A Note on Circular Geodesics and Phase Transitions of Black Holes

The circular motion of charged test particles in the gravitational field of a Reissner-Nordstr\"{o}m black hole in Anti de Sitter space-time is investigated, using a set of independent parameters, such as charge Q, mass M and cosmological constant $\Lambda= -3/l^2$ of the space-time, and charge to mass ratio $\epsilon=q/m$ of the test particles. Classification of different spatial regions where circular motion is allowed, is presented, showing in particular, the presence of orbits at special limiting values, $M=4/\sqrt{6} Q$ and $l=6 Q$. Thermodynamically, these values are known to occur when the black hole is on the verge of a second order phase transition, there by, giving an interesting connection between thermodynamics and geodesics of black holes. We also comment on the possibility of such a connection for black holes in flat spacetime in a box.


Introduction
Study of test particle motion in black holes in flat and curved space-times, such as Anti de Sitter space-times has received constant attention over the years. The motion of particles in the vicinity of a black hole is the first probe towards an understanding of the physics of gravitational objects in general relativity. Null geodesics, for instance give important information about the structure of the spacetime under consideration. It is well known that, for Schwarzschild spacetime, the geodesics equations are integrable, and those solutions are well known for a long time [1]. However, when the cosmological constant [2] is included, the situation is quite different, and the usual formulae of time delay and light deflection of Schwarzschild metric get modified. Besides, if we are interested in a universe filled or dominated by a cosmological constant, the usual expression for gravitational lensing in Friedmann-Robertson-Walker universes get modified too, and all the expressions need to be reanalyzed to test the cosmological observations [3][4][5][6][7]. Furthermore, the analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild Anti de Sitter, Reissner-Nordstrom, and Reissner Nordström (anti) de Sitter (RNAdS), spacetimes were found in [8], giving the complete set of solutions and a classification of the possible orbits in these geometries in terms of the Weierstrass functions. Various aspects of circular orbits in flat and RNAdS space-times, including photon spheres have been widely studied as well [9]- [38].
Thermodynamics and phase transitions of black holes, in particular in Anti de Sitter spacetimes has been a rich field of study, which has recently been boosted by a modified approach of extended phase formalism. Various novel aspects of black holes such as, P-V criticality and thorough investigation of critical regime have been made possible, in the scenario where the cosmological constant is dynamical [39]- [50]. These and other studies, show a phase structure that includes a line of first order phase transitions ending in a second order transition point [51][52][53][54][55]. In this context, apart from the Hawking-Page transition (connecting black holes in AdS to large N gauge theories at finite temperature), Van der Waals transition has captured the attention recently. Holographic interpretations for the later transition is still emerging and discussed for instance, in [39,44,56,57], where it is interpreted not as a thermodynamical transition but, instead, as a transition in the space of field theories (labeled by N, the number of colors in the gauge theory). Thus, varying the cosmological constant in the bulk corresponds to perturbing the dual CFT, triggering a renormalization group flow. This flow is captured in the bulk by Holographic heat engines with black holes as working substances [57]. Various aspects of this correspondence are being actively studied both from the gravity point of view as well as for potential applications to the dual gauge theory side, see [39]- [66] and references there in.
In this letter, we study the connection of circular motion of test particles in charged black holes in AdS space-time to their thermodynamics and phase transitions. Motivations are as follows. Recently, in [55,59], it was argued that black holes at criticality have a scaling symmetry, as a result of which all the parameters of the black hole, such as, Mass, Cosmological constant etc., become functions only of the charge of the black holes. Several interesting features emerge in this critical hole, especially in the limit of large charge, such as, the emergence of Rindler space-time and a fully decoupled geometry in the near horizon limit. Critical black holes are characterized by specific values taken by Charge, Mass and cosmological constant, which for RN black hole in AdS turn out to be: l 2 cr = 36Q 2 and M cr = m cr /2 = 4/ √ 6Q [55,59]. It is then interesting to ask whether such precise values can be obtained from study of circular orbits. The motivation for this question comes from a novel classificiation of circular geodesics put forward in [38], where, certain limiting values of black hole parameters depending on charge Q, mass M together and charge to mass ratio = q/m of the test particles arise. The authors found special values such as Q/M = 1/2, √ 13/5, 2/3 which are like boundaries for existence of circular motion [38]. It is then natural to ask, whether such a classification exists and in new limiting values occur, once the cosmological constant l is included; as a new parameter, such as, l/Q is now available. A more important question though is, whether the classification of circular orbits can give special limiting values correponding to those of critical black holes. In this letter, we point out that, extending the analysis presented in [38] with the inclusion of a cosmological constant and ignoring any back reaction effects, the critical values correponding to black holes can indeed be obtained. Let us also note that connections between photon spheres and phase transitions in extended thermodynamics have recently been discussed in [67], where it was argued that photon sphere radius and impact parameter show oscillatory behaviour close to the critical point. We extend this observation to include charged test particles as well, where we also discuss a modified impact parameter suitable for geometries with cosmological constant.
The plan of the rest of the paper is as follows. Section-1 is mostly introductory with few new expressions, where discussion of charged and neutral particles in RN AdS space-time is set up, in a way suitable to explore classification of orbits. In Section-2, we start by discussing the classification of orbits, identifying the case corresponding to a critical black hole. We then discuss the radius and impact parameter of circular geodesics of charged particles in terms of thermodynamic quantities of black holes in AdS. Second-3 contains conclusions.

