Should be pointed out that this is a critique of common popsci journalism tropes and not a fancy new research result. Anyone who has taken a graduate level class in General Relativity would have been able to tell you the same.
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1. I must admit to a lack of sufficient GR education to feel confident in this, but I think that one of the issues that made physicists unwilling to accept the idea of black holes when they were first postulated was that there ended up being a division by zero in the mathematics.
They don't just have the same name, they are the same thing.
A Schwarzschild black hole has both: a removable singularity at the event horizon that is just an artefact of a particular choice of coordinates and a true non-removable mathematical singularity at r=0 where curvature really does go to infinity. It also wouldn't be much of an issue in classical physics, because this singularity is always hidden from outside observers, so the mathematical weirdness there can't screw with your normal predictions in space outside the black hole. The problems start once you consider quantum mechanics, because any such singularity will break unitarity (a fancy way of saying that probabilities must add up to 1), which means your theory as a whole can no longer make predictions. This has opened a whole can of worms with a bunch of solution attempts, which are all sadly untestable for the foreseeable future.
How is this any different than classical? Isn't it still just an ~impossibility hidden behind an event horizon in either model?
My point is, it’s not super meaningful to argue whether a black hole has an inside.
So does spacetime exist in some frames of reference but not others because those frames disagree on the radius of the apparent event horizon?
Also note that in general an event horizon doesn’t require a singularity.
What I’m trying to say is that there is nothing special about the region of space near the event horizon.
Approximately everything in nature rotates. Including black holes. Schwarzschild blockholes do not rotate. Rotating black holes are much more complicated and don't necessarily shield their singularity behind an event horizon.
why does it matter that it is not 'visible' for anyone?
If a problem is not able to influence anything, even in theory, then by definition, it cannot possible influence any testable predictions we have.
Well, the Ricci curvature scalar blows up to infinity, which is obviously unphysical.
https://en.wikipedia.org/wiki/Scalar_curvature#Relation_betw...
Please say what you specifically believe is unphysical about the situation — what trajectory reaches the singularity in finite time and why specifically is that unphysical?
My understanding is that you have a cusp singularity that is actually an infinite spike, ie, distance to the singularity is unbounded; that is, no matter how small a circle/sphere around the singularity, you have an infinite diameter. And so you will need to be much more explicit about where the problem lies.
You say that and yet this thread is full of people arguing about it, and there's an entire Wikipedia article on this: https://en.wikipedia.org/wiki/Gravitational_singularity.
In fact, that article says:
> No complete and precise definition of singularities exist in the theory of general relativity,
So which is it? It can't both be trivial to any grad student but also an open question. And things like naked singularities aren't proven to not exist either.
Also, general relativity is a classical, geometric-only theory. It seems obvious that better understanding what a black hole's singularity is would require quantum mechanics because the singularity is effectively what's "left over" of the physical material once you go beyond a neutron star.
What do you mean by not exist? If you postulate the right black hole with a naked singularity, it would have a naked singularity.
> It seems obvious that better understanding what a black hole's singularity is would require quantum mechanics
If you postulate a classical black hole, it won't require quantum mechanics to understand.
Yea, okay.
I'm sorry but this is blowing my mind. What???
Also read Nick Gorkavyi: The Oscillating Universe: Einsteinian Cosmology of Black Holes and Gravitational Waves
Really what it means is that past the event horizon you can use the direction in space between you and the singularity as a way to measure time, specifically the amount of time left before you reach the singularity. It's not so mind blowing when you interpret it that way now is it? You can imagine many things in ordinary life that you use to measure time without claiming that time has literally swapped places with it. On a road trip, the number of kilometres to your exit tells you how long you have left, that's using space as a proxy for time... big deal. The notable difference between a road trip and a black hole is that on a road trip you could stop for a break, you could maybe take a detour, you could decide to go back home... and these would all break your use of space as a proxy for measuring time. Well with a blackhole you can't do any of those things, there is no going back, there is no detour, the relationship between the spatial direction towards the singularity and time is fixed and causal and there's nothing you can do about it.
