Perturbations of Extremal Kerr Spacetime:
Analytic Framework and Late-time Tails
We develop a complete and systematic analytical approach to field perturbations of the extreme Kerr spacetime based on the formalism of Mano, Suzuki and Takasugi (MST) for the Teukolsky equation. We give analytical expressions for the radial solutions and frequency-domain Green function in terms of infinite series of special functions. As an application, we compute the leading late-time behavior due to the branch point at zero frequency of scalar, gravitational, and electromagnetic field perturbations on and off the event horizon.
The prominence of black holes in modern physics and astrophysics makes study of their perturbations of essential importance. After pioneering work of Regge, Wheeler, Zerilli, and Moncrief Regge and Wheeler (1957); Zerilli (1970); Moncrief (1974) on perturbations of spherically symmetric black holes, the rotating case was cracked by Teukolsky Teukolsky (1973), who discovered a separable master equation for scalar (spin-0), fermion (spin-1/2), electromagnetic (spin-1), and gravitational (spin-2) perturbations of the Kerr metric. While Teukolsky’s equation can be solved numerically (say, in the time domain), it is of interest to develop complimentary analytical techniques. Leaver Leaver (1986a), who adopted a separation of variables approach by going to the frequency domain, found analytical solutions of the radial equation in terms of infinite series involving special functions. Mano, Suzuki, and Takasugi (MST) Mano et al. (1996a, b) cleverly reformulated Leaver’s solutions to produce a practical method of computing observable quantities such as the gravitational waveform from an extreme mass-ratio binary inspiral. With the advent of computer algebra programs capable of efficiently manipulating and computing special functions, this “MST method” has gained in popularity to become competitive with, and in many ways superior to, direct numerical solution of the linearized perturbation equation. For example, the MST method has been used for calculating the self-force, post-Newtonian coefficients and gauge-invariant quantities, the retarded Green function, the quantum correlator, the renormalized expectation value of the quantum stress-energy tensor and radiation emission in Schwarzschild and Kerr spacetimes in van de Meent (2016); Zhang et al. (2013); Casals et al. (2013); Buss and Casals (2018); Levi et al. (2017); Bini and Damour (2014); of particular relevance to this paper, it has also been used to calculate the late-time tail to high-order in Schwarzschild and Kerr spacetimes in Casals and Ottewill (2015); Casals et al. (2016a).
Since the original formulation for spin-field perturbations of vacuum, asymptotically flat, non-extremal black holes Mano et al. (1996a, b) (compiled in a review in Sasaki and Tagoshi (2003)), the MST method has been extended in a variety of ways to encompass charged black holes and/or cosmological boundary conditions Suzuki et al. (1999, 1998, 2000). However, to the best of our knowledge, there has been no prior work on extremal black holes. The extension is not straightforward since the coincidence of the inner and outer horizons converts a pair of regular singular points into a single irregular singular point, changing the character of the series solutions. We side-step the difficulty by starting with a functional expansion adapted to extremal Kerr and develop a Leaver-MST method accordingly. This provides a practical and efficient formulation for analytic evaluation of integer-spin perturbations.
