# Thermodynamics of the Weyl Geometric Gravity Black Holes

Muhammad F. A. R. Sakti<EMAIL_ADDRESS>High Energy Physics Theory
Group, Department of Physics, Faculty of Science, Chulalongkorn University,
Bangkok 10330, Thailand,  Piyabut Burikham<EMAIL_ADDRESS>High Energy
Physics Theory Group, Department of Physics, Faculty of Science, Chulalongkorn
University, Bangkok 10330, Thailand,  Tiberiu Harko
<EMAIL_ADDRESS>Department of Theoretical Physics, National
Institute of Physics and Nuclear Engineering (IFIN-HH), Bucharest, 077125
Romania,  Department of Physics, Babes-Bolyai University, Kogalniceanu Street,
Cluj-Napoca 400084, Romania,  Astronomical Observatory, 19 Ciresilor Street,
Cluj-Napoca 400487, Romania,

###### Abstract

We consider the thermodynamic properties of an exact black hole solution
obtained in Weyl geometric gravity theory, by considering the simplest
conformally invariant action, constructed from the square of the Weyl scalar,
and the strength of the Weyl vector only. The action is linearized in the Weyl
scalar by introducing an auxiliary scalar field, and thus it can be
reformulated as a scalar-vector-tensor theory in a Riemann space, in the
presence of a nonminimal coupling between the Ricci scalar and the scalar
field. In static spherical symmetry, this theory admits an exact black hole
solution, which generalizes the standard Schwarzschild-de Sitter solution
through the presence of two new terms in the metric, having a linear and a
quadratic dependence on the radial coordinate, respectively. The solution is
obtained by assuming that the Weyl vector has only a radial component. After
studying the locations of the event and cosmological horizons of the Weyl
geometric black hole, we investigate in detail the thermodynamical (quantum
properties) of this type of black holes, by considering the Hawking
temperature, the volume, the entropy, specific heat and the Helmholtz and
Gibbs energy functions on both the event and the cosmological horizons. The
Weyl geometric black holes have thermodynamic properties that clearly
differentiate them from similar solutions of other modified gravity theories.
The obtained results may lead to the possibility of a better understanding of
the properties of the black holes in alternative gravity, and of the relevance
of the thermodynamic aspects in black hole physics.

###### pacs:

03.75.Kk, 11.27.+d, 98.80.Cq, 04.20.-q, 04.25.D-, 95.35.+d

###### Contents

  1. I Introduction
  2. II Weyl Geometric Gravity
    1. II.1 Field equations
    2. II.2 Metric and Field Ansatzs
  3. III The Weyl Geometric Black Hole
    1. III.1 General black hole solutions
    2. III.2 Metric of the Weyl geometric black hole
    3. III.3 The horizons of the Weyl geometric black hole
    4. III.4 Effective density and pressure
    5. III.5 Dark matter and the Weyl geometric black hole
  4. IV The thermodynamics of the Weyl geometric black hole on the event horizon
    1. IV.1 The black hole mass
    2. IV.2 The Hawking temperature
    3. IV.3 Entropy from the Noether charge
    4. IV.4 The thermodynamic volume
  5. V Thermodynamic stability
    1. V.1 The Helmholtz and Gibbs free energy
    2. V.2 Heat capacity
    3. V.3 Luminosity and evaporation time of Weyl geometric black holes
      1. V.3.1 The evaporation time
  6. VI Thermodynamics on the cosmological horizon
  7. VII Discussions and final remarks
  8. A Basics of Weyl geometry and Weyl geometric gravity

##  I Introduction

Einstein’s general theory was historically the first approach that succeeded
in giving a full description of gravitational phenomena in geometric terms
Einstein:1915 . The development of general relativity took place within the
framework of Riemannian geometry Riem , which provided a powerful mathematical
instrument necessary for the understanding of the gravitational phenomena.
Extremely important in the evolution of gravitational theories was the
variational derivation of the gravitational field equations, which was
realized by Hilbert Hilbert:1915 . General relativity gives a remarkably good
description of the gravitational effects in the Solar System, like, for
example, the perihelion precession of the planets, the bending of light by the
Sun, or the Shapiro time delay effect Will ; Marchi . One of the most
important, and at the same time intriguing, predictions of general relativity
is the existence of gravitational waves, which were detected recently GW1 ;
GW2 ; AgazieNanoGrav2023 ; AgazieNanoGrav2023L9 ; AfzalNanoGrav2023 .

However, despite these important achievements, on both small and large scales,
the theory of general relativity faces several important challenges. The
precise astronomical observations of the Type IA Supernovae led to an
unexpected discovery, namely, that the present-day Universe is in a state of
accelerated, de Sitter type expansion. The accelerated expansion is assumed to
be caused by a special form of matter, called dark energy R1 ; Per . The
cosmological constant $\Lambda$ introduced by Einstein in the field equations
Ein1  is a particular, and the simplest form of such a fluid. However, the
physical, or geometrical natures of the cosmological constant, and of the dark
energy are not yet known Wein ; AmendolaTsujikawa2010 ; JoycePhysRep2015 ;
JoyceAnnuRev2016 ; TawfikDahab2019 ; FrusciantePhysRep2020 . Another unsolved
problem, which is also a fundamental one, is the dark matter problem. This
problem has its origins in the surprising dynamics of the galactic rotation
curves Oks . General relativity cannot describe the behavior of hydrogen
clouds gravitating outside the baryonic matter distribution of the galaxies
without postulating the existence of an exotic form of matter, called dark
matter Bian .

The $\Lambda$CDM standard cosmological paradigm, based on general relativity
with the inclusion of a cosmological constant, is also confronted with several
significant challenges. An intriguing open problem in present day cosmology is
related to the deviations between the Hubble expansion rates as obtained from
the Cosmic Microwave Background (CMB) experiments, and the small redshift
determinations, based on the studies of the Type IA supernovae. These
dissimilarities in the values of the Hubble constant $H_{0}$ for $z=0$
represent the so called Hubble tension Val .

If the Hubble tension does indeed exist, and it is not only an observational
artefact, it strongly points towards the requirement of considering, and
analyzing in depth alternative theories of gravity. These new theories must go
beyond standard general relativity, and they may provide new and unexpected
solutions to the dark energy and dark matter problems.

Almost immediately after the correct field equations of general relativity
were written down, Schwarzschild obtained the first vacuum solution of the
field equations Sch , for a static, point-like mass distribution. This
solution led to the introduction of a new astrophysical concept, and of a new
type of objects, called black holes. The external solution of the Einstein
equations for a spinning object was obtained around sixty years later by Kerr
Kerr . Black holes are essentially very simple objects, characterized by the
presence of an event horizon, and of a central singularity. An important
result in astrophysics states the existence of a limiting mass for neutron
stars of the order of $3.2M_{\odot}$ Ruf . Recently, the Event Horizon
Telescope collaboration presented the first images of the supermassive black
hole M87* EH1 ; EH2 ; EH3 . These observations seem to point towards a Kerr-
like nature for the Sgr A* black hole. However, based on these observations,
one cannot explicitly eliminate the possibility of the existence of deviations
with respect to the predictions of general relativity.

The Schwarzschild and the Kerr solutions raised some fundamental, and
essentially still unsolved problems, related to the existence of singularities
in black hole solutions SH1 ; SH2 .

Three years after the final form of the gravitational field equations was
found, Weyl Weyl1 ; Weyl2  did introduce a generalization of the Riemannian
geometry. Weyl’s main goal was to formulate a unified theory of gravity and
electromagnetism. In vacuum the Maxwell equations of electromagnetism are
conformally invariant. Weyl suggested that the gravitational field equations
must have the same symmetry. The conformal invariance of the physical laws was
introduced in a general way by Weyl through the creation of a new geometry.
Weyl’s geometry is nonmetric, with the covariant derivative of the metric
tensor not vanishing identically, and thus
$\nabla_{\mu}g_{\alpha\beta}=Q_{\mu\alpha\beta}=\omega_{\mu}g_{\alpha\beta}$,
with $Q_{\mu\alpha\beta}$ representing the nonmetricity tensor, and
$\omega_{\mu}$ is the Weyl vector field. In his original theory Weyl assumed
that $\omega_{\mu}$ is the electromagnetic four-potential $A_{\mu}$. In Weyl’s
geometry, the length of vectors change during their parallel transport from
the point $x^{\mu}$ to $x^{\mu}+dx^{\mu}$. This property led Einstein to
strongly criticise the physical interpretation of Weyl’s theory. For a
presentation of the historical development of Weyl’s geometry, and of its
recent physical applications, see Weyl3 .

However, if one abandons the interpretation of the Weyl vector field as
corresponding to the electromagnetic potential $A_{\mu}$, Weyl’s geometry
becomes a very interesting generalization of Riemann’s geometry, with an
important potential for cosmological and gravitational applications.

A generalization of Weyl’s theory was proposed by Dirac Di1 ; Di2 , who
extended the original theory of Weyl by introducing a real scalar field
$\beta$, of weight $w(\beta)=-1$. The cosmological applications of Dirac’s
theory were considered in detail in Di3 , Di4 , and Isrcosm , respectively.
Other generalizations of the Weyl theory, and of its physical applications,
have been discussed in Ut1 ; Ut2 ; Ni .

A conformally invariant gravitational theory, formulated in Riemannian
geometry, with action $S_{Weyl}=-\frac{1}{4}\int
d^{4}x\sqrt{-g}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$, where
$C_{\mu\nu\rho\sigma}$ is the Weyl tensor, was proposed and thoroughly
investigated in M1 ; M2 ; M3 ; M4 ; M5 ; M6 . The theories with actions
constructed by using the Weyl tensor are called conformally invariant, or Weyl
type gravity theories. An exact vacuum solution of the Weyl gravity theory,
given by
$A(r)=1-3\beta\gamma-\frac{\beta\left(2-3\beta\gamma\right)}{r}+\gamma
r+kr^{2},$ where $\beta$, $\gamma$ and $k$ are constants, was found in M1 .
This solution may provide an alternative solution of the dark matter problem,
since it can explain the observational data without introducing a dark matter
component M1 . A metric similar in form to the exact Weyl gravity vacuum
solution was found in Pi  as a solution of the field equations of the dRGT
massive gravity theory.

