Geodesic motion around a supersymmetric AdS5 black hole

In this article the geodesic motion of test particles in the spacetime of a supersymmetric AdS$_5$ black hole is studied. The equations of motion are derived and solved in terms of the Weierstrass $\wp$, $\sigma$ and $\zeta$ functions. Effective potentials and parametric diagrams are used to analyze and characterize timelike, lightlike and spacelike particle motion and a list of possible orbit types is given. Furthermore, various plots of orbits are presented.


I. INTRODUCTION
The famous anti-de Sitter/conformal field theory (AdS/CFT) correspondence provides a relation between gravity and quantum field theory, in particular, Maldacena [1] connected compactifications of string theory on anti-de Sitter to a conformal field theory. Therefore, black holes, which are asymptotically anti-de Sitter, are very interesting to study.
A few years after Kerr [2] presented an asymptotically flat rotating black hole, Carter [3] came up with the first rotating asymptotically anti-de Sitter black hole. In five dimensions Hawking et al. [4] found an AdS black hole with two rotation parameters. Five dimensionial AdS black holes are especially interesting since the AdS 5 /CFT 4 correspondence is very well understood and CFT can be described as N = 4 SU(N ) super Yang-Mills theory. One of the first supersymmetric AdS 5 black hole solutions were found by Gutowski and Reall [5]. They considered the minimal D = 5 gauged supergravity theory described in [6], and found an asymptotically AdS 5 black hole parametrized by its mass, charge and two equal angular momenta. Many more black hole solutions in supergravity theories were found, see e.g. [7].
The motion of test particles is a useful tool to study black holes in various theories of gravity. The solutions of the equations of motion can be applied to calculate observable quantities like the shadow of a black hole or the periastron shift of a bound orbit. Geodesics also provide information on the structure of a spacetime. In the framework of AdS/CFT, geodesics correspond to two-point correlators [8]. In particular, spacelike geodesics with both endpoints on the boundary (i.e. escape orbits of particles with imaginary rest mass) are related to correlation functions. CFT correlators in the bulk describe observables on the AdS boundary.
The Hamilton-Jacobi formalism represents an efficient method to derive the equations of motion for test particles. In the four-dimesional Kerr spacetime, Carter [3] showed that the Hamilton-Jacobi equation for test particles separates. The resulting equations of motion can be solved analytically in terms of elliptic functions. In higher dimensions, or in spacetimes with a cosmological constant, the analytical solutions of the geodesic equations often require hyperelliptic functions [9][10][11][12][13][14]. The geodesics in a rotating supersymmetric black hole spacetime were analyzed in [15,16], where the complete analytical solution of the geodesics equations in the supersymmetric Breckenridge-Myers-Peet-Vafa (BMPV) [17] spacetime was presented. Here we will study the geodesic motion of test particles around the supersymmetric, asymptotically AdS 5 black hole of Gutowski and Reall [5].
The article is structured as follows. We derive the equations of motion in Section II and give a complete classification of the geodesics in Section III. In Section IV we solve the equations of motion analytically in terms of the Weierstraß ℘, σ and ζ functions. Finally we present some example plots of the orbits in Section V and conclude in Section VI.

II. THE SUPERSYMMETRIC ADS5 BLACK HOLE
The solution of a supersymmetric AdS 5 black hole is given by the metric [5] (1) * jens-christian.drawer@uni-oldenburg.de † saskia.grunau@uni-oldenburg.de arXiv:2002.05481v1 [gr-qc] 13 Feb 2020 This metric is characterized by its conserved quantities associated with symmetries of the conformal boundary, which was shown for asymtotically AdS spacetimes of dimension D ≥ 4 by Ashtekar and Das [18]. In this case the black hole's conserved quantities can be defined by an Ashtekar and Das mass an angular momentum with respect to φ = φ + 2 t l of a vanishing angular momentum with respect to ψ, an energy of and a charge of Here R 0 is the radial coodinate of the black hole's degenerate horizon, = ±1 is the sign of its angular momentum J and l is the AdS radius. Note that can be absorbed into l, consequently, in the following we set = 1 but examine the geodesic motion for arbitrary sign of l.
To solve the Hamilton-Jacobi equation of an uncharged particle we make the ansatz for the action S Here E is the particle's conserved energy, L and J are its conserved angular momenta along φ and ψ, respectively, τ is an affine parameter along the geodesic and δ is equal to 0 for light, equal to 1 for particles of positive mass and equal to −1 for particles of imaginary mass. The case δ = −1 corresponds to spacelike geodesics and is of relevance for AdS/CFT, if the geodesics' endpoints are on the boundary R → ∞. This is discussed in more detail in [14] and references therein.
Using this ansatz, one can separate the Hamilton-Jacobi equation (11) by terms in R and θ and introduce K as a separation constant known as Carter [19] constant. Furthermore, dimensionless quantities can be introduced by scaling with R 0 This is achieved by setting R 0 = 1 and canceling factors of 4 in front of L, J and K for convenience. With the help of the Mino [20] time γ given by R 2 dγ = dτ to remove the factor R 2 from all equations and the substitution r = R 2 , this yields five differential equations of motion dθ dγ The polynomial P and the function Θ are