As an application, we compute the late-time behavior (“tail”) of the perturbing field. In the non-extremal case, the tail arises from a branch point at zero frequency in the complex frequency plane Leaver (1986b); Hod (2000a); Casals et al. (2016a). In the extremal case there is an additional branch point at the so-called superradiant bound frequency that must be considered Glampedakis and Andersson (2001). In previous work Casals et al. (2016b); Gralla and Zimmerman (2017), we have computed the extremal Kerr tail from the branch point at the superradiant bound frequency, showing that the asymptotic decay of the perturbing field, whether it be metric, vector potential, or scalar, is off the horizon and on the horizon ( being advanced time). The difference in the rates on and off the horizon accounts for the blow-up of transverse derivatives at the horizon Aretakis (2015); Lucietti and Reall (2012), a phenomenon named after its discoverer, Aretakis Aretakis (2011a, b). In these calculations we used the method of matched asymptotic expansions (MAE) to compute the late-time behavior. In this work we show that the transfer function (i.e., the fixed frequency modes of the Green function) used in MAE calculation is recovered exactly by the leading-order term in the MST series. This unites previously distinct techniques, provides a more rigorous justification for MAE, and shows how it can be systematically corrected to arbitrary order. Moreover, in the MAE calculations the late-time rates due to the superradiant bound frequency were reported under the assumption that the tail due to the branch point at the origin is subleading. Here we justify this assumption, showing that the tail from the origin is on the future event horizon (see Eq.(130)), at future null infinity (see Eq.(127)) and at future timelike infinity (see Eq.(118)), where is Boyer-Lindquist time and is retarded time. Our results are for nonaxisymmetric, integer-spin field perturbations111The axisymmetric case is already considered in Casals et al. (2016b); Gralla and Zimmerman (2017) with the exception of the tail at future null infinity. Also, it should be straightforward to extend our results to half-integer values for the spin of the field, which are also of interest..
The rest of this paper is organized as follows. In Sec. II we introduce the Teukolsky equation and its Green function. In Sec. III we develop the MST formalism for extreme Kerr. In Sec. IV we apply the MST formalism to obtain the formal contribution to the Green function from the branch cut down from the origin and derive the corresponding leading-order tail. In Sec. V we obtain the formal contribution to the Green function from the branch cut down from the superradiant bound frequency. In Sec. VI we show that, in the limit to the superradiant bound frequency, the MST method recovers the MAE results.
We follow the notation and units of Sec.VIII.B of Ref. Leaver (1986a). In particular, we choose and the unusual choice for the mass of the black hole.
Ii Perturbations of Extreme Kerr
ii.1 Retarded Green function
Scalar (spin ), electromagnetic (), and gravitational (2) perturbations222Fermion () field perturbations also obey the Teukolsky equation. However, in this paper we assume integer , which simplifies some of the formulas. of an extreme Kerr black hole are governed by a single “master” partial differential equation first derived by Teukolsky Teukolsky (1973). Our main study concerns the retarded Green function of this equation, defined to vanish when is outside the causal future of . We employ Boyer-Lindquist coordinates outside the event horizon of the black hole and the Kinnersley tetrad Kinnersley (1969). We denote the mass of the black hole by and is the radius of the event horizon at extremality. We choose units where , and so . Instead of the Boyer-Lindquist radial coordinate we shall use the shifted radial coordinate
In the shifted Boyer-Lindquist coordinates, the retarded Green function satisfies the fundamental equation Teukolsky (1973)
where is the Teukolsky operator333In the metric signature , we adopt corresponding to minus the left-hand-side of Eq.(4.7) of Teukolsky (1973)..
and ensures the integration contour of the inverse Laplace transform is in the analytic region of the transfer function . We have taken and without loss of generality.
For real frequencies, the spin-weighted spheroidal harmonics form a complete set of eigenfunctions in ), with eigenvalues denoted by . Following Teukolsky (1973), it is also convenient to define the quantity . Our convention is to normalize the spin-weighted spheroidal harmonics such that
We give here some useful properties of the angular eigenfunctions and their eigenvalues. From the angular equation (Eq.4.10 Teukolsky (1973)), the following symmetries are manifest:
for the angular eigenvalue and
for the angular eigenfunction product.
In its turn, routine separation of variables reveals that the transfer function obeys the ordinary differential equation
with given by
Here, we have introduced a shifted frequency,
where is the horizon frequency. The value (i.e., ) corresponds to the so-called superradiant bound frequency (in the literature, this frequency is also called horizon frequency or critical frequency; we use these three terms indistinctively to denote ).
