# An information geometric probe for charged black holes in 4D Einstein-Gauss-
Bonnet gravity

Syed Masood1<EMAIL_ADDRESS><EMAIL_ADDRESS>Said Mikki1,2
<EMAIL_ADDRESS>1Zhejiang University/University of Illinois at Urbana-
Champaign Institute (the ZJU-UIUC Institute), Zhejiang University, $718$ East
Haizhou Road, Haining $314400$, China.  2Department of Electrical and Computer
Engineering, University of Illinois at Urbana-Champaign, Urbana IL $61801$,
USA

(August 28, 2024; August 28, 2024)

###### Abstract

Glavan and Lin [Phys. Rev. Lett. 124, 081301 (2020)] have recently proposed a
consistent model of Einstein-Gauss-Bonnet modified gravity in four spacetime
dimensions. This model predicts significant contributions of the Gauss-Bonnet
coupling parameter $\alpha$ to gravitational dynamics, while circumventing the
Lovelock theorem and avoiding Ostrogradsky instability. As a powerful
competitor to general relativity, the model has been examined on various
phenomenological grounds. Here, we employ a technique from thermodynamic
geometry to analyze the thermodynamic phase structure of a charged black hole
with a quantum gravity-inspired entropy relation in this novel modified
gravity scenario. Based on the sign and magnitude of thermodynamic curvature,
we demonstrate that while the theory does not significantly impact larger
black holes, it may lead to multiple phase transitions and accelerate the
formation of black hole remnants at short-distance scales compared to general
relativity. Our analysis focuses solely on the non-extremal geometry case
where $M>\sqrt{Q^{2}+\alpha}$, with $M$ and $Q$ representing the mass and
charge of the black hole, respectively. We believe that these results may
offer insights for testing the phenomenological consistency of the theory as a
potential alternative to the standard Einstein paradigm.

##  I Introduction

The pioneering works of Hawking Hawking (1974, 1975) and Bekenstein Bekenstein
(1973) laid the foundation for the thermodynamics of black holes, establishing
a crucial cornerstone in the modern understanding of our Universe. The
conventional Bekenstein-Hawking formulation necessitates an entropy-area
correspondence, mathematically represented by $S_{\rm BH}=A/(4\ell_{\rm
p}^{2})$, where $A$ is the horizon area and $\ell_{\rm p}$ the Planck
length.111In this paper, the natural units $c=G=\hbar=1$ are used throughout.
The fact that black hole entropy scales with its area rather than its volume
is the foundation of the holographic principle Susskind (1995); Bousso (2003).
This paradigm effectively describes black hole thermodynamic behavior in terms
of macroscopic parameters (e.g., mass, charge, and angular momentum). However,
it does not provide insight into the underlying thermodynamic degrees of
freedom. Consequently, despite being considered perfect thermal systems, black
hole thermodynamics remains not fully understood Davies (1978); Page (2005).
Drawing insights from Boltzmann’s principle, “If you can heat it, it has
microscopic structure”, which has greatly guided our understanding of the
microstructure of thermodynamic systems, we naturally confront the question:
is it possible to decipher some kind of micromolecules (whatever they may look
like!) that give rise to what the Bekenstein-Hawking formulation envisions for
a black hole? Our work attempts to address this question within a novel
modified gravity framework going beyond the Einstein paradigm.

It is well-known that Hawking evaporation Hawking (1974, 1975) is based on a
semiclassical foundation, where spacetime geometry is described by classical
Einstein equations. This process reduces a black hole’s size, and a quantum
gravity theory is needed to describe the geometry once a characteristic size
is reached. Quantum gravity theories predict a minimum measurable distance
scale in nature, beyond which classical geometry breaks down due to quantum
fluctuations Hossenfelder (2013). This naturally challenges the Bekenstein-
Hawking formulation Mann and Solodukhin (1998); Upadhyay  _et al._ (2018);
Pourhassan  _et al._ (2020). As the geometric description changes, so does the
Bekenstein-Hawking entropy, leading us to compute black hole microstates
Strominger and Vafa (1996).

The modification of Bekenstein-Hawking entropy is tied to the quantum gravity
scale in string theories or loop quantum gravity, and these modifications can
be generically referred to as quantum corrections. Based on microstate
counting, additional sub-leading terms emerge beyond the leading Bekenstein-
Hawking contribution, which can be either perturbative or non-perturbative in
nature. Perturbative quantum corrections manifest as logarithmic or algebraic
terms Rovelli (1996); Ashtekar  _et al._ (1998); Kaul and Majumdar (2000);
Dabholkar  _et al._ (2011); Mandal and Sen (2010), while non-perturbative
corrections take on exponential forms Ashtekar (1991); Ghosh and Mitra (2014);
Dabholkar  _et al._ (2015); Chatterjee and Ghosh (2020).

Among the notable non-perturbative quantum corrections, there is the formalism
based on AdS/CFT correspondence Maldacena (1998) and the use of Kloosterman
sums near black hole horizons Murthy and Pioline (2009); Dabholkar  _et al._
(2013, 2015). For a black hole with a classical geometry, i.e., large size,
all quantum corrections are suppressed, and only the leading Bekenstein-
Hawking term predominates in its thermodynamic description. On small scales
where quantum fluctuations dominate the black hole geometry, both perturbative
and non-perturbative terms become significant. However, perturbative terms
contribute less compared to the non-perturbative ones. Non-perturbative terms,
particularly those with exponential forms Dabholkar  _et al._ (2013, 2015);
Chatterjee and Ghosh (2020), play a crucial role in black hole thermodynamics
in the quantum regime, making them a highly nontrivial case to consider. There
is a wide range of literature dedicated to quantum corrections to black hole
entropy, offering varied contexts and rich perspectives on the problem. For a
comprehensive discussion, we refer the reader to the relevant references
Hemming and Thorlacius (2007); Gregory  _et al._ (2008); Rocha (2008);
Saraswat and Afshordi (2020); Mann and Solodukhin (1998); Sen (2012);
Govindarajan  _et al._ (2001); Pourhassan  _et al._ (2017, 2018, 2020, 2021a);
Upadhyay  _et al._ (2022); Ghaffarnejad and Ghasemi (2022); Biswas  _et al._
(2021); Pourhassan  _et al._ (2022a, b); Aounallah  _et al._ (2022).

Though quantum gravity predicts corrections to both the geometric structure of
black holes and their associated thermodynamics, it is reasonable on
phenomenological grounds to focus solely on the thermodynamic aspects. This
forms the core idea behind the working assumptions of the present study and
has precedents in the literature (e.g., see Refs. Masood A. S. Bukhari  _et
al._ (2023); Pourhassan  _et al._ (2022a, b); Iso and Okazawa (2011)). It is
important to recognize that thermodynamic behavior is highly contingent on the
specific models of gravity considered.