III. CLASSIFICATION OF THE GEODESICS
The properties of the geodesics are determined by the polynomial P in Eq. (19) and the function Θ in Eq. (20). The characteristics of Θ and P are given by the particle's constants of motion (energy, angular momenta, Carter constant, δ parameter) and the metric's (positive or negative) AdS radius. In this section, features of the function Θ and the polynomial P -and therefore the types of orbits -for various sets of constants of motion are studied. This is done analogously to [21], where geodesic motion of electrically and magnetically charged test particles in the Reissner-Nordström spacetime has been examined.

A. The θ motion
To obtain real values of θ, the requirement Θ ≥ 0 has to be met. From this it follows that K ≥ 0. The substitution ξ = cos θ turns Eq. (15) into dξ dγ 2 = Θ ξ with Θ ξ := aξ 2 + bξ + c , where a = −L 2 − K, b = 2LJ and c = K − J 2 . Since K ≥ 0 holds, it follows a ≤ 0. The zeros of the second degree polynomial Θ ξ correspond to angles which confine the particle's θ motion. Note that for vanishing L in Eqs. (15) and (17) the test particle's motion is planar in the 3-dimensional subspace given by the spherical coordinates (R, θ, ψ) as in the Schwarzschild case. The Θ ξ polynomial's discriminant is given by D = b 2 −4ac and can be expressed as D = 4Kκ with κ = K +L 2 −J 2 . Θ ξ describes a downward opened parabola with zeros and maximum at LJ L 2 +K , Kκ L 2 +K . A real solution θ implies real zeros of Θ ξ and thus requires D ≥ 0. While |L| ≥ |J| is sufficient, other cases require an upper limit of |J| given by J max = √ K + L 2 . For symmetric motion with respect to the equatorial plane L or J have to vanish. Other cases are depending on the sign of K − J 2 : (1) K < J 2 : The zeros of Θ ξ are either both positive or both negative, which confines the particle's motion to θ ∈ [0, π/2) for LJ > 0 and θ ∈ (π/2, π] for LJ < 0.
with θ ∈ [0, π/2] for LJ > 0 and θ ∈ [π/2, π] for LJ < 0. With the additional condition |L| = |J|, the orbit fills an entire hemisphere (see Fig. 1 (3) K > J 2 : One zero of Θ ξ is positive and one is negative, allowing the particle to cross the equatorial plane and θ ∈ [0, π]. Similarly to an effective potential, this behavior can be seen in Fig. 1, where the allowed area of motion with respect to L is shown for different choices of K.
In case of a double zero ξ 0 of Θ ξ , the particle's motion is confined to a cone of opening angle arccos ξ 0 , which simplifies Eqs. (16) and (17). This is possible for K = 0 or κ = 0 in three cases: Since the θ motion is not depending on the particle's mass parameter δ, all results hold for all particle types.
B. The r motion

Possible types of orbits
For a degenerate horizon at r = 1 the following types of orbits can be found for this spacetime: (1) Escape orbits (EO) with range [r 1 , ∞) and 1 < r 1 .