The important role the conformal transformations may play in cosmology was
considered by Penrose P1 , who introduced a cosmological model called
Conformal Cyclic Cosmology (CCC). The CCC theory assumes that during both the
de Sitter era, and the Big Bang, the spacetime is conformally flat. The
cosmological and physical implications of the CCC model were investigated in
P1a ; P2 ; P3 ; P4 ; P5 ; P6 ; P7 . In H1  ’t Hooft proposed to explain the
small-scale characteristics of gravity by a breaking of the conformal symmetry
invariance. In H2  a theory of gravity was introduced, in which the conformal
component of the metric is interpreted as a dilaton field, with black holes
being regular solitons, topologically trivial, without any horizons,
firewalls, or singularities.

Weyl geometry is the mathematical basis of the $f(Q)$ gravity theory, also
called the symmetric teleparallel gravity, first introduced in Q1 . In this
approach it is assumed that the nonmetricity $Q$ is the fundamental
geometrical quantity that describes gravity. The initial analysis of Q1  was
generalized in Q2 , and led to the creation of the $f(Q)$ gravity theory,
sometimes also called the nonmetric gravity theory. In the $f(Q)$ theory, the
action is defined as $S=\int{\left(f(Q)+L_{m}\right)\sqrt{-g}d^{4}x}$, where
$f$ is an arbitrary function of the nonmetricity. The cosmological and
astrophysical implications of the modified $f(Q)$ gravity theory have been
extensively investigated in Q2 ; Q3 ; Q4 ; Q5 ; Q6 .

A novel point of view on the Weyl geometry, and on its applications in
gravity, was introduced, and analyzed in Gh1 ; Gh2 ; Gh3 ; Gh4 ; Gh5 ; Gh6 ;
Gh7 ; Gh8 ; Gh9 ; Gh10 ; Gh11 . As a starting point, the simplest possible
gravitational action in Weyl geometry was adopted, constructed additively from
the square of the scalar Weyl curvature $\tilde{R}^{2}$, and the square of the
strength of the Weyl vector field $\tilde{F}_{\mu\nu}^{2}$. The important new
characteristic of the theory is the linearization of the quadratic term in the
Weyl scalar, by introducing an auxiliary scalar field. Thus, the vector-tensor
Weyl theory can be reformulated as a scalar-vector tensor theory, linear in
the Weyl, and the Ricci scalars. The quadratic Weyl geometric action has a
spontaneous symmetry breaking through a Stueckelberg mechanism, with the Weyl
gauge field becoming massive. Hence, the scalar field is described by a Proca
like action Gh1 ; Gh2 ; Gh3 . Through this symmetry breaking mechanism the
Weyl geometric action recovers the Einstein-Hilbert action of general
relativity, together with a positive cosmological constant of geometric nature
Gh1 ; Gh2 ; Gh3 .

Conformally invariant couplings of geometry and matter in Weyl geometric
gravity were analyzed in HT1 , leading to the formulation of the conformally
invariant $f\left(R,L_{m}\right)$ theory of gravity HT1 . The conformally
invariant $f\left(R,L_{m}\right)$ theory accurately describes the cosmological
data for the Hubble function up to a redshift of around $z\approx 3$. The
conformally invariant $f\left(R,L_{m}\right)$ theory of gravity was studied in
the Palatini formulation in HT2 .

In the Weyl geometric gravity theory black hole type solutions in spherical
symmetry were investigated in detail in HT3 , by using numerical and
analytical methods. The quantum thermodynamic properties of the obtained
classes of Weyl geometric black holes were also considered. Moreover, an exact
static spherically symmetric three parametric black hole solution was
obtained, which represents an extension of the Schwarzschild - de Sitter
solution of general relativity, in the presence of a scalar field, and of the
Weyl vector.

The exact solution of the Weyl geometric gravity was applied for the
description of the properties of the galactic rotation curves in HT4 . A
comparison of the predictions of the exact Weyl geometric gravity solution
with a small sample of galactic rotation curves was also performed. To take
into account the effects of the baryonic matter the existence of an explicit
breaking of the conformal invariance at the galactic level was assumed. An
extensive comparison between the predictions of the Weyl geometric dark matter
model and the observational data of the SPARC database was performed in HT5 ,
with the results confirming the possibility of a Weyl geometric interpretation
of the dynamics of galaxies as proposed in HT4 . The properties of the compact
objects in Weyl geometric gravity were investigated in HT6 .

Black holes have many fascinating properties, but perhaps the most interesting
ones are related to their thermodynamical behaviors. The black hole
thermodynamics is constructed through a subtle relationship uniting general
gravity, quantum field theory, thermodynamics, and statistical mechanics. It
gives a powerful insight into the nature and the properties of the black
holes, and can lead to the understanding of the possible profound relation
existing between quantum theory and gravitational physics. The investigations
of the black hole thermodynamics did begin with the study of Hawking Haw1 , in
which it was proved that the area of the event horizon of a black hole does
not decrease Haw1 . This important result is known as the area theorem.
Bekenstein introduced the concept of entropy, which is proportional to the
area of the black hole’s horizon Haw2 ; Haw3 . The four laws of black hole
thermodynamics were established in Haw4 , and they have a form similar to the
four laws of standard thermodynamics.

An important result in black hole physics was the suggestion of the existence
of the Hawking radiation Haw5 , a result which follows from the application of
the quantum field theory in curved spacetimes. Moreover, this analysis also
allows to establish the relation between the temperature of a black hole and
its surface gravity. For detailed reviews of black hole thermodynamics see Dav
; Wald1 ; Wald2 . For some recent investigations on the thermodynamical or
astrophysical properties of black holes see Piel ; Bambi , and Xu ,
respectively.

Hence, in the physics and astrophysics of black holes the study of their
thermodynamic properties is one of the most interesting subjects to study. It
is the goal of the present paper to consider the thermodynamical properties of
the Weyl geometric black holes HT3 ; HT4 , obtained as solutions of the vacuum
field equations of Weyl geometric gravity. In order to understand the
thermodynamic properties one needs first to analyze the horizons of the black
hole solution. It turns out that the Weyl geometric black hole can have two
event horizons, and one cosmological horizon, respectively. The mass of the
black is also an important indicator of its astrophysical and thermodynamical
properties. The Hawking temperature of the Weyl geometric black hole is
obtained in terms of its surface gravity. The entropies of the event and of
the cosmological horizons are fundamental quantities that allows the
understanding of the thermodynamic properties of the black holes. To obtain
the entropy of the Weyl geometric black hole we use the Noether charge
approach. The stability of the black hole is investigated by considering the
behavior of the Helmholtz and Gibbs free energy. The heat capacity of the Weyl
geometric black hole is also investigated in detail. The first law of
thermodynamics on the cosmological horizon for the Weyl geometric black hole
is also formulated. In each case the dependence of the thermodynamical
quantities on the parameters of the solution are fully investigated.

The present paper is organized as follows. We review the basics of the Weyl
geometric gravity in Section II, where the vacuum field equations in static
spherical symmetry are also written down. The exact solution of the field
equations is introduced in Section III, where the position of the horizons,
and the effective energy density and pressure associated to the solution are
discussed. A possible relation with the dark matter problem is also
considered. The thermodynamic properties of the Weyl geometric black hole on
the event horizon are investigated in Section IV. The black hole mass, Hawking
temperature, entropy and volume are considered in detail. The thermodynamic
stability of the Weyl geometric black hole is analyzed in Section V, by
considering the heat capacity and the Helmholtz and Gibbs energies,
respectively. The Hawking luminosity and the evaporation time of the Weyl
geometric black holes are estimated in Sect. V.3. The thermodynamics of the
cosmological horizon of the Weyl geometric black hole is analyzed in Section
VI. We discuss and conclude our results in Section VII. The fundamentals of
the Weyl geometry, necessary for the understanding of Weyl geometric gravity,
are presented in AppendixA.

##  II Weyl Geometric Gravity

In the present Section we will briefly review the theoretical formalism of the
Weyl geometric gravity, and we present the exact vacuum solution of this
theory. We also reformulate the static, spherically symmetric field equations
in terms of an effective density, and effective pressure. For further details
on the mathematical and physical aspects of Weyl geometric gravity we refer
the readers to HT4 .

###  II.1 Field equations

The Weyl geometric gravity theory is represented by the following action HT4

$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\bigg{[}-\frac{1}{12}\frac{\phi^{2}}{\xi^{2}}\left(R-3\alpha\nabla_{\mu}\omega^{\mu}-\frac{3}{2}\alpha^{2}\omega_{\mu}\omega^{\mu}\right)$
(1)
$\displaystyle-\frac{1}{4}\frac{\phi^{4}}{\xi^{2}}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\bigg{]}.\
$

This action has its physical origins in Weyl geometry, and it is obtained
through the linearization, with be help of an auxiliary scalar field, of the
action
$S=\int{\left(\tilde{R}^{2}-\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}\right)\sqrt{-g}d^{4}x}$,
where $\tilde{R}$ is the Weyl scalar, defined in the Weyl geometry, and
$\tilde{F}_{\mu\nu}$ is the strength of the Weyl vector $\omega_{\mu}$. By
$\phi$ we have denoted the scalar degree of freedom of the theory. For a brief
review of the basic concepts of Weyl geometry and of Weyl geometric gravity
see Appendix A.

Note that the effective gravitational constant in this theory is
$G=-3\xi^{2}/(4\pi\phi^{2})$. To omit the non-minimal coupling between gravity
and scalar field, one can work in the Einstein frame rather than in this
Jordan frame. However, in the present work we will work in the Jordan frame
only.

