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The Geometry of RN-Ads Fluids Physics Letters B 805 (2020) 135416 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb The geometry of RN-AdS fluids ∗ Joy Das Bairagya, Kunal Pal, Kuntal Pal, Tapobrata Sarkar Department of Physics, Indian Institute of Technology, Kanpur 208016, India a r t i c l e i n f o a b s t r a c t Article history: We establish the parameter space geometry of a fluid system characterized by two constants, whose Received 20 December 2019 equation of state mimics that of the RN-AdS black hole. We call this the RN-AdS fluid. We study the Accepted 5 April 2020 scalar curvature on the parameter space of this system, and show its equivalence with the RN-AdS black Available online 8 April 2020 hole, in the limit of vanishing specific heat at constant volume. Further, an analytical construction of the Editor: N. Lambert Widom line is established. We also numerically study the behavior of geodesics on the parameter space of the fluid, and find a geometric scaling relation near its second order critical point. © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction mal correspondence has been studied extremely well, for almost five decades now. Black holes are singular solutions of general relativity that may Note that a pressure term is missing in the above discussion. arise as the end stages of gravitational collapse. Understanding the For a long time it was believed that a pressure (and a volume) physics of black holes continues to be the focus of much attention. cannot be associated to black hole thermodynamics. This however With the nature of black hole microstates being elusive, a popu- changed a few years back with the proposal of Kastor, Ray and lar line of research is to understand the macroscopic properties of Traschen [1] that a (varying) cosmological constant can be identi- black holes, for example their thermodynamic properties, as these fied with the pressure of a black hole, with the conjugate volume often provide useful insights into the underlying coarse grained associated with the volume of the event horizon. In this formal- structure of black holes. More recently, geometric methods have ism, the black hole mass has to be identified with the enthalpy been applied to such studies, and it has been claimed in the litera- of the system, and this can be shown to satisfy the correspond- ture that the nature of interactions between black hole microstates ing Smarr relation. This is popularly termed as the extended phase can possibly be elucidated by these analyses. space formalism in black hole thermodynamics. Charged black The four laws of black hole mechanics which were formulated holes appearing in theories of gravity with a (negative) cosmolog- in the early 70’s, are formally identical with the four laws of ther- ical constant i.e. the Reissner-Nordstrom-anti-de-Sitter (RN-AdS) modynamics. For black holes having electric charge Q and angu- black holes, where this is primarily interesting will be the focus lar momentum J with the corresponding potentials being and of this work. Thermodynamics of RN-AdS black holes and their re- respectively, these are given as: (i) The surface gravity κ on semblance with van der Waals (vdW) systems were analyzed in the horizon of a stationary black hole of area A is a constant. (ii) The change in energy E of a black hole can be expressed as the pioneering works of Chamblin, Emparan, Johnson and Myers dE = κdA/8π + dQ + dJ. (iii) The horizon area can never de- [2], [3]. In these works, an equation of state that gave the temper- crease with time, i.e. dA/dt ≥ 0. (iv) It is impossible to have κ = 0. ature as a function of the charge and the electric potential, was Hawking formalized this by showing that black holes can radiate, used to study this behavior. In the context of thermodynamics of and that such radiation will have a temperature T = κ/2π . Then, the extended phase space, the results of [1]were used by Kubiznak the four laws above are precisely the laws of thermodynamics, and Mann [4]to derive a relation between the pressure P BH and with the entropy of the black hole identified with A/4. This for- its volume V of the RN-AdS black hole of charge Q and horizon radius r+, via its Hawking temperature T BH. This reads Corresponding author. 1 * k T hc¯ hcQ¯ 2 3V 3 E-mail addresses: [email protected] (J.D. Bairagya), [email protected] (K. Pal), B BH P BH = − + , r+ = , (1) [email protected] (K. Pal), [email protected] (T. Sarkar). 