Mirror Wormhole

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 Schwarzschild wormhole as a reflection? A possible interpretation of the full Schwarzschild geometry? The Schwarzschild wormhole is a ‘reflection’ of the normal Schwarzschild geometry. There is no white hole, just a black hole and its reflection.

 Kruskal-Szekeres spacetime diagram for the ‘mirror’ wormhole In the mirror interpretation of the Schwarzschild wormhole, an event $$( t_\textrm{K} , r_\textrm{K} )$$ (tK, rK) in the Kruskal diagram is considered to be identical to the event (-tK, -rK). $$( - t_\textrm{K} , - r_\textrm{K} )$$ Whereas time flows upward on the right, time flows downward on the left. In the mirror interpretation, the past is connected to the future at the antihorizon (red line from lower right to upper left), an interesting twist. However, no information can pass the antihorizon in either direction, so causality is not violated. There are other ways to complete the Kruskal diagram by taking a mirror image of the Schwarzschild geometry and gluing it to the original along the antihorizon. However, they are arguably not as cute as the way shown here.

 Objections Misner, Thorne & Wheeler (1973, “Gravitation”, p. 840) object to the mirror interpretation of the Schwarzschild geometry on the grounds that: (1) it produces a sort of “conical” singularity at the crossing point of the two horizons, $$t_\textrm{K} = r_\textrm{K} = 0$$ (where the two red lines cross in the Kruskal-Szekeres spacetime diagram); and (2) it leads to causality violations in which a person can meet themself going backward in time. The 1st objection is true. Spacetime at the crossing point of the two horizons has a non-standard ‘spin 2’ kind of structure. Just as rotating a spin 2 particle by $$180^\circ$$ in space leaves it unchanged, so also rotating spacetime around the horizon crossing point by $$180^\circ$$ leaves the spacetime unchanged. Is this cause to reject the mirror spacetime structure, or is there interesting physics here? The 2nd objection is false. The Kruskal-Szekeres spacetime diagram shows that the Universe and its mirror image are causally disconnected from each other. A person who starts in the Universe on the right cannot pass through the horizon (the red line slanting from bottom right to top left) to the mirror Universe on the left, and vice versa. However, if no signal can pass between the Universe and its mirror image, then there is no observational way to tell whether the mirror image is really there or not (unless there are observable consequences from quantum mechanical tunnelling between the Universe and its mirror image?). And if there are no observational consequences for any observer, then who knows and who cares.

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Updated 19 Apr 1998; converted to mathjax 3 Feb 2018