By varying the action (1) with respect to the metric tensor, we find the
gravitational field equation below

$\displaystyle\frac{\phi^{2}}{\xi^{2}}\left(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\right)-\frac{1}{4\xi^{2}}\phi^{4}g_{\mu\nu}+\frac{1}{\xi^{2}}\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)\phi^{2}$
$\displaystyle-\frac{3\alpha}{2\xi^{2}}\left(\omega^{\rho}\nabla_{\rho}\phi^{2}g_{\mu\nu}-\omega_{\nu}\nabla_{\nu}\phi^{2}-\omega_{\mu}\nabla_{\nu}\phi^{2}\right)+6F_{\rho\mu}F_{\sigma\nu}g^{\rho\sigma}$
$\displaystyle-\frac{3}{2}F^{2}_{\rho\sigma}g_{\mu\nu}+\frac{3\alpha^{2}}{4\xi^{2}}\phi^{2}\left(\omega_{\rho}\omega^{\rho}g_{\mu\nu}-2\omega_{\mu}\omega_{\nu}\right)=0.$
(2)

By taking the trace of Eq. (II.1) and defining $\Phi=\phi^{2}$, we find

$\Phi
R+3\alpha\omega^{\rho}\nabla_{\rho}\Phi+\Phi^{2}-\frac{3}{2}\alpha^{2}\Phi\omega_{\rho}\omega^{\rho}-3\Box\Phi=0.$
(3)

The variation of the action (1) with respect to the scalar field $\phi$ gives
rise the following equation of motion of $\Phi$

$R-3\alpha\nabla_{\rho}\omega^{\rho}-\frac{3}{2}\alpha^{2}\omega_{\rho}\omega^{\rho}+\Phi=0.\
$ (4)

By multiplying Eq. (4) by $\Phi$, and subtracting it from Eq. (3), the
generalized Klein-Gordon equation for the scalar field $\Phi$ is obtained as
given by

$\Box\Phi-\alpha\nabla_{\rho}(\Phi\omega^{\rho})=0.$ (5)

We also need the equation of motion for Weyl vector field that can be obtained
by varying the action (1) with respect to $\omega_{\mu}$. The variation will
result in

$4\xi^{2}\nabla_{\nu}F^{\mu\nu}-\alpha^{2}\Phi\omega^{\mu}+\alpha\nabla^{\mu}\Phi=0.$
(6)

###  II.2 Metric and Field Ansatzs

In this paper, we investigate the properties of an exact vacuum solution in
Weyl geometric gravity. We consider a static spherically symmetric
configuration, with the metric given in a general form by

$\ ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}d\Omega^{2},$ (7)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$. The solutions of
the field equations of Weyl geometric gravity are dependent on the choice of
the Weyl vector field $\omega_{\mu}$, that can be generally represented as
$\omega_{\mu}=(\omega_{0},\omega_{1},0,0)$. Without any loss of the
generality, in the following we assumed that $\omega_{0}=0$. For the case when
$\omega_{0}\neq 0$, the numerical study of the field equations was carried out
in Ref. HT3 .

From the assumption of the form on the Weyl vector it follows that
$F_{\mu\nu}\equiv 0$. Then, by employing this condition in the Weyl vector
field equation of motion (6), we obtain

$\Phi^{\prime}=\alpha\Phi\omega_{1}.\ $ (8)

Since we can write

$\Box\Phi=\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{-g}g^{\mu\nu}\frac{\partial\Phi}{\partial
x^{\nu}}\right),$ (9)

and

$\nabla_{\mu}\omega^{\mu}=\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{-g}\omega^{\mu}\right),$ (10)

respectively, Eq. (8) now becomes

$\displaystyle-\alpha\Phi\frac{1}{\sqrt{-g}}\frac{d}{dr}\left(\sqrt{-g}\omega^{1}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-g}}\frac{d}{dr}\left(\sqrt{-g}g^{11}\frac{d\Phi}{dr}\right)$
(11) $\displaystyle-\alpha\omega^{1}\frac{d\Phi}{dr}.$

Now we can write the gravitational field equations (II.1) as

$\displaystyle
e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=$
$\displaystyle\frac{e^{-\lambda}}{r^{2}}\bigg{(}\frac{r^{2}}{4}e^{\lambda}\Phi+2\frac{\Phi^{\prime}r}{\Phi}-\frac{3r^{2}}{4}\frac{\Phi^{\prime
2}}{\Phi^{2}}-\frac{r^{2}\lambda^{\prime}}{2}\frac{\Phi^{\prime}}{\Phi}+r^{2}\frac{\Phi^{\prime\prime}}{\Phi}\bigg{)},$
(12) $\displaystyle
e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=$
$\displaystyle\frac{e^{-\lambda}}{r^{2}}\left(-\frac{r^{2}}{4}e^{\lambda}\Phi-2\frac{\Phi^{\prime}r}{\Phi}-\frac{3r^{2}}{4}\frac{\Phi^{\prime
2}}{\Phi^{2}}-\frac{r^{2}\nu^{\prime}}{2}\frac{\Phi^{\prime}}{\Phi}\right),$
(13) $\displaystyle re^{-\lambda}\left(2\nu^{\prime\prime}+v^{\prime
2}-\lambda^{\prime}\nu^{\prime}+\frac{2\nu^{\prime}}{r}-\frac{2\lambda^{\prime}}{r}\right)=$
$\displaystyle-(4-2r\lambda^{\prime}+2r\nu^{\prime})\frac{\Phi^{\prime}}{\Phi}-r\left(e^{\lambda}\Phi+4\frac{\Phi^{\prime\prime}}{\Phi}-\frac{\Phi^{\prime
2}}{\Phi^{2}}\right).$ (14)

The left hand side of Eqs. (12), (13) and (14) comes from the Einstein tensor,
while the right hand side comes from the contributions of the Weyl geometry.
Hence, we can define the effective energy density $\rho$ and pressure $p$
associated to the scalar field and to the Weyl geometric effects by

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{e^{-\lambda}}{8\pi
r^{2}}\bigg{(}\frac{r^{2}}{4}e^{\lambda}\Phi+2\frac{\Phi^{\prime}r}{\Phi}-\frac{3r^{2}}{4}\frac{\Phi^{\prime
2}}{\Phi^{2}}-\frac{r^{2}\lambda^{\prime}}{2}\frac{\Phi^{\prime}}{\Phi}$ (15)
$\displaystyle+r^{2}\frac{\Phi^{\prime\prime}}{\Phi}\bigg{)}=\frac{1}{8\pi}\left[e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}\right],$
$\displaystyle p$ $\displaystyle=$ $\displaystyle-\frac{e^{-\lambda}}{8\pi
r^{2}}\left(\frac{r^{2}}{4}e^{\lambda}\Phi+2\frac{\Phi^{\prime}r}{\Phi}+\frac{3r^{2}}{4}\frac{\Phi^{\prime
2}}{\Phi^{2}}+\frac{r^{2}\nu^{\prime}}{2}\frac{\Phi^{\prime}}{\Phi}\right)$
(16) $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left[e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}\right].\
$

By adding Eq. (12) and (13), we find the evolution equation for the scalar
field, given by

$\frac{\Phi^{\prime\prime}}{\Phi}-\frac{\lambda^{\prime}+\nu^{\prime}}{2}\frac{\Phi^{\prime}}{\Phi}-\frac{3}{2}\frac{\Phi^{\prime
2}}{\Phi^{2}}-\frac{\lambda^{\prime}+\nu^{\prime}}{r}=0.$ (17)

##  III The Weyl Geometric Black Hole

In this Section we will introduce the exact solution of the static spherically
symmetric field equations of the Weyl geometric gravity theory, and we will
analyze some of its properties.

###  III.1 General black hole solutions

Generally, black hole solutions satisfy the relation $g_{tt}g_{rr}=-1$ between
the components of the metric tensor. This case occurs when in a solution to
Einstein’s equations the Ricci tensor and the matter energy - momentum tensor
have vanishing radial null-null component. On the other hand, the Ricci tensor
is proportional to $g_{\mu\nu}$ in the $t-r$ subspace. This also implies that
the radial pressure is the negative of the energy density ($p=-\rho$). Hence,
in this case, the black hole violates the strong energy condition. The
condition $g_{tt}g_{rr}=-1$ also holds if and only if the radial coordinate is
an affine parameter on the radial null geodesics JacobsonCQG2007 .

In a more general approach, one can find black hole solutions in Weyl
geometric gravity by assuming $g_{tt}g_{rr}\neq-1$. In our case, we can write
that

$\nu(r)+\lambda(r)=f(r),$ (18)

where $f(r)$ is an arbitrary function of radial coordinate. As a function of
the scalar field $\Phi$, one can find the function $f(r)$ from Eq. (20) as
given by

$f(r)=\int\frac{2\Phi^{\prime\prime}\Phi-3\Phi^{\prime
2}}{2\Phi+r\Phi^{\prime}}rdr.$ (19)

This is the general condition that must be satisfied by the scalar field for
obtaining black hole solutions in Weyl geometric gravity. Equivalently, we
obtain for $\Phi$ the differential equation

$\frac{\Phi^{\prime\prime}}{\Phi}-\frac{3}{2}\frac{\Phi^{\prime
2}}{\Phi^{2}}-\frac{f^{\prime}(r)}{2}\frac{\Phi^{\prime}}{\Phi}-\frac{f^{\prime}(r)}{r}=0.$
(20)

###  III.2 Metric of the Weyl geometric black hole

In this Subsection, we will briefly review the black hole solution found in
Ref. HT4 , which corresponds to the case when $g_{tt}g_{rr}=-1$. We consider
the following condition for the metric tensor potentials

$\nu(r)+\lambda(r)=0,~{}~{}\forall r>0.\ $ (21)

Using this assumption, from Eq. (20) we immediately obtain the differential
equation satisfied by $\Phi$

$\Phi^{\prime\prime}=\frac{3\Phi^{\prime 2}}{2\Phi},$ (22)

corresponding to the choice $f(r)=0$ in Eq. (19). The solution of the above
equation is

$\Phi(r)=\frac{C_{1}}{(r+C_{2})^{2}},\ $ (23)

where $C_{2}$ is just an arbitrary integration constant. The scalar field
satisfies the condition $\Phi(r)\rightarrow 0$ at infinity. One can also find
the Weyl vector field

$\omega_{1}=\frac{\Phi^{\prime}}{\alpha\Phi}=-\frac{2}{\alpha(r+C_{2})}.\ $
(24)

Then, from the gravitational field equation Eq. (12) one can find the metric
potentials as

$e^{-\lambda}=e^{\nu}=1-\delta+\frac{\delta(2-\delta)}{3r_{g}}r-\frac{r_{g}}{r}+C_{3}r^{2},$
(25)

where $\delta,r_{g},$ and $C_{3}$ are arbitrary constants and
$C_{2}=3r_{g}/\delta$.

This metric is the generalization of the Schwarzschild-de Sitter solution.
There is a Schwarzschild-type singularity at the center due to the presence of
the $r_{g}/r$ term in the metric. Generically, for $C_{3}<0$, there will be
black hole and cosmological type horizons. The second term in the solution is
the constant $\delta$ term, which appears to be like a global monopole term
YuNPB1994 . Similar metrics can also be obtained for the Kiselev black hole
KiselevCQG2003 , and in Rastall gravity SaktiAnnPhys2020 ; SaktiPhysDark2021 ,
by assuming some arbitrary values of the black hole’s parameters. Furthermore,
this metric potential form is also similar to the black hole solution in dRGT
gravity Burikham:2017gdm ; Burikham:2020dfi ; Ponglertsakul:2018rot ;
GhoshWongjunEPJC2016 . Nonetheless, we would like to emphasize that $C_{3}$
comes as an integration constant, related to the scalar field, a situation
which is different from the dRGT gravitational theory, where the cosmological
constant is present directly from the beginning in the action of the theory.
Note that the constants $C_{1},C_{2},C_{3}$ and $\delta$ are dependent on each
other through the following equation

$3C_{3}C_{2}^{2}+\frac{C_{1}}{4}=3-\delta,$ (26)

which is obtained when we solve Eq. (12) to obtain the metric potentials (25)
HT4 .

