KAM and Geodesic Dynamics of Blackholes

Jinxin Xue

Department of Mathematics and Yau Mathematical Sciences Center,Tsinghua University, Beijing 100084, China

Abstract. In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of bound geodesic dynamics of a perturbed blackhole background. The general theories apply mainly to two observable phenomena:the photon shell (unstable bound spherical orbits) and the quasi-periodic oscillations (QPO). We prove that there is a gap structure in the photon shell that can be used to reveal information of the perturbation.

Key words: KAM, photon shell, blackholes, Kerr.

1 Introduction

This paper is a companion paper of [31]. The main purpose is to study the stability of the geodesic dynamics of a blackhole background under perturbations.Schwarzschild and Kerr are among the important metrics. In this paper,we take the two metrics as examples to show how perturbative theory in Hamiltonian dynamics applies to yield interesting phenomena. Similar analysis also applies to many other solutions in general relativity. The geodesic dynamics of particles in either background is integrable and well-understood. However,when the metrics are perturbed slightly, the integrability is generically broken and chaotic motions occur. In reality, the blackholes modeled on either Schwarzschild and Kerr always undergo some perturbations, which makes a perturbative analysis necessary.

The mathematical model that we use to study the geodesic dynamics of the blackholes is the following map called integrable twist map:

whereνis assumed to be smooth andν′(I)>c>0 for some constantcand allI ∈[0,1]. The dynamics of this example is as follows. EachI-circle is invariant and the dynamics on it is a rotation byν(I). Whenν(I) is rational, then each orbit on theI-circle is periodic, and whenν(I) is irrational, then each orbit is dense on theI-circle. Whenφ0is slightly perturbed within the class of smooth symplectic maps, we have the following picture for the dynamics. First, Moser’s version of Kolmogorov-Arnold-Moser (KAM) theory implies that eachI-circle withν(I) a Diophantine number (see Definition A.1) is perturbed to a nearby smooth loop, called invariant circle, that remains invariant under the perturbed map and the dynamics on it can be conjugate to the original unperturbed rotation, provided the perturbation is sufficiently small in a smooth enough topology. For a fixed small perturbation of sizeε, the set of remaining invariant circles in Moser’s theorem has rotation numbers lying in a Cantor set (a nowhere dense closed set) of measure 1?O(√ε). Between two nearby invariant circles(not necessarily KAM curve),there is a gap region called Birkhoffinstability region where the dynamics is very chaotic.There is an Aubry-Mather theory developed for general twist maps which gives the existence of some special orbits in the Birkhoffinstability region, such as periodic orbits, heteroclinic orbits corresponding to rational rotation numbers and Cantor set like orbit corresponding to irrational rotation numbers. So Moser’s theorem allows us to find a large measure set of the phase space where the motion is regular(quasiperiodic), while Aubry-Mather theory allows us to find some special orbits in the gaps. We give an outline of the two theories in Section 2.2 and more details in Appendix.

We first locate the part of phase space with bounded motions. For massless particles moving on null geodesics in Schwarzschild or Kerr background,these bounded motions are called bound photon orbits, or fundamental photon orbits in literature.Each such orbit is moving on a sphere with a fixed radius and is unstable under radial perturbations. This is an observable feature of the blackhole, which lies on the edges of the blackhole shadows[22,25]. If we also consider the motion of massive particles on timelike geodesics in Schwarzschild or Kerr background, again we have unstable bound spherical orbits, similar to the lightlike case. For simplicity, we use the termphoton shellto call the set of bound spherical orbits that are unstable under radial perturbations, for both Schwarzschild and Kerr, and both lightlike or timelike geodesics. We show in this paper that both Moser’s theorem and the twist map theory apply to the study of the dynamics on the photon shell in a perturbed Kerr spacetime,but neither applies to perturbed Schwarzschild a priori without further information of the perturbation. In particular, Moser’s theorem implies in the Kerr case the existence of a gap structure(many Birkhoffinstability regions)on the photon shell exhibiting coexistence of regular and chaotic motions,which we expect is observable and can reveal information of the perturbation.

Furthermore, in the massive case, in addition to the photon shell, there are a lot more bound timelike geodesics. It can be observed that outside a blackhole there is a slowly rotating accretion disk emitting X-rays exhibiting some quasi-periodic oscillations (QPO) in its frequency (c.f. [8,16,18] etc). In the phase space this is a region around a stable circular orbit. We show both Moser’s theorem and twist map theory apply in this region for both Schwarzchild and Kerr cases to yield lots of quasiperiodic orbits as well as other special orbits.

Both Schwarzschild and Kerr metrics are written in coordinates (τ,r,θ,?) whereτis the coordinate time,ris the polar radius,θis the latitudinal angle and?is the azimuthal angle. We will use the following definition throughout the paper.

Definition 1.1.A metric is calledstationaryif it is independent of the coordinate time τ andaxisymmetricif it is independent of the azimuth angle ?.

To model a perturbed blackhole, it is not avoidable to make some simplifying assumptions. In this paper, we shall consider only stationary perturbations. It is known that there is no stationary solution to the vacuum Einstein equation other than Kerr (including Schwarzschild as a special case), though stationary perturbations do exist as solutions to the linearized vacuum Einstein equation at Kerr. When a spacetime is close to Kerr and evolves very slowly, it can then be modeled by a stationary perturbation and our analysis applies to give useful information over a long time.

