Defining the dimension of a singularity?

In summary: Because if you do, it would restrict the number of holes that you could find.Yes, there's no need to know about the dimensions of the singularity to find holes in the topological space.
  • #1
bcrowell
Staff Emeritus
Science Advisor
Insights Author
Gold Member
6,724
431
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...?
 
Physics news on Phys.org
  • #3
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
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
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
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.
 
  • #8
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
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?
 

FAQ: Defining the dimension of a singularity?

1. What is a singularity?

A singularity is a point in space-time where the laws of physics break down and traditional mathematical equations no longer apply. It is often associated with the center of a black hole or the beginning of the universe.

2. How is the dimension of a singularity defined?

The dimension of a singularity is defined by the number of coordinates needed to describe its properties. In general relativity, a singularity is considered to have zero dimensions as it is a point with no size or volume.

3. Can the dimension of a singularity change?

The dimension of a singularity is a theoretical concept and is not subject to change. It is considered to be a fundamental property of the singularity itself.

4. Are all singularities zero-dimensional?

In general relativity, yes, all singularities are considered to be zero-dimensional points. However, some theories, such as string theory, suggest that singularities may have higher dimensions.

5. How do singularities affect our understanding of the universe?

Singularities challenge our current understanding of the laws of physics and the nature of space and time. They also play a crucial role in theories such as the Big Bang and black holes, and further research is necessary to fully comprehend their impact on the universe.

Back
Top