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Beyond quasinormal modes: A complete mode decomposition of black hole perturbations

Paolo Arnaudo*, Javier Carballo†, and Benjamin Withers‡

  • *Contact author: p.arnaudo@soton.ac.uk
  • †Contact author: j.carballo@soton.ac.uk
  • ‡Contact author: b.s.withers@soton.ac.uk
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Phys. Rev. D 114, 024025 – Published 9 July, 2026

DOI: https://doi.org/10.1103/1bcl-q79x

Abstract

We show that retarded Green’s functions of black hole spacetimes can be expressed as a convergent mode sum everywhere in spacetime. At late times a quasinormal mode sum converges, while at early times a Matsubara (or Euclidean) mode sum converges. The two regions are separated by a light cone that scatters from the black hole potential. The Matsubara sum is a Fourier series on the Euclidean thermal circle associated with the early time region. We illustrate our results for the Pöschl-Teller model, the Bañados-Teitelboim-Zanelli black hole, and the Schwarzschild black hole. In the case of the Schwarzschild black hole, we express the branch cut contribution as a convergent sum of de Sitter quasinormal modes as Λ→0+ and exploit recent exact solutions to the Heun connection problem. In each case, we analytically show convergence by studying the asymptotics of residue sums and also provide numerical demonstrations.

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Phys. Rev. D 114, 024025– Published 9 July, 2026
  • Received 15 January 2026
  • Accepted 11 June 2026
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