1.1 The photon shell dynamics

We consider only stationary perturbations so that the particle’s energyE, that is dual toτ,remains to be a constant of motion, so we fix a valueEand restrict to the photon shell. The resulting system has two degrees of freedom with coordinates(θ,pθ,?,Lz)and we only need to study the latitudinal and azimuthal motions,wherepθandLz=p?are generalized momentum variables due toθand?respectively,andLzin particular is a constant of motion having the physical meaning of thezcomponent of the angular momentum. This subsystem describes a particle’s motion on the photon shell that has a vertical oscillation within a spherical band symmetric around the equator when orbiting around the center of the blackhole.

Our general strategy is to introduce action-angle coordinates (αθ,Jθ,α?,Lz)∈T×R+×T×R for theθ- and?-motions. In these coordinates the unperturbed Schwarzschild or Kerr Hamiltonian has the formH(Jθ,Lz), so the dynamics leavesJθ,Lzinvariant and the angles (αθ,α?) move on the torus T2linearly as

The Schwarzschild case is relatively simple, since each photon spherical orbit moves on a plane passing through the origin so is periodic and all the photon spherical orbits with a given total angular momentum have the same period. In this case the ratioνis constantly 1, so neither Moser’s theorem nor twist map theory applies without further knowledge of the perturbations. We shall study this in Section 3.2.

It turns out that both theories apply to Kerr. For Kerr, we can reparametrize the ratioνby the radiusrof the photon shell orbits and denote byRthe domain of definition forr(c.f. Definition 4.1). So the photon shell of the unperturbed Kerr is foliated bySr, that is a spherical strip symmetric around the equator consisting of all spherical orbits of radiusr∈R. Theorem 4.1 for precise statement.

The first natural question is thatif the photon shell exists under perturbation.More generally, it is an important open problemto prove or disprove the existence of photon shells for a given blackhole background. We refer readers to the works [7,19] for geometric results. Here we treat the photon shell as a dynamical object called normally hyperbolic invariant manifolds (NHIM, see Appendix B), which is a submanifold in the phase space, dynamically invariant and has certain expansion and contraction in the normal direction. It is remarkable that such an object is in general robust under perturbations (see Theorem B.1), so we indeed obtain that the photon shell as a dynamical object exists for stationary slightly perturbed Kerr.The setting is as follows, we consider the photon shell that is the set of unstable bound spherical orbits in the phase space by taking union of all these orbits together with their momentum and we obtain a four dimensional manifoldNparametrized by (θ,pθ,?,Lz) that is normally hyperbolic in the sense of Definition B.1. There are some subtlety when applying Theorem B.1, which we elaborate later in Section 4.3. The conclusion is that there is a small deformationNεofNthat is invariant under the perturbed Hamiltonian systemHε. By Theorem B.1(6), the perturbed NHIMNεremains to be symplectic andHε|Nεremains Hamiltonian. We can find a symplectomorphismkε:N →Nεso thatk?εHεis a Hamiltonian system onNand can be written as a small perturbation of the Kerr Hamiltonian restricted toN.So we apply Moser’s theorem and twist map theory to the perturbed Hamiltonian systemk?εHε:N →R.

Theorem 1.1.Consider null geodesics in a Kerr spacetime with mass M and angular momentum a. Then there exists an open set of parameter a/M such that the frequency ratio ν=ω?/ωθ as a function of r ∈R is smooth, strictly monotone and has a discontinuity point at some r?∈R with ν(r??)?ν(r?+)=2. Orbits on Sr with rr?are retrograde. Denote by U a small neighborhood of r?.

Let εhμνdxμdxν be a stationary perturbation of the above Kerr metric. Then if ε sufficiently small, we have the following for the dynamics of null geodesics in the perturbed system

1. There is a closed subset C of RU such that for each r ∈C, in the perturbed system,there is a set?Sr that is a small deformation of Sr,such that?Sr consists of periodic or quasiperiodic orbits.

2. Each interval in R(U ∪C)corresponds to a Birkhoffinstability region where the spheres Sr are broken (c.f. DefinitionC.2) and twist map theory (c.f. TheoremC.1(2.b),(3.b)and TheoremC.2) applies.

Inparticular,the gaps(instability regions)often occur aroundrationalfrequency rati osofthe formp/q,duetothe existenceofFourier modes exp(√?1k(pαθ+qα?)),k,p,q∈Z,in the perturbation. In general, the gap is large when the denominator of the rational number is small. We expect that this is an observable phenomenon and can be used to conclude the information of the perturbation. We consider it as an analogue of the Kirkwood gap in the asteroid belt in the solar system. Indeed,between the orbits of Mars and Jupiter in solar system, there is an asteroid belt consists of huge number of asteroids. However, the distribution of the asteroids there is not uniform. There are gaps due to resonances of orbital frequencies with the planets.

In Fig. 1, we see that there is a discontinuity in the frequency ratioν, which occurs at a point whereLzchanges sign and the motion of the particle becomes from prograde to retrograde. This phenomenon was observed and explained in [28,30].We give a elementary mathematical explanation in Section 4.4.2.

For the geodesic motion of massive particles,we have a similar theorem(c.f.Theorem 4.1). The main difference of the massive case and the massless case is that the dynamics on the photon shell of the massive case depends on the particle’sE,while the massless case does not. So in the massive case, we need to fix anEahead of time, then there is the similar gap structure for the photon orbits of all massive particles with the same energyE.

Figure 1: The frequency ratio ω?/ωθ with parameters a/M=0.8, μ2=0.

