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In Kruskal-Szekeres Coordinates for the Schwarzschild blackhole, we know that the wormhole "opens up" joining regions I and III between Kruskal times $T = -1$ (vertex of the green hyperbola below) and $T=+1$ (vertex of blue hyperbola).

enter image description here

Say person A is at $r = 1.4, t = 0$ in region I in the above diagram, then a light ray (45 degree line) shot from A is in the throat of the wormhole (it cannot transverse it to region III since it will hit the top hyperbola, the singularity, before transversing the wormhole to region III).

However, if person B is at $r = 1.4, t = 2$ in region I, then B is past Kruskal time $T = +1$ and is out of the throat, so a light ray cannot enter the throat from B.

Now, the Schwarzschild metric is time independent, so we could just let time elapse so that $B$ is at a new time, say $t' = 0$ (instead of $t = 2$), then with respect to this new time, $B$ is in the throat, so he can attempt to shot a light ray to transverse the wormhole (of course, failing for the same reason as $A$ did).

What is confusing to me right now is that by time shifting you can always consider yourself in the throat and hence try (but of course failing) to transverse it. Hence, the wormhole is "always there" (but of course, never transferable). Is this reasoning correct?

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  • $\begingroup$ Schwarzschild is the correct spelling. $\endgroup$
    – my2cts
    Sep 6, 2019 at 21:14
  • $\begingroup$ You might want to look at arxiv.org/abs/0902.1994 . (I'm not sure about the relation between terminology & coordinates, but the piece is on ER bridges.) Spatial constraints on passage thru wormholes are addressed in plain English in the video at arxiv.org/abs/0902.1994, in Susskind's "ER = EPR" lecture, most of the way toward its end. $\endgroup$
    – Edouard
    Sep 6, 2019 at 23:37

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There is a nice discussion of this kind of thing in Misner, Thorne, and Wheeler, p. 837. We can't really define whether an event is in the throat or not, or whether the wormhole exists or not in the "now" of an observer in an exterior region. All of these descriptions are contingent on an arbitrary choice of a spacelike surface to define "now." From any event in region I, we can draw the spacelike surface in such a way that it (1) intersects the past singularity, (2) extends to spacelike infinity in region III, or (3) intersects the future singularity. 1 means the wormhole hasn't formed yet, 2 means the wormhole exists "now," and 3 means the wormhole has already pinched off.

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  • $\begingroup$ If we consider for instance the black hole in the center of our galaxy and our "now" as observers, it is difficult to understand why we can not know the status of the (hypothetical) wormhole. Any speculative theory or interpretation trying to explain this? $\endgroup$ Sep 7, 2019 at 17:46
  • $\begingroup$ @MicheleGrosso: it is difficult to understand why we can not know the status of the (hypothetical) wormhole The wormhole is a feature of the maximal extension of the Schwarzschild spacetime, not of physical black holes formed by gravitational collapse. But for more detail, you might want to ask this as a separate question. $\endgroup$
    – user4552
    Sep 9, 2019 at 21:30

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