Lectures on Linear Stability of Rotating Black Holes
aa r X i v : . [ g r- q c ] O c t LECTURES ON LINEAR STABILITY OFROTATING BLACK HOLES
FELIX FINSTERNOVEMBER 2018
Abstract.
These lecture notes are concerned with linear stability of the non-extreme Kerr geometry under perturbations of general spin. After a brief reviewof the Kerr black hole and its symmetries, we describe these symmetries by Killingfields and work out the connection to conservation laws. The Penrose process andsuperradiance effects are discussed. Decay results on the long-time behavior of Diracwaves are outlined. It is explained schematically how the Maxwell equations and theequations for linearized gravitational waves can be decoupled to obtain the Teukolskyequation. It is shown how the Teukolsky equation can be fully separated to a systemof coupled ordinary differential equations. Linear stability of the non-extreme Kerrblack hole is stated as a pointwise decay result for solutions of the Cauchy problemfor the Teukolsky equation. The stability proof is outlined, with an emphasis on theunderlying ideas and methods.
Contents
1. Introduction 22. The Kerr Black Hole 23. Symmetries and Killing Fields 34. The Penrose Process and Superradiance 55. The Scalar Wave Equation in the Kerr Geometry 66. An Overview of Linear Wave Equations in the Kerr Geometry 96.1. The Dirac Equation 96.2. Massless Equations of General Spin, the Teukolsky Equation 117. Separation of the Teukolsky Equation 148. Results on Linear Stability and Superradiance 159. Hamiltonian Formulation and Integral Representations 1610. A Spectral Decomposition of the Angular Teukolsky Operator 1911. Invariant Disk Estimates for the Complex Riccati Equation 2012. Separation of the Resolvent and Contour Deformations 2113. Proof of Pointwise Decay 2214. Concluding Remarks 22References 23 Introduction
These lectures are concerned with the black hole stability problem. Since this is abroad topic which many people have been working on, we shall restrict attention tospecific aspects of this problem: First, we will be concerned only with linear stability.Indeed, the problem of nonlinear stability is much harder, and at present there areonly few rigorous results. Second, we will concentrate on rotating black holes. Thisis because the angular momentum leads to effects (Penrose process, superradiance)which make the rotating case particularly interesting. Moreover, the focus on rotatingblack holes gives a better connection to my own research, which was carried out incollaboration with Niky Kamran (McGill), Joel Smoller (University of Michigan) andShing-Tung Yau (Harvard). The linear stability result for general spin was obtainedtogether with Joel Smoller (see [30] and the survey article [29]). Before beginning,I would like to remember Joel Smoller, who sadly passed away in September 2017.These notes are dedicated to his memory.2.
The Kerr Black Hole
In general relativity, space and time are combined to a four-dimensional space-time, which is modelled mathematically by a Lorentzian manifold ( M , g ) of signature(+ − − − ) (for more elementary or more detailed introductions to general relativitysee the textbooks [1, 36, 44, 42]). The gravitational field is described geometrically interms of the curvature of space-time. Newton’s gravitational law is replaced by theEinstein equations R jk − R g jk = 8 πκ T jk , (2.1)where R jk is the Ricci tensor, R is scalar curvature, and κ denotes the gravitationalconstant. Here T jk is the energy-momentum tensor which describes the distributionof matter in space-time.A rotating black hole is described by the Kerr geometry . It is a solution of thevacuum Einstein equations discovered in 1963 by Roy Kerr. In the so-called Boyer-Lindquist coordinates, the Kerr metric takes the form (see [7, 37]) ds = ∆ U ( dt − a sin ϑ dϕ ) − U (cid:18) dr ∆ + dϑ (cid:19) − sin ϑU (cid:16) a dt − ( r + a ) dϕ (cid:17) , (2.2)where U = r + a cos ϑ, ∆ = r − M r + a , (2.3)and the coordinates ( t, r, ϑ, ϕ ) are in the range −∞ < t < ∞ , M + p M − a < r < ∞ , < ϑ < π, < ϕ < π . The parameters M and aM describe the mass and the angular momentum of the blackhole.In the case a = 0, one recovers the Schwarzschild metric ds = (cid:18) − Mr (cid:19) dt − (cid:18) − Mr (cid:19) − dr − r ( dθ + sin θ dϕ ) . In this case, the function ∆ has two roots r = 2 M event horizon r = 0 curvature singularity . ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 3
In the region r > M , the so-called exterior region , t is a time coordinate, whereas r , ϑ and ϕ are spatial coordinates. More precisely, ( ϑ, ϕ ) are polar coordinates, whereas theradial coordinate r is determined by the fact that the two-surface S = { t = t , r = r } has area 4 πr . The region r < M , on the other hand, is the interior region . In thisregion, the radial coordinate r is time, whereas t is a spatial coordinate. Since timealways propagates to the future, the event horizon can be regarded as the “boundary ofno escape.” The surface r = 2 M merely is a coordinate singularity of our metric. Thisbecomes apparent by transforming to Eddington-Finkelstein or Kruskal coordinates.For brevity, we shall not enter the details here.In the case a = 0, the singularity structure is more involved. The function U isalways strictly positive. The function ∆ has the two roots r := M + p M − a event horizon (2.4) r := M − p M − a Cauchy horizon . (2.5)If a > M , these roots are complex. This corresponds to the unphysical situation of anaked singularity. We shall not discuss this case here, but only consider the so-called non-extreme case M < a . In this case, the hypersurface r = r := M + p M − a again defines the event horizon of the black hole. In what follows, we shall restrictattention to the exterior region r > r . This is because classically, no informationcan be transmitted from the interior of the black hole to its exterior. Therefore, it isimpossible for principal reasons to know what happens inside the black hole. With thisin mind, it seems pointless to study the black hole inside the event horizon, becausethis study will never be tested or verified by experiments.We finally remark that in quantum gravity , the situation is quite different becauseit is conceivable that a black hole might “evaporate,” in which case the interior ofthe black hole might become accessible to observations. In physics, such questionsare often discussed in connection with the so-called information paradox, which statesthat the loss of information at the event horizon is not compatible with the unitarytime evolution in quantum theory. I find such questions related to quantum effectsof a black hole quite interesting, and indeed most of my recent research is devoted toquantum gravity (in an approach called causal fermion systems; see for example thetextbook [13] or the survey paper [20]). But since this summer school is devoted toclassical gravity, I shall not enter this topic here.3. Symmetries and Killing Fields
The Kerr geometry is stationary and axisymmetric. This is apparent in Boyer-Lindquist coordinates (2.2) because the metric coefficients areindependent of t : stationaryindependent of ϕ : axisymmetric . These symmetries can be described more abstractly using the notion of
Killing fields .We recall how this works because we need it later for the description of the Pen-rose process and superradiance. We restrict attention to the time translation sym-metry, because for for the axisymmetry or other symmetries, the argument is similar.
