Eigenvalues of the MOTS stability operator for slowly rotating Kerr black holes

Abstract

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 $a=0$.

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

$$\begin{aligned} \int _{-1}^1 P_{\ell _1}(z) P_{\ell _2}(z) P_{\ell _3}(z)\,dz = 2 \begin{pmatrix} \ell _1 &{} \ell _2 &{} \ell _3 \\ 0 &{} 0 &{} 0 \end{pmatrix}^2. \end{aligned}$$

(32)

In general the 3-j symbols are difficult to compute explicitly, but the following special case

$$\begin{aligned} \begin{pmatrix} 2 &{} \ell &{} \ell \\ 0 &{} 0 &{} 0 \end{pmatrix} = (-1)^{\ell +1} \sqrt{\frac{\ell (\ell +1)}{(2\ell +3)(2\ell +1)(2\ell -1)}} \end{aligned}$$

(33)

is easily obtained from [19, Eq. (C.23b)], so we have

$$\begin{aligned} \int _{-1}^1 P_2(z) P_\ell (z)^2\,dz = \frac{2\ell (\ell + 1)}{(2\ell +3)(2\ell +1)(2\ell -1)}. \end{aligned}$$

(34)

More general (and complicated) formulas exist for integrals of associated Legendre polynomials. For $m_3 = m_1 + m_2$ we have

$$\begin{aligned} \begin{aligned} \int _{-1}^1 P_{\ell _1}^{m_1}(z) P_{\ell _2}^{m_2}(z) P_{\ell _3}^{m_3}(z)\,dz&= 2 (-1)^{m_3} \sqrt{ \frac{(\ell _1 + m_1)!(\ell _2 + m_2)!(\ell _3 + m_3)!}{(\ell _1 - m_1)!(\ell _2 - m_2)!(\ell _3 - m_3)!}} \\&\quad \times \begin{pmatrix} \ell _1 &{} \ell _2 &{} \ell _3 \\ 0 &{} 0 &{} 0 \end{pmatrix} \begin{pmatrix} \ell _1 &{} \ell _2 &{} \ell _3 \\ m_1 &{} m_2 &{} -m_3 \end{pmatrix}, \end{aligned} \end{aligned}$$

(35)

see [10, eq. (30)]. Choosing $m_1 = m_2 = m_3 = 0$, we get (32) as a special case. The other case we need is $\ell _1 = \ell _2 = \ell$, $\ell _3 = 2$, $m_1 = m_2 = 1$ and $m_3 = 2$. Using

$$\begin{aligned} \begin{pmatrix} \ell &{} \ell &{} 2 \\ 1 &{} 1 &{} -2 \end{pmatrix} = (-1)^{\ell +1} \sqrt{\frac{3}{2}} \sqrt{\frac{\ell (\ell +1)}{(2\ell +3)(2\ell +1)(2\ell -1)}} \end{aligned}$$

(36)

together with (33), we find that

$$\begin{aligned} \int _{-1}^1 P_2^2(z) P_{\ell }^1(z)^2 \,dz = \frac{12 \ell ^2(\ell +1)^2}{(2\ell +3)(2\ell +1)(2\ell -1)}. \end{aligned}$$

(37)