Analysis of the radial motion
To obtain real values of r from Eq. (14), the requirement P ≥ 0 has to be met. Radial regions of physically allowed motion are separated from forbidden ones by the positive zeros of P, which correspond to the orbits' turning points.
Whenever the polynomial has a non negative double zero, that is, P(r) = 0 and dP dr (r) = 0 , a variation of parameters is expected to change the number of positive zeros. By plotting the zeros of the resultant of the two expressions in Eq. (23), one obtains parameter plots showing the boundaries between regions of 1, 2, 3 or 4 zeros of P. This is shown for parametric L-l, K-E and L-E diagrams in Fig. 2.
As P is a polynomial of second degree in E, one can define the two-part effective potential V ± (r) as the values of energy which yield P = 0, that is, P can be rearranged in the form Some effective potentials are shown in Figs. 3-5. Since one can factor out (r − 1) in V ± , V + and V − intersect on the horizon at E = 0, given that V ± is real. Additionally, for the requirement that time should always run forward, i.e., dt / dγ ≥ 0, Eq. (18) is treated in a similar way as for the effective potential and corresponding regions are shown as well.
Since P < 0 for δ = 1 and P > 0 for δ = −1 hold in the limit r → ∞, orbits of particles of positive mass are always bounded, particles of imaginary mass can exist in unbound orbits for arbitrary energy. More precisely, one finds for particles of non zero mass, i.e., δ = ±1, in the limiting case r → ∞ from P = 0 Similarly, for massless particles, i.e., δ = 0, it follows in the limit r → ∞ Since here V ∞ + > 0 and V ∞ − < 0 hold (for K > 0) and P < 0 for E = 0 in the limit r → ∞, for every set of parameters a physically forbidden region in the vicinity of E = 0 can be found. This behavior corresponds to the border of region (2) in Fig. 2(e) as varying the energy E to region (3) + allows for an additional unbound orbit.
In special cases, terminating orbits can be found. Due to the smoothness of the polynomial P, the condition P(0) ≥ 0 has to be fulfilled, which implies real V ± (0), since P(0) opens downward with respect to E. It follows giving K = 0 and E = Ll as first conditions. To allow for non trivial terminating orbits with non zero range, additionally P(r) > 0 has to hold on r ∈ (0, ε) with ε > 0. This implies that the lowest order non vanishing coefficient of P must be positive, which is a condition that is always met in case of δ = −1 but never possible in other cases. The influence of the choice of δ on the effective potential at the singularity around E = Ll is shown in Fig. 4.
Here parameters are chosen to allow for bound orbits behind the horizon for all particle masses δ. As regions with dt / dγ < 0 are shown as well, it can be seen that all depicted bound orbits cross regions where time is running backwards. For a particle of non zero mass this can be avoided by lowering its energy to the bottom of the potential well. A short-range terminating orbit can be found in Fig. 4(c) at E = Ll. Additionally, the two possible types of terminating orbits can be seen in Figs. 5(a) and 5(b).
With the results of the parametric diagrams and the effective potentials we summarize all combinations of zeros of P and show their corresponding types of orbits in Table I. Here the following connection of regions in the parametric diagrams and types of orbits was found: (1) Region (1): P has one non negative zero r 1 < 1, which corresponds to a TEO of type A. For particles of imaginary mass a TO of type A 0 is possible for an energy of E = Ll (see e.g. Fig. 5(a)). (2) Region (2): P has two positive zeros r 1 < 1 < r 2 , which corresponds to a MBO of type B.  Fig. 5(b)). (4) Region (4): P has four positive zeros r i .
(a) Region (4) + : It holds r 1 < 1 < r 2 , r 3 , r 3 , resulting in a MBO and a BO of type E. (b) Region (4) − : It holds r 1 , r 2 , r 3 < 1 < r 4 , resulting in a BO and a MBO of type F. Note that for every radially allowed orbit an angular momentum J according to Section III A can be chosen to allow for θ motion as well.     Table I and are described in Section III B 2. In cases (d)-(i) one double zero of P for E = 0 and all K or L can be found. Since no regions of different numbers of zeros are separated by these lines (see Fig. 3 for E = 0), they are omitted.  Table I) behind the horizon. For a more detailed description see Fig. 3. Additionally, the turnaround energy Eturn of the φ motion is shown as a dash-dotted line which mostly overlaps with the border of the hatched area corresponding to dt / dγ < 0.   Table I). (c): Effective potential V± of the r motion with parameters chosen to allow for a pointy petal BO of type E. For a more detailed description see Fig. 3. Additionally, the turnaround energy Eturn of the φ motion is shown as a dash-dotted line.