Note that our result can be combined with the result of Johnson,Lupsasca,Strominger et al.[17]to give a delicate picture of the fine structure of the neighborhood of the photon shell. It is called the photon ring the image on the observer’s screen of photon’s on nearly bound null geodesics. So the photon ring contains the photon shell. It was discovered in [17] that there is a subring structure outside the photon shell. Indeed, if we shoot a light ray very near, a distance from the shadow edge atδr0, it will circlen～?ln|δr0| times before falling into the black hole or escaping to infinity, due to the instability in the direction normal to the photon shell,whereγis the Lyapunov exponent. A light ray that completesnhalf orbits collectsntimes more photons along its path. This gives a ring on the screen subdivided into subrings labeled by the number of circulationsn, exponentially narrower when approaching the photon shell.

1.2 Quasi-periodic oscillations (QPO)

In the case of massive particles, the geodesic dynamics has lots of bounded motions other than the bound spherical orbits. These bounded motions occur around the stable spherical orbits, thus they undergo both radial oscillations and the latitudinal oscillations when spinning around the blackhole. They are supposed to be responsible for the QPO behavior in some theories.

We study the dynamics of these bound orbits using similar approaches as before.For simplicity,we consider perturbations that are both stationary and axisymmetric henceEandLzremain to be constants of motion. Then we can use coordinates(r,pr,θ,pθ) to study the radial and latitudinal dynamics of the Hamiltonian system.We introduce action-angle coordinates (αr,Jr,αθ,Jθ)∈T×R+×T×R+so that the Hamiltonian is a function ofJrandJθonly and the radial frequency?JrHand vertical frequency?JθHare called fundamental frequencies in [27]. The ratioν=?JrH/?JθHplays a similar role to the photon shell case as above. Again we can computeνto verify that the assumptions for both Moser’s theorem and twist map theory apply, so we have an analogue of Theorem 1.1 in both Schwarzschild and Kerr cases. We refer readers to Theorem 3.1 and Theorem 4.2 for more details.

If we consider non axisymmetric perturbations,then we should consider a Hamiltonian system of three degrees of freedom with coordinates (r,pr,θ,pθ,?,Lz). The KAM nondegeneracy condition can be studied similarly. We shall outline how this can be done. Action-angle coordinates for Kerr has been constructed in [27]. In general, we do not expect that KAM nondegeneracy condition holds globally in the part of phase space with bounded motions for Kerr, considering the isofrequency pairing results (c.f. [29] etc).

The paper is organized as follows. The main body of the paper consists mainly of introductory and conceptual arguments and statements,and we put all the technical ingredients to the Appendix. In Section 2, we give some preliminaries on nearly integrable Hamiltonian dynamics including Moser’s theorem and twist map theory.In Section 3,we study the perturbed Schwarzschild dynamics. In Section 4,we study the dynamics of perturbed Kerr. Finally, we have three appendices. In Appendix A, we give more information on Hamiltonian dynamics including Liouville-Arnold theorem and KAM theorem. In Appendix B we introduce the theorem of normally hyperbolic invariant manifolds,and in Appendix C,we introduce the Aubry-Mather theory of twist maps.

2 Preliminaries on Moser’s theorem and twist maps

In this section, we give some preliminaries on Moser’s version of KAM theory and the twist map theory. We refer readers to Appendix A and C for more details.

2.1 Energetic reduction

We first show how to get an nonautonomous Hamiltonian system of one degree of freedom from an autonomous one with two degrees of freedom.

This procedure is called the energetic reduction (Section 45 of [1]). For the nonautonomous 1-periodic Hamiltoniany1, we can take its time-1 map, which is a map defined onT?T1. In case whenHis integrable, i.e., does not depend onx1, x2explicitly, the reduced time-1 map has the form (1.1) by restricting its domain.

2.2 Dynamics of twist maps

Twist map is defined as follows.

Definition 2.1.A diffeomorphism φ:T1×[0,1]→T1×[0,1]preserving the boundary is called an area preserving twist map if the following holds:

1. φ is symplectic, i.e., φ?ω=ω, ω=dx∧dy;

2. for each x∈T, the map π1φ:{x}×[0,1]→T1is a local homeomorphism, where π1means the projection to the first(T1)component.

One example of a twist map is the map(1.1). There is a version of KAM theorem developed for twist maps by Moser[21]that we shall mainly apply later in the paper.

Definition 2.2(Diophantine number).A number α ∈Ris called aDiophantine number, if there exist C>0and τ>0such that

We denote α∈DC(C,τ).

Another important example of twist maps is the standard map defined as follows:Φ: T2→T2via

wherekis a parameter.

For a general nonlinear twist map (not necessarily nearly integrable), we can introduce a variational principle and define globally minimizing orbits(see Definition C.1)which has the notion of rotation number playing a similar role asν(I)as in(1.1).In Aubry-Mather theory, the globally minimizing orbits are classified according to their rotation numbers, denoted byMρ. We give an overview of this theory in Appendix C.

Combining Moser’s theorem and Aubry-Mather theory, we get the following picture for the dynamics of a twist map. Letφεbe a smooth twist map and whenε=0, the mapφ0is of the form (1.1). Then forεsufficiently small, we have

1. Moser’s theorem implies that for Diophantine irrational rotation numbers(ρ∈DC(C,τ)forC>0 independent ofε)the setMρis a invariant curve restricted on which the mapφεis conjugate to an irrational rotation, i.e., there exists a diffeomorphismh:Mρ →T such thathφε=h+ρ.

2. For each irrational rotation numberρ, the setMρis either a homotopically nontrivial invariant curve(may or may not be the one in Moser’s theorem),or a Denjoy minimal set (c.f. Theorem C.1(2.b)).

3. For rational rotation numbersρ, it may happen that there exists a homotopically nontrivial curve consists of periodic orbits with the same rotation numberρ∈Q. However, typically, such curve does not exist andMρconsists of one single periodic orbit with two homoclinic orbits approaching it in both the future and the past.