F. FINSTER
Given τ ∈ R , we consider the mappingΦ τ : M → M , ( t, x ) ( t + τ, x )(where x stands for the spatial coordinates ( r, ϑ, ϕ )). The fact that the metric coef-ficients are time independent means that Φ τ is an isometry , defined as follows. Thederivative of Φ τ (i.e. the linearization; it is sometimes also denoted by (Φ τ ) ∗ ) is amapping between the corresponding tangent spaces, D Φ τ | p : T p M → T Φ τ ( x ) M . Being an isometry means that g p ( u, v ) = g Φ τ ( p ) (cid:0) D Φ τ | p u, D Φ τ | p v (cid:1) for all u, v ∈ T p M . Let us evaluate this equation infinitesimally in τ . We first introduce the vector field K by K := ddτ Φ τ (cid:12)(cid:12) τ =0 . Choosing local coordinates, we obtain in components (cid:0) D Φ τ | p u (cid:1) a = ∂ Φ aτ ( p ) ∂x i u i , where for clarity we denote the tensor indices at the point Φ τ ( x ) by a and b . We thenobtain 0 = ddτ g Φ τ ( p ) (cid:0) D Φ τ | p u, D Φ τ | p v (cid:1)(cid:12)(cid:12)(cid:12) τ =0 = ddτ (cid:16) g ab (cid:0) Φ τ ( p ) (cid:1) ∂ Φ aτ ( p ) ∂x i u i ∂ Φ bτ ( p ) ∂x j v j (cid:17)(cid:12)(cid:12)(cid:12) τ =0 = ∂ k g ( u, v ) K k + g (cid:0) u i ∂ i K, v (cid:1) + g (cid:0) u, v j ∂ j K (cid:1) . Choosing Gaussian coordinates at p , one sees that this equation can be written covari-antly as 0 = g (cid:0) ∇ u K, v (cid:1) + g (cid:0) u, ∇ v K (cid:1) , where ∇ is the Levi-Civita connection. This is the Killing equation , which can also bewritten in the shorter form0 = ∇ ( i K j ) := 12 (cid:0) ∇ i K j + ∇ j K i (cid:1) . (3.1)A vector field which satisfies the Killing equation is referred to as a Killing field . Weremark that if the flow lines exist on an interval containing zero and τ , then theresulting diffeomorphism Φ τ is indeed an isometry of M .A variant of Noether’s theorem states that Killing symmetries, which describe in-finitesimal symmetries of space-time, give rise to corresponding conservation laws. For geodesics , these conservation laws are obtained simply by taking the Lorentzian innerproduct of the Killing vector field and the velocity vector of the geodesic. Indeed,let γ ( τ ) be a parametrized geodesic, i.e. ∇ τ ˙ γ ( τ ) = 0 . Then, denoting the metric for simplicity by h ., . i p := g p ( ., . ), we obtain ddτ (cid:10) K ( γ ( τ )) , ˙ γ ( τ ) (cid:11) γ ( τ ) = (cid:10) ∇ τ K ( γ ( τ )) , ˙ γ ( τ ) (cid:11) γ ( τ ) + (cid:10) K ( γ ( τ )) , ∇ τ ˙ γ ( τ ) (cid:11) γ ( τ ) = (cid:10) ∇ τ K ( γ ( τ )) , ˙ γ ( τ ) (cid:11) γ ( τ ) = ∇ i K j (cid:12)(cid:12) γ ( τ ) ˙ γ i ( τ ) ˙ γ j ( τ ) = 0 , ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 5 ϑr r r es E in E out ∆ E Figure 1.
Schematic picture of the ergosphere (left) and the Penroseprocess (right).where in the last step we used the Killing equation (3.1). We thus obtain the conser-vation law (cid:10) K ( γ ( τ )) , ˙ γ ( τ ) (cid:11) γ ( τ ) = const , which holds for any parametrized geodesic γ ( τ ) and any Killing field K .4. The Penrose Process and Superradiance
In the Kerr geometry, the two vector fields ∂ t and ∂ ϕ are Killing fields. The corre-sponding conserved quantities are E := (cid:10) ∂∂t , ˙ γ ( τ ) (cid:11) γ ( τ ) energy (4.1) A := (cid:10) ∂∂ϕ , ˙ γ ( τ ) (cid:11) γ ( τ ) angular momentum . (4.2)Let us consider the energy in more detail for a test particle moving along the geodesic γ .In this case, γ ( τ ) is a causal curve (i.e. ˙ γ ( τ ) is timelike or null everywhere), andwe always choose the parametrization such that γ is future-directed (i.e. the timecoordinate γ ( τ ) is monotone increasing in τ ). In the asymptotic end (i.e. for large r ),the Killing field ∂ t is timelike and future-directed. As a consequence, the inner productin (4.1) is strictly positive. This corresponds to the usual concept of the energy beinga non-negative quantity. We point out that this result relies on the assumption thatthe Killing field ∂ t is timelike. However, if this Killing field is spacelike, then the innerproduct in (4.1) could very well be negative. In order to verify if this case occurs, wecompute (cid:10) ∂∂t , ∂∂t (cid:11) = g = ∆ U − a sin ϑU = 1 U (cid:16) r − M r + a cos ϑ (cid:17) , where we read off the corresponding metric coefficient in (2.2) and simplified it us-ing (2.3). Computing the roots, one sees that the Killing field ∂ t indeed becomes nullon the surface r = r es := M + p M − a cos ϑ , (4.3)the so-called ergosphere . Comparing with the formula for the event horizon (2.4), onesees that the ergosphere is outside the event horizon and intersects the event horizonat the poles ϑ = 0 , π (see the left of Figure 1). The region r < r < r es is the so-called ergoregion .The ergosphere causes major difficulties in the proof of linear stability of the Kerrgeometry. These difficulties are not merely technical, but they are related to physicalphenomena, as we now explain step by step. The name ergosphere is motivated fromthe fact that it gives rise to a mechanism for extracting energy from a rotating blackhole. This effect was first observed by Roger Penrose [38] and is therefore referred F. FINSTER to as the
Penrose process . In order to explain this effect, we consider a spaceship ofenergy E in which flies into the ergoregion (see the right of Figure 1), where it ejectsa projectile of energy ∆ E which falls into the black hole. After that, the spaceshipflies out of the ergoregion with energy E out . Due to energy conservation, we knowthat E in = E out + ∆ E . By choosing the momentum of the projectile appropriately,one can arrange that the energy ∆ E is negative. Then the final energy E out is largerthan the initial energy E in , which means that we gained energy. This energy gain doesnot contradict total energy conservation, because one should think of the energy asbeing extracted from the black hole (this could indeed be made precise by taking intoaccount the back reaction of the space ship onto the black hole, but we do not havetime for entering such computations). Therefore, the Penrose process is similar to theso-called “swing-by” or “gravitational slingshot,” where a satellite flies close to a planetof our solar system and uses the kinetic energy of the planet for its own acceleration.The surprising effect is that in the Penrose process, one can extract energy from theblack hole, although the matter of the black hole is trapped behind the event horizon.The wave analogue of the Penrose process is called superradiance . Instead of thespaceship one considers a wave packet flying in the direction of the black hole. Thewave propagates as described by a corresponding wave equation (we will see such waveequations in more detail later). As a consequence, part of the wave will “fall into” theblack hole, whereas the remainder will pass the black hole and will eventually leavethe black hole region. If the energy of the outgoing wave is larger than the energy ofthe oncoming wave, then one speaks of superradiant scattering. This effect is quitesimilar to the Penrose process. However, one major difference is that, in contrast tothe Penrose process, there is no freedom in choosing the momentum of the projectile.Instead, the dynamics is determined completely by the initial data, so that the onlyfreedom is to prepare the incoming wave packet. As we shall see later in this lecture,superradiance indeed occurs for scalar waves in the Kerr geometry.5. The Scalar Wave Equation in the Kerr Geometry
In preparation of the analysis of general linear wave equations, we begin with thesimplest example: the scalar wave equation. It has the useful property that it is ofvariational form, meaning that it can be derived from an action principle. Indeed,choosing the Dirichlet action S = ˆ M g ij ( ∂ i φ ) ( ∂ j φ ) dµ M , (where dµ M = p | det g | d x is the volume measure induced by the Lorentzian metric),demanding criticality for first variations gives the scalar wave equation0 = (cid:3) φ := ∇ i ∇ i φ . The main advantage of the variational formulation is that Noether’s theorem relatessymmetries to conservation laws. Another method for getting these conservation laws,which is preferable to us because it is closely related to the notion of Killing fields,is to work directly with the energy-momentum tensor of the field. Recall that inthe Einstein equations (2.1), the Einstein tensor on the left is divergence-free as aconsequence of the second Bianchi identities. Therefore, the energy-momentum tensoris also divergence-free, ∇ i T ij = 0 . (5.1) ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 7 r N Ω N ν ν Figure 2.