Even though Einstein’s general relativity has made remarkable progress in
aligning with observational data—including notable achievements such as the
detection of gravitational waves Abbott  _et al._ (2016a, b) and imaging black
hole shadows Akiyama  _et al._ (2019)—physicists have long speculated about
its inadequacy in addressing certain fundamental issues in the Universe. These
issues include the formation of singularities in black holes, the existence of
dark energy and dark matter, and the development of a viable theory of quantum
gravity. As a consequence, a myriad of alternative models to general
relativity has been proposed Capozziello and De Laurentis (2011); Berti  _et
al._ (2015); Shankaranarayanan and Johnson (2022). A notable class of these
alternative gravity models posits higher-curvature corrections to the
Einstein-Hilbert action. Among the most promising is Einstein-Gauss-Bonnet
(EGB) gravity, which originates from the work of Lanczos Lanczos (1938) and
was further generalized by Lovelock Lovelock (1971, 1972). The hallmark of EGB
theories is that they do not yield any nontrivial contributions to
gravitational dynamics unless coupled with additional fundamental fields, such
as a dilaton field Blázquez-Salcedo  _et al._ (2016); Konoplya  _et al._
(2020); Maselli  _et al._ (2015); Ayzenberg  _et al._ (2014). Moreover, EGB
theories predict field equations that are quadratic in the metric tensor (or
curvature), thereby avoiding Ostrogradsky instability Ostrogradsky (1850).
This quadratic nature of the curvature grants EGB theories a unique privilege
among modified gravity models, as string theory predicts a quadratic
contribution next to leading order terms in the classical Einstein equations
Zwiebach (1985); Gross and Witten (1986); Gross and Sloan (1987). The view
that the Gauss-Bonnet term only contributes in 4D when coupled with a scalar
field was challenged by Glavan and Lin in a novel gravitational model Glavan
and Lin (2020). They demonstrated that a specific rescaling of the Gauss-
Bonnet coupling constant in the action results in nontrivial contributions to
gravitational dynamics even in four spacetime dimensions, bypassing the
Lovelock theorem Lovelock (1971, 1972); Lanczos (1938). This is achieved
without the need for additional field degrees of freedom. As a new
phenomenological competitor to General Relativity in 4D, this theory has been
scrutinized in various aspects, including the model’s consistency Lu and Pang
(2020); Hennigar  _et al._ (2020); Gürses  _et al._ (2020), quasinormal modes
and shadows Konoplya and Zinhailo (2020); Kumar and Ghosh (2020), geodesic
motion Guo and Li (2020), and black hole thermodynamics Wei and Liu (2020);
Hosseini Mansoori (2021); Eslam Panah  _et al._ (2020), among others. For a
comprehensive overview of 4D-EGB gravity and its associated phenomenology,
interested readers can refer to the review article by Fernandes et al.
Fernandes  _et al._ (2022).

Our goal is to gain insight into this novel gravitational theory by exploring
the thermodynamics of a charged black hole with entropy modified by
exponential quantum corrections. We employ Ruppeiner’s thermodynamic geometry
Ruppeiner (2014) to compute the thermodynamic scalar curvature. We then
discuss the combined role of the Gauss-Bonnet coupling parameter and the
parameter quantifying quantum corrections to the black hole’s entropy via a
suitable choice of charge. Given that our charged black hole possesses
multiple horizons and contributions from the Gauss-Bonnet coupling parameter
(generally very small), in addition to its charge, a differentiated extremal
and non-extremal geometry naturally follows. We adopt a canonical ensemble-
like framework with negligible charge contributions so that the size of our
black hole system is primarily dictated by its mass. This choice is motivated
by the need for a small black hole size to apply quantum corrections to its
entropy, which is only achievable by selecting an extremely small charge.

This article is organized as follows. In Sec. II, we provide an overview of
4D-EGB gravity and black hole solutions within this framework, along with
quantum corrections to the entropy. In Sec. III, we compute the heat capacity
of the system and discuss its phase transitions and stability. Sec. IV is
devoted to computing the thermodynamic curvature and discussing its
implications. Finally, Sec. VI provides a summary and conclusion of our
results.

##  II  Conceptual aspects

###  II.1 The ABC of 4D-EGB Gravity

In standard general relativity, the 4D Einstein-Hilbert action is written as

$\displaystyle S_{\rm EH}=\int\mathrm{d}^{4}x\sqrt{-g}R,$ (1)

where $R$ is the Ricci curvature scalar, and $g$ the determinant of metric
tensor $g_{\mu\nu}$. Lovelock theorem Lovelock (1971, 1972); Lanczos (1938)
asserts that general relativity is the unique theory of gravity in four
dimensions, provided certain conditions are met. These conditions include
diffeomorphism invariance, metricity, and second-order equations of motion. In
higher-dimensional spacetime, the action that satisfies these three conditions
is the Einstein-Gauss-Bonnet (EGB) action, given by

$\displaystyle S_{\rm
EGB}=\int\mathrm{d}^{D}x\sqrt{-g}\left(R+\alpha\mathcal{G}\right),$ (2)

where
$\mathcal{G}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
is the Gauss-Bonnet invariant. Here, $R^{\mu\nu}$ and $R^{\mu\nu\rho\sigma}$
are the Ricci and Riemann tensors, respectively. By varying the action in Eq.
(2) with respect to metric tensor $g_{\mu\nu}$, we get the following field
equations of gravity:

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\alpha
H_{\mu\nu}=T_{\mu\nu},$ (3)

where

$H_{\mu\nu}=2RR^{\mu\nu}-4R_{\mu\sigma}R^{\sigma}_{\nu}-4R_{\mu\sigma\nu\rho}R^{\sigma\rho}-2R_{\mu\sigma\nu\delta}R^{\sigma\rho\delta}_{\nu}-\frac{1}{2}\mathcal{G}g_{\mu\nu},$
(4)

and $T_{\mu\nu}$ is the stress-energy tensor. Until recently, it was well
known that in ordinary 4D spacetime, the fact that $\mathcal{G}$ is a total
derivative implies it has no contribution to the dynamics. However, the idea
proposed in Ref. Glavan and Lin (2020) suggested a way to circumvent this
conclusion through a novel rescaling of the coupling constant by

$\displaystyle\alpha\rightarrow\frac{\alpha}{D-4},$ (5)

which, after taking the limit $D\rightarrow 4$, produces nontrivial
contributions of the Gauss-Bonnet term to the dynamics.

###  II.2 Black holes in 4D-EGB theory

The novel theory of 4D-EGB gravity thus predicts a line element of the form
Glavan and Lin (2020); Fernandes  _et al._ (2022)

$\displaystyle\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+\frac{1}{f(r)}\mathrm{d}r^{2}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}{\theta}\mathrm{d}\phi^{2}\right)$
(6)

with

$f(r)=1+\frac{r^{2}}{2\alpha}\left[1\pm\sqrt{1+\left(\frac{8\alpha
M}{r^{3}}\right)}\right].$ (7)

There are two branches of solutions for the above metric: the plus sign
indicates the Gauss-Bonnet branch, while the minus sign represents the general
relativity branch. The physical viability of these branches can be established
by examining the limits of $\alpha$ and $r$ in the solutions. Let’s consider
the far-field limit $r\rightarrow\infty$. For the general relativity branch
from Eq. (7), this yields the following:

$\displaystyle f(r)=1-\frac{2M}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right).$
(8)

Note that this is the usual Schwarzschild solution. Now for the Gauss-Bonnet
branch, we have

$\displaystyle
f(r)=1+\frac{2M}{r}+\frac{r^{2}}{\alpha}+\mathcal{O}\left(\frac{1}{r^{2}}\right),$
(9)

which is obviously not asymptotically flat and hence represents an unphysical
scenario. Furthermore, the condition $\alpha\rightarrow 0$ for the general
relativity branch gives a well-defined limit

$\displaystyle f(r)=1-\frac{2M}{r}+\mathcal{O}\left(\alpha\right),$ (10)

while for Gauss-Bonnet branch, one obtains

$\displaystyle
f(r)=1+\frac{2M}{r}+\frac{r^{2}}{\alpha}+\mathcal{O}\left(\alpha\right),$ (11)

which is not a well-defined limit. Note also that the mass term for the Gauss-
Bonnet branch has the wrong sign Glavan and Lin (2020); Fernandes  _et al._
(2022). Hence, this branch is discarded as a viable physical solution. On
these grounds, we only consider the general relativity branch in our analysis.

