Gravitational-Bumblebee perturbations: Exact decoupling and isospectrality

TL;DR

This paper derives exact decoupled equations for metric and bumblebee field perturbations in a Schwarzschild background, showing Lorentz violation affects vector modes but not gravitational spectra, with potential observational signatures.

Contribution

It provides the first exact decoupling of perturbations in a Lorentz-violating bumblebee gravity model, revealing isospectrality preservation despite Lorentz violation.

Findings

Decoupled four master equations for perturbations.

Lorentz violation modifies vector mode speeds.

Gravitational modes remain isospectral despite Lorentz violation.

Abstract

In this paper, we present the exact decoupling of the full metric and bumblebee field perturbations in a Schwarzschild-like background. The coupled system reduces to four decoupled master equations, revealing in each parity sector a Schwarzschild-like gravitational sector and a Lorentz-violating Maxwell-like vector sector. While Lorentz violation modifies the propagation speed of the emergent vector modes, we demonstrate that the gravitational master modes exhibit a ``dynamical immunity'' to the non-minimal Lorentz-violating coupling, and that the odd- and even-parity perturbations remain strictly isospectral. Our work provides a rare example in which Lorentz-violating couplings reshape the field reconstruction while leaving the gravitational ringdown spectrum intact. This mismatch in propagation speeds suggests a possible timing signature of bumblebee vector dynamics in black hole perturbations, offering a theoretical route to testing spontaneous Lorentz symmetry breaking in the era of multi-messenger astronomy.