The Teukolsky–Starobinsky constants: facts and fictions

Consider the angular ODE

$$\begin{align}& -\frac{1}{\mathrm{sin}\theta }\frac{\mathrm{d}}{\mathrm{d}\theta }\left(\mathrm{sin}\theta \frac{\mathrm{d}}{\mathrm{d}\theta }\right){{\Xi}}_{m{\Lambda}}^{[{\pm}s],\enspace (\nu )}(\theta )\\ & \qquad +\left(\frac{{(m{\pm}s\enspace \mathrm{cos}\enspace \theta )}^{2}}{{\mathrm{sin}}^{2}\enspace \theta }+{\nu }^{2}\enspace {\mathrm{sin}}^{2}\enspace \theta +2\nu ({\pm}s)\mathrm{cos}\enspace \theta \right){{\Xi}}_{m{\Lambda}}^{[{\pm}s],\enspace (\nu )}(\theta )\\ & \qquad ={\Lambda}\cdot {{\Xi}}_{m{\Lambda}}^{[{\pm}s],\enspace (\nu )}(\theta ),\end{align} \tag{ 2.1 }$$

where θ ∈ (0, π), $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, $m-s\in \mathbb{Z}$, $\nu \in \mathbb{R}$ and ${\Lambda}\in \mathbb{R}$. An asymptotic analysis leads us to conclude that for a solution to (2.1) there is a unique set of real numbers ${a}_{i}^{[{\pm}s]}$, i = 1, ..., 4, such that

$$\begin{equation}{{\Xi}}_{m{\Lambda}}^{[{\pm}s],\enspace (\nu )}={a}_{1}^{[{\pm}s]}{{\Xi}}_{\mathrm{norm},1}^{[{\pm}s]}+{a}_{2}^{[{\pm}s]}{{\Xi}}_{\mathrm{norm},2}^{[{\pm}s]}={a}_{3}^{[{\pm}s]}{{\Xi}}_{\mathrm{norm},3}^{[{\pm}s]}+{a}_{4}^{[{\pm}s]}{{\Xi}}_{\mathrm{norm},4}^{[{\pm}s]},\end{equation} \tag{ 2.2 }$$

where ${{\Xi}}_{\mathrm{norm},i}^{[{\pm}s]}$, for i = 1, ..., 4, encode the two linearly independent behaviors that solutions may take at the regular singular points θ = 0, π [Olv73, chapter 5]:

• If |m| ≠ s, take ${{\Xi}}_{\mathrm{norm},1}^{[{\pm}s]}{(1-\mathrm{cos}\enspace \theta )}^{-\frac{m{\pm}s}{2}}$ and ${{\Xi}}_{\mathrm{norm},2}^{[{\pm}s]}{(1-\mathrm{cos}\enspace \theta )}^{\frac{m{\pm}s}{2}}$ to be smooth as θ → 0 and normalized at the θ = 0 end;
• If |m| ≠ s, take ${{\Xi}}_{\mathrm{norm},3}^{[{\pm}s]}{(1+\mathrm{cos}\enspace \theta )}^{-\frac{m\mp s}{2}}$ and ${{\Xi}}_{\mathrm{norm},4}^{[{\pm}s]}{(1-\mathrm{cos}\enspace \theta )}^{\frac{m\mp s}{2}}$ to be smooth as θ → π and normalized at the θ = π end;
• If |m| = s, take ${{\Xi}}_{\mathrm{norm},4}^{[+s]}$, ${{\Xi}}_{\mathrm{norm},1}^{[+s]}$, ${{\Xi}}_{\mathrm{norm},3}^{[-s]}$ and ${{\Xi}}_{\mathrm{norm},2}^{[-s]}$ exactly as above; the definition of the remaining functions requires a logarithmic correction which we need not specify here.

We say that ${{\Xi}}_{m{\Lambda}}^{[{\pm}s],\enspace (\nu )}$ is a smooth (±s)-spin-weighted function if ${a}_{4}^{[{\pm}s]}={a}_{2}^{[{\pm}s]}=0$ when m > s, if ${a}_{3}^{[{\pm}s]}={a}_{1}^{[{\pm}s]}=0$ when m < −s and if ${a}_{3}^{[+s]}={a}_{2}^{[+s]}=0={a}_{4}^{[-s]}={a}_{1}^{[-s]}$ when |m| ⩽ s. These conditions ensure that ${\mathrm{e}}^{im\phi }{{\Xi}}_{m{\Lambda}}^{[{\pm}s],\enspace (\nu )}(\theta )$ are smooth sections of the Hopf bundle over ${\mathbb{S}}^{2}$ which transform in a particular way (according to the value of ±s) under rotation of the orthonormal frame in the tangent space of ${\mathbb{S}}^{2}$, see for instance [2019, section 2.2.1] and [Bey+14, CL78]. The space of such solutions is somewhat small, as the following proposition indicates:

Proposition 2.1 (Smooth spin-weighted spheroidal harmonics). Fix $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, let $m-s\in \mathbb{Z}$, and assume $\nu \in \mathbb{R}$. Consider the angular ODE (2.1) with the boundary condition that ${\mathrm{e}}^{im\phi }{S}_{m,\boldsymbol{\lambda }}^{[{\pm}s],\enspace (\nu )}$ is a non-trivial, normalized, smooth (±s)-spin-weighted function. There are countably many such solutions to this eigenvalue problem. Using l as an index, we write such solutions, also called (±s)-spin-weighted spheroidal harmonics with spheroidal parameter ν, as ${\mathrm{e}}^{im\phi }{S}_{ml}^{[{\pm}s],\enspace (\nu )}$ and denote the corresponding eigenvalues, which are real and independent of the sign chosen for s, by ${\mathbf{\Lambda }}_{sml}^{(\nu )}$. The parameter l is chosen so that $l-s\in \mathbb{Z}$, l ⩾ max{|m|, s} and ${\mathbf{\Lambda }}_{sml}^{(0)}=l(l+1)-{s}^{2}{\geqslant}s$. The following alternative notation will also be used:

$$\begin{equation}{\mathbf{\unicode{321}{}}}_{ml}^{(\nu )}{:=}{\mathbf{\Lambda }}_{sml}^{(\nu )}-2m\nu +s.\end{equation} \tag{ 2.3 }$$

Proof. We refer the reader to the careful treatment of the Sturm–Liouville theory for (2.1) for general $s\in \frac{1}{2}\mathbb{Z}$ in [HW74, Ste75], and [MS54] for the s = 0 case. □

Remark 2.2. In what follows, we often lighten the notation, replacing the spin-weighted spheroidal eigenvalues ${\mathbf{\Lambda }}_{sml}^{(\nu )}$ by Λ and similarly for Ł. However, non-bold characters Λ and Ł ≡ Λ − 2mν + s are not the same: they denote real parameters which are not constrained to be the eigenvalues identified in proposition 2.3 for some (m, l).

Given that (2.1) is analytic in the coefficient ν, the spin-weighted spheroidal eigenvalue ${\mathbf{\unicode{321}{}}}_{sml}^{(\nu )}$ admits an expansion in powers of ν as |ν| → ∞. Such expansions for s ≠ 0 go back to [BRW77], but they were completed and corrected by the work of the first author and collaborators:

Proposition 2.3 ([CO05, COW19]). Fix $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, a number m such that $m-s\in \mathbb{Z}$, a number l such that $l-\mathrm{max}\left\{\vert m\vert ,s\right\}\in {\mathbb{Z}}_{{\geqslant}0}$, and $\nu \in \mathbb{R}$. Let ${\mathbf{\unicode{321}{}}}_{sml}^{(\nu )}$ be as in proposition 2.1. Then, as ν → ∞, for any N > 0, we can find real constants A_k = A_k (s, l, m, ν), k ⩽ N, such that

$$\begin{equation}{\mathbf{\unicode{321}{}}}_{sml}^{(\nu )}=\sum\limits _{k=-1}^{N}\frac{{A}_{k}}{{\nu }^{k}}+O\left({\nu }^{-N-1}\right).\end{equation} \tag{ 2.4 }$$

The first two coefficients are

$$\begin{equation*}{A}_{-1}=2({q}_{sml}-m)\enspace ,\qquad {A}_{0}=-\frac{1}{2}\left[{({q}_{sml})}^{2}-{m}^{2}+1-2s\right],\end{equation*}$$

where, denoting by odd(⋅) a function on $\mathbb{Z}$ which is one if the argument is odd and zero otherwise, we may express q_sml as

$$\begin{equation}{q}_{sml}=\begin{cases}l+1-\mathrm{o}\mathrm{d}\mathrm{d}(l+m),\quad & \quad \text{if}\enspace l{\geqslant}\vert m+s\vert +s\\ 2l+1-(\vert m+s\vert +s),\quad & \quad \text{if}\enspace l{< }\vert m+s\vert +s\end{cases}.\end{equation} \tag{ 2.5 }$$

The following coefficients may be computed as follows. Let

$$\begin{align*}{Q}_{sml,n}^{+}& \equiv (2n+{q}_{sml}+s-\vert m-s\vert +1)(2n+{q}_{sml}-s+\vert m+s\vert +1),\\ {Q}_{sml,n}^{-}& \equiv (2n+{q}_{sml}+s+\vert m-s\vert -1)(2n+{q}_{sml}-s-\vert m+s\vert -1).\end{align*}$$

Then, for k ⩾ 1, we have

$$\begin{equation}{A}_{k}=\frac{1}{4}{Q}_{sml,0}^{+}{a}_{1,k}+\frac{1}{4}{Q}_{sml,0}^{-}{a}_{-1,k},\end{equation} \tag{ 2.6 }$$

where a_1,k and a_−1,k are computed from the following recursive relations for n ≠ 0:

$$\begin{equation}\begin{cases}_{n,\vert n\vert }& =-\frac{1}{16n}\left\{\begin{matrix}\hfill {Q}_{sml,n}^{-}{a}_{n-1,\vert n\vert -1},\hfill & \hfill \quad n{\geqslant}1\hfill \\ \hfill {Q}_{sml,n}^{+}{a}_{n+1,\vert n\vert -1},\hfill & \hfill \quad n{\leqslant}-1\hfill \end{matrix}\right\},\\ {a}_{n,\vert n\vert +1} & =\frac{1}{2}({q}_{sml}+n){a}_{n,\vert n\vert }-\frac{1}{16n}\left\{\begin{matrix}\hfill {Q}_{sml,n}^{-}{a}_{n-1,\vert n\vert },\hfill & \hfill n{\geqslant}1\hfill \\ \hfill {Q}_{sml,n}^{+}{a}_{n+1,\vert n\vert },\hfill & \hfill n{\leqslant}-1\hfill \end{matrix}\right\},\\ {a}_{n,j+1}& =\frac{1}{2}({q}_{sml}+n){a}_{n,j}-\frac{{Q}_{sml,n}^{+}}{16n}{a}_{n+1,j}-\frac{{Q}_{sml,n}^{-}}{16n}{a}_{n-1,j}+\sum\limits _{i{\geqslant}1}\frac{{A}_{i}}{4n}{a}_{n,i-j}\enspace ,\quad j{\geqslant}\vert n\vert +1,\end{cases}\end{equation} \tag{ 2.7 }$$

initialized by the choice a_0,0 = 1 and a_0,j = 0 if j ≠ 0. In particular, for k ⩽ 7, A_k are explicitly determined in [COW19, equations (3.15)–(3.21)].

