Null geodesics in five-dimensional Reissner-Nordstr\"om anti-de Sitter black hole

The study of the motion of photons around massive bodies is one of the most useful tools to know the geodesic structure associated with said gravitational source. In the present work, different possible paths projected in an invariant hyperplane are investigated, considering five-dimensional Reissner-Nordstr\"om anti-de Sitter black hole. Also, we study some observational test such as the bending of light and the Shapiro time delay effect. Mainly, we found that the motion of photons follows the hippopede of Proclus geodesic, which is a new type of trajectory of second kind, being the Lima\c{c}on of Pascal their analogue geodesic in four-dimensional Reissner-Nordstr\"om anti-de Sitter black hole.


Introduction
Extra-dimensional gravity theories have a long history that begins with an original idea propounded by Kaluza & Klein [1,2] as a way to unify the electromagnetic and gravitational fields, and nowadays finds a new realization within modern string theory [3,4]. The geodesics of massive test particles in higher dimensional black hole spacetimes has been studied in Refs. [5][6][7], and it was shown that a particular feature of Reissner-Nordström spacetimes is that bound and escape orbits traverse through different universes, and the study of the motion of particles in five-dimensional spacetimes has been performed in Refs. [8][9][10][11][12][13][14][15].
The spacetime that we consider in this study is a generalization of Reissner-Nordström anti-de Sitter (RNAdS) black hole to five dimensions, that are interesting in the context of the AdS/CFT correspondence [16][17][18]. Global fivedimensional Schwarzschild AdS solution was considered to describe a thermal plasma of finite extent expanding in a slightly anisotropic fashion [19]. Also, it was shown that four and five-dimensional charged black holes in AdS spacetime could be obtained by compactifications of the type IIB supergravity in eleven dimensions. The properties of Reissner-Nordstrom black hole in d-dimensional anti-de Sitter spacetime has been studied in Refs. [20,21], and the null geodesic structure of four-dimensional RNAdS black holes was analytically investigated in Ref. [22], where, concerning to the radial motion, it was shown that the photons arrive to the event horizon in a finite proper time, and infinite coordinate time, similar to the Schwarzschild case. Also, concerning to the angular motion of photons it was shown that there are five different kinds of motion for trapped photons depending on the impact parameter of the orbits, that corresponds to orbits where the photon arrives from infinity and falls into the event horizon, photons moving along the critical orbits that represent trajectories that come from infinity and fall asymptotically into a circle, photons falling from infinity arriving to some minimal distance and then going back to the infinity again, photon orbits described by Pascal Limaçon, which is an exclusive solution of black hole with the cosmological constant but it does not depend of the value of the cosmological constant, and finally confined orbits for the photons.
The aim of this work is to study the null geodesics in a five-dimensional charged black hole, and to see if it is possible to find orbits for the motion of photons different to the previously mentioned for a RNAdS spacetime. Here, we will find the null structure geodesic analytically, and interestingly enough we find a new kind of orbit called "Hippopede geodesics", that under our knowledge is the first time that has been reported in the literature.
It is worth mentioning that the same spacetime was considered in Ref. [8], where the null geodesics were studied from the point of view of the effective potential formalism and the dynamical systems approach. The radial and circular trajectories were investigated, and it was found that photons will trace out circular trajectories for only two distinct values of specific radius of the orbits. The dynamical systems analysis was applied to determine the nature of trajectories and the fixed points, and it was shown that the null geodesics have a unique fixed point and these orbits are terminating orbits. Also, the thermodynamics and the stability of the spacetime under consideration were studied via a thermodynamic point of view, and it was found special conditions on black hole mass and black hole charge where the black hole is in stable phase [23].
The paper is organized as follows. In section 2 we give a brief review of the spacetime considered. Then, in Sec. 3, we establish the null structure and we perform some test as the bending of light and the Shapiro time delay effect. Finally, we conclude in Sec. 4.
2 Five-dimensional Reissner-Nordström anti-de Sitter black holes Schwarzschild and Reissner-Nordström black hole solutions in d spacetime dimensions were presented by Tangherlini [24]. The five-dimensional Reissner-Nordström anti-de Sitter black holes are solutions of the equations of motion that arise from the action [21] where G 5 is the Newton gravitational constant in five-dimensional spacetime, R is the Ricci scalar, F 2 represents the electromagnetic Lagrangian, and Λ = −6/ 2 , is the cosmological constant where is the radius of AdS 5 space. The static and spherically symmetric metric that solves the field equation derived from the above action is given by where f (r) is the lapse function given by and dΩ 2 3 = dθ 2 + sin 2 θ dφ 2 + sin 2 θ sin 2 φ dψ 2 is the metric of the unit 3-sphere. Also, Q and M are the charge and the mass of the black hole, respectively. This spacetime allows two horizons (the event horizon r + , and the Cauchy horizon r − ), which are obtained from the equation f (r) = 0, or Now, with the change of variable x = r 2 − 2 /3, we obtain and the event and Cauchy horizons are given, respectively, by where ξ 0 = 2 α/3 and ξ 1 = 1 3 arccos − 3β 2 3 α 3 . Also, the extremal black hole is characterized by the degenerate horizon r ex = r + = r − , which is obtained when: Fig. 1, we plot curves for different values of Q that show the behaviour of the lapse function against r, we can observe that when the charge of the black hole Q increases we have a transition from a black hole to a naked singularity, passing by the extremal case.
Note that when Q = 0, the lapse function reduces to the five-dimensional Schwarzschild anti-de Sitter black hole, and the spacetime allows one horizon (the event horizon r + ) given by In Fig. 1, the blue line corresponds to the case Q = 0, we can observe that, for the same values of Λ and M , the event horizon is greater for a charged black hole than for uncharged.