From Eq. (26), we find $C_{1}=4(3-\delta-27C_{3}r_{g}^{2}/\delta^{2})$. The
scalar and vector fields then take the following forms

$\displaystyle\Phi(r)$ $\displaystyle=$
$\displaystyle\frac{4\left(3\delta^{2}-\delta^{3}-27C_{3}r_{g}^{2}\right)}{(\delta
r+3r_{g})^{2}},$ (27) $\displaystyle\omega_{1}(r)$ $\displaystyle=$
$\displaystyle-\frac{2\delta}{\alpha(\delta r+3r_{g})}.$ (28)

The vector field vanishes when $\delta=0$, while the scalar field remains
constant. At spatial infinity, both the scalar and the vector fields will
vanish.

###  III.3 The horizons of the Weyl geometric black hole

Within this work, we interpret the solution with the metric potentials given
by Eq. (25) as representing a hairy black hole solution. Due to the presence
of the constant that mimics a cosmological constant, the Weyl geometric black
hole possesses more than two horizons. In the following, we will redefine
$C_{3}=-1/l^{2}$. Hence, the horizons are the solutions of the third order
algebraic equation

$\Delta=(1-\delta)r-\frac{\delta(\delta-2)}{3r_{g}}r^{2}-r_{g}-\frac{r^{3}}{l^{2}}=0.$
(29)

The above equation has real roots (for $1/l^{2}>0$) if the following
conditions are satisfied

$-\frac{27r_{g}}{l^{2}}+2-12\delta+15\delta^{2}-5\delta^{3}+2\sqrt{(1-4\delta+2\delta^{2})^{3}}>0,\
$ $\displaystyle 0\leq\delta<1-\frac{2\sqrt{7}}{7}.\ $ (30)

The plot of the position of horizons versus the parameter $\delta$ is shown in
Fig. 1, for $1/l^{2}=10^{-2},r_{g}=1$. The cosmological horizon $r_{C}$ is
given by the green line in Fig. 1. The event horizon $r_{H}$ is represented by
the blue line. There is a third horizon, however, it is unphysical since it is
negative horizon ($r_{N}$) (the red line in Fig. 1). For cubic equation (29),
there are typical analytic expressions for $r_{H},r_{C},r_{N}$ which we will
not present here. Notably for $\delta>0$, the cosmological horizon $r_{C}$ can
extend far beyond the typical horizon at $r_{C}=l$ in $\delta=0$ case.

Figure 1: The positions of the horizons of the Weyl geometric black hole as a
function of $\delta$ for $1/l^{2}=10^{-2},r_{g}=1$. The green, blue, and red
lines represent the cosmological, event, and unphysical horizon respectively.

###  III.4 Effective density and pressure

The effective energy density and pressure which appear due to the contribution
from the scalar and vector fields in Weyl geometric gravity can be written for
the case of the exact black hole solution as

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{1}{24\pi
r^{2}}\left(3\delta+\frac{2(\delta-2)\delta}{r_{g}}r+\frac{9r^{2}}{l^{2}}\right),$
(31) $\displaystyle p$ $\displaystyle=$ $\displaystyle-\rho.\ $ (32)

Note that both the density and the pressure are divergent at the center for
nonzero $\delta$. At spatial infinity, the energy density approaches the “dark
energy” density $3/(8\pi l^{2})$. Thus, this black hole of Weyl geometric
gravity is different from the Schwarzschild-de Sitter solution in GR, since it
describes a physical configuration in which there are density and pressure
contributions from the Weyl scalar and vector fields.

The pressure vanishes at two different points that can be obtained by solving
the equation $p(R)=0$, which gives

$R_{\pm}=\frac{\delta^{2}-2\delta\pm\sqrt{-\displaystyle{\frac{27r_{g}^{2}\delta}{l^{2}}}+4\delta^{2}-4\delta^{3}+\delta^{4}}}{-\displaystyle{\frac{9r_{g}}{l^{2}}}}.$
(33)

Fig. 2 presents the plot of the effective pressure $p$ and the metric function
$e^{\nu}$ showing the position of the zero-pressure points in comparison to
the horizons of the black hole. The plots are obtained for $\delta=0.24,M=1/2$
and $1/l^{2}=10^{-2}$. With these values of the parameters, we can obtain
$p(R)=0$ at $R_{+}=0.948019$ and $R_{-}=17.9947$. We also have $r_{H}=1.12597$
and $r_{C}=8.87784$. Hence, for the given parameters,
$R_{+}<r_{H}<r_{C}<R_{-}$. The maximum effective pressure occurs at
$r=1.70455$, which is outside of $r_{H}$, but inside of $r_{C}$ and $R_{-}$.

Figure 2:  The metric tensor $e^{\nu}$ and the effective pressure $p$ of the
Weyl geometric black hole. The metric tensor component $e^{\nu}$ is shown as a
red line, while the pressure is represented by the blue line. The plots are
obtained for $\delta=0.24$, $M=1/2$ and $1/l^{2}=10^{-2}$.

###  III.5 Dark matter and the Weyl geometric black hole

In the original paper HT4 , the effective energy density and pressure
associated to the Weyl scalar and vector fields are assumed to represent in an
effective form the dark matter that can be found around the baryonic mass
distributions in galaxies. Therefore, one can compute the geometric mass that
contains the dark matter contribution by integrating the energy density over
the volume.

The effective geometric mass of the dark matter for the Weyl geometric black
hole solution is given by

$\displaystyle M_{G}(r)=\int_{0}^{r}4\pi\rho r^{\prime
2}dr^{\prime}=\frac{r}{2}\left[\delta+\frac{r^{2}}{l^{2}}-\frac{(2-\delta)\delta}{3r_{g}}r+\frac{r_{g}}{r}\right].$

Note that this mass is not the black hole mass.

We can find the accumulated mass at the points where the pressure vanishes by
inserting Eq. (33) into the geometric mass (III.5) to obtain

$\displaystyle M_{G\pm}$ $\displaystyle=$
$\displaystyle\frac{r_{g}}{2}+\frac{\delta\left(3\delta^{2}-6\delta\pm
2\sqrt{\delta^{2}(\delta-2)^{2}+27C_{3}r_{g}^{2}\delta}\right)}{54C_{3}r_{g}}$
$\displaystyle+\frac{\delta^{2}(\delta-2)^{2}\left(\delta^{2}-2\delta\pm\sqrt{\delta^{2}(\delta-2)^{2}+27C_{3}r_{g}^{2}\delta}\right)}{729C_{3}^{2}r_{g}^{3}}.$

We plot the effective geometric mass as a function of $R_{\pm}$ in Fig. 3. The
black hole mass is given by $r_{g}/2=0.5,$ while the geometric mass $M_{G}$
ranges between $0-4$ depending on $\delta,l^{2}$. The geometric mass can reach
more than 5 times of the black hole mass for $\delta\gtrsim 1.6$ at the
specific radius $R_{+}$. As distance grows the accumulated mass will continue
to increase since the density approaches constant dark energy density
$\displaystyle{\frac{3}{8\pi l^{2}}}$ asymptotically at large distance.

Figure 3: Effective geometric mass $M_{G}$ is plotted versus $R_{\pm}$.

It is interesting to consider two cases for the accumulated geometric mass,
corresponding to $R_{-}>R_{+}$, and $R_{-}=R_{+}$, respectively. When
$R_{-}>R_{+}$, we can find the mass difference between $R_{+}$ and $R_{-}$,
$\Delta M_{G}=M_{G-}-M_{G+}$ given by

$\Delta M_{G}=-\frac{2l^{4}\left[\delta^{2}(\delta-2)^{2}-\frac{27\delta
r_{g}^{2}}{l^{2}}\right]^{3/2}}{729r_{g}^{3}}.$ (36)

This mass difference acts like the mass of a “dark matter halo” surrounding
the Weyl geometric black hole.

On the other hand, for $R_{-}=R_{+}$ we find

$R=R_{-}=R_{+}=-\frac{\delta(\delta-2)l^{2}}{9r_{g}},$ (37)

and

$M_{G}=\frac{r_{g}}{2-\delta},$ (38)

respectively.

Note that we have also used the relation
$-27r_{g}^{2}+(4\delta-4\delta^{2}+\delta^{3})l^{2}=0$ from the vanishing
square-root term of Eq. (33). Since we already have (30) for $\delta$, it
follows that $R$ is always positive. In this case, the geometric mass does not
depend explicitly on $1/l^{2}$.

##  IV The thermodynamics of the Weyl geometric black hole on the event
horizon

In the present Section we will study the thermodynamics of the Weyl geometric
black hole. Since there is a cosmological constant term in the metric, in
order to satisfy the first law of thermodynamics, we need to also consider, in
our thermodynamical investigations, the pressure term coming from this
effective cosmological constant like contribution. The first law of black hole
thermodynamics thus takes the form

$dM=T_{H}dS_{BH}+VdP,$ (39)

where $P$ and $V$ are the thermodynamic pressure and volume terms related to
the effective cosmological constant $1/l^{2}$, respectively. We may consider
the cosmological constant as a dynamical variable, as a consequence of
considering the mass of the black hole as the enthalpy of the spacetime
CveticGibbonsPRD2011 ; DolanKastorPRD2013 . It has already been pointed out in
Ref. Alonso-SeranoPRD2022  that for general Weyl invariant theories of
gravity, due to the conformal invariance, in the first thermodynamic law of
black holes a cosmological constant related term must also exist.

###  IV.1 The black hole mass

The proper mass of the black hole is different from the mass function we have
introduced in Eq. (III.5). In the following we assume that the integration
constant $r_{g}$ in the Weyl geometric black hole solution is the
gravitational radius $r_{g}$ of the black hole, and thus it is proportional to
the black hole mass, i.e., $r_{g}=2M$ where $M$ is the black hole mass. This
choice is similar to the way the mass of the black hole is obtained in General
Relativity and Brans-Dicke theory XuLuYangJiangEntropy2023 .

