A Where exactly is the wormhole in the Kruskal-Szekeres diagram?

[PeterDonis]

What do you mean by ”seem like they entered the wormhole”?

Any such notion must rely on a choice of coordinates (for time slicing) and a convention for what counts as "inside the wormhole". The notion I implicitly used in my post is that we are using the Kruskal time slicing (and normalizing the coordinates in units of $2M$), so that the wormhole exists for the range $-1 < T < 1$, and that you are "inside the wormhole" at any event on your worldline that is in that time range in Kruskal coordinates and is at a value of $r$ that is less than or equal to $2M$. The reason I think this convention is reasonable is that, as I pointed out in an earlier post, all of the surfaces of constant Kruskal coordinate time in the range given above have the same "wormhole" topology as the $T = 0$ surface that contains the bifurcation 2-surface; the only difference between them is the size of the wormhole "throat" (i.e., the area of the 2-sphere of minimum area within the spacelike 3-surface).

One could of course adopt other conventions; for example, the convention implicitly used in post #2 and some of its follow-ups is that you are only "inside the wormhole" if you are at an event on the bifurcation 2-surface at the origin of Kruskal coordinates. Which means that no timelike path from either exterior region can ever be inside the wormhole by this convention (only timelike paths coming from the white hole region can reach the bifurcation surface).

Although it should be noted that first one would need to find an actual maximally extended Schwarzschild black hole

Yes, of course, which means that this thread in its entirety is almost certainly irrelevant to our actual universe, which almost certainly contains no such objects.

 

[Orodruin]

Any such notion must rely on a choice of coordinates (for time slicing) and a convention for what counts as "inside the wormhole".

Hence it is not really something the observer actually sees. Things like curvature invariants increase monotonically etc. Just checking.

 

[PeterDonis]

Hence it is not really something the observer actually sees.

If we want to take that point of view, then the observer never sees themselves inside the wormhole. Unless we want to say that being able to see incoming light from both exterior regions (which is an invariant) counts as being inside the wormhole--but then that would be true anywhere inside the black hole region.

 

[Ibix]

Any spacelike 3-surface which connects the two exterior regions counts as a constant time slice of the wormhole (at least for some time slicing)

Got it. Here's the Kruskal diagram from Wikipedia (originally by @DrGreg, modified by a user called Vilnius - note that regions III and IV are labelled the other way around to the version in the OP):

If I draw a horizontal line across the diagram, each point is representative of a sphere. As I move my finger along the line, the radius of the associated sphere falls until it reaches a minimum on the $T$ axis, then starts to grow again. But each sphere is "inside" the previous one in the sense that I have to pass through the sphere I'm on if I want to get from a previous one to one I haven't touched yet, so if I try to draw an embedding of this it forces me to draw a classic wormhole neck.

You can see from the diagram that the sphere radius on axis depends on where I draw my horizontal line. As Peter said, it is maximised at $2GM$ at the origin (the diagram uses units of $2GM$, so labels this $r=1$) and falls to zero at either singularity. So, as Peter said, the "neck" of the wormhole is narrower the further away from the X I draw my horizontal line - and it, in fact, falls to zero radius if I draw the line grazing either singularity. Any further away, and my line passes through two disjoint regions of spacetime and a region in the diagram that doesn't represent anywhere in the spacetime between them. That is, the wormhole collapsed (or hasn't formed yet if I'm at the bottom of the diagram).

The diagram shows a black line starting on the $X$ axios and curving into the upper singularity. This is the worldline of a test body falling into the black hole and striking the singularity. This one doesn't make it through the wormhole. The interesting thing about Kruskal diagrams, though, is that you can pick a different time origin and the whole thing rearranges rather like a boosted Minkowski diagram - the spokes "rotate" and the hyperbolae stay put. That means that by a different choice of origin I can make that test particle pass through the neck or not - in fact, there's no invariant sense in which the particle does or does not pass through the wormhole. It's kind of a moot point since it hits the singularity.