Conservation law corresponding to a Killing symmetry.Now let K be a Killing field. Contracting the energy-momentum tensor with theKilling field gives a vector field, u i := T ij K j . The calculation ∇ i u i = (cid:0) ∇ i T ij (cid:1) K j + T ij ∇ i K j = 0(where the first summand vanishes according to the conservation law (5.1), whereasthe second summand is zero in view of the Killing equation (3.1) and the symmetry ofthe energy-momentum tensor) shows that this vector field is divergence-free. There-fore, integrating the divergence of u over a space-time region Ω and using the Gaußdivergence theorem, we conclude that the flux integral of u through the surface ∂ Ωvanishes. The situation we have in mind is that the set Ω is the region between twospacelike hypersurfaces N and N (see Figure 2). Assuming that the vector field u hassuitable decay properties at spatial infinity (in the simplest case that it has spatiallycompact support), we obtain the conservation law0 = ˆ Ω ∇ i u i dµ M = ˆ N T ij ν i K j dµ N − ˆ N T ij ν i K j dµ N , (5.2)where ν is the future-directed normal on N / and dµ N / is the volume measure cor-responding to the induced Riemannian metric.In the Kerr geometry in Boyer-Lindquist coordinates, the Dirichlet action takes theexplicit form S = ˆ ∞−∞ dt ˆ ∞ r dr ˆ − d (cos ϑ ) ˆ π dϕ L ( φ, ∇ φ )with L ( φ, ∇ φ ) = − ∆ | ∂ r φ | + 1∆ (cid:12)(cid:12) (( r + a ) ∂ t + a∂ ϕ ) φ (cid:12)(cid:12) − sin ϑ | ∂ cos ϑ φ | − ϑ (cid:12)(cid:12) ( a sin ϑ∂ t + ∂ ϕ ) φ (cid:12)(cid:12) . Considering first variations, the scalar wave equation becomes " ∂∂r ∆ ∂∂r − (cid:26) ( r + a ) ∂∂t + a ∂∂ϕ (cid:27) + ∂∂ cos ϑ sin ϑ ∂∂ cos ϑ + 1sin ϑ (cid:26) a sin ϑ ∂∂t + ∂∂ϕ (cid:27) φ = 0 . (5.3)Using the formula for the energy-momentum tensor T ij = ( ∂ i φ )( ∂ j φ ) −
12 ( ∂ k φ ) ( ∂ k φ ) g ij , F. FINSTER rr r rt t reflecting sphere∆ E ∆ EE in E out E in E out E ∞ Figure 3.
The black hole bomb (left) and wave propagation in theKerr geometry (right).the conserved energy becomes E := ˆ N t T ij ν j ( ∂ t ) j dµ N t = ˆ N t T i ( ∂ t ) j dµ N t (5.4)= ˆ ∞ r dr ˆ − d (cos ϑ ) ˆ π dϕ E (5.5)with the “energy density” E = (cid:18) ( r + a ) ∆ − a sin ϑ (cid:19) | ∂ t φ | + ∆ | ∂ r φ | + sin ϑ | ∂ cos ϑ φ | + (cid:18) ϑ − a ∆ (cid:19) | ∂ ϕ φ | . Using (2.3), one sees that the factor in front of the term | ∂ ϕ φ | is everywhere positive.However, the factor in front of the term | ∂ t φ | is negative precisely inside the ergo-sphere (4.3). This consideration shows that, exactly as for point particles (4.1), theenergy of scalar waves may again be negative inside the ergosphere.What does the indefiniteness of the energy tell us? We first point out that it does not imply that superradiance really occurs, because in order to analyze superradiance, onemust study the dynamics of waves. Instead, it only means that there is a possibilityfor superradiance to occur. In technical terms, the indefiniteness of the energy leads tothe difficulty that energy conservation does not give us control of the Sobolev norm ofthe wave. A possible scenario, which does not contradict energy conservation, is thatthe amplitude of the wave grows in time both inside and outside the the ergosphere.It is a major task in proving linear stability to rule out this scenario.The basic difficulty can be understood qualitatively in more detail in the scenario ofthe so-called black hole bomb as introduced by Press and Teukolsky [39] and studied byCardoso et al [5]. In this gedanken experiment, one puts a metal sphere around a Kerrblack hole (as shown schematically on the left of Figure 3. We consider a wave packetof energy E in inside the sphere flying towards the black hole. Part of the wave willcross the event horizon, while the remainder will pass the black hole. As in the abovedescription of superradiance, we assume that the energy ∆ E of the wave crossing theevent horizon is negative. Then the energy E out of the outgoing wave is larger than the ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 9 energy E in of the incoming wave. The outgoing wave is reflected on the metal sphere,becoming a new wave which again flies towards the black hole. If it can be arrangedthat the new incoming wave has the same shape as the original wave, this processrepeats itself, generating in each step a certain positive energy. In this scenario, theenergy density inside the metal sphere would grow exponentially fast in time. Whenthe energy density gets too large, the metal sphere would explode, explaining thename “black hole bomb.” For clarity, we point out that in this mechanism one alwaysassumes that the total energy extracted from the black hole is much smaller than thetotal rotational energy of the black hole, so that the back reaction on the black holeneed not be taken into account.The black hole bomb suggests that, putting a metal sphere around the black holecould lead to an instability, which would become manifest in an explosion of the metalsphere. The point of interest in connection to the stability problem for rotating blackholes is that a very similar scenario might occur even without the metal sphere: Weagain consider a wave packet flying towards the black hole. Again, part of the wavewith energy ∆ E crosses the event horizon, whereas the remainder of energy E out passesthe black hole. The point is that only part of this wave will reach null infinity. Anotherpart will be backscattered by the gravitational field and will again fly towards the blackhole. Therefore, except for the “energy loss” E ∞ by the part of the wave propagatingto null infinity, we are again in the scenario of the black hole bomb where the processrepeats itself, potentially leading to an exponential increase in time of the amplitudeof the wave.Clearly, this picture is oversimplified because, instead of wave packets, one mustconsider waves which are spread out in space, leading to a nonlocal problem. Never-theless, in this picture it becomes clear why the problem of linear stability of rotatingblack holes amounts to a quantitative question: Can the initial wave packet be ar-ranged such that the “energy gain” − ∆ E is larger than the “energy loss” E ∞ ? Ifthe answer is yes, a rotating black hole should be unstable, and it should decay byradiation of gravitational waves to a Schwarzschild black hole. It is the main goal ofthese lectures to explain why this does not happen, i.e. why rotating black holes arelinearly stable. Before we can enter this problem in mathematical detail, we need tointroduce linear wave equations and review a few structural results.6. An Overview of Linear Wave Equations in the Kerr Geometry
The Dirac Equation.
