# Is the Schwarzschild wormhole (Einstein-Rosen Bridge) time dependent?

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).

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?

• Schwarzschild is the correct spelling. Sep 6, 2019 at 21:14
• 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. Sep 6, 2019 at 23:37