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Citation: Liu, H. Geometric Analysis
of Black Hole with Primary Scalar
Hair. Symmetry 2024,16, 1505.
https://doi.org/10.3390/sym16111505
Academic Editor: Hung T. Diep
Received: 2 October 2024
Revised: 29 October 2024
Accepted: 1 November 2024
Published: 9 November 2024
Copyright: © 2024 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
symmetry
S
S
Article
Geometric Analysis of Black Hole with Primary Scalar Hair
Haotian Liu
School of Mathematics, Monash University, Clayton, VIC 3168, Australia; shuaigepangzi@163.com
Abstract: Within the novel context of primary scalar hair black holes, this article explores the
fascinating subject of black hole thermal stability. Thermodynamic stability is the main subject of
our investigation, which involves measuring the bound points, divergence points, black hole mass,
thermal temperature, and specific heat capacity. In addition, we determine the scalar curvatures
of thermodynamic geometries like Ruppeiner, Weinhold, Hendi-Panahiyah-Eslam-Momennia, and
geometrothermodynamics formulations inside the framework of primary scalar hair black holes and
delve into their complexities. Improving our knowledge of fundamental scalar hair black holes, this
study sheds light on the intricate thermal geometric properties of these objects.
Keywords: primary scalar hair black hole; thermodynamics; phase transition; thermal geometries;
Hawking temperature
1. Introduction
Thermodynamics of black holes (BHs) has been an exciting and practical area of study,
yielding intriguing findings [
1
–
7
]. The relationship between geometrical characteristics like
surface gravity and horizon area and thermodynamics values like entropy and temperature
is well-established [
6
,
8
]. Hawking radiation emission makes BHs thermally unstable,
according to semi-classical analysis [
9
]. Therefore, BHs undergo a full heat loss process
in which their temperature rises as their size falls. Among the many thermodynamical
aspects of BHs that have been revealed is their heat stability. We can demonstrate the
system’s integrity following a small change in thermodynamic parameters, since thermal
stability is a fundamental thermodynamic property. When investigating a BH’s phase
structure close to its critical point, several methods are at our disposal [
10
]. Specific heat is
a famous and formal way to analyze the thermal stability of BH. System thermal stability
is physically represented by the positive conduct of heat capacity [
11
,
12
]. Researching
the BH phase transition relies heavily on the heat capacity [
12
–
16
]. Heat capacity and
divergence of heat capacity are two fundamental kinds of phase transitions. A system’s
heat capacity is the amount of energy needed to raise or lower its temperature by a specific
percentage. A shift in stability is typically indicated by a change in the black hole’s heat
capacity. Phase transitions from stable to unstable or unstable to stable phases are usually
indicated by a change in the sign of the heat capacity. The negative heat capacity of a
Schwarzschild BH, for instance, indicates a thermodynamic instability that causes the
black hole to be thermally unstable when subjected to Hawking radiation [
5
,
17
]. As the
system approaches to a critical point, the heat capacity diverges, which is a common sign
of a second-order (continuous) phase transition, in which thermodynamic variables vary
gradually. Critical events resembling those observed in fluids and other systems are linked
to these locations in black hole thermodynamics. One can study the details about these
claims in references [10,18].
In a completely different context, a number of writers have developed a geometrical
method for thermodynamics and phase transitions. The differential manifold was first
proposed by Hermann [19] as a subspace in equilibrium that is entangled with thermody-
namic phase space by a natural contact structure. Here, Weinhold proposed an alternative
Symmetry 2024,16, 1505. https://doi.org/10.3390/sym16111505 https://www.mdpi.com/journal/symmetry
Symmetry 2024,16, 1505 2 of 16
geometrical method [
20
] by posing a thermodynamic equilibrium state metric and employ-
ing the conformal mapping mechanism from Riemannian to thermodynamic space, which
likewise incorporates the geometry of thermodynamic fluctuation. The positive definite
line interval in this geometry is connected to two attractive equilibrium states. If we use the
Gaussian approximation, the first one describes the distribution of probabilities for thermo-
dynamic fluctuations. A scalar curvature of the thermodynamic properties resulting from
this geometry is a written record of the connection underlying the microscopic statistical
basis [
21
,
22
]. Therefore, the system’s correlation volume and divergence at the critical point
are related to the scalar curvature.