In these lectures I shall not enter the details of the Diracequation, although most of my work has been concerned with or related to the Diracequation. I only want to explain why the analysis for the Dirac equation is much easier than for other wave equations.The Dirac equation describes a relativistic quantum mechanical particle with spin.In order to keep the setting as simple as possible, we work in coordinates and localtrivializations of the spinor bundle (which has the advantage that we do not need toeven define what the spinor bundle is). Then the Dirac wave function ψ ( x ) ∈ C hasfour complex components, which describe the spinorial degrees of freedom of the wavefunction. The Dirac equation reads (cid:0) iγ j ( x ) ∂ j + B − m (cid:1) ψ = 0 . Here m is the rest mass of the Dirac particles, and the four matrices γ j encode theLorentzian metric via the anti-commutation relations (cid:8) γ j ( x ) , γ k ( x ) (cid:9) = 2 g jk ( x ) 11 C , where the anti-commutator is defined by (cid:8) γ j , γ k (cid:9) := γ j γ k + γ k γ j . The multiplication operator B involves the so-called spin coefficients, which, in analogyto the Christoffel symbols of the Levi-Civita connection, are formed of first partialderivatives of the Dirac matrices. We do not need the details here and refer instead tothe explicit formulas in [12] or [21, Chapter 3].Coming from quantum mechanics, the Dirac equation has additional structureswhich allow for the probabilistic interpretation of the wave function. In particular,there is a quantity which can be interpreted as the probability density as seen by anobserver, and the spatial integral of this probability density is equal to one, for anyfixed time of the observer. This probability integral is described mathematically asfollows. The spinors at a space-time point x ∈ M are endowed with an indefinite innerproduct of signature (2 , ≺ ψ | ψ ≻ x . For any solution ψ of theDirac equation, the pointwise expectation value of the Dirac matrices with respect tothis inner product defines a vector field J k ( x ) := ≺ ψ ( x ) | γ k ψ ( x ) ≻ x . This vector field is the so-called
Dirac current . The structure of the Dirac equationensures that this vector field is always non-spacelike and future-directed. Moreover,as a consequence of the Dirac equation, this vector field is divergence-free, i.e. ∇ k J k ( x ) = 0(where ∇ is again the Levi-Civita connection); this is referred to as current conserva-tion . Integrating this equation over a region Ω between two spacelike hypersurfaces(as shown in Figure 2), one obtains the conservation law ˆ N J i ν i dµ N = ˆ N J i ν i dµ N (6.1)(here we again assume that the Dirac wave function has suitable decay propertiesat spatial infinity). In view of this conservation law and the linearity of the Diracequation, one can normalize the Dirac solutions such that the integral in (6.1) equalsone. Then the integrand in (6.1) has the interpretation as the probability densityfor an observer for whom the spacelike hypersurface N (or N ) describes space. Wepoint out that the probability density is non-negative simply as a consequence of thefact that the current vector is non-spacelike and future-directed, and that the normalis timelike and future-directed. In particular, the probability density is non-negativeeven inside the ergosphere; note also that, in contrast to (5.2), the integrand in (6.1)does not involve a Killing field.These structures coming from the probabilistic interpretation of the Dirac equationare a major simplification for analyzing the long-time dynamics of Dirac waves inthe Kerr geometry. Namely, the current integral (6.1) can be used to define a scalarproduct on the solutions of the Dirac equation by( ψ | φ ) t := ˆ N t ≺ ψ | γ j φ ≻ x ν j dµ N t , ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 11 where N t is the surface of constant time in Boyer-Lindquist coordinates outside theevent horizon, and where we restrict attention to Dirac solutions with suitable decayproperties on the event horizon and near spatial infinity (for example wave functionswith spatially compact support outside the event horizon). Taking the completion,we obtain the Hilbert space ( H t , ( . | . ) t ) of Dirac solutions. The conservation law (6.1)means that the time evolution operator U t,t : H t → H t from time t to time t is aunitary operator. Since the Kerr geometry is stationary, we can canonically identifythe Hilbert space H t with H t by time translation of the wave functions. Moreover,the unitary time evolution can be written as U t,t = e − i ( t − t ) H , (6.2)where the so-called Dirac Hamiltonian H is a self-adjoint operator on the Hilbert space(the self-adjointness extension can be constructed in general using Stone’s theorem orChernoff’s method [8]). In this way, the long-time dynamics can be related to spectralproperties of a self-adjoint operator on a Hilbert space . No superradiance phenomenaoccur.More details on the Dirac equation and the above method can be found in my jointpapers with Niky Kamran, Joel Smoller and Shing-Tung Yau [14, 16, 15]. For thegeneral method of constructing self-adjoint extensions, the more recent paper [22] maybe useful.6.2. Massless Equations of General Spin, the Teukolsky Equation.
After theshort excursion to quantum mechanics, we now return to classical waves . The wavesof interest are scalar waveselectromagnetic wavesgravitational waves . Scalar waves were already considered in Section 5; they are studied mainly because oftheir mathematical simplicity. The waves of physical interest are electromagnetic andgravitational waves. Note that all these waves are massless .All the above wave equations can be described in a unified framework due to Teukol-sky [43]. We now explain schematically how the Teukolsky formulation works. Sincethe involved computations are quite lengthy, we cannot enter the details but refer in-stead to the textbook [7]. The Teukolsky equation is derived in the
Newman-Penroseformalism , which we now briefly introduce. In this formalism, one works with a doublenull frame , i.e. with a set of vectors ( l, n, m, m ) of the complexified tangent space withinner products h l, n i = 1 , h m, m i = − , whereas all other inner products vanish, h l, l i = h n, n i = h m, m i = h m, m i = 0 . In the example of Minkowski space in Cartesian coordinates ( t, x, y, z ), one can choose l and n as two null vectors, for example l = 1 √ (cid:16) ∂∂t + ∂∂x (cid:17) and n = 1 √ (cid:16) ∂∂t − ∂∂x (cid:17) . The orthogonal complement of these two vectors is the two-dimensional spacelike planespanned by the vectors ∂ y and ∂ z . Therefore, the only way to obtain additional null vectors is to complexify by choosing for example m = 1 √ (cid:16) ∂∂y + i ∂∂z (cid:17) and m = 1 √ (cid:16) ∂∂y − i ∂∂z (cid:17) . Likewise, on a Lorentzian manifold, the vectors ( l, n, m, m ) form a basis of the complex-ified tangent space. The Lorentzian inner product h ., . i is extended to the complexifiedtangent space as a bilinear form (not sesquilinear; thus no complex conjugation is in-volved). The double null frame is well-suited for the analysis of the vacuum Einsteinequations (indeed, the Kerr solution was discovered in the Newman-Penrose formal-ism).Next, one combines the tensor components in the double null frame in complex-valued functions. In the example of the Maxwell field , this works as follows. Theelectromagnetic field tensor F ij has six real components (three electric and three mag-netic field components). One combines these six real components to the three complexfunctions Ψ = F lm , Ψ = 12 (cid:0) F ln + F mm (cid:1) , Ψ = F nm . (6.3)Then the homogeneous Maxwell equations dF = 0 and ∇ k F jk = 0give rise to a first-order system of partial differential equations for Ψ = (Ψ , Ψ , Ψ ).For the gravitational field , one considers similarly the Weyl tensor C ijkl . Linearizedgravitational waves are described by linear perturbations of the Weyl tensor in theNewman-Penrose frame. Denoting the linear perturbation of the Weyl tensor by W ,its ten real components are combined to the five complex functionsΨ = − W lmlm , Ψ = − W lnlm , Ψ = − W lmmn Ψ = − W lnmn , Ψ = − W lmnm . Linearizing the second Bianchi identities R ij ( kl ; m ) = 0 gives a first-order system ofpartial differential equations for Ψ = (Ψ , . . . , Ψ ). In this formulation, the connectionto the spin can be obtained simply by counting the number of degrees of freedom.In quantum mechanics, a wave function of spin s has 2 s + 1 complex components.Therefore, we obtain the correct number of degrees of freedom if we setelectromagnetic waves: spin s = 1gravitational waves: spin s = 2 . We remark that the connection to the spin is more profound than merely counting thenumber of degrees of freedom, but we have no time to explain how this works. For ourpurposes, it suffices to take the spin s as a parameter which characterizes the masslesswave equation by counting the number of components of the Newman-Penrose wavefunction Ψ = (Ψ , . . . , Ψ s ).We write the first-order system of partial differential equations for electromagneticwaves or linearized gravitational waves symbolically as D Ψ ...Ψ s = 0 . (6.4)Working with this first-order system is not convenient for larger spin because thenumber of equations gets large, and the equations are coupled in a complicated way. ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 13