4. Whenever there is a region bounded by two neighboring invariant circles,there exist orbits crossing the gap and visiting any two neighborhoods of the boundary circles.

5. In a neighborhood of an elliptic periodic point of least periodq(Dφqεhas no eigenvalues offthe unit circle), the above (1)-(4) apply also when the twist condition is satisfied. Such a neighborhood is called an elliptic island.

3 The Schwarzschild spacetime

3.1 Schwarzschild dynamics

The Schwarzschild metric is as follows

We chooseμ2=0 for massless particles andμ2=1 for massive particles, and the geodesics in the former case is called a lightlike geodesic and the latter timelike.3.1.1 The radial dynamics, critical points and homoclinic orbits

We next analyze the radial dynamics. Setting

Figure 2: The Schwarzschild effective potential.

3.1.2 The normally hyperbolic invariant manifold

Denote byrc=3Min theμ2=0 case andrc=r?in theμ2=1 case. If we fix the constantp2τ=E2=2V(rc), which is a function ofL2andM, then we can treat the HamiltonianHin (3.2) as a system of three degrees of freedom depending on the variablesr, pr, θ, pθ, ?, p?. The four dimensional submanifold

is in variant under the dynamics. The Hamiltonian system restricted toNis a Hamiltonian system of two degrees of freedom of the form

where we useLzin place ofp?to signify its physical meaning of the third component of the angular momentum.

3.2 Dynamics on the photon shell

The system(3.6)has two degrees of freedom and is integrable in the sense of Theorem A.1. Note that this is the case for bothμ2=0 andμ2=1. The nontrivial part of the Hamiltonian (3.6) is the square of the total angular momentum, denoted by

The other action variable is taken to beLzand their dual angular variables are denoted byαθ,α? ∈T1respectively, introduced to preserve the symplectic form

In the new coordinates the HamiltonianH:=L2(Jθ,Lz): R+×T1×R×T1→R is independent of the angular variablesαθ,α?. Then the Hamiltonian equation becomes

The following result show that the mapφ0does not satisfy the twist condition,so neither Moser’s theorem nor the twist map theory applies to a small perturbation ofφ0a priori.

Proposition 3.1.The Hamiltonian HN written in action-angle coordinates satisfies

Note that this fixes the constantL2. Then we get

If we minimizeAc(μ) among allμ∈M, we get that the only choice is

Following the argument of [6], we get thatβis differentiable in the radial direction.The Legendre transform gives

attaining the sup in the definition ofβ. By the first item of the Proposition, we get thatα(c)=f(Jθ+Lz) for some convex functionf. Then forLzfixed, the differentiability ofβin the radial direction implies thatfis strictly convex.

The first item reflects the fact that all orbit on the photon sphere lies on a plane passing through the origin hence is periodic and they all have the same period on a fixed energy level. So neither Moser’s Theorem 2.1 nor twist map theory can be applied at this moment. However, when a perturbation is added, it may introduce certain twistness,which require further more careful analysis depending on the form of the perturbation.

3.3 KAM and QPO for Schwarzschild

In the case ofμ2=1, we have other bound orbits than those on the photon shell. In particular, orbits withr=r+are stably circular. In a neighborhood of this stable circular orbit, there are a lot of bound orbits.

3.3.1 KAM and twist maps theory for the QPO

We introduce the following bounded part of the phase space where Liouville-Arnold theorem (Theorem A.1) applies. LetCbe a large constant and define

In this setB(C), we note that both the radialr-component and the latitudinalθcomponent are lying in potential wells around the local minimums, so nearby orbit will have small oscillations in ther- andθ-components (c.f. Fig. 2).

Suppose for simplicity that we consider a stationary and axisymmetric perturbation of the Schwarzchild metric so that the perturbation has no dependence on?andτthenEandLzare constants of motion. Fixing a value forEandLzsatisfying the definition ofB(C), then the resulting Hamiltonian system has two degrees of freedom with coordinates (r,pr,θ,pθ).

We next introduce the action-angle coordinates and work out the frequencies of the oscillations. The action variablesJrandJθare defined as follows (c.f. (3.2))

wherer?

Theorem 3.1.Let εhμνdxμdxν be a stationary and axisymmetric perturbation to the Schwarzschild metric. Then there exist an open set of parameters(E,Lz), a neighborhood D of the point(r,pr,θ,pθ)=(r+,0,π/2,0)and ε0such that Moser’s Theorem2.1and the twist map theory are applicable to the perturbed system in D and for all0<|ε|<ε0.

Proof.Letfbe the map relating the two sets of constants of motion (Jr,Jθ) and(H,L2):f(Jr,Jθ)=(H,L2), whereHis the Schwarzchild Hamiltonian, then we haveDf·Df?1=I, which is more explicitly

Figure 3: The ratio ωr/ωθ for Schwarzschild with parameters E=0.97 and Lz=3.

Here we use the fact thatJθis independent ofHas we see from the expression ofJθ. This implies

So we get

which is a function ofL2from the above formulas. Fig. 3 is plotted using this formula. Since we have

3.3.2 KAM nondegeneracy and further derivatives

The last Theorem 3.1 uses Moser’s version of KAM theorem. If we would like to verify the KAM nondegeneracy conditions in Section A.2, we have to calculate?ω, where?is the derivative with respect toJr, Jθ. In this section, for the sake of completeness,we show how to take further derivative ofωand clarify some subtleties.Since we do not use it in the proof of the main theorem, readers are suggested to skip this section when first reading.