Remark 2.4. Due to the symmetries of the angular ODE (2.1), the high frequency limit ν → −∞ follows from proposition 2.3 by replacing m → −m.

In this section, we will require the notation

$$\begin{equation}{\hat{\mathcal{L}}}_{n}^{{\pm}}\equiv \frac{\mathrm{d}}{\mathrm{d}\theta }{\pm}\left(\frac{m}{\mathrm{sin}\theta }-\nu \enspace \mathrm{sin}\theta \right)+n\mathrm{cot}\theta ,\end{equation} \tag{ 2.8 }$$

where θ ∈ (0, π), $n,m\in \frac{1}{2}\mathbb{Z}$, $\nu \in \mathbb{R}$.

Proposition 2.5 (Angular TS identities). Fix $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, m such that $m-s\in \mathbb{Z}{\backslash}\left\{0\right\}$, ${\Lambda}\in \mathbb{R}$ and $\nu \in \mathbb{R}$. Then, if ${{\Xi}}_{m{\Lambda}}^{[{\pm}s],\enspace (\nu )}$ solve (2.1) and admit the decomposition (2.2),

$$\begin{equation}\begin{aligned}\left(\prod\limits _{k=0}^{2s-1}{\hat{\mathcal{L}}}_{s-k}^{+}\right){{\Xi}}_{m{\Lambda}}^{[+s],\enspace (\nu )}& ={a}_{1}^{[+s]}{\mathfrak{B}}_{s}^{(1)}{{\Xi}}_{\mathrm{norm},1}^{[-s]}+{a}_{2}^{[+s]}{\mathfrak{B}}_{s}^{(7)}{{\Xi}}_{\mathrm{norm},2}^{[-s]}\\ & ={a}_{3}^{[+s]}{\mathfrak{B}}_{s}^{(4)}{{\Xi}}_{\mathrm{norm},3}^{[-s]}+{a}_{4}^{[+s]}{\mathfrak{B}}_{s}^{(6)}{{\Xi}}_{\mathrm{norm},4}^{[-s]},\\ \left(\prod\limits _{k=0}^{2s-1}{\hat{\mathcal{L}}}_{s-k}^{-}\right){{\Xi}}_{m{\Lambda}}^{[-s],\enspace (\nu )}& ={a}_{1}^{[-s]}{\mathfrak{B}}_{s}^{(3)}{{\Xi}}_{\mathrm{norm},1}^{[+s]}+{a}_{2}^{[-s]}{\mathfrak{B}}_{s}^{(5)}{{\Xi}}_{\mathrm{norm},2}^{[+s]}\\ & ={a}_{3}^{[-s]}{\mathfrak{B}}_{s}^{(2)}{{\Xi}}_{\mathrm{norm},3}^{[+s]}+{a}_{4}^{[-s]}{\mathfrak{B}}_{s}^{(8)}{{\Xi}}_{\mathrm{norm},4}^{[+s]},\end{aligned}\end{equation} \tag{ 2.9 }$$

where the products on the left-hand side are replaced by the identity if s = 0 and, if s ≠ 0, have index k increasing from right to left. Here, ${\mathfrak{B}}_{s}^{(i)}={\mathfrak{B}}_{s}^{(i)}(\nu ,m,{\Lambda})$ for i = 1, ..., 8. Indeed, if s = 0, ${\mathfrak{B}}_{s}^{(i)}=1$ for i = 1, ..., 8. For s ≠ 0, we easily obtain

$$\begin{equation*}{(-1)}^{2s}{\mathfrak{B}}_{s}^{(2)}={\mathfrak{B}}_{s}^{(6)}={\mathfrak{B}}_{s}^{(1)}={(-1)}^{2s}{\mathfrak{B}}_{s}^{(5)}={2}^{s}\prod\limits _{j=0}^{2s-1}\left(m+s-\frac{3}{2}j\right);\end{equation*}$$

the other ${\mathfrak{B}}_{s}^{(i)}$ can be computed explicitly in terms of the first s coefficients of the asymptotic expansions of ${{\Xi}}_{\mathrm{norm},1}^{[+s]}$, ${{\Xi}}_{\mathrm{norm},3}^{[+s]}$, ${{\Xi}}_{\mathrm{norm},2}^{[-s]}$ and ${{\Xi}}_{\mathrm{norm},4}^{[-s]}$, which in turn can be explicitly computed in terms of (ν, m, Λ).

Proof. The result follows from differentiating the asymptotic formulas for ${{\Xi}}_{\mathrm{norm},i}^{[{\pm}s]}$, i = 1, ..., 4. In doing so, it can be useful to note that

$$\begin{equation*}\prod\limits _{k=0}^{2s-1}{\hat{\mathcal{L}}}_{s-k}^{{\pm}}={\left(\mathrm{sin}\theta \right)}^{2s}{\left(\frac{{\hat{\mathcal{L}}}_{s}^{{\pm}}}{\mathrm{sin}\theta }\right)}^{2s}.\end{equation*}$$

For further details, we refer the reader to an analogous proof, in the setting of the radial Teukolsky–Starobinsky identities, in [TdC20, proposition 2.14]. □

We are ready to define the angular Teukolsky–Starobinsky constants:

Definition 2.1 (Angular TS constants). Fix $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, m such that $m-s\in \mathbb{Z}$, ${\Lambda}\in \mathbb{R}$, and $\nu \in \mathbb{R}$. Consider the operator

$$\begin{equation*}\left(\prod\limits _{j=0}^{2s-1}{\hat{\mathcal{L}}}_{s-j}^{\mp }\right)\left(\prod\limits _{k=0}^{2s-1}{\hat{\mathcal{L}}}_{s-k}^{{\pm}}\right)={\left(\mathrm{sin}\theta \right)}^{2s}{\left(\frac{{\hat{\mathcal{L}}}_{s}^{\mp }}{\mathrm{sin}\theta }\right)}^{2s}{\left(\mathrm{sin}\theta \right)}^{2s}{\left(\frac{{\hat{\mathcal{L}}}_{s}^{{\pm}}}{\mathrm{sin}\theta }\right)}^{2s},\end{equation*}$$

with indices j, k increasing from right to left on the product, and the latter being replaced by the identity if s = 0. If solutions of the angular ODE (2.1) with spin ±s are eigenfunctions of the above operator corresponding to the same eigenvalue, the eigenvalue is denoted by ${\mathfrak{B}}_{s}={\mathfrak{B}}_{s}(a\omega ,m,{\Lambda})$ and it is called the angular Teukolsky–Starobinsky constant.

Remark 2.6. Note that, in definition 2.1, we do not constrain Λ to be a spin-weighted spheroidal eigenvalue, as defined in proposition 2.1. In what follows, if we do take ${\Lambda}={\mathbf{\Lambda }}_{sml}^{(\nu )}$ and $\mathrm{\unicode{321}{}}={\mathbf{\unicode{321}{}}}_{sml}^{(\nu )}$ for some l, then we write ${\mathfrak{B}}_{s}={\mathfrak{B}}_{s}(\nu ,m,l)$.

By direct computation, we can check that an angular Teukolsky–Starobinsky constant exists at least for low values of Teukolsky spin:

Lemma 2.7. For any $s\in \left\{0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3\right\}$, there exists an angular Teukolsky–Starobinsky constant. Moreover, it is given by:

$$\begin{align}{\mathfrak{B}}_{0}(\nu ,m,{\Lambda})& =1,\\ -{\mathfrak{B}}_{\frac{1}{2}}(\nu ,m,{\Lambda})& =\mathrm{\unicode{321}{}},\\ {\mathfrak{B}}_{1}(\nu ,m,{\Lambda})& ={\mathrm{\unicode{321}{}}}^{2}+4m\nu -4{\nu }^{2},\\ -{\mathfrak{B}}_{\frac{3}{2}}(\nu ,m,{\Lambda})& ={\mathrm{\unicode{321}{}}}^{2}\left(\mathrm{\unicode{321}{}}+1\right)-16\nu \mathrm{\unicode{321}{}}(\nu -m)+16{\nu }^{2},\\ {\mathfrak{B}}_{2}(\nu ,m,{\Lambda})& ={\mathrm{\unicode{321}{}}}^{2}{(\mathrm{\unicode{321}{}}+2)}^{2}+40\nu {\mathrm{\unicode{321}{}}}^{2}(m-\nu )+48\nu \mathrm{\unicode{321}{}}(m+\nu )+144{\nu }^{2}{(m-\nu )}^{2},\\ -{\mathfrak{B}}_{\frac{5}{2}}(\nu ,m,{\Lambda})& ={\mathrm{\unicode{321}{}}}^{2}{\left(\mathrm{\unicode{321}{}}+3\right)}^{2}\left(\mathrm{\unicode{321}{}}+4\right)+16m\nu \mathrm{\unicode{321}{}}(3+\mathrm{\unicode{321}{}})(8+5\mathrm{\unicode{321}{}})m\\ & \quad -16{\nu }^{2}\left(\mathrm{\unicode{321}{}}(-12+\mathrm{\unicode{321}{}}(2+5\mathrm{\unicode{321}{}}))\right)+1024(1+\mathrm{\unicode{321}{}}){\nu }^{2}{m}^{2}\\ & \quad +1024{\nu }^{3}(1-2\mathrm{\unicode{321}{}})m+1024{\nu }^{4}(-2+\mathrm{\unicode{321}{}}),\\ {\mathfrak{B}}_{3}(\nu ,m,{\Lambda})& ={\mathrm{\unicode{321}{}}}^{2}{(\mathrm{\unicode{321}{}}+4)}^{2}{(\mathrm{\unicode{321}{}}+6)}^{2}+4m\nu \mathrm{\unicode{321}{}}(4+\mathrm{\unicode{321}{}})(360+7\mathrm{\unicode{321}{}}(36+5\mathrm{\unicode{321}{}}))\\ & \quad +4{\nu }^{2}\left(-\mathrm{\unicode{321}{}}(4+\mathrm{\unicode{321}{}})(-120+7\mathrm{\unicode{321}{}}(4+5\mathrm{\unicode{321}{}}))\right.\\ & \quad \left.+4(900+\mathrm{\unicode{321}{}}(1140+259\mathrm{\unicode{321}{}})){m}^{2}\right)\\ & \quad +32{\nu }^{3}m\left(300-260\mathrm{\unicode{321}{}}-259{\mathrm{\unicode{321}{}}}^{2}+450{m}^{2}\right)\\ & \quad +16{\nu }^{4}\left(100+\mathrm{\unicode{321}{}}(-620+259\mathrm{\unicode{321}{}})\right)\\ & \quad -43200{\nu }^{4}m(m-\nu )-14400{\nu }^{6}.\end{align} \tag{ 2.10 }$$

