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Trapping of waves and null geodesics for rotating black holes

We present dynamical properties of linear waves and null geodesics valid for Kerr and Kerr-de Sitter black holes and their stationary perturbations. The two are intimately linked by the geometric optics approximation. For the nullgeodesic flow the key property is the r-normal hyperbolicity of the trapped set and for linear waves it is the distribution of quasi-normal modes: the exact quantization conditions do not hold for perturbations but the bounds on decay rates and the statistics of frequencies are still valid.

The basic dynamical properties of rotating (Kerr) black holes can be considered on the level of null geodesics and on the level of waves, that is, solutions to wave equations for the corresponding metrics. The exact Kerr solution is completely integrable [1] in the sense of dynamical systems and it has a regular trapped set of null geodesics. For the Schwarzschild (i.e. non-rotating) case this set is the phase space of the two-dimensional photon sphere. It becomes a smooth manifold in the phase space when the black hole starts rotating. In this letter we describe an important feature of the trapping: r-normal hyperbolicity. It roughly means that the flow is hyperbolic in directions normal to the trapped set in ways stronger than the flow on the (invariant and symplectic) trapped set. This property, unlike complete integrability, is known to be stable under perturbations [2] and it allows a more robust description of the trapping of null geodesics and of waves, the latter described using quasi-normal modes [3]. It shows that many features studied recently using WKB methods available in the completely integrable case [4][5][6] persist for perturbations. The r-normal hypebolicity of black hole dynamics [7] has not been discussed in the physics literature but the importance of normal hyperbolicity in molecular dynamics has been explored [8].
Following [4,5,7,9,11,12,18] we show how dynamical features (such as the maximal and minimal expansion rates) and statistical properties of the distribution of quasi-normal modes (QNMs) depend on a = J/M c, the rescaled angular momentum. The results are valid for almost the entire range of a's, including for instance the recently discovered [13] rapidly rotating black hole with a 0.84. For the exact Kerr (or Kerr-de Sitter) black hole the counting law for QNMs and their maximal and minimal decay rates determine the mass and the angular momentum.

II. THE TRAPPED SET OF NULL GEODESICS
We are interested in null geodesics and the wave equation g u = 0, where g is the Kerr(-de Sitter) metric [1,4,7]. The Kerr-de Sitter metric is a generalization of Kerr to the case of a positive cosmological constant Λ. For a black hole of mass M > 0, rotating with speed a projected to the coordinates (x, y, z) = (ξϕ, θ, ξ θ ). The vertical singular interval in front corresponds to θ = 0, with the symmetrical interval in the back corresponding to θ = π: the coordinates (θ, ϕ) are singular at that point. The additional coordinate, not shown in the figures, r is a function of ξϕ and ξt only. When a = 0, r = 3M , but it gets larger to the left (ξϕ > 0) and smaller to the right (ξϕ < 0) when a > 0. When a = 1 we see the flattening in the (θ, ξ θ )-plane at extremal values of ξϕ: the trapped set touches the event horizon r = 1 which results in lack of decay, and some QNMs have null imaginary parts [6]. Dynamically and invariantly this corresponds to the vanishing expansion rates -see Fig. 4 the space slice is X = (r + , r C ) × S 2 , where r C < ∞ when Λ > 0, and r C = ∞ when Λ = 0. The behaviour of g for r near r C is dramatically different in the two cases: for Λ > 0 the metric is hyperbolic (infinity is large) and for Λ = 0 it is Euclidean (infinity is small). For solutions to the wave equation that produces differences in time decay and in the behaviour at low energies [15][16][17].
The geodesic flow can be described on the cotangent bundle T * (R × X) by the Hamiltonian (here ξ denotes the momenta, that is, the fiber variables in the cotangent bundle; the subscripts refer to the corresponding base variables with (θ, ϕ) the spherical coordinates) The function G is the dual metric to g; it is also the principal symbol of g in the sense that g = G(t, x, ∂ t /i, ∂ x /i) modulo a first order differential operator. The limiting radii r + , r C solve ∆ r = 0.
The trapped set consists of null geodesics that stay away from r = r + , r = r C for all times. The variables (θ, ϕ, ξ θ ) evolve according to the Hamiltonian flow of G θ , and ξ t , ξ ϕ are conserved. The trapping depends on the evolution of (r, ξ r ) according to the flow of G r , which is essentially the one dimensional motion for a barrier top potential. Under the assumptions that either a = 0, 9ΛM 2 < 1 or Λ = 0, |a| < M , and for nearby values of M, Λ, a [18, Prop.3.2], [19] the trapped set K is given by K = {G = 0, ξ r = 0, ∂ r G r = 0, ξ = 0}. For a = 0, ∂ r G r = 0 gives r = 3M , the radius of the photon sphere. For a = 0, a more careful analysis is required, but K is still a smooth submanifold of the charateristic set {G = 0}. Moreover, it is symplectic in the sense that the spatial symplectic form dξ r ∧ dr + dξ θ ∧ dθ + dξ ϕ ∧ dϕ is nondegenerate on the surfaces K ∩ {t = const}.
Let C + ⊂ {G = 0} be the positive light cone and ϕ t : C + → C + the geodesic flow parametrized by t. The rnormal hyperbolicity condition asserts the existence of a C r splitting T K C + = T K ⊕ V + ⊕ V − , invariant under the flow and such that for some constants ν > 0, C > 0, This means that the maximal expansion rates (Lyapunov exponents) on the trapped set are r-fold dominated by the expansion and contraction rates in the directions transversal to the trapped set. As shown in [2],[11, §5.2] r-normal hyperbolicity is stable under perturbations: when G is a time independent Hamiltonian such that G is close to G in C r near K, the flow for G is rnormally hyperbolic in the sense that the trapped set K has C r regularity, is symplectic and (1) holds. For Kerr(de Sitter) metrics the flow is r-normally hyperbolic for all r as shown in [7,12,18], essentially because the flow on K is completely integrable. Key dynamical quantities are the minimal and maximal expansion rates 0 < ν min ≤ ν max , characterized by inequalities true for all ε > 0, a constant C ε depending on ε, For Kerr(-de Sitter) metrics, the quantities ν min , ν max are obtained by taking the minimum and maximum of averages of the local expansion rate A schematic comparison between QNMs in a completely integrable case and in the r-normally hyperbolic case. The former lie on a fuzzy lattice and are well approximated by WKB construction based on quantization conditions [4][5][6]. When trapping is r-normally hyperbolic and the pinching condition νmax < 2νmin holds, the quasi-normal modes are still localized to a strip with dynamically determined bounds, and their statistics are given by the Weyl law (6).
on the Liouville tori of the flow of G θ on the trapped set.