Circular Motion of Test Particles
The action for Einstein-Maxwell system in the presence of a negative cosmological constant Λ = −3/l 2 is 1 The black hole space-time is Reissner-Nordström like solution with metric where dΩ 2 is the metric on S 2 and A t is the gauge potential. The motion of a test particle of charge q and mass µ moving in a RN background (1.2) is described by the following Lagrangian density and equations of motion, respectively: where = q/µ is the specific charge of the test particle and F αβ ≡ A α,β − A β,α . Since the Lagrangian density (1.4) does not depend explicitly on the variables t and φ, the following two conserved quantities exist where L and E are respectively the angular momentum and energy of the particle as measured by an observer at rest at infinity. We restrict ourselves to the study of equatorial trajectories with θ = π/2, we getṙ which describes the motion inside an effective potential V given as: At the turning point, we set (V = E/µ), the point where the kinetic energy of the particle vanishes.  Considering the special case of V + , for later comparison with [14,15], conditions for circular orbits are 2 dV dr

L=15
Angular momentum is found to be: and Energy is obtained as The above expressions in equations (1.10) and (1.12) go over to the ones in [14,15] as the cosmological constant is set to zero. The radius of an orbit where particle is located at rest as seen by an observer at infinity, i.e., L = 0, dV dr = 0, is plotted in figure-(3), as the cosmological constant is varied and the radius of the last stable circular orbit in figure-(4). Numerically, it is possible to find the minimum radius for stable circular orbits by the condition ∂ 2 V /∂r 2 = 0, but we do not pursue it here. It is interesting to point out the case of a charged particle moving in the Coulomb potential in AdS, U (r) = Q r with the effective potential: where Q < 0. The Coulomb approximation is useful for understanding the system when the radial coordinate r takes higher values. The circular orbits are situated at r = r c , where Unlike the case of flat space [14,15], angular momentum expression is not a constant in AdS and circular orbits do not exist for any arbitrary value of r and/or L. Angular momentum and Energy are given as The zero angular momentum orbits (L = 0, dV /dr = 0, ZAMPs) are given by where A 5 = 9l 4 Q 2 2 + 81l 8 Q 4 4 − 12l 6 Q 6 6 . The Energy at this value of radius is . Analogous expressions for charged particle motion in full AdS black holes are not available analytically.
We now go on to discuss case of neutral particles. Values of Angular momentum and Energy for neutral particles in RN AdS black holes are given below. We find the specific angular momentum of the test particle on a circular orbit of radius r to be: and the corresponding energy is: (1.20) As in the case of charged particle, orbit corresponding to a particle at rest as noted by an observer at infinity is: In the present case, this is plotted in figure-5.

