Quasinormal Modes of a Massive Scalar Field in Slowly Rotating Einstein-Bumblebee Black Holes

TL;DR

This work studies quasinormal modes of a massive scalar field in slowly rotating Lorentz-violating Einstein-Bumblebee black holes, comparing a Newman-Janis–generated rotating solution with a second-order slow-rotation spacetime. It derives the master equation for scalar perturbations to second order in the rotation and computes QNM frequencies using both the continued fraction method and the matrix method, validating the two approaches. The analysis reveals how spin, Lorentz-violating parameter $\ell$, and scalar mass $\tilde{\mu}$ jointly shape the QNM spectrum, with a notable increased sensitivity to negative $\ell$ and lifted degeneracy at second order, including distinct differences between the NJA and slow-rotation pictures. The results offer insights into Lorentz-violating gravity phenomenology around rotating black holes and motivate future work on Proca fields and thermodynamic/topological aspects of these solutions.

Abstract

In this study, we examine the impacts of black hole spin, Lorentz-violating parameter, and the scalar field's mass on quasinormal modes (QNMs) of rotating Einstein-Bumblebee black holes, including computations up to the second-order expansion in rotation parameters. We investigate two classes of Lorentz-violating rotating black holes: one constructed via the Newman-Janis algorithm and the other obtained by solving the field equations through a series expansion. Within the slow-rotation approximation framework, we derive the master equations governing a massive scalar field and compute the corresponding QNM frequencies numerically using both the continued fraction method and the matrix method. The numerical results indicate that the QNM frequencies exhibit increased sensitivity to negative $\ell$ variations, which reduces the influence of the field mass parameter $\tildeμ$. Meanwhile, the spectral "cube" of NJA black holes shows slight compression for $m>0$ with $ \ell>0 $ and expansion for $m<0$ with $ \ell>0 $ compared to another black holes, where $m$ is approximately proportional to the spin parameter at first order, while richer structures and lifted degeneracy emerge at second order.

Figures (5)

• Figure 1: The percentage error between CFM and MM results is analyzed. (i) and (ii) represent real and imaginary components, respectively. Data points are sampled at intervals of 0.01 for the dimensionless parameters $\tilde{a}$, $\tilde{\mu}$ and $\ell$.
• Figure 2: Complex scalar frequencies corresponding to the $l=m=n=0$ mode, shown for various values of the Lorentz-violation parameter and field mass.
• Figure 3: Complex scalar frequencies corresponding to the $l=1, m=\pm1, n=0$ modes, shown for various values of the spin, Lorentz-violation parameter and field mass.
• Figure 4: Complex scalar frequencies corresponding to the $l=2, m=\pm1, n=0$ modes, shown for various values of the spin, Lorentz-violation parameter and field mass.
• Figure 5: Complex scalar frequencies corresponding to the $l=1, m=\pm2, n=0$ modes, shown for various values of the spin, Lorentz-violation parameter and field mass.