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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?

ziggy
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  • 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

3 Answers3

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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:

KS black hole

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

John Rennie
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    What do you mean by "the geometry is undefined" – ziggy Jul 20 '15 at 06:14
  • 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 06:16
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    @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 06:17
  • @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 06:21
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    This is wrong. The dimensionality on a spacetime diagram can be different depending on the coordinate system chosen. –  Aug 12 '17 at 22:50
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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 .

anna v
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  • 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 05:57
  • 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 06:14
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    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 07:13
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    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. –  Aug 12 '17 at 22:46
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    @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 05:33
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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.

Harry Johnston
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  • 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
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    @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
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    @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
  • I'm sorry Harry, but in the real world a concentration of energy in the guise of a massive star conditions the surrounding space, altering its properties, this effect diminishing with distance. As a result space is "neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitational potentials g${μν}$)"_. See the Einstein digital papers for this kind of thing, and beware of popscience that confuses mathematical abstraction with reality. – John Duffield Jul 21 '15 at 07:18
  • The relevance is that this "odd way to put it" is how Einstein put it. Some of what you read about this sort of thing is presented as Einstein's relativity, but it's very different to what Einstein actually said. – John Duffield Jul 21 '15 at 20:00
  • We've learnt a lot since then, but that's not the point. Right or wrong, nothing you've said is relevant to the question or to my answer. Or are you just objecting to my use of the phrase "in the real world" to mean "with general relativity turned back on"? I think that's over-critical, but OK, I'll rephrase it. – Harry Johnston Jul 21 '15 at 20:09
  • I didn't mean to be overly critical Harry, I raised the point because of what Anna V said. Sorry to make a big deal of it. – John Duffield Jul 21 '15 at 20:43
  • @JohnDuffield: no worries. I'm much happier with the revised version anyway. – Harry Johnston Jul 21 '15 at 20:53
  • Have an upvote for being such a nice guy. :) – John Duffield Jul 21 '15 at 21:06
  • This answer is completely wrong. 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. The singularity in a Schwarzschild spacetime does not "[exist] over a period of time." It is spacelike. Furthermore, it is not possible to define the dimensionality of a singularity without a boundary construction, and different boundary constructions give inconsistent answers. –  Aug 12 '17 at 22:49
  • @BenCrowell, but I wasn't talking about the singularity in a Schwarzschild spacetime; I said, "imagine the analogous situation without general relativity". As for the question of whether the dimensionality of the singularity is meaningful, see John's answer. – Harry Johnston Aug 13 '17 at 00:13
  • @Ben, ... oh, I see you're insisting John's answer is wrong, too. I think I'm going to need to see a reference on that; so far as I can see, the dimensionality of the singularity should be uniquely defined (in an informal sense, at least) by the metric around it. In the Schwartzchild metric, if you form an equivalence class for geodesics based on whether they converge or not, you should be left with only one parameter. I don't see how a different choice of coordinates can change that. What goes wrong? – Harry Johnston Aug 13 '17 at 00:33
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    See https://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. –  Aug 13 '17 at 04:11
  • @Ben, I really think that's an overly sophisticated analysis given the level at which this question was asked. Note that my answer says only that the singularity is usually informally described as one-dimensional, and provides an intuitive explanation of why that is, I do not claim that it actually is one-dimensional - to answer that for sure we'd need quantum gravity anyway. – Harry Johnston Aug 13 '17 at 04:50