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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.

loglo
2009-Apr-19, 11:54 AM
Dr Rocket,
Regarding point 4 - Space-time is modeled as (pseudo)-Riemannian under GR and flat space is Euclidean. If I recall correctly, you can have flat space and a curved space-time at the same time, and the curvature tensor would be non-zero.

Kwalish Kid
2009-Apr-19, 12:16 PM
DrRocket,

The cosmological principle is only applicable as an approximation to the universe. It is not intended to apply to small scales, where the universe is clearly not homogeneous and isotropic. One uses the cosmological principle to create a model of the universe that is a good approximation to the behaviour of systems at very large scales. Often, there are corrections considered for the actual deviation from the cosmological principle that the universe exhibits. For example, the 1999 paper from the Supernova Cosmology Project considers the difference to their results if there were significant inhomogeneities in the path to the supernovae they observed. (The results were relatively unchanged.)

DrRocket
2009-Apr-19, 02:13 PM
Dr Rocket,
Regarding point 4 - Space-time is modeled as (pseudo)-Riemannian under GR and flat space is Euclidean. If I recall correctly, you can have flat space and a curved space-time at the same time, and the curvature tensor would be non-zero.

Thanks for that observation, I have edited the original post to make note that the leaves of the foliation inherit a Riemannian metric from the Lorentzian metric of the full space-time.

But the basic question, more clearly formulated as a result of your observation, remains. The real issue relates to the space-like slices.

Ken G
2009-Apr-19, 06:23 PM
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 ?

I'm not sure I see the fundamental issue you are having, it sounds like your reference to analytic functions is a reference to their property that if you knew one exactly in any neighborhood, you would know it everywhere. But even if so, that does not mean that being very close to having zero curvature over large scales requires that it cannot deviate strongly from zero on short scales.

DrRocket
2009-Apr-20, 12:02 AM
I'm not sure I see the fundamental issue you are having, it sounds like your reference to analytic functions is a reference to their property that if you knew one exactly in any neighborhood, you would know it everywhere. But even if so, that does not mean that being very close to having zero curvature over large scales requires that it cannot deviate strongly from zero on short scales.

I think that you jogged my thoughts in the right direction. The answer to my original questions is that my logic was faulty. Here's why.

The issue was to what extent the manifold that describes reality (at least within general relativity) could be accurately described in the large by some other manifold. Or, given some description of one of the leaves of the foliation used by cosmologists, could it be perturbed analytically to produce the true manifold of reality.

I was thinking in terms of theorems that describe the behavior of analytic functions at infinity -- in particular Liouville's theorem that states that a bounded analytic function is constant. BUT Liouville's theorem applies only to complex-analytic functions, it does not apply to real-analytic functions. Therefore I believe that one could, on an open manifold of constant curvature have an analytic perturbation that produces the "real" manifold. I don't have a proof of this, but the line of thinking that I was pursuing to say it is impossible won't work, and I don't think any line will. It is easy to do this in the context of differentiable functions.

As an aside, and you probably know this already, the result that you noted, that knowing the value of an analytic function on a neighborhood determines the value everywhere, is true for both real-analytic and complex-analytic functions. It is a standard theorem in complex variable theory, and the real case follows by simply noting that any real-analytic function extends to a complex-analytic function in some open set (just use the defining local power series and put in complex numbers). You in fact can use a stronger result -- you don't need an open set, all you need is a set of points with a cluster point in the domain of definition). Liouville's theorem doesn't follow for real-analytic functions because while you can extend a real-analytic function from the line to an open subset of the complex numbers being bounded on the real line doesn't make you bonded over the complex numbers.