III. DEFINITION AND DISCRETENESS OF QUASI-NORMAL MODES
The scattering resonances, called quasi-normal modes (QNMs) in the context of black holes [3], replace eigenmodes when one switches from closed systems to open systems -see [20] for a recent experimental discussion. They are the frequencies ω of oscillating solutions to the wave equation g (e −iωt v(x)) = 0, which continue smoothly across the event horizons.
Solutions to g u = 0 are expected to have expansions valid in a suitable sense [5,10]. The fact that QNMs ω k form a discrete set in the lower half-plane is nontrivial but it is now rigorously known in the case of Kerr-de Sitter and its perturbations [4,9,10,12,21].
In the simpler Schwarzschild-de Sitter case we indicate the reason for it as follows. In the Regge-Wheeler coordinates, quasi-normal modes, ω, can be studied using the Schrödinger operator P (ω) = −∂ 2 x + ω 2 V 1 (x) + ( + 1)V 2 (x), where the potentials V 1 and V 2 are real analytic and satisfy C . (When Λ = 0 then V 2 ∼ x −2 as x → +∞ and that creates problems at low energies [15,16].) A number ω ∈ C is a QNM if there exists an angular momentum and a nonzero solution u(x) to the equation P (ω)u = 0 satisfying the outgoing condition: near x = ±∞, e ∓iV±ωx u(x) is a smooth function of e −A±|x| . For fixed , it follows by standard one-dimensional methods that the set of all corresponding ω is discrete.
Showing that as → ∞, quasi-normal modes corresponding to do not accumulate in compact sets is more delicate: we need to know that if |ω| ≤ R and is large enough depending on R, then ω cannot be a QNM corresponding to . Assume the contrary and let u(x) be the corresponding solution to the equation P (ω)u = 0. We fix large X > 0 independently of , to be chosen later. The potential V 2 is everywhere positive, therefore for large enough depending on R, X, Re(ω 2 V 1 (x) + ( + 1)V 2 (x)) > 0 for x ∈ [−X, X]. If u satisfied a Dirichlet or Neumann boundary condition at ±X, then integration by parts would give the impossible statement that u = 0 on [−X, X], finishing the proof: and the terms under the integral are all nonnegative. While u does not satisfy a boundary condition at ±X (after all, this 'boundary' is completely artificial), it suffices to prove the boundary inequalities ± Re(u (±X)u(±X)) < 0. ( To prove (3), we cannot use integration by parts on the whole R, since u does not lie in L 2 (R) and moreover the real part of our potential may become negative as x → ±∞. We instead use the methods of complex analysis and real analyticity of V 1 , V 2 . The characterization of u as a mode can be strengthened to say that e ∓iV±ωx u(x) is an analytic function of e −A±|x| , which means that for X large enough, we can extend u(x) to a holomorphic function in {| Re z| ≥ X}, and this extension is Floquet periodic: u(z + 2πi/A ± ) = e ∓2πV±ω/A± u(z), ± Re z ≥ X. Now, consider the restriction of u to the vertical lines {Re z = ±X}, w ± (y) := u(±X + iy), y ∈ R, and note that it solves the differential equation The key difference between (4) and the equation P (ω)u = 0 is that the potential V 2 (±X + iy) is no longer real-valued. For instance, if V 2 were equal to e −A±|x| , then V 2 (±X + iy) would equal e −XA± e ∓iA±y , only taking real values when y ∈ πA −1 ± Z. This means that the equation (4) is elliptic (in the semiclassical sense, where we treat ∂ y as having same order as ) except at a discrete set of points in the phase space T * R. Further analysis shows that w ± (y) is concentrated in phase space near y ∈ 2πA −1 ± Z, η = ∓ V 2 (±X), in particular implying |(∂ y ± i V 2 (±X))w ± (0)| ≤ C 1/4 |w ± (0)|, where ω k ( ) are the numerically computed resonances in the first band corresponding to the angular momentum [14] and νmin,νmax are minimal and maximal expansion rates. The agreement is remarkable when increases.