Black Hole Phase Transitions in AdS from Circular Orbits
Last section was devoted for obtaining the key quantities based on which a classification of circular orbits of null and charged test particles in the background of black holes in AdS would be presented. Before we proceed, it is useful to summarize a brief account of the thermodynamics side of the black holes with which the connection will be made. Towards this end, in this section, we first define the limit in which the charged black hole in AdS in D = 4 given in eqn. (1.2), becomes critical. We start by writing down the mass and charge parameters corresponding to this black holes as: 2) The temperature T = f (r + )/4π of the black hole can found to be [61][62][63][64][65] T = 1 4π and the entropy is S = ω 2 r 2 + /4. Recently, extended phase description of thermodynamics of black holes in AdS has received considerable attention [58]. Here, the cosmological constant is treated as dynamical [39] and mass M is actually the enthalpy H(S, p) of the system [39,47]. Here, the length scale l is replaced by the pressure p as: 4) and the thermodynamic volume is V = ω 2 r 3 + /3. Let us note that instead of r + , one can use entropy or volume, as they are not independent for static black holes (not the case in general, such as, for rotating black holes etc..). Thus, from eqn. (2.3), we obtain At the second order critical point [51][52][53][54][55], thermodynamic quantities scale with respect to charge q as where entropy and thermodynamic volume are S cr = 6πq 2 and V cr = 8 √ 6πq 3 . Understanding the microscopic physics at the above critical point thorough the study of how an N -particle system moving in the background of the black holes drives itself to criticality is interesting. The idea of a coupled system being driven to criticality is not new and is being pursued actively in recent statistical mechanics literature [66], as pointed out in the context of gravitational systems in [59]. Thus, the study of circular geodesics of test particles in the critical region of black holes is quite interesting from the point of view of critical heat engines as well. Now, the metric for a critical black hole can be obtained by inserting the the critical quantities given in equation (2.6). From the critical black hole mass parameter m cr and critical cosmological scale l cr the metric function for our critical black hole is [59]: All the thermodynamic quantities in the critical regime scale with the only free parameter available in this limit, i.e., charge Q, which can be taken to be large. It would be nice to know the effect of large charge limit on the geodesics of test particles. The connection of charge to mass ratio of test particles in the background of critical black holes is being actively pursued [59,68].

Critical Point of Phase Transition from Classification of Circular Orbits
We now investigate the conditions for circular orbits corresponding to a general charged black hole in AdS, with metric given in eqn. (1.2). Instead of the general approach of using the effective potential, we follow the alternate methodology of [38], where we first express the conditions for circular motion based on parameters M/Q, l/Q and the charge to mass ratio of the test particle = q/µ. This is to be contrasted with the case in flat space [38], where only the ratio M/Q is available. The physical values of the parameters are obtaned from the conditions for circular motion. Some of the parameters in AdS are plotted in figure-(6) for null particles, which are similar to flat space results, although the horizon radii r ± now correspond to black hole in AdS. Thus, demanding positivity of the expressions for the energy and angular momentum of charged particles (k = 1) given in equations (1.10) and (1.12) and analogous expressions for neutral particles, it is useful to introduce the parameters, which are same as the flat space expressions discussed in [38]. We have in addition new parameters which appear when the cosmological constant is present: (2.12)     Figure 6: Plot contains radii rγ ± ≡ [3M ± (9M 2 − 8Q 2 )]/2 and r± are the horizons of charged black in AdS and r * . Timelike circular orbits exist only for r > rγ + , whereas r = rγ + represents a null hypersurface. Circular motion inside the regions r < r * and r ∈ (rγ − , rγ + ) is possible only along spacelike geodesics.
The emergence of new parameters r l and r ± l in AdS space time, in the presence of cosmological constant is quite interesting, and leads to the classification of orbits giving different ranges of parameters, than those in flat space-time. In particular, let us note that based on Table 3, the point in the parameter space with values l/Q = 6 and M/Q = 2 2 3 , exactly corresponds to the ones used in the metric function of critical black holes in equations (2.7) and (2.8). This points to the fact that classification of circular orbits around charged black holes in AdS, contains information about the values taken by parameters of black holes at the critical point of phase transition. This connection of circular geodesics with phase transitions quite intriguing and might give a novel understanding of thermodynamics of black holes in AdS from the motion of test particles. Let us note that a detailed study of the classification of circular orbits and the corresponding classification for naked singularities in AdS is quite interesting to explore further (see Section-3 for further remarks). There are also few other parameters such as, two further radii r ± s and also h , l etc., whose explicit analytical expressions are not known. r ± s for instance correspond to zero angular momentum circular orbits given by the conditions L = 0, dV /dr = 0. It would be nice to explore these issues further.

