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The Teukolsky--Starobinsky constants: facts and fictions

The Teukolsky Master Equation describes the dynamics of massless fields with spin on a Kerr black hole. Under separation of variables, spin-reversal for this equation is accomplished through the so-called Teukolsky--Starobinsky identities. These identities are associated to the so-called Teukolsky--Starobinsky constants, which are spin-dependent. We collect some properties of the Teukolsky--Starobinsky constants and dispel some myths present in the literature. We show that, contrary to popular belief, these constants can be negative for spin larger than 2. Such fields thus exhibit a novel form of energy amplification which occurs for non-superradiant frequencies.


Introduction
Since as early as the 1950s [RW57], the study of perturbations of stationary black holes has been a central theme of research in classical General Relativity. In 4 spacetime dimensions and in vacuum, the paradigmatic and, conjecturally, only examples of stationary black hole solutions are the rotating black holes in the Kerr family [Ker63], parametrized by mass M > 0 and specific angular momentum |a| ≤ M .
While studying stability of Kerr, Teukolsky [Teu73] introduced what became to be known as the Teukolsky Master Equations. In Boyer-Lindquist coordinates (t, r, θ, φ) ∈ R × (r + ≡ M + √ M 2 − a 2 , ∞) × S 2 , these are given by (1.1) where ∆ = r 2 − 2M r + a 2 , ρ 2 = r 2 + a 2 cos 2 θ and g a,M the covariant scalar wave operator on the Kerr background. Here, ±s denotes a spin parameter, making α [±s] a spin-weighted function; this notion, and hence (1.1), is well-defined for any s ∈ 1 2 Z ≥0 , see [PR84]. The Teukolsky Master Equations (1.1) play a crucial role in characterizing perturbations of Kerr black holes. For s = 0, they reduce to the scalar wave equation on Kerr. For s = 1 and 2, they describe the dynamics of gauge-invariant electromagnetic and curvature quantities under the linearized Maxwell and Einstein equations, respectively, in the Newman-Penrose formalism [NP62]. Some half-integer spin cases can also be interpreted physically: s = 1/2 corresponds to the Dirac equation for (massless) neutrinos; the case s = 3/2 is known as the Rarita-Schwinger equation. The case s > 2 of the Teukolsky Master Equations (1.1), although less often considered, is of mathematical interest in its own right and might play a role in higher-spin theories [Vas96].
In this paper, we focus on separable solutions of (1.1) (see [Car68;Teu73]): for ω ∈ R, m − s ∈ Z and l ∈ Z ≥max{|m|,s} , these take the form [±s], aω ml satisfy ODEs, respectively referred to as the angular ODE, given below as (2.1), and the radial ODE, given below as (3.1).
As discovered by Starobinsky [SC74] and Teukolsky [TP74] for s = 1, 2 and later generalized to all s ∈ 1 2 Z ≥0 [KMW89], the angular and radial Teukolsky ODEs of spin ±s exhibit a curious property: on applying a certain first order differential operator s times to a solution to the ODE with spin +s, one obtains a solution to the ODE with spin −s, and vice-versa. Though seldom remarked in the classical literature, each of the radial Teukolsky-Starobinsky operators of spin ±s acts differently on the ingoing and outgoing components of the radial solutions, and a similar statement is true in the angular setting, see Propositions 2.5 and 3.1 below. Nevertheless, by applying these operators in succession to a solution of the angular or radial ODE, one can check that, at least for |s| ≤ 3, the exact same solution is recovered up to a constant: respectively, the angular Teukolsky-Starobinsky constant, denoted by B s = B s (aω, m, l), and the radial Teukolsky-Starobinsky constant, denoted by C s = C s (a, M, ω, m, l) .
It is on the radial Teukolsky-Starobinsky constant, C s , that we focus on for the rest of this introduction.
In the classical literature, see e.g. [KMW89; Cha90; KMW92], C s is frequently denoted as the square of a complex number. Taking this claim literally, we would conclude To justify (×), some authors point to the fact that α [−s] and ∆ s α [+s] satisfy complex conjugate equations. Unfortunately, this argument is incorrect: (×) does not follow from the fact that these radial ODEs are complex conjugates of each other. In fact, we show that (×) is manifestly false for general spin s:

Fact 1 (TS constant sign I). The radial Teukolsky-Starobinsky constant can be negative: for any
The proof of Fact 1 is short and elementary: we require only the high frequency expansions for spinweighted spheroidal angular eigenvalues, on which C s depends (see Lemma 3.3), which were obtained 1 in recent work of the first author and collaborators [CO05; COW19]. We note here the importance of the assumption a = 0: it is well-known that C s (a = 0, M, ω, m, l) ≥ 1, at least if |s| ≤ 3. The implications of Fact 1 are quite surprising. To explain these, recall that the Teukolsky-Starobinsky identities serve to define an energy identity for α [±s] , which will thus depend on the constants C s ; this observation goes back to [TP74] but the reader may find a rigorous statement below in Lemma 4.1. In the case s = 0, corresponding to the scalar wave equation, where C s=0 ≡ 1 plays no role in the energy, it is well-known that energy amplification occurs if and only if the frequency parameters (ω, m, l) are superradiant, i.e. such that ( In light of the results present here for the sign of C s , it is natural to try to understand whether C s (a, M, ω, m, l) = 0 for some real (ω, m, l). Such frequencies are known as real algebraically special [Wal73; Cha84]. Since C s (a, M, ω = 0, m, l) ≥ 1, at least for s ≤ 3, it follows from Fact 1 that Fact 4 (AS frequencies I). There are real algebraically special frequencies for s > 2: for any |a| ∈ (0, M ], s ∈ 5 2 , 3 =⇒ ∃ (ω, m, l) real such that C s (a, M, ω, m, l) = 0 .
In the case s ≤ 2, Fact 3 gives one hope of ruling out algebraically special frequencies. We show that these hopes are well-founded, thus answering a question raised in [Wal73]: Fact 5 (AS frequencies II). There do not exist real algebraically special frequencies for s ≤ 2: for any |a| ∈ [0, M ], To the best of our knowledge, prior to this work, this result had only been noted in the s = 2 case, in work of the second author and collaborators, in [TdC20; SRTdC20]. We contrast Fact 5 with the following result Indeed, for fixed a = 0, we show that the limit when |ω| → ∞ is algebraically special for some (l, m) if s ≤ 3/2. In the case s = 2, no such limit can be algebraically special: the comparison with the angular Teukolsky-Starobinsky constant gives C 2 (a, M, ω, m, l) ≥ 144M 2 ω 2 and it is the latter term which ensures that C 2 has a positive lower bound.
To conclude this introduction, we remark that we fully expect that our Facts 1, 2 and 4 can be generalized to higher s ∈ 1 2 Z ≥0 . Furthermore, we expect Facts 3, 5 and 6 to also hold for the Kerr-(anti-)de Sitter black hole spacetimes, for which there is a generalization of the Teukolsky-Starobinsky identities and constants [Tor88] (see also the more recent [DS13]). 310200/2017-2. R.TdC. acknowledges support from EPSRC (United Kingdom), grant EP/L016516/1, and thanks Simon Becker for helpful suggestions concerning the analysis of the angular ODE in this paper and Yakov Shlapentokh-Rothman for numerous discussions. This work makes use of the Black Hole Perturbation Toolkit.

The angular Teukolsky-Starobinsky constants
In this section, we introduce the angular ODE corresponding to the Teukolsky equation (1.1). We then define the angular Teukolsky-Starobinsky constants and characterize their sign.