The null structure
In order to obtain a description of the allowed motion in the exterior spacetime of the black hole, we use the standard Lagrangian formalism [25][26][27], so that, the corresponding Lagrangian associated with the line element (2) reads where L Ω is the angular Lagrangian: L Ω =θ 2 + sin 2 θφ + sin 2 θ sin 2 φψ 2 , and the dot indicates differentiation with respect to an affine parameter λ along the geodesic. Since the Lagrangian (8) does not depend on the coordinates (t, ψ), they are cyclic coordinates and, therefore, the corresponding conjugate momenta π q = ∂L/∂q are conserved. Explicitly, we have where E is a positive constant that depicts the temporal invariance of the Lagrangian, which cannot be associated with energy because the spacetime defined by the line element (2) is not asymptotically flat, whereas the constant L stands the conservation of angular momentum, under which it is established that the motion is performed in an invariant hyperplane. Here, we claim to study the motion in the invariant hyperplane θ = φ = π/2, soθ =φ = 0 and, from Eq. (11), Therefore, using the fact that L = 0 for photons together with Eqs. (10) and (11), we obtain the following equations of motion dr dt dr dψ where the effective potential V 2 (r) is defined by The effective potential for five-dimensional Schwarzschild anti-de Sitter black hole is obtained by setting Q = 0 in the above equation.

Radial motion
For the radial motion the condition L = 0 holds, which immediately yields to a vanishing effective potential, V 2 = 0. Consequently, the equations governing this kind of motion are and where the sign + (−) corresponds to massless particles moving toward the spatial infinite (event horizon). Assuming that photons are placed at r = r i when t = λ = 0, a straightforward integration of Eq. (17) yields which is plotted in Fig. 2. We observed that respect to the affine parameter the photons arrive at the horizon in a finite affine parameter, and when the photons move in the opposite direction, they require an infinity affine parameter to arrive to infinity, which does not depend with the charge of the black hole. This behaviour is essentially the same as that reported for the 4-dimensional counterpart [22]. On the other hand, Eq. (18) can be arranged and then integrated leading to the following expression where the functions t j (r) are given explicitly by with the corresponding constants, Thus, an observer located at r i will measure an infinite time for the photon to reach the event horizon, which also occurs in 3+1 dimensions. Nevertheless, when the test particles move in the opposite direction, they require a finite coordinate time for arrive to infinity given by the relation All previously described by Eqs. (19) and (20) is shown in Fig. 2. It is interesting to note that the behaviour given in (28) also appears in Lifshitz space-times [28,29], where it was argued that this corresponds to a general behavior of these manifolds [30], and also occurs in the three-dimensional rotating Hořava-AdS black hole [31].
On the other hand, for Q = 0 Eq. (19) is valid. However, the solution for the coordinate time is given by and Also, in the asymptotic region, r → ∞, the time to arrive at infinity reduces to It is possible to observe in Fig 2 that an observer located at r i will measure an infinite coordinate time for the photon to reach the event horizon, and it does not depend on the charge of the black hole. However, when the photons move in the opposite direction, they require a finite coordinate time to arrive to infinity, which decreases with the charge of the black hole.

