🚀 go-pugleaf

In article <1992Dec13.215743.11645@galois.mit.edu> jbaez@riesz.mit.edu  
(John C. Baez) writes:
> In article <398@moene.indiv.nluug.nl> toon@moene.indiv.nluug.nl (Toon  
Moene) writes:
> >In article <1992Dec2.232639.13222@galois.mit.edu> jbaez@riesz.mit.edu   
(John C. Baez) writes:
> >> In article <1992Dec2.014924.8076@netcom.com> raicu@netcom.com (Eugen   
Raicu) writes:
> >> >In principle, can there be any mass distribution that would result 
> >> >in a space-time which intersects itself? 
 
> >> Not as far as I can tell, if I understand the question.  The point is
> >> that spacetime is not embedded in some higher-dimensional space so it
> >> can't "bump into itself".  This is precisely why it's nice to have an
> >> intrinsic definition of manifolds, by the way!

> >Now that's interesting - I'm sure I remember from my Differential 
> >Geometry course that ANY manifold can be embedded in Rn. A five minutes 
> >worth of flapping through the pages of my DG book* tells me that I have 
> >to modify this to: ANY manifold that has a COUNTABLE base (so-called 
> >seperable manifold).

> Of course decent manifolds *can* be embedded in R^n - it's a nice
> theorem that's often useful - but there is no canonical way of doing so,
> and there is no reason to think that our actual spacetime is in fact so
> embedded.   

Sorry for all these inclusions :-), but, ehh, is there any reason to  
believe that a space-time structure describing a *physical* system would  
not be a seperable manifold ? If it's so obvious, you could probably  
provide me with some pointers to literature (Note: I only do GR in my  
precious spare time - normally I just worry about the weather :-)

> >> Of course there are mathematically possible spacetime geometries with
> >> "wormholes," that is, "handles."  This might be the closest thing to
> >> what you are wondering about.  

> >Yep, but do they represent physical space-time structures ?

> Nobody knows but there have been millions of papers by distinguished
> physicists exploring the possibility and its possible consequences.

> "Wait a minute!" you cry.  "How come they're exploring the possible
> consequences of wormholes but not the possible consequences of spacetime
> being embedded in a higher-dimensional R^n?"  This is primarily a matter
> of esthetics.  The prevailing view is that the former idea is pretty and
> the latter one is ugly.  I could try to explain the reasons for this
> view, but we just had a long discussion of embedded vs. intrinsic
> approaches to differential geometry, so I won't.

Well, ok, I know (and appreciate) that one. I, however, still don't  
understand why it is a question *whether* the manifold describing *our*  
space-time structure wouldn't be embedded in some R^n. I read the  
aforementioned theorem as saying: Either you prove that your manifold  
isn't seperable, or it *is* embedded in some R^n. Whether there is a  
constructive proof of it doesn't interest me. Besides, it bothers me that  
one needs this to be able to prove or disprove that a manifold can 'bump  
into itself'. This somehow contradicts your last paragraph.

--
Toon Moene (toon@moene.indiv.nluug.nl)
Kantershof 269, 1104 GN  Amsterdam, The Netherlands
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