Consider the angular ODE
where θ ∈ (0, π), s ∈ 1 2 Z ≥0 , m − s ∈ Z, ν ∈ R and Λ ∈ R. An asymptotic analysis leads us to conclude that for a solution to (2.1) there is a unique set of real numbers a [ norm,i , 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 Ξ to be smooth as θ → π and normalized at the θ = π end; • if |m| = s, take Ξ norm,2 exactly as above; the definition of the remaining functions requires a logarithmic correction which we need not specify here.
We say that Ξ sml . The parameter l is chosen so that l − s ∈ Z, l ≥ max{|m|, s} and Λ (0) sml = l(l + 1) − s 2 ≥ s. The following alternative notation will also be used: Proof. Since ν ∈ R, the result follows from standard Sturm-Liouville theory, see [MS54] for further details.

Remark 2.2.
In what follows, we often lighten the notation, replacing the spin-weighted spheroidal eigenvalues Λ (ν) sml by Λ and similarly for L. However, non-bold characters Λ and L ≡ 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 L (ν) sml 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: sml be as in Proposition 2.1. Then, as ν → ∞, for any N > 0, we can find real constants (2.4) The first two coefficients are where, denoting by odd(·) a function on Z which is one if the argument is odd and zero otherwise, we may express q sml as The following coefficients may be computed as follows. Let Then, for k ≥ 1, we have where a 1,k and a −1,k are computed from the following recursive relations for n = 0: sml,n a n−1,|n|−1 , n ≥ 1 Q + sml,n a n+1,|n|−1 , n ≤ −1 , a n,|n|+1 = 1 2 (q sml + n)a n,|n| − 1 16n initialized by the choice a 0,0 = 1 and a 0,j = 0 if j = 0.
Proposition 2.5 (Angular TS identities). Fix s ∈ 1 2 Z ≥0 , m such that m − s ∈ Z\{0}, Λ ∈ R and ν ∈ R. Then, if Ξ [±s], (ν) mΛ solve (2.1) and admit the decomposition (2.2), norm,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, B (i) the other B Proof. The result follows from differentiating the asymptotic formulas for Ξ [±s] norm,i , i = 1, . . . , 4. In doing so, it can be useful to note that 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:

Examples of angular Teukolsky-Starobinsky constants
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 ∈ {0, 1 2 , 1, 3 2 , 2, 5 2 , 3}, there exists an angular Teukolsky-Starobinsky constant. Moreover, it is given by: 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. 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 −B 1/2 (ν = 0, m, l = s = 1/2) = 1.
Remark 2.8. We expect that the angular Teukolsky-Starobinsky constant can be defined for all s ∈ 1 2 Z ≥0 . However, no proof has been given in the literature.

Properties of the angular Teukolsky-Starobinsky constants
where the integral on the right hand side is non-negative.
Remark 2.10. We note that the non-negativity of (−1) 2s 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.
In spite of the angular Teukolsky-Starobinsky constants' positivity, they approach arbitrarily small values, at least in the cases considered in Lemma 2.7.
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.

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.
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.

The radial Teukolsky-Starobinsky constants
In this section, we introduce the radial ODE corresponding to the Teukolsky equation (2.1) and define the radial Teukolsky-Starobinsky constants. The last subsection contains the proof of Facts 1 to 6.

The radial ODE
We consider the radial ODEs where M > 0, |a| ≤ M , r ∈ (r + , ∞), s ∈ 1 2 Z ≥0 , m such that m − s ∈ Z, Λ ∈ R, and ω ∈ R. An asymptotic analysis of (3.1) shows that a solution to (3.1) admits a unique set of complex numbers a [ where α

[±s]
I + and α [±s] I − are solutions of (3.1) with outgoing and ingoing, respectively, boundary conditions as r → ∞ which are normalized at r = ∞, and where α H − are solutions of (3.1) with ingoing and outgoing, respectively, boundary conditions as r → r + which are normalized at r = r + . Concretely, • α

The radial Teukolsky-Starobinsky identities and constants
In this section, we will require the notation where M > 0, |a| ≤ M , r ∈ (r + , ∞), n, m ∈ 1 2 Z and ω ∈ R. We quote from [TdC20, Proposition 2.14] the following: (3.1) and admit the decomposition (3.2), I + + a [+s] the remaining C H ∓ , 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:

Examples of radial Teukolsky-Starobinsky constants
By direct computation, we can check that a radial Teukolsky-Starobinsky constant exists at least for low values of Teukolsky spin: where B s may be read off from (2.10). In the above, if aω = 0, and L = L (aω) sml corresponds to a spinweighted spheroidal eigenvalue with spheroidal parameter ν = aω for some l, then C s (a, M, ω, m, l) ≥ 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.
Alternatively, in light of the radial Teukolsky-Starobinsky identities of Proposition 3.1, if the radial Teukolsky-Starobinsky constant exists, we must have Hence, one may compute C s from one of C s , C s . 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]).
We remark that the formulas for C s when s ≤ 3, given here in Lemma 3.3, have been obtain before in [Cha90] and [KMW89].

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 ∈ 1 2 Z ≥0 . These authors claim that non-negativity of C s may be viewed as a consequence of ∆ s 2 (1±1) α [±s], aω mΛ satisfying complex conjugate equations, or of the radial Teukolsky-Starobinsky identities being generated by complex conjugate operators, somehow implies non-negativity of the Teukolsky-Starobinky constant. To the best of our knowledge, the underlying rationale is that there is an analogy with the angular setting, where the relation between L ± n and L ∓ −n+1 in (2.11) leads to the conclusion of Lemma 2.9. However, (2.11) establishes an adjointness relation between the angular operators L ± n and −L ∓ −n+1 in the space of (real-valued) smooth spin-weighted functions. In contrast, the relation between D ± n and D ∓ n 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 2 . Indeed, for some weight w(r) : (r + , ∞) → [0, ∞), for f and h sufficiently regular complex-valued functions of r that the boundary terms in the following vanish, we have so the adjoint of D ± 0 will be −D ± 0 − d dr log w. This makes the notation used in classical references such as [Cha83] (see also the more recent [Teu15]) rather unfortunate if one is looking to extrapolate from the contents of this book to s > 2.
The upshot is that there is, in fact, no hope of establishing an analogue of Lemma 2.9 by a similar method. For we could establish where α [±s] is chosen to have unit L 2 w norm, only if we were to prove and (3.6) clearly cannot hold for any positive weight w. It follows that, if C s is indeed non-negative for some s, a different proof strategy should be sought.

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]), Indeed, one has F s ≡ 0 for s ≤ 3 2 and F 2 = 144 , F 5 2 = 1152( L + 2) , F 3 = 576 (3 L + 10) 2 + 100aω(m − aω) .
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: sml is a spin-weighted spheroidal eigenvalue for some l ∈ Z ≥max{|m|,s} , then C s (a, M, ω, 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, 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: Numerically, using [BHP], we find b ≈ 150 is enough.

A myth debunked: negative values for s ≥ 5 2
Not only is the pervasive argument outlined in Section 3.4.1 or in [KMW89; Cha90; KMW92] purportedly asserting non-negativity of 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 F s (aω, m, l):  While for s = 5 2 , 3 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

On non-superradiant amplification for s > 2 fields
In this section, we consider the implications of the properties of the radial Teukolsky-Starobinsky constants on the energy associated to the Teukolsky radial ODE (3.1).

Energy for the Teukolsky equation
We begin by discussing a notion of energy compatible with (1.1) for any s ∈ 1 2 Z ≥0 with the stationary Kerr Killing field fails to produce a conservation law unless s = 0. However, as (r 2 + a 2 ) If s is half-integer, Here, we take C Proof. The proof is sketched in the paragraph above, but we encourage the reader to see [And+17, Section IIB] and [SRTdC20] for details.

Energy for the system of linearized Maxwell or Einstein equations
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:  We note that it is only under the notion of energy in Lemma 4.3, seldom encountered in the physics literature, that amplification does not occur for non-superradiant frequencies for fields of any spin.