2 2 2 2 4 2lpr+ 8πlpr+ 8πlpr+ 4π https://doi.org/10.1016/j.physletb.2020.135416 0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 J.D. Bairagya et al. / Physics Letters B 805 (2020) 135416 where kB is the Boltzmann’s constant, h¯ is Planck’s constant, lp shown to be the same as the RN-AdS black hole, and we thus ar- is the Planck length, and c is the speed of light. Note that, upon gue why it might be possible to glean insight into the extended 2 2 4 restoring dimensions, Q ≡ Q G/(4π0c ), with G being New- phase space thermodynamics of such black holes via the RN-AdS ton’s constant and 0 the permittivity of vacuum, carries dimen- fluids, rather than vdW systems. We further study geodesics on sions of length squared. A large number of papers subsequently the parameter manifold arising in our fluid system, and compute appeared that studied such relations in a variety of examples. On a geometric scaling relation near criticality. These are shown to be qualitatively similar to the vdW fluid and we also comment on the other hand, the vdW equation of state of a fluid with pressure the case where c vanishes, which offers a ready comparison with P is given by eq. (76.7) of [5]as v the RN-AdS black hole. The scalar curvature gives us an analytical 2 handle to compute the Widom line, and we show that this has a NkB T N a P = − , (2) = − 2 different nature for the case cv 0, compared to the case of van- V s Nb V s ishing cv . where N denotes the number of particles, and V s the volume of This paper is organized as follows. In the next section 2, we the fluid. Now using V = V s/N which denotes the volume per briefly recall Ruppeiner’s geometric formalism, and point out some molecule (so that the molecular density is ρ = 1/V ), and intro- physical features that have to be respected in such analyses. Next, −4 in section 3, we establish the geometry of RN-AdS fluids and com- ducing a further V s dependence in the virial expansion, we can ment on their relation to the RN-AdS black hole. The Widom line write a modified form of eq. (2) that is formally identical to eq. (1), is also computed here. Section 4 studies geodesics on the param- and given by eter manifold of RN-AdS fluids and the corresponding black holes. kB T a d Next, in section 5, we define and compute a geometric critical P = − + , (3) V V 2 V 4 exponent for the fluid system. Section 6 ends with our conclu- sions. = = = 2 with P P BH, T T BH, and we identify V 2lpr+ as the vol- = ume per molecule of the fluid system, where the constants a 2. Ruppeiner’s geometry ¯ 2 = ¯ 2 6 hclp/(2π) and d 2hcQ lp/π carry dimensions of energy times volume, and energy time volume cubed, respectively. Note that in Ruppeiner’s formulation of the geometry of the parameter man- the original vdW equation of state P = k T /(V − b) − a/V 2, vdW B ifold of thermodynamic and statistical systems starts with assign- the constant b takes care of the fact that the volume per molecule ing the system a fixed volume V , which is in equilibrium with a cannot be lower than a certain finite cutoff, given by b. Although reservoir of volume V . The entropy of a such statistical system is it might seem that this condition is relaxed here, we will see later c that there is indeed such a minimum volume allowed by the sys- given by Boltzmann’s formula, tem, which is pressure dependent. S = k log , (4) It is known for some time that the vdW system (and for that B matter any system in thermodynamic equilibrium) is amenable where is the number of the microscopic states of the corre- to a geometric analysis. Such analyses began with the work of sponding thermodynamic system and kB is the Boltzmann con- Ruppeiner [6], was extended to the case of quantum systems by stant. For a system of n independent variables (i.e. tuning tuning Provost and Vallee [7], and is by now an established tool for the parameters) xα, α = 1, 2, ···n, the probability of finding its state study of phase transitions in classical and quantum systems. The between (x1, ··· , xn) and (x1 + dx1, ··· , xn + dxn) is proportional geometry is that of the parameter manifold, with coordinates be- ing the tunable parameters of the theory such as the temperature to the number of microstates and the density, on which one can define a Riemannian metric.
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