Asymptotic distribution of quasi-normal modes
for Kerr–de Sitter black holes
We establish a Bohr–Sommerfeld type condition for quasi-normal modes of a slowly rotating Kerr–de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multipicity modes at (Schwarzschild–de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also we prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.
Quasi-normal modes (QNMs) of black holes are a topic of continued interest in theoretical physics: from the classical interpretation as ringdown of gravitational waves [C] to the recent investigations in the context of string theory [HoHu]. The ringdown111Here is an irresistible quote of Chandrasekhar “…we may expect that any intial perturbation will, during its last stages, decay in a manner characteristic of the black hole itself and independent of the cause. In other words, we may expect that during these last stages, the black hole emits gravitational waves with frequencies and rates of damping that are characteristic of the black hole itself, in the manner of a bell sounding its last dying notes.” plays a role in experimental projects aimed at the detection of gravitational waves, such as LIGO [Ab]. See [KoSch] for an overview of the vast physics literature on the topic and [BeCaSt, KoZh1, KoZh2, YoUcFu] for some more recent developments.
In this paper we consider the Kerr–de Sitter model of a rotating black hole and assume that the speed of rotation is small; for , one gets the stationary Schwarzschild–de Sitter black hole. The de Sitter model corresponds to assuming that the cosmological constant is positive, which is consistent with the current Lambda-CDM standard model of cosmology.
A rigorous definition of quasi-normal modes for Kerr–de Sitter black holes was given using the scattering resolvent in [Dy1]. In Theorem 1 below we give an asymptotic description of QNMs in a band of any fixed width, that is, for any bounded decay rate. The result confirms the heuristic analogy with the Zeeman effect: the high multiplicity modes for the Schwarzschild black hole split.
Theorem 2 confirms the standard interpretation of QNMs as complex frequencies of exponentially decaying gravitational waves; namely, we show that the solutions of the scalar linear wave equation in the Kerr–de Sitter background can be expanded in terms of QNMs.
In the mathematics literature quasi-normal modes of black holes were studied by Bachelot, Motet-Bachelot, and Pravica [Ba1, Ba2, BaMoBa, Pr] using the methods of scattering theory. QNMs of Schwarzschild–de Sitter metric were then investigated by Sá Barreto–Zworski [SáBaZw], resulting in the lattice of pseudopoles given by (0.3) below. For this case, Bony–Häfner [BoHä] established polynomial cutoff resolvent estimates and a resonance expansion, Melrose–Sá Barreto–Vasy [MeSáBaVa] obtained exponential decay for solutions to the wave equation up to the event horizons, and Dafermos–Rodnianski [DaRo1] used physical space methods to obtain decay of linear waves better than any power of .
Quasi-normal modes for Kerr–de Sitter were rigorously defined in [Dy1] and exponential decay beyond event horizons was proved in [Dy2]. Vasy [Va] has recently obtained a microlocal description of the scattering resolvent and in particular recovered the results of [Dy1, Dy2] on meromorphy of the resolvent and exponential decay; see [Va, Appendix] for how his work relates to [Dy1]. The crucial component for obtaining exponential decay was the work of Wunsch–Zworski [WuZw] on resolvent estimates for normally hyperbolic trapping.
We add that there have been many papers on decay of linear waves for Schwarzschild and Kerr black holes — see [AnBl, BlSt, DaRo2, DaRo3, DoSchSo1, DoSchSo2, FiKaSmYa, FiKaSmYaErr, Ta, TaTo, To] and references given there. In that case the cosmological constant is 0 (unlike in the de Sitter case, where it is positive), and the methods of scattering theory are harder to apply because of an asymptotically Euclidean infinity.