In all the above examples, if ν = 0 and $\mathrm{\unicode{321}{}}={\mathbf{\unicode{321}{}}}_{sml}^{(\nu )}$ corresponds to a spin-weighted spheroidal eigenvalue with spheroidal parameter ν = 0 for some l, then ${(-1)}^{2s}{\mathfrak{B}}_{s}(\nu =0,m,l){\geqslant}1$.

Proof. The proof of existence is similar to the case of the radial constants, in lemma 3.3 to come, so we do not present it here, though see the attached Mathematica notebook for the computations (https://stacks.iop.org/CQG/38/165016/mmedia). For the final statement, it is easy to see that, if ν = 0, only the first term of each expression in (2.10) remains. The least value is attained by $-{\mathfrak{B}}_{1/2}(\nu =0,m,l=s=1/2)=1$. □

Remark 2.8. We expect that the angular Teukolsky–Starobinsky constant can be defined for all $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$. However, no proof has been given in the literature.

If they exist, ${\mathfrak{B}}_{s}$ are non-negative (integer s) or non-positive (half-integer s):

Lemma 2.9 (Sign of angular TS constants). If an angular Teukolsky–Starobinsky constant exists for a certain (ν, m, Λ) satisfying the constraints in definition 2.1, then ${(-1)}^{2s}{\mathfrak{B}}_{s}(\nu ,m,{\Lambda}){\geqslant}0$.

Proof. To show that one has ${(-1)}^{2s}{\mathfrak{B}}_{s}{\geqslant}0$, recall the integration by parts identity of [Cha83, section 68, lemma 4] (earlier in [TP74]): for f and h sufficiently regular functions of θ,

$$\begin{equation}{\int }_{0}^{\pi }h\left({\mathcal{L}}_{n}^{{\pm}}f\right)\mathrm{sin}\theta \mathrm{d}\theta =-{\int }_{0}^{\pi }f\left({\mathcal{L}}_{-n+1}^{\mp }h\right)\mathrm{sin}\theta \mathrm{d}\theta .\end{equation} \tag{ 2.11 }$$

Without loss of generality, let Ξ^[±s] be real solutions to (2.1) normalized to have unit L² norm. Then, by (2.11), assuming existence of the constant,

$$\begin{align}{\mathfrak{B}}_{s}& ={\int }_{0}^{\pi }{{\Xi}}^{[{\pm}s]}\prod\limits _{j=0}^{2s-1}{\mathcal{L}}_{s-j}^{\mp }\prod\limits _{k=0}^{2s-1}{\mathcal{L}}_{s-k}^{{\pm}}{{\Xi}}^{[{\pm}s]}\mathrm{sin}\theta \mathrm{d}\theta \\ & ={(-1)}^{2s}{\int }_{0}^{\pi }{\left(\prod\limits _{k=0}^{2s-1}{\mathcal{L}}_{s-k}^{{\pm}}{{\Xi}}^{[{\pm}s]}\right)}^{2}\mathrm{sin}\theta \mathrm{d}\theta ,\end{align} \tag{ 2.12 }$$

where the integral on the right-hand side is non-negative. □

Remark 2.10. We note that the non-negativity of ${(-1)}^{2s}{\mathfrak{B}}_{s}$ provides nontrivial constraints on the values that Λ can take in terms of m and ν; see also lemma 3.3 for concrete examples.

We may go further when we let ${\Lambda}={\mathbf{\Lambda }}_{sml}^{(\nu )}$ be a spin-weighted spheroidal eigenvalue for some l:

Lemma 2.11 (Zeros of angular TS constants). Fix $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, a number m such that $m-s\in \mathbb{Z}$, a number l such that $l-\mathrm{max}\left\{\vert m\vert ,s\right\}\in {\mathbb{Z}}_{{\geqslant}0}$, and $\nu \in \mathbb{R}$. If an angular Teukolsky–Starobinsky constant exists for such (ν, m, l), then ${(-1)}^{2s}{\mathfrak{B}}_{s}(\nu ,m,l){ >}0$.

Proof. If the angular Teukolsky–Starobinsky constant exists, then in proposition 2.5, we must have

$$\begin{equation*}{\mathfrak{B}}_{s}^{(3)}=\frac{{\mathfrak{B}}_{s}}{{\mathfrak{B}}_{s}^{(1)}}\enspace ,\qquad {\mathfrak{B}}_{s}^{(4)}=\frac{{\mathfrak{B}}_{s}}{{\mathfrak{B}}_{s}^{(2)}}\enspace ,\qquad {\mathfrak{B}}_{s}^{(7)}=\frac{{\mathfrak{B}}_{s}}{{\mathfrak{B}}_{s}^{(5)}}\enspace ,\qquad {\mathfrak{B}}_{s}^{(8)}=\frac{{\mathfrak{B}}_{s}}{{\mathfrak{B}}_{s}^{(6)}}.\end{equation*}$$

We deal first with the case |m| > s, where we follow the strategy of the proof of [TdC20, lemma 2.19], an analogous result for the radial ODE. Suppose m < −s and ${\mathfrak{B}}_{s}=0$; for a spin-weighted spheroidal harmonics, we have ${a}_{3}^{[{\pm}s]}={a}_{1}^{[{\pm}s]}=0$, but by (2.9), ${a}_{2}^{[-s]},{a}_{4}^{[+s]}=0$ too, which would force ${S}_{ml}^{[{\pm}s],\enspace (\nu )}\equiv 0$. Now suppose m > s and ${\mathfrak{B}}_{s}=0$; for a spin-weighted spheroidal harmonics, we have ${a}_{4}^{[{\pm}s]}={a}_{2}^{[{\pm}s]}=0$, but by (2.9), this implies ${a}_{1}^{[+s]},{a}_{3}^{[-s]}=0$ too, which leads to another contradiction.

Finally, suppose −s ⩽ m ⩽ s, so that a spin-weighted spheroidal harmonic has ${a}_{3}^{[+s]}={a}_{2}^{[+s]}=0={a}_{4}^{[-s]}={a}_{1}^{[-s]}$. On the other hand, from the formula (2.12), we find that ${\mathfrak{B}}_{s}=0$ if and only if (2.9) vanish, in which case one must also have ${a}_{1}^{[+s]}={a}_{4}^{[+s]}=0$, given that ${\mathfrak{B}}_{s}^{(1)},{\mathfrak{B}}_{s}^{(6)}\ne 0$, and that ${a}_{2}^{[-s]}={a}_{3}^{[-s]}=0$, given that ${\mathfrak{B}}_{s}^{(3)},{\mathfrak{B}}_{s}^{(2)}\ne 0$. We conclude ${S}_{ml}^{[{\pm}s],\enspace (\nu )}\equiv 0$, which is a contradiction.

Hence, ${\mathfrak{B}}_{s}(\nu ,m,l)\ne 0$. By lemma 2.9, the conclusion follows. □

In spite of the angular Teukolsky–Starobinsky constants' positivity, they approach arbitrarily small values, at least in the cases considered in lemma 2.7.

Lemma 2.12. Fix $s\in \left\{\frac{1}{2},1,\frac{3}{2},2,\frac{5}{3},3\right\}$. Then, there are some pairs (l, m), where $m-s\in \mathbb{Z}$ and $l-\mathrm{max}\left\{\vert m\vert ,s\right\}\in {\mathbb{Z}}_{{\geqslant}0}$, for which we have, as ν → ∞,

$$\begin{equation}{(-1)}^{2s}{\mathfrak{B}}_{s}(\nu ,m,l)=O({\nu }^{-N})\enspace ,\quad \forall \enspace N{ >}0.\end{equation} \tag{ 2.13 }$$

Proof. For s = 1/2, the leading order term vanishes if and only if

$$\begin{equation*}{q}_{\frac{1}{2},ml}-m=0{\Leftrightarrow}l=m,\quad m{\geqslant}\frac{1}{2}.\end{equation*}$$

For s = 1, ${\mathfrak{B}}_{1}=({A}_{-1}^{2}-4){\nu }^{2}+O(\nu )$, where the leading order term vanishes if and only if any one of the following holds

$$\begin{equation*}{q}_{1,ml}-m={\pm}1{\Leftrightarrow}\left\{\begin{matrix}\hfill l=m,\hfill & \hfill \quad \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}1\hfill \\ \hfill l=m+1,\hfill & \hfill \quad \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}0\hfill \end{matrix}\right..\end{equation*}$$

