Lorentz Violation without Gravitational Ringdown Modification in Einstein-Bumblebee Black Holes

Introduction: Lorentz symmetry breaking is expected to leave dynamical imprints on black holes, and gravitational ringdown provides one of the most direct probes of such effects. A key question is therefore whether Lorentz-violating couplings necessarily modify the gravitational quasinormal-mode spectrum. In Einstein-Bumblebee theory, this issue must be addressed by perturbing both the metric and the bumblebee field consistently.

Methods: We study the full linear perturbations of a Schwarzschild-like black hole in Einstein-Bumblebee theory. By imposing the fixed-norm constraint and treating the metric and bumblebee perturbations on the same footing, we develop a hierarchical decoupling procedure for the coupled gravito-vector system.

Results: The coupled system reduces to four decoupled master equations. In each parity sector, the perturbations separate into a Schwarzschild-like gravitational sector and a Lorentz-violating Maxwell-like vector sector. Although Lorentz violation changes the effective propagation speed of the vector modes, the gravitational master equations retain exactly the Regge-Wheeler and Zerilli forms. As a result, the gravitational ringdown spectrum remains unmodified, and the odd- and even-parity gravitational perturbations remain strictly isospectral. The reconstruction formulas show that Lorentz-violating effects are still present, but are encoded in the relation between the master variables and the original metric-bumblebee perturbations. The Maxwell-like vector sector, by contrast, exhibits nontrivial quasinormal-mode shifts that cannot be obtained by a simple rescaling of the Schwarzschild electromagnetic spectrum.

Conclusions: These results show that Lorentz violation can affect black hole perturbations without modifying the gravitational ringdown spectrum. The mismatch between the gravitational and vector propagation speeds may provide a possible timing signature of bumblebee vector dynamics in multi-messenger observations.