Next, we investigate the impact of $\alpha$ on the metric function and horizon
radius. Solving Eq. (7) for $r$ yields

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-\alpha},$ (12)

where $r_{+}$ represents the event horizon radius of the black hole, while
$r_{-}$ is the Cauchy horizon. For $\alpha\rightarrow 0$, this yields the
usual Schwarzschild solution.

We graphically illustrate the metric function $f(r)$ and the horizon radius
$r_{+}$ against Gauss-Bonnet parameter $\alpha$ in Fig.1.

(a)

(b)

Figure 1: Impact of $\alpha$ on (a) the metric function $f(r)$, where it
indicates finiteness of the metric coefficient at $r\rightarrow 0$ compared to
Einstein gravity ($\alpha=0$) , and (b) horizon radius $r_{+}$ depicting how
$\alpha$ shrinks the black hole.

It can be readily observed from Fig. 1(a) that $f(r)$ for the 4D-EGB black
hole is finite near the origin $r\rightarrow 0$. However, the metric
coefficients being finite at the origin $r\rightarrow 0$ do not eliminate the
central singularity, as the Kretschmann curvature scales as
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\propto r^{-3}$ Fernandes  _et al._
(2022). Interestingly, in general relativity for a Schwarzschild black hole,
the Kretschmann scalar scales as $r^{-6}$ at any radius $r$. This implies that
the introduction of the Gauss-Bonnet coupling parameter $\alpha$ weakens the
black hole singularity. One can also see from Fig. 1(b) that for a given black
hole mass $M$, the introduction of $\alpha$ reduces the black hole size
compared to the general relativity case. This subtle effect arises purely from
higher curvature corrections to Einstein gravity, providing a physical picture
of the nontrivial contributions of $\alpha$ to gravitational dynamics.
Additionally, we note that there is another singularity at a radius where
$r^{3}=-8\alpha M$, where the expression under the square root in Eq. (7)
becomes zero. However, it is important to note that the negative values of
$\alpha$ are strictly constrained. According to Refs. Guo and Li (2020);
Konoplya and Zinhailo (2020), the range of $\alpha$ is $-32M^{2}\leq\alpha\leq
4M^{2}$. In this work, we focus exclusively on positive values of $\alpha$,
following the approach in the original study Glavan and Lin (2020).

Charged black hole solutions in 4D-EGB gravity have also been found, for which
the following relation holds Fernandes (2020)

$f(r)=1+\frac{r^{2}}{2\alpha}\left[1-\sqrt{1+4\alpha\left(\frac{2M}{r^{3}}-\frac{Q^{2}}{r^{4}}\right)}\right].$
(13)

For the non-extremal geometry scenario, under which $M>\sqrt{Q^{2}+\alpha}$,
the zeros of $f(r)$ give rise to two horizons located at

$r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}-\alpha},$ (14)

where $r_{+}$ is the black hole horizon, and $r_{-}$ the inner Cauchy horizon.
One can work with a timelike Killing vector $\xi^{\mu}$, which allows us to
define the surface gravity $\kappa$ of the black hole as
$\kappa=\nabla_{\mu}\xi^{\mu}\nabla_{\nu}\xi^{\nu}=\frac{1}{2}\left[\frac{\mathrm{d}f(r)}{\mathrm{d}r}\right]_{r=r_{+}}$.
This surface gravity corresponds to a black hole temperature of
$\frac{\hbar\kappa}{2\pi}$. Utilizing $r_{+}$ from Eq. (14), this yields:

$\displaystyle T_{\rm
BH}=\frac{\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{3}\left[\sqrt{\frac{\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{4}+8\alpha
M\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)-4\alpha
Q}{\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{4}}}-1\right]-2\alpha
M}{\alpha\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{2}\sqrt{\frac{\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{4}+8\alpha
M\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)-4\alpha
Q}{\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{4}}}}.$ (15)

The quantity $T_{\rm BH}$ is plotted in Fig. 2, illustrating that the Gauss-
Bonnet coupling parameter $\alpha$ reduces the black hole temperature at
smaller scales, exhibiting a behavior similar to that of the charge $Q$.
Notably, we did not introduce any corrections to $T_{\rm BH}$ due to $\eta$,
see Eq. (17). The reason for this is that we use a modified entropy formula
based on quantum gravity principles Chatterjee and Ghosh (2020) without
invoking any quantum geometry. Consequently, the metric function $f(r)$ in Eq.
(7) remains unchanged, and the definition of temperature $(T_{\rm
BH}\propto\mathrm{d}f(r)/\mathrm{d}r)$ follows from this function. This
approach has been discussed in some earlier works Pourhassan  _et al._ (2022a,
b); Masood A. S. Bukhari  _et al._ (2023).

Figure 2: Black hole temperature $T_{\rm BH}$ vs the mass $M$. The Gauss-
Bonnet coupling $\alpha$ parameter tends to make the black hole colder on
smaller scales, mimicking the role of charge in Reissner-Nordström geometry.

###  II.3 Entropy in the quantum regime

One of the earliest attempts to understand the microscopic degrees of freedom
for black hole entropy originates from the string theory approach Strominger
and Vafa (1996). However, as previously mentioned, approaches to quantum
gravity and string theory through microstate counting predict additional
subleading perturbative or nonperturbative contributions to the original
Bekenstein-Hawking term. Nevertheless, all these approaches yield only the
Bekenstein-Hawking contribution for larger geometries, with the extra terms
becoming significant only in the quantum regime. A more robust and fundamental
approach would be to quantize the gravitational action and deduce the
resulting thermodynamic behavior. However, this task is exceedingly difficult
due to the mathematical complexity involved and the uncertaintiy about the
ultimate physical assumptions underlying quantum gravity. Yet one can adopt a
more pragmatic approach by considering only the quantum corrections to
entropy, thereby exploring black hole thermodynamics in the quantum regime. In
this context, the Jacobson framework Jacobson (1995) and its connections to
thermal fluctuations Faizal  _et al._ (2017) may serve as a motivational
basis.

The perturbative contributions to the quantum corrections of black hole
entropy have the following general form Ashtekar  _et al._ (1998); Kaul and
Majumdar (2000); Pourhassan  _et al._ (2018, 2020):

$\displaystyle S_{\rm p}=\mathcal{A}\ln{\left(\frac{A}{4\ell_{\rm
p}^{2}}\right)}+\frac{4\mathcal{B}\ell_{\rm p}^{2}}{A}+\cdots,$ (16)

where as before $A$ denotes horizon area of the black hole while $\mathcal{A}$
and $\mathcal{B}$ are some constants related to the quantum gravity scale. Now
the non-perturbative corrections read as Dabholkar  _et al._ (2013, 2015);
Chatterjee and Ghosh (2020):

$\displaystyle S_{\rm np}=\eta e^{-A/4\ell_{\rm p}^{2}}.$ (17)

Here, $\eta$ is a positive parameter measuring the scale of the non-
perturbative contribution to the black hole’s entropy. With that, the total BH
entropy of the black hole would be $S_{\rm BH}=S_{0}+S_{\rm p}+S_{\rm np}$.
There is an intriguing aspect to black holes in 4D-EGB theory: the entropy
already includes logarithmic contributions (albeit perturbative) from
classical geometry considerations, as outlined in Ref. Fernandes (2020).
Taking note of this, after incorporating exponential corrections Chatterjee
and Ghosh (2020), we express the total entropy of the black hole as