But, as discovered by Teukolsky, the system of equations can be decoupled such asto obtain a second-order partial differential equation for one complex-valued function.This decoupling works schematically as follows: One chooses a Newman-Penrose nullframe where l and n are aligned with the repeated principal null directions of the Weyltensor (in this frame, the Newman-Penrose components of the Weyl tensor satisfy theequations ψ = ψ = ψ = ψ = 0, with ψ being the only non-zero component).Multiplying the linear first-order system (6.4) in this frame by a suitable first-orderdifferential operator D , we obtain the equation0 = D D Ψ ...Ψ s = T · · · ∗ ∗ · · · ∗ ∗ ... ... . . . ... ... ∗ ∗ · · · ∗ ∗ · · · T s Ψ Ψ ...Ψ s − Ψ s , where the stars stand for differential operators which we do not need to specify here.The point is that this procedure generates zeros in the first and last row of the matrix,giving rise to decoupled equations for the first and and last components of the Newman-Penrose wave function Ψ, T Ψ = 0 = T s Ψ s . (6.5)Once the solution Ψ or Ψ s is known, all the other components of Ψ can be obtainedby employing the so-called Teukolsky-Starobinsky identities , which have similarities tothe “ladder operator” for the harmonic oscillator used for obtaining the excited statesfrom the ground state. With this in mind, in what follows we restrict attention to theequations for Ψ or Ψ s in (6.5). After detailed computations for the electromagneticfield and for linearized gravitational fields, in both cases one ends up with the sameequation, except for a parameter s describing the spin. We thus obtain the Teukolskyequation (sometimes called Teukolsky master equation; we use the form of the equationas given in [45]) (cid:18) ∂∂r ∆ ∂∂r − (cid:26) ( r + a ) ∂∂t + a ∂∂ϕ − ( r − M ) s (cid:27) − s ( r + ia cos ϑ ) ∂∂t + ∂∂ cos ϑ sin ϑ ∂∂ cos ϑ + 1sin ϑ (cid:26) a sin ϑ ∂∂t + ∂∂ϕ + is cos ϑ (cid:27) (cid:19) φ = 0 . (6.6)For s = 1, this equation describes the first component Ψ of the Newman-Penrose wavefunction (Ψ , Ψ , Ψ ) for electromagnetic waves, whereas the parameter value s = − . Likewise, setting s = 2 gives the first component Ψ of theNewman-Penrose wave function (Ψ , . . . , Ψ ) for gravitational waves, whereas s = − . By direct inspection, one sees that setting s = 0 gives backthe scalar wave equation (5.3). We remark that setting s = gives the massless Diracequation [43], and s = gives the massless Rarita-Schwinger equation [32].We close with a remark on gravitational perturbations. As outlined above, ourmethod is to consider perturbations of the Weyl tensor. Alternatively, one could con-sider perturbations of the metric (indeed, this was historically the first approach, goingback to the stability analysis by Regge and Wheeler [40]). Working with metric per-turbations has the disadvantage that infinitesimal coordinate transformations also leadto perturbations of the metric, which however have no geometric significance. In otherwords, when working with metric perturbations, the diffeomorphism invariance leads to a gauge freedom which is not easy to handle. This was the original motivation forTeukolsky and Press to consider instead perturbations of geometric quantities like theNewman-Penrose components of the Weyl tensor, leading to the Teukolsky framework.However, for some applications (for example in order to include matter models or todescribe nonlinear waves) it is necessary to work with metric perturbations. Therefore,working in the Teukolsky formulation, the following question remains: Given a linearperturbation of the Weyl tensor, how can it be realized by a metric perturbation? Thisis an interesting and in general difficult question which we cannot analyze here (seehowever [46] and the references therein).7. Separation of the Teukolsky Equation
The Teukolsky equation (6.6) has the remarkable property that it can be com-pletely separated into a system of ordinary differential equations (ODEs): Due to thestationarity and axisymmetry, we can separate the t - and ϕ -dependence with the usualplane-wave ansatz φ ( t, r, ϑ, ϕ ) = e − iωt − ikϕ φ ( r, ϑ ) , (7.1)where ω is a quantum number which could be real or complex and which correspondsto the “energy”, and k ∈ Z / s is half an oddinteger, then so is k ). Substituting (7.1) into (6.6), we see that the Teukolsky operatorsplits into the sum of radial and angular parts, giving rise to the equation( R ω,k + A ω,k ) φ = 0 , where R ω,k and A ω,k are given by (for details see [30, Section 6]) R ω = − ∂∂r ∆ ∂∂r − (cid:16) ω ( r + a ) + ak − i ( r − M ) s (cid:17) − isrω + 4 k aω (7.2) A ω = − ∂∂ cos ϑ sin ϑ ∂∂ cos ϑ + 1sin ϑ (cid:16) − aω sin ϑ + k − s cos ϑ (cid:17) . (7.3)We can therefore separate the variables r and ϑ with the multiplicative ansatz φ ( r, ϑ ) = R ( r ) Θ( ϑ ) , (7.4)to obtain for given ω and k the system of ODEs R ω,k R λ = − λ R λ , A ω,k Θ λ = λ Θ λ . (7.5)Solutions of the coupled system (7.5) are referred to as mode solutions .We point out that the last separation (7.4) is not obvious because it does not cor-respond to an underlying space-time symmetry. Instead, as discovered by Carter forthe scalar wave equation [6], it corresponds to the fact that in the Kerr geometrythere exists an irreducible quadratic Killing tensor (i.e. a Killing tensor which is not asymmetrized tensor product of Killing vectors). The separation constant λ is an eigen-value of the angular operator A ω,k and can thus be thought of as an angular quantumnumber. In the spherically symmetric case a = 0, this separation constant goes overto the usual eigenvalues λ = l ( l + 1) of the total angular momentum operator. ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 15 Results on Linear Stability and Superradiance
Being familiar with the structure of the different linear wave equations, we cannow state our results on stability and superradiance. The problem of linear stability of black holes amounts to the question whether solutions of the corresponding linearwave equations decay for large times. In order to put our results into context, wepoint out that the problem of linear stability of black holes has a long history. It goesback to the study of the Schwarzschild black hole by Regge and Wheeler [40] whoshowed that an integral norm of the perturbation of each angular mode is boundeduniformly in time. Decay of these perturbations was first proved in [31]. More detailedestimates of metric perturbations in Schwarzschild were obtained in [9, 34]. For theKerr black hole, linear stability under perturbations of general spin has been an openproblem for many years, which was solved in the dynamical setting in [30] (for relatedresults obtained with different methods see [35, 3, 2, 10] and the references in thesepapers). A key ingredient to our proof is the so-called mode stability result obtained byWhiting [45], who proved that the Teukolsky equation does not admit solutions whichdecay both at spatial infinity and at the event horizon and increase exponentially intime.We consider the Cauchy problem for the Teukolsky equation. Thus we seek a solu-tion φ of the Teukolsky equation (6.6) for given initial data φ | t =0 = φ and ∂ t φ | t =0 = φ . Being a linear hyperbolic PDE, the Cauchy problem for the Teukolsky equation hasunique global solutions. Also, taking smooth initial data, the solution is smooth forall times. Our task is to show that solutions decay for large times. In order to avoidspecifying decay assumptions at the event horizon and at spatial infinity, we restrictattention to compactly supported initial data outside the event horizon, φ , φ ∈ C ∞ (cid:0) ( r , ∞ ) × S (cid:1) . (8.1)Since the Kerr geometry is axisymmetric, the Teukolsky equation decouples intoseparate equations for each azimuthal mode. Therefore, the solution of the Cauchyproblem is obtained by solving the Cauchy problem for each azimuthal mode andtaking the sum of the resulting solutions. With this in mind, we restrict attention tothe Cauchy problem for a single azimuthal mode, i.e. φ ( r, ϑ, ϕ ) = e − ikϕ φ ( k )0 ( r, ϑ ) , φ ( r, ϑ, ϕ ) = e − ikϕ φ ( k )1 ( r, ϑ ) (8.2)for given k ∈ Z /
2. The main result of [30] is stated as follows:
Theorem 8.1.
