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We know that black holes can collide and merge, but if they each have a singularity at the core, how can the two singularities merge and become one? Wouldn't the mere fact of a second singularity in range of another make it not a singularity?

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Hypothetically, Yes. Realistically, No. As James mentioned a singularity isn't a physical object but rather a point in space time. In theory a wormhole is just two singularities which combined.

Orangeman
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