How many dimensions does a singularity have? [duplicate]

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?

marked as duplicate by Ben Crowell, honeste_vivere, Yashas, peterh says reinstate Monica, Kyle KanosAug 14 '17 at 10:22

• Note, that there are higher dimensional singularities. For instance, a rotating black hole has a circle-like singularity, not point-like. – M.Herzkamp Jul 20 '15 at 14:00

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.

• What do you mean by "the geometry is undefined" – ziggy Jul 20 '15 at 6:14
• But the diagram suppresses two space dimensions, so aren't the singularities three-dimensional then? (Just like the event horizon, though it appears one-dimensional in the diagram as well.) – Nathan Reed Jul 20 '15 at 6:16
• @ziggy: for example consider the Big Bang singularity and ask what is its dimensionality. At the Big Bang every point in spacetime was in the same place. Does that mean it's zero dimensional or three dimensional? There is no answer to this because at the Big Bang the dimensionality is undefined. See this post for more on this. – John Rennie Jul 20 '15 at 6:17
• @NathanReed: yes the angular coordinates $\theta$ and $\phi$ are not shown, but at the singularity all values of $\theta$ and $\phi$ coincide i.e. the proper distance between different values of $\theta$ and $\phi$ goes to zero as we approach the singularity. That's why I claim it's one dimensional. But you could argue that it is actually three dimensional. The point is that at singularities the dimensionality is not defined. – John Rennie Jul 20 '15 at 6:21
• This is wrong. The dimensionality on a spacetime diagram can be different depending on the coordinate system chosen. – Ben Crowell Aug 12 '17 at 22:50

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 .

• so a singularity can have any number of dimensions as long as all the values of the dimensions are zero? – ziggy Jul 20 '15 at 5:57
• Not the value of the dimensions, the value of the variable that is assigned to that dimension. It is a clear cut mathematical concept. The x axis is mapped from -infinity to + infinity ( two singular points at the very large positve and negative limit ) on the real numbers, but in between the numbers are real. x is the variable assigned to the x axis dimension – anna v Jul 20 '15 at 6:14
• Ziggy : the point singularity is zero-dimensional, and it's a "mathematical concept". Like it doesn't exist. I don't think a black hole has a point-singularity in the middle, because I think the coordinate singularity at the event horizon isn't some mere artefact. IMHO Kruskal-Szekeres coordinates are based on the schoolboy error of putting a stopped observer in front of a stopped clock and claiming he sees it ticking normally. – John Duffield Jul 20 '15 at 7:13
• This answer would be fine for discussing the dimensionality of a singularity in a function defined on a space with a predefined, non-singular background metric. It's completely incorrect for a metric singularity. – Ben Crowell Aug 12 '17 at 22:46
• @BenCrowell I hope that you are writing up an answer which shows the difference between singularities of functions defined in space and "metric singularity" that makes the above answers wrong enough for a downvote. After all the site is about sharing and increasing knowledge. – anna v Aug 13 '17 at 5:33

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.

• In the real world, there aren't any light cones. And in spacetime, there is no motion. As for space and time exchanging places, if you could describe how that works in the real world, I'd be interested to hear about it. – John Duffield Jul 20 '15 at 16:06
• @JohnDuffield: I'm not sure what you mean by "there aren't any light cones". Light cones are just a way to visualize the metric, and the metric is certainly a part of the real world. As for space and time changing places, that's not rigorous of course. When the horizon forms, the coordinate that we would naively consider to be the distance from the nominal singularity becomes timelike instead of spacelike. – Harry Johnston Jul 20 '15 at 21:05
• Take a walk outside, look up at the clear night sky, and point out a light cone. You can't, because a light cone is an abstract thing. Space is real, motion is real, your measurements are real. But the metric is an abstract thing too. So is spacetime actually. It's a mathematical space and it "works". But the map is not the territory. We live in a world of space and motion, not a block universe with worldlines running through it. I say all this as a guy who "roots for relativity" and is forever referring to Einstein. – John Duffield Jul 20 '15 at 22:13
• @JohnDuffield: that's fine, if you want to look at it that way. But the question was about this abstract idea of physics, not about the "real world" by your definition, so I really don't think your objections are relevant. – Harry Johnston Jul 20 '15 at 23:36
• See physics.stackexchange.com/a/144458/4552 . I think the part that most directly contradicts your claim is the part about how different boundary constructions give different dimensionalities for a Schwarzschild singularity. – Ben Crowell Aug 13 '17 at 4:11