Motion of particles on a Four-Dimensional Asymptotically AdS Black Hole with Scalar Hair

Motivated by black hole solutions with matter fields outside their horizon, we study the effect of these matter fields in the motion of massless and massive particles. We consider as background a four-dimensional asymptotically AdS black hole with scalar hair. The geodesics are studied numerically and we discuss about the differences in the motion of particles between the four-dimensional asymptotically AdS black holes with scalar hair and their no-hair limit, that is, Schwarzschild AdS black holes. Mainly, we found that there are bounded orbits like planetary orbits in this background. However, the periods associated to circular orbits are modified by the presence of the scalar hair. Besides, we found that some classical tests such as perihelion precession, deflection of light and gravitational time delay have the standard value of general relativity plus a correction term coming from the cosmological constant and the scalar hair. Finally, we found a specific value of the parameter associated to the scalar hair, in order to explain the discrepancy between the theory and the observations, for the perihelion precession of Mercury and light deflection.


I. INTRODUCTION
Hairy black holes are interesting solutions of Einstein's Theory of Gravity and also of certain types of Modified Gravity Theories. The first attempts to couple a scalar field to gravity was done in an asymptotically flat spacetime finding hairy black hole solutions [1] but these solutions violated the no-hair theorems because they were not physically acceptable as the scalar field was divergent on the horizon and stability analysis showed that they were unstable [2]. Then, by introducing a cosmological constant hairy black hole solutions with a minimally coupled scalar field and a self-interaction potential in asymptotically dS space were found, but it was unstable [3,4]. Also, a hairy black hole configuration was reported for scalar field non-minimally coupled to gravity [5], but perturbation analysis showed the instability of the solution [6,7]. In the case of a negative cosmological constant, stable solutions were found numerically for spherical geometries [8,9] and an exact solution in asymptotically AdS space with hyperbolic geometry was presented in [10] and generalized later to include electric charge [11]. Then, a generalization to non-conformal solutions was discussed in [12]. Further hairy solutions in the presence of a cosmological constant were reported in [13][14][15][16][17] with various properties. On the other hand, by introducing a coupling of a scalar field to Einstein tensor that acts as an effective cosmological constant [18,19] a hairy black hole solution was presented [20] Additionally, spherically symmetric hairy black hole solutions with scalar hair were found [21]. Additionally, there are also very interesting recent developments in Observational Astronomy. High precision astronomical observations of the supermassive black holes may pave the way to experimentally test the no-hair conjecture [22]. Also, there are numerical investigations of single and binary black holes in the presence of scalar fields [23].
On the other hand, the recent developments in string theory and specially the application of the AdS/CFT principle to condense matter phenomena like superconductivity (for a review see [24]), triggered the interest of further study of the behavior of matter fields outside the black hole horizon [25,26]. In this context, the gauge/gravity duality is a principle which relates strongly coupled systems to their weak coupled gravity duals. One of the most well studied system is the holographic superconductor. In its simplest form, the gravity sector is a gravitating system with a cosmological constant, a gauge field and a charged scalar field with a potential (for a review see [27]). The dynamics of the system defines a critical temperature above which the system finds itself in its normal phase and the scalar field does not have any dynamics. Below the critical temperature the system undergoes a phase transition to a new configuration. From the gravity side this is interpretated as the black hole to acquire hair while from boundary conformal field theory site this is interpretated as a condensation of the scalar field and the system enters a superconducting phase.
It is known that all solar system observations, such as light deflection, the perihelion shift of planets, the gravitational time-delay among other are described within Einstein's General Relativity. The study of geodesics has been performed under several black hole geometries. For instance, see [28][29][30][31][32][33][34][35] for the motion of particles on AdS space-time. In this work, motivated by black hole solutions with matter fields outside their horizon, we study their effect in the motion of massless and massive particles in the background of a four-dimensional asymptotically AdS black hole with scalar hair [16]. These hairy black holes solutions are characterized by a self-interacting potential that asymptotically tends to the cosmological constant, and the scalar field is regular everywhere outside the event horizon and null at spatial infinity. The geodesics are studied numerically and we discuss about the differences in the motion of particles between the four-dimensional asymptotically AdS black holes with scalar hair and their no-hair limit, that is, Schwarzschild AdS black holes. Also, we study classical tests such as perihelion precession, deflection of light and gravitational time delay in order to determine the contribution that arises from the scalar hair.
The paper is organized as follows. In Section II we give a brief review of the fourdimensional asymptotically AdS black holes with scalar hair that we will consider as background. In Section III we study the motion of massless and massive particles, and we perform some classical tests such as perihelion precession, deflection of light and gravitational time delay. Finally in Section IV we conclude.

