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General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the equirectangular projection has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.

So, is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate?

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Need for a coordinate-independent definition

multiple coordinates may refer to a single point

Normally, the way we define this kind of thing in GR is that we have an atlas, and the atlas is made of charts. Each chart is required to be invertible, so no, we can't have multiple coordinates that refer to a single point. In any case, when we define dimensionality on a topological space, we do it in a coordinate-independent way. E.g., one can use the Lebesgue covering dimension.

A singularity in the metric versus a singularity on a well-defined metric background

Suppose I have a two-dimensional space with coordinates $(u, v )$, and I ask you whether $S = \{(u, v )|v = 0\}$ is a point or a curve, while refusing to divulge what metric I have in mind. You’d probably say $S$ was a curve, and if the metric was $ds ^2 = du ^2 +dv ^2$, you'd be right. On the other hand, if the metric was $ds ^2 = v^2du ^2 +dv ^2$, $S$ would be a point. This was an example where there were two possible metrics we could imagine. At a singularity, it’s even worse. There is no possible metric that we can extend to the singularity.

Hawking and Ellis have a nice discussion along these lines, including an example similar to the one above, in section 8.3, "The description of singularities," p. 276:

[The singularity theorems] prove the occurrence of singularities in a large class of solutions but give little information as to their nature. To investigate this in more detail, one would need to define what one meant by the size, shape, location and so on of a singularity. This would be fairly easy if the singular points were included in the space-time manifold. However it would be impossible to determine the manifold structure at such points by physical measurements. In fact there would be many manifold structures which agreed for the non-singular regions but which differed for the singular points.

After presenting the example, they say:

In the first case the singularity would be a three-surface, in the second case a single point.

Not a point or set of points

So is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate?

Well, technically it's none of the above. A singularity in GR is like a piece that has been cut out of the manifold. It's not a point or point-set at all. Because of this, formal treatments of singularities have to do a lot of nontrivial things to define stuff that would be trivial to define for a point set. For example, the formal definition of a timelike singularity is complicated, because it has to be written in terms of light-cones of nearby points.

Boundary constructions don't provide an answer

There are some possible heuristics you might use in order to describe the singularity as if it were a point-set and talk about its dimensionality as if it were a point-set. You can draw a Penrose diagram. On a Penrose diagram, a horizontal line represents a spacelike 3-surface, with one dimension being shown explicitly on the diagram and the other two being because rotational symmetry is implicit. If you look at the Penrose diagram for a Schwarzschild black hole, the singularity looks like a horizontal line. It's not a point-set, but if it were, it would clearly be spacelike, and it sort of looks like it would be a 3-surface. This is very different from what most people would probably imagine, which would be that it's a 1-dimensional timelike curve, like the world-line of an electron.

If you try to develop this heuristic into something more rigorous, it basically doesn't work. This program is referred to as "boundary constructions." Reviews are available on this topic (Ashley, Garcia-Parrado). There are a number of more or less specific techniques for constructing a boundary, with an alphabet soup of names including the g-boundary, c-boundary, b-boundary, and a-boundary. As someone who is not a specialist in this subfield, the impression I get is that this is an area of research that has turned out badly and has never produced any useful results, but work continues, and it is possible that at some point the smoke will clear. As a simple example of what one would like to get, but doesn’t get, from these studies, it would seem natural to ask how many dimensions there are in a Schwarzschild black hole singularity. Different answers come back from the different methods. For example, the b-boundary approach says that both black-hole and cosmological singularities are zero-dimensional points, while in the c-boundary method (which was designed to harmonize with Penrose diagrams) they are three-surfaces (as one would imagine from the Penrose diagrams).

May depend on the type of black hole

People have studied GR in more than 3+1 dimensions, and, e.g., in 4+1 dimensions you can get things like "black rings." If you look at how these are actually described in the literature, people seem to find it more convenient to talk about the topology of the horizon, rather than the dimensionality of the singularity: http://relativity.livingreviews.org/Articles/lrr-2008-6/fulltext.html . This is presumably because the formalism is ill-suited to talking about the dimensionality of the singularity.

Another example is the Kerr metric for a rotating black hole. The singularity is commonly described as a ring. But in this review article, there is a discussion of the singularity on p. 8 and again on p. 28. In both spots, there are scare quotes around "ring." Again I think this is because we can't really answer geometrical questions about the singularity, because it isn't a point-set, and therefore you can't say what the metric looks like there.

A strong curvature singularity criterion doesn't provide an answer

Another approach is to look at what happens to matter that goes into the formation of a black hole through gravitational collapse, or to a hypothetical cloud of test particles that falls into an eternal black hole. If the matter is crushed to zero volume, then it might make sense to interpret this as evidence that the singularity should be thought of as having zero volume.