C. Static orbits
It has been shown in [22] that some axisymmetric rotating spacetimes possess a ring in the equatorial plane, on which stationary particles remain stationary with respect to an asymptotic static observer. As this is possible in higher dimensions as well (see [22] and references therein), we check whether this is the case for the spacetime at hand. Hence, parameters have to be chosen to allow for an extremum of V ± at r st and a double zero in Eq. (22) at cos θ st . Additionally, the right hand sides of Eqs. (16) and (17) have to vanish. We use cos θ st = LJ L 2 +K from Eq. (22), under conditions discussed in Section III A, together with Eq. (17), from which the requirement K = 0 and |J| ≤ |L| can be derived. With this, a turnaround energy E turn can be defined by solving Eq. (16) for E yielding E turn (r) = (r − 1)Ll 2r 2 + 2r − 1 .
This is also shown in Figs. 4 and 5, since here K = 0 holds. To obtain a stationary point, E turn has to intersect V ± in an extremum of V ± . In this spacetime, we could not find such intersections. However, pointy petal BOs (in front of the horizon), semi BOs (behind the horizon) as well as MBOs, where the particle is periodically at rest, are found at arbitrary intersections of E turn and V ± for particles of positive mass. In case of a pointy petal BO, an effective r potential and φ turnaround energy is shown in Fig. 5(c).

IV. SOLUTION OF THE GEODESIC EQUATIONS
In this section we solve the equations of motion (14)-(18) analytically.
A. Solution of the θ equation Eq. (21) can be solved by an elementary function. In case of a < 0 and D > 0, this leads to as a solution of Eq. (15). Here γ 0 and θ 0 are the initial values of γ and θ, respectively, we set γ θ 0 = arcsin 2a cos θ0+b √ D and "±" denotes the sign of dθ dγ (γ 0 ).

B. Solution of the r equation
By substituting r = 1 x + r P , with r P chosen to be a zero of P, one can simplify Eq. (14) to a differential equation of the type dx dγ 2 = P x 3 with a polynomial P x 3 := 3 i=0 b i x i of third order on its right hand side (this step is not necessary for δ = 0). The following substitution x = 1 b3 4y − b2 3 transforms this to dy dγ 2 = 4y 3 − g 2 y − g 3 =: P y 3 , with coefficients This elliptic differential equation is solved by the Weierstraß ℘ function [23], which leads to (33)
Again the substitution ξ = cos θ is used, this time upon B. Together with Eq. (21), one gets for the θ part Here the "±" indicates the sign of dθ dγ (γ 0 ). Upon the two fractions we apply partial fraction decompositions The first part can be easily integrated with the substitution y = ξ − 1, the second part with y = ξ + 1. The third part is equal to ∓L dγ, as can be seen from Eq. (21). This yields where Θ y,i := Furthermore, c i < 0 was assumed. Now we integrate the r dependent part A of Eq. (16). We use the two substitutions applied in Section IV B and recall Eq. (32) to identify A = A(r(y(γ))). A partial fraction decomposition on A(y) is performed to get where K 0 , K 1 and K 2 are constants of the partial fraction decomposition and p 1 = (l 2 +rP+2)b2−3b3 12(l 2 +rP+2) and p 2 = are poles of first order. According to [24] 1 , the elliptic integrals of third kind ν ν0 dν ℘(ν)−p can be solved in terms of the Weierstraß σ and ζ functions by using addition theorems. One obtains with ν : Here ν i is a Weierstraß transformed pole p i , which solves ℘(ν i ) = p i for i = 1, 2 in the fundamental parallelogram of ℘(ν). The solution of Eq. (34) is given by the sum of this expression, Eq. (37) and the initial value φ 0 .

D. Solution of the ψ equation
To obtain the ψ motion from Eq. (17), we proceed similarly to the solution of the θ part of the φ motion in Section IV C. Again, with the substitution ξ = cos θ and partial fraction decompositions, one gets Using the definitions of Eq. (38) we find with the initial value ψ 0 of ψ E. Solution of the t equation The integral of the right hand side of Eq. (18) can be derived similarly to that of A(r) in Section IV C. Here the partial fraction decomposition leads to with H 0 , H 1 , H 2 and H 3 given by the partial fraction decomposition and p 1 and p 2 are poles of first and second order, respectively, that are identical to those in Eq. (39). The integral of the first three terms yields an expression of the type obtained in Eq. (40). Again, according to [24] 2 , the elliptic integral of type ν ν0 dν (℘(ν)−p) 2 can be solved in terms of the Weierstraß σ and ζ functions. One obtains with ν := γ − γ 0 and the initial value t 0 of t (44)