Consider a non-extreme Kerr black hole of mass M and angular mo-mentum aM with M > a > . Then for any s ≥ and any k ∈ Z / , the solutionof the Teukolsky equation with initial data of the form (8.1) and (8.2) decays to zeroin L ∞ loc (( r , ∞ ) × S ) . This theorem establishes in the dynamical setting that the non-extreme Kerr blackhole is linearly stable.Our method of proof uses an integral representation of the time evolution operatorinvolving the radial and angular solutions of the separated system of ODEs (7.5).Such an integral representation was derived earlier for the scalar wave equation in [17],and it was used for proving decay in time [18]. Moreover, in [19] it was proven inthe dynamical setting that superradiance occurs for scalar waves. We now explain this result. Superradiance for scalar waves in the Kerr geometry was first studiedby Zel’dovich and Starobinsky [47, 41] on the level of modes. More precisely, theycomputed the transmission and reflection coefficients for the radial ODE in (7.5). Theabsolute value squared of these coefficients can be interpreted as the energy flux of theincoming and outgoing waves, respectively. Comparing these fluxes, one obtains therelative energy gain. Starobinsky computed the relative gain of energy to about 5%for k = 1 and less than 1% for k ≥ ◮ The Teukolsky equation for s = 0 is not of variational form , i.e. it cannot beobtained as the Euler-Lagrange equation of an action. ◮ As a consequence, we cannot apply Noether’s theorem to obtain conserved quan-tities. In particular, there is no conserved energy , being an integral of an energydensity. This means that, in contrast to the situation described for the Diracequation in Section 6.1, the time evolution cannot be described by a unitary op-erator on a Hilbert space. As a consequence, we cannot use the spectral theoremfor selfadjoint or unitary operators on Hilbert spaces. ◮ A related difficulty is that the coefficients of the first derivative terms in theTeukolsky equation for s = 0 are complex . Such complex potentials in a waveequation usually describe dissipation, implying that (depending of the sign of thedissipation terms) the solutions typically decay or increase exponentially in time.This means that, in order to show that the solution of the Teukolsky equationdecays for large times, one must carefully control the signs and the size of thecomplex coefficients by quantitative estimates. ◮ In the separation of variables (7.5), both the radial and angular differential opera-tors R ω,k and A ω,k depend on the separation constants k and ω . As a consequence,it is not at all obvious if and how for given initial data one can decompose thecorresponding solution of the Cauchy problem into a superposition of mode so-lutions. An obvious difficulty is that, for such a mode decomposition , one wouldhave to know the separation constant ω , which in turn can be specified only if wealready know the full dynamics of the wave.9. Hamiltonian Formulation and Integral Representations
In order to analyze the dynamics of the Teukolsky wave, it is useful to work withcontour integrals of the resolvent of the Hamiltonian, as we now outline. In prepara-tion, we must rewrite the Teukolsky equation in Hamiltonian form. To this end, weintroduce the two-component wave functionΨ = p r + a (cid:18) φi∂ t φ (cid:19) and write the Teukolsky equation as i∂ t Ψ = H Ψ , (9.1) ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 17 where H is a second-order spatial differential operator. We consider H as an operatoron a Hilbert space H with the domain D ( H ) = C ∞ (cid:0) ( r , ∞ ) × S , C (cid:1) . It would be desirable to represent H as a self-adjoint operator on a Hilbert space H ,because it would then be possible to apply the spectral calculus and write the timeevolution operator similar as for the Dirac equation in the form (6.2). Unfortunately,this procedure does not work here, as can be understood as follows. As already men-tioned at the end of the previous section, the Teukolsky equation is not of variationalform, implying that there is no conserved energy. If there were a conserved bilinearform h Ψ | Φ i on the solutions, then the calculation0 = ∂ t h Ψ | Φ i = h ˙Ψ | Φ i + h Ψ | ˙Φ i = i (cid:0) h H Ψ | Φ i − h Ψ | H Φ i (cid:1) would imply that the Hamiltonian were symmetric with respect to this bilinear form.But, having no conserved energy, there is also no bilinear form with respect to whichthe Hamiltonian is symmetric. In order to avoid confusion, we remark that there is aconserved physical energy, which in the example of a Maxwell field could be written inthe form (5.4) with T ij the energy-momentum tensor of the Maxwell field. However,this energy involves all the components of the field tensor or, in other words, allthe components of the Newman-Penrose wave function in (6.3). Since the Teukolskyequation only gives Ψ or Ψ , we would have to compute the other components usingthe Teukolsky-Starobinsky identities. As a consequence, the resulting formula forthe Maxwell energy would involve higher derivatives of the Teukolsky wave function,making the situation very complicated. This is why we decided not to use the physicalenergy in our construction.We conclude that we shall treat the operator H as a non-symmetric operator on aHilbert space. In order to get an idea for how to work with non-symmetric operators,it is helpful get a motivation from the finite-dimensional setting. Thus let A be a linearoperator on a finite-dimensional Hilbert space H . Clearly, this operator need not bediagonalizable, because Jordan chains may form. Nevertheless, one can get a spectralcalculus by working with contour integrals: Lemma 9.1.