Angular motion
Now we study the motion with L = 0, so we put our attention in Eq. (15), which, after using (16), is conveniently written as where b ≡ L/E is the impact parameter and B is the anomalous impact parameter, which is a typical quantity of the Anti-de Sitter spacetimes [26]. As a first approach, it is necessary to perform a qualitative analysis of the effective potential. So, we can observe in Fig. 3 the existence of a maximum potential placed at Note that for Q = 0, r u (Q = 0) = 2 √ 2M , and it is greater than r u for RNAdS; however, the maximum value of the potential E u (Q = 0) is smaller than E u for RNAdS. Next, based on the impact parameter values and Fig. 3, we present a brief qualitative description of the allowed angular motions for photons in RNAdS.
-Capture zone: If 0 < b < b u , photons fall inexorably to the horizon r + , or escape to infinity, depending on initial conditions, and its cross section, σ, in these geometry is [?] -Critical trajectories: If b = b u , photons can stay in one of the unstable inner circular orbits of radius r u . Therefore, the photons that arrive from the initial distance r i (r + < r i < r u , or r u < r i < ∞) can asymptotically fall to a circle of radius r u . The affine period in such orbit is and the coordinate period is -Deflection zone. If b u < b < b , the photons come from a finite distance r i (r + < r i < r u or r u < r i < ∞) to a distance r = r D (which is solution of the equation V (r D ) = E), then return to one of the two horizons. This photons are deflected. Note that photons with b ≥ b = are not allowed in this zone.
-Second kind and Hippopede geodesic . If b u < b < ∞, the return point is in the range r + < r < r u , and then the photons plunge into the horizon. However, when b = b a special geodesic can be obtain, known as Hippopede of Proclus.

Bending of light
Now, in order to obtain the bending of light we consider Eq. (35), which can be written as So, in order to obtain the return points, we solve the P(r) = 0. Thus, by performing the change of variable y = r 2 +B 2 /3, P(y) = y 3 −αy −β, wherẽ and the deflection distance r D is given by (43) and the return point r F is where χ 0 = 2 α/3 and χ 1 = 1 3 arccos 3β 2 3 α 3 . Then, after a brief manipulation, and performing the change of variable r = B 4x + 1/3 it is possible to integrate Eq. (40), given the following expression where the invariants are given by Therefore, by integrating the Eq. (45) and then solving for r leads to where ω D = ℘ −1 (r 2 D /4B 2 − 1/12). In Fig. 4 we show the behaviour of the bending of light. We can observe that the deflection angle is greater, when the black hole is uncharged. On the other hand, note that the above equations are straightforward obtained for five-dimensional Schwarzschild anti-de Sitter spacetime.

The deflection angle
It is known that photons can escape to infinity during a scattering process. So, by considering r(ψ)| ψ=0 = r D is the shortest distance to the black hole at which the deflection happens, and assuming that the incident photons are coming from infinity and escaping to infinity, we obtain the deflection angle asα = 2φ ∞ − π: The evolution of the deflection angle has been plotted in Fig. 5 which has an asymptotic behavior as E → E u . We can observe that the deflection angle takes an infinity value when the E = E u , such E u increases when the charge of the black hole increases.