For s = 3/2, the leading order term of $-{\mathfrak{B}}_{3/2}=({A}_{-1}^{2}-16){A}_{-1}{\nu }^{3}+O({\nu }^{2})$ vanishes if and only if any one of the following holds

$$\begin{equation*}{q}_{\frac{3}{2},ml}-m=0,{\pm}2{\Leftrightarrow}\left\{\begin{matrix}\hfill l=m,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}3/2\hfill \\ \hfill l=m+1,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}1/2\hfill \\ \hfill l=m+2,\hfill & \hfill \enspace \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}-1/2\hfill \end{matrix}\right..\end{equation*}$$

For s = 2, ${\mathfrak{B}}_{2}=({A}_{-1}^{2}-4)({A}_{-1}^{2}-36){\nu }^{4}+O({\nu }^{3})$, and the leading order term vanishes if and only if any one of the following holds

$$\begin{equation*}{q}_{2,ml}-m={\pm}1,{\pm}3{\Leftrightarrow}\left\{\begin{matrix}\hfill l=m,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}2\hfill \\ \hfill l=m+1,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}1\hfill \\ \hfill l=m+2,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}0\hfill \\ \hfill l=m+3,\hfill & \hfill \enspace \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}-1\hfill \end{matrix}\right..\end{equation*}$$

For s = 5/2, the leading order term of ${\mathfrak{B}}_{\frac{5}{2}}$ vanishes if and only if any one of the following holds

$$\begin{equation*}{q}_{\frac{5}{2},ml}-m=0,{\pm}2,{\pm}4{\Leftrightarrow}\left\{\begin{matrix}\hfill l=m,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}5/2\hfill \\ \hfill l=m+1,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}3/2\hfill \\ \hfill l=m+2,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}1/2\hfill \\ \hfill l=m+3,\hfill & \hfill \enspace \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}-1/2\hfill \\ \hfill l=m+4,\hfill & \hfill \enspace \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}-3/2\hfill \end{matrix}\right..\end{equation*}$$

For s = 3, the leading order term of ${\mathfrak{B}}_{3}$ vanishes if and only if any one of the following holds

$$\begin{equation*}{q}_{3,ml}-m={\pm}1,{\pm}3,{\pm}5{\Leftrightarrow}\left\{\begin{matrix}\hfill l=m,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}3\hfill \\ \hfill l=m+1,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}2\hfill \\ \hfill l=m+2,\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}1\hfill \\ \hfill l=m+3\hfill & \hfill \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}0\hfill \\ \hfill l=m+4,\hfill & \hfill \quad \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}-1\hfill \\ \hfill l=m+5,\hfill & \hfill \quad \text{as}\enspace \text{long}\;\text{as}\;m{\geqslant}-2\hfill \end{matrix}\right..\end{equation*}$$

To obtain (2.13) for some N > 0, we need to compute the coefficients A_k of the asymptotic expansion (2.4) up to k = N − 1 + 2s. The formulas in [COW19, equation (3.15)–(3.21)] for A_k up to k = 7, for instance, yield that (2.13) holds for N ⩾ 7 + 1 − 2s for all (s, l, m) identified in the preceding paragraph.

In the cases

$$\begin{equation}\begin{gathered}s=\frac{1}{2}\enspace ,\enspace \enspace l=m{\geqslant}1/2\enspace ;\quad s=1\enspace ,\enspace \enspace l=1\enspace ,\enspace \enspace m=0\enspace ;\quad s=\frac{3}{2}\enspace ,\enspace \enspace l=\frac{3}{2}\enspace ,\enspace \enspace m=\frac{1}{2};\\ s=2\enspace ,\enspace \enspace l=2\enspace ,\enspace \enspace m=-1\enspace ;\quad s=\frac{5}{2}\enspace ,\enspace \enspace l=\frac{5}{2}\enspace ,\enspace \enspace m=-\frac{3}{2}\enspace ;\quad s=3\enspace ,\enspace \enspace l=3\enspace ,\enspace \enspace m=-2;\end{gathered}\end{equation} \tag{ 2.14 }$$

the formulas [COW19, equations (3.15)–(3.21)] in fact imply that A₋₁ = ±2 and that A_k = 0 for k = 0, ..., 7. By the recursive formulas (2.6) and (2.7), we must have that A_k = 0 for all k ⩾ 1. Hence, for such (s, m, l), (2.13) holds for all N > 0, as stated. □

Remark 2.13. Computer-based symbolic computations suggest that, in fact, all of the triples (s, m, l) identified in the first paragraph of the proof of lemma 2.12 verify the conclusion of the lemma, rather than just the smaller set in (2.14). This is confirmed by numerical analysis, see figure 1. However, to establish such a result would require a much more detailed analysis of the recursive relations (2.6) and (2.7) than we pursue here.

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In the case s = 1, numerical computations suggest that the superpolynomial decay with ν identified in lemma 2.12 for some (l, m) is actually exponential, see figure 1.

We consider the radial ODEs

$$\begin{align}& \left[{{\Delta}}^{\mp s}\frac{\mathrm{d}}{\mathrm{d}r}\left({{\Delta}}^{{\pm}s+1}\frac{\mathrm{d}}{\mathrm{d}r}\right)+\frac{{[\omega ({r}^{2}+{a}^{2})-am]}^{2}-2i({\pm}s)(r-M)[\omega ({r}^{2}+{a}^{2})-am]}{{\Delta}}\right]{\times}{\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }(r)\\ & \qquad +\left({\pm}4is\omega r-{\Lambda}\mp s+2\enspace \mathrm{a}\mathrm{m}\omega \right){\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }(r)=0\enspace ,\end{align} \tag{ 3.1 }$$

where M > 0, |a| ⩽ M, r ∈ (r₊, ∞), $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, m such that $m-s\in \mathbb{Z}$, ${\Lambda}\in \mathbb{R}$, and $\omega \in \mathbb{R}$. An asymptotic analysis of (3.1) shows that a solution to (3.1) admits a unique set of complex numbers ${a}_{{\mathcal{H}}^{+}}^{[{\pm}s]}$, ${a}_{{\mathcal{H}}^{-}}^{[{\pm}s]}$, ${a}_{{\mathcal{I}}^{+}}^{[{\pm}s]}$ and ${a}_{{\mathcal{I}}^{-}}^{[{\pm}s]}$ such that

$$\begin{equation}{\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }={a}_{{\mathcal{H}}^{+}}^{[{\pm}s]}\cdot {\alpha }_{{\mathcal{H}}^{+}}^{[{\pm}s]}+{a}_{{\mathcal{H}}^{-}}^{[{\pm}s]}\cdot {\alpha }_{{\mathcal{H}}^{-}}^{[{\pm}s]}={a}_{{\mathcal{I}}^{+}}^{[{\pm}s]}\cdot {\alpha }_{{\mathcal{I}}^{+}}^{[{\pm}s]}+{a}_{{\mathcal{I}}^{-}}^{[{\pm}s]}\cdot {\alpha }_{{\mathcal{I}}^{-}}^{[{\pm}s]},\end{equation} \tag{ 3.2 }$$

where ${\alpha }_{{\mathcal{I}}^{+}}^{[{\pm}s]}$ and ${\alpha }_{{\mathcal{I}}^{-}}^{[{\pm}s]}$ are solutions of (3.1) with outgoing and ingoing, respectively, boundary conditions as r → ∞ which are normalized at r = ∞, and where ${\alpha }_{{\mathcal{H}}^{+}}^{[{\pm}s]}$ and ${\alpha }_{{\mathcal{H}}^{-}}^{[{\pm}s]}$ are solutions of (3.1) with ingoing and outgoing, respectively, boundary conditions as r → r₊ which are normalized at r = r₊. Concretely,

• ${\alpha }_{{\mathcal{I}}^{+}}^{[{\pm}s]}{\text{e}}^{-\text{i}\omega r}{r}^{-2iM\omega +1{\pm}2s}$ and ${\alpha }_{{\mathcal{I}}^{-}}^{[{\pm}s]}{\text{e}}^{\text{i}\omega r}{r}^{2iM\omega +1}$ are smooth functions of 1/r as r → ∞ which are normalized at r = ∞;
• If |a| < M, ${\alpha }_{{\mathcal{H}}^{+}}^{[{\pm}s]}{(r-{r}_{+})}^{i\frac{2M{r}_{+}}{{r}_{+}-{r}_{-}}(\omega -m{\omega }_{+}){\pm}s}$ and ${\alpha }_{{\mathcal{H}}^{-}}^{[{\pm}s]}{(r-{r}_{+})}^{-i\frac{2M{r}_{+}}{{r}_{+}-{r}_{-}}(\omega -m{\omega }_{+})}$ are smooth as r → r₊ and normalized at r = r₊;
• If |a| = M, ${\alpha }_{{\mathcal{H}}^{+}}^{[{\pm}s]}{(r-M)}^{2iM\omega {\pm}2s}{\text{e}}^{-\text{i}\frac{2{M}^{2}}{r-M}(\omega -m{\omega }_{+})}$ and ${\alpha }_{{\mathcal{H}}^{-}}^{[{\pm}s]}{(r-M)}^{-2iM\omega }{\text{e}}^{\text{i}\frac{2{M}^{2}}{r-M}(\omega -m{\omega }_{+})}$ are smooth in 1/(r − M) as r → M and normalized at r = M.

In this section, we will require the notation

$$\begin{equation}{\hat{\mathcal{D}}}_{n}^{{\pm}}\equiv \frac{\mathrm{d}}{\mathrm{d}r}{\pm}i\left(\frac{\omega ({r}^{2}+{a}^{2})}{{\Delta}}-\frac{am}{{\Delta}}\right)+\frac{2n(r-M)}{{\Delta}},\end{equation} \tag{ 3.3 }$$

where M > 0, |a| ⩽ M, r ∈ (r₊, ∞), $n,m\in \frac{1}{2}\mathbb{Z}$ and $\omega \in \mathbb{R}$.

