RN-AdS流体的参数空间几何与RN-AdS黑洞等价性探索

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本文主要探讨了RN-AdS流体的几何特性，这是一种由两个常数特征的流体系统，其状态方程设计灵感来源于Reissner-Nordström-AdS (RN-AdS) 黑洞。RN-AdS黑洞性质的独特性被用于构建这个流体模型，以便研究在恒定体积条件下，其热性质的变化。首先，作者们详细分析了该流体系统在参数空间上的几何结构。他们着重于计算和理解标量曲率，这是一种度量空间弯曲程度的重要物理量。他们发现，当流体的比热在特定的体积保持恒定时，其标量曲率与RN-AdS黑洞表现出惊人的相似性。这一结果揭示了两种系统在特定物理条件下的深层次联系。其次，文章引入了Widom线的概念，这是一种在相变理论中至关重要的概念，用于描述系统的相界面。通过建立Widom线的分析构造，研究者能够更好地理解和预测流体在不同参数下的相行为，特别是在第二临界点附近。接着，作者采用数值方法来深入探究流体参数空间中的测地线行为。测地线是描述物体在流体参数空间中沿最短路径移动的轨迹，这对于理解流体动力学行为及其稳定性至关重要。他们在数值模拟中观察到，在流体的二阶临界点附近，存在显著的几何比例关系，这可能暗示着临界点附近的特殊物理现象。这篇文章不仅提供了RN-AdS流体的几何形状和其与RN-AdS黑洞的关联，还通过分析Widom线和测地线的行为，探讨了流体相变和动力学响应。这些研究对于理解复杂流体系统的性质以及黑洞物理与流体物理学之间的潜在联系具有重要意义。此外，由于该研究是开放获取的，它有助于促进学术交流和进一步的理论发展。

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:

Received 20 December 2019

Accepted 5 April 2020

Available online 8 April 2020

Editor: N. Lambert

We establish the parameter space geometry of a ﬂuid system characterized by two constants, whose

equation of state mimics that of the RN-AdS black hole. We call this the RN-AdS ﬂuid. We study the

scalar curvature on the parameter space of this system, and show its equivalence with the RN-AdS black

hole, in the limit of vanishing speciﬁc heat at constant volume. Further, an analytical construction of the

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.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Black holes are singular solutions of general relativity that may

arise as the end stages of gravitational collapse. Understanding the

physics of black holes continues to be the focus of much attention.

With the nature of black hole microstates being elusive, a popu-

lar

line of research is to understand the macroscopic properties of

black holes, for example their thermodynamic properties, as these

often provide useful insights into the underlying coarse grained

structure of black holes. More recently, geometric methods have

been applied to such studies, and it has been claimed in the litera-

ture

that the nature of interactions between black hole microstates

can possibly be elucidated by these analyses.

The four laws of black hole mechanics which were formulated

in the early 70’s, are formally identical with the four laws of ther-

modynamics.

For black holes having electric charge Q and angu-

lar

momentum J with the corresponding potentials being  and

 respectively, these are given as: (i) The surface gravity κ on

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

dE = κdA/8π + dQ + dJ. (iii) The horizon area can never de-

crease

with time, i.e. dA/dt ≥ 0. (iv) It is impossible to have κ = 0.

Hawking formalized this by showing that black holes can radiate,

and that such radiation will have a temperature T = κ/2π . Then,

the four laws above are precisely the laws of thermodynamics,

with the entropy of the black hole identiﬁed with A/4. This for-

Corresponding author.

E-mail addresses: joydas@iitk.ac.in (J.D. Bairagya), kunalpal@iitk.ac.in (K. Pal),

kuntal@iitk.ac.in (K. Pal),

tapo@iitk.ac.in (T. Sarkar).

mal correspondence has been studied extremely well, for almost

ﬁve decades now.

Note that a pressure term is missing in the above discussion.

For a long time it was believed that a pressure (and a volume)

cannot be associated to black hole thermodynamics. This however

changed a few years back with the proposal of Kastor, Ray and

Traschen [1] that a (varying) cosmological constant can be identi-

ﬁed

with the pressure of a black hole, with the conjugate volume

associated with the volume of the event horizon. In this formal-

ism,

the black hole mass has to be identiﬁed with the enthalpy

of the system, and this can be shown to satisfy the correspond-

ing

Smarr relation. This is popularly termed as the extended phase

space formalism in black hole thermodynamics. Charged black

holes appearing in theories of gravity with a (negative) cosmolog-

ical

constant i.e. the Reissner-Nordstrom-anti-de-Sitter (RN-AdS)

black holes, where this is primarily interesting will be the focus

of this work. Thermodynamics of RN-AdS black holes and their re-

semblance

with van der Waals (vdW) systems were analyzed in

the pioneering works of Chamblin, Emparan, Johnson and Myers

[2], [3]. In these works, an equation of state that gave the temper-

ature

as a function of the charge and the electric potential, was

used to study this behavior. In the context of thermodynamics of

the extended phase space, the results of [1]were used by Kubiznak

and Mann [4]to derive a relation between the pressure P

and

its volume V of the RN-AdS black hole of charge Q and horizon

radius r

, via its Hawking temperature T

. This reads

−

8πl

hcQ

8πl

, r



4π



, (1)

https://doi.org/10.1016/j.physletb.2020.135416

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