If electrons can be pointlike particles, why can't black holes?

[Feynstein100]

As you no doubt have heard countless times, GR's prediction that a black hole should collapse to zero size is considered problematic because it would imply infinite density, which isn't physically possible. And yet, over on the other side in QM, electrons seem to be considered pointlike particles with no actual size without any problems. So why the inconsistency?
Of course, I do understand that the electron is a single particle whereas a black hole is a collection of multiple particles. Thus, what applies to an electron might not necessarily apply to a black hole but that should not matter imo. The fact that electrons are sizeless means that being sizeless is not prohibited by the laws of the universe and thus theoretically attainable by any object.

 

[Orodruin]

As you no doubt have heard countless times, GR's prediction that a black hole should collapse to zero size is considered problematic because it would imply infinite density, which isn't physically possible.

No. I have not heard this countless times, not in any reputable source at least. It sounds like popular scientific mumbo jumbo. Please provide a reputable source.

 

[Ibix]

GR's prediction that a black hole should collapse to zero size

That's not correct. The singularity in a black hole is not a point in space - it's closer to a moment in time. This is very different from an electron.

And the problem is not to do with infinite density, really, but the fact that the singularity isn't really part of spacetime and you can reach that in finite time. The model therefore cannot predict what happens when that happens, but doesn't forbid it - which is a failure of the model.

 

[Feynstein100]

The singularity in a black hole is not a point in space - it's closer to a moment in time

Hmm I don't understand. Any point in spacetime, by virtue of being in a 4D manifold, would have 4 co-ordinates, right? How can the singularity be closer to a moment in time? I mean, what does that even mean? That it doesn't exist in space but only exists in time?

but the fact that the singularity isn't really part of spacetime

Again, I don't understand what makes the black hole a singularity. I thought it was because it had zero size. But you mentioned that that's not true. So what exactly are the criteria for being a singularity?

 

[Orodruin]

Hmm I don't understand. Any point in spacetime, by virtue of being in a 4D manifold, would have 4 co-ordinates, right? How can the singularity be closer to a moment in time? I mean, what does that even mean? That it doesn't exist in space but only exists in time?

A moment in time is a spatial hypersurface in spacetime. For example everywhere at a fixed time coordinate in Minkowski space. A location is a timelike curve, it can be made to have fixed spatial coordinates. The singularity is more like a moment in time than a location in space.

Again, I don't understand what makes the black hole a singularity.

It is not a singularity. The black hole solution has a spacelike singularity.

I thought it was because it had zero size. But you mentioned that that's not true. So what exactly are the criteria for being a singularity?

Certain curvature invariants diverge.

Overall, it seems to me that you have taken a deep dive into popular scientific treatments of this and then try to apply your takeaways to draw conclusions. That will never work out well. If you want to really understand what is going on and be able to draw your own conclusions, you will need to study an actual textbook on the subject.

 

[Ibix]

How can the singularity be closer to a moment in time?

The singularity is not a point, it's a line. And it is your future if you go into a black hole. Normally moments in time are three-surfaces (all of 3d space at one instant), but in this particular case two of the dimensions go to zero size as you approach the singularity. That's the "singular" bit. Fundamentally, though, it's a moment in time just like Monday morning, and just as inevitable if you enter the hole.

Worth noting that inside the black hole, the Schwarzschild radial coordinate is timelike, decreasing into the future.

Again, I don't understand what makes the black hole a singularity.

A black hole isn't a singularity. It's a region of spacetime that contains a singularity (that's not the only rule, but it'll do here - formally defining a black hole can get quite complicated). The singularity is a place where the measures of spacetime curvature go to infinity. That doesn't necessarily mean zero size.

So what exactly are the criteria for being a singularity?

A singularity is a place where you can draw a worldline that you can't extend any further. It just stops. That's usually (always?) accompanied by one or more measures of curvature going to infinity. It doesn't mean zero size.

 

[Ibix]

As Orodruin comments, if you want to reason about black holes you need a textbook. Popsci descriptions, even from reputable sources, just won't do the job.

 

[Dale]

How can the singularity be closer to a moment in time? I mean, what does that even mean?

See this picture

This is a Kruskal–Szekeres diagram of a standard Schwarzschild black hole. You can read this diagram like a normal spacetime diagram, light cones go at 45 degrees up to the left and right, and timelike worldlines always have a slope more vertical than that.