We quote from [TdC20, proposition 2.14] the following:

Proposition 3.1 (Radial TS identities). Fix M > 0, |a| ⩽ M and $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, m such that $m-s\in \mathbb{Z}$, ${\Lambda}\in \mathbb{R}$. Then, if α^[±s] solve (3.1) and admit the decomposition (3.2),

$$\begin{align}{{\Delta}}^{s}{\left({\hat{\mathcal{D}}}_{0}^{+}\right)}^{2s}\left({{\Delta}}^{s}{\alpha }^{[+s]}\right)& ={a}_{{\mathcal{I}}^{+}}^{[+s]}{\mathfrak{C}}_{s}^{(1)}{\alpha }_{{\mathcal{I}}^{+}}^{[-s]}+{a}_{{\mathcal{I}}^{-}}^{[+s]}{\mathfrak{C}}_{s}^{(7)}{\alpha }_{{\mathcal{I}}^{-}}^{[-s]}\\ & ={a}_{{\mathcal{H}}^{+}}^{[+s]}{\mathfrak{C}}_{s}^{(4)}{\alpha }_{{\mathcal{H}}^{+}}^{[-s]}+{a}_{{\mathcal{H}}^{-}}^{[+s]}{\mathfrak{C}}_{s}^{(6)}{\alpha }_{{\mathcal{H}}^{-}}^{[-s]},\\ {\left({\hat{\mathcal{D}}}_{0}^{-}\right)}^{2s}{\alpha }^{[-s]}& ={a}_{{\mathcal{I}}^{+}}^{[-s]}{\mathfrak{C}}_{s}^{(3)}{\alpha }_{{\mathcal{I}}^{+}}^{[+s]}+{a}_{{\mathcal{I}}^{-}}^{[+s]}{\mathfrak{C}}_{s}^{(5)}{\alpha }_{{\mathcal{I}}^{-}}^{[+s]}\\ & ={a}_{{\mathcal{H}}^{+}}^{[-s]}{\mathfrak{C}}_{s}^{(2)}{\alpha }_{{\mathcal{H}}^{+}}^{[+s]}+{\alpha }_{{\mathcal{H}}^{-}}^{[+s]}{\mathfrak{C}}_{s}^{(8)}{\alpha }_{{\mathcal{H}}^{-}}^{[+s]},\end{align} \tag{ 3.4 }$$

where the products on the left-hand side are replaced by the identity if s = 0 and, if s ≠ 0, have indices increasing from left to right. Here, ${\mathfrak{C}}_{s}^{(i)}={\mathfrak{C}}_{s}^{(i)}(a,M,\omega ,m,{\Lambda})$ for i = 1, ..., 8. Indeed, if s = 0, ${\mathfrak{C}}_{s}^{(i)}=1$ for i = 1, ..., 8. For s ≠ 0, we easily obtain

$$\begin{align*}{\mathfrak{C}}_{s}^{(2)}& =\prod\limits _{j=0}^{2s-1}\left[-4iM{r}_{+}(\omega -m{\omega }_{+})+(s-j)({r}_{+}-{r}_{-})\right]\\ & \quad {\times}\begin{cases}1& \text{if}\;\;\vert a\vert =M\\ {({r}_{+}-{r}_{-})}^{-2\vert s\vert }& \text{if}\;\;\vert a\vert {< }M\end{cases},\\ {\mathfrak{C}}_{s}^{(1)}& ={(2i\omega )}^{2s}\enspace ,\quad {\mathfrak{C}}_{s}^{(5)}={(-2i\omega )}^{2s}\enspace ,\quad \\ {\mathfrak{C}}_{s}^{(6)}& =\prod\limits _{j=0}^{2s-1}\left[4iM{r}_{+}(\omega -m{\omega }_{+})+(s-j)({r}_{+}-{r}_{-})\right];\end{align*}$$

the remaining ${\mathfrak{C}}_{s}^{(i)}$ can be computed explicitly in terms of the first s coefficients in the asymptotic expansions of ${\alpha }_{{\mathcal{I}}^{\mp }}^{[{\pm}s]}$ and ${\alpha }_{{\mathcal{H}}^{\mp }}^{[\mp s]}$, which in turn can be explicitly computed in terms of (a, M) and (ω, m, Λ).

We are now ready to define the radial Teukolsky–Starobinsky constants:

Definition 3.1 (Radial TS constants). Fix $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, m such that $m-s\in \mathbb{Z}$, ${\Lambda}\in \mathbb{R}$, M > 0, |a| ⩽ M and $\omega \in \mathbb{R}$. Consider the operator

$$\begin{equation*}{{\Delta}}^{s}{\left({\hat{\mathcal{D}}}_{0}^{\mp }\right)}^{2s}\left[{{\Delta}}^{s}{\left({\hat{\mathcal{D}}}_{0}^{{\pm}}\right)}^{2s}\right]\equiv \prod\limits _{j=0}^{2s-1}\left({{\Delta}}^{1/2}{\hat{\mathcal{D}}}_{j/2}^{\mp }\right)\prod\limits _{k=0}^{2s-1}\left({{\Delta}}^{1/2}{\hat{\mathcal{D}}}_{k/2}^{{\pm}}\right),\end{equation*}$$

with indices j, k increasing from right to left on the product, and the latter being replaced by the identity if s = 0. If ${{\Delta}}^{\frac{s}{2}(1{\pm}1)}{\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }$, where ${\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }$ solve the radial ODE (3.1) of spin ±s, are eigenfunctions of the above operator corresponding to the same eigenvalue, the eigenvalue is denoted by ${\mathfrak{C}}_{s}={\mathfrak{C}}_{s}(a,M,\omega ,m,{\Lambda})$ and it is called the radial Teukolsky–Starobinsky constant.

Remark 3.2. Note that, in definition 3.1 and lemma 3.3 below, we once again do not constrain Λ to be an eigenvalue of the angular ODE (2.1) with spheroidal parameter ν = aω. In what follows, if we do take ${\Lambda}={\mathbf{\Lambda }}_{sml}^{(a\omega )}$ and $\mathrm{\unicode{321}{}}={\mathbf{\unicode{321}{}}}_{sml}^{(a\omega )}$ for some l, then we write ${\mathfrak{C}}_{s}(a,M,\omega ,m,l)$.

By direct computation, we can check that a radial Teukolsky–Starobinsky constant exists at least for low values of Teukolsky spin:

Lemma 3.3. For any $s\in \left\{0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3\right\}$, there exists a radial Teukolsky–Starobinsky constant. Furthermore, it can be computed explicitly, for instance:

$$\begin{equation}\begin{aligned}{\mathfrak{C}}_{0}(a,M,\omega ,m,{\Lambda})& =1,\\ {\mathfrak{C}}_{\frac{1}{2}}(a,M,\omega ,m,{\Lambda})& =-{\mathfrak{B}}_{\frac{1}{2}}(a\omega ,m,{\Lambda}),\\ {\mathfrak{C}}_{1}(a,M,\omega ,m,{\Lambda})& ={\mathfrak{B}}_{1}(a\omega ,m,{\Lambda}),\\ {\mathfrak{C}}_{\frac{3}{2}}(a,M,\omega ,m,{\Lambda})& =-{\mathfrak{B}}_{\frac{3}{2}}(a\omega ,m,{\Lambda}),\\ {\mathfrak{C}}_{2}(a,M,\omega ,m,{\Lambda})& ={\mathfrak{B}}_{2}(a\omega ,m,{\Lambda})+144{M}^{2}{\omega }^{2},\\ {\mathfrak{C}}_{\frac{5}{2}}(a,M,\omega ,m,{\Lambda})& =-{\mathfrak{B}}_{\frac{5}{2}}(a\omega ,m,{\Lambda})+1152(\mathrm{\unicode{321}{}}+2){M}^{2}{\omega }^{2}\\ {\mathfrak{C}}_{3}(a,M,\omega ,m,{\Lambda})& ={\mathfrak{B}}_{3}(a\omega ,m,{\Lambda})+576\left[{(3\mathrm{\unicode{321}{}}+10)}^{2}+100a\omega (m-a\omega )\right]{M}^{2}{\omega }^{2},\end{aligned}\end{equation} \tag{ 3.5 }$$

where ${\mathfrak{B}}_{s}$ may be read off from (2.10). In the above, if aω = 0, and $\mathrm{\unicode{321}{}}={\mathbf{\unicode{321}{}}}_{sml}^{(a\omega )}$ corresponds to a spin-weighted spheroidal eigenvalue with spheroidal parameter ν = aω for some l, then ${\mathfrak{C}}_{s}(a,M,\omega ,m,l){\geqslant}1$.

Proof. There are several ways of showing existence of the radial Teukolsky–Starobinsky constants. One option is to use definition 3.1, i.e. to apply the operators in the products above to radial functions one by one and using the radial ODE (3.1) to trade second order derivatives of those functions by first and zeroth order terms (see for instance [Cha83, sections 70 and 81] for s = 1, 2). With the aid of a standard laptop, this naive approach allows one to verify existence of the constant beyond the upper bound s = 3 of the statement; indeed, the reader can check this with the Mathematica notebook attached.

Alternatively, in light of the radial Teukolsky–Starobinsky identities of proposition 3.1, if the radial Teukolsky–Starobinsky constant exists, we must have

$$\begin{equation*}{\mathfrak{C}}_{s}^{(3)}=\frac{{\mathfrak{C}}_{s}}{{\mathfrak{C}}_{s}^{(1)}}\enspace ,\qquad {\mathfrak{C}}_{s}^{(4)}=\frac{{\mathfrak{C}}_{s}}{{\mathfrak{C}}_{s}^{(2)}}\enspace ,\qquad {\mathfrak{C}}_{s}^{(7)}=\frac{{\mathfrak{C}}_{s}}{{\mathfrak{C}}_{s}^{(5)}}\enspace ,\qquad {\mathfrak{C}}_{s}^{(8)}=\frac{{\mathfrak{C}}_{s}}{{\mathfrak{C}}_{s}^{(6)}}.\end{equation*}$$

Hence, one may compute ${\mathfrak{C}}_{s}$ from one of ${\mathfrak{C}}_{s}^{(3)},{\mathfrak{C}}_{s}^{(4)},{\mathfrak{C}}_{s}^{(7)},{\mathfrak{C}}_{s}^{(8)}$. As the latter are computable from the recursive formulas which yield the first s coefficients of certain asymptotic series for solutions of (3.1), this method is less computationally demanding than the previous one and has been suggested earlier in [Fiz09] (see also the companion paper [Fiz10]).