In this paper we will carry out an in-depth analysis of the transfer function . For that purpose, we first define some solutions of the homogeneous version of the radial equation (8) that it satisfies.
ii.2 Radial Teukolsky equation
The homogeneous version of the radial equation (8),
where is a radial function, is the key equation for frequency-domain perturbations of extreme Kerr. This second-order, linear ordinary differential equation has rank-1 irregular singular points at infinity (, i.e., ) and at the horizon (, i.e., ). This classifies it as a doubly-confluent Heun equation Ronveaux and Arscott (1995). This is in contrast with the radial equation in subextremal Kerr, which instead possesses two regular singular points (at the Cauchy and event horizons) and only one irregular singular point (at ), thus classifying it as a confluent Heun equation. The main purpose of this paper is to, first, develop analytic techniques for solving Eq. (12) and, second, use these techniques to obtain late-time tails of perturbations of extreme black holes.
In applications, it is natural to consider four different solutions to (12), which, following Leaver’s notation, we denote and , defined according to boundary conditions imposed at the event horizon and infinity 444 It is worth noting that the asymptotics for and in, respectively, Eqs.(13) and (14) are only true boundary conditions (i.e., specify the solutions uniquely when giving specific values to the incidence coefficients) when ; the values of the solutions for are determined by analytic continuation from . The reason is that, for , the incidence waves become exponentially dominant solutions in those asymptotic regions: for and for ; any exponentially subdominant solutions that may be part of the “boundary conditions” in those regions are not unambiguosly determined by the asymptotic expressions. . In our notation, “” refers to the horizon/infinity, while “” means purely outgoing/incoming. For example, corresponds to the solution with purely outgoing radiation at infinity, while corresponds to radiation entering the black hole but none emerging from the white hole. Based on these properties, and following a more standard notation in the literature, we shall also denote by the upgoing radial solution and by the ingoing radial solution . The two notations are interchangeable. Mathematically, the in and up functions satisfy the following boundary conditions:
where , and are, respectively, complex-valued incidence, reflection and transmission coefficients of the ingoing/upgoing solutions.
The other homogeneous radial solutions and obey boundary conditions such that they are purely outgoing from the horizon and purely ingoing from infinity, respectively (see Eqs.(153) and (154a)). Also following standard notation in the literature, we shall denote by the outgoing radial solution . It satisfies
where is its transmission coefficient.
Clearly, either set or forms a complete set of linearly independent solutions of the homogeneous radial equation.
It shall prove useful to also normalize radial solutions and coefficients in a different way. We adopt the notation of placing a hat over a radial function or coefficient to indicate that quantity normalized via the corresponding transmission coefficient:
In particular, it follows from Eq.(13) that
In both of the asymptotic expressions in (17) and (18), applying the transformation is equivalent to complex-conjugating them. Similarly, using the symmetries in Eq.(6), it is easy to see that applying on the radial operator in Eq.(12) is also equivalent to complex-conjugating it. It then follows that
Similarly, and are all complex-conjugated under . This is the main reason for choosing the normalization as in the hatted radial quantities.
ii.3 Transfer function
The method we adopt for constructing the transfer function involves a set of linearly independent homogeneous solutions of the radial differential equation (12). The radial solutions yielding the retarded Green function of the Teukolsky equation are the above “in” and “up” solutions, which correspond to solutions of the Teukolsky equation having no radiation coming out of the white hole or from past null infinity, respectively. The transfer function corresponding to the retarded Green function is thus given by
where , , is the constant scaled Wronskian,
We use the notation
for the actual Wronskian, where and are any two solutions of the homogeneous radial equation. We may equivalently express the transfer function in terms of the hatted quantities as
where is the scaled Wronskian,
In the next-to-last equality in Eq.(25) we have evaluated the radial solutions for and in its last equality we have evaluated them for . We here give other scaled Wronskian identities which will be useful for our purposes. From the asymptotics in Eqs.(13), (14) and (15), it is straightforward to find
The last Wronskian identity that we give is
We note that if is a solution of the radial Teukolsky equation for spin , then, because the radial operator is not self-adjoint, is not a solution of the equation (for any spin), but is a solution of the equation for spin .