Region I is the region outside the black hole, and region II is the interior of the black hole, with the boundary between I and II being the event horizon. The singularity is the thick blue line at the top of region II. Notice that the singularity goes from horizontal to less than 45 degree slope. That is the characteristic shape of a moment in time, not a position in space. Specifically, the singularity is a moment in time that is to the future of all events inside the event horizon.

 

[Feynstein100]

The singularity is a place where the measures of spacetime curvature go to infinity. That doesn't necessarily mean zero size.

You've lost me here. Doesn't the spacetime curvature of a region depend on the density of matter-energy in that region? As far as I know, the only way to get infinite curvature is to have infinite mass-energy density, which again implies zero size. I don't see how you can have one without the other.

A singularity is a place where you can draw a worldline that you can't extend any further. It just stops.

So I'm guessing that's the problem? Worldlines aren't supposed to stop. They're stop to extend to the infinite future. So the fact that they stop at the singularity is a problem. However, doesn't this only apply for Schwarzschild black holes, which don't exist in nature? All black holes have some angular momentum, thus making them Kerr black holes. Would rotation solve the problem of world lines terminating? Although, that in itself raises an interesting question. That while Schwarzschild black holes don't exist in nature, it is possible to artificially convert a Kerr black holes into a Schwarzschild one by stealing its angular momentum. What would happen in that case? Hmm

As Orodruin comments, if you want to reason about black holes you need a textbook. Popsci descriptions, even from reputable sources, just won't do the job.

I do agree and apologize for my ignorance. However, I believe that, at least for physics anyway, even though the mathematical details might be complicated, the underlying principle/logic itself should be able to be expressed in plain English. If you can't do that, then it kind of implies that you don't really understand it.

 

[Feynstein100]

Specifically, the singularity is a moment in time that is to the future of all events inside the event horizon.

Yes, I am aware of this but what does this mean physically? To me, this seems no different than the instances where you solve a quadratic equation and get a negative value for distance/time. A mathematical artifact that doesn't correspond to physical reality.

 

[Feynstein100]

The singularity is not a point, it's a line. And it is your future if you go into a black hole. Normally moments in time are three-surfaces (all of 3d space at one instant), but in this particular case two of the dimensions go to zero size as you approach the singularity. That's the "singular" bit. Fundamentally, though, it's a moment in time just like Monday morning, and just as inevitable if you enter the hole.

Worth noting that inside the black hole, the Schwarzschild radial coordinate is timelike, decreasing into the future.

A black hole isn't a singularity. It's a region of spacetime that contains a singularity (that's not the only rule, but it'll do here - formally defining a black hole can get quite complicated). The singularity is a place where the measures of spacetime curvature go to infinity. That doesn't necessarily mean zero size.

A singularity is a place where you can draw a worldline that you can't extend any further. It just stops. That's usually (always?) accompanied by one or more measures of curvature going to infinity. It doesn't mean zero size.

And going back to my original question, why is this not a problem for the electron? If it indeed has zero size but non-zero mass, that would imply infinite mass density and thus infinite spacetime curvature, making the electron a singularity. Obviously that's not the case. So, why not? That's what I was interested in finding out originally.

 

[Ibix]

Doesn't the spacetime curvature of a region depend on the density of matter-energy in that region?

No. For example, spacetime near Earth is a vacuum, but is curved due to the Earth nearby.

As far as I know, the only way to get infinite curvature is to have infinite mass-energy density,

Actually, black hole solutions like the Schwarzschild black hole are vacuum everywhere.

I don't see how you can have one without the other.

That's the problem with not using the maths.

All black holes have some angular momentum, thus making them Kerr black holes

Kerr black holes also have singularities, albeit with a more complex structure. They are also "moments in time", again with some caveats.

Would rotation solve the problem of world lines terminating?

No.

Although, that in itself raises an interesting question. That while Schwarzschild black holes don't exist in nature, it is possible to artificially convert a Kerr black holes into a Schwarzschild one by stealing its angular momentum. What would happen in that case? Hmm

It is in principle possible to spin-down a black hole. However, a black hole that you do this to is neither Kerr nor Schwarzschild, since those solutions both have fixed angular momentum. And the problem of singularities existing within is an extremely general one - they're inevitable in an extremely broad class of spacetimes that includes any kind of black hole that doesn't contain exotic matter.

However, I believe that, at least for physics anyway, even though the mathematical details might be complicated, the underlying principle/logic itself should be able to be expressed in plain English. If you can't do that, then it kind of implies that you don't really understand it.

No, it implies that a language developed by apes to tell each other where the tasty fruit is and coordinate stabbing mammoths isn't great at describing the interiors of black holes. Scientists don't use maths for fun. It's the only language we have to talk about these things well enough to understand them.