Unfortunately this doesn't necessarily give a definitive answer either. One can define something called a strong curvature singularity (s.c.s.), defined as one for which a geodesic is incomplete at affine parameter $\lambda=0$, with $\lim_{\lambda\rightarrow0}\lambda^2R_{ab}v^av^b\ne0$, where $v^a$ is the tangent vector. The volume of a cloud of test particles goes to zero as it approaches such a singularity, the interpretation being that infalling matter is crushed, not just spaghettified. A Schwarzschild spacetime's singularity is not an s.c.s., because it's a vacuum spacetime, so the Ricci tensor vanishes. That is, there is only spaghettification, not crushing. A cloud of test particles maintains an exactly constant volume as it falls in.

However, a completely different situation could possibly exist during the collapse leading to the formation of an astrophysical black hole. During the collapse, there is infalling matter present, so the Ricci tensor need not vanish. Indeed, it appears that in some fairly realistic models of gravitational collapse, the singularity, during the period of collapse, is a timelike (locally naked) singularity (Joshi), meaning that it is completely different in character from the spacelike singularity of an eternal black hole such as a Schwarzschild black hole. It appears that in such calculations, the density of matter does blow up at the singularity, which suggests that it may be an s.c.s. during formation.

Difficulties in the case of the Schwarzschild spacetime

When we think about a black hole, we usually imagine by default the kind of eternal black hole described by the Schwarzschild spacetime. Some significant difficulties occur even in this simplest possible case. As noted above, the singularity may have a completely different character than the singularity occurring during the gravitational collapse of an astrophysical black hole, and this leads to suspicion that by considering the Schwarzschild case, we are omitting essential considerations.

Furthermore, we have the no-hair theorem, which states that for a stationary electrovacuum spacetime having an event horizon, there is only one class of solutions, which can be parametrized by three variables: mass, charge, and angular momentum. This defines a clear sense in which the singularity of a stationary black hole has no physical properties. If it did have such properties, those would have to be limited to the list of three properties described by the no-hair theorem. However, these are not properties of the singularity but rather properties of some large region of spacetime as measured by a distant observer, who can't even say whether the singularity exists "now." (It may in fact appear to such an observer that infalling matter takes infinite time to pass through the horizon.)

The chances for a more definitive answer might be better in the case of a naked singularity. Such a singularity can exist in both the past light cone and the future light cone of an observer, so one can imagine doing experiments on it and finding the results. In this sense it may be more likely to have measurable properties.

References

Ashley, "Singularity theorems and the abstract boundary construction," https://digitalcollections.anu.edu.au/handle/1885/46055

Garcia-Parrado and Senovilla, "Causal structures and causal boundaries," http://arxiv.org/abs/gr-qc/0501069

Joshi and Malafarina, "All black holes in Lemaitre-Tolman-Bondi inhomogeneous dust collapse," https://arxiv.org/abs/1405.1146

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  • $\begingroup$ Can you outline the differences between the various boundary constructions, from a physical point of view? The obvious approach is to form an equivalence class of the geodesics that intersect the singularity, based on whether they converge or not, and it's hard to see what other choices would be reasonable. Also, could you expand on why you say that the Penrose diagram approach gives you a 3-surface rather than a line? That seems wrong. $\endgroup$ – Harry Johnston Aug 15 '17 at 4:08
  • $\begingroup$ @HarryJohnston: My answer gives links to review articles on boundary constructions. There's no way to explain the whole topic in an SE answer. Also, could you expand on why you say that the Penrose diagram approach gives you a 3-surface rather than a line? That seems wrong. I'm not sure what expansion would be helpful. What seems wrong to you? $\endgroup$ – Ben Crowell Aug 30 '17 at 23:53
  • $\begingroup$ Looking at the first reference, most of the constructions don't look all that hard to summarize: the g-boundary uses geodesics (the same as my suggestion) whereas the b-boundary uses parallel transport and the c-boundary uses causality relationships. The a-boundary I don't know. The b-boundary is clearly dysfunctional, and according to the paper it is believed (though unproven) that the g-boundary and c-boundary are equivalent, at least in the simpler cases. So at this point I'm not entirely convinced that classifying the Schwartzchild singularity is as hard as your answer suggests. :-) $\endgroup$ – Harry Johnston Aug 31 '17 at 2:05
  • $\begingroup$ As for the Penrose diagram, that looks like a spacelike line to me, not a three-surface, for the same reason that the center of a sphere is a point and not a two-surface. The radial coordinates converge. $\endgroup$ – Harry Johnston Aug 31 '17 at 2:11
  • $\begingroup$ I agree with @Harry Johnston, that's above the OP and the context in which the question was asked, along with it's other shortcomings. Kerchhoff's principle: your failure to consider "quantum gravity theory" en.wikipedia.org/wiki/… (an unsolved problem) means by definition your answer is probably incorrect: "Such a theory is required in order to understand problems involving the combination of very high energy and very small dimensions of space, such as the behavior of black holes, ...". - quel dommage. $\endgroup$ – Rob Feb 8 '18 at 6:11
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Is black hole singularity a single point?

General relativity [(GR or GTR)] expressed in terms of differential geometry [three-dimensional Euclidean space] ... lets you to do interesting things with the coordinates: multiple coordinates may refer to a single point.