IV. DISTRIBUTION OF QUASI-NORMAL MODES
The distribution of QNMs ω k can now be studied in the more general stable setting of r-normally hyperbolic trapped sets. Three fundamental issues are (a) distribution of decay rates, that is of the imaginary parts of QNMs; (b) asymptotics of the counting function; (c) expansion of waves in terms of QNMs.
(a) We can bound the decay rates from below whenever the trapped set is normally hyperbolic, without requiring the stronger r-normal hyperbolicity assumption. The bound [22] is given by Im ω k < −(ν min −ε)/2. In the case of r-normal hyperbolicity and under the pinching condition we get more detailed information [11]: there are no QNMs with −(ν min − ε) < Im ω k < −(ν max + ε)/2. That means that the modes with least decay are confined to a band shown in Fig.5. In the completely integrable case this follows from WKB constructions [4-6] -see Fig.5 but this structure persists under perturbations. For a recent experimental investigation of the distribution of decay rates and the relation to classical dynamics see [23].  (6) holds. We note that as a increases the QNMs split (see Fig. 5) and hence we do expect the counting function to decrease, in agreement with the behaviour of the volume. From volume and the gap giving the decay rate one can read off a and M .
H (for instance the Dirichlet Laplacian on a bounded domain), below energy λ 2 satisfies the Weyl law For open systems QNMs replace real energy levels and the counting becomes much more tricky [20]. In the case of exact Kerr(-de Sitter) black holes the WKB constructions can be used to show that the number N QNM (λ) of QNMs satisfying |ω k | ≤ λ, Im ω k ≥ −(ν min − ε) satisfies N QNM (λ) ∼ cλ 2 . The constant c has a geometric interpretation: in a scattering problem the phase space T * X is replaced by the trapped set K. Under the assumptions of r-normal hyperbolicity (1) and pinching (5) where the volume is taken using the symplectic form on K ∩ {t = const} [18,Thm 3]. We note that just as dim X = 1 2 dim T * X in the exponent of the Weyl law, here 2 = 1 2 dim(K ∩ {t = const}): the effective phase space is now the trapped set. For exact Kerr(-de Sitter) metrics with several values of Λ the volume as function of a is shown in Fig. 7. The volume is finite provided that (1 − Λa 2 /3) 3 > 9ΛM 2 , see Fig. 2.
(c) The expansion (2) is rigorously established for slowly rotating black holes [5,10] and heuristically it is one of the motivations for studying quasi-normal modes [3]. For rapidly rotating black holes, or for their perturbations satisfying (5) a more robust version can be formulated using a projector onto the space of quasi-normal modes in the first band shown in Fig. 5. The solution to the wave equation g u = 0 with initial data localized near frequency λ 1 can be decomposed as u = u QNM + u DEC where g u QNM (t), g u DEC (t) = O(λ −∞ ) for 0 ≤ t ≤ T log λ. This means that both terms solve the wave equation approximately at high energies for logarithmic time. We then have, again for 0 ≤ t ≤ T log λ, where strictly speaking errors O(λ −∞ ) u(0) E should be added to the right hand sides. The norm · E is the standard energy norm in any sufficiently large compact subset of X (in the case of exact Kerr-de Sitter, we can take (r + + δ, r C − δ) × S 2 ) -see [18,Thm 2]. The term u QNM (t) corresponds to the part of the solution dominated by the QNMs in the first band and it has the natural decay properties dictated by these QNMs.

V. CONCLUSIONS
We have shown that for Kerr-de Sitter metrics and their perturbations quasi-normal modes are rigorously defined and form a discrete set in the lower half plane, provided that (1 − Λa 2 /3) 3 > 9ΛM 2 > 0, see Fig. 2. This is due to the size of infinity when Λ > 0 and the compactness of the trapped set at finite energies.
If one neglects the issues of long time decay and of behaviour at low energies, then the results are also valid in the case of Λ = 0.
Thanks to the r-normal hyperbolicity of the trapped set, the main features of the distribution of quasimodes are preserved for perturbations: the decay rates are bounded from below in terms of the minimal expansion rate (Im ω k ≤ −(ν min −ε)/2) and under the pinching conditions, the least decaying modes are confined to a strip where they satisfy a counting law (6) -see Fig. 5. We should stress that normally hyperbolic behavior (unlike r-normal hyperbolicity) is often unstable under perturbations as shown by examples of hyperbolic quotients where a small perturbation can change the dimension of the trapped set [24, Fig.1], leading to a fractal Weyl law -see [20] for recent experiments on that.
The r-normal hyperbolicity is valid for all rotating black holes but the pinching condition (5) needed for the finer results (6) and (7) fails in the case of very fast rotation -see Fig. 3.