The phrasing used is used almost certainly to evoke some kind of voodoo mind-blowing mystery that completely disappears when you get down to the more strict formalism.
That's not correct. There is a relationship between the radial coordinate r you are at and the time it will take you, by your clock, to reach the singularity (at least assuming you are freely falling), but that relationship can't be described the way you are describing it.
To put the issue with what you say as starkly as possible: at any event inside the black hole's horizon, there are spacelike curves in every direction from you that will hit the singularity! So "the direction in space between you and the singularity" is not even well-defined.
To be clear, you are right that Susskind's statement is very misleading (I posted my own criticism of it upthread).
You can avoid a coordinate, for example by choosing not to go there, or revisit another one repeatedly.
A black hole on the other hand doesn't have that: you cannot revisit old locations - attempting to do so moves you closer to the singularity.
Also, the region inside the inner horizon of Kerr spacetime is widely considered to be not physically reasonable, not just because of the closed timelike curves, but because the inner horizon itself is unstable--there is an infinite blueshift there which, it is believed, would cause it to be destroyed by the first tiny bit of incoming matter or radiation.
Note that, once you're inside the horizon, you can't "turn around" and go back outside again. You're inside the hole for good.
And once you're inside the hole, yes, no matter which direction in space you move, you're moving "towards" the singularity. But a better way to look at it is that the singularity is a moment of time, not a place in space. You're moving "towards" the singularity in the same sense as you're moving "towards" next Tuesday. You can't stop moving towards next Tuesday by changing which direction in space you move. The same is true for the singularity once you're inside the hole's horizon.
Unless you went in butt-first, but the path of the photons would have changed and would now be going toward the black hole, and everything would look probably all smushed together.
The reason this phenomenon has a spooky-sounding name is that it also affects whether two objects can be causally connected. If you can only ever move closer to the center of the black hole, then there are (conceivably) other objects inside the event horizon that you can never have a causal relationship with.
But it doesn't mean that space and time literally switch places.
If Susskind's book does in fact say that, it's extremely disappointing to me, because, as a number of other GR textbooks will tell you (e.g., Misner, Thorne & Wheeler and Wald, the two great classic GR textbooks), the "switch places" is an artifact of a particular choice of coordinates (Schwarzschild coordinates), and does not represent anything physical. So it's not something that should be relied on. (Not to mention the confusion it causes when pop science sources repeat the statement and then draw all manner of wrong conclusions from it.)
The part about being "a surface in time" might be all right, assuming that by that he means "a surface representing a moment in time, not a place in space"--in more technical language, a spacelike surface. That is correct, and it's an invariant that does not depend on any choice of coordinates. But that invariant fact can be described without having to talk about the "switch places" thing at all.
(If anyone has the book, it is chapter 6 section "Interchange of Space and Time Dimensions at the Horizon" and the following section points out the singularity is a time (and you can't escape it (in a Schwartzschild model at least) just like you can't escape time). I'm sorry if my wording is still incorrect.).
That's good. However:
> Interchange of Space and Time Dimensions at the Horizon
This still seems misleading to me, because "Dimensions" makes it seem like it's not just an artifact of coordinates--but it is.
If you really want to get a picture of what is happening, you can look at Eddington-Finkelstein coordinates. In particular at a light cone field diagram around a collapsing shell of matter that turns into a black hole. Then this whole stuff suddenly makes sense without even going into the math. You don't just see how an event horizon can form out of nothing, you also see how gravity starts to bend your causal forward light cone (i.e. all points in spacetime with events that you could interact with in the future) inward in such a way that you will necessarily always fall closer to the center of the mass once you pass a certain line (aka the event horizon). No need to deal with those weird infinities or points in time suddenly lying on a different axis.
The great Roger Penrose (the same guy who also came up with some of the most confusing diagrams) published a beautiful, simple overview of exactly this stuff in Scientific American: https://www.wkbpic.com/wkbx/SA/1972/1972-05-01.pdf (starting on page 38)
Still one of the best things you can read if you don't just want the math.
