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Quasinormal modes of Proca and Maxwell fields in d-dimensional Schwarzschild-AdS black holes

David C. Lopes1,*, Tiago V. Fernandes2,†, and José P. S. Lemos3,‡

  • *Contact author: david.d.lopes@tecnico.ulisboa.pt
  • †Contact author: tiago.vasques.fernandes@tecnico.ulisboa.pt
  • ‡Contact author: joselemos@ist.utl.pt
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Phys. Rev. D 114, 084034 – Published 13 October, 2026

DOI: https://doi.org/10.1103/myw7-h3cd

Abstract

Proca and Maxwell fields in d-dimensional Schwarzschild black holes with anti-de Sitter (AdS) asymptotics are investigated through their linear perturbations and associated quasinormal modes (QNMs) with Dirichlet boundary conditions at infinity. We begin by reviewing how the Proca field equations in d-dimensional spherically symmetric spacetimes reduce to three radial wavelike equations, one fully decoupled equation and two mutually coupled equations. We also demonstrate explicitly how the Maxwell equations emerge from the zero-mass limit of the Proca system. Several analytical properties of the corresponding QNM spectrum are examined. To compute the QNM frequencies, we employ two complementary numerical methods particularly suited to asymptotically AdS spacetimes. Using these techniques, we determine the QNMs modes of Proca field perturbations in four-, five-, six-, and seven-dimensional Schwarzschild-AdS backgrounds. As a new result, we find numerically that scalar-type Maxwell perturbations in large d≥5 Schwarzschild-AdS black holes exhibit purely imaginary low-frequency modes, analogous to those found in vector-type gravitational perturbations. The presence of such modes is especially relevant within the AdS/CFT correspondence, as they correspond to the linearized hydrodynamic regime in the dual conformal field theory. We further analyze the influence of the Proca mass on the QNM spectrum, also emphasizing how Maxwell modes are recovered in the massless limit. The dependence of the spectrum on the black hole radius is explored. In addition, analytic expressions for the QNM frequencies of vector-type and monopole Proca perturbations, as well as Maxwell modes, are derived for small d-dimensional Schwarzschild-AdS black holes by matching asymptotic expansions using an intermediate region. These analytic results show good agreement with the numerical findings, confirming, in particular, the existence of purely imaginary low-frequency scalar-type Maxwell modes in large d≥5 Schwarzschild-AdS spacetimes.

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Phys. Rev. D 114, 084034– Published 13 October, 2026
  • Received 14 May 2026
  • Accepted 10 August 2026
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