$\displaystyle S_{\rm exp}=S_{0}+\eta e^{-S_{0}},$ (18)

where $S_{0}=A+\alpha\log{A}$ is the original entropy reported in Ref.
Fernandes (2020). Given that $A=r_{+}^{2}$ (omitting proportionality
constants), we expand Eq. (18) as follows:

$\displaystyle S_{\rm exp}$
$\displaystyle=S_{0}+\eta\exp{\left(-S_{0}\right)}$
$\displaystyle=\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{2}+2\alpha\log{\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)}$
$\displaystyle+\eta\exp{\left[-\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)^{2}-2\alpha\log{\left(M+\sqrt{M^{2}-Q^{2}-\alpha}\right)}\right]}.$
(19)

(a)

(b)

Figure 3: Variation of black hole entropy with respect to (a) exponential
parameter $\eta$, and (b) Gauss-Bonnet coupling $\alpha$. Exponential
contributions become dominant on quantum scales compared to Bekenstein-Hawking
term.

Fig. 3 illustrates the distinctive nature of the entropy curves corresponding
to different values of $\alpha$ (the main 4D-GBG parameter) and $\eta$ (the
quantum correction scale). For larger sizes, all curves tend to coincide,
reflecting the dominance of the Bekenstein-Hawking term. It’s important to
note that our black hole system features bifurcate horizons and exhibits
distinct non-extremal and extremal geometric descriptions corresponding to the
cases $M>\sqrt{Q^{2}+\alpha}$ and $M=\sqrt{Q^{2}+\alpha}$, respectively.
Beyond the extremal limit, the black hole singularity becomes naked, a
scenario typically forbidden by the cosmic censorship conjecture Penrose
(1969). Due to the nonperturbative nature of exponential corrections in the
quantum regime, the plots exhibit a sudden jump near the extremal limit
whenever $M=\sqrt{Q^{2}+\alpha}$. It’s noteworthy that the entropy reaches a
large value at this point but does not diverge. If one imagines the extremal
limit as the endpoint of Hawking evaporation, as will become apparent from the
heat capacity analysis in the next section, a black remnant is formed there.
Upon further examination of the plots, we observe that $\eta$ contributes to a
larger entropy [Fig. 3(a)], whereas $\alpha$ tends to hasten the end of
evaporation (for a fixed $Q$) by shifting the extremal limit
$M=\sqrt{Q^{2}+\alpha}$ each time $\alpha$ is changed [Fig. 3(b)].
Additionally, from Fig. 3(b), we notice that for the $\alpha=0$ case, the jump
in entropy occurs precisely at $M=Q$, a characteristic of the Reissner-
Nordström geometry in general relativity.

As one might discern, the quantum corrections to the entropy become prominent
near the extremal limit $M=\sqrt{Q^{2}+\alpha}$, strongly indicating a precise
definition of the black hole geometric scales and the applicability of quantum
corrections to the entropy. The conventional perspective of Hawking
evaporation assumes the black hole diminishes in size, involving the entirety
of its horizon radius $r_{+}$, which naturally encompasses a combination of
all three parameters in our case: $M$, $Q$, and $\alpha$. This fact is well-
established concerning charged black holes Hiscock and Weems (1990), where the
charge-to-mass ratio evolves as the black hole continues its evaporation
towards the extremal limit. However, our definition assumes a canonical
ensemble framework where $M$ fluctuates while $Q$ and $\alpha$ remain
constant. This implies that the entire evaporation process occurs via $M$,
which governs the black hole’s geometric scales. However, it’s important to
exercise caution regarding the magnitudes of all three parameters $M$, $Q$,
and $\alpha$, as their values must satisfy the quantum gravity scale in
relation to entropy corrections. Here, we enforce the condition that $Q$ and
$\alpha$ are extremely small, providing nearly negligible contributions.
Consequently, whenever $M\rightarrow\sqrt{Q^{2}+\alpha}$, i.e., the extremal
limit, our black hole tends to possess a quantum description; otherwise, it
behaves as a classical system for all $M>\sqrt{Q^{2}+\alpha}$. This
description would place the black hole in a coexisting phase of classical and
quantum descriptions corresponding to a characteristic value of $M$. As we
will later observe, this particular value of $M$ corresponds to the first root
of the heat capacity $C_{Q}$ (Fig. 4). With that said, the quantum corrections
to the entropy via Eq. (18) naturally follow.

##  III  Black hole stability via heat capacity

The thermodynamic stability of black holes can be explored through the
examination of various thermodynamic potentials, depending on the chosen
ensemble approach. For instance, in the canonical ensemble approach, one can
define the heat capacity $C_{Q}$ of the system, which in our case corresponds
to a constant charge $Q$, and is given by the following expression:

$\displaystyle C_{Q}=T\left(\frac{\partial S_{\rm exp}}{\partial
T}\right)_{Q}=T\left(\frac{\partial S_{\rm exp}/\partial M}{\partial
T/\partial M}\right)_{Q}.$ (20)

Substituting Eq. (18) into Eq. (20), the following formula is obtained:

$\displaystyle C_{Q}$
$\displaystyle=\frac{2e^{-r_{+}^{2}}\left(Q^{2}+\alpha-2Mr_{+}-1\right)\left(-r_{+}^{4}-8\alpha
Mr_{+}+4\alpha Q\right)\left(Y-2\alpha
M\right)\left[e^{r_{+}^{2}}\left(Q^{2}+\alpha-2Mr_{+}\right)+\eta\right]}{r_{+}^{2}\left[A+B+C+D+2\left(-r_{+}^{4}-8\alpha
Mr_{+}+4\alpha Q\right)\left(2\alpha M-Y\right)\right]},$

where, for brevity, we introduce the quantities

$\displaystyle Y$ $\displaystyle=r_{+}^{3}\left(\sqrt{\frac{8\alpha
Mr_{+}-4\alpha Q}{r_{+}^{4}}+1}-1\right),$ $\displaystyle A$
$\displaystyle=3\left(r_{+}^{4}+8\alpha Mr_{+}-4\alpha
Q\right)\left(1-\sqrt{\frac{8\alpha Mr_{+}-4\alpha
Q}{r_{+}^{4}}+1}\right)r_{+}^{3},$ $\displaystyle B$ $\displaystyle=4\alpha
r_{+}^{3}\left[\alpha+(Q-2)Q+2Mr_{+}\right]\sqrt{\frac{8\alpha Mr_{+}-4\alpha
Q}{r_{+}^{4}}+1},$ $\displaystyle C$
$\displaystyle=2\alpha\sqrt{M^{2}-Q^{2}-\alpha}\left(r_{+}^{4}+8\alpha
Mr_{+}-4\alpha Q\right),$ $\displaystyle\ D$
$\displaystyle=4\alpha\left[\alpha+(Q-2)Q+2Mr_{+}\right]\left[2\alpha
M-r_{+}^{3}\left(\sqrt{\frac{8\alpha Mr_{+}-4\alpha
Q}{r_{+}^{4}}+1}-1\right)\right].$

(a)

(b)

(c)

Figure 4: Heat capacity $C_{Q}$ as a function of black hole mass $M$ for $Q=1$
and Gauss-Bonnet coupling (a) $\alpha=0.1$ and (b) $\alpha=0.2$. The second
zero of $C_{Q}$ occurs whenever $M=\sqrt{Q^{2}+\alpha}$.