From Eqs.(24) and (21), it follows that the symmetry of Eq.(19) carries over to the transfer function modes: . Applying to the exponential factors in the integrand in Eq.(3) is also equivalent to complex-conjugating them. If the whole integrand in Eq.(3) transformed in this way, then the -modes of the retarded Green function , as well as itself, would be real-valued. However, under the transformation , Eq.(7) shows that the angular factor not only becomes complex-conjugated but also undergoes (or, equivalently, it undergoes ). This means that is generally not real-valued (although it is real-valued on the equator and everywhere for ): complex-conjugating it is equivalent to taking . Mathematically, the fact that is not real-valued for can be traced back to the fact that the Teukolsky operator is not self-adjoint for .
The asymptotic theory of Laplace transforms Doetsch (1974) relates the late-time behaviour of a function of time, say as , to the asymptotics of its Laplace transform, , near its uppermost singular point , or points if more than one happen to lie on the same abscissa, in the complex- plane. As discussed in Ref.Casals et al. (2016b), the late-time decay of the master field obeying the Teukolsky equation is determined by the asymptotics of the transfer function near its uppermost singular point(s). In extreme Kerr, there is numerical evidence Richartz (2016); Burko and Khanna (2017); Casals and Longo () suggesting that these points are branch points at the superradiant bound frequency () and at zero frequency (). However, to the best of our knowledge, mode stability has not been demonstrated for extremal Kerr for generic perturbations, though rigorous linear stability results exist for axisymmetric perturbations Aretakis (2012). Short of performing an exhaustive spectral analysis of the transfer function, we cannot say with full certainty that the uppermost singular points are the mentioned branch points and we leave this for a future study. We therefore proceed on the assumption that an asymptotic analysis about the branch points at the superradiant bound frequency and at zero frequency determines the decay. We finally note that, although the angular eigenfunctions and eigenvalues also possess branch points in the complex- plane Oguchi (1970); Hunter and Guerrieri (1982); Barrowes et al. (2004), they do not play any physical role, and vanish upon summation over and Casals et al. (2016a).
Iii MST Method
In this section we provide series representations for the various solutions to the homogeneous radial Teukolsky equation and their radial coefficients. At the end, we shall also obtain the asymptotic behaviour as of quantities required by these series.
We note that most of the analytical MST expressions in this section have been numerically validated Longo (2018); Casals and Longo (). Specifically, these works have obtained numerical agreement of different expressions for the same quantity and/or compared values with expressions from the MST formalism in subextremal Kerr using near-extremal values of the angular momentum of the black hole.
iii.1 Series representations for the radial solutions
We here derive analytical expressions for the four solutions . Following Leaver and MST, our basic approach is to seek solutions to the radial equation in the form of infinite series of special functions. Solutions of the radial equation were found in terms of series representations in the pioneering analytical work of Leaver Leaver (1986a). We obtain new representations by rewriting Leaver’s series solutions, Eqs. (191) and (192) Leaver (1986a), in a more calculationally-practical way inspired by MST.
iii.1.1 Series coefficients and renormalized angular momentum
In order to calculate the incidence and reflection coefficients of the ingoing solution, one must match to a linear combination of and (which form a linearly-independent set) in some overlap region. Starting with Leaver’s radial solutions in Eqs. (191) and (192) Leaver (1986a), the immediate difficultly is that the coefficients in the series representation of (and in the series representation of ) are generally different from the coefficients in the series representation of . The main advantage of the MST method, developed in Mano et al. (1996a, b); Sasaki and Tagoshi (2003), is that it relates the two corresponding sets of coefficients in subextremal Kerr. In the same spirit, in extremal Kerr, we transform both of Leaver’s coefficients and to new coefficients . The upshot of this transformation is that the same coefficients then appear in the series representations for all solutions. This will allows us, in particular, to efficiently compute the Wronskian needed for the transfer function Eq.(24).
First, we transform from Leaver’s coefficients in Eq. (191) Leaver (1986a), to some new coefficients , thereby mapping the index to a new index via
where we have introduced
The parameter is the so-called renormalized angular momentum parameter, which we describe further down.
