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Characteristic Interaction Potential of Black Hole Molecules from The Characteristic interaction potential of black hole molecules from the microscopic interpretation of Ruppeiner geometry Shao-Wen Wei1,2,3∗, Yu-Xiao Liu1,2,3†, and Robert B. Mann4 ‡ 1Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, People’s Republic of China, 2Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, People’s Republic of China, 3Academy of Plateau Science and Sustainability, Qinghai Normal University, Xining 810016, P. R. China 4Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 (Dated: August 26, 2021) Ruppeiner geometry has been found to be a novel promising approach to uncover the microstruc- ture of fluid systems and black holes. In this work, combining with the micro model of the Van der Waals fluid, we shall propose a first microscopic interpretation for the empirical observation of Ruppeiner geometry. Then employing the microscopic interpretation, we disclose the potential microstructure for the anti-de Sitter black hole systems. Of particular interest, we obtain the micro- scopic interaction potentials for the underlying black hole molecules. This significantly strengthens the study towards to the black hole nature from the viewpoint of the thermodynamics. PACS numbers: 04.70.Dy, 04.60.-m, 05.70.Ce I. INTRODUCTION space [11–14]. Motivated by the interpretation of the cosmological constant as a thermodynamic pressure [15, Classically a black hole is a purely absorbing object; 16], four-dimensional charged AdS black holes were found it cannot emit particles. However Hawking showed that to exhibit a small-large black hole phase transition, fully when quantum mechanical effects are considered near the analogous to the liquid-gas phase transition of a Van der horizon, black holes can indeed emit particles, with an Waals (VdW) fluid [14]. Later studies also showed that average energy flux given by [1] this small-large black hole phase transition is ubiquitous, taking place in most black hole systems [17]. dE ∞ Γω = dω, (1) dt 0 eωβ 1 Z − where Γ is the greybody factor of the radiation. This It is well known that phase transitions result from the suggests that a black hole emits particles at a temper- coexistence and competition among different constituents ature TH = 1/β known as the Hawking temperature, of a physical system. Phase transitions thereby provide which depends on the black hole mass, charge and other insight into the properties of a given system’s microstruc- parameters. Furthermore, the corresponding thermody- 3 ture. An effective method, Ruppeiner geometry [18], has namic entropy is S = kBc /4~G with the horizon A A been developed to explore this idea. It has been suc- area. The four laws of black hole thermodynamics were cessfully applied to many fluid systems, including VdW subsequently established [1–3] and it is now extensively fluids, and many interesting microscopic properties have believed that black holes are thermodynamic systems. In been uncovered, shedding new light on thermodynamic contrast to an ordinary fluid system, black hole entropy is systems. Applying this approach, in recent work [19– arXiv:2108.07655v2 [gr-qc] 25 Aug 2021 proportional to the horizon area of the black hole rather 21] we proposed that the constituent degrees of freedom than its volume. This brings new peculiar properties to of black holes have a molecular character. Specifically, black holes. we found that under certain circumstances repulsive mi- An outstanding challenge is that of understanding the crostructure interactions could exist for black holes, in microstructure of a black hole that underlies this ther- strong contrast to VdW fluids [20, 21]. However the ori- modynamics. Different theories and models have been gin of the repulsive/attractive interactions remains to be proposed, including string theory [4–7], fuzzball models solved. [8, 9], and pierced horizons [10], each with its own limi- tations. Over the past two decades there has been considerable interest in black hole phase transitions in anti-de Sitter The present work focuses on this significant problem. We first provide a microscopic interpretation of Rup- peiner geometry, and then apply it to black hole systems. ∗Corresponding author. E-mail: [email protected] We also find the explicit potentials these underlying black †E-mail: [email protected] hole molecules experience. We take the following units ‡ E-mail: [email protected] G = c = kB = lP = 1. 