Fix the mass of the black hole and the cosmological constant . (See Section 1.1 for details.) Then there exists a constant such that for and each , there exist constants 222As in [Dy1], the indices next to constants, symbols, operators, and functions do not imply differentiation. such that the set of quasi-normal modes satisfying
coincides modulo with the set of pseudopoles
(Since the set of QNMs is symmetric with respect to the imaginary axis, one also gets an asymptotic description for negative. Also, by [Dy1, Theorem 4], all QNMs lie in the lower half-plane.) Here is a complex valued classical symbol333Here ‘symbol’ means a microlocal symbol as in for example [Tay, Section 8.1]. For the proofs, however, we will mostly use semiclassical symbols, as defined in Section 2.1. of order 1 in the variables, defined and smooth in the cone . The principal symbol of is real-valued and independent of ; moreover,
The pseudopoles (0.2) can be computed numerically; we have implemented this computation in a special case and compared the pseudopoles with the QNMs computed by the authors of [BeCaSt]. The results are described in Appendix B. One should note that the quantization condition of [SáBaZw] was stated up to error, while Theorem 1 has error ; we demonstrate numerically that increasing the order of the quantization condition leads to a substantially better approximation.
Another difference between (0.2) and the quantization condition of [SáBaZw] is the extra parameter , resulting from the lack of spherical symmetry of the problem. In fact, for each pole in (0.3) has multiplicity ; for this pole splits into distinct QNMs, each corresponding to its own value of , the angular momentum with respect to the axis of rotation. (The resulting QNMs do not coincide for small values of , as illustrated by (0.4)). In the physics literature this is considered an analogue of the Zeeman effect.
Since the proof of Theorem 1 only uses microlocal analysis away from the event horizons, it implies estimates on the cutoff resolvent polynomial in (Proposition 1.3). Combining these with the detailed analysis away from the trapped set (and in particular near the event horizons) by Vasy [Va], we obtain estimates on the resolvent on the whole space (Proposition 1.2). These in turn allow a contour deformation argument leading to an expansion of waves in terms of quasinormal modes. Such expansions have a long tradition in scattering theory going back to Lax–Phillips and Vainberg — see [TaZw] for the strongly trapping case and for references.
For Schwarzschild-de Sitter black holes a full expansion involving infinite sums over quasinormal modes was obtained in [BoHä] (see also [ChrZw] for simpler expansions involving infinite sums over resonances). The next theorem presents an expansion of waves for Kerr–de Sitter black holes in the same style as the Bony–Häfner expansion:
Under the assumptions of Theorem 1, take such that for some , every QNM has . (Such exists and can be chosen arbitrarily large, as the imaginary parts of QNMs lie within of those in (0.3).) Then for large enough depending on , there exists a constant such that every solution to the Cauchy problem on the Kerr–de Sitter space
where is the space slice, is the time variable, and is a small constant (see Section 1.1 for details), satisfies for ,
the outer sum is over QNMs , is the algebraic multiplicity of as a pole of the scattering resolvent, and are finite rank operators mapping . Moreover, for large enough (that is, for all but a finite number of QNMs in the considered strip), , has rank one, and
Here is a constant depending on , but not on ; therefore, the series (0.7) converges in for .
The proofs start with the Teukolsky separation of variables used in [Dy1], which reduces our problem to obtaining quantization conditions and resolvent estimates for certain radial and angular operators (Propositions 1.5 and 1.6). These conditions are stated and used to obtain Theorems 1 and 2 in Section 1. Also, at the end of Section 1.2 we present the separation argument in the simpler special case , for convenience of the reader.
In the spherically symmetric case , the angular problem is the eigenvalue problem for the Laplace–Beltrami operator on the round sphere. For , the angular operator is not selfadjoint; however, in the semiclassical scaling it is an operator of real principal type with completely integrable Hamiltonian flow. We can then use some of the methods of [HiSj] to obtain a microlocal normal form for ; since our perturbation is , we are able to avoid using analyticity of the coefficients of . The quantization condition we get is global, similarly to [VũNg]. The proof is contained in Section 3; it uses various tools from semiclassical analysis described in Section 2.