Roughly the general public (including me) knows that gravity is meant to have some effect on the apparent passage of time, so it seems significant but under explained what it means to be in a region of space where all possible directions lead to the singularity.
It's not just that it's always in your future in the sense that you can't avoid it. It's that the reason you can't avoid it is that it's a moment of time, not a place in space. You can't avoid it for the same reason you can't avoid tomorrow. And which direction in space you move has no effect on whether or not you reach the singularity for the same reason it has no effect on whether or not you reach tomorrow.
But it is just a mathematical artefact of weirdly chosen coordinates. In reality, the singularity is still just a point in space (or a line in spacetime), except that inside the event horizon all paths you are allowed to travel lead to it. There's no need for this whole "space turns into time" notion apart from the fact that you are guaranteed to hit it in *your* future as a local observer. And in Eddington-Finkelstein coordinates you can easily see that globally, things simply hit the spatial coordinates of the singularity at certain slices of coordinate time. Other coordinate systems make this whole process seem much more weird than it is.
No, that's not correct. The fact that the singularity is always in your future inside the horizon is an invariant, independent of any choice of coordinates.
> the singularity is still just a point in space
No, it's not. A point in space would be a timelike line in spacetime. But the singularity is a spacelike line in spacetime. That's a moment of time, not a place in space.
> There's no need for this whole "space turns into time" notion
That's true; that notion is an artifact of Schwarzschild coordinates. But that does not imply the other claims you are making.
> the spatial coordinates of the singularity
I'm not sure what you mean by this. It's true that, since the singularity is a spacelike line, you can treat a coordinate that varies along it as a "spatial" coordinate marking different spatial points on the singularity. But the singularity itself is a moment of time (as above, a spacelike line), so it is not a "place", and it does not have a particular set of "spatial coordinates". A spatial coordinate marking different points along the singularity is marking different points in space at a moment of time.
That also makes your use of the term "spatial coordinates" questionable, as I already pointed out. The fact that the line r = 0 is vertical in an Eddington-Finkelstein spacetime diagram does not mean it's automatically a "point in space" or that r inside the horizon is automatically a "spatial coordinate". You need to look at the actual physics, not just the surface appearance of the diagram.
Because it is. Remember: space, not spacetime. Hence the remark in brackets in the original comment and my reminder to read precisely in the other one. And in Eddington Finkelstein it is most obvious that it is a point in space (i.e. it has spatial coordinate r=0 where r has the metric signature of a spatial dimension) that you can hit at various points in (global) time (and actually also in free falling observer time, but let's ignore that since it is not immediately obvious). You can literally trace incoming light rays crossing the event horizon and hitting the singularity at r=0 at a certain points in time in the diagram. This stuff is really not that weird once you choose less confusing coordinates. It only gets weird once you start asking what local observers can actually see, because from their perspective their relation to all other coordinates in spacetime gets really messy. That's probably where 95% of the confusion among laypeople comes from. But for that you can still resort to other coordinates which show it much better.
You are quite correct that, since the singularity is a line in spacetime, different incoming light rays (or free-falling observers, for that matter) can hit it at different points. Depending on how you choose your coordinates, you can set it up so that those points have different "time" coordinates. But that doesn't make the singularity a point in space. It means you're running up against relativity of simultaneity--whether or not different events on a spacelike line (or more generally a spacelike surface) happen at the same time depends on your choice of coordinates. You can, in fact, choose coordinates in which all events on the singularity happen at the same time (for a "time" coordinate that is genuinely timelike--see below). The standard Penrose chart does that, for example.
You are also correct that a good choice of coordinates can make it easier to see certain properties of a spacetime geometry. But it can also make it harder to see other properties. In this case, your choice of Eddington-Finkelstein coordinates is making it harder for you to see why your claim that the singularity is a point in space is wrong, and why the things I said above are true.