To conclude, we note that, for ω = 0, a glance at the formulas gives ${\mathfrak{C}}_{s}(a,M,\omega =0,m,l)={(-1)}^{2s}{\mathfrak{B}}_{s}(a\omega =0,m,l){\geqslant}1$, from lemma 2.7. □

We remark that the formulas for ${\mathfrak{C}}_{s}$ given here in lemma 3.3 match those obtained before in [Cha90] and [KMW89], although in the latter papers they appear under the somewhat misleading notation |C_s |².

3.4.1. Why the complex-conjugation argument for non-negativity is false

This section examines the argument in [KMW89, KMW92], subsequently picked up by other authors, purportedly showing non-negativity of the Teukolsky–Starobinky constant for general $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$. These authors claim that non-negativity of ${\mathfrak{C}}_{s}$ may be viewed as a consequence of ${{\Delta}}^{\frac{s}{2}(1{\pm}1)}{\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }$ satisfying complex conjugate equations and the radial Teukolsky–Starobinsky identities being generated by complex conjugate operators somehow imply non-negativity of the Teukolsky–Starobinky constant.

No justification is given for why the authors believe this implication should be true. One potential argument would be to try draw an analogy with the angular setting, where the relation between ${\mathcal{L}}_{n}^{{\pm}}$ and ${\mathcal{L}}_{-n+1}^{\mp }$ in the ODE (2.1) and in the identities (2.11) leads to the conclusion of lemma 2.9. However, (2.11) establishes an adjointness relation between the angular operators ${\mathcal{L}}_{n}^{{\pm}}$ and $-{\mathcal{L}}_{-n+1}^{\mp }$ in the space of (real-valued) smooth spin-weighted functions. In contrast, the relation between ${\mathcal{D}}_{n}^{{\pm}}$ and ${\mathcal{D}}_{n}^{\mp }$ is one of complex-conjugation.

It is a fact of life that, in a space of complex-valued functions, as solutions to the radial ODE (3.1) are bound to lie in, the complex conjugate and the adjoint of an operator are not necessarily the same ⁵ . Indeed, for some weight w(r): (r₊, ∞) → [0, ∞), and for f and h sufficiently regular complex-valued functions of r that the boundary terms in the following vanish, we have

$$\begin{equation*}{\int }_{{r}_{+}}^{\infty }\overline{h}\enspace {\mathcal{D}}_{0}^{{\pm}}fw\enspace \mathrm{d}r=-{\int }_{{r}_{+}}^{\infty }f\overline{\left({\mathcal{D}}_{0}^{{\pm}}+\frac{\mathrm{d}}{\mathrm{d}r}\enspace \mathrm{log}\enspace w\right)h}w\enspace \mathrm{d}r,\end{equation*}$$

so the adjoint of ${\mathcal{D}}_{0}^{{\pm}}$ will be $-{\mathcal{D}}_{0}^{{\pm}}-\frac{\mathrm{d}}{\mathrm{d}r}\enspace \mathrm{log}\enspace w$. This makes the notation used in classical references such as [Cha83] (see also the more recent [Teu15]), where ${\mathcal{D}}_{0}^{-}$ is written as ${({\mathcal{D}}_{0}^{+})}^{{\dagger}}$ thus suggesting an adjointness relation, rather unfortunate.

The upshot is that there is, in fact, no hope of establishing an analogue of lemma 2.9 by a similar method to the one used there. Indeed, let us try to run the argument backward to see where it goes wrong. Copying the proof of lemma 2.9, we would like to show that there is some weight w(r): (r₊, ∞) → [0, ∞) such that for any α^[±s] is chosen with unit ${L}_{w}^{2}$ norm,

$$\begin{align*}0{\leqslant}{\mathfrak{C}}_{s}(a,M,\omega ,{\Lambda})& ={\int }_{{r}_{+}}^{\infty }\overline{{\alpha }^{[{\pm}s]}}\prod\limits _{j=0}^{2s-1}\left({{\Delta}}^{1/2}{\mathcal{D}}_{j/2}^{\mp }\right)\prod\limits _{k=0}^{2s-1}\left({{\Delta}}^{1/2}{\mathcal{D}}_{k/2}^{{\pm}}\right){\alpha }^{[{\pm}s]}w\enspace \mathrm{d}r,\\ & \enspace {\Longleftarrow}\enspace {\int }_{{r}_{+}}^{\infty }\overline{{\alpha }^{[{\pm}s]}}\prod\limits _{j=0}^{2s-1}\left({{\Delta}}^{1/2}{\mathcal{D}}_{j/2}^{\mp }\right)\prod\limits _{k=0}^{2s-1}\left({{\Delta}}^{1/2}{\mathcal{D}}_{k/2}^{{\pm}}\right){\alpha }^{[{\pm}s]}w\enspace \mathrm{d}r\\ & ={\int }_{{r}_{+}}^{\infty }{\left\vert \prod\limits _{k=0}^{2s-1}\left({{\Delta}}^{1/2}{\mathcal{D}}_{k/2}^{{\pm}}\right){\alpha }^{[{\pm}s]}\right\vert }^{2}w\enspace \mathrm{d}r.\end{align*}$$

The latter equality would hold only if we could prove

$$\begin{equation}{\int }_{{r}_{+}}^{\infty }\overline{h}{{\Delta}}^{1/2}{\mathcal{D}}_{n/2}^{{\pm}}fw\enspace \mathrm{d}r={\int }_{{r}_{+}}^{\infty }f{{\Delta}}^{1/2}\overline{{\mathcal{D}}_{s-\frac{n+1}{2}}^{\mp }h}w\enspace \mathrm{d}r,\end{equation} \tag{ 3.6 }$$

for suitably regular f and h; but there is no choice of w ⩾ 0 which would make (3.6) hold. It follows that, if ${\mathfrak{C}}_{s}$ is indeed non-negative for some s, a different proof strategy should be sought.

3.4.2. Saving and improving non-negativity for s ⩽ 2

As we have noted that the angular Teukolsky–Starobinsky constants have a definite sign (lemma 2.9), it is natural to try to compare the explicit expressions for such constants with those of the radial ones, i.e. lemmas 2.7 and 3.3 in the case where the spheroidal parameter in the angular ODE is ν = aω. As is clear from our (3.5) (see also [KMW92]),

Lemma 3.4. Fix $s\in \left\{0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3\right\}$, m such that $m-s\in \mathbb{Z}$, ${\Lambda}\in \mathbb{R}$, M > 0, |a| ⩽ M and $\omega \in \mathbb{R}$. Then, there is a real ${\mathfrak{F}}_{s}={\mathfrak{F}}_{s}(a\omega ,m,{\Lambda})$ such that

$$\begin{equation*}{\mathfrak{C}}_{s}(a,M,\omega ,m,{\Lambda})={(-1)}^{2\vert s\vert }{\mathfrak{B}}_{s}(a\omega ,m,{\Lambda})+{\mathfrak{F}}_{s}(a\omega ,m,{\Lambda}){M}^{2}{\omega }^{2}.\end{equation*}$$

Indeed, one has ${\mathfrak{F}}_{s}\equiv 0$ for $s{\leqslant}\frac{3}{2}$ and

$$\begin{equation*}{\mathfrak{F}}_{2}=144\enspace ,\qquad {\mathfrak{F}}_{\frac{5}{2}}=1152(\mathrm{\unicode{321}{}}+2)\enspace ,\qquad {\mathfrak{F}}_{3}=576\left[{(3\mathrm{\unicode{321}{}}+10)}^{2}+100a\omega (m-a\omega )\right].\end{equation*}$$

Lemma 3.4 indeed yields, in the restricted case s ⩽ 2, non-negativity of the Teukolsky–Starobinsky constant, as correctly noted in Teukolsky's original paper on the identities [TP74]. Indeed, by lemma 2.11, it even yields positivity:

Lemma 3.5 (Positivity of radial TS constant for $\mathbf{s}\hspace{2pt}\mathbf{{\leqslant}}\hspace{2pt}\mathbf{2}$). Fix $s\in \left\{0,\frac{1}{2},1,\frac{3}{2},2\right\}$, m such that $m-s\in \mathbb{Z}$, ${\Lambda}\in \mathbb{R}$, M > 0, |a| ⩽ M and $\omega \in \mathbb{R}$. Then, ${\mathfrak{F}}_{s}(a\omega ,m,{\Lambda}){\geqslant}0$ hence ${\mathfrak{C}}_{s}(a,M,\omega ,m,{\Lambda}){\geqslant}0$.

Furthermore, if ${\Lambda}={\mathbf{\Lambda }}_{sml}^{(a\omega )}$ is a spin-weighted spheroidal eigenvalue for some $l\in {\mathbb{Z}}_{{\geqslant}\mathrm{max}\left\{\vert m\vert ,s\right\}}$, then ${\mathfrak{C}}_{s}(a,M,\omega ,m,l){ >}0$. In particular, there are no real algebraically special frequencies (ω, m, l) for any Kerr parameters (a, M).

Remark 3.6. For s > 2, ${\mathfrak{F}}_{s}$ depends nontrivially on Λ. Without more constraints on Λ in terms of the black hole parameters (a, M) and frequencies ω and m, one cannot hope to investigate the validity of lemma 3.5 for s > 2.

In fact, for s ⩽ 2, we can use lemma 3.4 and the last statement in lemma 3.3 to obtain a more precise statement when Λ = Λ is a spin-weighted spheroidal eigenvalue:

Lemma 3.7 (Lower bound for radial TS constant for $\mathbf{s}\hspace{2pt}\mathbf{=}\hspace{2pt}\mathbf{2}$). Fix s = 2, $m\in \mathbb{Z}$, $l\in {\mathbb{Z}}_{{\geqslant}\mathrm{max}\left\{\vert m\vert ,2\right\}}$, M > 0, |a| ⩽ M and $\omega \in \mathbb{R}$. The radial Teukolsky–Starobinsky constant ${\mathfrak{C}}_{2}$ admits a positive lower bound, i.e. there is a b > 0 such that

$$\begin{equation*}\underset{(a,M,\omega ,m,l)}{\mathrm{inf}}{\mathfrak{C}}_{2}(a,M,\omega ,m,l){\geqslant}b{ >}0.\end{equation*}$$

Numerically, using [BHP], we find b ≈ 150 is enough.