The black hole’s heat capacity is computed and displayed in Fig. 4. It’s
important to note that a negative heat capacity indicates an unstable
thermodynamic phase, and vice versa Davies (1978). We observe from the plots
that regardless of the Gauss-Bonnet parameter $\alpha$ and the correction
parameter to the entropy $\eta$, the heat capacity $C_{Q}$ is negative for
this charged black hole on larger scales [Fig. 4(c)]. The scenario unfolds
differently as the black hole geometry shrinks due to Hawking evaporation.
$C_{Q}$ either tends to become more negative [Fig. 4(a), $\eta=0,50$, and
$80$], or transitions to positive values via $C_{Q}=0$ [Fig. 4(a), $\eta=150$,
and $200$] before encountering an infinite discontinuity at a characteristic
mass value $M$. This distinct behavior of $C_{Q}$ arises for specific choices
of $\eta$ and $\alpha$, and may be absent in other cases [see Fig. 4(b)]. We
also observe that compared to the original uncorrected case ($\eta=0$), which
indicates a single root for $C_{Q}$, all curves with $\eta\neq 0$ possess two
zeros of $C_{Q}$. The transition of $C_{Q}$ from negative to positive values
indicates that the black hole phase changes from being unstable to a stable
one, reflecting a second-order phase transition in charged black holes Davies
(1978). This transition closely resembles familiar thermodynamic phase
transitions in non-gravitational systems, such as ferromagnetic to
paramagnetic, conductor to superconductor, liquid-crystal phase transitions,
etc.

The zeros of $C_{Q}$ are typically interpreted as critical points that help
distinguish between positive and negative temperature solutions Hendi and
Momennia (2018). In this case, the second zero of $C_{Q}$ occurs at the
extremal limit where black hole evaporation ceases, corresponding to
$M=\sqrt{Q^{2}+\alpha}$. The first zero of $C_{Q}$ marks the onset of the
phase transitions and represents the point where our earlier discussion
regarding the definition of classical and quantum geometry for the black hole,
in relation to entropy corrections, becomes relevant. This endpoint related to
$M=\sqrt{Q^{2}+\alpha}$ may signify the formation of a black hole remnant, and
is independent of $\eta$ in our case. However, it manifests explicit
dependence on $\alpha$, which we have numerically computed and indicated in
terms of certain characteristic values of $M$ for $\alpha=0.1$ and $0.2$ in
Figs. 4(a) and (b), respectively. Each time $\alpha$ takes new values, the
extremal limit and corresponding shift in the last root of $C_{Q}$ is
observed. Therefore, it is apparent that compared to general relativity,
4D-EGB theory predicts the formation of remnants much earlier. This
underscores the significance of 4D-EGB gravity for short-distance or high-
energy scales. Meanwhile, determining the (in)stability of this remnant is
challenging within our approach, as we only explore up to the extremal limit
of the black hole geometry. However, we will demonstrate in Section IV that
thermodynamic geometry offers a more comprehensive framework to address this
issue.

##  IV  Black hole phase structure via information geometry

###  IV.1 Basic tenets of Ruppeiner geometry

Information or thermodynamic geometry, or in short, geometrothermodynamics, is
a powerful tool for understanding phase transitions and the stability of
systems undergoing fluctuations around thermal equilibrium. In this approach,
a parameter space, akin to Riemannian geometry in gravitational systems, is
defined, spanning some extensive quantities of the system, which later aids in
defining a Ricci-like curvature Ruppeiner (1995). The initial impetus came
from Weinhold Weinhold (1975a, b), who defined the metric by taking the
Hessian of the internal energy with respect to the other extensive variables
of the system. This was followed by a rigorous approach by Ruppeiner Ruppeiner
(1979, 1981, 2014), who employed entropy instead of internal energy. While
these methods have been primarily developed and rigorously applied in various
well-known fluctuating systems such as quantum liquids, magnetic systems,
Ising models, and so on Ruppeiner (1979, 1981); Janyszek and Mrugala (1989);
Ruppeiner (2014), their scope has nevertheless transcended these more familiar
non-gravitational thermodynamic systems. They have provided deeper insights
into exotic gravitational systems, such as black holes Ruppeiner (2014). It is
anticipated that such a construction may potentially offer insights into the
microscopic structure of black hole thermodynamics, which is typically absent
in the conventional Bekenstein-Hawking formalism.

Denoting the internal energy by $M$, Weinhold metric has the form
$g_{\mu\nu}^{W}=\partial_{\mu}\partial_{\nu}M(S,N^{i}),$ where $S$ is the
entropy, while $N^{i}$ are all other extensive quantities indexed by $i$.
These quantities may include volume, internal energy etc. Each combination of
$\mu,\nu=0,1,2,\dots$ represents one of these quantities. With this, Weinhold
line element is given by
$\mathrm{d}s_{W}^{2}=g_{\mu\nu}^{W}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}.$
Likewise, we have Ruppeiner metric given by
$g_{\mu\nu}=-\frac{\partial^{2}}{\partial x^{\mu}\partial x^{\nu}}S.$

Several other information geometric methods have been developed recently, such
as Quevedo Quevedo (2007); Quevedo and Sanchez (2008) and the Hendi-Panahiyan-
Eslam-Panah-Momennia (HPEM) metric Hendi  _et al._ (2015), which have also
proven useful for exploring black hole thermodynamics. However, our focus here
is on employing the Ruppeiner formalism, for which we will now provide a
detailed derivation in our specific context in this work.

Let’s start with the standard Boltzmann entropy formula $S=k_{\rm
B}\ln\Omega$, with $\Omega$ being the microstate count
$\Omega=\exp{\left(S/k_{\rm B}\right)}.$ Next, we consider a parameter space
comprising $x^{0}$ and $x^{1}$ that define the black hole. As fluctuations
occur in the system, we can estimate the probability of finding the black hole
system within the intervals $x^{0}+\mathrm{d}x^{0}$ and
$x^{1}+\mathrm{d}x^{1}$ as
$P(x^{0},x^{1})\mathrm{d}x^{0}\mathrm{d}x^{1}=\Lambda\Omega(x^{0},x^{1})\mathrm{d}x^{0}\mathrm{d}x^{1}$,
where $\Lambda$ is a normalization constant. Using the expression for $\Omega$
given above, we may write $P(x^{0},x^{1})\propto\exp{\left(S/k_{\rm
B}\right)}$ and $S(x^{0},x^{1})=S_{\rm BH}(x^{0},x^{1})+S_{\rm
E}(x^{0},x^{1}),$ with $S_{\rm BH}$ and $S_{\rm E}$ respectively being the
black hole and environment entropies. If there is a small fluctuation in the
equilibrium entropy around the point $x_{0}^{\mu}$ (with $\mu,\nu=0,1$), one
can Taylor expand the total entropy around $x_{0}^{\mu}$ as follows:

$\displaystyle S$ $\displaystyle=S_{0}+\frac{\partial S_{\rm BH}}{\partial
x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{\rm bh}^{\mu}+\frac{\partial
S_{\rm E}}{\partial x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{\rm
E}^{\mu}$ $\displaystyle+\frac{1}{2}\frac{\partial^{2}S_{\rm BH}}{\partial
x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{\rm
bh}^{\mu}\Delta x_{\rm bh}^{\nu}+\frac{1}{2}\frac{\partial^{2}S_{\rm
E}}{\partial x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta
x_{\rm E}^{\mu}\Delta x_{\rm E}^{\nu}+\cdots,$ (21)

where $S_{0}$ is the equilibrium entropy at $x_{0}^{\mu}$. Now, if we assume a
closed system where the extensive parameters of the black hole ($x_{\rm
bh}^{\mu}$) and the environment ($x_{\rm E}^{\mu}$) have a conservative
additive nature, such that $x_{\rm bh}^{\mu}+x_{\rm E}^{\mu}=x_{\rm
total}^{\mu}=\text{constant}$, then we can write:

$\displaystyle\frac{\partial S_{\rm BH}}{\partial
x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{\rm bh}^{\mu}=-\frac{\partial
S_{\rm E}}{\partial x^{\mu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta x_{\rm
E}^{\mu}.$ (22)

We therefore have

$\displaystyle\Delta S$ $\displaystyle=\frac{1}{2}\frac{\partial^{2}S_{\rm
BH}}{\partial x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta
x_{\rm bh}^{\mu}\Delta x_{\rm bh}^{\nu}+\frac{1}{2}\frac{\partial^{2}S_{\rm
E}}{\partial x^{\mu}\partial x^{\nu}}\bigg{|}_{x^{\mu}=x_{0}^{\mu}}\Delta
x_{\rm E}^{\mu}\Delta x_{\rm E}^{\nu}.$ (23)

As the entropy of the environment is almost equal to the total entropy, i.e.,
$S_{\rm E}\sim S$, the corresponding fluctuations in $S_{\rm E}$ are
negligible. Consequently, we are left with only the black hole system, such
that $P(x^{0},x^{1})\propto\exp{\left(-\frac{1}{2}\Delta l^{2}\right)},$ with
$\Delta l^{2}$ given by $\Delta l^{2}=-\frac{1}{k_{\rm B}}g_{\mu\nu}\Delta
x^{\mu}\Delta x^{\nu}.$ Setting $k_{\rm B}=1$, one has $\Delta
l^{2}=g_{\mu\nu}\Delta x^{\mu}\Delta x^{\nu}$, where
$g_{\mu\nu}=-\frac{\partial^{2}}{\partial x^{\mu}\partial x^{\nu}}S_{\rm BH}.$
Given that probability is a dimensionless scalar, $\Delta l^{2}$, as given up,
is a dimensionless, positive definite, invariant quantity. This line element
mimics the line element in black holes and is usually considered a quantifying
measure of the thermodynamic length between two fluctuating black hole
microstates. Quoting Ruppeiner Ruppeiner (2014): “Thermodynamic states are
further apart if the fluctuation probability is less.” This principle
resonates with Le Chatelier’s principle, which ensures the local stability of
thermodynamic systems. Dropping the subscript $\rm BH$, we write
$g_{\mu\nu}=-\frac{\partial^{2}}{\partial x^{\mu}\partial x^{\nu}}S$, as the
metric of Ruppeiner geometry. Based on this metric, we can now compute the
associated curvature scalar in the same fashion as one usually does in
Riemannian geometry. Given the Christoffel connections are
$\Gamma_{\mu\nu}^{\sigma}=\frac{1}{2}g^{\sigma\rho}\left(\partial_{\nu}g_{\rho\mu}+\partial_{\mu}g_{\rho\nu}-\partial_{\rho}g_{\mu\nu}\right)$,
along with the Riemann tensor
$R_{\rho\mu\nu}^{\sigma}=\partial_{\nu}\Gamma_{\rho\mu}^{\sigma}-\partial_{\mu}\Gamma_{\rho\nu}^{\sigma}+\Gamma_{\rho\mu}^{\delta}\Gamma_{\delta\nu}^{\sigma}-\Gamma_{\rho\nu}^{\delta}\Gamma_{\delta\mu}^{\sigma}$,
one can define Ricci tensor and scalar as
$R_{\mu\nu}=R_{\mu\sigma\nu}^{\sigma}$ and $R=g^{\mu\nu}R_{\mu\nu}$. Here, the
Ricci curvature is Carroll (2019)

$\displaystyle R$
$\displaystyle=-\frac{1}{\sqrt{g}}\bigg{[}\frac{\partial}{\partial
x^{0}}\left(\frac{g_{01}}{g_{00}\sqrt{g}}\frac{\partial g_{00}}{\partial
x^{1}}-\frac{1}{\sqrt{g}}\frac{\partial g_{11}}{\partial x^{0}}\right)$
$\displaystyle+\frac{\partial}{\partial
x^{1}}\left(\frac{2}{\sqrt{g}}\frac{\partial g_{01}}{\partial
x^{0}}-\frac{1}{\sqrt{g}}\frac{\partial g_{00}}{\partial
x^{1}}-\frac{g_{01}}{g_{00}\sqrt{g}}\frac{\partial g_{00}}{\partial
x^{0}}\right)\bigg{]},$ (24)

where $g=\det{g_{\mu\nu}}=g_{00}g_{11}-g_{01}^{2}$.

###  IV.2 Computing the Ruppeiner thermodynamic curvature $R_{C}$

Employing the same technique for the Ruppeiner metric
$g_{\mu\nu}=-\partial_{\mu}\partial_{\nu}S(M,N^{i})$, with a $2$-dimensional
state space of non-diagonal $g_{\mu\nu}$, the line element is given by

$\displaystyle\mathrm{d}s_{R}^{2}=g_{MM}\mathrm{d}M^{2}+2g_{MQ}\mathrm{d}M\mathrm{d}Q+g_{QQ}\mathrm{d}Q^{2},$
(25)

with the metric $g_{\mu\nu}$ specified as
$g_{00}=g_{MM},g_{01}=g_{MQ},g_{10}=g_{QM},g_{11}=g_{QQ},$ where we have used
$M$ and $Q$ as the extensive variables as they are the most natural choice for
our charged black hole. These components of $g_{\mu\nu}$ can be computed by
expressing the metric in terms of derivatives of the entropy with respect to
the extensive variables as follows:

$\displaystyle g_{MM}=-\frac{\partial}{\partial M}\left(\frac{\partial
S}{\partial M}\right),\ g_{MQ}=-\frac{\partial}{\partial
M}\left(\frac{\partial S}{\partial Q}\right),$ (26) $\displaystyle
g_{QM}=-\frac{\partial}{\partial Q}\left(\frac{\partial S}{\partial
M}\right),\ g_{QQ}=-\frac{\partial}{\partial Q}\left(\frac{\partial
S}{\partial Q}\right),$ (27)

They can be explicitly evaluated, leading to the following expressions:

$\displaystyle g_{MM}$
$\displaystyle=-\frac{1}{\left(M^{2}-Q^{2}-\alpha\right)^{3/2}}\Bigg{[}2e^{-\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}\bigg{\\{}e^{\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}\left(2\sqrt{M^{2}-Q^{2}-\alpha}-M\right)$
$\displaystyle\hskip
85.35826pt+\eta\left[4M^{3}+4M^{2}\sqrt{M^{2}-Q^{2}-\alpha}-2\left(\alpha+Q^{2}+1\right)\sqrt{M^{2}-Q^{2}-\alpha}+M\left(-4\alpha-4Q^{2}+1\right)\right]\bigg{\\}}\Bigg{]}$
$\displaystyle g_{MQ}$
$\displaystyle=\frac{2Q}{\left(M^{2}-Q^{2}-\alpha\right)^{3/2}}\Bigg{[}-\alpha+\eta
e^{-\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}\bigg{\\{}\alpha+2\Big{[}4M^{4}-5M^{2}\left(\alpha+Q^{2}\right)-3M\left(\alpha+Q^{2}\right)\sqrt{M^{2}-Q^{2}-\alpha}$
$\displaystyle\hskip
85.35826pt+4M^{3}\sqrt{M^{2}-Q^{2}-\alpha}+\left(\alpha+Q^{2}\right)^{2}\Big{]}+Q^{2}\bigg{\\}}-Q^{2}\Bigg{]}$
$\displaystyle g_{QM}$
$\displaystyle=\frac{2Q}{\left(M^{2}-Q^{2}-\alpha\right)^{3/2}}\Bigg{[}-\alpha+\eta
e^{-\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}\bigg{\\{}\alpha+2\Big{[}4M^{4}-5M^{2}\left(\alpha+Q^{2}\right)-3M\left(\alpha+Q^{2}\right)\sqrt{M^{2}-Q^{2}-\alpha}$
$\displaystyle\hskip
85.35826pt+4M^{3}\sqrt{M^{2}-Q^{2}-\alpha}+\left(\alpha+Q^{2}\right)^{2}\Big{]}+Q^{2}\bigg{\\}}-Q^{2}\Bigg{]}$
$\displaystyle g_{QQ}$
$\displaystyle=\frac{2}{\left(M^{2}-Q^{2}-\alpha\right)^{3/2}}\Bigg{[}M^{3}+M^{2}\sqrt{M^{2}-Q^{2}-\alpha}-\left(\alpha+Q^{2}\right)\sqrt{M^{2}-Q^{2}-\alpha}+\eta
e^{-\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}\bigg{\\{}-M^{3}\left(4Q^{2}+1\right)$
$\displaystyle\hskip
85.35826pt+\left[\left(2Q^{2}+1\right)\left(\alpha+Q^{2}\right)-M^{2}\left(4Q^{2}+1\right)\right]\sqrt{M^{2}-Q^{2}-\alpha}+M\left[\alpha+4Q^{2}\left(\alpha+Q^{2}\right)\right]\bigg{\\}}-\alpha
M\Bigg{]},$

along with the determinant

$\displaystyle g$
$\displaystyle=-\frac{4e^{-2\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}}{\left(M^{2}-Q^{2}-\alpha\right)^{5/2}}\Bigg{[}\Big{(}e^{\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}-\eta\Big{)}e^{\left(\sqrt{M^{2}-Q^{2}-\alpha}+M\right)^{2}}\bigg{\\{}4M^{5}-M^{3}\left(9\alpha+7Q^{2}\right)-M^{2}\left(7\alpha+5Q^{2}\right)$
$\displaystyle\times\sqrt{M^{2}-Q^{2}-\alpha}+\left(\alpha+Q^{2}\right)\left(2\alpha+Q^{2}\right)\sqrt{M^{2}-Q^{2}-\alpha}+4M^{4}\sqrt{M^{2}-Q^{2}-\alpha}+M\left(\alpha+Q^{2}\right)\left(5\alpha+3Q^{2}\right)\bigg{\\}}$
$\displaystyle+\eta\bigg{\\{}32M^{7}-4M^{5}\left(18\alpha+18Q^{2}+1\right)+M^{3}\left[\alpha(50\alpha+9)+50Q^{4}+(100\alpha+7)Q^{2}\right]+M^{2}\left[\alpha(26\alpha+7)+26Q^{4}+(52\alpha+5)Q^{2}\right]$
$\displaystyle\times\sqrt{M^{2}-Q^{2}-\alpha}-\left(\alpha+Q^{2}\right)\left[2\alpha(\alpha+1)+2Q^{4}+(4\alpha+1)Q^{2}\right]\sqrt{M^{2}-Q^{2}-\alpha}+32M^{6}\sqrt{M^{2}-Q^{2}-\alpha}-4M^{4}$
$\displaystyle\times\left(14\alpha+14Q^{2}+1\right)\sqrt{M^{2}-Q^{2}-\alpha}-M\left(\alpha+Q^{2}\right)\left[5\alpha(2\alpha+1)+10Q^{4}+(20\alpha+3)Q^{2}\right]\bigg{\\}}\Bigg{]}.$

We are now in a position to compute the thermodynamic curvature

$\displaystyle R_{C}$
$\displaystyle=-\frac{1}{\sqrt{g}}\left[\frac{\partial}{\partial
M}\left(\frac{g_{MQ}}{g_{MM}\sqrt{g}}\frac{\partial g_{MM}}{\partial
Q}-\frac{1}{\sqrt{g}}\frac{\partial g_{QQ}}{\partial
M}\right)+\frac{\partial}{\partial Q}\left(\frac{2}{\sqrt{g}}\frac{\partial
g_{MQ}}{\partial M}-\frac{1}{\sqrt{g}}\frac{\partial g_{MM}}{\partial
Q}-\frac{g_{MQ}}{g_{MM}\sqrt{g}}\frac{\partial g_{MM}}{\partial
M}\right)\right],$ (28)

where $g=\det{g_{\mu\nu}}=g_{MM}g_{QQ}-g_{MQ}^{2}$.

Figure 5: Thermodynamic curvature $R_{C}$ vs black hole size: impact of (a)
quantum corrections $\eta$ and (b) Gauss-Bonnet coupling parameter $\alpha$;
zoomed-in view of (c) $\eta=20$ plot in (a) and (d) extremal limit
$M=\sqrt{Q^{2}+\alpha}.$

In this framework, $R_{C}$ has the following interpretation: a zero curvature
indicates that the system is non-interacting and in an ideal gas-like
configuration, providing no additional information about black hole
micromolecules. A positive $R$ signifies repulsive interactions, indicating an
unstable system, while a negative $R$ represents attractive interactions and
thus a stable system. Divergences in $R$ correspond to phase transitions
Ruppeiner (2014). From this interpretation of $R$, one might wonder if $R$
would be very large for a black hole due to its infinite density. This point
is emphasized in Ref. Ruppeiner (2014), where it is suggested that the
gravitational degrees of freedom in a black hole system might possess some
non-statistical description because all the gravitating material has been
compressed into the central singularity. Thus, thermodynamic curvature
represents some kind of non-gravitational interactions among the black hole
constituents at its surface, arising effectively from the underlying
gravitational degrees of freedom.

##  V Discussion of the Results

Since the expression for thermodynamic curvature $R_{C}$ from Eq. (28) is
quite lengthy, we refrain from writing it down here and instead provide a
graphical analysis of our results. The results are plotted in Fig. 5 for
various choices of the parameters $\eta$ and $\alpha$. A common feature of the
plots is immediately evident: $R_{C}=0$ for larger black hole sizes,
regardless of the values of $\eta$ and $\alpha$. The deviations from zero
begin at smaller scales, where quantum corrections to the entropy become
significant. The zero curvature for larger sizes indicates Ruppeiner flatness,
suggesting that the black hole resides in a non-interacting phase, akin to an
ideal gas. This feature is peculiar to charged black holes in Einstein gravity
across all spacetime dimensions Aman  _et al._ (2003); Aman and Pidokrajt
(2006); Masood A. S. Bukhari  _et al._ (2023), and interestingly, it is also
observed in 4D-EGB theory. For a fixed $Q$ and $\alpha$, $R_{C}$ transitions
from zero at larger $M$ values to either positive or negative values for
smaller $M$, supplemented with divergences, depending on the values of $\eta$
and $\alpha$. In particular, two divergences exist in $R_{C}$ as shown in Fig.
5(a): one at larger $M$ (positive divergence) and another at the extremal
limit of the geometry (negative divergence), occurring at
$M=\sqrt{Q^{2}+\alpha}$. Meanwhile, between these two divergences, the black
hole also passes through negative $R_{C}$ values [see Fig. 5(c)]. This implies
that while the black hole is in an ideal state for larger sizes, it first
becomes stable in the quantum regime before attaining instability with
positive $R_{C}$.

