††thanks<EMAIL_ADDRESS>
# Phase transition and entropic force of de Sitter black hole in massive
gravity

Yubo Ma1, Yang Zhang1, Lichun Zhang1, Liang Wu1, Ying Gao2, Shuo Cao3, Yu
Pan4∗ 1Institute of Theoretical Physics, Shanxi Datong University, Datong,
037009, China
2School of Mathematical and Statistical, Shanxi Datong University, Datong,
037009, China
3Department of Astronomy, Beijing Normal University, Beijing, 100875, China
4School of Science, Chongqing University of Posts and Telecommunications,
Chongqing, 400065, China

###### Abstract

It is well known that de Sitter(dS) black holes generally have a black hole
horizon and a cosmological horizon, both of which have Hawking radiation. But
the radiation temperature of the two horizons is generally different, so dS
black holes do not meet the requirements of thermal equilibrium stability,
which brings certain difficulties to the study of the thermodynamic
characteristics of black holes. In this paper, dS black hole is regarded as a
thermodynamic system, and the effective thermodynamic quantities of the system
are obtained. The influence of various state parameters on the effective
thermodynamic quantities in the massive gravity space-time is discussed. The
condition of the phase transition of the de Sitter black hole in massive
gravity space-time is given. We consider that the total entropy of the dS
black hole is the sum of the corresponding entropy of the two horizons plus an
extra term from the correlation of the two horizons. By comparing the entropic
force of interaction between black hole horizon and the cosmological horizon
with Lennard-Jones force between two particles, we find that the change rule
of entropic force between the two system is surprisingly the same. The
research will help us to explore the real reason of accelerating expansion of
the universe.

##  I Introduction

In recent years, the study of thermodynamic properties of black holes has
aroused great interest. Especially, after treating the cosmological constant
as the thermodynamic pressure, the first law of black hole thermodynamics in
extended phase space has been derived. Comparing the state parameters of black
holes with that of the Van der Waals equation, the critical phenomena of
various AdS space-time black holes have been studied. The influence of various
parameters on the phase transition is discussed. Some progress has been made
in the study of black hole phase transition (David Kubiznak et al., 2012;
Robie A. Hennigar et al., 2017; Aruna Rajagopal et al., 2014; Hendi et al.,
2017, 2017, 2015; Panahiyan et al., 2019; Rong-Gen Cai et al., 2013, 2016;
Jia-Lin Zhang et al., 2015, 2015; Hendi et al., 2016; Wei Xu et al., 2014; Wen
et al., 2014; Wilson G. Brenna et al., 2015; Rabin Banerjee et al., 2017,
2011, 2012; Meng-Sen Ma et al., 2017, 2017; Hendi et al., 2017; Dayyani et
al., 2017; De-Cheng Zou et al., 2017; Peng Cheng et al., 2016; Mozhgan Mir et
al., 2017; Zixu Zhao et al., 2014; Ren Zhao et al., 2013; Zeinab et al., 2018;
Run Zhou et al., 2019; Toledo et al., 2019). For de Sitter space-time, space-
time has black hole horizon and cosmological horizon. Both horizons have
Hawking radiation, and the radiation temperatures of the two horizons are
generally different. Therefore dS black holes do not meet the requirements of
thermal stability, which brings certain difficulties to the study of the
thermodynamic properties of black hole. In recent years, the study of the
thermodynamic properties of de Sitter space-time has attracted extensive
attention (Saoussen Mbarek et al., 2019; Fil Simovic et al., 2019; Brian P.
Dolan et al., 2013; Hendi et al., 2017; Sekiwa et al., 2006; David Kubiznak et
al., 2016; James McInerney et al., 2016; Miho Urano et al., 2009; Sourav
Bhattacharya et al., 2013; Cai et al., 2002, 2002; Yu-Bo Ma et al., 2018;
Azreg-Aiou et al., 2015; Zhang et al., 2016; Li-Chun Zhang et al., 2014; Hui-
Hua Zhao et al., 2014; Huaifan Li et al., 2017; Li-Chun Zhang et al., 2016;
Yu-Bo Ma et al., 2018). Because in the early period of inflation, our universe
is a quasi de Sitter space-time, and the constant term introduced in the study
of de Sitter space-time is the contribution of vacuum energy, which is also a
form of material energy. If the cosmological constant corresponds to dark
energy, our universe will evolve into a new de Sitter phase. In order to
construct the whole history of universe evolution, we should have a clear
understanding of the classical and quantum properties of the de Sitter space-
time.