For asymptotically de Sitter solutions another way to compute the black hole
mass has been described in Ref. DolanCQG2019 . In this approach, the invariant
definition of the mass for a black hole in de Sitter spacetime does not rely
on taking the limit $r\rightarrow\infty$. All what is necessary for the the
calculation of the mass is the existence of a time-like Killing vector at some
point $r<r_{C}$, in a region of spacetime containing a 2-sphere surrounding
the mass. The mass of the black hole can then be calculated by integrating
over a sphere of any radius, as long as it thoroughly surrounds the mass, and
is within a region containing the time-like Killing vector. Similar to the
Schwarzschild-de Sitter solution DolanCQG2019 , one can then find the mass of
the Weyl geometric black hole as $M=r_{g}/2$.

In terms of the event horizon, we can solve Eq. (29) to find the black hole
mass as a function of $r_{H}$,

$\displaystyle M_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\Bigg{[}3\left(-\frac{r_{H}^{3}}{l^{2}}-\delta
r_{H}+r_{H}\right)$ (40) $\displaystyle\pm$
$\displaystyle\left.\sqrt{9\left(-\frac{r_{H}^{3}}{l^{2}}-\delta
r_{H}+r_{H}\right)^{2}-12r_{H}^{2}\delta(\delta-2)}\right].\ $

Note that in terms of the event horizon the mass has two branches. Thus, the
mass could be negative when one adopts the solution $M_{-}$, if the value in
the square root is bigger than the other terms. Hence, for this case, we need
the condition $\delta>2$ in order for the mass to be positive. However,
$\delta$ has a constraint given by Eq. (30), where the maximum value of
$\delta$ is $0.244071$. Therefore the negative sign of the mass does not obey
this condition, and we will not consider this case further.

In addition to the black hole mass, the geometric mass on the event horizon
can also be computed. Interestingly enough, by using Eq. (III.5) and (29) at
$r=r_{H}$, a very simple formula for the geometric mass on the event horizon
can be obtained for the Weyl geometric black hole

$M_{G}|_{r=r_{H}}=\frac{r_{H}}{2}.\ $ (41)

This condition is identical to the Schwarzschild radius $r_{H}=2M_{G}$ for the
Schwarzschild black hole.

The black hole mass $M\equiv M_{+}$, plotted as a function of $r_{H}$ for
varying $\delta$ and $1/l^{2}$, is shown in Fig. 4. A comparison with the
geometric mass is also presented. There is a maximum mass $M$ that can be
obtained from $dM/dr_{H}=0$, given by

$M_{\rm
max}=\frac{\sqrt{(\delta+1)^{3}l^{6}}+(1-2\delta)\sqrt{\delta+1}l^{3}}{6\sqrt{3}l^{2}}.$
(42)

The position of the horizon for the maximum mass is obtained as

$r_{H}^{M}=\sqrt{\frac{(\delta+1)l^{2}}{3}}.$ (43)

The maximum mass is simply the mass of black hole in the Nariai limit where
event horizon and cosmic horizon merge. This is shown in Fig. 5.

Figure 4:  The mass of the Weyl geometric black hole plotted as a function of
$r_{H}$, for different values of the parameters. The orange line depicts the
geometric mass versus $r_{H}$. Figure 5: The Weyl geometric black hole mass
versus $r_{H}$ and $r_{C}$.

For $\delta=0$, there is only one black hole mass given by

$M=\frac{1}{2}\left(r_{H}-\frac{r_{H}^{3}}{l^{2}}\right).$ (44)

This is exactly the mass for Schwarzschild-de Sitter black hole expressed in
terms of $r_{H}$.

###  IV.2 The Hawking temperature

For a generic spherically symmetric spacetime given by (7), if both
$g_{00}=e^{\nu}$ and $1/g_{rr}=-e^{-\lambda}$ have the same zero at $r_{H}$,
it can be shown that the Hawking temperature $T_{H}$ is related to the surface
gravity

$\kappa_{H}=\frac{g^{\prime}_{00}}{2}\sqrt{(-g_{00}g_{rr})^{-1}}\Big{|}_{r_{H}},$
(45)

computed on the event horizon of the black hole, by the relation,

$T_{H}=\frac{\kappa_{H}}{2\pi}.$ (46)

$\kappa_{H}$ is the surface gravity .

To show this, we perform the coordinate transformation

$g_{rr}dr^{2}\to\frac{dr^{\prime 2}}{g_{00}(r(r^{\prime}))}.$ (47)

Then the Hawking temperature is given by the conventional formula

$T_{H}=\frac{g^{\prime}_{00}(r^{\prime})}{4\pi}=\frac{g^{\prime}_{00}(r)\sqrt{(-g_{00}g_{rr})^{-1}}}{4\pi},$
(48)

where, in the new coordinates,

$\kappa_{H}=\frac{1}{2}g^{\prime}_{00}(r^{\prime})=\frac{1}{2}g^{\prime}_{00}(r)\sqrt{(-g_{00}g_{rr})^{-1}},$
(49)

at $r_{H}$. This ends the proof of this result. Here, the surface gravity is
defined as the acceleration experienced by an object at the event horizon of
the black hole.

Alternatively, in terms of the Killing vector, the surface gravity is given by

$\kappa=\sqrt{-\frac{1}{2}\nabla^{\nu}\xi^{\mu}\nabla_{\nu}\xi_{\mu}}.\ $ (50)

Here $\xi^{\mu}$ denotes the Killing vector possessed by the spacetime metric.
For the spherically symmetric spacetime (7), we have $\xi^{\mu}=(1,0,0,0)$.
Hence, the non-vanishing components of the covariant derivatives of the
Killing vector are

$\displaystyle\nabla^{t}\xi^{r}\nabla_{t}\xi_{r}=\nabla^{r}\xi^{t}\nabla_{r}\xi_{t}=-\frac{{\nu^{\prime}}^{2}}{4}e^{\nu-\lambda}.$
(51)

Thus, we obtain the surface gravity and the Hawking temperature on the event
horizon as

$\kappa_{H}=\frac{\nu^{\prime}}{2}e^{(\nu-\lambda)/2}\bigg{|}_{r=r_{H}},~{}~{}~{}T_{H}=\frac{\nu^{\prime}}{4\pi}e^{(\nu-\lambda)/2}\bigg{|}_{r=r_{H}}.\
$ (52)

This is the same result as presented in (48). Under the condition that both
$e^{\nu}$ and $e^{-\lambda}$ contain the same zeros at $r=r_{H}$, and by using
the l’Hopital rule, we find the relation
$\nu^{\prime}|_{r_{H}}=-\lambda^{\prime}|_{r_{H}}$. Hence, one can also
rewrite the equation for the Hawking temperature as

$T_{H}=\frac{1}{4\pi}\left(-\nu^{\prime}\lambda^{\prime}e^{\nu-\lambda}\right)^{1/2}\bigg{|}_{r=r_{H}}.$
(53)

This is exactly the equation for the Hawking temperature obtained from the
tunneling method SiahaanEPJC2016 . Note that this formula is valid generically
regardless of the vacuum condition $\nu+\lambda=0$, as long as both metric
tensor components $g_{00}$ and $g_{rr}$ have the same zeros at the horizon.

For the Weyl geometric black hole, the Hawking temperature is

$T_{H}=\frac{1}{4\pi}\left[\frac{r_{g}}{r^{2}_{H}}-2\frac{r_{H}}{l^{2}}-\frac{\delta(\delta-2)}{3r_{g}}\right].\
$ (54)

The Hawking temperature of the Weyl geometric black holes is not identical to
the temperature of a black hole in dRGT gravity GhoshWongjunEPJC2016 , when we
insert the mass formula (40).

The plot of the Hawking temperature is presented in Fig. 6.

Figure 6:  Hawking temperature as a function of $r_{H}$ for various values of
the parameters of the Weyl geometric black hole. The red line depicts the
Hawking temperature for the basic numerical values of the parameters. The
green and the blue lines are the temperatures obtained by varying $1/l^{2}$.
The yellow and the orange curves represent the Hawking temperatures when we
vary $\delta$.

We will also see that the entropy of the Weyl geometric black holes is not
similar to that of the black holes in the dRGT gravity, due to the presence of
the non-minimal coupling between gravity and the scalar field.

For $\delta=0$, Eq. (54) reduces to

$T_{H}=\frac{1}{4\pi}\left(\frac{r_{g}}{r^{2}_{H}}-\frac{2r_{H}}{l^{2}}\right).\
$ (55)

By using the expression of the mass as given by Eq. (44), it is easy to show
that the above equation gives exactly the Hawking temperature of the
Schwarzschild-de Sitter black hole.

###  IV.3 Entropy from the Noether charge

The Noether charge formalism provides a systematic and powerful way to
calculate conserved quantities, e.g., the entropy, without being tied to the
specific details of the gravitational theory. In particular, this formalism it
is useful in understanding black holes, and their thermodynamic properties.
The main ingredient of the Noether charge formalism is the diffeomorphism
invariance.

For general Weyl invariant theories, the Noether charge formalism was
introduced in in Ref. Alonso-SeranoPRD2022 . Likewise, in Weyl geometric
gravity, there is an effective term behaving like the cosmological constant in
the conserved charge. As a consequence, there is a term related to the
cosmological constant in the first law of thermodynamic formula, as given in
Eq. (39).

Interestingly, the Weyl geometric gravity theory, described by the Lagrangian
(1), contains the Ricci scalar coupled to a scalar field, similarly to the
case of the Brans-Dicke theory. We may interpret the other terms in the
Lagrangian as the matter contribution. In the Brans-Dicke theory, the entropy
does not satisfy the exact area law, since the scalar field and gravity are
coupled non-minimally VollickPRD2007 . The entropy expression has additional
scalar field terms. One may remove these factors by working in the Einstein
frame. However, this is not the case that we will investigate within this
paper.

We can choose $\xi^{2}$ in a particular way so that we can write the
Lagrangian in (1) in the simple form

$\mathcal{L}=\frac{\Phi}{16\pi}R+\mathcal{L}_{M},$ (56)

where $\mathcal{L}_{M}$ represents all the other terms in Eq. (1), and which
can be interpreted as the Lagrangian of an effective form of matter. The
matter terms do not contribute to the entropy directly through the Noether
charge. Hence, the non-vanishing contribution to the entropy in the Noether
charge formalism comes only through the coupling of the gravitational field
with the scalar, and which is given by VollickPRD2007

$Q=\int_{\partial\Sigma}Q^{\mu\nu}d\sigma_{\,u\nu},$ (57)

where

$\displaystyle Q^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\Phi}{16\pi}(\nabla^{\mu}\xi^{\nu}-\nabla^{\nu}\xi^{\mu}),$
(58) $\displaystyle d\sigma_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sqrt{-g}\varepsilon_{\mu\nu\theta\phi}\text{det}\left(\frac{\partial(x^{\mu},x^{\nu})}{\partial(\theta,\phi)}\right)d\theta
d\phi.$ (59)

$d\sigma_{\mu\nu}$ is the surface element of a closed two-dimensional surface
of the event horizon, and $\partial\Sigma$ is a two-sphere of an arbitrary
radius, which we choose to be the horizon.

