Finitely supertranslated Schwarzschild black hole and its perturbations

Abstract

A finitely supertranslated Schwarzschild black hole possesses nontrivial super-Lorentz charges compared with the standard one. This may impact the quasinormal modes of the black hole. Since the Einstein's equations are generally covariant, the quasinormal modes of a supertranslated black hole can be obtained by supertranslating the familiar results for a standard black hole. It turns out that the supertranslated quasinormal modes can be obtained by simply shifting the retarded time of the standard modes by an angle-dependent function parameterizing the supertranslation. Therefore, the supertranslated quasinormal modes oscillate at the same frequencies and decay at the same rates as the corresponding standard ones. The supertranslated metric is time translation invariant, but does not explicitly respect spherical symmetries, although it is implicitly rotationally symmetric. So the supertranslated perturbations can still be written as linear combinations of the eigenfunctions of the generalized angular momentum operators for the underlying rotational symmetry. With a suitably defined asymptotic parity transformation, any perturbation can be decomposed into the even and odd parity parts. Then, one may conclude that the isospectrality still holds. To detect such supertranslated quasinormal modes, one has to place multiple gravitational wave interferometers around the supertranslated black hole, and measure the differences in the time shifts between interferometers. Gravitational lensing may also be helpful in the same spirit.