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In order to derive ERB from Schwartzchild Metric, we replace $(r - 2M)$ by $u^2$ - which means after you cross the horizon the metric becomes the same as it was before the horizon. That means you never have to travel the interior of the black hole to reach the other side. Am I right?

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There are no timelike or lightlike worldlines that cross the bridge.

A spacelike/tachyonic worldline can cross the bridge without passing through the interior. In Kruskal-Szekeres coordinates, it crosses at $X=T=0$ (the point on the diagram where the event horizons intersect). $u=0$ in your coordinates corresponds to $X=T=0$, and positive and negative $u$ correspond to regions I and III.

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The event horizon in Schwarzschild metric does not depend on the coordinates, so it should not matter in what coordinates is ERB derived. ERB in Schwarzschild metric connects parallel universes through interior of the black hole.

Also, you cannot reach the other side, because the bridge will collapse quicker than you can pass through, even if you move at the speed of light.

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