We study the eigenvalues of the MOTS stability operator for the Kerr black hole with angular momentum per unit mass \(|a| \ll M\). We prove that each eigenvalue depends analytically on a (in a neighbourhood of \(a=0\)), and compute its first nonvanishing derivative. Recalling that \(a=0\) corresponds to the Schwarzschild solution, where each eigenvalue has multiplicity \(2\ell +1\), we find that this degeneracy is completely broken for nonzero a. In particular, for \(0 < |a| \ll M\) we obtain a cluster consisting of \(\ell \) distinct complex conjugate pairs and one real eigenvalue. As a special case of our results, we get a simple formula for the variation of the principal eigenvalue. For perturbations that preserve the total area or mass of the black hole, we find that the principal eigenvalue has a local maximum at \(a=0\). However, there are other perturbations for which the principal eigenvalue has a local minimum at \(a=0\).
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The authors would like to thank José Luis Jaramillo for helpful comments and discussions on this problem. G.C. and H.K. acknowledge the support of NSERC grants RGPIN-2017-04259 and RGPIN-2018-04887, respectively.
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Appendix A: Wigner 3-j symbols
Appendix A: Wigner 3-j symbols
Here we review some properties of the Wigner 3-j symbols (as used above in the proof of Theorem 1.2), following the presentation of [19, Appendix C]. It is well known that the integral of three Legendre polynomials can be written in terms of Wigner 3-j symbols as
In general the 3-j symbols are difficult to compute explicitly, but the following special case
is easily obtained from [19, Eq. (C.23b)], so we have
More general (and complicated) formulas exist for integrals of associated Legendre polynomials. For \(m_3 = m_1 + m_2\) we have
together with (33), we find that
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Bussey, L., Cox, G. & Kunduri, H. Eigenvalues of the MOTS stability operator for slowly rotating Kerr black holes. Gen Relativ Gravit 53, 16 (2021). https://doi.org/10.1007/s10714-021-02786-3
- Marginally outer trapped surfaces
- Kerr spacetime
- Analytic perturbation theory