Proof. First note that, once a and (s, m, l) are fixed, ${\mathbf{\unicode{321}{}}}_{ml}^{[s],\enspace (a\omega )}$ is continuous in ω (see, for instance, [MS54] or [HW74]), hence ${\mathfrak{C}}_{2}(a,M,\omega ,m,l)$ is also continuous in ω.

At ω = 0, ${\mathfrak{C}}_{2}(a,M,0,m,l)={(-1)}^{4}{\mathfrak{B}}_{2}(0,m,l)={(24)}^{2}=576{ >}0$. Hence, by continuity in ω, there is a δ > 0 such that ${\mathfrak{C}}_{2}(a,M,\omega ,m,l){\geqslant}1$ for |Mω| ⩽ δ. On the other hand, if |Mω| > δ, by the non-negativity of ${\mathfrak{B}}_{2}$ (lemma 2.9) ${\mathfrak{C}}_{2}{\geqslant}{\mathfrak{F}}_{2}{M}^{2}{\omega }^{2}{ >}144{\delta }^{2}$. This concludes the proof. □

However, lemma 3.7 does not extend to s < 2. Though always positive, it follows from lemma 2.12 that the radial Teukolsky–Starobinksy constants for those spins asymptotically approach 0 as ω → ∞ if a ≠ 0 is fixed and suitable (l, m) are chosen:

Lemma 3.8. Fix M > 0, 0 < |a| ⩽ M and $s\in \left\{\frac{1}{2},1,\frac{3}{2}\right\}$. Then, there are some pairs (l, m), where $m-s\in \mathbb{Z}$ and $l-\mathrm{max}\left\{\vert m\vert ,s\right\}\in {\mathbb{Z}}_{{\geqslant}0}$, for which we have, as ω → ∞,

$$\begin{equation}{\mathfrak{C}}_{s}(a,M,\omega ,m,l)=O(\vert M\omega {\vert }^{-N})\enspace ,\quad \forall \enspace N{ >}0.\end{equation} \tag{ 3.7 }$$

3.4.3. A myth debunked: negative values for $s{\geqslant}\frac{5}{2}$

Not only is the pervasive argument outlined in section 3.4.1 or in [Cha90, KMW89, KMW92] purportedly asserting non-negativity of ${\mathfrak{C}}_{s}$ for all s incorrect, but its conclusion is also false. Lemma 2.12 is our starting point to debunk this myth; we now consider the contribution of ${\mathfrak{F}}_{s}(a\omega ,m,l)$:

Lemma 3.9 (Negativity for $\mathbf{s}\hspace{2pt}\mathbf{{\geqslant}}\hspace{2pt}\frac{\mathbf{5}}{\mathbf{2}}$). Fix M > 0, 0 < |a| ⩽ M and $s\in \left\{\frac{5}{2},3\right\}$. Then, there are some pairs (l, m), where $m-s\in \mathbb{Z}$ and $l-\mathrm{max}\left\{\vert m\vert ,s\right\}\in {\mathbb{Z}}_{{\geqslant}0}$, for which there is an A > 0 such that, as ω → ∞,

$$\begin{equation*}{\mathfrak{C}}_{s}(a,M,\omega ,m,l)=-A\vert M\omega {\vert }^{2\vert s\vert -2}+O(\vert M\omega {\vert }^{2\vert s\vert -3}).\end{equation*}$$

Proof. We use proposition 2.3 once more, appealing to the proof of lemma 2.12. Indeed, if s = 5/2, ${\mathfrak{F}}_{\frac{5}{2}}=2304({q}_{\frac{5}{2},ml}-m)+O(1)$, and we note that

$$\begin{align*}{q}_{\frac{5}{2},ml}-m=-2& {\Rightarrow}{\mathfrak{C}}_{\frac{5}{2}}=-4608\frac{a}{M}\vert M\omega {\vert }^{3}+O(\vert M\omega {\vert }^{2}),\\ {q}_{\frac{5}{2},ml}-m=-4& {\Rightarrow}{\mathfrak{C}}_{\frac{5}{2}}=-9216\frac{a}{M}\vert M\omega {\vert }^{3}+O(\vert M\omega {\vert }^{2}).\end{align*}$$

For s = 3, ${\mathfrak{F}}_{3}=576\left[36{({q}_{3,ml}-m)}^{2}-100\right]{(a\omega )}^{2}+O(\vert M\omega \vert )$, and we note that

$$\begin{equation*}{q}_{3,ml}-m={\pm}1{\Rightarrow}{\mathfrak{B}}_{3}=-36864\frac{{a}^{2}}{{M}^{2}}\vert M\omega {\vert }^{4}+O(\vert M\omega {\vert }^{3});\end{equation*}$$

see also figure 2. As shown in the proof of lemma 2.12, the above conditions may be realized for pairs (l, m) satisfying the constraints of the present lemma. □

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While for $s=\frac{5}{2},3\enspace {\mathfrak{C}}_{s}$ may take negative values for some (ω, m, l), they can also take positive values, for instance if ω = 0 (see lemma 3.3). Appealing once more to continuity in ω (see proof of lemma 3.7), we thus conclude

Lemma 3.10. Fix M > 0, 0 < |a| ⩽ M and $s\in \left\{\frac{5}{2},3\right\}$. There exist real algebraically special frequencies, i.e. real (ω, m, l), such that ${\mathfrak{C}}_{s}(a,M,\omega ,m,l)=0$.

We begin by discussing a notion of energy compatible with (1.1) for any $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$ with the stationary Kerr Killing field fails to produce a conservation law unless s = 0. However, as ${({r}^{2}+{a}^{2})}^{\frac{1}{2}}{{\Delta}}^{{\pm}\frac{s}{2}}{\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }$ satisfy complex conjugate ODEs, see (3.1), the Wronskian

$$\begin{equation*}\left({\Delta}\frac{\mathrm{d}}{\mathrm{d}r}({{\Delta}}^{s}{\alpha }_{m{\Lambda}}^{[+s],\enspace a\omega }){{\Delta}}^{-s}\overline{{\alpha }_{m{\Lambda}}^{[-s],\enspace a\omega }}-{\Delta}\frac{\mathrm{d}}{\mathrm{d}r}\left(\overline{{\alpha }_{m{\Lambda}}^{[-s],\enspace a\omega }}\right){\alpha }_{m{\Lambda}}^{[+s],\enspace a\omega }\right)\end{equation*}$$

is independent of r, and hence is conserved. From the Teukolsky–Starobinsky identities, given a solution ${\alpha }_{m{\Lambda}}^{[+s],\enspace a\omega }$ to (3.1) with spin +s, we may generate a ${\alpha }_{m{\Lambda}}^{[-s],\enspace a\omega }$ which solves (3.1) with spin −s, and vice-versa, to plug into this conservation law. Multiplying through by ω makes each of the terms of the conservation law into an energy, see already remark 4.2 and [SRTdC20, section 5.3]. Thus, we obtain:

Lemma 4.1 (TS energy identity). Fix M > 0, |a| ⩽ M, $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, $\omega \in \mathbb{R} {\backslash} \left\{0,m{\omega }_{+}\right\}$ and ${\Lambda}\in \mathbb{R}$. Suppose there exists a radial Teukolsky–Starobinsky constant ${\mathfrak{C}}_{s}(a,M,\omega ,m,{\Lambda})$ as given in definition 3.1.

Let ${\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }$ be a solution to (3.1), and let its decomposition (3.2) be characterized by

$$\begin{align*}& {a}_{{\mathcal{H}}^{-}}^{[{\pm}s]}=0\enspace ,\qquad \\ & {~{a}}_{{\mathcal{H}}^{+}}^{[{\pm}s]}\equiv {a}_{{\mathcal{H}}^{+}}^{[{\pm}s]}\begin{cases}{(2M{r}_{+})}^{1/2}\left\{\begin{matrix}\hfill 1\enspace ,\hfill & \hfill \vert a\vert =M\hfill \\ \hfill {({r}_{+}-{r}_{-})}^{{\pm}s}\enspace ,\hfill & \hfill \vert a\vert {< }M\hfill \end{matrix}\right\}& \text{if}\enspace s\;\text{integer}\\ \left\{\begin{matrix}\hfill {(2M{r}_{+})}^{-1/2}\enspace ,\hfill & \hfill \vert a\vert =M\hfill \\ \hfill {({r}_{+}-{r}_{-})}^{{\pm}s-1/2}\enspace ,\hfill & \hfill \vert a\vert {< }M\hfill \end{matrix}\right\}& \text{if}\enspace s\;\text{half}-\text{integer}\end{cases}.\end{align*}$$

Then, it satisfies the energy identity

$$\begin{equation*}1={\mathfrak{R}}^{[{\pm}s]}(a,M,\omega ,m,{\Lambda})+{\mathfrak{T}}^{[{\pm}s]}(a,M,\omega ,m,{\Lambda}),\end{equation*}$$

where ${\mathfrak{R}}^{[{\pm}s]}(a,M,\omega ,m,{\Lambda})$ and ${\mathfrak{T}}^{[{\pm}s]}(a,M,\omega ,m,{\Lambda})$ are called the reflection and transmission coefficients, respectively, and are given as follows. If s is an integer,

$$\begin{equation*}\begin{aligned}{\mathfrak{T}}^{[-s]}& \equiv \frac{\omega -m{\omega }_{+}}{\omega }\frac{{\mathfrak{C}}_{s}^{(10)}}{{(2\omega )}^{2s}}\frac{{\left\vert {~{a}}_{{\mathcal{H}}^{+}}^{[-s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[-s]}\right\vert }^{2}}\enspace ,\qquad {\mathfrak{R}}^{[-s]}\equiv \frac{{\mathfrak{C}}_{s}}{{(2\omega )}^{4s}}\frac{{\left\vert {a}_{{\mathcal{I}}^{+}}^{[-s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[-s]}\right\vert }^{2}};\\ \text{if}\;\;\text{further}\enspace {\mathfrak{C}}_{s}\ne 0\enspace ,\qquad {\mathfrak{T}}^{[+s]}& \equiv \frac{\omega -m{\omega }_{+}}{\omega }\frac{{(2\omega )}^{2s}}{{\mathfrak{C}}_{s}^{(9)}}\frac{{\left\vert {~{a}}_{{\mathcal{H}}^{+}}^{[+s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[+s]}\right\vert }^{2}}\enspace ,\qquad {\mathfrak{R}}^{[+s]}\equiv \frac{{(2\omega )}^{4s}}{{\mathfrak{C}}_{s}}\frac{{\left\vert {a}_{{\mathcal{I}}^{+}}^{[+s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[+s]}\right\vert }^{2}}.\end{aligned}\end{equation*}$$