As a geometrical tool, thermodynamic scalar curvature elucidates the system’s macro-
scopic structure and establishes a connection to its microscopic structure through the appli-
cation of Gaussian fluctuation. As discussed in the aforementioned research articles [
23
,
24
],
thermodynamic scalar curvature provides insight into the character of microscopical in-
teraction. Important insights into phase structure, divergency, and critical phenomena for
BHs have been proposed using this geometric framework [
25
–
31
]. Additionally, the scalar
curvature provides thermal stability to the system [
20
,
32
]. A BH thermodynamic system in
general relativity is similar to the general thermodynamic system in classical physics. All
four of thermodynamics’ postulates, including the zero law and three more, form the basis
of the research. That BHs can radiate particles close to the event horizon via gravitational
interactions was initially hypothesized by Hawking [
4
]. The physics and thermodynamic
systems of several kinds of general relativity BH solutions have been extensively researched
because of their importance [
4
]. According to Hawking and Page [
5
], the phase transi-
tion of Schwarzschild AdS BH is studied. The lack of a core singularity makes regular
BHs highly intriguing. References [
33
,
34
] discuss research on the phase transitions and
thermodynamic properties of these BHs. Some other works on BH thermodynamics are
also addressed in the literature [
32
,
35
–
39
]. Weinhold and Ruppeiner geometries reveal
features of thermodynamic stability and critical phenomena through metrics based on
internal energy and entropy, respectively. The Hendi-Panahiyah-Eslam-Momennia (HPEM)
approach extends these analyses to more complex BH models, while geometrothermody-
namics (GTD) provides a unified framework that captures the full thermodynamic behavior.
Utilizing these methods allows for a comprehensive analysis of BH thermodynamics, par-
ticularly in understanding phase transitions, stability, and complex behavior in advanced
BH models [40–43].
Because of the potential impact of scalar fields produced by BH, the BH with primary
scalar hair configuration must be considered in discussions of thermodynamics and optical
characteristics. We probe the effects of scalar fields on optical traits by studying the unique
geometry of a BH with main scalar hair. A BH can only be completely described by its mass
and spin, as stated in the “no-hair” theorem. When things are brought into a BH, no-hair
reveals this [44–46]. A fascinating sign that transcends GR would be to look into the hair.
This paper is novel since it investigates thermodynamic geometry in the context of
main scalar hair black holes. According to [
47
], who studied scalar fields, black hole
configurations with them offer more flexibility, which could mean stability under some
circumstances. Though scalar hair is known to affect a black hole’s thermodynamic and
physical properties, there are few thermodynamic studies that deal with stability and geom-
etry in scalar hair black holes. Our work contributes to the field by investigating thermal
stability using a variety of thermodynamic metrics and geometric methods. In contrast to
earlier methods that have relied on individual thermodynamic metrics, our study evaluates
stability conditions from several thermodynamic angles by combining different geometric
frameworks, such as GTD and the HPEM metric. This work describes the critical behavior
caused by scalar hair and compares the behavior of several geometries to reveal possible
differences in stability. By demonstrating how scalar hair brings novel features of thermal
stability, which may vary substantially from black holes devoid of scalar fields, these results
enrich our knowledge of black hole thermodynamics. In addition to shedding new light,
Symmetry 2024,16, 1505 3 of 16
this method points the way toward potential future research directions in the field of black
hole thermodynamics.
Following this strategy, we continue our investigation throughout the article. We
cover the primary scalar hair BH model’s fundamental formalism in Section 2. The topic
of thermal stability and phase transition of solutions through heat capacity is covered in
Section 3. The geometry of thermodynamics is covered in Section 4, while the outcomes
and a summary of our research are found in Section 5.
2. A Short Summary of Primary Scalar Hair Black Hole
The geometry of the primary scalar hair BH is examined in this section. The case study
beyond the Horndeski theories is used to create primary scalar hair BH physics [
44
,
48
].
Under the condition
ϕ−→ −ϕ
, parity symmetry is established, and under the condition
ϕ−→ ϕ+
, shift symmetry is provided.