We can now compute the conserved charge $Q$ for the spacetime metric (7). The
non-vanishing components of the covariant derivative of the Killing vector are
already given in Eq. (51). And the element of the two-dimensional surface is

$d\sigma_{tr}=e^{(\nu+\lambda)/2}r^{2}\sin^{2}\theta d\theta\wedge
d\phi|_{r=r_{H}}.$ (60)

Using these expressions into the integral form of $Q$, we obtain

$Q|_{r=r_{H}}=\frac{\Phi_{H}e^{\frac{\nu-\lambda}{2}}\nu^{\prime}}{4}r_{H}^{2}.\
$ (61)

Then the entropy of the Weyl geometric black hole is given by

$S_{BH}=\frac{2\pi Q|_{r=r_{H}}}{\kappa_{H}}=\Phi_{H}\pi r_{H}^{2}.$ (62)

Note that $\Phi_{H}$ is the value of the scalar field on the event horizon.
Yet, the entropy (62) is the general entropy for the Lagrangian (56). It
differs from the area law for the entropy by a scalar factor $\Phi_{H}$ at the
horizon. The spacetime metric of the Weyl geometric black hole is identical to
the black hole solution in dRGT theory. However, we obtain a different entropy
from the dRGT theory, which was computed in Ref. GhoshWongjunEPJC2016 ,
because the Lagrangian is totally different from the Lagrangian of the dRGT
gravity. The plot of the black hole entropy versus $r_{H}$ is shown in Fig. 7.
The entropy is saturated in the large $r_{H}$ limit. The physical region of
the spacetime, however, is limited to $r_{H}\leq r_{C},T_{H}\geq 0$, before
the entropy is saturated.

Figure 7:  The Weyl geometric black hole entropy as a function of $r_{H}$. The
red line depicts the entropy for the basic parameters. The green and the blue
lines show the entropies obtained by varying $1/l^{2}$. The yellow and the
orange lines represent the entropies obtained by varying $\delta$. The
vertical lines correspond to the values of $r_{H}$ where $T_{H}=0$.

It would be also interesting to study the entropy of the Weyl geometric black
hole in the Einstein frame, in which the scalar field is not coupled directly
to the Ricci scalar. We leave this issue for future study.

For the special case when $\delta=0$, the scalar field on the event horizon is

$\Phi_{H}=\frac{12}{l^{2}}.\ $ (63)

The entropy is then

$S_{BH}=\frac{12\pi r_{H}^{2}}{l^{2}}.\ $ (64)

The entropy (64) is proportional to the black hole horizon area, yet with an
extra factor rather than $1/4$. Hence, we can see that the entropy will vanish
for $1/l^{2}=0$. This entropy does not reproduce the entropy of the
Schwarzschild solution for a vanishing cosmological constant-like term. The
entropy expression (64) is obviously different from the one obtained from the
Schwarzschild-de Sitter solution in GR.

In Weyl geometric gravity, in order to obtain the GR limit, one needs to break
the conformal symmetry, e.g., by introducing a spontaneous symmetry breaking,
which can be achieved by giving $\Phi$ a vacuum expectation value
$\Phi_{0}(r)$ Ghilencea:2021lpa , which satisfies (23) for the spherically
symmetric vacuum. In the conformal-breaking vacuum, the gravitational constant
will vary with $r$ as $G=1/\Phi_{0}(r)$.

In the GR limit, the entropy of the Weyl geometric black hole for $\delta\to
0$ will reduce to the Schwarzschild-de Sitter black hole entropy $S=\pi
r_{H}^{2}/G$, with the identification $G=\Phi_{0}(r_{H})$.

###  IV.4 The thermodynamic volume

Let’s recall that the Weyl geometric black hole needs to satisfy the
thermodynamic law (39). We have already computed the Hawking temperature and
black hole entropy. Now we need to compute the thermodynamic volume, and its
conjugate, which correspond to the constant $1/l^{2}$. For the Weyl geometric
black hole, we can assume the Smarr formula

$\frac{1}{2}M=T_{H}S_{BH}-PV.$ (65)

In the presence of the cosmological constant $\Lambda=8\pi\rho_{\Lambda}$, the
spacetime has negative pressure $P=-\rho_{\Lambda}$, related to the
cosmological constant via the relation

$P=-\frac{\Lambda}{8\pi}=-\frac{3}{8\pi l^{2}}.$ (66)

This is actually the value of the effective pressure at spatial infinity. The
thermodynamic volume $V$ is then determined from the Smarr formula (65),

$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{\pi
r_{H}}{9}\Bigg{[}-3(\delta-1)l^{2}+3r_{H}^{2}(4\Phi_{H}-1)$ (67)
$\displaystyle+$
$\displaystyle\sqrt{r_{H}^{4}+18(\delta-1)l^{2}r_{H}^{2}-3(\delta-3)(\delta+1)l^{4}}$
$\displaystyle\times$ $\displaystyle(1-2\Phi_{H})\Bigg{]}.$

The plot of the thermodynamic volume is shown in Fig. 8.

Now we can consider a simpler case corresponding to $\delta=0$. In this case,
the thermodynamic volume is given by

$V=\frac{2}{3}\pi r_{H}^{3}\left(\frac{36}{l^{2}}-1\right)-\frac{2}{3}\pi
l^{2}r_{H}\left(\frac{12}{l^{2}}-1\right).$ (68)

For the Schwarzschild-de Sitter solution, the thermodynamic volume is given by
$V=\frac{4}{3}\pi r_{H}^{3}$. The thermodynamic volumes are very different in
the two theories. For the special case $l^{2}=12$, the thermodynamic volume of
the Weyl geometric black hole becomes $V=\frac{4}{3}\pi r_{H}^{3}$, identical
to the volume of the Schwarzschild-de Sitter black hole.

Figure 8:  The thermodynamic volume as a function of $r_{H}$. The red line
represents the thermodynamic volume for the basic parameters. The green and
blue curves depict the volumes for varying $1/l^{2}$. The yellow and orange
curves show the volumes for varying $\delta$.

##  V Thermodynamic stability

One can study the thermodynamic stability of the black holes by using the
Helmholtz and Gibbs free energy. When there is no particle transfer, the
Helmholtz free energy should be considered, while the Gibbs free energy is
more suitable when there is a particle exchange. On the other hand, to study
the stability of the thermal equilibrium between the black hole, and its
surroundings, we can consider the heat capacity of the black hole. The heat
capacity is also an important avenue for investigating critical phenomena, and
the phase structures of the black hole DaviesProcRSocA1977 . More
specifically, when the heat capacity diverges or changes sign, it indicates a
phase transition, and the breakdown of the equilibrium thermodynamic
description. It is worth noting that the local stability does not imply the
global one. On the other hand, it is important to note that if a black hole is
stable globally, or it is in a global thermodynamic equilibrium, it must be
also stable locally.

###  V.1 The Helmholtz and Gibbs free energy

Since $M$ is identified with enthalpy $U+PV$ ($U$ is the energy), the Gibbs
and Helmholtz energy are given by

$\displaystyle G$ $\displaystyle=$ $\displaystyle M-T_{H}S_{BH},$ (69)
$\displaystyle F$ $\displaystyle=$ $\displaystyle
G-PV=\frac{3}{2}M-2T_{H}S_{BH},\ $ (70)

respectively.

As compared to the vacuum case, the global thermodynamic stability of a black
hole is determined by the condition $F\leq 0$. In this case, $F=0$ is the
condition for the first-order phase transition from the vacuum state to the
black hole. For the Weyl geometric black hole, in terms of $r_{H}$, the
Helmholtz free energy is given by

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{r_{H}}{24l^{2}}\Bigg{[}3(8\Phi-3)r_{H}^{2}-9(\delta-1)l^{2}+(3-4\Phi)$
(71) $\displaystyle\times$
$\displaystyle\sqrt{9r_{H}^{4}+18(\delta-1)l^{2}r_{H}^{2}-3(\delta-3)(\delta+1)l^{4}}\Bigg{]}.$

and the Gibbs free energy is

$\displaystyle G$ $\displaystyle=$
$\displaystyle\frac{r_{H}}{12l^{2}}\Bigg{[}3(2\Phi-1)r_{H}^{2}-3(\delta-1)l^{2}-(\Phi-1)$
(72) $\displaystyle\times$
$\displaystyle\sqrt{9r_{H}^{4}+18(\delta-1)l^{2}r_{H}^{2}-3(\delta-3)(\delta+1)l^{4}}\Bigg{]}.$

Note that the vacuum state without a black hole in this background is obtained
when $\delta=r_{g}=r_{H}=0$, which naturally implies $F=0$. Fig. 9 presents
the plots of the Helmholtz free energy $F$ and of the Gibbs free energy $G$
versus $r_{H}$. From the range of parameters we choose, it turns out that the
Weyl geometric black hole has negative free energy associated with the event
horizon only for $r_{H}\ll 1$.

Figure 9: The Helmholtz ($F$) and the Gibbs ($G$) free energy versus $r_{H}$
for the Weyl geometric black hole.

###  V.2 Heat capacity

Now we will study the thermodynamic stability locally by using the heat
capacity. The heat capacity at fixed pressure (fixed $l^{2}$) on the event
horizon can be computed from the relation

$C_{P}=\frac{dM}{dT_{H}}=\frac{dM}{dr_{H}}\Big{/}\frac{dT_{H}}{dr_{H}}.\ $
(73)

The plot of the heat capacity is given in Fig. 10. When $C_{P}$ is positive,
the thermal equilibrium with the surrounding environment is stable, while the
negative values implies thermodynamical instability. The Weyl geometric black
hole is locally stable in some ranges of sufficiently large $r_{H}$, and there
are no divergences. Generically, there is a maximum and minimum of heat
capacity for the Weyl black hole. The maximum (minimum) $C_{P}$ occurs in the
positive (negative) $C_{P}$ region. For Schwarzschild-dS-like black hole
(black line), there is also the maximum heat capacity at the Nariai limit
above which the black hole spacetime ceases to exist. Local instability for
small black holes only implies that these black holes will evaporate by means
of the Hawking radiation.

