Do non-black hole singularities have observable properties?

[bcrowell]

I hope the topic of this post is not too philosophical to be appropriate here.

Some recent discussions on PF have helped to crystallize my view of how classical GR treats singularities, and black hole singularities in particular. However, I'm not sure to what extent these ideas generalize to singularities that are not black hole singularities.

Singularities are not geometrical objects, in the sense that we can't even define basic geometrical properties for them such as how many dimensions they have: https://www.physicsforums.com/threads/boundary-construction-for-of-b-h-and-b-b-singularities.833399/ . "There is no there there."

Black hole singularities are also not physical objects. A physical object would have dynamical properties. If black hole singularities were to have dynamical properties, then the no-hair theorems would limit them to mass, charge, and spin. But these are not properties of the singularity but of some large region of spacetime as measured by a distant observer. The distant observer can't even say whether the black-hole singularity already exists "now."

I feel pretty secure in these conclusions, but do the conclusions about the dynamical properties of black hole singularities extend as well to non-black hole singularities? Does the big bang singularity have dynamical properties? Would a timelike or naked singularity? A conical singularity?

MTW has a nice discussion at p. 457 that explains why, for example, we can't define the total electric charge of a closed universe -- which we might have imagined we could do because the amount of matter in such a universe is finite. Does this extend to a statement that we can never define the mass, charge, or spin of any big bang singularity?

There's a famous (mis)quote from John Earman to the effect that if we had a timelike singularity, anything at all could come out of it, including green slime or my lost socks. Does this extend to a statement that we have no way of characterizing anything about such a singularity? Here we have a failure of global hyperbolicity, which is sort of a breakdown of the entire enterprise of physics. In the absence of global hyperbolicity, it's hard to see how we can define any sane properties for anything, except perhaps in some local region of spacetime that is at a safe distant from the misbehaving regions.

 

[fresh_42]

I have some difficulties with the term singularity. What makes it one? Merely the fact that our descriptions fail? If so, it might rather be a lack of knowledge than a lack of physics. No one really can know what happens beyond the Schwarzschild horizon. Considering the Big Bang I'ld like to consider it as a similar act as virtual pair production out of vacuum. I just can't tell why it doesn't happen more often. And here comes another difficulty along: what if time is quantized? Will we get stuck at the Planck scale?
Aren't the papers in the links you've provided not attempts to fill that singularity gap in knowledge by searching for sound theories of explanation? Although I don't buy the holographic model yet.

 

[bcrowell]

I have some difficulties with the term singularity. What makes it one?

The standard definition is geodesic incompleteness.

 

[fresh_42]

Metric spaces can be completed. And as this definition is geometric shouldn't we regard the fabric of space as a projective space with black holes as kind of infinite points? Maybe I'm too much used to handle seemingly strange facts as, e.g. from the axiom of choice arise. But entanglement isn't less strange. I just hope we will find a model that describes black holes in a sufficient way. As been said some of your links try exactly this. And I remember someone - unfortunately I forgot who - stated that the Big Bang might have been more like a tiny buckle, i.e. no singularity but the maximum of a process that allows the definition of a "before", a kind of a crashed previous universe.
Maybe black holes are just Alexandroff extensions.

As long as GR isn't integrated in GUT we cannot know which modifications or extension have to be made.

Sorry if I'm annoying.

 

[Nugatory]

As long as GR isn't integrated in GUT we cannot know which modifications or extension have to be made.

That's true, but in the meantime there's no shortage of interesting and difficult open questions in the unmodified and unextended theory we already have. If I'm understanding bcrowell properly, he's asking one of those questions.

 

[fresh_42]

That's true, but in the meantime there's no shortage of interesting and difficult open questions in the unmodified and unextended theory we already have. If I'm understanding bcrowell properly, he's asking one of those questions.

Yes, and his links are some attempts of answers. I'm curious, too, and want to know. My questions might seem naive, I'm no physicist. On the other hand that allows me questions like: geodesic "incompleteness" - so let's complete it, or suggesting a Alexandroff extension, which conceivably might supply at least a possible way of considering it.
And to be honest: My experience is there's no better way of deepen an understanding than being forced to explain it. (Of course as long as the questions aren't too dumb. If so feel free to tell me.)

 

[PeterDonis]

geodesic "incompleteness" - so let's complete it

You can't; that's the definition of geodesic incompleteness (that geodesics exist in the spacetime that cannot be extended past some finite value of their affine parameter).

 

[Demystifier]

MTW has a nice discussion at p. 457 that explains why, for example, we can't define the total electric charge of a closed universe --

They actually say that "the charge is trivially zero". And this indeed is the correct conclusion. Maxwell equations in a closed universe imply that the total charge must be zero. This simple physical result can be represented as a mathematically rigorous theorem in differential topology. It does not say that the charge "cannot be defined". Just the opposite, it is well defined and highly constrained to only one possible value - zero.

 

[fresh_42]

that's the definition of geodesic incompleteness

Unless stated otherwise, well-defined and properly, "incompleteness" is just a missing limit of a Cauchy sequence. Whether on a geodesic (shortest distance between 2 points) or not. It doesn't say anything about its nature, its topological neighborhood or its capability to be completed.

 

[bcrowell]

Unless stated otherwise, well-defined and properly, "incompleteness" is just a missing limit of a Cauchy sequence. Whether on a geodesic (shortest distance between 2 points) or not. It doesn't say anything about its nature, its topological neighborhood or its capability to be completed.

