Abstract
We conduct an analysis of the quasi-normal 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 $\tau$-function for the Painlev\'e V transcendent. We show good 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 Painlev\'e V and III transcendents.
Highlights
The Teukolsky master equation [1] governs linear perturbations of the Kerr metric
The method related these physical properties of the solutions of Fuchsian equations, and confluent limits, like (1) and (2), to the monodromy properties associated to the equation, relying on the Riemann-Hilbert map between the parameters of the equation and the monodromy data
The difference between the roots found in the literature are shown in Fig. 2, with excellent agreement throughout, and the discrepancies increasing as one approaches the extremal limit, where one expects the method based solely on the continued fraction expansion of the accessory parameter to fail
Summary
The Teukolsky master equation [1] governs linear perturbations of the Kerr metric. For vacuum perturbations, its solutions can be written as combinations of products of solutions of two ordinary differential equations. The asymptotic nature of these expansions, brings up technical problems, both in the expansions and in the numerical analysis Quasinormal modes present such a challenge: the non-local boundary conditions involved requires consideration of general complex frequencies. In [18,19], an alternative scheme was put forward that allowed for calculation of both the eigenvalue problem as well as scattering coefficients, building in previous work [20,21] The method related these physical properties of the solutions of Fuchsian equations, and confluent limits, like (1) and (2), to the monodromy properties associated to the equation, relying on the Riemann-Hilbert map between the parameters of the equation and the monodromy data. We include in the Appendix the frequencies and eigenvalues found for some modes in the extremal case
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
