Chaotic particle motion around a homogeneous circular ring
KKEK-Cosmo-255KEK-TH-2227
Chaotic particle motion around a homogeneous circular ring
Takahisa Igata ∗ KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization,Tsukuba 305-0801, Japan (Dated: August 18, 2020)We consider test particle motion in a gravitational field generated by a homogeneouscircular ring placed in n -dimensional Euclidean space. We observe that there exist no stablestationary orbits in n = 6 , , . . . ,
10 but exist in n = 3 , , n = 3, we show that the separation of variables of the Hamilton-Jacobiequation does not occur though we find no signs of chaos for stable bound orbits. Sincethe system is integrable in n = 4, no chaos appears. In n = 5, we find some chaotic stablebound orbits. Therefore, this system is nonintegrable at least in n = 5 and suggests that thetimelike geodesic system in the corresponding black ring spacetimes is nonintegrable. ∗ [email protected] a r X i v : . [ g r- q c ] A ug I. INTRODUCTION
Circular ring structure appears in many areas of physics, from elementary particles to theuniverse. For example, planetary rings in astronomy are Newtonian gravitational phenomena inwhich ring shape appears clearly. In a strong gravity regime, the ring structure appears in anaccretion disk of a compact object, and therefore, such system is modeled by a relativistic solutionwith a ring source [1–3]. In a regime where gravity is extreme, there exist ring-shaped singularitiesat the center of the Kerr black hole [4]. Even in higher-dimensional spacetimes, which are activelystudied in relativity and particle physics [5], a ring appears as a black hole or a fundamental objectsuch as a closed string. A typical example in relativity is a black ring spacetime, an exact solutionto the 5D vacuum Einstein equation, of which the horizon topology and the central singularitiesare ring-shaped [6].In the gravitational fields of these rings, particle dynamics is basic for understanding the phe-nomena occurring in the system. The dynamics of particles in a gravitational field generated bya homogeneous circular ring source in 3D space was numerically analyzed in detail in Ref. [7]; inparticular, they focused on periodic orbits and classified them. Since particle motion constrainedto the 2D plane on which the ring lies is integrable because of the conservation of energy and an-gular momentum, the complexity of periodic orbits is relatively low. On the other hand, periodicorbits that deviate from the symmetric plane are relatively complicated, which leads the authors tospeculate that such nature comes from nonintegrability of the system. Note that the integrabilityof this system is nontrivial and has not yet been concluded. Certainly, separation of variables ofthe equation of motion is unlikely to occur because the Newtonian potential includes the completeelliptic integral of the first kind. However, since the separability of an equation of motion is asufficient condition for its integrability [8], we cannot conclude that it is nonintegrable just becauseit does not occur. Answering the question of whether this system is integrable or not is one of themotivations for this study.Now let us recall why the integrability of a test particle system is so significant. We call asystem integrable if there are as many Poisson commutable conserved quantities as or more thanthe system’s degrees of freedom. This nature relates to the predictability of a system becauseif it is nonintegrable, trajectories may exhibit chaotic behavior. Such trajectories are generallycomplicated and sensitive to changes of initial conditions. On the other hand, the predictability ispreserved if a system is integrable. Then we can also use constants of motion to learn about thesymmetry of systems and backgrounds. In fact, the so-called hidden symmetry of the Kerr blackhole spacetime was discovered using a nontrivial constant found in the proof of the integrability ofthe geodesic equation [9–11]. This is known today to be the fundamental quantity that characterizesthe Kerr geometry. In other words, clarifying the integrability is an effective way to discover thesystem’s hidden symmetry.The Newtonian potential sourced by a homogeneous circular ring in 4D space appears naturallyin the Newtonian limit of the black ring solution. The equation of motion with this potentialis separable and, therefore, integrable [12]. This property must be due to the simplicity of thepotential form compared to the 3D case. It is noteworthy that a massive particle system (i.e.,timelike geodesic system) on the singly rotating black ring spacetime [13–17], which restores theNewtonian potential in the weak gravity limit, exhibits chaos, i.e., the geodesic equation is non-integrable [18]. As suggested in this example, the integrability of particle systems tends to berecovered in the Newtonian limit. Other known examples are that the timelike geodesic system inthe Schwarzschild spacetime is integrable while its Newtonian limit, the Kepler problem, is super-integrable, and that in a static dihole spacetime is chaotic [19, 20] while its Newtonian limit, theEuler’s 3-body problem, is integrable [21]. Thus, it is quite natural to speculate that if a particlesystem is chaotic in the corresponding Newtonian gravitational field, the chaotic nature will alsoappear in a geodesic system on a relativistic gravitational field.The Newtonian potential due to a ring source is known to have a parity of spatial dimension; itcontains complete elliptic integrals when n is odd but has a simpler structure when n is even [22–24]. Based on this property and the above observations, let us make the following conjecture: A particle system moving in a potential sourced by a homogeneous circular ring in n -dimensionalEuclidean space is nonintegrable if n is odd and is integrable if n is even. If it is true, we canpredict that timelike geodesics in black ring spacetimes with an even number of spatial dimensionbehave chaotically.The purpose of this paper is to verify the above conjecture. We use the Poincar´e map as anindicator of chaos. Therefore, we first identify a region where there are stable stationary orbitsfor each spatial dimension. These orbits are so fundamental as to be comparable to stable circularorbits and are important regardless of its integrability. Increasing an energy from the level of astable stationary orbit, we inevitably find stable bound orbits in its vicinity. We consider theemergence of chaotic nature by evaluating the Poincar´e section for the stable bound orbits.This paper is organized as follows. In Sec. II, after presenting the explicit form of the Newto-nian gravitational potential sourced by a homogeneous circular ring, we derive conditions for theexistence of stationary orbits in terms of an effective potential of particle dynamics and clarifycriteria for determining whether a stationary orbit is stable or unstable. In Sec. III, we show theregions where stable stationary orbits exist in each dimension according to the prescriptions devel-oped in Sec. II. We apply the Poincar´e map method to stable bound orbits that appear associatedwith stable stationary orbits and attempt to determine the chaotic nature of particle dynamics.Section IV is devoted to a summary and discussions.
II. FORMULATION
We consider the dynamics of a particle moving in a Newtonian gravitational potential sourcedby a homogeneous circular ring in n -dimensional Euclidean space E n ( n ≥ g ij be theEuclidean metric, which is given by g ij d x i d x j = d ζ + ζ d ψ + d ρ + ρ dΩ n − , (1)where i , j are 1 , , . . . , n , and ( ζ, ψ ) are polar coordinates in 2D plane, and ( ρ, φ , . . . , φ n − ) arespherical coordinates in the remaining ( n − n − is the metric on theunit ( n − ≤ ρ < ∞ for n ≥ −∞ < ρ < ∞ for n = 3. Let R be theradius of a homogeneous circular ring and M be the total mass. Then the Newtonian potentialsourced by the ring is given by [24]Φ n ( r ) = − GM ( n − r n − F (cid:18) , n − , z (cid:19) , (2)where G is the gravitational constant, and F is the hypergeometric function, and z and r ± aredefined by z = 1 − r − r , (3) r ± = (cid:112) ( ζ ± R ) + ρ , (4)respectively. The range of z is restricted in 0 ≤ z <
1. Some specific forms of Φ n for relativelysmall values of n can be written as follows:Φ ( r ) = − GMπ K ( z ) r + , (5)Φ ( r ) = − GM r + r − , (6)Φ ( r ) = − GM π E ( z ) r + r − , (7)Φ ( r ) = − GM r + r − ) (cid:18) r + r − + r − r + (cid:19) , (8)Φ ( r ) = − GM πr − (cid:20) − r − r K ( z ) + 2 r − r + (cid:18) r − r (cid:19) E ( z ) (cid:21) , (9)Φ ( r ) = − GM r + r − ) (cid:18) r r − + 23 + r − r (cid:19) , (10)Φ ( r ) = − GM πr − (cid:20)(cid:18) r − r + + 7 r − r + 8 r − r (cid:19) E ( z ) − r − r (cid:18) r − r (cid:19) K ( z ) (cid:21) , (11)Φ ( r ) = − GM r + r − ) (cid:18) r r − + 35 r + r − + 35 r − r + + r − r (cid:19) , (12)where K ( z ) is the complete elliptic integrals of the first kind, and E ( z ) is the complete ellipticintegrals of the second kind. Whether n is even or odd makes considerable difference to the shapeof Φ n [22–24].We consider the dynamics of a particle moving in Φ n . Let m be mass of a particle and p i becanonical momenta. The Hamiltonian is given in the form H n = 12 m ( p ζ + p ρ ) + V n , (14) V n = L mζ + Q mρ + m Φ n , (15)where L = p ψ , and Q = γ ab p a p a (for n ≥ γ ab is the inverse of the metric on the unit ( n − a , b label ( φ , . . . , φ n ). Both L and Q are constants of motion associated withaxial symmetry in the ( ζ, ψ )-plane and spherical symmetry in the remaining ( n − n = 3, the term Q / (2 mρ ) in V disappears. We call V n theeffective potential in what follows. Since H n does not depend on time explicitly, it is also constant,which coincides with the conserved energy E , i.e., H n = E . (16) The convention of the complete elliptic integrals of the first and second kind is K ( z ) = (cid:90) π/ d θ (cid:112) − z sin θ , E ( z ) = (cid:90) π/ (cid:112) − z sin θ d θ. (13) Now we focus on stationary orbits, i.e., particle orbits in which ζ and ρ coordinates remainconstant. These orbits appear when initial conditions are given to stay at an extremum point of V n . From the equations of motion and the energy conservation law (16), we obtain the conditionsfor the existence of stationary orbits ∂ ζ V n = 0 , (17) ∂ ρ V n = 0 , (18) V n = E . (19)In the remainder of this section, we analyze these conditions for n ≥ n = 3, which isformulated differently from these cases, will be analyzed separately in Sec. III A. Solving Eq. (17)for L and Eq. (18) for Q , we obtain L = L := − GM m ζ r n +2+ (cid:20) R ( R + ρ − ζ ) F (cid:18) , n , z (cid:19) − ( ζ + R ) r F (cid:18) , n − , z (cid:19)(cid:21) , (20) Q = Q := GM m ρ r n +2+ (cid:20) RζF (cid:18) , n , z (cid:19) + r F (cid:18) , n − , z (cid:19)(cid:21) . (21)Note that stationary orbits exist only in the region where these squared angular momenta do nottake negative values. The energy of a particle in a stationary orbit is given by E = V n | L = L ,Q = Q . (22)Next, let us consider conditions for stability of stationary orbits. A stable stationary orbit isa state in which a particle on the orbit can remain in its vicinity even if a small perturbation isapplied. In this state, a particle stays at a local minimum point of V n . To determine that theextremum point of V n is a local minimum or not, we use the determinant and the trace of theHessian of V n , h ( ζ, ρ ; L , Q ) = det ∂ ζ V n ∂ ζ ∂ ρ V n ∂ ρ ∂ ζ V n ∂ ρ V n , (23) k ( ζ, ρ ; L , Q ) = tr ∂ ζ V n ∂ ζ ∂ ρ V n ∂ ρ ∂ ζ V n ∂ ρ V n . (24)Using these quantities, we define a region D n in the ζ - ρ plane by D n = (cid:8) ( ζ, ρ ) | L ≥ , Q ≥ , h > , k > (cid:9) , (25)where h and k are h and k evaluated at a stationary point, respectively, i.e., h = h ( ζ, ρ ; L , Q ) , (26) k = k ( ζ, ρ ; L , Q ) . (27)This gives the region where stable stationary orbits exist. III. STABLE STATIONARY ORBITS AND CHAOTIC ORBITSA. n = 3 We consider stable stationary orbits of a particle moving in the Newtonian potential Φ . Theexplicit form of the effective potential V is given by V ( r ) = L mζ − GM mπ K ( z ) r + , (28)where the term proportional to Q in Eq. (15) does not exist. In this case, the condition (18) takesthe form ∂ ρ V = 2 GM mπ ρE ( z ) r − r + = 0 . (29)Since E ( z ) > ≤ z <
1, this holds only on the symmetric plane, ρ = 0. The condition (17)restricted on ρ = 0 yields L = L = GM m ζ π (cid:18) K ( z ) ζ + R + E ( z ) ζ − R (cid:19) , (30)where z = 4 Rζ/ ( ζ + R ) . The squared angular momentum L is not negative in the range R < ζ < ∞ . (31)Therefore, the stationary orbits in Φ exist only on ρ = 0 within the range (31). Note that all ofsuch stationary orbits are circular orbits.Let us further restrict the inequality (31) to the range where stable circular orbits are allowedto exist. We introduce the determinant h ( ζ, ρ ; L ) and the trace k ( ζ, ρ ; L ) of the Hessian of V [see Eqs. (23) and (24)] and define h and k by h := h ( ζ, L ) = 4 G M m E ( z ) π ζ ( ζ − R ) (cid:18) K ( z ) − R ( ζ − R ) E ( z ) (cid:19) , (32) k := k ( ζ, L ) = 2 GM mπζ (cid:18) K ( z ) ζ + R + E ( z ) ζ − R (cid:19) , (33)respectively. Both of these are positive in the range ζ ISCO < ζ < ∞ , (34)where ζ ISCO /R = 1 . · · · and is determined by solving h = 0 [24, 25]. After all, the stablestationary orbits in n = 3 exist in the region D = { ( ζ, ρ ) | ρ = 0 , ζ ISCO < ζ < ∞} , (35)and all of them are stable circular orbits. We refer to ζ ISCO as the radius of the innermost stablecircular orbit (ISCO), as in the case of black hole spacetimes.We use these results to consider the integrability of this system. In general, when a particlein a stable stationary orbit gains some positive energy, it moves away from the local minimum ofthe effective potential. However, if a potential contour at an acquired energy level has still closedshape, then the particle remains confined in a finite region of the vicinity of the local minimumpoint. We call such orbits stable bound orbits. They provide information about chaotic natureof particle motion through the method of the Poincar´e map. In Figs. 1, we show typical stablebound orbits in V (upper panels) and the Poincar´e sections (lower panels), where L is chosen sothat the local minimum point of V coincides with the point ( ζ, ρ ) = ( ζ , E is chosen sothat the contour of V = E (red solid curves) is closed and almost a separatrix. The black solidcurves show the contours of V . The blue solid curves show particle trajectories with energy E ,which are confined inside each red closed curve. Though we have chosen three different parametersets in Figs 1-(a)–1-(c), all of these trajectories appear to be some sort of Lissajous figures, whichis a sign when stable bound orbits are not chaotic. In fact, the corresponding Poincar´e sectionsfor various initial conditions with fixing L and E draw closed curves, as seen in the lower panelsof Figs. 1, where the section is placed at constant- ζ plane, and phase space coordinates ( ρ, p ρ ) arerecorded when a particle passes through the section with p ζ >
0. Therefore, within the presentanalysis, we do not find any chaotic nature.In nonintegrable systems, as is known in, e.g., the H´enon-Heiles system [26], the degree of chaosoften increases if a particle in a stable bound orbit approaches a separatrix. It is worth notingthat in our case, although particles approach separatrices, no chaos has emerged. However, thevisualization of chaotic nature in this way may be hindered because the existence of the ISCOprevents stable bound orbits from being close enough to the ring.The fact that signs of chaos are hard to capture may mean that this system is integrable. Letus discuss this possibility below. One of the powerful methods for analyzing the integrability of - - - ρ - - p ρ - - ρ - - - p ρ - ρ - - - p ρ - - - ζ ρ - - ζ ρ - - - ζ ρ (a) ⇣ = 1 . , E = . ⇣ = 2 . , E = . ⇣ = 2 . , E = . FIG. 1. Typical shapes of stable bound orbits in Φ (upper panels) and Poincar´e sections with the sameenergy and angular momenta but different initial positions and velocities (lower panels). Units in which R = 1, m = 1, and GM = 1 are used. The local minimum point of V is located at ( ζ, ρ ) = ( ζ ,
0) in eachcase. In the upper panels, the black and the red solid curves show contours of V ; in particular, the redcorresponds to V = E . Each blue solid curve shows a stable bound orbits with energy E . Each point in thelower panels show a value ( ρ , p ρ ) of a particle that passes through a constant- ζ surface with p ζ >
0. Thirtyorbits with different initial conditions are superposed in each plot. equations of particle motion is the Hamilton-Jacobi method because a sufficient condition for theintegrability is the separation of variables of the Hamilton-Jacobi equation. It is known that theseparability is closely related to the existence of the rank-2 Killing tensors [8]. Therefore, even inour present case, clarifying the existence of nontrivial constants of motion associated with rank-2Killing tensors is a useful way to learn about the integrability. As a starting point of our discussion,we adopt a rank-2 reducible Killing tensor, i.e., a linear combination of the flat metric tensor andthe symmetric tensor products of Killing vectors, K ij = α g ij + (cid:88) A =1 6 (cid:88) B =1 α AB ξ ( iA ξ j ) B , (36)0where α and α AB are constants, and ξ i = ( ∂/∂x ) i , ξ i = ( ∂/∂y ) i , ξ i = ( ∂/∂z ) i , (37) ξ i = y ( ∂/∂z ) i − z ( ∂/∂y ) i , ξ i = z ( ∂/∂x ) i − x ( ∂/∂z ) i , ξ i = x ( ∂/∂y ) i − y ( ∂/∂x ) i (38)are the Killing vector in E , which are represented by the standard Cartesian coordinates ( x, y, z ) =( ζ cos ψ, ζ sin ψ, ρ ). We assume α AB = α ( AB ) because the antisymmetric part of α AB does notcontribute to K ij . Let us focus on a quadratic quantity in p i written by K ij as C = K ij p i p j + K, (39)where K is a scalar function, and without loss of generality, we have assumed that C does notcontain the first-order term of p i because, even assuming that it is included, it eventually disappearsin the following analysis. In the remainder of this section, we use units in which m = 1. If C is aconstant of motion, then the pair of K ij and K must satisfy the Killing hierarchy equations [27, 28] g ij ∂ i K kl − K ij ∂ i g kl = 0 , (40) ∂ i K = 2 K ij ∂ j Φ , (41)where K ij = g ik K kj (see a brief review in Appendix A). Our Killing tensor (36) is a solution to thefirst equation (40), which is the rank-2 Killing tensor equation in E . Our next task is to clarifywhether there is a nontrivial solution to the second equation (41) for K with K ij in Eq. (36) as asource. From the conditions for K to be integrable, ∂ [ i ∂ j ] K = 0, both Φ and K ij must satisfy thefollowing relation: ∂ [ i ( K j ] k ∂ k Φ ) = 0 , (42)which leads to the restriction of the components of α AB as α AB = α α α α
00 0 α α α α α α . (43)Therefore, the integrability condition for K restricts the form of K ij as K ij = α g ij + α ξ i ξ j , (44)1where we have assume α = 0 because we can rescale α . Using the restricted form (44) as thesource of Eq. (41), we obtain K = 2 α Φ , (45)where we have removed a constant term. Finally, we find that C consists of the sum of the knownconserved quantities, C = 2 α H + α L , (46)which is not independent from H and L . From these results, we conclude that the separationof variables of the equation of motion does not occur. Note that, however, this result does notnecessarily mean that the system is nonintegrable. For example, there may exist a constant ofmotion that is higher-order in p i more than rank-2 [29] or nonpolynomial form [30]. We needfurther analysis to clarify the integrability of this system, which is an important task for thefuture. B. n = 4 We consider stable stationary orbits in the Newtonian potential Φ . The explicit form of theeffective potential V is given by V = L mζ + Q mρ − GM m r + r − . (47)As formulated in Eqs. (20) and (21), the two squared angular momenta for stationary orbits aregiven by L = GM m ζ ( ζ + ρ − R ) r r − , (48) Q = GM m ρ ( ζ + ρ + R ) r r − . (49)The squared angular momentum Q does not take a negative value everywhere, while L is notnegative in the range of ζ + ρ ≥ R or ζ = 0, and hence only in which the stationary orbits exist.At the points where L vanishes, the gravitational force in the ζ direction is just balanced. FromEq. (22), the energy in stationary orbits is given by E = − GM mR ( R + ρ − ζ )2 r r − . (50)2 ζ ρ FIG. 2. Region D , the allowed region for stable stationary orbits in Φ . Units in which R = 1 are used.The circular ring source is located at ( ζ, ρ ) = (1 , D . The blue solid anddashed curve are the boundaries of D and are determined by L = 0 and h = 0, respectively. The region D is colored in blue when E < E > Furthermore, h and k in Eqs. (26) and (27) reduces to h = 16 G M m R r r − (cid:2) ( ζ + ρ ) ( R − ζ + ρ ) − R ( R − ζ )( R + ρ ) (cid:3) , (51) k = 4 GM m ( ζ + ρ ) r r − , (52)respectively. The positivity of k does not make any restriction to D because it is not negativeeverywhere, while the positivity of h restricts D . Figure 2 shows the numerical plot of D , whichis drawn by the shaded region. The solid blue curve denotes the boundary of D determined by L = 0, i.e., ζ + ρ = R , and the dashed blue curve the boundary of D determined by h = 0.The region D coincides with the region of stable stationary orbits allowed in the asymptoticallyfar from the thin black ring in 5D spacetime; on the other hand, a difference appears in theirvicinity [15]. We find that a stable stationary orbit exists arbitrary close to the Newtonian ring,but in the black ring, the last stable orbit appears, which does not reach the horizon.As was shown in Ref. [12], the Hamilton-Jacobi equation of this system causes the separationof variables in the spheroidal coordinate system, and hence this system is integrable. In relationto the recent work on the Newtonian analogue of the Kerr black hole [31], our potential Φ isconsistent with the time-time metric component of the 5D singly rotating Myers-Perry black hole(see Appendix B). This implies that the integrability of the particle system in Φ is closely related3to the integrable property of the timelike geodesic equation in the 5D black hole. Whether or notthere is a further correspondence in particle dynamics, etc., other than the integrability remainsan open question. C. n = 5 We consider stable stationary orbits in the Newtonian potential Φ . The effective potential in n = 5 is given by V = L mζ + Q mρ − GM m π E ( z ) r + r − . (53)At an extremum point of V , the squared angular momenta L and Q take the form L = GM m ζ πr r − (cid:2) r − ( R − ζ + ρ ) K ( z ) − (cid:2) r r − + 8 ζ ( R − ζ − ρ ) (cid:3) E ( z ) (cid:3) , (54) Q = 2 GM m ρ πr r − (cid:2) r + r − ) E ( z ) − r − K ( z ) (cid:3) , (55)respectively, and the energy E is E = − GM m πr − r (cid:2) R + 2 R ( ρ − ζ ) − ζ + ρ ) E ( z ) + r − ( ζ + ρ − R ) K ( z ) (cid:3) . (56)Using Eqs. (54)–(56) and h and k defined in Eqs. (26) and (27), we obtain D , where stablestationary orbits exist, as shown in Fig. 3. The region D is drawn by the shaded region, wherethe energy E is negative in blue shaded region and is positive in orange shaded region. The innerboundary of D denoted by a solid blue curve is determined by L = 0. Here corresponds to abalance point of the gravitational force in the ζ direction. The outer boundary of D denoted by adashed blue curve is determined by h = 0. In contrast to D and D , which indicate unboundedregions, D is distributed in a bounded region near the source.Now, we investigate the chaotic nature of this system by using stable bound orbits as inSec. III A. Each upper panel in Figs. 4 draws a certain stable bound orbit (blue curve) with initialconditions at E = 0, where angular momenta L and Q are chosen so that V takes a local minimumpoint at ( ζ, ρ ) = ( ζ , ρ ). Black and red solid curves are contours of V , and the red is V = 0.Each of the lower panels in Figs. 4 depicts Poincar´e sections for stable bound orbits of particleswith the same E , Q , and L but different initial positions and velocities. Each section is placed in aplane where ζ is constant, and phase space coordinates ( ρ, p ρ ) are recorded when a particle passesthrough the section with p ζ >
0. In Figs. 4-(a), we find a stable bound orbit in the vicinity ofthe axis of symmetry, which shows a Lissajous-like pattern. However, the corresponding Poincar´e4 ζ ρ FIG. 3. Region D , the allowed region for stable stationary orbits in Φ . Units in which R = 1 are used.The circular ring source is located at ( ζ, ρ ) = (1 , D . The blue solid anddashed curves are the boundaries of D and are determined by L = 0 and h = 0, respectively. The region D is colored in blue when E < E > sections show that although some of plotted points lie on closed curves on the ρ - p ρ plane, some ofthese structures are broken. As the contour of V = 0 approaches the ring such as in Figs. 4-(b), astable bound orbit no longer shows a pattern like the Lissajous figure, and the structure of closedcurves in Poincar´e sections is broken for many initial conditions. Their properties are more pro-nounced for stable bound orbits in the vicinity of the ring, such as in Figs. 4-(c). These resultsindicate chaotic nature, and therefore, we conclude that this system is a nonintegrable system. D. n ≥ We consider stable stationary orbits in Φ n of the case n ≥
6. According to the prescription inSec. II, some numerical searches for n = 6 , , . . . ,
10 show that the region D n does not exist in the ζ - ρ plane, D n = ∅ for n = 6 , , . . . , . (57)With this result, there are inevitably no stable bound orbits for n = 6 , , . . . ,
10. Since we cannotuse the method of the Poincar´e map without stable bound orbits, we need to use other criteria fordetermining chaos to conclude the integrability in these cases. The result (57) leads us to expectthe absence of stable stationary orbits for particles in Φ n for n ≥ ζ ρ ρ - - - p ρ ζ ρ ρ - - p ρ ζ ρ ρ - p ρ (a) ( ⇣ , ⇢ ) = (0 , . , E = 0 (b) ( ⇣ , ⇢ ) = (0 . , . , E = 0 (c) ( ⇣ , ⇢ ) = (0 . , . , E = 0 FIG. 4. Typical shapes of stable bound orbits in Φ (upper panels) and Poincar´e sections with the sameenergy and angular momenta but different initial positions and velocities (lower panels). Units in which R = 1, m = 1, and GM = 1 are used. The values ( ζ , ρ ) denote the location of the local minimum point of V in each case. The black and red solid curves of each upper panel are contours of the effective potential V , which take 1 , − , − (black), and 0 (red). The blue curves show stable bound orbits with E = 0.The plot in the ρ - p ρ plane in each lower panel show the Poincar´e sections. Thirty orbits with different initialconditions are superposed in each plot. IV. SUMMARY AND DISCUSSIONS
We have considered the dynamics of particles moving in a gravitational potential sourced bya homogeneous circular ring in n -dimensional Euclidean space. In each dimension below n = 11,we have clarified the regions where stable stationary orbits exist. In n = 3, all of such orbits arestable circular orbits and exist only on the symmetric plane outside the ISCO radius, which islarger than the ring radius. In n = 4, there are no stable stationary orbits on the symmetric plane,but rather in an unbounded region connected to the axis of symmetry. In n = 5, stable stationaryorbits exist in an bounded region connected to the axis of symmetry and do not exist at infinity.In n = 6 , , . . . ,
10, no stable stationary orbits exist in whole region. These results would predict a6region of stable stationary/bound orbits of massive particles in the far region from thin black ringsin n ≥
4. At least in n = 4, the region of the existence of stable stationary orbits revealed in theNewtonian mechanics are consistent with those in the asymptotic region of the known black ringsolution.Furthermore, using stable bound orbits that appear associated with stable stationary orbits,we have analyzed chaotic nature of particle dynamics in n = 3 and 5, in which cases system’sintegrability is unknown so far. We have not found any chaotic nature in n = 3 by means ofthe Poincar´e map. It should be noted that this result does not guarantee the integrability of thesystem. However, by showing that there are no nontrivial constants of motion associated withany rank-2 Killing tensors in E , at least we have clarified that the separation of variables of theHamilton-Jacobi equation does not occur. If this system is integrable, the proof of integrabilitymust be achieved not by the separation of variable but by finding a constant of motion morethan second-order in momentum or a non-polynomial constant. On the other hand, in n = 5, thePoincar´e sections show a sign of chaos, indicating that the system is nonintegrable.Our results suggest that the system of a freely falling particle (i.e., timelike geodesic) in 6Dblack ring spacetimes is nonintegrable. At the same time, they strongly suggest that there are nohidden symmetries, such as the Killing tensors. Therefore, finding 6D black ring solutions basedon the ansatz that assumes a hidden symmetry would not work well.Our conjecture in the introduction holds so far for n = 4 and 5. We should further discussthe appearance of chaos for odd dimensions (i.e., n = 3 , , , . . . ) and should reveal integrabilityfor even dimensions (i.e., n = 6 , , , . . . ). Since there are various characterizations of chaos, itis important to check the chaos in several different indicators, not only in the Poincar´e map. Forexample, in the current system with periodic motions, it may be useful to evaluate homoclinictrajectories and chaos analytically using the Melnikov method (see, e.g., Ref. [32]). This is aninteresting issue for the future. ACKNOWLEDGMENTS
This work was supported by Grant-in-Aid for Early-Career Scientists from the Japan Societyfor the Promotion of Science (JSPS KAKENHI Grant No. JP19K14715).7
Appendix A: Integrability condition of the Killing hierarchy equation
We review the condition for the existence of a constant of particle motion that is quadratic ina momentum [27, 28]. Let us focus on particle motion under some scalar potential force. We useunits in which particle mass m is unity in this section. Then the Hamiltonian generally takes theform H = 12 g ij p i p j + Φ( r ) , (A1)where g ij is the inverse metric of the background space, and p i are canonical momenta, and Φ( r )is a potential. We introduce a dynamical quantity C in the form of a second-order polynomial ofmomenta, C = K ij p i p j + K, (A2)where, without loss of generality, we have assumed that C does not contain a first-order term ofmomentum. Even assuming that it is included, that term eventually disappears in the discussionbelow.If C is a constant of motion, then the Poisson bracket of H and C must disappear: { H, C } = ∂H∂p i ∂C∂x i − ∂H∂x i ∂C∂p i (A3)= ( g ij ∂ i K kl − K ij ∂ i g kl ) p j p k p l + ( g ij ∂ i K − K ij ∂ i Φ) p j (A4)= 0 . (A5)Since p i in Eq. (A4) can be any value of the on-shell, the coefficients for each order of momentamust disappear. As a result, we obtain the Killing hierarchy equation as shown in Eqs. (40) and(41). Appendix B: Newtonian analogue of a singly rotating Myers-Perry black hole
The metric of the Myers-Perry black hole that rotates in a single plane is given in the Boyer-Lindquist coordinates by g µν d x µ d x ν = − d t + µr D − Σ (d t − a sin θ d φ ) + Σ∆ d r + Σd θ + ( r + a ) sin θ d φ + r cos θ dΩ D − , (B1)8where µ and a are mass and spin parameters, respectively, andΣ = r + a cos θ, ∆ = r + a − µr D − , (B2)where we use units in which G = 1 and c = 1 (see, e.g., Ref. [5]). We define a Newtonian potentialΨ from the time-time component of the metric (B1) asΨ = − g tt − µ r D − Σ . (B3)In the oblate spheroidal coordinates, r = aξ, (B4) θ = cos − η, (B5)we obtain Ψ as Ψ = − µ a D − ξ D − ( ξ + η ) . (B6)In making the further coordinate transformation, z = aξη, (B8) ρ = a (cid:112) (1 + ξ )(1 − η ) , (B9)where ξ ∈ [0 , ∞ ) and η ∈ [ − , ξ = R + ρ + z − a a , (B10) η = R − ρ − z + a a , (B11)where R = (cid:112) ( ρ + z − a ) + 4 a z = (cid:112) [( ρ − a ) + z ][( ρ + a ) + z ] , (B12)finally we obtain the following form of Ψ:Ψ = − ( D − / µR ( R + ρ + z − a ) ( D − / . (B13) The new coordinates are related to the Boyer-Lindquist coordinates as ρ = (cid:112) r + a sin θ, z = r cos θ. (B7) D = 4, i.e., in the case of the Kerr black hole, under the complex π/ a → ia , the potential Ψ corresponds to that of the Euler’s 3-body problem with equalmass m = m = M/ | D =4 = − GM/ (cid:112) ( z + a ) + ρ − GM/ (cid:112) ( z − a ) + ρ , (B14)where µ = 2 GM . In the viewpoint of the separability of the equations of particle motion, theEuler’s 3-body problem is closely related to the particle system of the Kerr spacetime (see recentprogress in Ref. [31]). In D = 5, we can find that the potential Ψ reduces to the formΨ | D =5 = − µ R . (B15)This corresponds to the Newtonian potential of a homogeneous circular ring with radius a placed inthe 4D Euclidean space (see, e.g., Ref. [24]) without any complex transformation of the parameter.As known in Ref. [12], the equation of motion of a particle moving in this potential is integrable.This fact seems to be closely related to the integrability of the timelike geodesic equation of the 5Dsingly rotating Myers-Perry black hole spacetime. For D ≥
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