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  • Open Access

Teukolsky master equation and Painlevé transcendents: Numerics and extremal limit

Bruno Carneiro da Cunha and João Paulo Cavalcante
Phys. Rev. D 104, 084051 – Published 13 October 2021
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  • INTRODUCTION
  • MONODROMY PROPERTIES AND BOUNDARY…
  • ANGULAR AND RADIAL SYSTEMS
  • RESULTS FOR GENERIC a
  • THE a→M LIMIT
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We conduct an analysis of the quasinormal modes for generic spin perturbations of the Kerr black hole using the isomonodromic method. The strategy consists of solving the Riemann-Hilbert map relating the accessory parameters of the differential equations involved to monodromy properties of the solutions, using the τ-function for the Painlevé V transcendent. We show excellent accordance of the method with the literature for generic rotation parameter a<M. In the extremal limit, we determined the dependence of the modes with the black hole temperature and establish that the extremal values of the modes are obtainable from the Painlevé V and III transcendents.

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    • Received 16 July 2021
    • Accepted 13 September 2021

    DOI:https://doi.org/10.1103/PhysRevD.104.084051

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Classical black holesClassical solutions in field theoryConformal field theoryFluids & classical fields in curved spacetimeQuantum fields in curved spacetime
    Gravitation, Cosmology & AstrophysicsParticles & Fields

    Authors & Affiliations

    Bruno Carneiro da Cunha* and João Paulo Cavalcante†

    • Departamento de Física, Universidade Federal de Pernambuco, 50670-901, Recife, Brazil

    • *bruno.ccunha@ufpe.br
    • †joaopaulocavalcante@hotmail.com.br

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    Issue

    Vol. 104, Iss. 8 — 15 October 2021

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    Images

    • Figure 1
      Figure 1

      Real (left) and imaginary (right) parts of the fundamental quasinormal frequency for s=−2, ℓ=2 and m=2 (top), m=1 (middle) and m=0 (bottom). The continuous line shows the numerical results from (58) and the dashed refers to the results obtained with the continuous fraction method.

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    • Figure 2
      Figure 2

      The real (left) and imaginary (right) parts of the relative difference between the isomonodromic method based and the continued fraction method for the s=−2, ℓ=2 and m=2. We note that the agreement worsens as a→M, where the expansion parameter t of τV is smaller.

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    • Figure 3
      Figure 3

      The near-extremal behavior for the fundamental quasinormal frequency for s=0, ℓ=m=1 (top) and s=−2, ℓ=m=2 (bottom). Both refer to the extremal behavior described in Sec. 5a. In the top case, the α0 parameter is real, and near-horizon corrections to the real part of the frequency are of higher order in ν. In the bottom, α0 is imaginary and corrections to both real and imaginary parts are linear.

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    • Figure 4
      Figure 4

      The near-extremal behavior for the fundamental quasinormal frequency for s=−1, ℓ=2 and m=1. We note the roughly quadratic dependence for small ν, which is consequence of the first near-confluent correction to the τIII-function involved.

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