To complete the proof of the angular quantization condition, we need to extract information about the joint spectrum of and from the microlocal normal form; for that, we formulate a Grushin problem for several commuting operators. The problem that needs to be overcome here is that existence of joint spectrum is only guaranteed by exact commutation of the original operators, while semiclassical methods always give errors. This complication does not appear in [HiSj, HiSjVũNg] as they study the spectrum of a single operator, nor in earlier works [Ch, VũNg] on joint spectrum of differential operators, as they use spectral theory of selfadjoint operators. Since this part of the construction can be formulated independently of the rest, we describe Grushin problems for several operators in an abstract setting in Appendix A.
The radial problem is equivalent to one-dimensional semiclassical potential scattering. The principal part of the potential is real-valued and has a unique quadratic maximum; the proof of the quantization condition follows the methods developed in [CdVPa, Ra, Sj1]. In [CdVPa], the microlocal behavior of the principal symbol near a hyperbolic critical point is studied in detail; however, only self-adjoint operators are considered and the phenomenon that gives rise to resonances in our case does not appear. The latter phenomenon is studied in [Ra] and [Sj1]; our radial quantization condition, proved in Section 4, can be viewed as a consequence of [Ra, Theorems 2 and 4]. However, we do not compute the scattering matrix, which simplifies the calculations; we also avoid using analyticity of the potential near its maximum and formulate the quantization condition by means of real microlocal analysis instead of the action integral in the complex plane. As in [Ra], we use analyticity of the potential near infinity and the exact WKB method to relate the microlocal approximate solutions to the outgoing condition at infinity; however, the construction is somewhat simplified compared to [Ra, Sections 2 and 3] using the special form of the potential.
It would be interesting to see whether our statements still hold if one perturbs the metric, or if one drops the assumption of smallness of . Near the event horizons, we rely on [Dy1, Section 6], which uses a perturbation argument (thus smallness of ) and analyticity of the metric near the event horizons. Same applies to Section 4.2 of the present paper; the exact WKB construction there requires analyticity and Proposition 4.2 uses that the values defined in (4.9) are nonzero, which might not be true for large . However, it is very possible that the construction of the scattering resolvent of [Va] can be used instead. The methods of [Va] are stable under rather general perturbations, see [Va, Section 2.7], and apply in particular to Kerr–de Sitter black holes with satisfying [Va, (6.12)].
A more serious problem is the fact that Theorem 1 is a quantization condition, and thus is expected to hold only when the geodesic flow is completely integrable, at least on the trapped set. For large , the separation of variables of Section 1.2 is still valid, and it is conceivable that the global structure of the angular integrable system in Section 3.2 and of the radial barrier-top Schrödinger operator in Section 4.1 would be preserved, yielding Theorem 1 in this case. Even then, the proof of Theorem 2 no longer applies as it relies on having gaps between the imaginary parts of resonances, which might disappear for large .
However, a generic smooth perturbation of the metric supported near the trapped set will destroy complete integrability and thus any hope of obtaining Theorem 1. One way of dealing with this is to impose the condition that the geodesic flow is completely integrable on the trapped set. In principle, the global analysis of [VũNg] together with the methods for handling nonselfadjoint perturbations developed in Section 3 and Appendix A should provide the quantization condition in the direction of the trapped set, while the barrier-top resonance analysis of Section 4.3 should handle the transversal directions. However, without separation of variables one might need to merge these methods and construct a normal form at the trapped set which is not presented here.
Another possibility is to try to establish Theorem 2 without a quantization condition, perhaps under the (stable under perturbations) assumption that the trapped set is normally hyperbolic as in [WuZw]. However, this will require to rethink the contour deformation argument, as it is not clear which contour to deform to when there is no stratification of resonances by depth, corresponding to the parameter in Theorem 1.
