Probing soft hairs in a multi-black-hole system
Although Hawking, Perry and Strominger showed that an asymmetric shock wave absorbed by a black hole would modify the metric as if it undergoes a BMS super-translation, the resulting black hole remains diffeomorphic to the original one. Here we conduct a different thought experiment involving 3 black holes arranged in a triangle, each sending a null beam to the black hole on the right, and show that 3 black holes would undergo different BMS super-translations, i.e. being implemented with different soft hairs. We also show that there exists no BMS transform that can map the final state of the system back to the initial state. In other words an observer could not find a frame in which all 3 black holes are Schwarzschild while the universe is asymptotically flat along every direction. Since the matter distribution of the final state is identical to that of the initial state, we end up with an inevitable conclusion that the soft hairs are observable in a multi-black-hole system.