SCALAR HAIR
The hairy black hole that we consider is solution of the Einstein-Hilbert action with a negative cosmological constant and a neutral scalar field minimally coupled to the curvature having a self-interacting potential V (φ) [16]. The action is given by being the self-interacting potential Here, the cosmological constant is incorporated in the potential, that is, Λ = V (0) (V (0) < 0). Where, Λ = −6l −2 /κ, being l the length of the AdS space and κ = 8πG N , with G N the Newton constant. This potential has a global maximum at φ = 0. The equations of motion are where the energy momentum tensor T (φ) µν for the scalar field is and the Klein-Gordon equation of the scalar field reads The following metric is solution of the theory defined by (1) where and the scalar field is In expression (6), dσ 2 k is the metric of the spatial 2-section, which can have positive, negative or zero curvature, and the coordinates are defined in the ranges 0 < r < ∞, −∞ < t < ∞, 0 ≤ θ < π, 0 ≤ φ < 2π. For the lapsus function k = 1, 0, −1 parametrizes the curvature of the spatial 2-sections and F , G are constants being proportional to the cosmological constant and to the mass respectively. It was shown, that for spherical horizons k = 1 there is no phase transition of the hairy asymptotically AdS black holes to Schwarzschild AdS black hole. However, for hyperbolic horizons k = −1 there exists a phase transition only for negative masses, and the hairy black hole dominates for small temperatures, while for large temperatures the topological black hole would be preferred, for more details see [16].
In the next section we perform a numerical analysis of the geodesics by considering the hairy black hole solution. So, without loss of generality, we consider the following values for the parameters: k = 1, ν = −1, F = 1 and G = 2. Thus, in order to show that these parameters yield a hairy black hole solution we plot in Fig. 1 the behavior of the metric function f (r) that changes sign for r = 1.15 signaling the presence of an horizon. Also we plot the behavior of the Ricci scalar R(r), the principal quadratic invariant of the Ricci tensor R µν R µν (r), and the Kretschmann scalar R µνλτ R µνλτ (r) and we observe that there is not Riemann curvature singularity outside the horizon. Also, we observe that the Riemann curvature singularities are covered by the horizon. Therefore, the choice of parameters mentioned above gives a hairy black hole solution which is asymptotically AdS.

III. GEODESICS
In order to find the geodesics of the space-time described by (6), we will solve the Euler-Lagrange equations for the variational problem associated with this metric. The Lagrangian associated to the metric (6) is given by whereq = dq/dτ , and τ is an affine parameter along the geodesic that we choose as the proper time. Since the Lagrangian (10) is independent of the cyclic coordinates (t, φ), then their conjugate momenta (Π t , Π φ ) are conserved and the equations of motion readṡ where Π q = ∂L/∂q is the conjugate momenta to the coordinate q. The above equation can be written asΠ which yields Π θ = a 2θ , and Π φ = a 2 sin 2 θφ .
Now, without loss of generality, we consider that the motion is developed in the invariant plane θ = π/2 andθ = 0, which is characteristic of the central fields. With this choice, Eqs. (16) and (15) become where E and L are dimensionless integration constants associated to each of them. So, inserting equations (16) into equation (10) we obtain where V (r) is the effective potential given by where m is the test mass. Finally, by normalization, we shall consider that m = 1 for massive particles and m = 0 for photons.