Figure 10: The heat capacity of the Weyl geometric black holes versus $r_{H}$.

For a continuous $F$ as well as for its first derivatives, the divergence or
discontinuity of the heat capacity implies a second-order phase transition,
since heat capacity is related to the second derivative of the free energy
DaviesProcRSocA1977 . For the Weyl geometric black hole, there are no
divergences in the heat capacity, and thus no second-order phase transition
can take place.

###  V.3 Luminosity and evaporation time of Weyl geometric black holes

The luminosity of the black hole can be approximated using the Stephan-
Boltzmann law, or by explicit calculation of evaporation rates BambiEPJC2018 .
It is given by

$L=\frac{dM}{dt}=\sigma A_{H}T_{H}^{4},$ (74)

where $\sigma$ is a Stefan–Boltzmann-like constant that depends on the black
hole mass and on the particle content of the theory. $A_{H}$ is the area of
the black hole on the horizon, which is given by

$A_{H}=\int^{2\pi}_{0}\int^{\pi}_{0}\left(g_{\theta\theta}g_{\phi\phi}\right)^{1/2}d\theta
d\phi.$ (75)

For the Weyl geometric black hole, $A_{H}=4\pi r_{H}^{2}$. Hence, the black
hole luminosity can be written as

$L=\frac{\sigma
r_{H}^{2}}{64\pi^{3}}\left(\frac{2M}{r_{H}^{2}}-\frac{2r_{H}}{l^{2}}-\frac{\delta(\delta-2)}{8M}\right)^{4}.$
(76)

We show the plot of the luminosity divided by $\sigma$ in Fig. 11, where we
have used the mass function (40). The effects of the parameter $1/l^{2}$ and
$\delta$ become important, and distinctive, in the large $r_{H}$ region.

Figure 11: Luminosity of the Weyl geometric black holes versus the radius of
the event horizon $r_{H}$.

####  V.3.1 The evaporation time

From Eq. (74), we can also compute the evaporation time of the black hole. It
is given by

$t_{\rm Evap}=\int_{0}^{M}\frac{d\hat{M}}{\sigma A_{H}T_{H}^{4}}.$ (77)

Figure 12: Evaporation time $t_{\rm Evap}$ of the Weyl geometric black hole
versus black hole mass $M$.

We show the plot of the evaporation time $t_{\rm Evap}$ of Weyl geometric
black hole versus the mass $M$ in Fig. 12. The evaporation time of Weyl
geometric black hole for the given parameter increases with the black hole
mass. In the Nariai limit, where $r_{H}\to r_{C}$, the Hawking temperature
becomes zero, and $t_{\rm Evap}\to\infty$.

##  VI Thermodynamics on the cosmological horizon

Generically, Weyl geometric black holes could have a cosmological horizon and
we can compute the thermodynamic quantities on this horizon. The thermodynamic
quantities on the cosmological horizon are obtained by considering the event
horizon as the boundary GomberoffPRD2003 . On the other hand, the
thermodynamic quantities on the event horizon are obtained when the
cosmological horizon is considered as the boundary.

The plot of the Weyl geometric black hole mass $M$ versus $r_{H},r_{C}$, is
shown in Fig. 5. For each $M$ there could possibly be two horizons up until
the maximum mass where $r_{H}=r_{C}$ at the Nariai limit. This limit
corresponds to the largest possible Weyl geometric black hole.

The first law of the thermodynamics on the cosmological horizon takes a
similar form with that on the event horizon. It is given by

$dM_{C}=T_{C}dS_{C}+V_{C}dP_{C}.\ $ (78)

First, the mass on the cosmological horizon is obtained as

$\displaystyle M_{C}$ $\displaystyle=$ $\displaystyle-M$ (79)

We can see the main difference from the mass on the event horizon, i.e., the
overall sign of $M_{C}$ becomes negative GomberoffPRD2003 . The Hawking
temperature computed from Eq. (53), and the entropy, obtained from Eq. (62),
are

$\displaystyle T_{C}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\left(\frac{2M}{r^{2}_{C}}-2\frac{r_{C}}{l^{2}}-\frac{\delta(\delta-2)}{6M}\right),$
(80) $\displaystyle S_{C}$ $\displaystyle=$ $\displaystyle\Phi_{C}\pi
r_{C}^{2},\ $ (81)

where $\Phi_{C}=\Phi(r_{C})$. The temperature of the cosmological horizon is
negative since the entropy at the cosmological horizon increases when the
black hole mass decreases Klemm:2004mb ; Cvetic:2018dqf . Finally, the
pressure and the thermodynamic volume from the Smarr formula are given by

$P_{C}=\frac{3}{8\pi l^{2}},$ (82) $\displaystyle V_{C}$ $\displaystyle=$
$\displaystyle\frac{1}{9}\pi\Big{[}-3(\delta-1)l^{2}r_{C}+3r_{C}^{3}\left(4\Phi_{C}-1\right)+\left(1-2\Phi_{C}\right)$
$\displaystyle\times$
$\displaystyle\sqrt{-3(\delta-3)(\delta+1)l^{4}r_{C}^{2}+18(\delta-1)l^{2}r_{C}^{4}+9r_{C}^{6}}\Big{]}.$

For a system containing two different horizons, the event and cosmological
horizons, the total entropy is the sum of the entropies from each horizon
UranoSaidaCQG2006

$S_{tot}=S_{BH}+S_{C}.\ $ (84)

Finally, the Helmholtz and Gibbs free energy associated with the cosmological
horizon are plotted in Fig. 13.

Figure 13: The Helmholtz ($F$) and the Gibbs ($G$) free energy of the
cosmological horizon of the Weyl geometric black hole versus $r_{C}$.

The thermodynamic stability of a spacetime with two horizons occurs when the
total free energy,

$F_{T}=F+F_{C},G_{T}=G+G_{C},$ (85)

is negative, as shown in Figs. 14 and 15, respectively. In the Nariai limit
$r_{H}=r_{C}$, since $M=-M_{C},T_{H}=-T_{C},S=S_{C}$, the total free energies
vanish.

Figure 14:  The total Helmholtz ($F_{T}$) and Gibbs ($G_{T}$) free energy for
the Weyl geometric black hole versus $r_{H}$ and $r_{C}$ for fixed $\delta$.
Figure 15: The total Helmholtz ($F_{T}$) and Gibbs ($G_{T}$) free energy for
the Weyl geometric black hole versus $r_{H},r_{C}$ for fixed $1/l^{2}$.

##  VII Discussions and final remarks

Weyl geometric gravity is an interesting extension of general relativity,
which implements the requirement of conformal invariance at both the action
and field equations level. The starting point is the action constructed in
Weyl geometry which contains only two terms. An important result of the theory
is the extraction of a scalar degree of freedom through the linearization of
the action in terms of the Weyl (Ricci) scalar, and the formulation of then
theory as an equivalent scalar-vector-tensor theory. The theory admits an
exact vacuum solution in spherical symmetry, corresponding to a particular
choice of the Weyl vector, assumed to have only a radial component. As a
result the strength of the Weyl vector $\tilde{F}_{\mu\nu}$ identically
vanishes. Moreover, the Weyl vector becomes the gradient of the scalar field,
$\omega_{1}=(1/\alpha)d\ln\Phi/dr$. Thus, the solution is not given in the
proper Weyl geometry, but in a subclass of it, called integrable Weyl geometry
WIG . However, it is important to point out that Weyl Integrable Geometry does
not reduce to Riemann geometry, since the group of geometrical automorphisms
also contains the group of conformal transformations Scholz1 .

In the field equations of the Weyl geometric black hole solution, due to the
effects of the Weyl geometry, some extra terms, induced by the presence of the
scalar field, do appear. These new terms can be interpreted as describing the
energy density and pressure of an effective fluid located around the black
hole. Hence, the Weyl geometric black hole is surrounded by a material cloud,
which may be interpreted physically as forming an accretion disk, or acting as
an effective dark matter, as suggested in HT4 . Hence, black holes in Weyl
geometric theory contain scalar hair, which determines the strength of the
gravitational interaction. Due to the presence of the scalar field, an
effective gravitational constant is generated in the spacetime of the black
hole, which changes with the distance from the center of the black hole, even
after the spontaneous conformal symmetry breaking of the vacuum.

In analyzing the thermodynamic properties of the Weyl geometric black hole we
have first discussed the existence, and the borders for horizons of this black
hole. The equation giving the position of the horizons is given by Eq. (29), a
third order algebraic equation. For the existence of three real roots the
condition (30) must be satisfied. The positions and the nature of the event
horizon are strongly dependent on the solution parameter $\delta$, whose
variation in a relatively small range induces significant changes in the
nature and position of the horizons.

The extra terms in the gravitational field equations, generated by the
presence of the scalar field, can be interpreted as generating a fluid with
negative pressure. The total mass, expressed by Eq. (III.5), contains the
contribution of the ordinary (baryonic) mass $r_{g}/2$, plus the contribution
of the scalar field, which increases due to the presence of the terms linear,
quadratic and cubic in $r$. However, dark matter cannot extend beyond the zero
pressure region (if we define “matter” as having non-negative pressure), which
gives for the total mass of dark matter the finite expression (LABEL:eq:Mmax).
It is interesting to note that the total mass of the dark matter is dependent
on the baryonic mass.

We have also studied the thermodynamic characteristics of the Weyl geometric
black hole, and we have obtained the explicit expressions of the basic
thermodynamic quantities, as well as their dependence on the parameters of the
solution.

While the general expression of the Hawking temperature of the evaporating
Weyl geometric black hole remains similar to the ones for black hole in
general relativity, since both can be obtained from the metric tensor
components, the entropy has a new scalar field factor, in addition to the
standard area law.

By appropriately defining the thermodynamic volume corresponding to the
thermodynamic pressure generated due to the presence of the cosmological
constant term, the first law of black hole thermodynamic of Weyl geometry
black hole can be established. The heat capacity for small black holes is
negative, but smoothly changes to positive values for large black holes,
suggesting a continuous phase transition between unstable and stable black
holes in the presence of a surrounding, even when the spacetime is
asymptotically de Sitter.

The analyses of the Helmholtz and Gibbs free energy support the thermodynamic
picture obtained from the study of the heat capacity - small Weyl black holes
are unstable, and they become smoothly stable as they grow in sizes.

