Spacetime - formal description of No Rip/Tear

[Julian M]

Could someone point me in the direction of the relevant differential geometry/topology terminology/definitions/explanation etc. to express the idea that spacetime cannot be "torn".

Methinks it's "diffeomorphic invariance" but, even if it is, a few nice words and/or an example (or two...) of what's allowable and what isn't would be welcome.

And then, if it's not asking too much, how would I go about proving that two manifolds were inequivalent?

Specific example (based on spacetime surgery for creating wormholes):

1. Take a standard simply connected spacetime manifold M
2. Consider the two situations
2a excise a pair of spheres (radius 1) centred at x₁ = 1 and x₂ = 10 and identify the surfaces of the two spheres.
2b ditto, but at x₁ = 1 and x₂ = 20

Being very specific that the x values are coordinate values and not distances (i.e. you can't expand/shrink the interval because the two x₂s are different points) , are 2a and 2b inter-convertible (if so, how etc.)?

I hope this is a sensible question...

Thanks, Julian

 

[atyy]

And then, if it's not asking too much, how would I go about proving that two manifolds were inequivalent?

From Chapter 3 of http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html: "The notion of two spaces being diffeomorphic only applies to manifolds, where a notion of differentiability is inherited from the fact that the space resembles R^n locally. But "continuity" of maps between topological spaces (not necessarily manifolds) can be defined, and we say that two such spaces are "homeomorphic," which means "topologically equivalent to," if there is a continuous map between them with a continuous inverse. It is therefore conceivable that spaces exist which are homeomorphic but not diffeomorphic; topologically the same but with distinct "differentiable structures." In 1964 Milnor showed that S7 had 28 different differentiable structures; it turns out that for n < 7 there is only one differentiable structure on Sn, while for n > 7 the number grows very large. R^4 has infinitely many differentiable structures.

For methods to see if two manifolds with metric are isometric, see http://books.google.com/books?id=YuTzf61HILAC&dq=olver+invariants&source=gbs_navlinks_s

Ray d'Inverno has a nice page about the isometry question here http://www.personal.soton.ac.uk/jav/karlhede.html .

 

[Julian M]

Thanks atyy -

I have the Sean Carroll (the information in quote you gave has, I think, ended up in the Wikipedia article on Diffeomorphism), and have just looked at the (scary) d'Inverno stuff. The Olver (where I could see bits) is obviously relevant too... so they are all, I suppose, perfectly good answers to the equivalency question in all generality.

I think however they're too sophisticated for me, and probably too heavy-duty for the problem at hand, but the reply is appreciated.

 

[JesseM]

Is the "no tear" idea related to the idea that if you foliate spacetime into a series of spacelike surfaces, the topology should never change from one surface to another?

 

[Julian M]

Good question JesseM

My first thought was: yes, I think I am saying that in classical relativity I thought the topology should be constant on each spacelike hypersurface from minus to plus timelike infinity. In which case, part of the question is "what's the proper way of describing such topological invariance in the language of GR?" Which rule(s) say(s) you can/can't do this?

But then I thought: I'm not sure whether the topology is the problem. I have a vague recollection that there is a certain type of manifold surgery that can change the topology, and thus the question is - is that sort of thing allowed in GR? If (not), why (not)?

The problem is, IIRC, it is (mathematically) possible to shrink one hole down to nothing and at the moment of disappearance to create another infinitesimally small hole somewhere else. The topology is unchanged but the holes are in different places. Well, that's part of the problem: another part is that usually manifolds have distances rather than specific locations, but spacetime does (it has "events", if not utterly empty).

The attachment shows a red and a green wormhole with the standard embedding. The question is can you turn red <-> green without changing distances. I don't think that can be done in GR but I don't have a sound basis for that judgement.