Since the thermodynamic quantities corresponding to the two horizons in de
Sitter space-time are all functions of mass $M$, charge $Q$ and cosmological
constant $\Lambda$. There must be certain relation between the thermodynamic
quantities corresponding to the two horizons. Considering whether the
corresponding thermodynamic quantity in the de Sitter space-time has the
thermodynamic characteristics of AdS black hole after the correlation of black
hole horizon and universe horizon; What is the relationship between the
entropy of de Sitter space and the corresponding entropy of two horizons? What
is the relationship between the entropic force caused by the interaction of
two horizons and the ratio of the position of the two horizons? These problems
are very important to study the stability and evolution of de Sitter space-
time. Therefore, it is worth to establish a self-consistent relationship
between the thermodynamic quantities of de Sitter space-time. In
(Hinterbichler et al., 2012), the authors suggest a massive gravity which
overcomes the traditional problems and yield an avenue for addressing the
cosmological constant naturalness problem. Fierz and Pauli firstly provided
the possibility of a massive graviton (Fierz et al., 1939, 1939). Further, van
Dam and Veltman (van Dam et al., 1970) and Zakharov (Zakharov et al., 1970)
had found the linear theory which was coupled to a source, they found that
even when the mass of the graviton approaches to zero, the prediction of the
theory is different from that of the linear theory. Later, Boulware-Deser
ghost theories had been studied, which were nonlinear and ghostlike
instability massive gravity theories (Boulware et al., 1972, 1972). In the
absence of such instability, significant progress has been made in
establishing the theory of massive gravity (de Rham et al., 2010, 2011; B.
Eslam Panah et al., 2019, 2019, 2020; S. H. Hendi et al., 2018). The most
straightforward way to construct the massive gravity theories is to simply add
a mass term to the GR action, giving the graviton a mass in such a way that GR
is recovered as mass vanishes. Recently, a charged BTZ black holes has been
studied in (Hendi et al., 2016, 2017), which consider a massive gravity. The
massive BTZ black holes in the presence of Maxwell and Born-Infeld
electrodynamics in asymptotically (A)dS spacetimes was studied in (Hendi et
al., 2015). In (Majhi et al., 2017; Cai et al., 2015; Jianfei Xu et al., 2015;
Upadhyay et al., 1), thermodynamics and entropy of AdS black hole in massive
gravity were obtained, and the phase transition was discussed.

In this paper, on the basis of effective thermodynamic quantity of de Sitter
black hole in massive gravity (DSBHMG), the thermodynamic characteristics of
de Sitter space-time are discussed. The condition of the phase transition of
DSBHMG space-time is found, and the influence of each parameter on the
critical point is analyzed. It is noted that DSBHMG space-time has a second-
order phase transition similar to that AdS black hole when the parameters meet
certain conditions, except that the heat capacity at constant volume of DSBHMG
is not zero. By calculating the entropic force of the interaction between the
two horizons, the changing law of entropic force with the ratio of the
position of two horizons is given. By comparing the curve with that of
Lennard-Jones force (David C. Johnston et al., 2) between two particles, it is
found that the curve of entropic force changing with the position ratio of two
horizons is similar to the curve of Lennard-Jones force changing with the
distance between two particles, which has the same change rule. This discovery
provides a new way to study the internal cause of accelerating expansion of
the universe.

This paper is arranged as follows: for the continuity of this paper, in the
second part, we briefly introduce the corresponding thermodynamic quantities
of black hole horizon and cosmological horizon in DSBHMG and the effective
thermodynamic quantities of DSBHMG. In the third part, we discuss the critical
phenomena of DSBHMG. In the fourth part, we discuss the entropic force of
interaction between the two horizons, and obtain the entropic force and the
Lennard-Jones force between two particles has a similar law of change. The
fifth part of the paper discusses the conclusion. (We use the units
$G=h=k_{B}=1$).

