Beyond quasinormal modes: A complete mode decomposition of black hole perturbations
Open accessAbstract
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 lightcone which scatters from the black hole potential. The Matsubara sum is a Fourier series on the Euclidean thermal circle associated to the early time region. We illustrate our results for Pöschl-Teller, BTZ, and Schwarzschild. In the case of Schwarzschild, we express the branch cut contribution as a convergent sum of de Sitter quasinormal modes as $Λ\to 0^+$, 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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Cites
Scalar, electromagnetic, and Weyl perturbations of BTZ black holes: Quasinormal modes
Cardoso, V., & Lemos, J. P. S. (2001) · Physical Review D · doi:10.1103/physrevd.63.124015 · Cited by 372 · see citing works
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- Authors
- Arnaudo, P., Carballo, J., & Withers, B.
- Affiliations
- University of Southampton (GB)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2026-07-09
- Volume
- 114
- Issue
- 2
- ISSN
- 2470-0010 (ISSN-L)
- 2470-0029
- OpenAlex
- W7164353361
- Cited by
- 5
- References
- 1
- Citations by year
- 2026: 5
- FWCI
- 14.15
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D/Physical review. D. (publishedVersion)
- ePrints Soton (University of Southampton) (acceptedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University) (cc-by)
- Funders
- Royal Society (URF\R\231002, RF\ERE\210267)
- Science and Technology Facilities Council (ST/X000583/1)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Pulsars and Gravitational Waves Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Astrophysical Phenomena and Observations
- Keywords
- black hole perturbations, quasinormal modes, modal decomposition, black hole stability, gravitational waves
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