From Eqs. (186) and (187) of Ref. Leaver (1986a), it follows that the new coefficients satisfy the bilateral recurrence relation
where . Note that is merely ‘’, with replaced by in Leaver’s Eq. (187) Leaver (1986a). Here we have defined
We note that, under , transforms to and transforms to . We choose the normalization and therefore , , directly follows.
We now turn to the Leaver’s coefficients in Eq. (191) Leaver (1986a). We define the partner coefficients via
(with ) so that they satisfy the same recurrence relations as the , which can be readily checked from the recurrence relations in Eq.188 Leaver (1986a) satisfied by the coefficients. In our construction, we will place a symmetry condition on the normalizations, but they are otherwise freely chosen. We naturally choose , from which , , follows. We have thus expressed both sets of (Leaver) coefficients and in terms of just one set of (MST) coefficients: .
The bilateral (i.e., with ) recurrence relations Eq.(31) may be solved in the following way. Define
and then express them as the following continued fractions:
Once and have been obtained (either numerically to within a certain prescribed precision or analytically up to a certain order in an expansion parameter), respectively, and , then one can obtain , , by starting from given a certain choice for as a normalization choice. Similarly, one can obtain , .
To investigate the convergence of the continued fractions we carry out the large- asymptotics of the series coefficients . In order to find the large- behaviour of the solutions of the recurrence relations Eq. (31), we apply theorem 2.3 of Ref. Gautschi (1967). We find that there exist a pair of solutions, say and , which satisfy
and another pair, say and , which satisfy
Since , is said to be a minimal solution (which is unique up to a normalization) and a dominant solution as ; similarly, is said to be a minimal solution and a dominant solution as . The continued fraction in Eq.(35) [resp. Eq.(36)] converges when applied to a minimal solution as [resp. ] (see theorem 1.1 of Ref. Gautschi (1967)). The renormalized angular momentum parameter is determined by the consistency requirement that the minimal solution as coincides with the minimal solution as , i.e., that . This means that the value of is fixed via the following condition:
Equivalently, one may impose the condition
This is a transcendental equation for where and may be obtained from the continued fractions in Eq.(35). One is free to choose the value of in Eqs.(39) and (40). The series coefficients shall henceforth denote the corresponding unique minimal solution of the recurrence relations Eq.(31) as with the normalization choice .
We next note the following properties and symmetries exhibited by the MST construction and which relate to the renormalized angular momentum parameter 555We note that the parameter appears in other analyses of black hole perturbations which do not use the MST formalism. See, for example, Castro et al. (2013) in subextremal Kerr, where is related to the monodromy of the upgoing radial solution at the irregular singular point . See also Sec.III.5.:
The MST formalism is fundamentally invariant under . That is, a mode of any MST-derived physical quantity should be invariant under . The reason is that was introduced as a parameter in the radial ODE through the combination ‘’ only. This leads, in particular, to the symmetry
Applying and to all coefficients , and is equivalent, from their definitions, to complex-conjugating them.666For , we use the first two properties in Eq.(6). This implies, from Eq.(40), that applying to is equivalent to complex-conjugating . We note that these transformation properties are, however, not true if lies on a branch cut of ; in that case, is not necessarily real when is real.
It follows from the above property that, for real, complex-conjugation of can be achieved by applying the MST symmetries of and addition of an integer to .
The series coefficients , and are all invariant777For , we use the last property in Eq.(6). under and , and, therefore, so is .