The positive region for $R_{C}$, accompanied by a positive divergence,
indicates two unstable phases for the black hole on either side of the
divergent point where a phase transition occurs. The two unstable phases
coexist around $R_{C}\rightarrow\infty$ and may generally be different in
nature. Each time the magnitude of quantum corrections to the entropy,
quantified by $\eta$, increases, the positive divergence in $R_{C}$ shifts to
larger $M$, as depicted in Fig. 5(a). However, the second divergence still
occurs at the extremal limit for all curves, as evident from Fig. 5(d). As
$\alpha$ increases while $Q$ and $\eta$ remain constant [Fig.5(b)], the
shifting of the negative divergence of $R_{C}$ occurs along the horizontal
axis towards larger $M$, indicating that the black hole stops evaporating
earlier than in the previous configuration. This is expected because each time
$\alpha$ increases, the term $\sqrt{Q^{2}+\alpha}$ grows, shifting the
extremal limit of the black hole geometry to larger sizes.

We also observe from our plots that the black hole remains in the negative
$R_{C}$ region as it approaches the extremal limit, indicating that in all
situations, the black hole is stable in the quantum regime. The negative
divergence of $R_{C}$ at the extremal limit naturally coincides with the zero
of the heat capacity at extremal limit [see Fig. 4]. Hence, we may conclude
from this coincidence that the Ruppeiner geometry provides a good description
of the interacting micromolecules of the black hole.

An interesting observation can be drawn from the above description. The
extremal limit is the point where the black hole terminates its evaporation
and ends up as a remnant. This occurs when the black hole attains zero
temperature, as shown earlier in Fig. 2. Though it is difficult to determine
whether the remnant is stable or unstable from this analysis, as the heat
capacity $C_{Q}$ and $R_{C}$ become unphysical beyond extremal limit, it is
reasonable to make the following observation. In ordinary thermodynamics, as
the temperature drops towards absolute zero, the interactions generally freeze
out, leaving behind only quantum statistical interactions like those in ideal
Fermi or Bose gases Ruppeiner and Sturzu (2023). The extremal limit of the
black hole in our case also corresponds to zero temperature, where $R_{C}$
diverges to negative infinity. This may reflect a situation where black hole
microstates freeze out, similar to those of an ideal Bose gas, forming a
stable remnant.

Although we made efforts to interpret our results based on graphical behavior
of computed formulas, it remains worth attempting to to understand the
physical mechanisms behind the thermodynamic properties exhibited by black
holes on smaller scales. To this end, it is important to note that black hole
thermodynamics is perhaps not very well understood, especially at the
microscopic level Davies (1978); Page (2005). Adding the complications due to
extra curvature corrections from Gauss-Bonnet theory and quantum scale
modifications, it becomes increasingly challenging to precisely quantify this
thermodynamic behavior. This complexity arises from the combined effects of
the parameters involved in the process, as demonstrated in Fig. 5. However, we
believe that certain peculiarities regarding $\alpha$ and $\eta$ can still be
inferred from these observations. First, note from Eq. 14 (or Eq. 12) that the
size dynamics of the black hole are primarily dictated by its mass $M$, which
usually dominates over other properties (e.g., charge $Q$). The effect of
$\alpha$ on black hole size is thus pronounced only on smaller scales, as it
has been tightly constrained Fernandes  _et al._ (2022). We previously
highlighted the $Q$-like character of $\alpha$ in that it diminishes the size
of the black hole, a situation similar to the Reissner-Nordström geometry in
Einstein gravity. This reduction in black hole size has potential implications
for its area and surface gravity, which in turn affect its thermodynamic
properties such as entropy and temperature. It seems plausible to assume that
in such scenarios, the thermodynamic degrees of freedom may be influenced. The
Ruppeiner flatness associated with larger black holes might be due to the
presence of an equal number of repulsive and attractive interactions. For
larger black holes, the extensive area provides equal opportunities for both
attractive and repulsive micromolecules to interact. This situation may change
at quantum scales, where one type of interaction may dominate, causing the
black hole to become selective in its thermodynamic configuration. It appears
that the black hole favors attractive micromolecules until it reaches the
extremal limit, effectively diminishing the influence of repulsive ones. This
may also indicate the emergence of a new kind of interaction at such scales.
Given the singular nature of $R_{C}$, there might be connections to quantum
gravity, where infinities in physical parameters are prevalent. It is
conjectured that a robust quantum geometrothermodynamic approach could either
avoid these complications or provide insights to further understand these
phenomena.

In passing, it’s worth noting that the conventional understanding of Ruppeiner
geometry is rooted in classical gravity and the associated thermodynamic
fluctuations in equilibrium configurations. However, on quantum scales, some
form of non-equilibrium description for black hole thermodynamics is expected
to emerge Pourhassan  _et al._ (2021b, 2022c); Iso and Okazawa (2011);
Pourhassan  _et al._ (2021a). Here, we incorporated quantum corrections to the
entropy without modifying the geometry of the black hole. This should render
the Ruppeiner geometric analysis performed here somewhat effective, as it is
based on a quantum-corrected entropy.

##  VI  Conclusion

Four-dimensional Einstein-Gauss-Bonnet (4D-EGB) gravity is a novel theory that
extends the Einstein paradigm by incorporating higher-order curvature
corrections into gravitational dynamics in four spacetime dimensions. This
theory achieves these contributions while circumventing Lovelock’s theorem. In
this work, we aimed to evaluate the phenomenological aspects of this novel
gravitational theory through a detailed analysis. We conducted a thermodynamic
geometric analysis based on the Ruppeiner formalism for a charged black hole,
aiming to unveil its thermodynamic phase structure while incorporating non-
perturbative quantum corrections to the black hole entropy. Our findings
indicate that for large black hole sizes, 4D-EGB exhibits behavior similar to
standard general relativity. However, it may signal various types of phase
transitions in the quantum regime, contingent upon the Gauss-Bonnet coupling
parameter and the magnitude of quantum corrections to the entropy. A striking
feature is that our black hole system exhibits a stable regime on quantum
scales, where microstates tend to freeze out, resembling a typical Bose gas,
as the black hole geometry approaches the extremal limit, coinciding with zero
temperature. This scenario may suggest the emergence of a stable remnant, the
formation of which is influenced by the strength of the Gauss-Bonnet coupling
parameter as it accelerates the coming into being of such remnant.

The analysis can be extended in several ways. Our model is arguably the
simplest, considering a charged black hole, which is the most straightforward
generalization beyond an uncharged one. Results can be extended to encompass
all black hole geometries in the Kerr-Newman family, and later, to include
additional matter-energy distributions around the black holes. Considering the
intriguing thermodynamic behavior attributed to negative cosmological
constant, which are analogized with thermodynamic pressure Kubiznak  _et al._
(2017), extending our formalism to study the corresponding phase structures
would be intriguing. This, along with several other ideas, presents possible
avenues for future research directions, which will be explored in subsequent
studies.

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