Defining the dimension of a singularity?

1. Jul 4, 2011

bcrowell

Staff Emeritus
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...?

2. Jul 6, 2011

bcrowell

Staff Emeritus
Bump.

3. Jul 6, 2011

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.

4. Jul 6, 2011

Hurkyl

Staff Emeritus
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

5. Jul 7, 2011

bcrowell

Staff Emeritus
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?

6. Jul 7, 2011

Hurkyl

Staff Emeritus
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.

7. Jul 7, 2011

bcrowell

Staff Emeritus
Last edited: Jul 7, 2011
8. Jul 7, 2011

Hurkyl

Staff Emeritus
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)

9. Jul 7, 2011

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?