[Paper Review] Characteristic interaction potential of black hole molecules from the microscopic interpretation of Ruppeiner geometry

Shao-Wen Wei, Yu-Xiao Liu+1 more|arXiv (Cornell University)|Aug 17, 2021

Black Holes and Theoretical Physics21 references4 citations

TL;DR

This paper provides the first microscopic interpretation of Ruppeiner geometry for anti-de Sitter black holes by modeling their constituent molecules using a Lennard-Jones-type potential. It derives explicit interaction potentials—particularly a short-range repulsive interaction for charged AdS black holes—linking negative/positive Ruppeiner curvature scalars to attractive/repulsive molecular forces, thereby resolving the origin of such empirical observations in black hole thermodynamics.

ABSTRACT

Ruppeiner geometry has been found to be a novel promising approach to uncover the microstructure 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 microscopic 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.

Motivation & Objective

• To resolve the long-standing question of why negative Ruppeiner curvature scalar corresponds to attractive interactions and positive curvature to repulsive ones in thermodynamic systems.
• To extend the microscopic interpretation of Ruppeiner geometry—previously applied to fluid systems—into the realm of black hole thermodynamics.
• To derive explicit interaction potentials for black hole molecules in anti-de Sitter spacetime, particularly for charged AdS black holes.
• To clarify the origin of repulsive interactions in small charged AdS black holes, which were previously observed empirically but not microscopically explained.
• To establish a framework applicable beyond AdS black holes, given the invariance of Ruppeiner curvature under thermodynamic coordinate transformations.

Proposed method

• Adopt the Van der Waals fluid model with Lennard-Jones potential as a microscopic reference for fluid-like behavior in black hole systems.
• Use the Ruppeiner geometry formalism to relate thermodynamic curvature scalar to intermolecular interactions, assuming that negative curvature implies attraction and positive curvature implies repulsion.
• Construct a modified Lennard-Jones potential with a shifted radial dependence: $\phi = 4\phi_0\left[\left(\frac{r_0}{r+r_0}\right)^{12} - \left(\frac{r_0}{r+r_0}\right)^6\right]$, to model black hole molecules without a hard cutoff at the potential well.
• Fix the potential parameters $\phi_0$ and $r_0$ by matching the location of the potential well to the zero point of the normalized Ruppeiner curvature scalar, as observed in prior work.
• Relate the black hole's specific volume $v$ to the well depth and location via $v = 4\pi r_{\rm min}^3 / 3$, ensuring consistency with the thermodynamic state.
• Derive the expression $\phi_0 \approx 0.0003388 / Q$ by equating the well's location to the curvature scalar zero point at $v = 2Q(8\sqrt{2}\pi/3)^{1/3}$, enabling quantitative modeling of the interaction potential.

Experimental results

Research questions

• RQ1 What is the microscopic origin of the empirical correlation between Ruppeiner curvature scalar sign and intermolecular interaction type (attractive/repulsive) in black hole systems?
• RQ2 How can the Ruppeiner geometry of black holes be interpreted in terms of a fundamental molecular model, analogous to the Van der Waals fluid?
• RQ3 What explicit form does the interaction potential between black hole molecules take, particularly for charged AdS black holes?
• RQ4 Why do charged AdS black holes exhibit repulsive interactions at small sizes, while Schwarzschild AdS black holes do not?
• RQ5 Can the molecular potential model be generalized to other black hole systems beyond AdS spacetime, given the invariance of Ruppeiner curvature under thermodynamic coordinate transformations?

Key findings

• The authors derive a novel Lennard-Jones-type potential $\phi = 4\phi_0\left[\left(\frac{r_0}{r+r_0}\right)^{12} - \left(\frac{r_0}{r+r_0}\right)^6\right]$ that models black hole molecules without a hard cutoff at the potential well, enabling a continuous description of short-range repulsion.
• The potential exhibits a short-range repulsive interaction for $r < r_{\rm min} = (2^{1/6} - 1)r_0$, which is responsible for the observed positive Ruppeiner curvature scalar in small charged AdS black holes.
• The parameter $\phi_0$ is quantitatively fixed as $\phi_0 \approx 0.0003388 / Q$ by matching the potential well location to the zero point of the normalized Ruppeiner curvature scalar in the charged AdS black hole case.
• The model explains the absence of repulsive interactions in Schwarzschild AdS black holes due to the lack of charge-induced modification in the potential structure.
• The microscopic origin of repulsive interactions is attributed to two factors: the finite size of the constituents and the presence of a short-range repulsive core in the potential, even without a hard cutoff.
• The framework is generalizable to other black hole systems, as the Ruppeiner curvature scalar is invariant under thermodynamic coordinate transformations, enabling future application to asymptotically flat black holes and phase transition studies.