Re: topological equivalence?

ajiko wrote: 
Start with a curved GR world. [...]
Now take that curved space and flatten it out. [...]


There are two major problems with this:

 1. It is only possible for manifolds with topologies consistent
    with a flat metric. In particular, FRW manifolds with positive
    spatial curvature have spatial topology S^3 and this simply is
    not possible.

 2. The GR manifold is spaceTIME, and you are discussing only the
    "space part" -- that's a problem as it can depend on how
    you foliate spacetime into space and time.

There are lots of minor problems, too. For instance, any singularities will result in holes punched into your flat manifold -- in the curved manifold they make sense, but in a flat manifold they don't. And the foliation is subtle but important: for instance, I doubt a manifold with two black holes that inspiral and merge can be manipulated this way, not for topological reasons, but because the foliation you envision is probably not possible.


 Compact
the space where it is curving so that it becomes flat.  Black holes would
require a deep reach to pull the space back into the small volume that we
imagine it to be in.

But what do you do with the singularity inside? It is topologically significant. Some singularities are topologically non-trivial....



The new space would then be described with a different kind of space-time
metric.  There would be a space-time scaling field that maps the original
space-time to the manipulated space-time (or vice-versa).

Except for singular points. And ONLY for manifolds for this this is topologically possible. For instance, for a spatial manifold with topology S^3 this is NOT possible, because there can be no such "scaling field" (the problem is the requirement that it be a field).


Tom Roberts	tjroberts@xxxxxxxxxx
.

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