G2
,
G4
, and
F4
are the three parameterised functions
of the scalar-field kinetic term X=−1
2∂µϕ∂µϕ. The course of action is as follows:
S[gµν,ϕ] = 1
2κZd4xp−gG2(X)−ϕµνϕµν +G4(X)R+G4(X)[(□ϕ)2] + F4(X)ϵµνρσϵα βγ
σϕµϕαϕνβ ϕργ . (1)
As the Horndeski functionals are provided by the below equation, the under con-
sideration theory is parameterized by the two coupling constants
λ
and
η
, which have
dimensions (length)4and (length) , respectively:
G2=−8η
3λX2,G4=1−4η
3X2,F4=η. (2)
This theory is invariant under
λ−→ −λ
; therefore, for convenience of notation,
λ
will
always have a positive sign. There is no restriction on the sign of
η
. The theory admits a
homogeneous solution with two integration constants,
M
and
q
, where
q
is not referred
to as a primary hair since it does not appear in the metric and does not give rise to any
additional charge [46]. The metric function f(r)then appears as:
f(r) = 1−2M
r+ηq4π/2 −arctan(r/λ)
r/λ+1
1+ (r/λ)2, (3)
However, the scalar field is stated as:
ϕ(t,r) = qt +ψ(r),(ψ′(r))2=q2
f2(r)1−f(r)
1+ (r/λ)2, (4)
where prime states the derivative related to
r
in the above equation. The kinetic term linked
to the scalar field appears in the following form:
X=q2/2
1+ (r/λ)2. (5)
The charge
q
and principal scalar hair, in contrast to stealth solutions, have a significant
impact on the metric by modifying it from its GR from
M
, one of the
two autonomous
inte-
gration constants in the solution that signifies the Arnowitt-Deser-Misner (ADM) mass [
45
].
Scalar hair vanishes at
q=
0, resulting in a well-known Schwarschild solution [
48
]. Space-
time flatness is achieved when
q
and
M
disappear. The fundamental scalar hair BH in its
spherically symmetric static form is this [46]:
ds2=−f(r)dt2+dr2
f(r)+r2(dθ2+sin2θdϕ2), (6)
Here, the metric function f(r)is represented as:
f(r) = 1−2M
r+2λ2ηq4
r2+O1
r4. (7)
Symmetry 2024,16, 1505 4 of 16
The above lapse function is just like the Reissner-Nordstrom solution with ADM mass
Mand main scalar hair q, which scales like electromagnetic charge.
The geometric mass of BH can be found by setting f(r) = 0 [40,41]:
M=ηλ2q2+ηq2r2
0+r2
0
2r0
. (8)
The entropy can be changed by the introduction of components depending on the
strength and dispersion of the scalar field when scalar hair is present, but the BH’s mass,
charge, and angular momentum remain unchanged. Consequently, the scalar field’s prop-
erties determine the Bekenstein entropy, which implies that identical BHs can have varying
entropies due to variations in the distribution and strength of scalar hair. We are here
using the Bekenstein entropy instead of the corrected entropy acting as the thermodynamic
quantity: In four-dimensional spacetime, the Einstein-Hilbert action is written as [5]:
SEH =1
16πGZd4xp−gR. (9)
with
R
being the Ricci scalar,
g
being the metric’s determinant, and
G
being Newton’s
gravitational constant. If we want to calculate the BH thermodynamics in Euclidean
spacetime (imaginary time
τ=it
), we need to switch to the Euclidean action. Hence,
the IE-function of geometry is:
IE=1
16πGZd4x√gR, (10)
where
g
and
R
are currently located in spacetime according to Euclidean geometry. In order
to prevent a conical singularity at the horizon, the temperature and the Euclidean time
coordinate
τ
of a BH must be periodic. At
β=1
T
, the inverse temperature of the BH is
defined by this periodicity. The fourth step is contribution to the horizon and on-shell
Euclidean action. The primary factor impacting the evaluation of
IE
on the BH solution
(on-shell) is a horizon surface term. The action becomes proportional to the horizon area
A
for a BH [5]:
IE=A
4G. (11)
Linking the action of Euclidean space to entropy, a system’s entropy
S
in thermal
equilibrium is connected to the Euclidean action according to thermodynamics:
S=βE−IE. (12)
To calculate the Bekenstein-Hawking entropy for a BH in thermal equilibrium, we use
the energy contribution
βE
and the gravitational action contribution
IE=A
4G
. The entropy
is directly provided by the gravitational action IE; therefore, we obtain:
S=A
4G. (13)
If a BH has main scalar hair, then the entropy is not directly proportional to the domain
area. Alternatively, the scalar field can influence the BH’s entropy calculation by changing
its surrounding geometry. In certain altered theories, like scalar-tensor or Horndeski gravity,
the entropy-area relationship incorporates contributions from the scalar field. To account
for the effect of scalar hair on the thermodynamics of the BH, this can involve adding or
modifying variables. The Bekenstein-Hawking entropy equation
S=kc3A
4G¯h
could need
to be tweaked because of the scalar field’s interaction with spacetime geometry and the
Bekenstein-Hawking haircut. Here is one way to express the BH entropy [5]:
S=πr2
0. (14)
Symmetry 2024,16, 1505 5 of 16
Here is the first law of thermodynamics [42,43]:
dM =TdS. (15)
Thermodynamic temperature from first law of thermodynamics is given by:
T=r2
0ηq2+1−ηλ2q2
4πr2
03/2 . (16)
The thermal temperature of the BH is used to categorize the physical and non-physical
solutions to the BH. The non-physical solutions are represented by the region where
T<
0,
whereas the physical solutions are presented by
T>
0 [
49
,
50
]. The temperature along the
event horizon for different parameter values of
q
,
η
,
λ
is shown in Figure 1, illustrating the
direct correlation between these parameters and the thermal behavior of the BH. Reduced
thermal temperature is the direct result of raising
q
,
η
,
λ
. This is because, after a short
distance, the thermal temperature becomes positive around the radius of the horizon.
