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Peer-reviewedCC BY 4.0The European Physical Journal C

Scalar quasinormal modes of Schwarzschild–anti-de Sitter black holes: spectral analysis and a formal coefficient deformation

  • Davide Batic1
  • Alan S. Cornell2
  • Denys Dutykh1
  1. 1Khalifa University of Science and Technology
  2. 2University of Johannesburg
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Abstract

We study quasinormal modes (QNMs) of a minimally coupled massless scalar field on four-dimensional Schwarzschild–anti-de Sitter (SAdS) black holes using a Chebyshev spectral method. After compactifying the exterior domain, the radial problem is formulated as a quadratic matrix pencil in the dimensionless frequency. For the standard Dirichlet, or vanishing-field, boundary condition at the conformal AdS boundary, we reproduce the known scalar spectra across small, intermediate, and large black holes, including long overtone sequences and the expected approach to pure-AdS normal modes in the small black hole limit. We also impose a one-parameter relation between the slow and fast asymptotic coefficients. For a massless scalar in four dimensions, the slow branch lies outside the standard finite canonical energy domain. Because we neither construct a renormalized variational principle nor prove well-posedness of an evolution containing this branch, we present only the formal spectrum of the deformed radial problem. Each non-Dirichlet value examined yields an additional eigenvalue with positive imaginary part and hence, an exponentially growing separated formal mode. This does not establish a physical instability of the standard scalar theory. The interpolation parameter is representation-dependent. In a black-hole-independent conformal coordinate, the corresponding response coupling scales with the cube of the dimensionless horizon radius, so equal intermediate parameter values for different black-hole sizes do not represent the same boundary coupling.

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Cite

BibTeXRIS
BibTeX
@article{batic2026scalar,
  title = {{Scalar quasinormal modes of Schwarzschild–anti-de Sitter black holes: spectral analysis and a formal coefficient deformation}},
  author = {Davide Batic and Alan S. Cornell and Denys Dutykh},
  journal = {The European Physical Journal C},
  year = {2026},
  month = oct,
  doi = {10.1140/epjc/s10052-026-16449-7},
  url = {https://link.springer.com/article/10.1140/epjc/s10052-026-16449-7},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Scalar quasinormal modes of Schwarzschild–anti-de Sitter black holes: spectral analysis and a formal coefficient deformation
AU  - Davide Batic
AU  - Alan S. Cornell
AU  - Denys Dutykh
PY  - 2026
DA  - 2026/10/29
JO  - The European Physical Journal C
DO  - 10.1140/epjc/s10052-026-16449-7
UR  - https://link.springer.com/article/10.1140/epjc/s10052-026-16449-7
AB  - We study quasinormal modes (QNMs) of a minimally coupled massless scalar field on four-dimensional Schwarzschild–anti-de Sitter (SAdS) black holes using a Chebyshev spectral method. After compactifying the exterior domain, the radial problem is formulated as a quadratic matrix pencil in the dimensionless frequency. For the standard Dirichlet, or vanishing-field, boundary condition at the conformal AdS boundary, we reproduce the known scalar spectra across small, intermediate, and large black holes, including long overtone sequences and the expected approach to pure-AdS normal modes in the small black hole limit. We also impose a one-parameter relation between the slow and fast asymptotic coefficients. For a massless scalar in four dimensions, the slow branch lies outside the standard finite canonical energy domain. Because we neither construct a renormalized variational principle nor prove well-posedness of an evolution containing this branch, we present only the formal spectrum of the deformed radial problem. Each non-Dirichlet value examined yields an additional eigenvalue with positive imaginary part and hence, an exponentially growing separated formal mode. This does not establish a physical instability of the standard scalar theory. The interpolation parameter is representation-dependent. In a black-hole-independent conformal coordinate, the corresponding response coupling scales with the cube of the dimensionless horizon radius, so equal intermediate parameter values for different black-hole sizes do not represent the same boundary coupling.
SN  - 1434-6052
SN  - 1434-6044
ER  -