We have

The expressions in the last matrix are all computable from the definitions ofJr, Jθ.

To obtain the second order derivatives, we cannot differentiate the first order derivatives in (3.10) for the technical reason that further derivative will introduce divergent improper integrals. This problem can be resolved as follows. We consider only the caseJrand the caseJθis similar. Note that the singularity originates from the fact thatP(r±)=0 will appear in the denominator,which we should avoid. The actionJris the area enclosed by the curve Graphpr. Letr?be the minimum ofαr?2L2+2α. Then we split the integral

into two pieces

We remark that ?pandr(pr) are both dependent on the variablesE2andL2while ?ris not. The derivatives of this expression does not involve improper integrals hence we can take its second derivative to yield an expression forDω.

3.3.3 Three fundamental frequencies

In Theorem 3.1 we consider stationary and axisymmetric perturbations so we have only two fundamental frequenciesωrandωθto consider. In general, when a perturbation depends on?, i.e., nonaxisymmetric, but not onτ, we have to treat the Hamiltonian system with as one with three degrees of freedom, in which case, we cannot apply Moser’s Theorem 2.1 or twist map theory, so we have to verify the KAM nondegeneracy assumptions in Section A.2 if we wish to obtain KAM type dynamics. Again readers are suggested to skip this section when first reading, since here we only show how to get the formulas for?ω, where?means the derivative with respect to the action variablesJr, Jθ, J?. We introduce the action variablesJr, Jθas before and the extraJ?is chosen to beLz.

Then there is a mapFrelating the two set of conserved quantities:

SinceDFis nondegenerate, the nondegeneracy condition is then reduced to that of(?E,?L2,?Lz)ω.

4 The Kerr spacetime

In this section, we study the dynamics of perturbed Kerr spacetime. The Kerr spacetime in the standard Boyer-Lindquist coordinates has the form

is a constant of motion calledCarter constant.

We next introduce the Hamiltonian formalism. We treat the Kerr metric as twice a Lagrangian and get the corresponding Hamiltonian via the formal Legendre transform The system has four independent and commuting constants of motionH, E, Lz, Q.

4.1 The vertical dynamics

Introducing a time reparametrizationdλ=Σ(r,θ)?1dtcalled Mino time,we determine the vertical dynamics by the equation

So the dynamics can be visualized as a particle moving in the potential well ofU(θ).

The case ofLz=0 differs drastically fromLz/=0 case. Indeed, ifLz=0 andQ>(μ2?E)a2, the orbit goes through the south and north poles, whileLz/=0, as will see in the following, the orbit is bounded away from the south and north poles.

Figure 4: Graph of U.

We first consider the caseμ2=1 and suppose(μ2?E2)>0,then we haveU(θ)≥0 is an infinitely deep single well potential andQ ≥0 (see Fig. 4). The minimum minU=0 is attained atθ=π/2. We also haveU(π/2+θ)=U(π/2?θ)forθ∈[0,π/2)andU(θ)→∞forθ→0,π.Letθ?<θ+be two root of the equation Θ(θ)=0 withQ>0.Then the orbit on the photon sphere with conserved quantitiesQ, E, Lzoscillates within the spherical stripθ ∈[θ?,θ+]. The explicit solution ofθ(λ) is obtained as follows. Denoting by

Lemma 4.1.

In this critical caseQ=0 and the double well case, using the general principle of[31],in a generically periodically perturbed Kerr metric,separatix splitting will be created and chaotic motion will occur,which implies that there exists orbit tunneling from one well to the other. In each of the above cases,solutionsθ(λ)of the equation of motion can be found explicitly (c.f. Section 63 of [5]).

4.2 The photon shell and the radial dynamics

Similar to the photon sphere in Schwarzschild case, the Kerr spacetime also admits unstable spherical orbits. However, the dynamics of these orbits in the Kerr spacetime differs drastrically from the Schwarzschild case. Such an orbit in the Schwarzschild case lies on a fixed plane through the origin thus is periodic and all these orbits form a sphere. However, in the Kerr case, such an orbit is no longer confined to a plane but may oscillate in a spherical strip and may be quasiperiodic.Moreover, depending on the parametersE, Lz, Q, the radius varies, so the union of all such orbits forms a ring calledphoton shell. The photon shell for Kerr was first studied by [30] in the extreme casea=Mfor timelike geodesics. The lightlike case was studied by [28]. The equatorial case ofQ=0 was studied by [13].

4.2.1 Determining the photon shell

An orbit in the photon shell has constant radial component. So from the radial equation of (4.2), the following equations should be satisfied

Figure 5: Graphs of ? and q in the μ2=1 case.

in theμ2=1 case. The formulas can be found in Section 63 and 64 of [5] and we have discarded an unphysical solution in each case.

The functions?andqin (4.9) are plotted in Fig. 5(a) with dataM=1, a=0.8,E=1 and Fig. 5(b) withM=1, a=0.8, E=0.92 in the case ofμ2=1. For nearby choices of parameters the picture is qualitatively similar since the functions are continuous inE2anda. In the massless case, the RHS of (4.8) has no dependence onE. The picture of?andqin (4.8) is qualitatively similar to Fig. 5(a) fora=0.8.

Givenr=rc, it determines the values of?=?c,q=qcsuch thatrcsolves Eq. (4.7)with parameters?candqc. So for this choice of parameters?c, qc, the radiusrcis a double root ofR(r)=0, thusRcan be put in the form

2. In the case of μ2=0, the admissible set R:=R(M,a)radii of bound spherical orbits is defined as the set of rc satisfying q(rc)≥0in(4.8)(see Lemma4.2below).

