Maximally extended Schwarzschild spacetime

[Ibix]

I have a very quick question about the maximally extended Schwarzschild spacetime. I know you can't reach regions III and IV from I and II, and vice versa. But can you see in? If I'm in region I and I look down, the null paths reaching me originated in the white hole singularity. Likewise in region II null paths from "below" me originate somewhere in region III. So I should be able to see (briefly!) parts of that other spacetime. Or would it all be redshifted into oblivion?

 

[Orodruin]

I and III are completely equivalent. Anything originating in IV will eventually end up in one of the other regions. Observers from I and III could perfectly well meet up inside the BH.

 

[PeterDonis]

I know you can't reach regions III and IV from I and II, and vice versa.

The labeling of the regions is actually not completely consistent in the literature. The labeling I'm most used to is the one used in the Insights article I wrote on the Schwarzschild geometry:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-3/#toggle-id-1

In this labeling, region I is the "normal" exterior region (the one that "we", the people in our universe, occupy), region II is the black hole, region III is another exterior region (another "universe", not the one we occupy), and region IV is the white hole. With this labeling, regions I and III can be reached from region IV, region II can be reached from regions I and III, and region IV can't be reached from anywhere.

As far as "seeing" is concerned, from region I you can "see" the portion of regions I and IV that are in your past light cone, and nothing of the other regions. From region III you can "see" the portion of regions III and IV that are in your past light cone, and nothing of the other regions. And from region II you can see portions of all four regions that are in your past light cone.

should be able to see (briefly!) parts of that other spacetime.

No, you can't. As the Kruskal diagram in the Insights article linked to above makes clear, there are no null paths from region III to I or vice versa.

Also, region III is not "another spacetime" with respect to region I. It's just another region of the same spacetime.

 

[Orodruin]

No, you can't. As the Kruskal diagram in the Insights article linked to above makes clear, there are no null paths from region III to I or vice versa.

That part of his post referred to an observer that had already fallen into the BH from our universe. No need for a path to reach I from III.

 

[PeterDonis]

That part of his post referred to an observer that had already fallen into the BH from our universe.

Ah, ok.

Or would it all be redshifted into oblivion?

For anyone falling into the black hole, region II, from region I, light signals coming in from Region I would be redshifted, more and more as the observer gets closer to the singularity. But I don't think light signals coming in from region III (the "other universe") would be redshifted for such an observer; they might even be blueshifted.

 

[Ibix]

The labeling of the regions is actually not completely consistent in the literature.

It's always the simple stuff. I'd fixed Dr Greg's labelling in my head and not noticed that Carroll's lecture notes swap the III and IV labels. Coupled with the Schwarzschild t coordinate increasing down the page in the left hand region I'd ended up with some rather confused notions, to which I was leading with the question above.

To summarise what I think I now understand: In the white hole timelike and null worldlines must exit into one or other of the asymptotically flat regions. Timelike and null worldlines in the asymptotically flat regions may remain in the region or fall into the black hole. So, if we could start in the lower wedge, I could shake hands with my evil twin, then we could follow different paths into the left and right wedges, live similar (although rather boring given that this is a vacuum spacetime) lives, then plummet into the black hole and meet up again so that I could admire his rather fine goatee before we hit the black hole singularity.

 

[Ibix]

Also, region III is not "another spacetime" with respect to region I. It's just another region of the same spacetime.

Indeed - sloppy wording on my part.

 

[Orodruin]

In the white hole timelike and null worldlines must exit into one or other of the asymptotically flat regions.

Technically, a time-like worldline could pass through the origin of the Kruskal diagram and go directly from white to black without ever passing through either of those regions.

So, if we could start in the lower wedge, I could shake hands with my evil twin, then we could follow different paths into the left and right wedges, live normal lives, then plummet into the black hole and meet up again so that I could admire his rather fine goatee before we hit the black hole singularity.

This could indeed be possible. There would be nothing stopping you from doing this. (Apart from your evil twin breaking your agreement and deciding to continue living forever in the mirror universe.) Given a point on your world-line where you decide to go into the black hole, there would also be a latest point at your twin's world-line where he must do the same in order for you to be able to meet up, i.e., your future light cones from these respective events would need to overlap, which is not a foregone conclusion.

I'd fixed Dr Greg's labelling in my head and not noticed that Carroll's lecture notes swap the III and IV labels.

I usually try to avoid this by calling them "our universe", "black hole", "mirror universe", and "white hole" instead. This leaves less room for confusion.

 

[martinbn]

In the diagram that looks like a hexagon, the null lines are straight 45 deg to the left and to the right. So you can draw them, follow them, and see where the can go. Time-like ones have to stay in the null wedge/cone at any point i.e. they cannot tilt too mach to the left or right, so you can draw them as well.

 

[Ibix]

I usually try to avoid this by calling them "our universe", "black hole", "mirror universe", and "white hole" instead. This leaves less room for confusion.

Indeed. I'd adopted a similar convention myself by the end of #6. Care still needed when reading others' work, though.

Thanks for the other points - in summary the only guaranteed thing is leaving the white hole, and not leaving the black hole if you enter it.

In the diagram that looks like a hexagon

I'm not sure what you mean by the diagram that looks like a hexagon. I'm familiar with the Kruskal diagram, as in the Insight Peter linked to in #3, which has the light cone properties you mention, but I wouldn't have called that hexagonal.

 

[Orodruin]

I'm not sure what you mean by the diagram that looks like a hexagon.

He is talking about a Penrose diagram. This should also be discussed in Carrol if I don’t misremember.

Thanks for the other points - in summary the only guaranteed thing is leaving the white hole, and not leaving the black hole if you enter it.

And not being able to reach the mirror universe or the white hole from our universe.

 

[Ibix]

He is talking about a Penrose diagram. This should also be discussed in Carrol if I don’t misremember.

It is. I hadn't thought of it as a hexagon, although I see it is. I'm not sure of the advantage over the Kruskal diagram.

 

[PeterDonis]

In the white hole timelike and null worldlines must exit into one or other of the asymptotically flat regions. Timelike and null worldlines in the asymptotically flat regions may remain in the region or fall into the black hole.

Yes.

 

[Orodruin]

It is. I hadn't thought of it as a hexagon, although I see it is. I'm not sure of the advantage over the Kruskal diagram.

You cannot draw all of the spacetime on a finite piece of paper using Kruskal coordinates. The Penrose diagram is also a tool you can use for other spacetimes.

 

[PeterDonis]

I'm not sure of the advantage over the Kruskal diagram.

To expand on what @Orodruin said, a Penrose diagram let's you see structure "at infinity" that you can't see in a Kruskal diagram. For example, only in a Penrose diagram does it become apparent that there is more than one "infinity" (for an asymptotically flat spacetime there are five: future timelike infinity, future null infinity, spacelike infinity, past null infinity, and past timelike infinity). This becomes important when you are looking at causal structure.