The other interesting thing is that if you pick interior Schwarzschild coordinates, the spatial plane become the hyperbolae of constant $r$ - that is, the spheres I encounter as I move along those lines have equal radii - there is no wormhole in these coordinates! That's actually a cheat. Because those coordinates only cover the upper and lower wedges, I'm able to shuffle all my wormholiness off into the regions the coordinates don't cover - in particular the origin. There's no way to wish the wormhole away completely - it's a fact of the geometry, so any coordinate system that covers the whole diagram must include lines that extend to $X=\pm\infty$ and therefore necessarily have smaller radius spheres in the middle than either end. The very best you can do, I think, is Schwarzschild coordinates, where the planes with wormhole-like topology are the spacelike straight lines that pass through the origin, where the coordinates are singular and I can't describe it.

I hadn't quite grasped that before. It's fascinating how much complexity lurks inside the simplest non-trivial solution to the EFEs.

 

[PeterDonis]

there's no invariant sense in which the particle does or does not pass through the wormhole

Yes: you have to pick an arbitrary boundary for what counts as "through the wormhole", and a Kruskal boost can change whether a particular worldline does or does not pass inside the boundary.

you can pick a different time origin and the whole thing rearranges rather like a boosted Minkowski diagram - the spokes "rotate" and the hyperbolae stay put

I'm not sure "pick a different time origin" is the correct way to describe what is being done. What you are doing is a "time translation" along the integral curves of the Killing vector field that is timelike in the exterior regions. ("Time translation" is also something of a misnomer because the KVF is not timelike everywhere; it's null on the two horizons and spacelike in the black hole and white hole regions.) In Schwarzschild coordinates this KVF is $\partial_t$. This KVF indeed works like the boost KVF in Minkowski spacetime, except that its norm varies with $r$ in a different way than the boost KVF in Minkowski spacetime varies with $x$. But the integral curves and boost action look the same.

Also, though, you can't shift the spacetime origin of this KVF the way you can shift the spacetime origin of the boost KVF in Minkowski spacetime--or, to put it in what might be a better way, there is not just one "boost KVF" in Minkowski spacetime, there are an infinite number of them, depending on which point in spacetime you pick as the "bifurcation point" of the boost (the point where the two Killing horizons generated by the boost meet). In Schwarzschild spacetime, you can't do that; there is only one possible spacetime origin of the boost, and that origin is not a spacetime point but a 2-sphere, the one represented by the origin of the Kruskal diagram. All this is why I'm not sure "pick a different time origin" is a good description for the Schwarzschild case.

 

[Ibix]

I'm not sure "pick a different time origin" is the correct way to describe what is being done.

Carroll develops Kruskal-Szekeres coordinates starting as a transformation from Schwarzschild exterior coordinates. I was thinking of picking a different zero for the original Schwarzschild coordinate time and repeating the same development, which generates boosted coordinates (so I should have said "different Schwarzschild coordinate time origin"). Transporting along the KVF that's not associated with spherical symmetry is a better and coordinate free description of that, yes.

And I agree your point that unlike the origin of Einstein $x-t$ coordinates on Minkowski spacetime, the choice of origin of K-S $X-T$ coordinates is not free - it's got to lie on the center of the spherical symmetry and half way between the singularities. And that is a sphere and not an event, as you say - in fact, it's the neck of the wormhole at its maximum radius, as discussed earlier. The boost freedom is just a freedom to pick which direction from an event on that sphere that we call the $X$ axis, analogous to the freedom to pick a zero meridian (through Greenwich or Paris or whatever) on the Earth.

 

[PeterDonis]

I was thinking of picking a different zero for the original Schwarzschild coordinate time and repeating the same development, which generates boosted coordinates

Yes, that's one way of describing what the boost does.

The boost freedom is just a freedom to pick which direction from an event on that sphere that we call the $X$ axis, analogous to the freedom to pick a zero meridian (through Greenwich or Paris or whatever) on the Earth.

Yes.