V. THE ORBITS
To plot the spatial coordinates (φ, ψ, θ, r) of the set of analytical solutions for the particle's motion in a Cartesian coordinate system, a coordinate transformation has to be chosen. The angular line element dΩ 2 = 1 On the other hand, the angular line element of flatspace in biazimuthal coordinates (ϕ 1 , ϕ 2 , ϑ, R) is A comparison of Eqs. (45) and (46) suggests the coordinate transformation which, together with the biazimuthal coordinates and r = R 2 , transforms (φ, ψ, θ, r) to (x, y, z, w) by Furthermore, we choose a projection in the 3-dimensional Cartesian subspace of the coordinates (x, y, z) by setting w = 0. This preserves the notion of a horizon on the surface of a sphere with radius R = 1, but as a consequence we have to set ψ = −φ, so only analytical solutions of φ, θ, and r are used. Another consequence is that the θ motion on the interval [0, π] is now mapped to the upper hemisphere only, thereby, as the horizon is shown in the following as a complete sphere of radius 1, the lower half sphere is of no physical relevance.
The figures below show the particle's motion for various sets of parameters. The first orbit of a particle of positive mass is of type E and is shown in Fig. 6, its effective potential can be found in Fig. 3(d). For similar parameters (L was decreased) a lightlike orbit of type C is shown in Fig. 7. Both of those orbits pass the equatorial plane. An orbit that is confined to one hemisphere has to satisfy K < J 2 , as was discussed in Section III A. A lightlike orbit of this kind is shown in Fig. 8 for type C.
In case of K = 0 and κ > 0, one obtains an orbit confined to a cone of fixed opening angle arccos J |L| . This is shown in Fig. 9 for type C. A two-world escape orbit of type A can be found in Fig. 10(a). This is obtained by increasing the energy slightly above the effective potential's maximum while the other parameters are unchanged compared to Fig. 7. By the increase in energy, the many-world periodic bound orbit and the escape orbit merged into one two-world escape orbit. Now we present orbits of particles with imaginary mass that reach the boundary at infinity and therefore are of relevance for AdS/CFT. A terminating orbit of type A 0 , with its effective potential depicted in Fig. 5(a), can be found in Fig. 10(c). In Fig. 11 we show a terminating orbit and a two-world escape orbit of type D 0 of a particle with imaginary mass in an effective potential very similar to the one shown in Fig. 5(b).
The pointy petal bound orbit of type E of a particle that is periodically at rest is depicted in Fig. 12, its effective potential in Fig. 5(c). Here parameters are chosen by determining the intersection of the effective potential V ± and turnaround energy E turn while θ is, for simplicity, chosen to be constant, as discussed in Section III C.
Finally, a bound orbit behind the horizon for a particle of positive mass (type F) is shown in Fig. 13. This requires small K and large L. Its effective potential is very similar to the one shown in Fig. 4(a) but its trajectory is not confined to a cone of fixed opening angle θ.

VI. CONCLUSION
In this article the spacetime of a supersymmetric AdS 5 black hole was studied by analyzing the geodesics (elliptic) equation of motion and deriving its analytical solutions in terms of the Weierstraß ℘, σ and ζ functions. Effective potentials and parametric diagrams were used to classify possible types of orbits, which are characterized by the particle's energy, angular momenta, Carter constant and mass parameter as well as the metric's AdS radius.
We showed that timelike orbits are always bounded and thus do not reach the AdS boundary. For lightlike and spacelike geodesics multiple types of orbits with a boundary at infinity and therefore with relevance for AdS/CFT were found. This, for spacelike geodesics and for a specific energy, includes the possibility of a terminating orbit. Bound orbits behind the horizon (for large angular momentum L) and many-world periodic bound orbits are possible independently of the particle's mass. However, stable bound orbits outside of the horizon are only possible for particles of positive mass.
Future work might focus on extending the equations of motion to particles with electric and magnetic charge. Additionally, the orbits' observables, e.g., in case of a bound orbit, its periastron shift or, in case of a lightlike escape orbit, its light deflection or the black hole's shadow, might be calculated similarly as in [13] by making use of the analytical solutions.      The trajectory lies on a cone with opening angle θ < π/2. The corresponding effective potential of the r motion can be found in Fig. 5(a). The spheres and dashed circles show the horizon at R = 1. The trajectory lies on a cone with opening angle θ < π/2. The sphere and dashed circle show the horizon at R = 1. The corresponding effective potential of the r motion is very similar to the one shown in Fig. 5(b). The corresponding effective potential and turnaround energy of the r and φ motion, respectively, can be found in Fig. 5(c). The particle periodically stops at the outer turning points.  Fig. 4(a).