Let A be a linear operator A on a Hilbert space H of dimension n < ∞ .Then e − itA = − πi ‰ Γ e − iωt (cid:0) A − ω ) − dω , (9.2) where Γ is a contour which encloses the whole spectrum of A with winding numberone.Proof. If A is diagonalizable, we can choose a basis where A is diagonal, A = diag( λ , . . . , λ n ) . In this case, (9.2) is obtained immediately by carrying out the contour integral foreach matrix entry with the help of the Cauchy integral formula.The case that A is not diagonalizable can be obtained by approximation, notingthat the diagonalizable matrices are dense and that both sides of (9.2) are continuouson the space of matrices (endowed with the topology of C n · n ). (cid:3) Motivated by this formula for matrices, we can hope that the Cauchy problem for theequation (9.1) with initial data Ψ could be solved with the Cauchy integral formula by Ψ( t ) = − πi ‰ Γ e − iωt (cid:0) H − ω ) − Ψ dω , (9.3)where Γ is a contour which encloses all eigenvalues of H (note that this formula holdsfor any matrix H , even if it is not diagonalizable). It turns out that in our infinite-dimensional setting, this formula indeed holds. The first step in making sense of thisformula is to localize the spectrum of H and to make sure that the resolvent existsalong the integration contour. To this end, we choose the scalar product on H as asuitable weighted Sobolev scalar product in such a way that that the operator H − H ∗ is bounded, i.e. k H − H ∗ k ≤ c c >
0. Then we prove that the resolvent R ω := ( H − ω ) − exists if ω lies outside a strip enclosing the real axis (see [30, Lemma 4.1]): Lemma 9.2.
For every ω with | Im ω | > c , the resolvent R ω = ( H − ω ) − exists and is bounded by k R ω k ≤ | Im ω | − c . When forming contour integrals, one must always make sure to stay outside thestrip | Im ω | ≤ c , making it impossible to work with closed contours enclosing the spec-trum. But we can work with unbounded contours as follows (see [30, Corollary 5.3]): Proposition 9.3.
For any integer p ≥ , the solution of the Cauchy problem for theTeukolsky equation with initial data Ψ | t =0 = Ψ ∈ D ( H ) has the representation Ψ( t ) = − πi ˆ C e − iωt ω + 3 ic ) p (cid:16) R ω (cid:0) H + 3 ic (cid:1) p Ψ (cid:17) dω , (9.4) where C is the contour C = (cid:8) ω (cid:12)(cid:12) Im ω = 2 c (cid:9) ∪ (cid:8) ω (cid:12)(cid:12) Im ω = − c (cid:9) (9.5) with counter-clockwise orientation. Here the factor ( ω + 3 ic ) − p gives suitable decay for large | ω | and ensures that theintegral converges in the Hilbert space H .The representation (9.4) gives an explicit solution of the Cauchy problem in termsof a Cauchy integral of the resolvent. Unfortunately, this representation does notimmediately give information on the long-time dynamics of the Teukolsky wave. Thisshortcoming can be understood immediately from the fact that the factor e − iωt in theintegrand increases exponentially for large times because | e − iωt | = e Im ωt = e ± ct . Inorder to bypass this shortcoming, our strategy is to move the contour onto the realaxis. Once this has been accomplished, the integral representation (9.4) simplifies toa Fourier transform, Ψ( t ) = ˆ ∞−∞ e − iωt ˆΨ( ω ) dω . The decay of such a Fourier transform can be obtained from the
Riemann-Lebesguelemma , stating that ˆΨ ∈ L ( R , dω ) = ⇒ lim t →±∞ Ψ( t ) = 0 ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 19 (where the wave functions are evaluated pointwise in space). One of the difficulties inmaking this strategy work is to prove that the contour can indeed be moved onto thereal axis. This makes it necessary to show that the Hamiltonian has no spectrum awayfrom the real axis. We did not succeed in proving this result using operator theoreticmethods. Instead, our method is to first make use of the separation of variables,making it possible rule out the spectrum in the complex plane using Whiting’s modestability result [45].10.
A Spectral Decomposition of the Angular Teukolsky Operator
Following the strategy we just outlined, our next task is to employ the separationof variables in the integrand of the integral representation (9.4). Regarding the an-gular equation (7.5) as an eigenvalue equation, we are led to considering the angularoperator A ω in (7.3) as an operator on the Hilbert space H k := L ( S ) ∩ { e − ikϕ Θ( ϑ ) | Θ : (0 , π ) → C } with dense domain D ( A ω ) = C ∞ ( S ) ∩ H k . Unfortunately, the parameter ω is not realbut lies on the contour (9.5). As a consequence, the operator A ω is not symmetric,because its adjoint is given by A ∗ ω = A ω = A ω . The operator A ω is not even a normal operator, making it impossible to apply thespectral theorem in Hilbert spaces. Indeed, A ω does not need to be diagonalizable,because there might be Jordan chains. On the other hand, in order to make use ofthe separation of variables, we must decompose the initial data into angular modes.This can be achieved by decomposing the angular operator into invariant subspaces ofbounded dimension, as is made precise in the following theorem (see [28, Theorem 1.1]): Theorem 10.1.
Let U ⊂ C be the strip | Im ω | < c . Then there is a positive integer N and a family of bounded linear operators Q ωn on H k defined for all n ∈ N ∪ { } and ω ∈ U with the following properties: (i) The image of the operator Q ω is an N -dimensional invariant subspace of A k . (ii) For every n ≥ , the image of the operator Q ωn is an at most two-dimensionalinvariant subspace of A k . (iii) The Q ωn are uniformly bounded in L( H k ) , i.e. for all n ∈ N ∪ { } and ω ∈ U , k Q ωn k ≤ c for a suitable constant c = c ( s, k, c ) (here k · k denotes the sup -norm on H k ). (iv) The Q ωn are idempotent and mutually orthogonal in the sense that Q ωn Q ωn ′ = δ n,n ′ Q ωn for all n, n ′ ∈ N ∪ { } . (v) The Q ωn are complete in the sense that for every ω ∈ U , ∞ X n =0 Q ωn = 11 (10.1) with strong convergence of the series. Invariant Disk Estimates for the Complex Riccati Equation
In order to locate the spectrum of A ω , we use detailed ODE estimates. The opera-tors Q ωn are then obtained similar to (9.3) as Cauchy integrals, Q ωn := − πi ‰ Γ n s λ dλ , n ∈ N , where the contour Γ n encloses the corresponding spectral points, and s λ = ( A ω − λ ) − is the resolvent of the angular operator. What makes the analysis doable is the factthat A ω is an ordinary differential operator. Transforming the angular equation in (7.5)into Sturm-Liouville form (cid:18) − d du + V ( u ) (cid:19) φ = 0 , (11.1)(where u = ϑ and V ∈ C ∞ ((0 , π ) , C ) is a complex potential), the resolvent s λ can berepresented as an integral operator whose kernel is given explicitly in terms of suitablefundamental solutions φ D L and φ D R , s λ ( u, u ′ ) = 1 w ( φ D L , φ D R ) × ( φ D L ( u ) φ D R ( u ′ ) if u ≤ u ′ φ D L ( u ′ ) φ D R ( u ) if u ′ < u , (11.2)where w ( φ D L , φ D R ) denotes the Wronskian.The main task is to find good approximations for the solutions of the Sturm-Liouvilleequation (11.1) with rigorous error bounds which must be uniform in the parameters ω and λ . These approximations are obtained by “glueing together” suitable WKB, Airyand parabolic cylinder functions. The needed properties of these special functions arederived in [26]. In order to obtain error estimates, we combine several methods:(a) Osculating circle estimates (see [28, Section 6])(b) The T -method (see [27, Section 3.2])(c) The κ -method (see [27, Section 3.3])The method (a) is needed in order to separate the spectral points of A ω (gap estimates).The methods (b) and (c) are particular versions of invariant disk estimates as derivedfor complex potentials in [25] (based on previous estimates for real potentials in [23]and [18]). These estimates are also needed for the analysis of the radial equation, seeSection 12 below. We now explain the basic idea behind the invariant disk estimates.Let φ be a solution of the Sturm-Liouville equation (11.1) with a complex poten-tial V . Then the function y defined by y = φ ′ φ is a solution of the Riccati equation y ′ = V − y . (11.3)Conversely, given a solution y of the Riccati equation, a corresponding fundamentalsystem for the Sturm-Liouville equation is obtained by integration. With this in mind,it suffices to construct a particular approximate solution ˜ y and to derive rigorous errorestimates. The invariant disk estimates are based on the observation that the Riccatiflow maps disks to disks (see [25, Sections 2 and 3]). In fact, denoting the center of ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 21 the disk by m ∈ C and its radius by R >
0, we get the flow equations R ′ = − R Re mm ′ = V − m − R . Clearly, this system of equations is as difficult to solve as the original Riccati equa-tion (11.3). But suppose that m is an approximate solution in the sense that R ′ = − R Re m + δRm ′ = V − m − R + δm , with suitable error terms δm and δR , then the Riccati flow will remain inside the diskprovided that its radius grows sufficiently fast, i.e. (see [25, Lemma 3.1]) δR ≥ | δm | . This is the starting point for the invariant disk method. In order to reduce the numberof free functions, it is useful to solve the linear equations in the above system of ODEsby integration. For more details we refer the reader to [25, 27].12.
