The "ringdown" that was detected after the merger of two black holes by LIGO in 2015 has been compared to the damping of a rung bell.
Why doesn't the merged black hole ring perpetually? What force dampens the ringing?
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Gravitational waves carry energy, momentum, and angular momentum away to infinity, thereby damping the ringing and allowing the merged hole to settle down into a Kerr metric. The amount of energy radiated can be enormous... a significant fraction of the mass-energy of the merging black holes. For the first LIGO event, three solar masses was lost as gravitational radiation!
The ringdown can be studied using perturbation theory around a Kerr metric. The analysis does not typically involve “forces” because there is no matter to experience a force. However, one can think of the radiation as causing some kind of “recoil” causing the spacetime of the hole to settle down.