Spectroscopy of bumpy black holes: non-rotating case

Recent detections of gravitational-wave ringdowns have made black hole quasinormal modes a powerful tool in studying predictions of general relativity(GR). Understanding this spectrum in a broad class of beyond-GR(bGR) theories and general environments of GR black holes remains vital. We compute the quasinormal mode spectrum of a static, "bumpy" Schwarzchild spacetime with a modified multipole structure. These bumps preserve the axisymmetry of the spacetime and are parametrized by small multipole potentials in Weyl coordinates, recovering the Schwarzchild solution when the potentials vanish. Our calculation uses the recently developed modified Teukolsky formalism that provides decoupled differential equations for the radiative Weyl scalars ψ₀ and ψ₄ in bGR theories. We use these equations and the eigenvalue perturbation method to solve for the quasinormal mode shifts for the even- and odd-parity modes. We find a bumpy black hole spacetime breaks the isospectrality found in GR. Additionally, we discuss the connection between a spacetime's multipoles and its spectrum. Our calculation provides an avenue to parametrize an observable GR violation in terms of a spacetime's multipoles.