Re: The Instantaneous Creation of Infinite Space

From: davidoff404 (davidoff404_AT__at__yahoo.com)
Date: 06/28/04 

Date: Mon, 28 Jun 2004 21:13:00 +0100

Perfectly Innocent wrote:
> The 3-space in which we live is one of an infinite variety of possible
> geometries. http://arxiv.org/PS_cache/gr-qc/pdf/9804/9804006.pdf. No
> mortal can identify the right spatial form. The Big Bang creation
> story comes in two incomprehensible versions. No human knows which
> version is correct. This thread is a request that knowledgeable
> mathematicians answer questions as simply as possible on the very
> essence of mathematical cosmology and what is knowable and provable.
>

Fine then. Take a compact 3-manifold M (without boundary). Define {Riem}
as the space of Riemannian 3-geometries admissable on M. Define the
quotient space {Sup(M)} as

{Sup(M)} = Riem / {3-Diff(M)}

where {3-Diff(M)} denotes 3-diffeomorphisms of the elements of M. Go one
further and define {Conf.Sup(M)} as

{Conf.Sup(M)} = Riem / ( {3-Diff(M)}x{Conf(g)} )

where {Conf(g)} denotes the set of suitably smooth conformal
transformations of 3-metrics. Define a reparamatrisation-invariant
action which uses the Yamabe theorem to include a volume term, and work
out the dynamical equations for g_{ij} and pi^{ij}. We now have a
conformal system of freely-specifiable initial data. Include a
cosmological constant in the action, or any other sources you wish, and
work out the equations of motion for the canonical variables.

Et cetera.