These findings based on positive temperature show that our calculated results depict that
the BH system is physically and thermally stable.
q=0.5
q=0.6
q=0.7
q=0.8
1 2 3 4 5
-0.2
-0.1
0.0
0.1
0.2
r0
T
η=0.5
η=0.6
η=0.7
η=0.8
1 2 3 4 5
-0.2
-0.1
0.0
0.1
0.2
r0
T
λ=0.5
λ=0.6
λ=0.7
λ=0.8
1 2 3 4 5
-0.2
-0.1
0.0
0.1
0.2
r0
T
Figure 1. The three plots show the variation of thermodynamic parameter
T
against horizon radius
r0
under different conditions. Left panel: different values of
q
with fixed
η=
0.5,
λ=
0.5. Middle
panel: different values of
η
with fixed
λ=
0.5,
q=
0.5 Right panel: different values of
λ
with fixed
η=0.5, q=0.5.
3. Thermal Stability of Solution
The intriguing subject of BH’s thermal stability can be uncovered by studying the
positive and negative behavior of the specific heat capacity
C
as it fluctuates. The BH is
in a stable area when
C
is positive, while an unstable condition is predicted by negative
values. We can learn a lot about how to understand phase transitions by looking at the
disappearance of heat capacity [
51
–
55
]. These geometric methods have a general criterion
for thermal stability is the positive definiteness of the thermodynamic metric, which is
discussed by the eigenvalues of the Hessian matrix directly linked with the thermodynamic
potential (internal energy and Bekenstein-Hawking entropy). If all eigenvalues are positive,
the system is thermodynamically stable. Conversely, the presence of negative eigenvalues
or negative curvature, particularly in GTD, indicates thermodynamic instability or the
occurrence of a phase transition. The heat capacity estimate for BH with primary scalar
hair looks like this:
C=−2πr2
0r2
0ηq2+1−ηλ2q2
r2
0(ηq2+1)−3ηλ2q2. (17)
For different values of
q
,
η
,
λ
, the transitional phase of the heat capacity is presented
in Figure 2. A root point and a heat capacity divergence are shown on the graph, which
represent the physical limits and critical points of phase transition, respectively. At the
outset, heat capacity points to a stable area for small event horizon
r0
values, but it changes
to a globally unstable area and stays unstable after that. It should be noted that the
parameters
q
,
η
,
λ
determine the key points for phase transition, which might vary
depending on values. In light of the foregoing, it may be useful to examine where the
Symmetry 2024,16, 1505 6 of 16
concept of heat capacity came from and how it has evolved. We find the bound points,
which are the roots of the heat capacity, as shown below [56]:
r0{±},B.P=±√ηλq
pηq2+1, (18)
and divergence points are:
r0{±},D.P=±√3√ηλq
pηq2+1. (19)
q=0.5
q=0.6
q=0.7
q=0.8
0.5 1.0 1.5 2.0
-20
-10
0
10
20
r0
C
η=0.5
η=0.6
η=0.7
η=0.8
0.5 1.0 1.5 2.0
-20
-10
0
10
20
r0
C
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.5 1.0 1.5 2.0
-20
-10
0
10
20
r0
C
Figure 2. The three plots show the variation of heat capacity
C
against horizon radius
r0
under
different conditions. Left panel: different values of
q
with fixed
η=
0.5,
λ=
0.5. Middle panel:
different values of
η
with fixed
λ=
0.5,
q=
0.5 Right panel: different values of
λ
with fixed
η=0.5, q=0.5.
The bound points are denoted by the subscript B.P, whereas the divergence points are
shown by the subscript D.P. There are two potential roots of the heat capacity. They need to
meet the criterion
λ≥
0 in order to be real roots. B.P and D.P as functions of
q
,
η
,
λ
that
decrease and increase depending on the values of parameters. The positive and negative
bound points and divergence points can be seen in Figures 3and 4.