EG: The equirectangular projection has a whole line at the top and bottom that correspond to North and South poles; or a single coordinate may refer to multiple points, for example by using inversive geometry the origin refers to all the points infinitely far away.

The apparent (longitudinal) coordinate singularity at the 90 degree latitude in spherical coordinates is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity, e.g. by replacing the latitude/longitude representation with an n-vector representation.

A spherical object like the Earth, or a black hole that rotates, becomes an oblate spheroid due to it's rotation. Mapping an oblate spheroid to an equirectangular projection will result in distortion.

See also JPL's "Transformations from an oblate spheroid to a plane and vice versa: The equations used in the cartographic projection program MAP2".

Gravity around a black hole is so powerful that it affects both space and time, for that reason a metric (to use your word, map projection) is used to describe the space, and not Euclidean differential geometry. In order to take gravity into account, physicists use the theory of general relativity, which is formulated in the mathematics of a non-Euclidean geometry.

A few articles discussing these calculations are:

Other reading:

You are misusing compactification to gain an understanding of a complicated subject, this is only OK if you don't want to devote a lot of time to learn a difficult subject and only desire a rudimentary understanding.


So is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate?

A single point is akin to a pointlike particle, a massless particle of zero size; an infinite number of them would weight: $\infty \times 0$. Without weight there would be no gravity and without gravity there would be no black hole.

Even zero sized particles can have ill defined boundaries, and many clumped together make exact analysis and estimation of the center; in order to provide a location, a difficult task. Because of their zero dimension many can occupy the same location, passing through each other, or move minutely apart creating an extended object.

A black hole on the other hand has mass, which means size, and while we can't see them either, their minimum size is far larger than a point particle. They are a singularity (have no size) in the sense that they are so massive that they have warped space in their vicinity to a point, much like a fun-house mirror can make you a different shape yet your true shape and size remains unchanged.

In principle, a black hole can have any mass equal to or above the Planck mass (about 22 micrograms). To make a black hole, one must concentrate mass or energy sufficiently that the escape velocity from the region in which it is concentrated exceeds the speed of light.

In reference to their mass Manasse Mbonye, in "A black hole cast on a non-commutative background" (link above), says this:

"Noting that the black hole mass spectrum covers a large parameter space, with a conservative estimate of ∼42 orders of magnitude (from say a 1g primordial black hole to a 10$^{42}$ g), this suggests an equally large spectrum in the central black hole density. Thus, in theory the density function could take on any values since neither theory nor observation puts a hard upper bound on black hole mass, yet.".

Before suggesting 22 micrograms could be compressed to create a physical singularity (not a gravimetric one), a zero dimensional point, consider how large it is compared to a point particle.

Point particles can and do pass right through you all the time. If you inhaled 22 micrograms of botulinum toxin type H it would be enough to kill you (1692 / your weight in kg) times over.

A small amount of mass is much larger than a point and so is a black hole. Black holes only occupy such a small amount of space because they (bend) compress the area to tiny dimensions. The space surrounding such a large mass (often over 10 times the mass of our Sun) occupying such a small area is a point (a singularity). A black hole would not be a point sized dimension if it's gravity could be turned off, if it occupied a significantly larger area (reduced density), or it's mass (weight) where enormously reduced.

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  • $\begingroup$ for that reason a metric (to use your word, map projection) is used to describe the space, and not differential geometry. This doesn't make sense. A metric is a tool used in differential geometry. $\endgroup$ – Ben Crowell Feb 8 '18 at 0:53
  • $\begingroup$ A black hole on the other hand has mass, which means size Not true. An electron has mass, but as far as we know it's pointlike. $\endgroup$ – Ben Crowell Feb 8 '18 at 0:54
  • $\begingroup$ @Ben Crowell - First comment reply. Corrected by adding a word. In English we have context, which remains current through other sentences without being explicitly re-stated. I've now left it less confusing. You understand that the answer is simplicity based, based upon context in which it was asked. -- I'll address your second comment and scrutinize your answer when I have some time available (you might want to check it). You are arguing about a point mass (with mass) vs. a pointlike particle without mass; a en.m.wikipedia.org/wiki/Dimensionless_quantity -- Arnold Quote: I'll be back. $\endgroup$ – Rob Feb 8 '18 at 4:18
  • $\begingroup$ @BenCrowell One should keep clear in mind that GR is a classical theory, and that elementary particles are quantum mechanical entitities. One cannot use quantum mechanical particles to argue GR points until gravity is really quantized ( unless you want to enter the mathematical terminology of strings where gravity is quantized but not yet modeling measurements, and most people would not be able to follow) . In classical physics there are no particles with zero mass and as far as I can see this answer is within classical physics. $\endgroup$ – anna v Feb 8 '18 at 4:30
  • $\begingroup$ Thanks @anna v - I've made a smart comment on his answer above. When I get a break I'll revisit this. $\endgroup$ – Rob Feb 8 '18 at 5:30

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