Defining the dimension of a singularity?

[bcrowell]

In general, how do you define the dimension of a singularity? E.g., we think of a Schwarzschild singularity as pointlike, so that its world-line is one-dimensional, and on a conformal diagram we represent it as a spacelike line, which seems to make sense.

In point-set topology, we have definitions of dimension like the Lebesgue covering dimension and the inductive dimension, but this doesn't seem to help in the case of a singularity, which isn't actually part of the manifold.

If you define a singularity by saying that a spacetime has a singularity if there are incomplete geodesics, then maybe you need to define the dimension of the singularity by saying something about the dimensionality of the set of incomplete geodesics...?

 

[bcrowell]

Bump.

 

[PAllen]

Is there a way to formalize a limiting argument? For example, for the spherically symmetric case, whether one use Schwarzschild or Kruskal coordinates, you can define a 4-tube outside of which there is no singularity. You can decrease the volume of this tube arbitrarily. The limit as cross section goes to zero of tube is a line. A similar argument for a ring singularity might lead to a sheet or 2 d singularity.

 

[Hurkyl]

Homology and homotopy are tools for classifying the "holes" in a topological space. You can talk in some sense about the 'dimension' of a hole by what groups detect it.

For example, if I remove the origin from R³, this shows up in the second homology group, more or less because the hole can be enclosed in a sphere.

Removing an entire ball from R³ results in the same^* topological space, of course.

However, if I remove the z-axis, the hole is detected by the first homology group, more or less because a circle can wrap around the hole.

If I remove the entire xy plane, the hole is now detected by the zeroth homology group, more or less because a pair of points can be separated by the hole.



Of course, what holes are present in spatial slices -- or if there are any holes at all -- depends very much on how you chop space-time up into slices.




*: Meaning homeomorphic

 

[bcrowell]

Thanks for the replies!

Hurkyl, I'm not familiar with homology and homotopy. Would you suggest WP as a first stop to learn about them, or some other online resource?

 

[Hurkyl]

My learning of the subject is rather hodge-podge so I can't give a personal recommendation.

I've heard several people recommend Hatcher's Algebraic Topology which is available online.

 

[bcrowell]

OK, I'll try that and WP. Thanks!

Seems like a fun subject. I have to wrap my mind around the house with two rooms: http://sketchesoftopology.wordpress.com/2010/03/25/bings-house/

 

[Hurkyl]

Now that I think of it, I should add the caveat that the part of algebraic topology I'm aware of (and what I think is covered in the text) is dealing with the topological information.

I can't predict if there's any geometric information you would find useful. (e.g. questions like "what is the surface area of a hole?" or issues of things being time-like vs space-like, when they would make sense)

 

[martinbn]

Hurkyl, are sure that for the computation of the homotopy and homology groups you don't already need to know enough about the singularities, including their dimensions?