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.5
0.0
0.5
a
r0{-}& r0{+}, BP
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
λ
r0{-}& r0{+}, BP
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.4
-0.2
0.0
0.2
0.4
q
r0{-}& r0{+}, BP
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
q
r0{-}& r0{+}, BP
Figure 3. Cont.
Symmetry 2024,16, 1505 7 of 16
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
η
r0{-}& r0{+}, BP
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.4
-0.2
0.0
0.2
0.4
η
r0{-}& r0{+}, BP
Figure 3. Bound points
r0{±}
, B.P along horizon radius
r0
. Here, we fix
η=
0.5,
λ=
0.5,
q=
0.5
(first-row left graph along a for different values of
q
given in legend, first-row right graph along
λ
for different values of
η
given in legend, second row both graph along
q
left different for values of
η
and right for different values of
λ
given in legend, third row both graph along
η
left for different
values of qand right for different values of λgiven in legend).
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-1.0
-0.5
0.0
0.5
1.0
λ
r0{-}& r0{+}, DP
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-1.0
-0.5
0.0
0.5
1.0
λ
r0{-}& r0{+}, DP
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
q
r0{-}& r0{+}, DP
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-1.0
-0.5
0.0
0.5
1.0
q
r0{-}& r0{+}, DP
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
η
r0{-}& r0{+}, DP
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.5
0.0
0.5
η
r0{-}& r0{+}, DP
Figure 4. Divergence points
r0{±}
, D.P along horizon radius
r0
. Here, we fix
η=
0.5,
λ=
0.5,
q=
0.5
(first row both graph along
λ
left for different values of
q
right for different values of
η
given in
legend, second row both graph along
q
left different for values of
η
and right for different values of
λ
given in legend, third row both graph along
η
left for different values of
q
and right for different
values of λgiven in legend).
4. Thermodynamic Geometries
Thermodynamic studies of BHs have shown that, in many cases, the exact causes
of the system’s mass, temperature, and heat capacity anomalies are not known. Many
efforts have been undertaken in the last few decades to integrate various geometric ideas
into standard thermodynamics. The thermodynamic phase space is best understood as
Symmetry 2024,16, 1505 8 of 16
a differential manifold with an intrinsic contact structure, where a separate subspace of
equilibrium states is present. According to many observations, the geometric framework
appears to quantify the system’s underlying statistical mechanics. In systems without
statistical mechanical interactions (e.g., ideal gas), the Ruppeiner geometry is flat, and a
non-zero Ruppeiner curvature is suggestive of a phase transition [
57
]. Numerous studies
support these conclusions [
58
,
59
]. Furthermore, the Ruppeiner metric is connected to
the Weinhold metric, which defines the Hessian of energy (mass) [
32
,
60
]. Here, we will
explore the intriguing realm of thermodynamical geometries obtained for primary scalar
hair BHs by Weinhold and Ruppeiner [
49
]. We still do not have a full picture of the
microstructure of BHs, even though we have studied their thermodynamics a lot. The good
news is that geometric analysis of thermodynamic systems provides a way to investigate
the microstructure of BHs [57].
The reason why this invariant approach should be preserved is that the entropy
difference and the probability of fluctuation between two states should be independent of
the system’s coordinates. We introduce a new and valuable component for our discussion,
the thermodynamic length, because the portion of
gµν∆Eµ∆Eν
resembles a Riemannian
metric representing a distance between thermodynamic states [61]:
∆ℓ2≡gµν∆Eµ∆Eν, (20)
that is definitely positive. Riemannian metrics in the space of thermodynamic states are well-
defined, and the relation above is no exception. Given this space and two thermodynamic
states, the physical meaning of the distance between them is crystal clear: it is a measure
of probability in thermodynamics. Two states are thought to be closer together if there
is a higher possibility of a fluctuation occurring between them. As an aside, in general
relativity (GR), the square of the thermodynamic length is usually measured in meters
squared, although in Riemannian geometry, the square of the length is inversely related
to volume.
The Figures 3and 4are different shape of the metric can be investigated in other
coordinate systems, but the value of
∆ℓ2
must remain constant regardless of these systems.