Thermodynamics from Circular Orbits
In this subsection, we extend the relationship between photon spheres and thermodynamics of black holes recently pointed out in [67], to the case of charged particles. In [67], it was noted that the photon radius and impact parameter, both show oscillatory behaviours, promising to be order parameters for the small to large black hole phase transition. The critical exponents corresponding to these parameters were also found there. Here, we point that the radius of circular orbits of charged particles continue to show oscillatory behaviours. Regarding connection of impact parameter with thermodynamics, we point out that the in AdS, it is useful to have a redefined impact parameter depending on the cosmological constant [36], with out change in the results obtained in [67]. From equation (1.7), we have The motion of the test particles, say photons, is permitted in a region where V (r) is nonnegative. Since the effective potential V (r) in the limit r → ∞ becomes a photon can be at the spatial infinity r → ∞. Thus, in the presence of a cosmological constant, it is useful to define an anamolous impact parameter B as where the standard impact parameter in flat space-time used in [67] is b = L/E = r/ Y (r).
In fact, the classification of various orbits in AdS space-time in [36] is completely based on the above definition of impact parameter, given in terms of B, where the nature of orbits crucially depends on whether B < 0 or B > 0. The temperature in the reduced parameter space 3 is The radius of circular orbits for time-like and null particles in AdS space-time is actually exactly same as in flat space-time, as given in equations (2.9) and (2.10). Rewriting the same in terms of thermodynamic variables, we have Combining with (2.16), we can plot the reduced temperature as a function of the photon sphere radiusr ms for varying pressure and charge-to-mass ratio, which is shown in Figure-  This of course reduces to b msc ≈ 4.78Q when = 0 [67]. From figure-(7-(b)), one notes oscillatory plots that forP <1, where as the oscillations dissapear forP ≥1. This indicates a first-order phase transition as first pointed out in [67]. The change in the radius of orbits before and after the transition can be computed and this is plotted figure- (8) for different values of specific charge. The behavior persists in the plots ofT as a function of B ps for non-zero . A comparative plot of minimum impact parameter b and anamolous parame ter B are given in figure-(8)), both showing oscillatory behavior.

Remarks
In this paper, we presented a classification of circular orbits of charged and null particles in the background of charged black holes in AdS. In particular, we showed that the classification based on charge Q, mass M and cosmological constant l, together with the charge-to-mass ratio of test particles, reveals a special point in the parameter space. The values taken by the parameters at this point exactly coincide with those coming independingly from thermodynamic considerations on the black hole side, namely the critical point of a phase transition. The connection between circular geodesics of particles and phase transitions of black holes in AdS is counter intuitive and deserves further study, in particular generalizations to other non-trivial black holes. Classification of circular geodesics in flat space-time considered in [38] also gave certain limiting values of parameters and it would be interesting to know if anything special happens, e.g., thermodynamically at these values. We also note that it would be interesting to perform a full classification of circular geodesics for all values of specific charge, not just for black holes, but also for naked singularities in AdS. These works are in progress.