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I've been reading up on some material about black holes and Einstein-Rosen bridges. Generally it is said that a black hole is defined by the event horizon (boundary in space where the gravitational force is so strong towards the singularity, even light cannot escape it) and the singularity itself (I'm not sure if this is scientifically correct, if not, I'd love to learn the correct terminology to properly explain this).

Furthermore, an Einstein-Rosen bridge is a connection between two black holes. A connection shortcutting the fabric of spacetime. My question however is; If two black holes formed an Einstein-Rosen bridge, a "bridge between" them would be created, as is often depicted in illustrations. However, if these two black holes are "connected" via this bridge, where do both their singularities go? As I understand it, black holes "collapse" into singularities at some point, so how can two singularities exist, if there is a connection between two black holes and no "points" where the black holes "collapse" into. (I hope this question is somewhat clear)

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Have a look here. You'll find a Penrose diagram of a black hole spacetime with two asymptotic regions ("outside" the BHs), two event horizons and two interiors where worldlines from either side approach a common spacelike Schwarzschild singularity. The wormhole connecting the two sides only has spacelike paths connecting the exterior regions.

(Note that this is a "toy" spacetime, not one that you get by following the evolution of, say, a collapsing star.)

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