A number of thermodynamic coordinate systems have the form of the Ruppeiner metric,
which will be determined with this in mind:
u,Ei
, As already explained, working with
systems where the coordinates are extended parameters, denoted as
u,E1,E2, . . . , Er1
,
results in the following [32,61]:
gR
µν =−1
kB
∂2s
∂Eµ∂,Eν, (21)
This is the Hessian matrix representing the thermodynamic entropy (14). Additionally,
the thermodynamic stability requires that
gR
µν
be positively charged. The extensive and
intense characteristics are used to define an alternative coordinate system. For this situation,
we obtain by using the aforementioned equations and the transfer function
∆Eµ=∂Eµ
∂xα∆xα
:
∆ℓ2=−1
kB
∆Iµ∆Eµ. (22)
Furthermore, in order to fully express the metric using the intense parameters, we employ:
∆Eµ=∂Eµ
∂Iα∆Iα. (23)
and plugging this into Equation (22) produces:
∆ℓ2=1
kB
∂2ϕ
∂Iµ∂Iν∆Iµ∆Iν, (24)
where
ϕ
is the Legendre transformation of the entropy with respect to all extensive system
parameters (except volume) and is expressed as [61]:
Symmetry 2024,16, 1505 9 of 16
ϕI0,I1, . . . , Ir=s− IµEµ, (25)
The intense parameters, while working in the entropy representation, can be deter-
mined in the following way:
nI0,Iio=1
T,−µi
T, (26)
for each component of the fluid, where
µi
represents its chemical potential. By plugging
the given relation into Equation (22), we may obtain:
∆E0=∆u=T∆s+
r
∑
i=1
µi∆Ei,
∆I0=−1
T2∆T,
∆Ii=µi
T2∆T−1
T∆µi, 1 ≤i≤r,
this becomes as
∆ℓ2=1
kBT∆T∆s+1
kBT
r
∑
i=1
∆µi∆Ei. (27)
In the coordinate system
T,E1,E2, . . . , Er
, we can determine the shape of the metric.
In order to do this, we employ the subsequent relation:
∆s=∂s
∂T∆T+
r
∑
i=1
∂s
∂Ei∆Ei,
∆µi=∂µi
∂T∆T+
r
∑
i=1
∂µi
∂Ei∆Ei,
as well as the Maxwell relation
∂s
∂ρi=−∂µi
∂T, (28)
and substituting Equation (28) into Equation (27), we obtain
∆ℓ2=1
kBT
∂s
∂T(∆T)2+1
kBT
r
∑
i,j=1
∂µi
∂Ej∆Ei∆Ej. (29)
First, we will look at the basic features of theories of thermodynamic geometry. A met-
ric space is important to the equilibrium state space of a thermodynamic system, and this
idea bridges the gap between thermodynamics and statistical mechanics. The Weinhold
geometry, as mentioned in [32], can be expressed in terms of mass M:
gij W=∂i∂jM(S,η,λ,q). (30)
The metric for primary scalar hair BH can be stated as:
ds2
W=MSSdS2+Mqq dq2+2Mqs dqdS, (31)
whose matrix form is given by:
MSS MSq
MqS Mqq . (32)
If we have the matrix and the equations, we can calculate the curvature scalar of the
Weinhold metric
R(wein)
. To determine the covariant metric tensor, Christoffel symbols,
Symmetry 2024,16, 1505 10 of 16
Riemann tensor, and Ricci tensor, we employed the Hessian matrix MAPLE program.
The non-zero components of the Ricci tensor are:
R11 =−1
8−q2ηS+3q2ηπλ2−S−S+3πλ2S+πλ2
S(−3q2ηS2+6q2ηSπλ2+q2ηπ2λ4−S2−Sπλ2)2,
R12 =1
2
q−S+πλ2−S+3πλ2ηS+πλ2
(−3q2ηS2+6q2ηSπλ2+q2ηπ2λ4−S2−sπλ2)2,
R22 =−S+πλ22S−S+3πλ2η
(−3q2ηS2+6q2ηSπλ2+q2ηπ2λ4−S2−Sπλ2)2.
The Ruppeiner has been computed by combining the Ricci scalar with temperature.
This scalar’s mathematical expression looks like this:
R(wein) = 2√ππλ2+SS3/2S−3πλ2
(π2ηλ4q2−3ηq2S2+6πηλ2q2S−S2−πλ2S)2. (33)
Upon careful examination of Figure 5, an interesting finding becomes apparent: there
is no singularity seen in the curvature scalar of the Weinhold geometry of the fundamental
scalar BH. This suggests that the Weinhold metric has interesting information about the
phase transition (no change between the negative and positive areas), and that its high
negative value would lead to a strong attractive interaction in the BH microstructure [
62
].