Separation of the Resolvent and Contour Deformations
The next step is to use the spectral decomposition of the angular operator in The-orem 10.1 in the integral representation of the solution of the Cauchy problem. Morespecifically, inserting (10.1) into (9.4) givesΨ( t ) = − πi ˆ C ∞ X n =0 e − iωt ω + 3 ic ) p (cid:16) R ω Q ωn (cid:0) H + 3 ic (cid:1) p Ψ (cid:17) dω . (12.1)At this point, the operator product R ω Q ωn can be expressed in terms of solutions ofthe radial and angular ODEs (7.5) which arise in the separation of variables (see [30,Theorem 7.1]). Namely, the operator Q nω maps onto an invariant subspace of A ω of dimension at most N , and it turns out that the operator product R ω Q ωn leavesthis subspace invariant. Therefore, choosing a basis of this invariant subspace, thePDE ( H − ω ) R ω Q nω = Q nω can be rewritten as a radial ODE involving matrices ofrank at most N . The solution of this ODE can be expressed explicitly in terms of theresolvent of the radial ODE. In order to compute this resolvent, it is useful to alsotransform the radial ODE into Sturm-Liouville form (11.1). To this end, we introducethe Regge-Wheeler coordinate u ∈ R by dudr = r + a ∆ , mapping the event horizon to u = −∞ . Then the radial ODE takes again theform (11.1), but now with u defined on the whole real axis. Thus the resolvent can bewritten as an integral operator with kernel given in analogy to (11.2) by s ω ( u, v ) = 1 w ( ´ φ, ` φ ) × ( ´ φ ( u ) ` φ ( v ) if v ≥ u ` φ ( u ) ´ φ ( v ) if v < u ,where ´ φ and ` φ form a specific fundamental system for the radial ODE. The solutions ´ φ and ` φ are constructed as Jost solutions, using methods of one-dimensional scatteringtheory (see [11] and [30, Section 6], [18, Section 3]).The next step is to deform the contour in the integral representation (12.1). Stan-dard arguments show that the integrand in (12.1) is holomorphic on the resolvent set (i.e. for all ω for which the resolvent R ω in (9.4) exists). Thus the contour may bedeformed as long as it does not cross singularities of the resolvent. Therefore, it iscrucial to show that the integrand in (12.1) is meromorphic and to determine its polestructure. Here we make essential use of Whiting’s mode stability result [45] whichstates, in our context, that every summand in (12.1) is holomorphic off the real axis. Inorder to make use of this mode stability, we need to interchange the integral in (12.1)with the infinite sum. To this end, we derive estimates which show that the summandsin (12.1) decay for large n uniformly in ω . Here we again use ODE techniques, in thesame spirit as described above for the angular equation (see [30, Section 10]). In thisway, we can move the contour in the lower half plane arbitrarily close to the real axis.Moreover, the contour in the upper half plane may be moved to infinity. We thusobtain the integral representation (see [30, Corollary 10.4])Ψ( t ) = − πi ∞ X n =0 lim ε ց ˆ R − iε e − iωt ( ω + 3 ic ) p (cid:16) R ω,n Q ωn (cid:0) H + 3 ic (cid:1) p Ψ (cid:17) dω . The remaining issue is that the integrands in this representation might have poleson the real axis. These so-called radiant modes are ruled out by a causality argument(see [30, Section 11]; for an alternative proof see [4]). We thus obtain the followingresult (see [30, Theorem 12.1]).
Theorem 12.1.
For any k ∈ Z / , there is a parameter p > such that for any t < ,the solution of the Cauchy problem for the Teukolsky equation with initial data Ψ | t =0 = e − ikϕ Ψ ( k )0 ( r, ϑ ) with Ψ ( k )0 ∈ C ∞ ( R × S , C ) has the integral representation Ψ( t, u, ϑ, ϕ )= − πi e − ikϕ ∞ X n =0 ˆ ∞−∞ e − iωt ( ω + 3 ic ) p (cid:16) R − ω,n Q ωn (cid:0) H + 3 ic (cid:1) p Ψ ( k )0 (cid:17) ( u, ϑ ) dω , (12.2) where R − ω,n Ψ := lim ε ց (cid:0) R ω − iε,n Ψ) . Moreover, the integrals in (12.2) all exist in theLebesgue sense. Furthermore, for every ε > and u ∞ ∈ R , there is N such that forall u < u ∞ , ∞ X n = N ˆ ∞−∞ (cid:13)(cid:13)(cid:13)(cid:13) ω + 3 ic ) p (cid:16) R − ω,n Q ωn (cid:0) H + 3 ic ) p Ψ ( k )0 (cid:17) ( u ) (cid:13)(cid:13)(cid:13)(cid:13) L ( S ) dω < ε . (12.3)13. Proof of Pointwise Decay
Theorem 8.1 is a direct consequence of the integral representation (12.2) in Theo-rem 12.1. Namely, combining the estimate (12.3) with Sobolev methods, one can makethe contributions for large n pointwise arbitrarily small. On the other hand, for eachof the angular modes n = 0 , . . . , N −
1, the desired pointwise decay as t → −∞ followsfrom the Riemann-Lebesgue lemma. For details we refer to [30, Section 12].14. Concluding Remarks
We first point out that the integral representation of Theorem 12.1 is a suitablestarting point for a detailed analysis for the dynamics of the solutions of the Teukolskyequation. In particular, one can study decay rates (similar as worked out for massive
ECTURES ON LINEAR STABILITY OF ROTATING BLACK HOLES 23
Dirac waves in [18]) and derive uniform energy estimates outside the ergosphere (simi-lar as for scalar waves in [24]). Moreover, using the methods in [19], one could analyzesuperradiance phenomena for wave packets in the time-dependent setting.Clearly, the next challenge is to prove nonlinear stability of the Kerr geometry.This will make it necessary to refine our results on the linear problem, for exampleby deriving weighted Sobolev estimates and by analyzing the k -dependence of ourestimates. Moreover, it might be useful to combine our methods and results withmicrolocal techniques (as used for example in the proof of nonlinear stability resultsin the related Kerr-De Sitter geometry [33]). Acknowledgments:
I would like to thank the organizers of the first “Domoschool –International Alpine School of Mathematics and Physics” held in Domodossola, 16-20July 2018, for the kind invitation. This article is based on my lectures delivered atthis summer school. I am grateful to Niky Kamran and Igor Khavkine for helpfulcomments on the manuscript.
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