up the excellent point that you can't really do a center of momentum for curved space. If you assume that the curved space can be embedded within a larger Euclidean space (Nash proved that in his last pre-breakdown paper, which was an amazing work), then it's easy to see that the center of mass of the curved piece is not very likely to fall within the piece itself. It's more likely to wind up somewhere "other" than the surface, e.g. for an ordinary balloon it falls at the center of the balloon! So, when I say "center of mass frame," I'm really using a sloppy shorthand for a procedure that would require following along the locally 3D surfaces to construct an overall average. Theorem: For any smooth manifold decorated with components that move locally in accordance to the accepted rules of mass, energy, and momentum conservation, and whose shape can be approximated as being invariant over the time scales considered for those motions, there exists a singular "manifold rest frame" that is motionless with respect both to (a) the overall form of the manifold if it is irregular, and (b) with respect to the SR energy minimum defined for the entire set of components moving within the manifold.

(I am dancing lightly over rotations in 4D; there are two orthogonal ones, makes life interesting. I'm also skipping over the issue of the two different forms of 4D holes, the equivalents of 1-spheres (rings) and of 2-spheres (balloons), on the assumption that such topologies are unlikely for a realistic universe.)

(All dimensional solids n>3 have more than one type of, each hole type having an equivalence to one of a series of lower dimensional spheres from the 1-sphere (ring) to the (m-2)-sphere. That's not counting erasures (m=n), splits (m=n-1), and internal voids (m=0) (Do they address that in string theory? If not, how do they specify or distinguish between the 7 or 8 hole types of the n=9 or n=10 spaces used in M-theory? Ah.. hmm... maybe I should make that into a real question?)

Addendum 2012-05-30.20:220 EST

Angular momentum

I skipped over angular momentum in defining the set of objects used to define the minimum energy SR frame. Unlike linear momentum, residual angular momentum in a set of object of course cannot be resolved simply by resetting the frame. Always interesting, angular momentum, since if it exists it necessarily "points" to additional matter that exists outside the current set, at least if you believe in absolute universal conservation of it (which I certainly do). Its energy can be quantified locally, though.

"Center of mass" in curved spaces

@LubošMotl and others brought up the excellent point that you can't really do a center of momentum for curved space. If you assume that the curved space can be embedded within a larger Euclidean space (Nash proved that in his last pre-breakdown paper, which was an amazing work), then it's easy to see that the center of mass of the curved piece is not very likely to fall within the piece itself. It's more likely to wind up somewhere "other" than the surface, e.g. for an ordinary balloon it falls at the center of the balloon! So, when I say "center of mass frame," I'm really using a sloppy shorthand for a procedure that would require following along the locally 3D surfaces to construct an overall average. Theorem: For any smooth manifold decorated with components that move locally in accordance to the accepted rules of mass, energy, and momentum conservation, and whose shape can be approximated as being invariant over the time scales considered for those motions, there exists a singular "manifold rest frame" that is motionless with respect both to (a) the overall form of the manifold if it is irregular, and (b) with respect to the SR energy minimum defined for the entire set of components moving within the manifold.

(I am dancing lightly over rotations in 4D; there are two orthogonal ones, makes life interesting. I'm also skipping over the issue of the two different forms of 4D holes, the equivalents of 1-spheres (rings) and of 2-spheres (balloons), on the assumption that such topologies are unlikely for a realistic universe.)

(All dimensional solids n>3 have more than one type of, each hole type having an equivalence to one of a series of lower dimensional spheres from the 1-sphere (ring) to the (m-2)-sphere. That's not counting erasures (m=n), splits (m=n-1), and internal voids (m=0) (Do they address that in string theory? If not, how do they specify or distinguish between the 7 or 8 hole types of the n=9 or n=10 spaces used in M-theory? Ah.. hmm... maybe I should make that into a real question?)

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Terry Bollinger
Terry Bollinger
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Energy conservation and center-of-mass inertial frames

For a given set of fast-moving objects in empty space, the maximum mass-energy that can be extracted by using only interactions between objects in the set (e.g., colliding all of them together) is identical to the relativistic mass-energy that an observer would see when residing in the center-of-mass inertial frame for that set. (You can trust me on that or prove it yourself; it's not difficult.)

This is why a proton colliding with earth at $\gamma=10$ velocity does not cause a catastrophic release of ten earth's worth of energy. Since the mass of the proton is entirely negligible in comparison to the earth, the mass-energy available by colliding all objects in the set {earth, proton} is very close to the mass of earth plus ten times the mass of the proton.

Thus for any set of objects that are isolated in space, the center-of-mass inertial frame is the only special relativity frame for which the observed mass-energy of all parts is identical to the available mass-energy. More energy can of course be extracted, but only by bringing in objects that were not in the original set. The center-of-mass inertial frame thus is both "distinguished" and unique with respect to that set of objects (only).

Now here's the fun part: Define the set to include all objects in the universe. Since you have just placed all your eggs in one basket, so to speak, there are no longer any objects to bring in from outside to alter the result, even in principle. Next, calculate for "distinguished" inertial frame for this set. (For a pretty good guess, try the CMB.)

My question is this: Doesn't the above argument imply that under the rules of special relativity the universe really does have a single, unique, and non-trivially distinguished inertial frame, that being its center-of-mass inertial frame?