DrRocket

2009-Apr-19, 02:02 AM

Edit: After a bit of prodding from Ken G the answer become clear -- My logic is faulty. And yes, I missed something. {Big surprise -- a whole bunch of smart guys have not made a relatively simple error.)

I am not a GR expert, but perhaps experts can answer my question. Bear with me as it takes a bit of work to state the question.

1. The cosmological principle is an idealization that assumes that the space-time manifold of the universe is homogeneous and isotropic.

2. With that assumption comes the ability to realize the space-time manifold as a 1-parameter family of space-like hypersurfaces of constant curvature sectional curvature, a foliation by spaces of constant curvature. It is this foliation that permits one to speak sensibly of the notion of "space" in the general context of space-time and to ask whether space is open or closed. Among the possiblities is curvature of zero -- which is normally assumed to be Euclidean space.

3. If I understand some of the writings of Penrose, there are at least some who believe that the geometric features of space-time are described by analytic functions. The curvature tensor would be one such feature.

4. If the curvature is analytic and if space-time is flat and Euclidean, even in the large, then should not it follow that the curvature tensor is actually zero everywhere ? (Edit: This observation also applies to each of the 3-dimensional space-like leaves of the folation, which inherits a Riemannian metric from the Lorentzian 4-dimensional space-time.)

5. It is pretty clear that space-time ) does not have a curvature tensor that identically zero -- we don't seem to be flying off the surface of the Earth into space. (Edit :This also applies to the spacial sub-manifolds) Gravity is real.

This seems to be a contradiction, that indicates a faulty assumption somewhere, or faulty reasoning on my part.

The question (s): Have I missed something? Is it impossible that analytic functions describe space-time ? Is the problem in the assumed cosmoligical principle itself, and the cosmological principle cannot be valid ? Or is it simply impossible to have an open universe ?

Thanks to Ken G for making me think about these issues. I am now confused on a higher plane.

I am not a GR expert, but perhaps experts can answer my question. Bear with me as it takes a bit of work to state the question.

1. The cosmological principle is an idealization that assumes that the space-time manifold of the universe is homogeneous and isotropic.

2. With that assumption comes the ability to realize the space-time manifold as a 1-parameter family of space-like hypersurfaces of constant curvature sectional curvature, a foliation by spaces of constant curvature. It is this foliation that permits one to speak sensibly of the notion of "space" in the general context of space-time and to ask whether space is open or closed. Among the possiblities is curvature of zero -- which is normally assumed to be Euclidean space.

3. If I understand some of the writings of Penrose, there are at least some who believe that the geometric features of space-time are described by analytic functions. The curvature tensor would be one such feature.

4. If the curvature is analytic and if space-time is flat and Euclidean, even in the large, then should not it follow that the curvature tensor is actually zero everywhere ? (Edit: This observation also applies to each of the 3-dimensional space-like leaves of the folation, which inherits a Riemannian metric from the Lorentzian 4-dimensional space-time.)

5. It is pretty clear that space-time ) does not have a curvature tensor that identically zero -- we don't seem to be flying off the surface of the Earth into space. (Edit :This also applies to the spacial sub-manifolds) Gravity is real.

This seems to be a contradiction, that indicates a faulty assumption somewhere, or faulty reasoning on my part.

The question (s): Have I missed something? Is it impossible that analytic functions describe space-time ? Is the problem in the assumed cosmoligical principle itself, and the cosmological principle cannot be valid ? Or is it simply impossible to have an open universe ?

Thanks to Ken G for making me think about these issues. I am now confused on a higher plane.