For example, inside the horizon, the Eddington-Finkelstein "time" coordinate that you are using is not timelike. It's spacelike. In other words, it's not actually a "time" coordinate (even though it's labeled as such). It is actually a "space" coordinate! You should be able to see this by observing that the singularity is a spacelike line, and in E-F coordinates it's a vertical line--i.e., the only coordinate that changes along it is the "time" coordinate. That means the "time" coordinate must actually be spacelike there.
And, for extra confusion, the r coordinate in Eddington-Finkelstein coordinates is also spacelike, even inside the horizon (unlike in Schwarzschild coordinates, where it becomes timelike). So in this chart there is no coordinate that is timelike inside the horizon! That means any timelike curve inside the horizon must have more than one coordinate in this chart that changes along it (in the simplest case, a radial timelike curve, both the "time" and r coordinates must change along the curve).
You said the singularity is a point in space. That's what I'm arguing against.
The rest was an attempt to try to help you understand the correct physics. Evidently it was wasted effort. I won't do it again.
> all you had to do was write down the explicit metric and point out exactly where it disagrees with what I said.
Sure, it's the one in terms of t' and r in the Wikipedia article on Eddington Finkelstein coordinates. [1]
> if you did, you would immediately see that your argumentation falls apart.
No, I see that yours does.
The article is using the timelike signature convention, so positive ds^2 is timelike and negative ds^2 is spacelike. Vertical lines in a spacetime diagram (like the one just a little bit below the metric, on the right, that shows the light cones) are intervals where only dt' is nonzero. It is obvious from the metric that for any r < 2M, i.e., anywhere inside the horizon, such intervals give a negative ds^2, since 1 - 2 GM / r is negative. So vertical lines, of which the singularity is one, are spacelike, and t' is a spacelike coordinate inside the horizon (just as r is). And a spacelike line cannot be a point in space. It can only be a moment of time. The fact that it is vertical on the diagram does not change that.
[1] https://en.wikipedia.org/wiki/Eddington%E2%80%93Finkelstein_...
It's a spacelike line on the Kruskal diagram, yes.
> The issue is that these diagrams are for eternal, static black holes
The full Kruskal diagram is, yes. But the essential features of the black hole portion of that diagram are still there in a black hole that forms by gravitational collapse of a massive body. That includes the singularity being a spacelike line, and there being spacelike curves inside the horizon that are infinitely long.
I agree that Eddington-Finkelstein coordinates can help with intuitions about this spacetime geometry as well.
It also is in Schwarzschild coords, so you've gained nothing with respect to the original issue from switching coordinates. Kruskal-Szeres really is only useful for demonstrating that the event horizon is not such a weird place, but it does nothing for the singularity at the center.
>the essential features of the black hole portion of that diagram are still there in a black hole that forms by gravitational collapse
The issue is even the limited diagram doesn't really show you that and the full one goes crazy with white holes. So not a good place if you don't want to confuse laypeople.
True--indeed, the statement that it's a spacelike line is an invariant, independent of any choice of coordinates. But it's a lot harder to see that in Schwarzschild coordinates.
> Kruskal-Szeres really is only useful for demonstrating that the event horizon is not such a weird place, but it does nothing for the singularity at the center.
I disagree, and I think many GR physicists would as well. Indeed, both classic GR textbooks, Misner, Thorne & Wheeler (1973) and Wald (1984) discuss Kruskal coordinates and their associated spacetime diagram (MTW in Box 31.2 and sections 31.5 and 31.5, Wald in section 6.4), and those discussions contain a good deal more than just showing that the horizon is "not such a weird place".
A Fleeting Detection of Gravitational Waves
https://physics.aps.org/story/v16/st19
Gravitational wave blues
https://aeon.co/essays/how-joe-weber-s-gravity-ripples-turne...
I have trouble really conceptualizing black hole physics, I just think of it as a mass so great that nothing, including light, can escape its gravity. Works for me.
