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We know that two black holes can merge to form a single black hole with the release of vast amounts of energy. Can the reverse happen? That is can it happen that large energy is supplied to a black hole and it splits into two?

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No. It would mean the the distance increase between the 2 parts, while they are inside the event horizon of each other. Nothing can move outward inside an event horizon, for it's the space itself than flows at speed c.

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Given that as far as anyone knows there is only one black hole in the centre of every galaxy, if it were to split into two it would split the galaxy. We know that galaxies often collide and can see one feeding into another, but surely there is no evidence of a split galaxy. It would look rather like a cell dividing wouldn't it?