If you want to propose a nonstandard definition of a singularity in GR, please start a separate thread for that. It's off topic here.

 

[bcrowell]

They actually say that "the charge is trivially zero". And this indeed is the correct conclusion. Maxwell equations in a closed universe imply that the total charge must be zero. This simple physical result can be represented as a mathematically rigorous theorem in differential topology. It does not say that the charge "cannot be defined". Just the opposite, it is well defined and highly constrained to only one possible value - zero.

You've given that quote without the context. Reading the context, they're very clearly saying that it's undefined. The end of that paragraph is: "These terms are undefined and undefinable. Words, yes; meaning, no."

I can see your point about charge in a closed universe, because clearly we can't have a solution to Maxwell's equations on such a background with, say, only one electron present and everything else being empty space. The field lines coming out of the electron can't go off to spatial infinity. By symmetry, they all have to converge somewhere, and that would be a charge +e at the antipodes.

But MTW is using this only as an illustrative example, and if you look at the context, they're making a broader point about measurement. In the example of trying to find the charge of a closed universe, we don't even have an operational definition that is valid on its face.

 

[fresh_42]

If you want to propose a nonstandard definition of a singularity in GR, please start a separate thread for that. It's off topic here.

No, Sir. I just did not understand why a singularity in GR differs from one in Math. You didn't say that there is a different usage earlier and I didn't know. Perhaps you can give me a hint where I can read about the GR version of it.

 

[PeterDonis]

Unless stated otherwise, well-defined and properly, "incompleteness" is just a missing limit of a Cauchy sequence.

And if you look at the definition of geodesic incompleteness in GR, you wil see that it is indeed "stated otherwise, well-defined and properly".

I just did not understand why a singularity in GR differs from one in Math. You didn't say that there is a different usage earlier and I didn't know.

Please note that this is an "A" thread, meaning "Advanced". That means graduate-level knowledge of the subject matter is assumed. PF has these labels on threads to make it easier to tell what level of knowledge is being discussed. If you don't have that level of knowledge, rather than jumping into an "A" thread, please start a separate thread at a lower level ("B" or "I").

Perhaps you can give me a hint where I can read about the GR version of it.

The definitive source in GR for things like this is Hawking & Ellis (which is the first reference on the Wiki page linked to below). The Wikipedia page here has a brief discussion of the topic (but it's Wikipedia, so not everything it says is going to be rigorously correct; it can only give you a flavor of what the topic involves):

https://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems

 

[bcrowell]

A possible counterexample to the hypothesis that singularities never have observable properties would be a Misner string. In Bonnor's interpretation, summarized here https://www.physicsforums.com/threads/what-is-a-misner-string.837180/#post-52575 , the Misner string is a topological defect with a parameter (the "NUT parameter") that is interpreted as half the angular momentum of the string per unit length. I guess that angular momentum would be an observable property, although it might be nontrivial to state how you would measure it operationally. Although Clement et al. recently proved that (contrary to what had been believed for 50 years) some such strings do not cause any geodesic incompleteness, I think that might be only for special values of the parameters that describe those spacetimes.

Bonnor, Proc Camb Phil Soc 66 (1969) 145 http://journals.cambridge.org/actio...587332E4.journals?fromPage=online&aid=2067080

Clement et al., http://arxiv.org/abs/1508.07622

Clement et al., http://arxiv.org/abs/1002.4342

 

[loislane]

we have no way of characterizing anything about such a singularity?

Not sure if this is a rhetorical question.

Here we have a failure of global hyperbolicity, which is sort of a breakdown of the entire enterprise of physics.

Is that so? You have a peer-reviewed reference for this statement? It would seem according to this site rules you shouldn't get away with this kind of statement unless you justify it(of course the rules don't mean much in the presence of "ingroup bias").

In the absence of global hyperbolicity, it's hard to see how we can define any sane properties for anything, except perhaps in some local region of spacetime that is at a safe distant from the misbehaving regions.

That's exactly what GR does. The EFE are local equations.

 

[PeterDonis]

You have a peer-reviewed reference for this statement?

Yes, he does: Hawking & Ellis.

It would seem according to this site rules you shouldn't get away with this kind of statement unless you justify it(of course the rules don't mean much in the presence of "ingroup bias").

Please note my comments in post #13 about this being an "A" level thread. Knowledge of concepts like global hyperbolicity and what Hawking & Ellis say about it is assumed in a thread at this level, and it's not necessary to give explicit references for statements which are common knowledge at the graduate level in the subject matter (which bcrowell's statements are).

 

[strangerep]

Unless stated otherwise, well-defined and properly, "incompleteness" is just a missing limit of a Cauchy sequence. Whether on a geodesic (shortest distance between 2 points) or not. It doesn't say anything about its nature, its topological neighborhood or its capability to be completed.

As others have hinted, this notion of "completeness" is different from the one you're familiar with from pure math.
See, e.g., Penrose–Hawking singularity theorems.

 

[strangerep]

[...] do the conclusions about the dynamical properties of black hole singularities extend as well to non-black hole singularities? [...]

This reminds me of a situation in QED. A classical electron is also singular in a sense, since it's Coulomb field diverges at short distances. Nevertheless, QED works a bit better (no IR divergences in the S-matrix) if one reformulates it to consider an electron together with its field as a basic entity (i.e., dressed electron). Afaict, nature doesn't care about our arbitrary distinction between source and field.