Everywhere else in the universe with mass and energy you can do what you want (sort of). An event horizon throws a hard shroud over that and drastically reduces opportunities: your free will to use mass and energy is significantly curtailed (you must head towards the singularity).
If you don’t have enough upward velocity to escape earths gravity, hitting the ground is also inevitable.
A black hole is interesting because you inexorably move towards the singularity - which is a defined location in spacetime, and also has a boundary - the event horizon.
So now your freedom of action is reduced: you must move towards the singularity, but you also can't actually move outside of the event horizon either.
The singularity in a rotating black hole is entirely different but the interior of classical Kerr (rotating) black holes is one of the most controversial if inconsequential topics in theoretical physics because there are reasons to believe (without real proof mind you) the Kerr solution is unstable inside the inner event horizon so that whatever happens in there is not what that theory says.
And of course black holes are quantum objects which might actually have an “interior” entirely different from the classical picture.
In a gravitational singularity spacetime breaks down. You could argue that time stops, but it’s also valid to argue that causality breaks down and we can no longer make any predictions about the future. Just because we don’t have a theory describing what could happen, doesn’t rule out that something could happen.
Some of the things the paper points out are not emphasized in other sources, which is probably why the authors chose to write it. But there is nothing in the paper that is in the least questionable or ill-defined; it's all standard General Relativity as applied to the Schwarzschild and Kerr black hole solutions.
People like to reduce papers to a simple hot take, but the paper is more than that, offers viewpoints that are non-standard and speculation about new possibilities.
It is in this sense that, AIUI, electrons are modeled as point particles.
Of course, that doesn’t mean that if we zoom in enough, probing at higher and higher energy scales, that it can’t turn out to have some non-zero fundamental size outside of just uncertainty in its center of mass position. I think string theory would say that at the string scale it would be a string.
But, AIUI, no experiment has shown it to have the kind of extent that would make it be called not a point particle (an extent in a sense beyond just uncertainty in COM position)
https://quicycle.com/understanding-electrons/
And the video essay on the subject https://www.youtube.com/watch?v=hYyrgDEJLOA (Huygens Optics: Williamson & Van der Mark electron model | Are electrons made of light?)
Although some physicists disagree, QM slants very anti-realist. There are no objects anywhere, no particles, no waves, only probabilistic interactions, some of which can be snapshotted into localised partially definite results.
So there are only interactions between probability distributions in space and time, and "particle-like events."
No pointy objects, and no need for them.
The agnostic view is it's just a mathematical model that makes accurate probabilistic predictions when measurements are made, which says nothing about what's really going on.
Of course treating particles as points is also mathematical.
History of the Universe : What Is Hidden In The Core Of A Neutron Star? - https://youtu.be/YoYjkNQ27T8
That video goes into it... without getting mathy at any point.
One of the bits that you're having trouble with is the compression of matter to a point. There's a theoretical type of black hole known as a kugelblitz - https://en.wikipedia.org/wiki/Kugelblitz_(astrophysics)
A kugelblitz is a theoretical astrophysical object predicted by general relativity. It is a concentration of heat, light, or radiation so intense that its energy forms an event horizon and becomes self-trapped. In other words, if enough radiation is aimed into a region of space, the concentration of energy can warp spacetime so much that it creates a black hole. This would be a black hole the original mass–energy of which was in the form of radiant energy rather than matter
I haven't watched the video, but if we're compressing electrons, neutrons, or other fermions, I imagine if we want to keep compressing that down to an arbitrarily small radius, won't we pretty quickly find it favorable to shift those fermions to something else, probably photons, to respect Pauli exclusion?
Really, I don't know enough physics to figure out the reason why it shouldn't always end up in this incorporeal energy-curving-space situation either way, if we're compressing arbitrarily far.
Finding a tame enough special case was how Hawking discovered his radiation.
It's how it was formulated, but the radiation is far too weak to measure to know it's a real thing on an actual black hole; from this calculator, a 1 solar mass black hole has a Hawking radiation power of 9e-29 W: https://www.vttoth.com/CMS/physics-notes/311-hawking-radiati...