The luminosity and evaporation time of Weyl black hole show certain deviation
from the standard general relativistic black holes, especially for the large
mass black holes.

We would like to point out that our results on the thermodynamics of Weyl
geometric gravity black holes have been obtained for a specific, exact
solution of the field equations, under some particular assumptions adopted for
the behavior of the auxiliary scalar field, and for the Weyl vector.
Therefore, as compared to all the possible spectrum of the black hole
solutions in Weyl geometric gravity they may be considered as having a
qualitative relevance only. But, even within this restricted level of
generality, the exact solution of the field equations of Weyl geometric
gravity may provide an important insight into the complex physical and
thermodynamic behavior of the Weyl geometric black holes, and of the novel and
interesting astrophysical and physical features related to them.

As the analysis of the present work has shown, Weyl geometric gravity black
holes have more thermodynamic variability, resulting from their basic
properties, as compared with the Schwarzschild black holes of general
relativity. This can be easily seen from the study of the Hawking temperature,
entropy or luminosity of the Weyl geometric black holes. These novel
properties do follow from the presence of the vector and scalar degrees of
freedom, and they reflect the complex mathematical structure of the theory,
described mathematically by very complicated, and strongly nonlinear field
equations. Thus, the physical effects associated with the Weyl vector field
and the scalar field, two new degrees of freedom of the theory, could also
lead to a new understanding of the laws of black hole thermodynamics, as well
as to complex interplay between thermodynamics, gravity, and quantum
mechanics. The possible quantum implications of the existence of the Weyl
geometric black holes will be considered in a future study.

## Acknowledgments

M. F. A. R. S. is supported by the Second Century Fund (C2F), Chulalongkorn
University, Thailand. PB is supported in part by National Research Council of
Thailand (NRCT) and Chulalongkorn University under Grant N42A660500. This
research has received funding support from the NSRF via the Program Management
Unit for Human Resources & Institutional Development, Research and Innovation
[grant number B39G660025]. The work of T.H. is supported by a grant from the
Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-
III-P4-ID-PCE2020-2255 (PNCDI III).

##  Appendix A Basics of Weyl geometry and Weyl geometric gravity

We define in a general sense Weyl conformal geometry as the equivalence
classes of ($g_{\mu\nu},\omega_{\mu}$) of the metric and of the Weyl gauge
field ($\omega_{\mu}$), related by the Weyl gauge transformations

$\displaystyle\hat{g}_{\mu\nu}\\!=\\!\Sigma^{d}\,g_{\mu\nu},\sqrt{\hat{-g}}=\\!\Sigma^{2d}\sqrt{-g},\hat{\omega}_{\mu}\\!=\\!\omega_{\mu}-\frac{1}{\alpha}\,\partial_{\mu}\ln\Sigma,$

where by $d$ we have denoted the Weyl weight (charge) of $g_{\mu\nu}$, while
$\alpha$ is the Weyl gauge coupling. In our present approach we consider only
the case $d=1$, but our results can be generalized to arbitrary $d$ by
rescaling the coupling as $\alpha\rightarrow\alpha\,\times d$.

The Weyl gauge field is included in the Weyl connection $\tilde{\Gamma}$,
which is obtained as the solution of the system

$\displaystyle\tilde{\nabla}_{\lambda}g_{\mu\nu}=-d\,\alpha\,\omega_{\lambda}g_{\mu\nu}$
(87)

with $\tilde{\nabla}_{\mu}$ defined with the help of
$\tilde{\Gamma}_{\mu\nu}^{\lambda}$ according to

$\displaystyle\tilde{\nabla}_{\lambda}g_{\mu\nu}=\partial_{\lambda}g_{\mu\nu}-\tilde{\Gamma}^{\rho}_{\mu\lambda}g_{\rho\nu}-\tilde{\Gamma}^{\rho}_{\nu\lambda}\,g_{\rho\mu}.$
(88)

From Eq.(87) it follows that Weyl geometry is non-metric. Still the covariant
derivative can be written as
$(\tilde{\nabla}_{\lambda}+d\,\alpha\,\omega_{\lambda})\,g_{\mu\nu}=0$, a form
similar to the Riemannian case. Hence, in Weyl geometry one can use the
Riemannian expressions in which the covariant derivatives $\nabla_{\lambda}$,
acting on the geometric and physical quantities are replaced by their Weyl-
geometric counterparts, as, for example, in
$\partial_{\lambda}\rightarrow\partial_{\lambda}+\rm{weight}\times\alpha\times\omega_{\lambda}$,
where ’weight’ denotes the Weyl charge.

The expression of $\tilde{\Gamma}$ from Eq. (87) is obtained by using the
cyclic permutations of the indices, and combining the resulting equations.
Thus we find

$\displaystyle\tilde{\Gamma}_{\mu\nu}^{\lambda}=\Gamma_{\mu\nu}^{\lambda}+\alpha\,\frac{d}{2}\,\Big{[}\delta_{\mu}^{\lambda}\,\omega_{\nu}+\delta_{\nu}^{\lambda}\,\omega_{\mu}-g_{\mu\nu}\,\omega^{\lambda}\Big{]},$
(89)

where by $\Gamma_{\mu\nu}^{\lambda}$ we have denoted the Levi-Civita
connection of the Riemann geometry

$\displaystyle\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}\,g^{\lambda\rho}(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}).$
(90)

It is important to note that $\tilde{\Gamma}$ is invariant under the group of
conformal transformations (A). Taking the trace of Eq. (89), and denoting
$\Gamma_{\mu}\equiv\Gamma_{\mu\lambda}^{\lambda}$ and
$\tilde{\Gamma}_{\mu}\equiv\tilde{\Gamma}_{\mu\lambda}^{\lambda}$,
respectively, we obtain

$\displaystyle\tilde{\Gamma}_{\mu}=\Gamma_{\mu}+2d\,\alpha\,\omega_{\mu}.$
(91)

Hence, the Weyl gauge field can be interpreted as describing the deviation of
the Weyl connection from the Levi-Civita connection. One can obtain the scalar
and the tensor curvatures of the Weyl geometry, using expressions similar to
those in Riemannian geometry, but with $\tilde{\Gamma}$ substituting $\Gamma$.
Thus

$\displaystyle\tilde{R}^{\lambda}_{\mu\nu\sigma}$ $\displaystyle=$
$\displaystyle\partial_{\nu}\tilde{\Gamma}^{\lambda}_{\mu\sigma}-\partial_{\sigma}\tilde{\Gamma}^{\lambda}_{\mu\nu}+\tilde{\Gamma}^{\lambda}_{\nu\rho}\,\tilde{\Gamma}_{\mu\sigma}^{\rho}-\tilde{\Gamma}_{\sigma\rho}^{\lambda}\,\tilde{\Gamma}_{\mu\nu}^{\rho},$
$\displaystyle\tilde{R}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{R}^{\lambda}_{\mu\lambda\sigma},$
$\displaystyle\tilde{R}$ $\displaystyle=$ $\displaystyle
g^{\mu\sigma}\,\tilde{R}_{\mu\sigma}.$ (92)

After some simple calculations one obtains

$\displaystyle\tilde{R}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}+\frac{1}{2}\,(\alpha
d)(\nabla_{\mu}\omega_{\nu}-3\,\nabla_{\nu}\omega_{\mu}-g_{\mu\nu}\,\nabla_{\lambda}\omega^{\lambda})$
$\displaystyle+\frac{1}{2}\,(\alpha
d)^{2}\,(\omega_{\mu}\omega_{\nu}-g_{\mu\nu}\,\omega_{\lambda}\omega^{\lambda}),$
$\displaystyle\tilde{R}_{\mu\nu}$ $\displaystyle-$
$\displaystyle\tilde{R}_{\nu\mu}=2\,d\alpha\,F_{\mu\nu},$
$\displaystyle\tilde{R}$ $\displaystyle=$ $\displaystyle
R-3\,d\,\alpha\,\nabla_{\mu}\omega^{\mu}-\frac{3}{2}\,(d\,\alpha)^{2}\,\omega_{\mu}\,\omega^{\mu}.$
(93)

In the above equations the right hand side is defined in the Riemannian
geometry, and hence $\nabla_{\mu}$ is defined via the Levi-Civita connection
($\Gamma$).

An important property of Weyl geometry is that $\tilde{R}$ transforms
covariantly under the transformations (A)

$\displaystyle\hat{\tilde{R}}=(1/\Sigma^{d})\,\tilde{R},$ (94)

a result which follows from the transformation rule of $g^{\mu\sigma}$, and
from the property of invariance of $\tilde{R}_{\mu\nu}$ under the conformal
transformations, since $\tilde{\Gamma}$ is also invariant. Then, it follows
immediately that the term $\sqrt{g}\,\tilde{R}^{2}$ is also Weyl gauge
invariant.

An important geometrical quantity is the strength of the Weyl vector field
$\tilde{F}_{\mu\nu}$, defined according to

$\tilde{F}_{\mu\nu}=\nabla_{\mu}\omega_{\nu}-\nabla_{\nu}\omega_{\mu}.$ (95)

The simplest conformally invariant gravitational Lagrangian density is given
by

$L_{W}=\left(\frac{1}{4!\,\xi^{2}}\tilde{R}^{2}-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}\right),$
(96)

where $\xi<1$ is the parameter of the perturbative coupling.

The Lagrangian $L_{W}$ can be linearized with the help of the substitution

$\tilde{R}^{2}\rightarrow-2\phi^{2}\,\tilde{R}-\phi^{4},$ (97)

where $\phi$ is an auxiliary scalar field. Then the Lagrangian becomes

$L_{W}=\left(-\frac{\phi^{2}}{12\xi^{2}}\tilde{R}-\frac{\phi^{4}}{4!\,\xi^{2}}-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}\right)\sqrt{-\tilde{g}}.$
(98)

Substituting into Eq. (98) the expression of $\tilde{R}$,

$\tilde{R}=R-3\alpha\nabla_{\mu}\omega^{\mu}-\frac{3}{2}\alpha^{2}\omega_{\mu}\omega^{\mu},$
(99)

we obtain the action of the Weyl geometric gravitational theory as given by

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\int\Bigg{[}-\frac{\phi^{2}}{12\xi^{2}}\Big{(}R-3\alpha\nabla_{\mu}\omega^{\mu}-\frac{3}{2}\alpha^{2}\omega_{\mu}\omega^{\mu}\Big{)}$
(100)
$\displaystyle-\frac{\phi^{4}}{4!\,\xi^{2}}-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}\Bigg{]}\sqrt{-g}\,d^{4}x,$

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