2 II. MICROSCOPIC INTERPRETATION OF 0.5 RUPPEINER GEOMETRY H1, 0L 0.0 Ruppeiner observed for various fluid systems that a 0 Φ negative/positive curvature scalar of the geometry was related to a respective attractive/repulsive interaction Φ -0.5 of the fluid molecules [18]. Employing this property, 1 Ruppeiner geometry has been used to examine the mi- J2 6 , -1N -1.0 crostructure of some complex fluid systems and black holes. Why this empirical observation works is still a 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 mystery. Since Ruppeiner geometry is closely related to rr0 phase transitions, we begin by providing a microscopic interpretation of Ruppeiner geometry for the VdW fluid FIG. 1: The Lennard-Jones potential. The red dashed curve model. and blue solid curve are for the short-range repulsive interac- Comparing to an ideal fluid, one needs to take into tion and longer-range attractive interaction, respectively. account the size of and interaction between the molecules for the VdW fluid, which are described by the parameters b and a, respectively. The equation of state reads freedom are left, and thus one may conjecture there is no interaction between these molecules. Furthermore, we 2 can expect that when this condition is violated repulsive NT N a T a 27 P = 2 = 2 , (2) interactions will be present. Second, when T < 32 Tc = V Nb − V v b − v a − − 4b , the pressure is negative for a given range of v, which where P , T , V , and v are the pressure, temperature, may lead to the repulsive interaction. total volume, and specific volume of the system, and N Making use of fluctuation theory, Ruppeiner geometry corresponds to the number of the molecules contained in has been used to great success in revealing the intrin- the fluid. The equation of state shows a first-order phase sic microstructure of black holes. In particular, the nor- transition, and admits a second order phase transition malized Ruppeiner curvature scalar was introduced and a 8a c c c 2 point at the critical point (P , T , v )=( 27b , 27b , 3b). shown for black holes to have universal behaviour near Such phenomena can be understood from a microscopic a critical point [20, 21]. Here we consider the relation viewpoint. As is well known, the interaction between two between the interactions and the normalized curvature fluid molecules is given by the Lennard-Jones potential scalar, which for a VdW fluid is given by [23] 2a(v b) ab + P v3 12 6 RN = − 2 . (5) r0 r0 − (2ab av + P v3) φ =4φ0 (3) r − r − Obviously, the denominator of RN is non-negative, van- illustrated in Fig. 1. The red dashed curve and blue ishing when the system approaches the critical point, −2 solid curve are for the short-range repulsive interaction which leads to RN (Tc T ) . Here we are concerned ∼ − and longer-range attractive interaction, respectively. At with the numerator of RN. Noting that the parameters a r = r0, we have φ=0, with the minimum of the potential and b are positive, we see that RN < 0 when v b. This 1 ≥ φ0 occurring at rmin =2 6 r0. A VdW fluid is described implies an attractive interaction, which is consistent with by− a “hard-core” model, meaning that the interatomic the microscopic potential model of the VdW fluid. distance between molecules cannot be smaller than a We can gain further insight by relaxing the require- molecular diameter d = rmin. Hence there is a cutoff ments on these parameters. First, we see that RN van- at the location of the well, and the short-range repulsive ishes when a = 0. Since a is a measure of molecular interaction denoted by the red dashed curve is excluded. interaction, vanishing a indicates there is no interaction, Furthermore, taking the mean field approximation, the consistent with empirical observations of Ruppeiner ge- characteristic parameters a and b in the equation of state ometry. As expected, from Fig. 1 and Eq. (4), pos- (2) have a microscopic interpretation [24] itive/negative a corresponds to a well/barrier, indicat- ing attractive/repulsive interactions respectively. Since 3 20πr0φ0 3 RN has opposite sign to a, a negative/positive curva- a = , b = √2r0. (4) 9√2 ture scalar indicates respective attractive/repulsive in- teractions. The absolute value of RN provides a measure where a > 0, indicating the attractive interaction domi- of the strength of these interactions. nates the fluid microstructure. Turning to the second term (v b), if b = v, all the We now clearly address two issues for a VdW fluid space of the fluid system is occupied− by the molecules, so that are often neglected. The first one is the volume no interactions are allowed. The corresponding curvature v b. When v = b, it means the molecules completely scalar vanishes, consistent with empirical observations of occupy≥ the space of the system. No spatial degrees of Ruppeiner geometry. If we do not impose a cutoff at 3 the location of the well in the Lennard-Jones potential, 0.05 shifting it to smaller r, then a short-range repulsive in- 0.00 teraction is present. There is a range of negative (v b), -0.05 − and RN becomes positive as expected.
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