Second kind trajectories and Hippopede geodesic
The spacetime allows second kind trajectories, when b u < b < ∞, where the return point is in the range r + < r < r u , and then the photons plunge into the horizon. However, a special geodesic can be obtained when the anomalous impact parameter B → ∞ (b = ). In this case, the radial coordinate is restricted to r + < r < r 0 , and the equation of motion (35) can be written as As it is expected, the deflection angle reaches its limit as E tends to E u which in this case is around 0.077 for RNAdS and 0.0725 for Q = 0. and the return points are: Thus, it is straightforward to find the solution of Eq. (50), which is given by which represents to the Hippopede of Proclus geodesic (see Fig. 6). This trajectory is a new type of orbits in five-dimensional RNAdS, and it does not depend on the value of the cosmological constant. It is worth mentioning that the analogue geodesic in four-dimensional RNAdS corresponds to the Limaçon of Pascal [22]. Also, when the spacetime is the five-dimensional Schwarzschild anti-de Sitter this geodesic is given by r = 2M cos[ψ], which describes a circumference with radius M , that is analogue to the cardioid geodesics found in four-dimensional Schwarzschild anti-de Sitter [26].

Critical trajectories and capture zone
In the case of b = b u , the particles can be confined on unstable circular orbits of the radius r u . This kind of motion is indeed ramified into two cases; critical trajectories of the first kind (CFK) in which the particles come from a distant position r i to r u and those of the second kind (CSK) where the particles start from an initial point d i at the vicinity of r i and then tend to this radius by spiraling. We obtain the following equations of motion for the aforementioned trajectories: where In Fig. 7, we show the behaviour of the CFK and CSK trajectories, given by Eq. (54). Note that for the second kind trajectories, r i must be replaced by d i in the constant C (56). On the other hand, for photons with an impact parameter less than the critical one (b u ), which are in the capture zone, they can plunge into the horizon or escape to infinity, with a cross section given by Eq. (37).

Shapiro time delay
An interesting relativistic effect in the propagation of light rays is the apparent delay in the time of propagation for a light signal passing near the Sun, which is a relevant correction for astronomic observations, and is called the Shapiro time delay. The time delay of Radar Echoes corresponds to the determination of the time delay of radar signals which are transmitted from the Earth through a region near the Sun to another planet or spacecraft and then reflected back to the Earth. The time interval between emission and return of a pulse as measured by a clock on the Earth is where r D as closest approach to the Sun. Now, in order to calculate the time delay we use Eq. (14), and by considering that dr/dt vanishes, thereby E 2 Thus, the coordinate time that the light requires to go from r D to r is given by So, at first order correction we obtain where Therefore, for the circuit from point 1 to point 2 and back the delay in the coordinate time is where Now, for a round trip in the solar system, we have (r D << r 1 , r 2 ) Note that the classical result of GR; that is, ∆t GR = 4M 1 + ln 4r1r2 . For a round trip from the Earth to Mars and back, we get (for r D r 1 , r 2 ), where r 1 ≈ r 2 = 2.25×10 8 Km is the average distance Earth-Mars. Considering r D , as closest approach to the Sun, like the radius of the Sun (R ≈ 696000 Km) plus the solar corona (∼ 10 6 Km), r D ≈ 1.696×10 6 Km, then, the time delay is ∆t GR ≈ 240 µ s. On the other hand, if we consider the limit M → M , Q = 0, and Λ = 0, in Eq. (66), we obtain ∆t ≈ 161ns. It is worth to mention that this value is more near to the value measure in the Viking mission, where the error in the time measurement of a circuit was only about 10 ns [32].

Final remarks
We considered the motion of photons in the background of five-dimensional Reissner-Nordström anti-de Sitter black holes, and we established the null structure geodesic. This spacetime is described by one Cauchy horizon and an event horizon. Concerning to the radial motion, we showed that as seen by a system external to photons, they will fall asymptotically to the event horizon. On the other hand, this external observer will see that photons arrive in a finite coordinate time to spatial infinite. Concerning to the angular motion, we found analytically orbit of first and second kind; and critical orbit. Interestingly, for second kind trajectory, we found that the motion of photons follows the hippopede of Proclus geodesic when the parameter of impact b takes the value b = , and it does not depend on the value of the cosmological constant, being the Limaçon of Pascal their analogue geodesic in four-dimensional Reissner-Nordström anti-de Sitter. On the other hand, we studied some observational test such as the bending of light, which have a similar behaviour that four-dimensional Reissner-Nordström antide Sitter, and the Shapiro time delay effect, where our results show that ∆t ≈ 161ns while that for GR ∆t GR ≈ 240 µ s.