The next analysis will include computing the conformal to the Weinhold as well as Rup-
peiner geometry. According to [
63
], the Ruppeiner metric for a thermodynamic system is
as follows:
ds2
RUP =ds2
W
T. (34)
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6
-500
-400
-300
-200
-100
0
r0
R(wein)
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6
-500
-400
-300
-200
-100
0
r0
R(wein)
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6
-500
-400
-300
-200
-100
0
r0
R(wein)
Figure 5. Weinhold geometry R(wein) along horizon radius
r0
. Here, we fix
η=
0.5,
λ=
0.5,
q=
0.5 (left
panel for different values of
q
,middle panel for different values of
η
, and right panel for different values
of λgiven in legend).
The expression for Ruppeiner geometry is stated as below:
R(RUP) = 8πS3S−3π λ2S+π λ2
(q2(S−π λ2)η+S)(−3q2ηS2+6q2ηSπ λ2+q2η π2λ4−S2−Sπ λ2)2. (35)
The curvature scalar of Ruppeiner geometry is not singular for various
q
,
η
,
λ
values,
as can be seen in Figure 6. Moreover, it is positive for the short-range of
r0
, and afterward,
it is negative. This is the motivation of the present study. The HPEM geometry follows as:
ds2=SMS
∂2M
∂q23(−MSSdS2+Mqq dq2). (36)
Symmetry 2024,16, 1505 11 of 16
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.1 0.2 0.3 0.4 0.5
-4000
-2000
0
2000
r0
R(RUP)
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.1 0.2 0.3 0.4 0.5
-4000
-2000
0
2000
r0
R(RUP)
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.1 0.2 0.3 0.4 0.5
-4000
-2000
0
2000
r0
R(RUP)
Figure 6. Ruppenier geometry
R(RUP)
along horizon radius
r0
. Here, we fix
η=
0.5,
λ=
0.5,
q=
0.5
(left panel for different values of
q
,middle panel for different values of
η
, and right panel for different
values of λgiven in legends).
The mathematical expression for the HPEM geometry scalar is calculated as:
R(HPEM) = S5/2 S−3πλ2π3/2 Sηq2+1−πηλ2q2
2η3(πλ2+S)4(S(ηq2+1)−3πηλ2q2)2.
By examining the singularity-free behavior of the HPEM curvature scalar, one can de-
rive valuable insights from Figure 7. Since the zero point of the HPEM metric’s divergence
of scalar curvature is near the heat capacity, we can derive useful information from this
formalism. In Figure 8, we examine the scalar behavior of GTD curvature. In such instances,
we found that the GTD curvature and the zero point of heat capacity are synchronous.
Here, the GTD geometry provides all the physical details of this structure. The GTD metric
can be expressed as:
ds2= (S MSS +qMqq )−MSS 0
0mqq. (37)
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-60
-50
-40
-30
-20
-10
0
r0
R(HPEM)
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-60
-50
-40
-30
-20
-10
0
r0
R(HPEM)
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-60
-50
-40
-30
-20
-10
0
r0
R(HPEM)
Figure 7. HPEM geometry
R(HPEM)
along horizon radius
r0
. Here, we fix
η=
0.5,
λ=
0.5,
q=
0.5
(left panel for different values of
λ
,middle panel for different values of
q
, and right panel for different
values of ηgiven in legends).
λ=0.5
λ=0.6
λ=0.7
λ=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-40
-30
-20
-10
0
r0
R(GTD)
q=0.5
q=0.6
q=0.7
q=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-40
-30
-20
-10
0
r0
R(GTD)
η=0.5
η=0.6
η=0.7
η=0.8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-40
-30
-20
-10
0
r0
R(GTD)
Figure 8. GTD geometry
R(GTD)
along horizon radius
r0
. Here, we fix
η=
0.5,
λ=
0.5,
q=
0.5 (left
panel for different values of
λ
,middle panel for different values of
q
, and right panel for different
values of ηgiven in legends).
From the above expression, one can obtain:
Symmetry 2024,16, 1505 12 of 16
R(GTD) = S−3πλ2−3πηλ2q2+8ηqS2+Sηq8πλ2+q+1
4(πλ2+S)(S(ηq2+1)−3πηλ2q2)2.
