Black hole perturbations in vector-tensor theories: The odd-mode analysis

Ryotaro Kase, Masato Minamitsuji, Shinji Tsujikawa, Ying Li Zhang

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


In generalized Proca theories with vector-field derivative couplings, a bunch of hairy black hole solutions have been derived on a static and spherically symmetric background. In this paper, we formulate the odd-parity black hole perturbations in generalized Proca theories by expanding the corresponding action up to second order and investigate whether or not black holes with vector hair suffer ghost or Laplacian instabilities. We show that the models with cubic couplings G3(X), where X=-AμAμ/2 with a vector field Aμ, do not provide any additional stability condition as in General Relativity. On the other hand, the exact charged stealth Schwarzschild solution with a nonvanishing longitudinal vector component A1, which originates from the coupling to the Einstein tensor GμνAμ Aν equivalent to the quartic coupling G4(X) containing a linear function of X, is unstable in the vicinity of the event horizon. The same instability problem also persists for hairy black holes arising from general quartic power-law couplings G4(X) β4 Xn with the nonvanishing A1, while the other branch with A1=0 can be consistent with conditions for the absence of ghost and Laplacian instabilities. We also discuss the case of other exact and numerical black hole solutions associated with intrinsic vector-field derivative couplings and show that there exists a wide range of parameter spaces in which the solutions suffer neither ghost nor Laplacian instabilities against odd-parity perturbations.

Original languageEnglish
Article number048
JournalJournal of Cosmology and Astroparticle Physics
Issue number2
Publication statusPublished - 2018 Feb 23
Externally publishedYes


  • astrophysical black holes
  • dark energy theory
  • modified gravity

ASJC Scopus subject areas

  • Astronomy and Astrophysics


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