> the space and time coordinates switch places as you cross the event horizon
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.
Kruskal-Szeres coordinates indeed get rid of the wonky coordinate stuff at the event horizon, but if you look at the corresponding diagrams, you'll just end up with the same confusion, because the singularity is still a point (or rather surface) in the future instead of a point in space. The issue is that these diagrams are for eternal, static black holes, which cause diagrams to have these weirdly stretched infinite regions that are quite useful for understanding details of the math, but are highly confusing to laypeople. In fact these diagrams make it look like you'll always fall into the black hole at t=infinity, no matter how far you are away, when in reality you could orbit a static black hole pretty close for eternity.
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.
This article is great but the references to gravitational waves as if they were an observed fact (in 1972) was mystifying to me. It led me down a rabbit hole to Joe Weber, who is name-checked at the end of the Penrose article.
This doesn't seem meaningfully different to me? Like the notion that the singularity is always in your future (because you can't escape it anymore past the event horizon) makes sense, the point of confusion is what are the implications?
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.
>the notion that the singularity is always in your future makes sense
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.
> it is just a mathematical artefact of weirdly chosen coordinates.
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.
Just draw the geometry in Eddington Finkelstein coordinates and you will see everything I wrote above is true at the technical level if you read precisely.
No, everything you wrote is not true, in Eddington Finkelstein or any other coordinates. You wrote that the singularity is a point in space. It's not, no matter what coordinates you choose. It's a line in spacetime, but it's a spacelike line, and a spacelike line cannot describe a point in space. It can only describe a moment of time. No choice of coordinates can change that. (Similar remarks apply to your claim that the singularity always being in your future once you're inside the horizon is an artifact of a coordinate choice. It's not. It's just as true in Eddington-Finkelstein coordinates, or any others.)
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.
>You wrote that the singularity is a point in space
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.
Sorry, you're just repeating the same wrong statement. I know you said "space", and I already explained that a spacelike line in spacetime cannot be a point in space. It can only be a moment of time.
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).
Sorry, you are still arguing against things I never said or that you desperately want to misinterpret in a disingenuous way. If I was actually wrong about anything I said, all you had to do was write down the explicit metric and point out exactly where it disagrees with what I said. But if you did, you would immediately see that your argumentation falls apart.
> you are still arguing against things I never said
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.
> the singularity is still a point (or rather surface) in the future instead of a point in space.
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".
Susskind's book does also mention that the event-horizon shenanigans are due to coordinates and not a physical thing. Certainly I'd trust what he says rather than me, so sorry if I was misleading.
(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.).
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.