Quasinormal modes and dynamical evolution of scalar fields in the Einstein-bumblebee theory with a cosmological constant

Hao Hu; Guoxiong Zhu

doi:10.5802/crphys.260

Research article

Quasinormal modes and dynamical evolution of scalar fields in the Einstein-bumblebee theory with a cosmological constant

Hao Hu ¹; Guoxiong Zhu ^2,³

¹ Department of Oncology, The Third Xiangya Hospital, Central South University, Changsha, 410013, Hunan, China
² Radiotherapy Department, Hunan Hospital of Integrated Traditional Chinese and Western Medicine, Changsha, 410006, Hunan, China
³ Institute of Nuclear Science and Technology, Sichuan University, Chengdu, 610064, Sichuan, China

Comptes Rendus. Physique, Volume 26 (2025), pp. 571-586

Abstract (Original language)
Résumé

In this paper, we investigate the quasi normal modes (QNMs) of static spherically symmetric black holes in the Einstein-bumblebee gravity model, taking into account the effects of the field mass and the cosmological constant. Through separation of the angular components, the scalar field perturbations outside the black hole are reduced to a purely radial main equation. We calculated the quasinormal modes using the matrix method and the WKB approximation, and also studied the dynamical evolution of the purely radial main equation using the finite difference method in the time domain. The eigenfrequencies of the waveforms from the time-domain evolution are fitted to cross-validate the frequency-domain results. The Lorentz-violating parameter $\ell $, cosmological constant $\Lambda $, and scalar field mass $\mu M$ affect QNMs. Specifically, increasing $\ell $ decreases both real and imaginary parts of the monopole modes, but in the dipole and quadrupole modes, the real part remains nearly unchanged while the imaginary part drops rapidly; rising $\Lambda $ reduces both parts of QNMs; increasing $\mu M$ raises the real part and lowers the imaginary part. Time-domain analysis confirms these findings, clarifying how Lorentz symmetry breaking impacts QNMs in de Sitter spacetime.

Dans cet article, nous étudions les modes quasi normaux (QNM) des trous noirs statiques sphériquement symétriques dans le modèle de gravité Einstein-bumblebee, en tenant compte des effets de la masse du champ et de la constante cosmologique. Grâce à la séparation des composantes angulaires, les perturbations du champ scalaire à l’extérieur du trou noir sont réduites à une équation principale purement radiale. Nous avons calculé les modes quasi normaux à l’aide de la méthode matricielle et de l’approximation WKB, et avons également étudié l’évolution dynamique de l’équation principale purement radiale à l’aide de la méthode des différences finies dans le domaine temporel. Les fréquences propres des formes d’onde issues de l’évolution dans le domaine temporel sont ajustées afin de valider les résultats dans le domaine fréquentiel. Le paramètre de violation de Lorentz $\ell $, la constante cosmologique $\Lambda $ et la masse du champ scalaire $\mu M$ affectent les QNM. Plus précisément, l’augmentation de $\ell $ diminue à la fois les parties réelles et imaginaires des modes monopôles, mais dans les modes dipôles et quadrupôles, la partie réelle reste pratiquement inchangée tandis que la partie imaginaire diminue rapidement ; l’augmentation de $\Lambda $ réduit les deux parties des QNM ; l’augmentation de $\mu M$ augmente la partie réelle et diminue la partie imaginaire. L’analyse dans le domaine temporel confirme ces résultats, clarifiant ainsi l’impact de la rupture de la symétrie de Lorentz sur les QNM dans l’espace-temps de Sitter.

Received: 2025-04-09
Revised: 2025-08-22
Accepted: 2025-09-01
Published online: 2025-09-22

DOI: 10.5802/crphys.260

Keywords: Lorentz violation, black hole, quasinormal modes
Mots-clés : Violation de Lorentz, trous noirs, modes quasi normaux

Author's affiliations:

Hao Hu ¹; Guoxiong Zhu ^{2, 3}

¹ Department of Oncology, The Third Xiangya Hospital, Central South University, Changsha, 410013, Hunan, China
² Radiotherapy Department, Hunan Hospital of Integrated Traditional Chinese and Western Medicine, Changsha, 410006, Hunan, China
³ Institute of Nuclear Science and Technology, Sichuan University, Chengdu, 610064, Sichuan, China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Hao Hu and Guoxiong Zhu},
     title = {Quasinormal modes and dynamical evolution of scalar fields in the {Einstein-bumblebee} theory with a cosmological constant},
     journal = {Comptes Rendus. Physique},
     pages = {571--586},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {26},
     doi = {10.5802/crphys.260},
     language = {en},
}

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Hao Hu; Guoxiong Zhu. Quasinormal modes and dynamical evolution of scalar fields in the Einstein-bumblebee theory with a cosmological constant. Comptes Rendus. Physique, Volume 26 (2025), pp. 571-586. doi: 10.5802/crphys.260

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