Figures 9and 10 illustrate how effectively the Weinhold and Ruppeiner approaches
work with heat capacity zeros. The resulting curvature scalar of the Ruppiner and Weinhold
metrics is shown in terms of the horizon radius
rh
in order to investigate the thermodynamic
phase transition. There is a physical limitation point and one zero point in the heat capacity
at
rh=
1.651. Figure 11 shows that the zero points of heat capacity coincide with the
divergence of the scalar curvature of the HPEM metric. Consequently, the HPEM formalism
can yield some valuable insights. Furthermore, learning about the scalar curvature of
thermodynamic geometry is essential since HPEM metrics can show how a BH’s principal
scalar hair diverges. The divergences of the BH with primary scalar hair coincide with the
GTD approach in Figure 12. Here, we highlight the impact of various spacetime parameter
values on the BH’s stability requirements. The figure shows that similar to the BH in the
preceding section, this likewise exhibits phase change. The zero point of heat capacity
coincides with the GTD scalar when
q
is set. Therefore, it is possible to glean some relevant
data from this situation.
Figure 9. Plot of Weinhold curvature scalar
R(Wein)
with heat capacity of fixed values
q=
0.5,
η=
2.3
and λ=1.5.
Figure 10. Plot of Ruppeiner curvature scalar
R(Rup )
with heat capacity of fixed values
q=
0.5,
η=1.5 and λ=1.5.
Symmetry 2024,16, 1505 13 of 16
Figure 11. Plot of HPEM curvature scalar
R(HPEM)
with heat capacity of fixed values
q=
2.5,
η=
1.2
and λ=0.5.
Figure 12. Plot of GTD curvature scalar
R(GTD)
with heat capacity of fixed values
q=
2.5,
η=
0.2
and λ=0.3.
5. Results and Conclusions
In this article, we examine the ansatz of a primary scalar hair BH and examine its
complete properties through graphical analysis. These characteristics include: geometric
mass; geometric interpretations of BHs in relation to the Weinhold, Ruppeiner, HPEM,
and GTD formalisms; phase transition via heat capacity; bond points; and divergence
points. The findings demonstrate the physical existence and thermodynamic stability of
BHs. It should be noted that these findings differ sufficiently from relevant previous work
in terms of phase transition points and stable-unstable zones, and are more precise. We
arrive at the following results summary:
The BH solution is stable since the thermal temperature is behaving positively. There
is an inverse relationship between the parameters
q
,
η
,
λ
and thermal temperature in
terms of behavior. Based on the parameters of
q
,
η
,
λ
, the heat capacity behavior in our
example shows that the BH model is thermally stable (positive) with an initially unstable
zone (negative). It also sheds light on the crucial sites for phase transition. The geometrical
composition of primary scalar hair BH has been determined using the Weinhold and
Ruppeiner formalism. It has been found that these geometries do not contain singularities,
behave attractively (negatively) in the thermodynamic system, and do not reveal anything
Symmetry 2024,16, 1505 14 of 16
about the phase transition. As a measure of phase transition, zero points of heat capacity
correlate with HPEM and GTD scalar curvature divergence.
The Weinhold and Ruppeiner methods reveal the features of thermodynamic stability
and critical phenomena through metrics based on internal energy and entropy, respectively.
The HPEM approach extends to the analysis of primary scalar hair BH, while GTD provides
a unified framework that captures the full thermodynamic behavior. Utilizing these meth-
ods allows us to comprehensively analyze the BH stability, particularly in understanding
the microstructure of primary scalar hair BH. Moreover, we have compared our results
with recent papers [
41
–
43
] and find that this approach plays an important role in physics
and thermodynamic systems of specific BH solutions in general relativity.
As proposed in [
64
], future research should broaden the scope of scalar fields’ examina-
tion in black hole thermodynamics to include non-minimally coupled and time-dependent
scalar fields, among others. By extending this model, we can learn more about the effects of
different scalar field features on the stability of black hole systems. Theoretical predictions
of scalar hair stability could be further empirically supported by comparisons with current
observational evidence, especially from the Event Horizon Telescope (EHT). We argue that
our results could be more applicable with empirical validation, building on frameworks [
65
]
that link black hole thermodynamic models with EHT observational data. It is possible that,
in the future, researchers will broaden their focus to include multi-hair black hole models
and more complicated black hole systems to see if the patterns of thermal stability seen in
primary scalar hair black holes are applicable to these or other scalar field configurations.
By using this route, thermodynamic geometry may be able to describe more phenomena
observed in the field of black hole astrophysics.
Funding: This research received no external funding.
Data Availability Statement: Data are contained within the article.
Acknowledgments: I am deeply grateful to Todd Oliynik for his invaluable guidance and insightful
contributions, which have significantly enhanced the quality of this manuscript.
Conflicts of Interest: The author declares no conflict of interest.
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