Stable bound orbits in black lens backgrounds

In contrast to five-dimensional Schwarzschild-Tangherlini and Myers-Perry backgrounds, we show that there are stable bound orbits of massive/massless particles in five-dimensional black lens backgrounds, in particular, the supersymmetric black lens with $L(2,1)$ and $L(3,1)$ topologies. We also show that in the zero-energy limit of massless particles, there exist stable circular orbits on the evanescent ergosurfaces.


I. INTRODUCTION
A stable bound particle orbit is an orbit where a particle keeps moving in a bounded spatial region without reaching infinity or singularities even if small perturbations are applied. Such an orbit often appears in astrophysical phenomena because the stability results in a relatively long duration. The essence of the mechanism can be seen from the dynamics of a massive particle in the Schwarzschild spacetime. A massive particle moving on a stable bound orbit is localized near a radial potential well, which is made by a balance between the gravitational potential −M m/r and the centrifugal potential l 2 /(2m 2 r 2 ), where M is the black hole mass, m and l are the mass and angular momentum of a particle, and r is the circumferential radius. The relativistic correction effect −M l 2 /(m 2 r 3 ) becomes dominated near the event horizon and causes proper relativistic phenomena such as perihelion shift and the innermost stable circular orbits. These can explain or predict events in the vicinity of black holes and other celestial objects, such as stellar orbital motion and accretion disks. Similarly, it was shown that stable bound orbits exist in the Kerr black hole spacetime [1].
For photons moving in the Schwarzschild spacetime and Kerr spacetime, it is very well-known that there exist unstable circular orbits but not stable ones. However, in the Kerr-Newman spacetime with relatively large electric charge, stable photon orbits exist on the horizon [2], and in the 2 4D Majumdar-Papapetrou spacetimes with two black holes, they appear even outside the horizon [3][4][5]. The existence of such stable bound photon orbits has a physically significant meaning because it implies instability of the background spacetime in the following sense: If they exist, many massless particles (not only photons but also gravitons, etc.) are stably trapped on the orbits and accumulate more and more in the finite region of the spacetime. This will cause so large backreaction to the background geometry that it will eventually break the background. On the other hand, from the wave perspective, linear waves localize in the vicinity of the trapping null geodesics resulting in a long timescale for the decay [6]. This phenomenon suggests the existence of nonlinear instabilities of the background spacetime [7].
It is now evident that even within the framework of vacuum Einstein gravity, there is a much richer variety of black hole solutions in higher dimensions. For instance, an asymptotically flat, stationary and biaxisymmetric five-dimensional black holes can have three types of topologies of the event horizon, a sphere S 3 , a ring S 1 × S 2 and lens spaces L(p, q) [8][9][10][11]. The corresponding solutions are, respectively, called a black hole, a black ring [12][13][14] and a black lens [15,16]. For the Schwarzschild-Tangherlini solution and Myers-Perry solution in five dimensions, it was shown that there are no stable circular orbits in equatorial planes [17][18][19][20][21]. For the black ring solutions, it was shown in contrast to black holes that there exist stable bound orbits [22][23][24]. For the black lens solutions, in our previous work [25], we numerically show examples of stable bound orbits of particles around the supersymmetric black lens with the horizon topology of L(2, 1) in the five-dimensional minimal supergravity.
In this paper, focusing on the supersymmetric black lenses with L(2, 1) and L(3, 1) topologies in the five-dimensional minimal supergravity, we give more detail about the existence of the stable bound orbits for the massive and massless particles which move around the horizon than in the previous paper. In our analysis, we consider such motion of particles as a two-dimensional potential problem, where a problem of whether a stable bound orbit exists for massive or massless particles is reduced to a simple problem of whether the two-dimensional effective potential has a negative or zero local minimum. We also discuss if there are stable bound orbits at infinity because for the five-dimensional Majumdar-Papapetrou solution with two black holes [26], they can exist at infinity when the separation of two black holes is large enough.
The rest of the paper is composed as follows: In the following Sec. II, we briefly review the supersymmetric black lens solution in the five-dimensional minimal supergravity. In Sec. III, we provide our formalism to show the existence of stable bound orbits. In Sec. IV, we show that there 3 are stable bound orbits for supersymmetric black lenses with L(2, 1) and L(3, 1) topologies. In Sec. V, we summarize our results and discuss possible generalizations of our analysis to other black hole backgrounds.

A. Solutions
In the five-dimensional minimal supergravity, the local metric and gauge potential 1-form of the supersymmetric black lens solutions take the form [15,16] where ds 2 M is the Gibbons-Hawking metric, which is given by where r i := |r − r i | = x 2 + y 2 + (z − z i ) 2 , r := (x, y, z) and r i := (0, 0, z i ). For the constants z i , we assume z 1 = 0 < z 2 < · · · < z n and H is a harmonic function with point sources at r = r i on E 3 . The vectors ∂/∂t, ∂/∂ψ and ∂/∂φ := x∂/∂y − y∂/∂x are Killing vectors, where ∂/∂φ is the coordinate basis in the standard polar coordinates (x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ).

4
The other functions and one forms are written as where z ji := z j − z i and As discussed in Ref. [16], from the requirements of regularity at r = r i (i = 2, . . . , n) and the absence of closed timelike curves around the horizon r = r 1 and the (n − 1) points r = r i (i = 2, . . . , n), the parameters (k i≥1 , l 1 , z i≥2 ) must satisfy for i = 2, . . . , n and the inequalities It is shown in Ref. [16] that when the parameters simultaneously satisfy these conditions, the point r = r 1 (= 0) denotes a null degenerate horizon whose spatial cross section is the lens space 5 L(n, 1), whereas the points r = r i (i = 2, . . . , n) give regular points, which correspond to the merely coordinate singularities like the origin of the Minkowski spacetime in the Gibbons-Hawking coordinates.

B. Evanescent ergosurface
The supersymmetric black lens admits the presence of evanescent ergosurfaces [15,16], which are defined as timelike hypersurfaces such that a stationary Killing vector field becomes null there and timelike everywhere except there. Reference [27] proved that on such surfaces, massless particles with zero energy (E = 0) relative to infinity move along stable trapped null geodesics. They exist at f = 0, which corresponds to For n = 2, they cross the points z = 2z 2 /3 and z = 2z 2 on the z axis. For n = 3, they cross the points z satisfying on the z axis. It turns out from simple computations that F (z) = 0 has only a single root on I + and I 1 , two roots on I 2 and no root on I − .

C. New coordinates
In the work of the geodesic motion of massive/massless particles around the black lenses, it is more convenient to use the coordinates (η, ξ, φ 1 , φ 2 ) defined by than the Gibbons-Hawking coordinates (r, θ, φ, ψ), where (φ 1 , φ 2 ) are the coordinates with 2π periodicity. It should be noted that the points r i = (0, 0, z i ) (i = 1, 2, . . . , n) on the z axis correspond to (η, ξ) = (η i , 0) (i = 1, 2, . . . , n) on the η axis in the new coordinates, where η i := 2 √ z i . The case n = 2 coincides with a black lens solution with the horizon topology of L(2, 1) in Ref. [15]. Equation (15) is simply written as and the inequalities (14) and (16) are, respectively, The blue-shaded regions D in Fig. 1 show the parameter region where the inequalities (21), (22) and (23) are simultaneously satisfied for l 1 = 1 under Eq. (21). In particular, we consider the case of k 1 = 0, which simplifies the conditions (21)-(23) as E. n = 3 case The case n = 3 describes a black lens with the horizon topology of L(3, 1), which has 4 independent parameters (k 1 , k 2 , k 3 , l 1 ). In this paper, for simplicity, we consider only the case of k 1 = 0, in which case the conditions (15) are written as and the inequalities (14) and (16) are reduced to, respectively, From Eqs. (27) and (28), z 2 and z 3 can be written as the functions of k 2 , k 3 and l 1 . When we normalize l 1 = 1 from (31), the inequalities (29) and (30) can be denoted by a certain region in a (k 2 , k 3 )-plane. This parameter region D which gives a black lens with L(3, 1) topology is drawn as the blue-colored region in Fig. 2.

III. OUR FORMALISM
Our method to find stable bound orbits is based on the previous work [25], where we considered the geodesic motion of particles around the black lenses as a two-dimensional potential problem. Now, we give the brief review as follows. The Hamiltonian of a free particle with the mass m is written as where p µ is the momentum. From the independence of H on the coordinates (t, φ 1 , φ 2 ), the momenta (p t , p φ 1 , p φ 2 ) are constants of motion, and we denote them as Then, the Hamiltonian can be written in terms of these constants as 8 The function U is the effective potential defined by where we have normalized two angular momenta by the energy as l φ 1 := L φ 1 /E and l φ 2 := L φ 2 /E.
Thus we can consider that the massive and massless particles move on the two-dimensional space For simplicity, we focus on the shape of the potential U on the z axis (i.e., θ = 0, π) of E 3 in the Gibbons-Hawking space, which corresponds to η = 0 and ξ = 0 in the coordinates (η, ξ).
The z axis is composed of the n + 1 intervals: . . , n − 1) and I + = {(η, ξ)|η > η n , ξ = 0}. On I + and I − , only the particles with the angular momenta of l φ 2 = 0 and l φ 1 = 0, respectively, are allowed to stay there, whereas on I i (i = 1, . . . n − 1) only the particles with the angular momenta of the special ratio of l φ 1 /l φ 2 = −(n − i + 1)/(n − i) are allowed to stay there because I i corresponds to the fixed points of the Killing isometry v := (n − i)∂/∂φ 1 + (n − i + 1)∂/∂φ 2 , and hence only the particle with a zero angular momentum of J := p µ v µ = (n − i)L φ 1 + (n − i + 1)L φ 2 = 0 can stay on the axis I i and otherwise the potential diverges on I i .
In the above formalism, we must remove the zero-energy limit E → 0 because l φ 1 and l φ 2 are divided by E. For this limit, it is better to use the potential U defined by instead of the potential U , though in the previous work [25], we considered the zero-energy as the limit l φ 1 , l φ 2 → ±∞ of U . The potential U takes a simple form of For this zero energy limit, massive particles are not allowed to move because U is nonnegative, whereas massless particles can move on U = 0, which corresponds to the intersection of two curves in the (η, ξ) plane, H = 0 and G : where Ω 2 := H(η 2 + ξ 2 )/4f , so that it is enough to compute the determinant and trace of ∂ i ∂ j U , which are written, respectively, as where we have used For the particles with J = 0 and (L φ 1 , L φ 2 ) = (0, 0), one of the two conditions, G = 0, is always satisfied at least on I ± and I i (i = 1, . . . , n − 1) because χ φ = ±1 on I ± and χ φ = 2n − 2i + 1 on I i (i = 1, 2, . . . , n − 1). Therefore, for such particles with zero energy, U has a zero local minimum at the intersection of the evanescent ergosurfaces and the axes I ± and I i (i = 1, . . . , n − 1). This is why on such surfaces, massless particles with zero energy move along stable trapped null geodesics.
In the following sections, we will discuss it from the contours of U in the (η, ξ) plane.

IV. STABLE BOUND ORBITS
A. A black lens with the topology L(2, 1) In the previous work, we have seen the existence of stable bound orbits for the black lens with This means that there are stable bound orbits for massive/massless particles.
Furthermore, we consider whether there can exist such stable bound orbits at infinity z → ∞ (η → ∞, ξ = 0) on I + . To this end, let us expand the potential U at z → ∞ on I + as where the constants U ∞ and U ∞ are denoted by

11
After simple computations, we find that 0 < −U (1) ∞ 1 and U ∞ > 0 are the necessary conditions that U has a local minimum at infinity because the local minimum is at z −2U (2) ∞ . Noting that the condition 0 < −U (1) ∞ 1 can be denoted by l φ 1 = ±2 2(2k 2 2 + l 1 ) ∓ (0 < 1), we can confirm that the leading term of U ∞ can be written as From the leading terms, we find that U ∞ < 0, which cannot satisfy the other condition U  Let us discuss whether there can exist stable bound orbits at the other infinity z → −∞ (η = 0, ξ → ∞) on the z axis. To do so, we expand the potential U at z → −∞ as where the constants U (1) The inequalities 0 < U (1) −∞ 1 and U −∞ > 0 are necessary for U to have a local minimum at the infinity z → −∞. The condition 0 < U (1) −∞ 1 can be denoted by l φ 2 = ±2 2(2k 2 2 + l 1 ) ∓ (0 < 1), and the leading term of U (2) −∞ can be written as From the leading terms, we find that U −∞ < 0 for any parameter l 1 , k 2 , which cannot satisfy the necessary condition for the existence of stable bound orbits in the asymptotic region z → −∞, at least, on the z axis.
Moreover, let us expand U near the horizon z = 0 as and then we find that the first and second terms are negative, which is due to the effect of the attraction by the horizon. From Fig. 4, U does not make a local minimum on the z axis.

I 1
Finally, we consider the potential U on the interval I 1 , where the particles with the angular momenta of the special ratio of l φ 1 /l φ 2 = −2 are allowed to stay. Figure 5 shows the typical behavior of the potential U with a local minimum for the same set of parameters (k 1 , k 2 , l 1 ) = (0, 10, 1) and  Moreover, let us consider whether there exist stable bound orbits at infinity z → ∞. To this end, let us expand the potential U at z → ∞ as where and U ∞ can be written as a certain quadratic equation for l φ 1 . It turns out from this asymptotic behavior that the inequalities 0 < −U (1) ∞ 1 and U ∞ > 0 are required so that U has a local minimum at infinity because the local minimum is at z −2U (2) where η 2 = 6.36 · · · , η 3 = 47.91 · · · and the origin (0, 0) corresponds to the horizon.
under whose condition the sign of U ∞0 > 0 does not overlap the parameter region D, which means that there are no stable bound orbits on the z axis in the asymptotic region z → ∞.
Next we study the existence of stable bound orbits near the endpoint z = z 3 of I + , near which the potential behaves as where U (−1) We find numerically that U (−1) 3+ > 0 is always satisfied within the parameter region D, so that the potential diverges to ∞ at z = z 3 . Moreover, when 0 < U (1) 3+ < 0, U has a negative local minimum near z = z 3 on the z axis.
Let us consider the geodesic motion of the massless particles with zero-energy limit E → 0, in which case it is more convenient to use the potential U rather than U . In Fig. 7, the left figure shows the typical shape of the effective potential U for the particles with the angular momenta L φ 2 = 0, where we put (k 1 , k 2 , k 3 , l 1 ) = (0, 10, −50, 1) and (L φ 1 , L φ 2 ) = (1, 0). The right figure shows the contours of U for the particles with the same angular momenta. The potential U has a zero local minimum on the evanescent ergosurface, where massless particles with zero energy are stably trapped. the same angular momenta. The potential has a local minimum, whose values are zero on the evanescent ergosurfaces.

I −
Next, let us see the shapes of the effective potential on I − . The left figure of Fig. 8 shows the typical shape of the potential U with a local minimum, where this corresponds to the particles with the angular momenta of (l φ 1 , l φ 2 ) = (0, 50) for (k 1 , k 2 , k 3 , l 1 ) = (0, 10, −50, 1). The right figure shows the contours of U for the particles with the same angular momenta. The red curve corresponding to U = 0 is not closed. The potential U does not have a local minimum on the ξ axis (on the z axis corresponding to z < 0), but as is seen from the right contour plots of U , there are the closed contours U = U 0 (U 0 < 0) which cross the ξ axis. Therefore, massive particles crossing the ξ axis inside the region U ≤ U 0 turn out to be stably bounded. Furthermore, massive particles can also stably bounded inside closed contours apart from the ξ axis.
We discuss the existence of stable bound orbits at the infinity z → −∞ on the z axis. Similarly we expand the potential U at z → −∞ as where and U −∞ > 0 are the necessary conditions for U to have a local minimum at the infinity. The condition 0 < U (1) −∞ 1 can be denoted by l φ 2 = ±2 2(2k 2 2 + 2k 2 l 3 + 2k 2 3 + l 1 ) ∓ (0 < 1), then the leading term of −∞ can be written as U −∞0 + O( ). However, we can see numarically that the region U (2) −∞0 > 0 does not has an overlap region with the parameter region D, which means that there are no stable bound orbits in the asymptotic region z → −∞ on the z axis.
Near the horizon z = 0, U behaves as The negativity of the first and second terms makes U diverge to −∞ at the horizon z = z 1 (= 0), which is a pure effect of gravity.
Moreover, to consider the zero energy limit for particles with the angular momenta L φ 1 = 0, let us see Fig. 9. The left figure shows the effective potential U for the particles with (L φ 1 , L φ 2 ) = (0, 1) under the parameter setting (k 1 , k 2 , k 3 , l 1 ) = (0, 10, −50, 1). The right figure shows the contours of U for the particles with the same angular momenta. The potential U does not have a local minimum for the absence of evanescent ergosurface on the z axis. Massless particles with zero energy are not stably trapped.

I 1
Let us discuss the existence of stable bound orbits at the interval I 1 on the z axis. When we expand the potential U at z = z 2 as The signature of U (1) 2− does not depend on angular momenta l φ 1 and l φ 2 of particles, and U (1) 2− < 0 is satisfied within the whole parameter region. Therefore U diverges to ∞ at z = z 2 on I 1 . On the other hand, near the horizon z = 0, U behaves as The negativity of the first and second terms makes U diverge to −∞ at the horizon z = z 1 (= 0), which is a pure effect of gravity. In order that U has a local minimum near the horizon, U must 19 have a local maximum because U diverges to ∞ at z = z 2 . Hence, U (1) 1+ must be negative, which can be realized for sufficiently large l φ 1 = | − 3l φ 2 /2| because for |l φ 2 | → ∞, (58) Figure 12 shows the typical shape of the effective potential U . The left figure shows the effective potential on the η axis (η 1 ≤ η ≤ η 2 ) [on the z axis (0 ≤ z ≤ z 2 )] for the particles with the angular momenta (l φ 1 , l φ 2 ) = (−1500, 1000) under the parameter setting (k 1 , k 2 , k 3 , l 1 ) = (0, 10, −50, 1) and the right figure shows the contours of U for the particles with the same angular momenta. One can see from these figures that U has a negative local minimum on I 1 and inside the red curve U = 0, both massive and massless particles are stably bounded (in the finite region U ≤ −m 2 /E 2 and U ≤ 0, respectively). To consider the geodesic motion of the massless particles with zero-energy limit E → 0, let us see Fig. 11. The left figure shows the typical effective potential U for the particles with the angular momenta L φ 1 /L φ 2 = −3/2 under the parameter setting (k 1 , k 2 , k 3 , l 1 ) = (0, 10, −50, 1) and

I 2
Finally, let us discuss the existence of stable bound orbits on I 2 . This type of an interval does not exist for the black lens with L(2, 1) topology. When we expand the potential U at z = z 2 , z 3 Regardless of the angular momenta l φ 1 and l φ 2 of particles, U 2− and U (1) 3+ are positive and negative, respectively, within the parameter region D. Therefore U diverges to ∞ at z = z 2 and z = z 3 on I 2 . As is already expected, U has a local minimum on I 2 . Figure 12 shows the typical shape of the effective potential U . The left figure shows the effective potential on the η axis (η 2 ≤ η ≤ η 3 ) [on the z axis (z 2 ≤ z ≤ z 3 )] for the particles with the angular momenta (l φ 1 , l φ 2 ) = (−2000, 1000) under the parameter setting (k 1 , k 2 , k 3 , l 1 ) = (0, 10, −50, 1) and the right figure shows the contours of U for the particles with the same angular momenta. One can see from these figures that U has a negative local minimum on I 2 and inside the red curve U = 0, both massive and massless particles are stably bounded (in the finite region U ≤ −m 2 /E 2 and U ≤ 0, respectively).
Moreover, to consider the zero energy limit for particles with the angular momenta L φ 1 /L φ 2 = −2, let us see Fig. 13. The left figure shows the effective potential U for the particles with (L φ 1 , L φ 2 ) = (−2, 1) under the parameter setting (k 1 , k 2 , k 3 , l 1 ) = (0, 10, −50, 1). The right figure shows the contours of U for the particles with the same angular momenta. The potential has two the contours of U for the particles with the same angular momenta. The potential has local minima whose values are zero at the evanescent ergosurfaces.

V. SUMMARY AND DISCUSSIONS
In this paper, we have analyzed the geodesics for the supersymmetric black lenses with L(2, 1) and L(3, 1) topologies in the five-dimensional minimal supergravity. We have reduced the geodesic motion of particles to a two-dimensional potential problem, and from the two-dimensional contour plots of the potential, we have clarified the existence of the stable bound orbits for the massive and massless particles. In the previous work [25] on the black lens with L(2, 1) topology, where one-dimensional potential was used, we found the stable bound orbits of particles on the z axis corresponding to two rotational axes (η axis and ξ axis). In this work, we have found that there also exist stable bound orbits in a place away from the z axis as well as on the axis. Moreover, we have discussed the range such that stable bound orbits exist on the z axis and have shown that there exist no stable bound orbits in the asymptotic region z → ±∞, at least, on the z axis.
This result suggests that there exists the outermost stable circular orbit as the boundary of stable circular orbits.
In particular, we have also discussed the geodesic motion for massless particles with zero energy.
We have proved that for such particles, the potential always has a local minimum at the intersection of the evanescent ergosurface and the curve G(η, ξ, L φ 1 , L φ 2 ) = 0 in the (η, ξ) plane, where, in particular, for particles with the angular momenta satisfying either condition L φ 2 = 0 or J = 0, the intersection exists at least, at the rotational axes. This means that such particles are stably trapped on the geodesics on the the evanescent ergosurface, which is exactly consistent with the mathematical result [27], which proved that on an evanescent ergosurface, massless particles with zero energy move along stable trapped null geodesics. As shown in Ref. [27], for the microstate geometry, the existence of evanescent ergosurfaces leads to some nonlinear instability. This result does not apply to the solution with a black hole horizon but the presence of stable bound orbits of particles with zero energy even in a black lens background may exhibit corresponding nonlinear instability.