4.2.2 The Hamiltonian restricted to the photon shell

When restricted to the photon shell

obtained from the Hamiltonian (4.5) by settingr=rc, pr=0. We should further eliminate therc-dependence by substitutingrcas a function ofLzby inverting the first equation in (4.8) or (4.9) on each of its monotone intervals. Thus we get a Hamiltonian system of two degrees of freedom with coordinates (θ,pθ,?,Lz)integrable in the Liouville-Arnold sense (Theorem A.1).

4.2.3 The radial dynamics

We look at the radial motion in (4.2). After a time rescaling

called the Mino time, the radial equation of motion becomes

Thus, the dynamics can be visualized as a particle moving in the potential well of?R(r) on the zero energy level.

Lemma 4.2.In the case of μ2=0, the only bound orbits outside the event horizon are on the photon shell, i.e., we have r=rc along each such orbit and ?R′′(rc)<0.

Figure 6: Graph of ?R.

in the case ofrc>M, which is clearly true. Ifrc=M, the inequality is still true if|a|

Letrc ∈R, then we have

See Fig. 6(b).μ2=1 case. We next consider the Consider an admissib Besides the bound spherical orbit, there is also a homoclinic orbit withr(λ) approachingrcasλ→±∞. We do not treat the dynamics of the homoclinic orbit in this paper but refer readers to [31] for more details.

4.3 Persistence of the photon shell

4.4 KAM and the photon shell dynamics

We next consider the dynamics on the photon shell in Kerr spacetime. Restricted to the NHIMN(c.f.(4.11)),the HamiltonianHNis(4.12)with two degrees of freedom.Each orbit has constant radius hence lies on a sphere. The azimuthal angle?rotates periodically in [0,2π) and the latitudinal angleθoscillates within a spherical band symmetric around the equator according to (4.6).

4.4.1 KAM and twist map nondegeneracy

We next introduce action-angle coordinates for the HamiltonianHN. In the equation Σ2˙θ2=Θ (c.f. (4.2)), we substitutepθ=Σ ˙θ(c.f. (4.4)) to yield=Θ. We consider both casesμ2=0 andμ2=1 together, and always assumeQ>0. Whenμ2=0,the Carter constantQcan be negative, which case can be considered similar to the procedure that we are doing now.

We next introduce the action variablesJ?:=p?=Lzand

whereθ?<θ+are two roots of the integrand (c.f. (4.6)). We further introduce their dual angular variablesαθ,α?∈T respectively using the symplectic form(c.f.Section 50 of [1]). In the new coordinates, we get that the Hamiltonian is a function ofJθandLzonly, and the the dynamics of (αθ,α?) on T2is a linear flow with frequency

Similarly as before, after performing an energetic reduction, to apply Moser’s Theorem 2.1 and the twist map theory, we need to show that the ratioν:=ω?/ωθhas no critical point.

Here we point out some essential differences between the massive case (μ2=1)and massless case (μ2=0). In the case ofμ2=0, we note that theEvariable can be eliminated by the rescaling

The RHS of Eq.(4.8)has no dependence on the particle’sE, nor does the Hamiltonian (4.12). Thus the dynamics, in particular the frequency ratioν, on the photon shell has no dependence onEso in the following it is enough to chooseE=1.However, for the massive case, the energyEdoes play a role in the frequency ratioν.

In the proof of the next theorem,we provide explicit formula for the ratioν. Leta,M,Ebe fixed constants. Restricted to the energy level

the ratioνdepends only on one parameter that can be chosen to be eitherJθorLz.To be more physically relevant, we introduce a new parametrization as follows.

In the Kerr case, the photon shell is foliated by bound unstable orbits with constant radius varying in an intervalR. Recall that in Eq.(4.9)we have expressedLz/EandQ/Eas functions ofrcwhich is the radius of the photo sphere. Since both(Lz,Q) and (Lz,Jθ) can be used as independent integrals of the HamiltonianHN,there is a mapfrelating the two sets of integrals:g(Q,Lz)=(Jθ,Lz). Thereforeν?gcan be considered as a function ofrcusing(4.9). We thus obtain a reparametrization of the ratioνin terms of the radiusrcof the slice of the photon shell.

As we have explained in the introduction, we obtain a perturbed Hamiltonian system of two degrees of freedom pulled back toNto which we apply KAM and twist map theory.

Theorem 4.1.Let εhμνdxμdxν be a stationary perturbation of the Kerr metric. In either of the following two settings:

1. In the massless case μ2=0, there exists an open set of parameter a/M;

2. In the massive case μ2=1, there exists an open set of parameters a, M, E,such that the following holds. Denote by U?R a small neighborhood(may be empty)of the point of discontinuity of ν?g(Q(rc),Lz(rc)). For ε sufficiently small, we have

1. There is a closed subset C of RU with measure|R(U ∪C)|=O(ε1/2)such that for each rc ∈C in the perturbed system there is a photon sphere on which all orbit is periodic or quasiperiodic with frequency ν?g(Q(rc),Lz(rc))and all orbit lies in Mν.

2. Each open interval in R(U ∪C)corresponds to a Birkhoffinstability region where the photon spheres are broken in the sense that Mν with

are all broken (see DefinitionC.2), and twist map theory (c.f. TheoremC.1(2.b),(3.b)and TheoremC.2) applies.Proof.Letfbe the map relating the two set of constants of motions (Jθ,Lz) and(H,Q), i.e.,f(Jθ,Lz)=(H,Q). FromDf·Df?1=I, we obtain

the first row ofDf. We next provide explicit formula for the derivatives in the above expression (noting thatUdepends onLzandLzis a function ofQ)

Figure 7: The frequency ratio ω?/ωθ in the μ2=1 case.

from (4.9) for theμ2=1 case (4.8) for theμ2=0 case. Thus the ratioω?/ωθis a function ofrcin both cases.