A. Time like geodesic
In order to observe the possible orbits, we plot the effective potential for massive particles (18) which is shown in Fig. (2). In the following, we describe the radial motion and the angular motion.

Radial motion
In this case L = 0. The particles always fall into the horizon from an upper distance determined by the constant of motion E = 30.88. This fact is due to the attractive force generated by the proportional term to the cosmological constant, see Fig. (2). In Fig. 3 we plot the proper (τ ) and coordinate (t) time as function of r for a particle falling from a finite distance with zero initial velocity, we can see that Schwarzschild AdS black hole.

Angular motion
For the angular motion we consider L > 0. The allowed orbits depend on the value of the constant E.
• If E = 61.6 the particle can orbit in a stable circular orbit at r s = 2.84, see Fig. 2.
• If E = 69.4 the particle can orbit in an unstable circular orbit at r u = 1.518. Also, there are two critic orbits that approximates asymptotically to the unstable circular orbit. First kind, the particle starts from the rest and a finite distance greater than the unstable radio, see Fig. 4. The second kind, the particle starts from a finite distance greater than the horizon, but smaller than the unstable radio, see Fig. 4.
• The planetary orbits are constrained to oscillate between an apoastro and a periastro.
We plot in Fig. 5 the planetary orbit for E = 65. We can observe that the particle completes an oscillation in an angle less than 2π contrary to the Schwarzschild AdS black hole, where the angle is greater than 2π [36].  It is possible to calculate the periods of the circular orbits (r c.o. ), which can be stable (r s ) or unstable (r u ) orbits using the constant of motion √ E and L, given by (16), which yields and where T τ is the period of the orbit with respect to the proper time and T t is the period of the orbit with respect to the coordinate time. It is worth to mention that the periods depend on the value of ν and in the limit ν → 0 these periods correspond to the periods of the circular orbits in the spacetime Schwarzschild AdS. On the other hand, for the stable circular orbits is possible to find the epicycle frequency, given by κ 2 = V ′′ (r s )/2.

Perihelion precession
Here, we follow the treatment performed by Cornbleet [37], which allows us to derive the formula for the advance of the perihelia of planetary orbits. The starting point is to consider the line element in unperturbed Lorentz coordinates together with line element (6). So, considering only the radial and time coordinates in the binomial approximation, the transformation gives We will consider two elliptical orbits, one the classical Kepler orbit in (r, t) space and a hairy AdS orbit in (r,t) space. Then, in the Lorentz space dA = R 0 rdrdφ = R 2 dφ/2, and hence dA dt which corresponds to Kepler's second law. For the hairy AdS case we have where dr is given by Eq. (23), and the binomial approximation for the radial function a(r) is a(r) ≈ r 1 + ν 2r .
So, we can write (25) as Therefore, applying the binomial approximation we obtain So, using this increase to improve the elemental angle from dφ to dφ. Then, for a single where we have neglected products of G, F and ν. The polar form of an ellipse is given by where ǫ is the eccentricity and l is the semi-latus rectum. In this way, plugging Eq. (30) into Eq. (29), we obtain (31) which at first order yields Therefore, the perihelion advance has the standard value of general relativity plus the correction term coming from cosmological constant and scalar hair. It is worth to mention that there is a (negative) discrepancy between the observational value of the precession of perihelion for Mercury, ∆φ Obs. = 5599.74 (arcsec/Julian − century) and the total ∆φ T otal = 5603.24 (arcsec/Julian − century), see [38]. Which, is possible attribute to the scalar hair correction, given ν = −0.359(Km).

B. Null geodesic
In the next analysis, we consider two kinds of motion, for L = 0 (radial motion), and L > 0 (angular motion) of the photons (m = 0).

