> 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.

▲

sigmoid10 6 hours ago | parent | next [-]

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.

▲

pdonis 5 hours ago | parent | next [-]

> 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.

▲

sigmoid10 4 hours ago | parent [-]

>It's a spacelike line on the Kruskal diagram

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.

	▲	pdonis 3 hours ago \| parent [-]
		> It also is in Schwarzschild coords 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".

▲

XorNot 5 hours ago | parent | prev [-]

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.

▲

pdonis 5 hours ago | parent | next [-]

> the singularity is always in your future

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.

▲

sigmoid10 4 hours ago | parent | prev [-]

>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.

	▲	pdonis 3 hours ago \| parent [-]
		> 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.

▲

kwoff 6 hours ago | parent | prev [-]

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.).

	▲	pdonis 5 hours ago \| parent [-]
		> Susskind's book does also mention that the event-horizon shenanigans are due to coordinates and not a physical thing. 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.