 

[haushofer]

I'm reading "black holes" by Cox and Forshaw, which is an excellent "popular science" book explaining among others the wormholes. I have a related question: in the figure from the book you see a Rindler observer (purple) in a conformal diagram, asymptotically (!) accelerating from +c to -c. Why does it leave and enter the null past and future in the diagram anyhow when it can never reach c exactly?

 

[haushofer]

I also understand how a time evolving wormhole can form now in a static solution: the original solution is only for r>2M and time and space switch roles inside the horizon. The fact that the geometry outside the horizon depends on r opens up this possibility in the analytic extension.

 

[Orodruin]

Why does it leave and enter the null past and future in the diagram anyhow when it can never reach c exactly?

It must. It has a null path as an asymptote. In the future there are null paths that will always lie above it and beliw it so it must end up at null infinity.

 

[PeterDonis]

I also understand how a time evolving wormhole can form now in a static solution: the original solution is only for r>2M

I'm not sure what you mean by this.

time and space switch roles inside the horizon.

No, they don't. This is a common pop science misconception, but it's still a misconception. Your worldline doesn't suddenly switch from timelike to spacelike if you fall through the horizon.

The Schwarzschild coordinates called $t$ and $r$ switch from timelike, spacelike to spacelike, timelike inside the horizon, but that's an artifact of those particular coordinates and has no physical meaning.

The fact that the geometry outside the horizon depends on r opens up this possibility in the analytic extension.

If by this you mean that $r$ becomes timelike in Schwarzschild coordinates inside the horizon, as above, that is a coordinate artifact and has no physical meaning.

The "time evolving wormhole" has already been explained in previous posts: use Kruskal coordinate time. The fact that the spacetime is static outside the horizon does not mean that all observers must see an unchanging spacetime geometry. It just means there is a particular family of observers that does: the ones that are "hovering" at constant $r > 2M$. There is no inconsistency between the spacetime being static and it having a "time evolving wormhole". One just has to be clear about exactly what "static" does and doesn't mean.

 

[Ibix]

Why does it leave and enter the null past and future in the diagram anyhow when it can never reach c exactly?

Well, they reach $c$ "at infinity", and the whole point of the conformal diagrams is that they include infinity as a real place on the diagram.

Note that the diagram that you've shown has each point on it being representative of a hemisphere in space, not a sphere. This is distinct from the Kruskal diagram where each point represents a sphere. You can draw just the right half of the diagram and have every point represent a spherical surface in spacetime, and this is a closer analogue to the Kruskal diagram.

 

[haushofer]

Well, they reach $c$ "at infinity", and the whole point of the conformal diagrams is that they include infinity as a real place on the diagram.

Note that the diagram that you've shown has each point on it being representative of a hemisphere in space, not a sphere. This is distinct from the Kruskal diagram where each point represents a sphere. You can draw just the right half of the diagram and have every point represent a spherical surface in spacetime, and this is a closer analogue to the Kruskal diagram.

Yeah, I guess that makes it clear. It's really different from thinking about diagrams in an "asymptotically flat" way. Thanks!

 

[haushofer]

I'm not sure what you mean by this.No, they don't. This is a common pop science misconception, but it's still a misconception. Your worldline doesn't suddenly switch from timelike to spacelike if you fall through the horizon.

The Schwarzschild coordinates called $t$ and $r$ switch from timelike, spacelike to spacelike, timelike inside the horizon, but that's an artifact of those particular coordinates and has no physical meaning.If by this you mean that $r$ becomes timelike in Schwarzschild coordinates inside the horizon, as above, that is a coordinate artifact and has no physical meaning.

The "time evolving wormhole" has already been explained in previous posts: use Kruskal coordinate time. The fact that the spacetime is static outside the horizon does not mean that all observers must see an unchanging spacetime geometry. It just means there is a particular family of observers that does: the ones that are "hovering" at constant $r > 2M$. There is no inconsistency between the spacetime being static and it having a "time evolving wormhole". One just has to be clear about exactly what "static" does and doesn't mean.

Yes, you're right, already in the choice of parametrization of the metric you stick to a subset of all possible observers. I was thinking about the "staticness" in an absolute way. Thanks!