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I have been told that no observer can ever cross the wormhole present in the Kruskal-Szekeres coordinates, connecting region 1 and 4 in the Kruskal-Szekeres diagram, usually called Einstein-Rosen bridge. The reason given is that, in order to cross the wormhole from region to 1 to 4 or vice-versa, the trajectory of the observer would have to be spacelike somewhere.

Could someone explain better this statement? I cannot understand the deep meaning of what I have been told. Would someone be able to show better this concept, by drawing a very simple diagram that shows the fact that the trajectory would have to become spacelike?

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The Kruskal-Szekeres coordinates have the reputation of being hard to understand, and to some degree this is justified as they are unintuitive compared to the Schwarzschild coordinates. However their huge advantage is that many phenomena can be explained simply by looking at the illustration of a maximally extended black hole in the KS coordinates:

KS depiction of black hole

Region 1 is our spacetime and Region 2 is the inside of the black hole. Region 4 is the inside of the white hole and Region 3 is the other universe. The Einstein-Rosen bridge connects Regions 1 and 3, i.e. a curve from region 1 to 3 passes through the event horizon, misses both singularities and emerges from the event horizon into region 3.

For our purposes the advantage of the KS diagram is that on this diagram all radial light rays move on straight lines with a gradient of 45º. I've drawn in one such light ray in blue, but any line at 45º shows the worldline of a light ray. Any massive object must move on a line that is everywhere steeper, i.e. nearer the vertical than 45º. Any line that is shallower, i.e. nearer the horizontal than 45º, shows an object moving faster than light.

And we can answer your question just by looking at the diagram - no maths is needed! It is (I hope) obvious that there is nowhere in Region 1 that a light ray can start then travel at 45º without hitting the singularity, and this means light cannot make it through the ER bridge. Massive particles travel at angles nearer the vertical, so they can't make it through the bridge either. The only way to get from Region 1 to Region 3 requires a curve at an angle shallower than 45º, and this requires travelling faster than light.

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  • $\begingroup$ Don't get me wrong, but this is must an explanation of a model; there is no physical answer to why this can't happen. $\endgroup$
    – Our
    Dec 1, 2018 at 7:44
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    $\begingroup$ @ourcanbektas : The Einstein-Rosen bridge is, itself, a model. So it's a question about a model, and thus any answer will and does involve that model. In a more "realistic" physical scenario, the whole left half (along the downward-to-the-right slanting diagonal) of this diagram likely doesn't even exist. $\endgroup$ Dec 1, 2018 at 8:40
  • $\begingroup$ @The_Sympathizer --Poplawski has a 2009 preprint on Arxiv, at arxiv.org/abs/0902.1994 , which started a long series of articles on a torsion-based cosmological model that most of those later articles relate to Einstein-Cartan Theory (or, in some cases, to ECSK, which is the version of the 1929 theory, worked out by Einstein in collaboration with the mathematician Cartan, that was modified some decades later by Sciama and Kibble). Poplawski's model relies on fermions having a tiny spatial extent, some orders of magnitude over Planck length. Does it seem more "physical" to you? $\endgroup$
    – Edouard
    Jun 12, 2022 at 12:11

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