The Schwarzschild Black Hole in an External Gravitational Tidal Field: the Quasinormal Spectrum

Abstract

A black hole need not ring down in isolation, since a nearby companion can subject it to an external gravitational tide. We calculate how a quadrupole tide modifies the Schwarzschild black hole resonances. Linearising the exact tidally distorted Schwarzschild solution in the tidal amplitude $A$, we derive the diagonal odd- and even-parity perturbation equations on the deformed background and compute the formal first order displacements of analytically continued Schwarzschild resonances. Each diagonal first order shift factorises as $\delta {\omega }_{n\ell m}=A{c}_{\ell m}{\kappa }_{n\ell }$, giving one reduced complex coefficient for each $(n,\ell )$, expressed as a ratio of contour integrals, multiplied by a universal Zeeman-like pattern that splits the $2\ell +1$ azimuthal multiplet. For the $\ell =2$ and $\ell =3$ sectors analysed explicitly, the odd- and even-parity shifts coincide, so Schwarzschild isospectrality survives at first order in the tidal amplitude. Quadratic tidal forces, by contrast, can break this degeneracy. Moreover, for a physical companion, the astrophysically dominant $(n,\ell ,m)=(0,2,2)$ mode oscillates faster and decays more slowly. Finally, we show that the eikonal shifts admit a geometric description in terms of the Penrose limit about the tidally deformed photon ring.

Comments: 1+33 pages, 1 figure