Assume, the universe is shaped as an analytic manifold. Such a manifold is locally flat, say R(3) and can always be embedded into a certain R(n). In case of big bang: the universe was a POINT. But a point of what surrounding R(n)? And if the manifold is compact and expanding, it is still a subset of R(n). What are the most common used models of these expanding manifolds, into what R(n)? If the universe is not compact, then it is hyperbolic or R(n) and then it cannot expand. If the singularaity was dipol like, the universe could be toroidal (compact)
very nice, this are 24 pages: eventually you can/may tell me, what and where Penrose writes about dipol shaped big bang singularity.
Thanks a lot
T. Schober
Dear Anton,
your question should focus attention between actual global structure of the space-time and the structure of eventual singular points ...
This point of view could be suggestive but, I think, too naive !
For example your statement:
If the universe is not compact, then it is hyperbolic or R(n) and then it cannot expand.'
can be contradicted ! In fact the one-point compactification of such a space-time can be obtained with a null-hypersurface that can be reached in an infinite time ... In other word, neither a global hyperbolic space-time can exclude an expansive universe !
On the other hand by considering that boundary conditions of our actual universe are encoded by its Planck epoch, it follows that in order to know such boundary conditions it is necessary to study Planck epoch with the quantum gravity. This is what I made in this paper:
http://arxiv.org/abs/1206.4856.
Therein it is possible to understand why universe expands and what dark matter/energy are
My best regards,
Agostino
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