Yes, I am aware of this but what does this mean physically?

You understand what a moment in time is, surely. The whole universe at exactly 6am tomorrow, for example. You can't avoid it because it's the future.

The singularity isn't exactly like 3am tomorrow, because the curvature invariants aren't infinite at 3am tomorrow, but it has the important characteristics: you can't avoid it and there's only so much you can do to delay it.

why is this not a problem for the electron?

If we had a theory of quantum gravity we could tell you, but we don't so we can't. General relativity assumes its sources of gravity are classical continuous matter, and attempts to create a theory that can handle quantum sources are ongoing. A theory of quantum gravity would also hopefully provide a correct explanation for what goes on inside black holes where GR says there are singularities.

 

[Orodruin]

You've lost me here. Doesn't the spacetime curvature of a region depend on the density of matter-energy in that region?

No, this is an incorrect understanding. Curvature is a result of the global stress-energy tensor distribution and any boundary conditions imposed, not necessarily at the event you are considering. Compare this to the electric field. You do not need to have charges at a particular point for there to be an electric field at that point. In fact, for a point charge, most of space has zero charge density and non-zero electric field.

In the case of the (fully extended) Schwarzschild solution, there is no stress-energy at all. It is a vacuum solution to Einstein's field equations with a singularity (which is not actually a part of the spacetime itself). The solution is parametrised by a parameter $M$, which is the total mass.

All black holes have some angular momentum, thus making them Kerr black holes.

Schwarzschild black holes do not exist in nature, but not for the reasons you think. Kerr black holes do not exist either. They are both vacuum solutions with asymptotically flat boundary conditions at spatial infinity. As such they make a lot of assumptions and have to have persisted forever - unlike real black holes.

I do agree and apologize for my ignorance. However, I believe that, at least for physics anyway, even though the mathematical details might be complicated, the underlying principle/logic itself should be able to be expressed in plain English. If you can't do that, then it kind of implies that you don't really understand it.

That's a cheap and unfortunately way too repeated cop-out. It is simply not true. English as a language is imprecise and not well suited for describing physics. So no, you can definitely understand things without being able to communicate it to laymen in plain English.

And going back to my original question, why is this not a problem for the electron? If it indeed has zero size but non-zero mass, that would imply infinite mass density and thus infinite spacetime curvature, making the electron a singularity. Obviously that's not the case. So, why not? That's what I was interested in finding out originally.

An electron is not a classical object so it makes no sense to think of it as a small billiard ball (or even a billiard ball with zero volume), which would be required to treat it with a classical theory such as general relativity. It is a quantum particle.

 

[Dale]

Yes, I am aware of this but what does this mean physically? To me, this seems no different than the instances where you solve a quadratic equation and get a negative value for distance/time. A mathematical artifact that doesn't correspond to physical reality.

Yes, and that is what makes it different than an electron, in the sense you were asking in the OP.

 

[Nugatory]

And going back to my original question, why is this not a problem for the electron? If it indeed has zero size but non-zero mass, that would imply infinite mass density and thus infinite spacetime curvature, making the electron a singularity. Obviously that's not the case. So, why not? That's what I was interested in finding out originally.

"Zero size" doesn't mean what it sounds like for a particle governed by the rules of quantum mechanics. It's tempting to imagine the electron as little tiny ball centered on some point in space, it has radius $r$ and therefore volume $V=\frac{4}{3}\pi r^3$, divide the mass by $V$ to get a density, then imagine shrinking $r$ to zero and watch the density go to infinity.... but that's our classical intuition for what "zero size, non-zero mass" means and it's not applicable here.

Unfortunately there is no classical analogy that works, the only options are to do the math yourself or take the word of someone else who has done it.

(OK, somewhat against my better judgment I'm going to give you a really hand-wavy statement of the problem: this hypothetical shrinking procedure assumes that $r=0$ defines a particular point in space. That's fine for a little tiny ball that has a center point somewhere, but not for quantum particles that don't even have a position in space when not measured. This is a total hand-wave, useful only as an attempt to point out the hidden classical assumption behind the mistaken notion of the infinite-density electron).

The geodesic incompleteness that is a problem at black hole singularities is the result of carrying the non-quantum model of spacetime all the way down to $r=0$. If we knew how to do quantum mechanics under conditions of extreme spacetime curvature we'd likely find a way of avoiding it. But that is so far an unsolved problem.

 

[PeterDonis]

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