Wormhole between two distant point in space, where are the two ends?

Question

I'm struggling with the wormhole solution of the Einstein equation. I can't understand how it may connect two distant points in the universe. It looks to me totally misinterpreted and I would like to understand what I get wrong. Let me provide a bit of background first so that we are all on the same page. The wormhole solution is derived by the Swartzchild solution $$ ds^2=-(1-2m/r)dt^2 + \frac{1}{1-2m/r}dr^2 + r^2d\Omega^2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1) $$ by performing the substitution $r=u^2 + 2m$ and passing from a coordinate system $(t,r,\theta,\phi)$ to the Kruskal coordinates $(t,u, \theta,\phi)$, one obtains the new proper interval (solution) $$ ds^2=-\frac{u^2}{u^2 +2m}dt^2 + 4(u^2 + 2m)du^2 + (u^2 + 2m)^2d\Omega \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2) $$ that should describe the wormhole. Let's see how. In (1) the $r$ coordinate is positive definite ($r>0$), while the new coordinate $u$ being definite as $u=\pm\sqrt{r - 2m}$ can now have both positive and negative values. One observes that both solution $u_+$ and $u_-$ equivalently satisfy the relation between $u$ and $r$ and therefore are equally valid. This creates the interpretation that one could travel along $u$ across the point $u=0$ ($r=2m$, i.e. the event horizon) and therefore moving from region with positive $u$ to region with negative $u$ and viceversa. The two region with positve and negative $u$ (i.e. $u_+$ and $u_-$) would be the two sides of the wormhole.

Up-until now everything seems to be kind of reasonable. What I really don't understand is how the two end of the wormhole may end up to be even several light-years apart. Let's call $O_r$ the reference frame described by the coordinate $r$, and $O_u$ the reference frame described by the coordinate $u$. Both reference frames would describe the same universe, just in two different way. In $O_r$ I have a single origin in $r=0$ and a single object that I decided to call wormhole close to that origin (r=2m). The reference frame is centered there at $r=0$, all the other coordinates $\theta$, $\phi$ are measured around that single point. Also in $O_u$ I seems to have a single origin in $u=0$, and by continuity it seems to be the same both for $u_+$ and $u_-$. But if the object we are describing is a wormhole connecting two points of the universe, the origin of $u_+$ and $u_-$ in some reference frame should be different, the two should be centered in two different point in space! (i) How is it possible that by a simple change of variable $r\leftrightarrow u$ (that is just some sort of non-linear rescaling) now I end up with two different ends displaced light year apart, and moreover two different origin of the reference frame? (ii) the relation between $u_+$, $u_-$ and $r$ is fixed, mathematically in the process we never did any consideration on the origin. They should be always linked to the same value of $r$, and therefore same origin $u=0$. What is the law connecting $O_{+}$ and $O_{-}$ to their displaced position in the universe and how is it connected to the wormhole solution?

Lets's assume that in $O_r$ there is a single origin and that the reference frames $O_+$ and $O_-$ have two different origins. What is the rule that explains the relationship between the origin of $O_r$, $O_u$, $O_+$ and $O_-$? How can I predict, given a wormhole in $O_r$, where the two ends (origins) of $O_+$ and $O_-$ would be? What is the phenomena that describes this shift of reference frames + and - and that justify the two different origin? If this is the "wormhole solution" shouldn't be able to describe it and predict it?

All three reference frames describe the same universe after all. In all three I should be able to describe the position of the two points connected via the wormhole.

What am I missing? Where are the wormhole ends? Isn't possible that u+ and u- would describe simply some weird phenomena happening the event horizon at $u=0$ ($r=2m$)? Or, better, just two identical solution like positive and negative solution of a wave equation, equally correct but both working in the same space, same origin, same everything and not a connection of two different point of the universe? I understand that the solution can be seen as connecting two sides of something, but I don't see how these sides can be mapped to two point very very far apart.

Answer 1

A Schwarzschild (zero angular momentum) black hole is not usually considered as related to a wormhole. Once a particle crosses the event horizon, it falls into the singularity in a finite proper time and is not expected to come out.

Wormholes are usually associated to Kerr (nonzero angular momentum) black hole. It has extra features compared to the unique event horizon of the Schwarzschild black hole, like an ergosphere, Cauchy horizon, and so on.

https://en.wikipedia.org/wiki/Kerr_metric#Kerr_black_holes_as_wormholes

Now I am not claiming to have a full understanding, but as it says in the relevant paragraph of the wiki article on black holes, if an object crosses the Cauchy horizon it might follow a trajectory that makes it exit the Kerr black hole, through another Cauchy horizon, in a region with a "normal" geometry. For observers in that region of space, it would look as if this object is emerging of a "white hole", of Kerr type, of course, not Schwarzschild type.

Now this region of space is a priori not related at all with the region one has started from. It may be in a totally different universe, one to which is impossible to reach by any other way but going through the two successive Cauchy horizons described in the above paragraph if the Wiki article.

But for all we know, it is possible that this Kerr "white hole" is in our Universe, but it might just be anywhere, in the same galaxy, or a totally different galaxy anywhere. We can mathematically describe the path "through two Cauchy horizons" but where the exterior of the second is with respect to the first one, through "normal" space is totally out of our knowledge. And quite possibly, it might well be in a "different" universe altogether.

This is what is referred to as a "wormhole" : entering a Kerr black hole here, going through a first Cauchy horizon, exiting through a second Cauchy horizon, but where ?

Maybe I should point out that a Kerr hole is not "intrinsically" black or white. It has a "past" Cauchy horizon from which an object might come out (as if from a "white hole") but also a "future" Cauchy horizon. So if we plunge into a Kerr "black" hole in our Universe and emerge into the "normal" space of a different one in through the "past" Cauchy horizon of a Kerr hole there, which would in this event behave as a "white hole" for external observers, one could decide afterwards to plunge again into the same Kerr hole. But it is not "going back". It would be treating this Kerr hole as a "black hole". We will reach its "future" Cauchy horizon, and if we reemerge in normal space, a priori it will be in a third Universe. Nothing at all guarantees we are going back !

Answer 2

You're misunderstanding a couple of things.

You refer to "the wormhole solution" of the field equations, but there is not just one such solution. There's no "the."

The spacetime you mathematically describe is not a traversible wormhole connecting two points in the same asymptotically flat region. It's the maximal extension of the Schwarzschild solution. This is a spacetime containing four regions: a black hole, a white hole, and two separate copies of Minkowski space, which are called regions I and III. An observer who falls into the black hole region can see light coming in from both I and III. However, that observer can't cross between I and III, and the event horizons are not two horizons in the same "universe" (i.e., the same space that is asymptotically Minkowski), they're event horizons in two separate universes, I and III.

To get a traversible, stable wormhole connecting two places in the same universe, you need exotic matter. The Schwarzschild spacetime is a vacuum solution. It doesn't contain any matter.