Black Hole Collision

Two black holes collide head-on. Will they coalesce into one black hole?

Answer

Yes. The two black holes should coalesce like two liquid drops. We need to ensure that the entropy afterward in the coalesced final state is greater than the entropy in the initial state. We do that by adding up the entropy in the two states, separate black holes versus one larger black hole with gravitational waves carrying away some energy and entropy in the final state.

The black hole entropy is proportional to the event horizon area, which grows as the mass to the fourth power. Suppose we take two black holes, one of mass M₁ and the other of mass M₂. Their initial total entropy is proportional to M⁴₁ + M⁴₂. If they merge and their final total mass is approximately M₁ + M₂, then their final total entropy is proportional to (M₁ + M₂)⁴, which you can verify is greater than the original total entropy, so the reaction will go. In cases where the final entropy is only slightly greater than the initial entropy for the black hole parts only, one may need to add in the entropy in the gravitational waves to ensure a bigger inequality

Thus a bigger mass means a bigger surface area, which means a bigger entropy than if the two smaller black holes remained apart. Simulations in 3-D of colliding black holes and their emitted gravitational waves can be seen on the Internet.

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