If s is half-integer,

$$\begin{equation*}\begin{aligned}{\mathfrak{T}}^{[-s]}& \equiv \frac{{\mathfrak{C}}_{s}^{(10)}}{{(2\omega )}^{2s}}\frac{{\left\vert {~{a}}_{{\mathcal{H}}^{+}}^{[-s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[-s]}\right\vert }^{2}}\enspace ,\qquad {\mathfrak{R}}^{[-s]}\equiv \frac{{\mathfrak{C}}_{s}}{{(2\omega )}^{4s}}\frac{{\left\vert {a}_{{\mathcal{I}}^{+}}^{[-s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[-s]}\right\vert }^{2}};\\ \text{if}\;\;\text{further}\enspace {\mathfrak{C}}_{s}\ne 0\enspace ,\quad {\mathfrak{T}}^{[+s]}& \equiv \frac{{(2\omega )}^{2s}}{{\mathfrak{C}}_{s}^{(9)}}\frac{{\left\vert {~{a}}_{{\mathcal{H}}^{+}}^{[+s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[+s]}\right\vert }^{2}}\enspace ,\qquad {\mathfrak{R}}^{[+s]}\equiv \frac{{(2\omega )}^{4s}}{{\mathfrak{C}}_{s}}\frac{{\left\vert {a}_{{\mathcal{I}}^{+}}^{[+s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[+s]}\right\vert }^{2}}.\end{aligned}\end{equation*}$$

Here, we take ${\mathfrak{C}}_{s}^{(9)}={\mathfrak{C}}_{s}^{(10)}=1$ when s = 0, ${\mathfrak{C}}_{s}^{(9)}=1$ and ${\mathfrak{C}}_{s}^{(10)}={\left[4M{r}_{+}(\omega -m{\omega }_{+})\right]}^{2}+{({r}_{+}-{r}_{-})}^{2}/4$ if s = ±1/2, and otherwise, using the shorthand notation ${\mathfrak{C}}_{s,j}{:=}{\left[4M{r}_{+}(\omega -m{\omega }_{+})\right]}^{2}+{(s-j)}^{2}{({r}_{+}-{r}_{-})}^{2}$,

$$\begin{equation*}{\mathfrak{C}}_{s}^{(9)}=\begin{cases}\prod\limits _{j=1}^{\vert s\vert }{\mathfrak{C}}_{s,j}\quad & \quad \text{if}\;\;s\in \mathbb{Z}\\ \prod\limits _{j=1}^{\vert s\vert -1/2}{\mathfrak{C}}_{s,j}\quad & \quad \text{if}\;\;s\in \left(\frac{1}{2}\mathbb{Z}\right){\backslash}\mathbb{Z}\end{cases}\enspace ,\qquad {\mathfrak{C}}_{s}^{(10)}=\begin{cases}\prod\limits _{j=0}^{\vert s\vert -1}{\mathfrak{C}}_{s,j}\quad & \quad \text{if}\;\;s\in \mathbb{Z}\\ \prod\limits _{j=0}^{\vert s\vert -1/2}{\mathfrak{C}}_{s,j}\quad & \quad \text{if}\;\;s\in \left(\frac{1}{2}\mathbb{Z}\right){\backslash}\mathbb{Z}\end{cases}.\end{equation*}$$

Proof. The proof is sketched in the paragraph above, but we encourage the reader to see [And+17, section 2.2] and [SRTdC20] for details. □

Remark 4.2. The notion of energy put forth in lemma 4.1 is consistent with previous literature: it matches that introduced in [TP74] for s = 1, 2, [Unr73] for s = 1/2 and [TS90] for s = 3/2.

If ${\mathfrak{T}}^{[{\pm}s]}{< }0$ and ${\mathfrak{R}}^{[{\pm}s]}{ >}0$, there is amplication in the energy reflected to future null infinity. As

$$\begin{equation*}{\mathfrak{T}}^{[{\pm}s]}(a,M,\omega ,m,{\Lambda}){< }0{\Leftrightarrow}\omega (\omega -m{\omega }_{+}){< }0\enspace ,\quad s\in {\mathbb{Z}}_{{\geqslant}0},\end{equation*}$$

we refer to this as superradiant amplification. On the other hand, if ${\mathfrak{T}}^{[{\pm}s]}{ >}0$ and ${\mathfrak{R}}^{[{\pm}s]}{< }0$, there is amplication in the energy transmitted into the future event horizon. As

$$\begin{equation*}{\mathfrak{R}}^{[{\pm}s]}(a,M,\omega ,m,{\Lambda}){< }0{\Leftrightarrow}{\mathfrak{C}}_{s}(a,M,\omega ,m,{\Lambda}){< }0,\end{equation*}$$

we refer to this as non-superradiant amplification. By lemmas 3.5 and 3.9, if we constrain Λ to be a spin-weighted spheroidal eigenvalue, ${\Lambda}={\mathbf{\Lambda }}_{sml}^{(a\omega )}$ for some l, we see that non-superradiant amplification occurs for some (ω, m, l) when s > 2, see figure 3.

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In the previous section, we considered a scattering problem under the evolution equation (1.1) alone. Consequently, the notion of energy in lemma 4.1 involves a single spin sign. If, for s = 1, 2, one considers a scattering problem under the entire system of linearized Maxwell or Einstein equations, respectively, then the natural notion of energy involves both spin signs. We extend this reasoning to other spins:

Lemma 4.3 (TS energy identity under TS correspondence). Fix M > 0, |a| ⩽ M, $s\in \frac{1}{2}{\mathbb{Z}}_{{\geqslant}0}$, $\omega \in \mathbb{R}{\backslash}\left\{0,m{\omega }_{+}\right\}$ and ${\Lambda}\in \mathbb{R}$. Suppose there exists a radial Teukolsky–Starobinsky constant ${\mathfrak{C}}_{s}(a,M,\omega ,m,{\Lambda})$ as given in definition 3.1. Further assume that the frequencies are such that ${\mathfrak{C}}_{s}(a,M,\omega ,m,{\Lambda})\ne 0$.

Let ${\alpha }_{m{\Lambda}}^{[{\pm}s],\enspace a\omega }$ be solutions to (3.1) which are related to each other by the radial Teukolsky–Starobinsky identities of proposition 3.1. Assume their decompositions (3.2) to be characterized by

$$\begin{align*}& {a}_{{\mathcal{H}}^{-}}^{[-s]}=0\enspace ,\qquad \\ & {~{a}}_{{\mathcal{H}}^{+}}^{[+s]}\equiv {a}_{{\mathcal{H}}^{+}}^{[+s]}\begin{cases}{(2M{r}_{+})}^{1/2}\left\{\begin{matrix}\hfill 1\enspace ,\hfill & \hfill \vert a\vert =M\hfill \\ \hfill {({r}_{+}-{r}_{-})}^{s}\enspace ,\hfill & \hfill \vert a\vert {< }M\hfill \end{matrix}\right\}\quad & \quad \text{if}\enspace s\;\;\text{integer}\\ \left\{\begin{matrix}\hfill {(2M{r}_{+})}^{-1/2}\enspace ,\hfill & \hfill \vert a\vert =M\hfill \\ \hfill {({r}_{+}-{r}_{-})}^{+s-1/2}\enspace ,\hfill & \hfill \vert a\vert {< }M\hfill \end{matrix}\right\}\quad & \quad \text{if}\enspace s\;\;\text{half}-\text{integer}\end{cases}.\end{align*}$$

Then, one has the energy identity

$$\begin{equation*}1={\mathfrak{R}}_{s}(a,M,\omega ,m,{\Lambda})+{\mathfrak{T}}_{s}(a,M,\omega ,m,{\Lambda}),\end{equation*}$$

where the reflection and transmission coefficients, ${\mathfrak{R}}_{s}(a,M,\omega ,m,{\Lambda})$ and ${\mathfrak{T}}_{s}(a,M,\omega ,m,{\Lambda})$ respectively, are given as follows:

$$\begin{align*}& {\mathfrak{R}}_{s}\equiv \frac{{\left\vert {a}_{{\mathcal{I}}^{+}}^{[-s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[+s]}\right\vert }^{2}}\enspace ;\quad \enspace s\enspace \text{integer,}\enspace \enspace \enspace {\mathfrak{T}}_{s}\equiv \frac{\omega -m{\omega }_{+}}{\omega }\frac{{(2\omega )}^{2s}}{{\mathfrak{C}}_{s}^{(9)}}\frac{{\left\vert {~{a}}_{{\mathcal{H}}^{+}}^{[+s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[+s]}\right\vert }^{2}}\enspace ;\quad \enspace s\enspace \text{half}-\text{integer,}\enspace \enspace \enspace \\ & {\mathfrak{T}}_{s}\equiv \frac{{(2\omega )}^{2s}}{{\mathfrak{C}}_{s}^{(9)}}\frac{{\left\vert {~{a}}_{{\mathcal{H}}^{+}}^{[+s]}\right\vert }^{2}}{{\left\vert {a}_{{\mathcal{I}}^{-}}^{[+s]}\right\vert }^{2}};\end{align*}$$

where we take ${\mathfrak{C}}_{s}^{(9)}$ to be the same as in lemma 4.1.

Remark 4.4. The notion of energy considered in lemma 4.3 matches that of the recent [Mas20, section 1.3.4] on scattering under the linearized Einstein vacuum equations around a = 0 in Kerr. Indeed, when considering this system, the two gauge-invariant curvature quantities satisfy the Teukolsky master equation (1.1) with spin ±2 together with a physical space version of the radial Teukolsky–Starobinsky identities of proposition 3.1, see [Mas20, equations (1.5) and (1.6)]. A similar situation arises when one considers the linearized Maxwell equations, as one may readily deduce from [Pas19, equations (3.11)–(3.16)].

We note that it is only under the notion of energy in lemma 4.3 that amplification does not occur for non-superradiant frequencies for fields of any spin.