How many dimensions does a singularity have?

Question

my understanding of a singularity is that it is a 'point' in the universe with an infinite energy density. I have also read on the internet that a singularity is one-dimensional.

This is a concept that troubles me. using my (very limited) knowledge of what dimensions are and how they work, I have made the following assumptions:

• we live in a three-dimensional world with length, width and height (excluding time which I am confused about and am unsure as to whether it is a dimension or not)
• so a two-dimensional object would have only length and width (sort of like a flat shape in an area question in a textbook).
• so a one-dimensional object would only have length and would be a sort of 'line'.

But a 'line' is not a 'point', therefore in order to have a pointlike singularity it would need to be zero-dimensional. another aspect of the one-dimensional 'linear' singularity that I find difficult to understand is that for the finite amount of energy in the universe to be infinitely dense, it cannot be spread out along one or more dimensions, but rather it must be contained in zero dimensions. Hence the energy is 'X' joules per zero metres so the density is undefined.

are singularities one or zero-dimensional and why/how?

if the above reasoning is wrong, why is it wrong?

Answer 1

There are a few complications with your question. For example you need to distinguish between coordinate singularities and curvature singularities. A coordinate singularity is a place where the coordinate system becomes singular. A good example of this is a black hole event horizon. A curvature singularity is a place where the spacetime curvature becomes infinite and/or the geometry is undefined. Examples of this are the centre of a black hole or the Big Bang. Also, when you talk about dimensionality you need to consider spacetime not just space.

From your question I would guess you're only interested in curvature singularities, but for the record coordinate singularities are generally straightforward. For example a black hole event horizon is a spatial surface so it is three dimensional - two space and one time dimensions. Likewise the cosmological horizon that will eventually develop as our universe approaches a de Sitter spacetime.

Curvature singularities have the complication that they're not part of the universe and the geometry isn't defined there. We model the universe as a manifold equipped with a metric. As a rough guide, the manifold determines the dimensionality and the metric gives a way to measure distances. But curvature singularities are not considered part of the manifold i.e. the manifold comprises everything except the singularities. So when you ask what is the dimensionality of a singularity that is a meaningless question because the spacetime geometry is undefined at a singularity.

But this is perhaps unnecessarily pedantic. For example we routinely draw singularities on spacetime diagrams like this diagram of a Schwarzschild black hole in Kruskal-Szekeres coordinates where the singularities (there are two of them!) are the red lines:

In this diagram the singularity is a line in spacetime i.e. a one dimensional object in spacetime.

Answer 2

The concept of a singularity is a mathematical concept, not a physics concept, and you have to get that clear in your understanding. As simple as the function 1/r has a singularity at r=0, i.e the number becomes infinite. If one variable is used the singularity is one dimensional, if r is the length of a vector the singularity comes when x,y,z are zero. So you can have many dimensions for a point singularity, or you could have line singularity if you construct your function appropriately, etc. For example for 1/x at x=0 the y,z plane is singular.

As simple as that. If you have a singularity in a function mapping on the real variables it means you get an infinite real number at that point.

Now mathematics used in physics is a modeling tool. There are experiments describing data in real numbers up to the measurement accuracy. For example the 1/r potential between two unlike charges has been fitted to data, but in physics the measurement error is the limit to how close one can come to the singularity.

When we discovered electrons and nuclei we found out that our classical mathematical model of how the 1/r potential behaves had to be changed, and quantum mechanics was invented to deal with the fact that there are no singularities , i.e. the electron does not deterministically drop on the nucleus (singularity) . Instead the Schrodinger equation applies and its solutions allow for the special orbits of the electrons about the nucleus .

Answer 3

I believe John's answer is correct. However, to get an intuitive (and I think not too far wrong) picture of why we usually call the singularity one-dimensional and not zero-dimensional, imagine the analogous situation without general relativity. Instead of a black hole, we'd just have the singularity; still a point of infinite density, just one that doesn't mess with the geometry. What is the dimension of this point?

If we think just in terms of space, it's zero-dimensional. But in spacetime it is one-dimensional, because it exists over a period of time. That's the key here.

When we take general relativity into account, we find that our usual coordinate systems don't work very well any more; space and time appear to have changed places, and odd stuff happens at the event horizon. But the singularity is still usually described as one-dimensional, at least informally.