We end this section by considering the branch points in the complex frequency plane of the angular eigenvalue and eigenfunction , the renormalized angular momentum , and the MST series coefficients . These quantities possess the following two types of branch points, which are, however, unphysical. Firstly, as stated earlier, and possess branch points in the complex frequency plane Oguchi (1970); Hunter and Guerrieri (1982); Barrowes et al. (2004). These “angular branch points”, in principle, carry over to and to . However, it has been shown that the angular branch points do not lie on the real axis when Hunter and Guerrieri (1982), and there is convincing numerical evidence that this is also the case for (for this, one can make use of the MATHEMATICA toolkit in BHP ()). Furthermore, it has been shown that the angular branch points do not contribute to the full Green function Casals et al. (2016a). Secondly, strong numerical evidence is presented in Longo (2018); Casals and Longo () that and do not possess any more discontinuities in the complex frequency plane aside from two harmless types. These are those due to the angular branch points and those due to discontinuities which can be removed by applying the above “MST symmetries” of addition of an integer to and/or transformation . In conclusion, any possible discontinuities in and do not carry over to any physical quantities. Therefore, when calculating later the discontinuity of the transfer function across the branch cuts from and from , and their corresponding late-time tails, we shall assume that no discontinuities from , , or contribute.
iii.1.2 Radial series
We give in Eqs.(155a) and (155b) the series representations of the radial solutions resulting from the transformation of the coefficients described in the previous subsection. For completeness, in App. A we also give the radial asymptotics of the solutions in Eqs.(191) and (192) Leaver (1986a) before applying this transformation of the coefficients.
Next, we use Eq.(125) of Ref. Leaver (1986a) to rewrite the Coulomb wavefunctions that appear in Eqs. (155a) and (155b) in terms of the irregular confluent hypergeometric function . This results in the following (MST) series representations for the various radial solutions:
We note that throughout the paper we make use of the symmetry Eq.(41) in the -sums of the MST series. We postpone giving explicit expressions for the normalization constants and until Sec. III.2. As mentioned in Sec. III.1.1, the renormalized angular momentum is chosen so that is the minimal solution of the bilateral recurrence relations Eq.(31) as . The choice of as a minimal solution guarantees Leaver (1986a) that the series in Eq.(III.1.2) converges everywhere except on and that the series in Eq.(III.1.2) converges everywhere except on .
It follows from Eqs.(III.1.2) and (III.1.2), together with the analytical properties of the irregular confluent hypergeometric function (as well as the prefactors and , respectively), that, in principle, is a branch point of the solutions and is a branch point of the solutions . As we shall see, these are indeed branch points of the radial solutions and they carry over to the transfer function.
We note that, in subextremal Kerr, while the corresponding solutions are similarly expressed in terms of the irregular confluent hypergeometric -functions, the corresponding solutions are instead expressed in terms of the regular hypergeometric -functions. This is a consequence of the aforementioned fact that the event horizon is a regular singular point of the radial equation in subextremal Kerr whereas it is an irregular singular point in extremal Kerr. As a consequence, the transfer function in subextremal Kerr only possesses a branch point at the origin, , which is responsible for the late-time decay of the linear perturbations Leaver (1986b); Hod (2000a); Casals et al. (2016a). The branch point at is a new feature of the extremal configuration and gives rise to the Aretakis phenomenon of field perturbations on the horizon of an extremal black hole Casals et al. (2016b); Gralla and Zimmerman (2017).
To simplify future calculations, we now give slightly more compact expressions for the radial solutions separately and obtain their transmission coefficients. For the ingoing and upgoing solutions, which are the ones of main interest here, the above expressions simplify a little further:
The reason for writing as the argument of but as that of is to make manifest their branch points at and respectively (we write the arguments of and merely out of notational consistency, not to denote any branch points in these functions). The transmission coefficients, defined via Eqs.(13) and (14), readily follow from Eqs.(44) and (45). Using Eq.13.2.6 DLMF (), we obtain
iii.2 Symmetric decomposition of the radial solutions
We now write the radial solutions in a form which is manifestly invariant under . For that purpose, we use Eq. 13.2.42 DLMF () to write the solutions in terms of the regular confluent hypergeometric function as888We note that here we choose to use the subindex “” for the solutions which, in the subextreme case, correspond to those in Sasaki and Tagoshi (2003) with subindex “” since in the extremal case both the ingoing and upgoing solutions have representations in terms of Coulomb wavefunctions.: