Stable bound orbits around a supersymmetric black lens

It is an issue of physical interest whether a black hole spacetime has bound orbits of particles. The existence of stable circular orbits is one of characteristic features in the four-dimensional black holes. For instance, let us consider the effective potential of the radial motion of a massive particle in Schwarzschild spacetimes. For the four dimensions, the centrifugal force and gravitational force make a local minimum of the potential corresponding to a stable circular orbit, whereas for the higher dimensions, the effective potential has no local minimum because near the event horizon the gravitational potential becomes more dominated than for four dimensions. This leads to the absence from stable circular orbits in higher-dimensional cases in contrast to the four-dimensional case.

It is an issue of physical interest whether a black hole spacetime has bound orbits of particles.
The existence of stable circular orbits is one of characteristic features in the four-dimensional black holes. For instance, let us consider the effective potential of the radial motion of a massive particle in Schwarzschild spacetimes. For the four dimensions, the centrifugal force and gravitational force make a local minimum of the potential corresponding to a stable circular orbit, whereas for the higher dimensions, the effective potential has no local minimum because near the event horizon the gravitational potential becomes more dominated than for four dimensions. This leads to the absence from stable circular orbits in higher-dimensional cases in contrast to the four-dimensional case.
In an asymptotically flat, stationary and bi-axisymmetric five-dimensional spacetime, the spatial cross section of an event horizon can have nontrivial topologies of a ring S 1 × S 2 and lens spaces L(p, q) as well as a sphere S 3 [1][2][3][4]. For the spherical topology, the exact solutions were found in five-dimensional Einstein theory [5,6] and five-dimensional minimal supergravity [7]. It was shown that in the vacuum black hole spacetime, there are no stable circular orbits in equatorial planes (rotational axes) [8][9][10][11]. Furthermore, for the supersymmetric black hole spacetime, the complete analytical solution of the geodesic equations was presented and the properties of massive and massless test particle motion were systematically studied [13,14]. For the ring topology, the corresponding solutions were found in both the theories [15][16][17]. Remarkably, in contrast to a spherical black hole, it was explicitly shown that the black ring admits the existence of stable bound orbits [18][19][20]. For the lens space topologies, the supersymmetric solutions with the horizon topologies of L(n, 1) were first constructed in the five-dimensional minimal supergravity [21,22].
Because it seems that the black lens cannot admit the separability of the geodesic equations, we cannot show the existence/nonexistence of stable bound orbits by merely solving a one-dimensional potential problem in the radial direction.
The three-dimensional lens spaces are mathematically defined as simply, quotients of S 3 by Z/p-actions. Therefore, as in a black hole spacetime with the horizon topology of S 3 , we expect intuitively that there may be no stable bound orbits in a black lens spacetime. However, in this paper, contrary to the expectation, we indeed show that there are stable bound orbits around a supersymmetric black lens. The metric of the supersymmetric black lens solution can be written as a timelike fiber bundle over the multi-centered Gibbons-Hawking space [23]. The essential difference from the supersymmetric black hole with the horizon topology of S 3 is the existence of centers outside the horizon, so-called nuts. The existence of the centers significantly changes the motion of particles orbiting around the horizon because the potential diverges by the centrifugal force effect of particles orbiting around them. This strongly suggests that there exist local minimums of the effective potential, which admits the existence of stable bound orbits of particles.
From the requirements of regularity at r = r i (i = 2, · · · , n) and the absence of closed timelike curves around the horizon and r = r i (i = 2, . . . , n), the parameters (k i≥1 , l 1 , z i≥2 ) must be subject for each i = 2, . . . , n and If the parameters are subject to these constraints, the point r = r 1 (= 0) corresponds to a null degenerate horizon with the spatial cross section of the lens space L(n, 1), whereas the points r = r i (i = 2, . . . , n) denote the coordinate singularities like the origin of the Minkowski spacetime.
The case n = 1 corresponds to the BMPV black hole, for which the inequality (11) and the condition (12) are not imposed. The case n = 2 coincides with the black lens solution with the horizon topology of L(2, 1) [21], in which case Eq. (12) is reduced to and the inequalities (11) and (13) become The shaded regions in Fig. 1-2 show the parameter region where all of the inequalities (14), (15) and (16) are simultaneously satisfied for the normalized l 1 by l 1 = 1 and l 1 = −1, respectively. The Hamiltonian of a free particle with the mass m is given by where p µ is the canonical momentum. It is obvious from independence of H on the coordinates (t, φ 1 , φ 2 ) that (p t , p φ 1 , p φ 2 ) are constants of motion, which we denote by (p t , p φ 1 , p φ 2 ) = (−E, L φ 1 , L φ 2 ). Then, the Hamiltonian can be written in terms of these constants as where U is the effective potential defined by with l φ 1 := L φ 1 /E and l φ 2 := L φ 2 /E. The massive and massless particles move on the two- negative local minimum, so that there exist stable bound orbits for massive particles. Futhermore, because the equation U = 0 has three roots z = z in , z out , z ∞ (z in < z out < z ∞ ) for z > z 2 ≈ 32.8 such that U > 0 for z < z in , U < 0 for z in < z < z out , U > 0 for z out < z < z ∞ and U < 0 for z > z ∞ there are stable bound orbits of the massless particles in the range of z in ≤ z ≤ z out . It is shown from these figures that as |l φ 1 | is larger, the width ∆z := z out − z in gets smaller, asymptotically approaches to zero for l φ 1 → ∞. This shows that there are stable circular orbits of massless particles for large |l φ 1 | because ∆z ≈ 0 means that such massless particles stay at z = const.
Next, we move on the behaviors of the effective potential on I − . In Fig. 4, the blue, orange and green curves correspond to (l φ 1 , l φ 2 ) = (0, 16), (0, 13.44...) and (0, 30), respectively, in the same choice of the parameters (k 1 , k 2 , l 1 ). At the horizon z = 0, the gravitational force makes the potential diverge to −∞. The potential U seems to have no local minimum but a local maximum, which takes zero for l φ 1 = 0, l φ 2 ≈ 13.45. This shows that there exist unbounded orbits of massless particles on I − (it seems that there exist no stable bound orbits of massive and massless particles as far as we have checked numerically). Finally, we consider the effective potential on the interval I 1 , which has the two centers at endpoints z = z 1 (= 0) and z = z 2 . Note that only the particles with the angular momenta of the special ratio of L φ 1 /L φ 2 = l φ 1 /l φ 2 = −2 are allowed to stay on I 1 because I 1 corresponds to the fixed points of the Killing isometry v := ∂/∂φ 1 + 2∂/∂φ 2 , and hence only the particle with a zero angular momentum of J := p µ v µ = L φ 1 + 2L φ 2 = 0 can stay on the axis I 1 (for the potential diverges, which means that I 1 corresponds to the potential barrier). Figure 5 shows effect of the gravitational force causes the potential to diverge to −∞. On the other hand, near the other center z = z 2 , due to the effect of the centrifugal force, the potential again increases monotonically and then diverges to ∞ except for N = 0 (l φ 1 = l φ 2 = 0). Thus, the potential also has a positive minimum on I 1 . This is the reason why in contrast to the higher-dimensional Schwarzschild spacetimes, there exist stable bound orbits of massless particles as well as massive particles in a black lens spacetime with multiple centers.
In this paper, we have numerically clarified the existence 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. It can be expected that the stable bound orbits also exist even for the more general topologies L(n, 1) (n ≥ 3) for the following reason. The supersymmetric black lens with the horizon topology L(n, 1) [22] has (n − 1) centers at r = r i (i = 2, . . . , n) outside the horizon, by which the z-axis of E 3 in the Gibbons-Hawking space is split into the (n + 1) intervals: and I + = {(x, y, z)|x = y = 0, z > z n }. For the same reason as in the case L(2, 1), only the particles with the angular momenta of the ratio l φ 1 /l φ 2 = −(n − i + 1)/(n − i) are allowed to stay on I i . At the centers z = z i and z = z i+1 on I i , the effective potential for particles with such angular momenta diverges to ∞ by the centrifugal force effect. Therefore, this effective potential is expected to have, at least, a single local minimum between adjacent two centers in the z-direction.
On the other hand, the effective potential for the particles staying on I i will also make a local minimum in the normal direction to the z-axis (i.e., in the θ direction) due to the gravitational attractive force of the black lens at z = z 1 (= 0). As a result, we will find stable bound orbits of massive/massless particles in each interval I i . This may lead to the existence of many stable bound orbits on the z-axis. We will analyze the more general case in a future paper.
The supersymmetric black lens admits the presence of evanescent ergosurfaces [21,22], which are defined as timelike hypersurfaces such that a stationary Killing vector field becomes null there and timelike everywhere except there. They appear at the points at which f = 0 corresponding to H = 0 (for example, for n = 2 and k 1 = 0, they exist at z = 2z 2 /3 and z = 2z 2 in the z-axis). Reference [24] proved that on such surfaces, massless particles with zero energy (E = 0) relative to infinity move along stable trapped null geodesics. In the above analysis, we remove such massless particles staying on the evanescent ergosurfaces because (l φ 1 , l φ 2 ) are divided by E (hence, for such particles, we must use a different effective potential, for instance, U = g φ 1 φ 1 L 2 φ 1 + g φ 2 φ 2 L 2 φ 2 + 2g φ 1 φ 2 L φ 1 L φ 2 ). However, we can know the motion of particles in the zero energy limit E → 0 from the potential U in the limit |l φ 1 | → ∞ or |l φ 2 | → ∞. It turns out from the right figure in Fig. 3 that for considerably large |l φ 1 | (= 40000), the potential U has the local minimum at z ≈ 65.6, which is just the position of the evanescent ergosurface on I + because 2z 2 ≈ 65.6. This means that there can be stable bound orbits of massless particles with zero energy (E = 0) staying on the evanescent ergosurface at z = 2z 2 because ∆z 0, whereas in the above analysis, we have shown that for massless particles with nonzero energy (E = 0), there are also stable bound orbits. Furthermore, it is shown in Ref. [24] that for the horizonless supersymmetric solutions, the presence of evanescent ergosurfaces makes some linear perturbation decay slowly and leads to nonlinear instability. This result does not directly apply to the supersymmetric black lens solution but the presence of stable bound orbits of particles with nonzero energy may exhibit corresponding nonlinear instability.