Quasinormal modes and AdS/CFT correspondence of a rotating BTZ-like black hole in the Einstein-bumblebee gravity

Quasinormal Modes and AdS/CFT Correspondence in Rotating BTZ-like Black Holes with Lorentz Symmetry Breaking

Introduction

The analytic treatment of quasinormal modes (QNMs) in black hole spacetimes is foundational for probing both gravitational wave data and theoretical questions in holography. This paper rigorously addresses QNMs for scalar, fermionic, and vector perturbations in the background of a rotating BTZ-like black hole constructed in the Einstein-bumblebee gravity framework, which includes a Lorentz symmetry breaking (LSB) parameter $\ell$. The work provides closed-form QNM frequencies, investigates their dependence on $\ell$ and black hole rotation $j$, and traces these results’ implications for the AdS/CFT correspondence, including the conformal weights of dual CFT operators.

Rotating BTZ-like Black Hole in Einstein-Bumblebee Gravity

The rotating BTZ-like solution considered arises from a three-dimensional Einstein-bumblebee theory, in which spontaneous Lorentz symmetry breaking is parameterized by a vector field with nonzero vacuum expectation value, giving rise to the parameter $\ell$ as a direct measure of the LSB sector coupling. The background metric possesses an AdS$_3$ asymptotic structure, with horizon radii and thermodynamic properties closely mirroring, but not identical to, those of the standard BTZ black hole. A crucial property is that while horizon radii are independent of $\ell$, Hawking temperature and QNM structure are not, a key differentiator from standard GR backgrounds.

Analytic Quasinormal Modes: Scalar, Fermion, and Vector Perturbations

Scalar Sector

The paper derives the Klein-Gordon equation in the BTZ-like geometry and reduces the radial mode equation to a hypergeometric form, enabling analytic expressions for the QNMs. The primary result is that $\ell$ enters only the imaginary parts of the QNM frequencies, leaving the real parts, which govern oscillatory behavior and correspond to angular momenta, unaffected and identical to their values in the standard BTZ spacetime. The principal decay timescales for the perturbations are thus modulated by LSB, with larger $\ell$ systematically reducing the inverse damping rate (i.e., slower decay), except in special cases under vector perturbations.

Figure 1: Variation of the imaginary parts of left- and right-moving quasinormal frequencies for fundamental scalar modes as a function of $\ell$ and rotation parameter $j$.

Fermionic Sector

For fermion fields, the Dirac equation is also reduced analytically. The structure and parameter dependence of the fermion QNMs closely mirror those of the scalar case: the imaginary part of the fundamental QNMs increases monotonically with $\ell$0, implying that LSB accelerates the decay of fermionic excitations (i.e., makes the system less dissipative as $\ell$1 decreases), while $\ell$2 introduces a branch asymmetry: increasing $\ell$3 increases the left-moving mode’s imaginary part and decreases the right-moving’s, clearly demonstrating parity-violating signatures in rotating backgrounds.

Figure 2: Imaginary parts of left- and right-moving fundamental fermionic QNMs versus $\ell$4 and $\ell$5.

Vector Sector

Maxwell field perturbations are solved via first-order and second-order equations. A notable result is the existence of special cases: the imaginary part of the left-moving fundamental QNM with positive mass and right-moving fundamental QNM with negative mass are independent of $\ell$6, in contrast to both the scalar and fermion sectors. Otherwise, for generic parameters and higher overtones, the qualitative dependence on $\ell$7 is consistent with the scalar and fermionic cases.

Figure 3: Imaginary parts of left- and right-moving fundamental vector QNMs as functions of $\ell$8 and $\ell$9.

AdS/CFT Correspondence and Conformal Weights

The holographic analysis applies the standard AdS$j$0/CFT$j$1 correspondence, relating left and right bulk QNMs to poles of thermal two-point functions of dual CFT operators. For all three field types, the QNMs can be cast in the universal form:

$j$2

where $j$3 are the left/right conformal weights, $j$4 is the overtone number, and $j$5 are the respective temperatures of left/right sectors determined by black hole parameters.

The analysis reaffirms that the key relation

$j$6

remains valid for all spins in the presence of LSB, where $j$7 is the conformal dimension and $j$8 the field spin.

• For the scalar, $j$9.
• For the fermion and vector, $\ell$0.

Thus, the main effect of LSB is to rescale the conformal dimensions through $\ell$1-dependent effective masses, directly affecting the CFT relaxation times $\ell$2 and the spectrum of poles in correlation functions.

Distinctive Features and Numerical Behavior

• Imprint of LSB ($\ell$3): Only the damping (imaginary) component of the QNMs is altered, with the direction and strength of the effect depending on mode and spin. For most modes, increased $\ell$4 enhances the inverse damping rate (slows decay), which has direct implications in gravitational wave "ringdown" observables.
• Rotational Asymmetry ($\ell$5): The black hole rotation $\ell$6 bifurcates imaginary part scaling between left/right branches, reflecting the interplay between rotation and LSB in parity-sensitive sectors.
• Conformal Weight Universality: Despite LSB, the characteristic CFT operator structure is preserved, with conformal weights governed by modified mass terms due to $\ell$7.

Theoretical and Practical Implications

The analytic results demonstrate that Lorentz-violating modifications to the gravitational sector, even if confined to the bumblebee effective vector, possess observable ramifications in the QNM spectrum, which is precisely the gravitational waveform regime probed in modern black hole spectroscopy. In the AdS/CFT context, the persistence of universal conformal weight structures suggests robustness of holographic dualities, but with modulated decay/thermalization timescales and operator weights as direct signatures of LSB physics.

The special $\ell$8-independence of certain fundamental vector QNMs indicates a delicate interplay between spin, mass, and symmetry breaking, meriting further investigation. These results also confirm that black hole perturbations in LSB backgrounds remain an effective tool for high-precision tests of gravitational and quantum field theoretic extensions of GR.

Conclusion

The study rigorously establishes that Lorentz symmetry breaking in Einstein-bumblebee gravity reparametrizes, but does not qualitatively destabilize, the QNM spectra and their AdS/CFT correspondence. LSB directly modulates the decay rates of black hole perturbations via the QNM imaginary parts and shifts the conformal dimensions of dual CFT operators, while preserving the universal structure expected from holography. The closed-form analytic results enable precise predictions relevant for both gravitational wave signal analysis and further theoretical developments in Lorentz-violating quantum gravity scenarios.