1.1. Kerr–de Sitter metric
First of all, we define Kerr–de Sitter metric and briefly review how solutions of the wave equation are related to the scattering resolvent; see also [Dy1, Section 1] and [Va, Section 6]. The metric is given by
Here and are the spherical coordinates on and take values in ; is the mass of the black hole, is the cosmological constant, and is the angular momentum;
The metric in the coordinates is defined for ; we assume that this happens on an open interval , where are two of the roots of the fourth order polynomial equation . The metric becomes singular at ; however, this apparent singularity goes away if we consider the following version of the Kerr-star coordinates (see [DaRo2, Section 5.1] and [TaTo]):
with the functions blowing up like as approaches . One can choose so that the metric continues smoothly across the surfaces , called event horizons, to
with is a small constant. Moreover, the surfaces are spacelike, while the surfaces are timelike for , spacelike for , and null for . See [Dy1, Section 1], [Dy2, Section 1.1], or [Va, Section 6.4] for more information on how to construct with these properties.
Let be the d’Alembert–Beltrami operator of the Kerr–de Sitter metric. Take for some , and furthermore assume that is supported in . Then, since the boundary of is spacelike and every positive time oriented vector at points outside of , by the theory of hyperbolic equations (see for example [DaRo2, Proposition 3.1.1] or [Tay, Sections 2.8 and 7.7]) there exists unique solution to the problem
We will henceforth consider the problem (1.2); the Cauchy problem (0.5) can be reduced to (1.2) as follows. Assume that solves (0.5) with some , . Take a function such that and ; then solves (1.2) with supported in and the norm of is controlled by .
is well-defined and holomorphic in . Here both and are functions on . Moreover, if is the Fourier–Laplace transform of , then
where is the stationary d’Alembert–Beltrami operator, obtained by replacing with in . (The factor will prove useful in the next subsection.) Finally, since is supported in , the function is holomorphic in the entire , and
for bounded by a fixed constant. Here , , is the semiclassical Sobolev space, consisting of the same functions as , but with norm instead of .
If was, say, an elliptic operator, then the equation (1.3) would have many solutions; however, because of the degeneracies occuring at the event horizons, the requirement that acts as a boundary condition. This situation was examined in detail in [Va]; the following proposition follows from [Va, Theorem 1.2 and Lemma 3.1] (see [Dy1, Proposition 1.2] for the cutoff version):
Fix . Then for large enough depending on , there exists a family of operators (called the scattering resolvent)
meromorphic with poles of finite rank and such that for solving (1.2), we have
Note that even though we originally defined the left-hand side of (1.5) for , the right-hand side of this equation makes sense in a wider region , and in fact in the entire complex plane if is smooth. The idea now is to use Fourier inversion formula
and deform the contour of integration to to get exponential decay via the factor. We pick up residues from the poles of when deforming the contour; therefore, one defines quasi-normal modes as the poles of .
Our ability to deform the contour and estimate the resulting integral depends on having polynomial resolvent estimates. To formulate these, let us give the technical
Let be a parameter and , , be a meromorphic family of operators, with Hilbert spaces. Let also be open and be a finite subset; we allow elements of to have multiplicities. We say that the poles of in are simple with a polynomial resolvent estimate and given modulo by , if for small enough, there exist maps and from to and the algebra of bounded operators , respectively, such that:
for each , is a pole of , , and is a rank one operator;
there exists a constant such that for each and, moreover,
In particular, every pole of in lies in the image of .
Fix and let be a parameter. Then for small enough (independently of ), the poles of in the region
are simple with a polynomial resolvent estimate and given modulo by
Here and are some constants and is a classical symbol:
The principal symbol is real-valued and independent of ; moreover,
Finally, if we consider as a family of operators between the semiclassical Sobolev spaces , then the constant in Definition 1.1 is independent of .
Theorem 1 follows from the here almost immediately. Indeed, since is independent of , each is homogeneous in variables of degree ; we can then extend this function homogeneously to the cone and define the (non-semiclassical) symbol
The series in (1.9) is over QNMs ; all but a finite number of them in the region are equal to for some given by (0.2) and the residue in this case is . Here and are taken from Definition 1.1. Now, by (1.4), we have
1.2. Separation of variables
First of all, using [Va, (A.2), (A.3)], we reduce Proposition 1.2 to the following444One could also try to apply the results of [DaVa] here, but we use the slightly simpler construction of [Va, Appendix], exploiting the fact that we have information on the exact cutoff resolvent.