In the massless case, the ratioω?/ωθis independent ofE. Fig. 1 is plotted with choice of parametera/M=0.8. In the massive case, we plot Fig. 7(a) with parametera=0.8, M=1, E=0.96 and Fig. 7(b) with parametera=0.8, M=1,E=0.92, corresponding to Figs. 5(a) and 5(b) respectively. We note that Fig. 7(a)is discontinuous which is related to the fact thatLzchanges sign, and we have that 7(b) is continuous since?does not change sign in Fig. 5(b). We shall explain the discontinuity in the next section. Also there is a critical point in Fig.7(b),so we can only apply KAM and twist map theory outside a small neighborhood of the critical point.

4.4.2 The point of discontinuity

There is an interesting feature in the graph in Fig. 1 and Fig. 7(a) that is the appearance of a discontinuity point. The point of discontinuity occurs at the place where?(rc)=0 in (4.9). This means that the orbit becomes from prograde to retrograde asrcrosses the zero ofLzwhen increasing. This phenomenon was noticed and explained in [28,30]. Let us give an elementary mathematical explanation.

Proof.We see from Section 4.1 that the vertical dynamics in theLz=0 case differs drastically from theLz/=0 case. WhenLz=0 and ˉQ:=Q?(μ2?E2)a2>0, the latitudinal motionθranges over [0,π], in particular, the orbit passes through the south and north pole. For|Lz|/=0 but sufficient small,the latitudinal angleθ∈[δ,π?δ]whereδsolves the equationQ?U(δ)=0. In particular, asLz →0 we have

The discontinuity originates from the second term on the RHS of (4.14). Another way to see it is to evaluate the quantity ??that is the change of the azimuthal angle during each period of theθ-motion. We have

We remark that for some choices of parametersa, E, the function?(r) in (4.9)may be strictly positive (see Fig. 5(b)), in which case the ratioνdoes not change sign.

4.5 KAM and QPO

In this section, we consider only the timelike case withμ2=1.

SupposeE2<1 and?Rattains its local max atr?(>rh) and a local min atr?>r?. Suppose also?R(r?)>0 and?R(r?)<0(imagine that we shift the graph of?Rin Fig. 6(b) upward slightly). LetC>0 be a large constant, and we define the following bounded part of the phase space where all Kerr orbits are bounded hence Liouville-Arnold theorem (Theorem A.1) applies

This is a neighborhood of the stable circular orbit. Denote byr?

Again, similar to the Schwarzchild case, we consider for simplicity perturbations that are stationary and axisymmetric, thusEandLzremain constants of motion in the perturbed system. Therefore we fix values ofLzandE, and introduce the action-angle coordinates (αr,Jr,αθ,Jθ)∈T×R+×T×R+as follows

Figure 8: The ratio ωr/ωθ for Kerr with parameters a=0.8, E=0.96, Lz=1.5.

Theorem 4.2.Let εhμνdxμdxν be a stationary and axisymmetric perturbation to the Kerr metric. Then there exists an open set of parameters a, E, Lz such that there are a neighborhood D of the point(r,pr,θ,pθ)=(r?,0,π/2,0)and ε0such that Moser’s Theorem2.1and the twist map theory are applicable to the perturbed system in D and for all0<|ε|<ε0.

Fig. 8 is plotted with parametersa=0.8, E=0.96, Lz=1.5.

An example of such perturbations is given by the Monko-Novikov metric [10],which is so complicated that we do not cite it here. In[10]the authors analyzed the 1:2 and 2:3 resonances and discovered that the dynamics of the perturbed Kerr is chaotic. The frequency ratioωr/ωθhas been studied by many authors, c.f. [4,27]etc.

Fig.8 is then plotted with this formula. This completes the proof of the theorem.

Finally, for the sake of completeness, we provide formula for the verification of the KAM nondgeneracy conditions in Section A.2. This part is similar to Section 3.3.2 so readers are suggested to skip it when first reading. We need to evaluate?ω

where?is the derivative with respect toJr, Jθ. We thus arrive at

We can then compute?ωsimilar to Section 3.3.2. If we are interested in perturbations also depending on the angle?, then we have a Hamiltonian system of three degrees of freedom and we have to consider three fundamental frequencies as in Section 3.3.3. The procedure is similar to the Schwarzschild case in Section 3.3.3,though the computations are more involved. We skip the details here.

Appendix A: introduction to nearly integrable Hamiltonian dynamics

In the Hamiltonian formalism of the classical mechanics, a smooth Hamiltonian functionHon a symplectic manifold (M,ω) is given, and defines a vector fieldXHthroughω(·,XH)=dHwhich defines a dynamical system by solving the ODE ˙x=XH(x).

A.1 Liouville-Arnold theorem

The classical Liouville-Arnold theorem gives important characterization of the integrable systems.

Theorem A.1(Liouville-Arnold).Let H1=H:M2n →Rbe a Hamiltonian and suppose there are H2,···,Hn:M →Rsatisfying

(a) {Hi,Hj}≡0, for all i,j=1,···,n, where {Hi,Hj}:=ω(XHi,XHj)is the Poisson bracket;

(b) the level set Ma:={(q,p)∈M|Hi(q,p)=ai, i=1,···,n} is compact;

(c) At each point of Ma, the n vectors DHi, i=1,···,n are linearly independent.Then

1. Mais diffeomorphic toTn=Rn/Zn and is invariant under the Hamiltonian flow of each Hi.

2. in a neighborhood U of Ma, there is a symplectic transformΦ(q,p)=(θ,I)such thatΦ(U)=Tn×(?δ,δ)n for some δ>0.