Radial motion
In this case, the master equation (17) can be written as where (+) stands for outgoing photons and (−) stands for ingoing photons. The solution of the above equation yields

Angular motion
In this case, the allowed orbits for photons depend on the value of the impact parameter b ≡ L/ √ E. Next, based on the impact parameter values shown in Fig.7, we give a brief qualitative description of the allowed angular motions for photons, described in the following: • Capture zone: If 0 < b < b u , photons fall inexorably to the horizon, and its cross section, σ, in this geometry is [39] • Critical trajectories: If b = b u (E u = 5.58), photons can stay in one of the unstable inner circular orbit of radius r u (r u = 1.5). Therefore, the photons that arrive from the initial distance r i (r + < r i < r u , or r u < r i < ∞) can asymptotically fall into a circle of radius r u , see Fig. 8.
• Deflection zone. If b u < b < b 0 = 1/ √ F , the photons can fall from the infinity to a minimum distance r d = 2.09 and can come back to the infinity. This photons are deflected, see Fig. 9. The other allowed orbits correspond to photons moving into the other side of the potential barrier, which plunges into the singularity. In the next section, we will focus on this topic.

Deflection of light
The deflection of light is important because the deflection of light by the Sun is one of the most important test of general relativity, and the deflection of light by galaxies is the mechanism behind gravitational lenses. The distance of the closest approach r 0 for the metric (6) can be defined by where b is the impact parameter. Now, by using the change of variables u = 1/r, the above equation can be written as which at first order and applying the binomial approximation wherever necessary, we obtain Following [40] we define So, solving for u yields where we have considered first order terms. Therefore (38) becomes This can be integrated to give and by considering ∆φ = 2 φ ∞ − π (see Fig. 9), it is possible to find the deflection angle accurately, and it reads: Therefore, the deflection of light is given by the standard value plus the correction term coming from cosmological constant (F ) and scalar hair (ν

Gravitational 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 ρ 0 is the closest approach to the Sun. Now, in order to calculate the time delay we use (17) and the coordinate timeṙ So, (17) can be written as By considering ρ 0 the closest approach to the Sun, dr/dt vanishes, so that Now, by inserting (47) in (46), the coordinate time which the light requires to go from ρ 0 to r is So, at first order correction we obtain Therefore, for the circuit from point 1 to point 2 and back, the delay in the coordinate time is ∆t : where (54) Therefore, as in the previous cases the time delay has the standard value of general relativity plus the correction term coming from cosmological constant and scalar hair.

IV. CONCLUDING COMMENTS
We have considered a four-dimensional asymptotically AdS black hole with scalar hair [16]. These solutions asymptotically give the Schwarzschild anti-de Sitter solution, and they are characterized by a scalar field with a logarithmic behavior, being regular everywhere outside the event horizon and null at spatial infinity, and by a self-interacting potential, which tends to the cosmological constant at spatial infinity. The equations for the geodesics were solved numerically in order to study their behavior. We note that radial motion results to be equivalent to the Schwarzschild AdS space-time [36]. Mainly, we have found that it is possible to find bounded orbits like planetary orbits in the background of a four-dimensional asymptotically AdS black holes with scalar hair. However, the periods associated to circular orbits are modified by the presence of the scalar hair. Besides, we have found that some classical tests such as perihelion precession, deflection of light and gravitational time delay have the standard value of general relativity plus a correction term coming from the cosmological constant and scalar hair. Finally, we found a specific value of the parameter associated to the scalar hair, in order to explain the discrepancy between the theory and the observations, for the perihelion precession of Mercury (ν = −0.359(Km)) and light deflection (ν = −0.386 (Km) for ∆φ Obs. = 1.98). Interestingly, these values are of the same order and sign. In furthering our understanding, it would be interesting to study the motion of massless and massive particles in a charged hairy black hole. Work in this direction is in progress.