Furthermore, by [Va, Proposition A.1] the family of operators coincides with the one constructed in [Dy1, Theorem 2], if the functions in (1.1) are chosen so that in . We now review how the construction of in [Dy1] works and reduce Proposition 1.3 to two separate spectral problems in the radial and the angular variables. For the convenience of reader, we include the simpler separation of variables procedure for the case at the end of this section.
First of all, the operator is invariant under the rotation ; therefore, the spaces of functions of angular momentum are invariant under both and . In [Dy1], we construct by piecing together the restrictions for all . Then, Proposition 1.3 follows from
Under the assumptions of Proposition 1.3, there exists a constant such that for each ,
Now, we recall from [Dy1, Section 1] that the restriction of to has the form666The operator of [Dy1] differs from our operator by the conjugation done in [Va, Appendix]; however, the two coincide in . , where
is a certain right inverse to , while is the inverse to ; we write . Recall that both and are meromorphic families of operators, as defined in [Dy1, Definition 2.1]; in particular, for a fixed value of , these families are meromorphic in with poles of finite rank. By definition of , a number is a pole of this operator if and only if there exists such that is a pole of both and .
Now, for small we put
the assumptions of Proposition 1.4(2) imply that , , and . Moreover, [Dy1, Proposition 3.4] suggests that under these assumptions, all values of for which is a pole of both and have to satisfy , for some constant .
Proposition 1.5 (Radial lemma).
Let be a fixed constant and put . Then the poles of as a function of , in the region
are simple with polynomial resolvent estimate (in the sense of Definition 1.1) and given modulo by
for some constant . The principal part of the classical symbol is real-valued, independent of and , and
In particular, for satisfying (1.12), every pole satisfies for some constant .
Proposition 1.6 (Angular lemma).
Let be a fixed constant. Then the poles of as a function of in the region
are simple with polynomial resolvent estimate and given modulo by
for some constant . The principal part of the classical symbol is real-valued, independent of , and
Moreover, , , and consequently, .
Proof of Proposition 1.4.
We let be the solution to the equation
We can see that this equation has unique solution by writing and examining the principal parts of the real-valued symbols for .
The idea now is to construct an admissible contour in the sense of [Dy1, Definition 2.3]; e.g. a contour that separates the sets of poles (in the variable ) of and from each other; then [Dy1, (2.1)] provides a formula for , which can be used to get a resolvent estimate.
We will use the method of proof of [Dy1, Proposition 3.4]. Take the contour introduced there, for , , and some large constant. Then we know that all angular poles are to the right of (in ). Moreover, the only radial poles to the right of lie in the domain and they are contained in the set for some constant , where is the radial pole corresponding to . In particular, those radial poles are contained in
for some constant .
Assume that is not a pole of ; then we can consider the admissible contour composed of and the circles , , enclosing , but none of the other poles of or . Using the meromorphic decomposition of at and letting its principal part be , we get
Here we only include the poles lying to the right of ; one might need to change in the definition of a little bit in case some comes close to . The integral in (1.17) is holomorphic and bounded polynomially in , by the bounds for given by Proposition 1.5, together with the estimates in the proof of [Dy1, Proposition 3.4].
Now, the poles of in are given by (1.15); let be the pole corresponding to and be the principal part of the corresponding meromorphic decomposition. Then the resolvent estimates on given by Proposition 1.6 together with (1.17) imply
Here is a family of operators holomorphic in and bounded polynomially in . Moreover,
with as above and being the product of a coefficient polynomially bounded in with ; this finishes the proof. ∎
Finally, let us present the simplified separation of variables for the case , namely the Schwarzschild–de Sitter metric:
Note that is just the round metric on the unit sphere. The metric decouples without the need to take Fourier series in ; the stationary d’Alembert–Beltrami operator has the form , where
Here is the Laplace–Beltrami operator on the round sphere; it is self-adjoint (thus no need for the contour integral construction of [Dy1, Section 2]) and is known to have eigenvalues