We do see analogous effects in physical analogues of black holes, but we don't know for sure that Hawking radiation actually comes off of actual black holes.
Worse, when they're small enough(!) to be luminous enough to actually observe, they should be hot enough to be spewing out a whole load of exotic nonsense particles (not just photons) that we don't really know how to model correctly with regards to Hawking radiation even if it is part of whatever ends up unifying QM and GR.
Given even the event horizon of a BH doesn't play well with QM, it's probably best to wait for some physicists to work out how to combine QM and GR better.
Pauli exclusion isn't an impenetrable force field - as you say, it's just often more favourable to do something else than to work around it. Consider an iron atom with however many electrons though - all those orbitals except the inner one are electrons working around Pauli.
I'm not a physicist either.
I was shown it at school, using microscope glass slides that we waved over a Bunsen burner running cold, to coat with soot. We then carefully etched parallel lines with a compass by hand. Our (~17 y/o) efforts were a bit random but we did get some smudgy banding results on the screen.
Your pair of razor blades is a great solution.
Common sense would tell you they can't exist at all because you can't compress atoms - you have lived your entire life with atoms being entirely incompressible for the practical purpose of anything you do.
Leaning on common sense to discuss fundamental physics has been wrong since round about the start of the practice of physics.
That's because a lot of the ordinary mass in the universe is ionised or in other weirder states.
The required density for that stuff goes down as the volume goes up, so a solar-system sized object (~77 AU radius) at mere normal sea-level (Earth) air density (1.2 kg/m^3) would just be one automatically.
Given that the maths of GR requires spacetime to not have a singularity*, and yet it predicts a singularity from benign starting conditions, I take this as a sign that GR is not correct.
But black holes are nonetheless an outcome, not a presumption.
* it's more complicated than that
The easy thing to miss, and blew my mind when I read it. is that general relativity is the concept of space-time, emphasis on the time, and this is also compressed by the mass, so if this singularity can actually occur it would also take an infinite amount of time to fall into it. So nothing can actually enter it. From the point of view of an astronaut(deliberately ignoring all the other relativistic implications) flying directly toward the event horizon. As you approach you will quickly see the rest of the universe age and die. and if hawking radiation is real the black hole will evaporate in front of you before you can reach it.
Where my imagination fails(above my pay grade) is in the face of infinity, what are the implications of infinite time compression?(everything happens at once?)
So time is localised? I’m not sure what localised time means but I’m hoping the question makes sense.
All objects within a given radius of the black hole (possibly modulo spin) would experience the same time dilation. Remote objects in the universe would not experience the dilation.
From the perspective of the astronaut falling toward the singularity, the rest of the Universe would age at an ever-increasing rate.
From the perspective of a remote observer, the astronaut falling toward the singularity would be experiencing time at an ever-decreasing rate.
The notion of relativity is that time-perception is relative, and dependent on acceleration, whether from motion (as on a spaceship) or from gravitational acceleration (as near a black hole). Objects in orbit around Earth, further from Earth's centre, and hence subject to reduced gravitational acceleration, age more quickly than objects on Earth's surface. This is actually measurable using atomic clocks, though the effect is quite small. It is sufficient that GPS satellites require time correction.
You could travel arbitrarily far into the future by getting close to a black hole's event horizon for a while without crossing it and then leaving, assuming you had the energy for it and you didn't get obliterated by all the mass and energy falling into the black hole in that timeframe.
See https://physics.stackexchange.com/questions/82678/does-someo...
But in the object's own time coordinates the math says it does hit the singularity. If you fell in you wouldn't die of old age before you hit it.
Of course I'm missing something here. I've taken QM and not GR so I would have this interpretation.
What do you mean by “particle” here? This kind of handwaving is fundamentally classical, and breaks down in the presence of quantum physics.
And if they didn't form a superconductor. I'm not sure why they would but if they did they would violate it. That's actually what makes superconductors superconducting - the really weird state where electron pairs act like bosons.