3. In the new coordinates, each Ki:=Hi?Φ?1is a function of I only so the Hamiltonian equation is

We refer readers to Section 50 of [1] for the proof and more details. The coordinates (θ,I) are called action-angle coordinates and can be constructed by standard recipe in [1].

A.2 The Kolmogorov-Arnold-Moser theorem

The Kolmogorov-Arnold-Moser theorem is an important stability result on the dynamics of nearly integrable systems.

Definition A.1.A vector α ∈Rn, n ≥2is calledDiophantineif there exist some C>0, τ>n?1such that for all k∈Zn{0}, we have

We denote α∈DC(C,τ).

For eachτ>n?1, the set∪C>0DC(C,τ) has full Lebesgue measure in Rn. For a fixedC, the set DC(C,τ) is a Cantor set of positive measure.

A version of the KAM theorem states as follows.

We refer readers to [26] for more details. When applying KAM theory to an energy level,we may replace the nondegeneracy condition detD2h(I)/=0 by Arnold’s isoenergetic nondegeneracy condition

(c.f. Appendix 8 of [1] and Chapter 6.3 of [2] for various versions of nondegeneracy conditions).

Appendix B: The theorem of normally hyperbolic invariant manifold

In this section we give the version of normally hyperbolic invariant manifold theorem that we used in the proof of our main theorem. The standard references are [9,15].

Definition B.1(NHIM).Let N ?M be a submanifold (maybe noncompact) invariant under f, f(N)=N. We say that N is a normally hyperbolic invariant manifold if there exist a constant C>0, rates0<λ<μ?1<1and an invariant (under Df)splitting for every x∈N

in such a way that

Here the Riemannian metric|·|can be any prescribed one, which may change the constant C but not λ, μ.

Theorem B.1.Suppose N is a NHIM under the Cr, r>1, diffeomorphism f:M →M. Denote

Then for any Cr f?that is sufficiently close to f in the C1norm,

1. there exists a NHIM N?that is a C? graph over N,

Appendix C: Aubry-Mather theory for twist mapsThe dynamics of twist maps are studied in details in Aubry-Mather theory. Let us summarize the main result and refer readers to [3] for more detailed proofs.The theory can be put in a very general setting. For the sake of concreteness, we consider the following setting. LetH:T?T1×T1→R be a nonautonomous 1-periodic Hamiltonian satisfying the following Tonelli conditions

Since the LagrangianLis 1-periodic in its first entry, we may lift each curveγ: R→T1to a curve ?γ: R→R by unfolding T1to its universal covering space R, so we can visualize the graph of ?γin R2.

The following theorem summarizes the main results of the twist map theory(c.f. [3]).

Theorem C.1.

1. Each globally minimizing curve γ has a well-defined rotation number

For each ρ, the set of globally minimizing curves with rotation number ρ is not empty, we introduce the set

We also use the same notation to denote its Legendre transform to T?T1.

2. If the rotation number ρ is irrational, Mρ is

(a) either an invariant circle (a homotopically nontrivial circle on T?T1invariant under the dynamics),

(b) or a Denjoy minimal set (a Cantor subset of a homotopically nontrivial circle on T?T1that is minimal in the sense of topological dynamics).

3. If the rotation number ρ is rational, Mρ always contains periodic orbits.

(a) It may be an invariant circle consists of periodic orbits of the same period.

(b) If Mρ is not an invariant circle, then in the gap between two neighboring periodic orbits?γ?,?γ+inR2, where {(γ±(n),˙γ±(n))}?Mρ, there is a globally minimizing curve γ with rotation number ρ whose lift is asymptotic to?γ?in the future and to?γ+in the past, and another asymptotic to?γ+in the future and to?γ?in the past.

Definition C.2.When Mρ is not a homotopically nontrivial invariant circle in T?T1, we say that Mρ isbroken. This includes case(2.b)and(3.b)in the above TheoremC.1.

LetRbe the region(called Birkhoffregion of instability)bounded byφ1-invariant homotopically non-trivial Jordan curves Γ?and Γ+withρ(Γ?)<ρ(Γ+),whereρ(Γ±)is the rotation number ofφ1|?！? Note that allMρ ?Rwithρ(Γ?)<ρ<ρ(Γ+) is broken. We cite the following theorem of Mather.

Theorem C.2([24]).

1. Suppose ρ(Γ?)<α,ω<ρ(Γ+). Then there is an orbit of φ1whose α-limit set lies in Mα and whose ω-limit set lies in Mω. Furthermore, if ρ(Γ?)(resp. ρ(Γ+))is irrational, then this conclusion still holds with the weaker hypothesis ρ(Γ?)≤α, (resp. α, ω≤ρ(Γ+)).

2. Consider for each i∈Za real number ρ(Γ?)≤ωi≤ρ(Γ+)and a positive number εi. Then there exists an orbit(···,Pj,···)and an increasing bi-infinite sequence of integers j(i)such that dist.(Pj(i),),Mω(i))<εi.

Acknowledgements

The author would like to express his deep gratitudes to Professor S.-T. Yau for suggesting the topic. He also would like to thank Mr. Yifan Guo for drawing all the figures. The author is supported by NSFC (Significant project No. 11790273)in China and Beijing Natural Science Foundation (No. Z180003).