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

[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!