##  II Thermodynamics of black holes in massive gravity

We consider $(3+1)$ -dimensional massive gravity with a Maxwell field (Cai et
al., 2015; Jianfei Xu et al., 2015; Upadhyay et al., 1; Hassan et al., 2011;
Allan Adams et al., 2015), the action is as follows

$S=\frac{1}{k^{2}}\int
d^{4}x\sqrt{g}(R-2\Lambda-\frac{1}{4}F^{2}+m^{2}\sum_{i}^{4}{c_{i}\mu_{i}}),$
(1)

where $\Lambda$ is the cosmological constant and $k=0$,$1$, or $-1$ ,
correspond to a Ricci flat, sphere, or hyperbolic horizon for the black hole,
respectively. Here $F_{\mu\nu}$ is the Maxwell field-strength tensor, $c_{i}$
are constants, and $\mu_{i}$ are symmetric polynomials of the eigenvalues of
the matrix $\sqrt{g^{\mu\nu}f_{\mu\nu}}$ where $f_{\mu\nu}$ is a fixed
symmetric tensor. The action admits a static black hole solution with the
space-time metric and reference metric as

$ds^{2}=-f(r)dt^{2}+f^{-1}dr^{2}+r^{2}h_{ij}dx^{i}dx^{j},\quad(i,j=1,2)$ (2)

where $h_{ij}dx^{i}dx^{j}$ is the line element for an Einstein space with
constant curvature, $f(r)$ is the metric function, which is written by (De-
Cheng Zou et al., 2017; Petarpa Boonserm et al., 2018).

$f(r)=k-\frac{\Lambda}{3}r^{2}-\frac{m_{0}}{r}+\frac{q^{2}}{4r^{2}}+\frac{c_{1}m^{2}}{2}r+m^{2}c_{2},$
(3)

The positions of black hole horizon and universe horizon satisfy the equation
$f(r_{+,c})=0$, thus, the mass $m_{0}$ can be expressed in terms of $r_{+,c}$
as

$M=\frac{m_{0}}{2}=\frac{\left(k+m^{2}c_{2}\right)r_{c}x(1+x)}{2\left(1+x+x^{2}\right)}+\frac{q^{2}(1+x)\left(1+x^{2}\right)}{8r_{c}x\left(1+x+x^{2}\right)}+\frac{r_{c}^{2}m^{2}c_{1}x^{2}}{4\left(1+x+x^{2}\right)},$
(4)

where $x=\frac{r_{+}}{r_{c}}$ and

$\frac{\Lambda}{3}r_{c}^{2}\left(1+x+x^{2}\right)=k-\frac{q^{2}}{4r_{c}^{2}x}+\frac{c_{1}}{2}m^{2}r_{c}(1+x)+m^{2}c_{2},$
(5)

The radiation temperature of black hole horizon and cosmological horizon is

$T_{+,c}=\pm\frac{f\left(r_{+,c}\right)}{4\pi}=\frac{1}{4\pi
r_{+,c}}\left(k-\Lambda
r_{+,c}^{2}-\frac{q^{2}}{4r_{+,c}^{2}}+m^{2}c_{1}r_{+,c}+m^{2}c_{2}\right).$
(6)

The thermodynamic quantities corresponding to the two horizons satisfy the
first law of thermodynamics

$dM=T_{+,c}dS_{+,c}+V_{+,c}dP+\mu_{+,c}dQ,$ (7)

where

$V_{+,c}=\frac{v_{2}}{3}r_{+,c}^{3},\quad P=-\frac{\Lambda}{8\pi}$ (8)

when the radiation temperature $T_{+}$ of the black hole horizon is equal to
that of the cosmological horizon $T_{c}$, the charge $Q$ and $\Lambda$ of the
system satisfies the equation

$\frac{1}{r_{+}}\left(k-\Lambda
r_{+}^{2}-\frac{q^{2}}{4r_{+}^{2}}+m^{2}c_{1}r_{+}+m^{2}c_{2}\right)=-\frac{1}{r_{c}}\left(k-\Lambda
r_{c}^{2}-\frac{q^{2}}{4r_{c}^{2}}+m^{2}c_{1}r_{c}+m^{2}c_{2}\right),$ (9)

we can get the radiation temperature $T$ when $T_{+}=T_{c}$.

$T=T_{+}=T_{c}=\frac{(1-x)}{2\pi
r_{c}(1+x)^{2}}\left[k+\frac{m^{2}c_{1}r_{c}\left(1+4x+x^{2}\right)}{4(1+x)}+m^{2}c_{2}\right].$
(10)

Recently, through the study of the thermodynamic characteristics of dS space-
time, one can get the thermodynamic volume of dS space-time is (Brian P. Dolan
et al., 2013; Li-Chun Zhang et al., 2016)

$V=\frac{4\pi}{3}\left(r_{c}^{3}-r_{+}^{3}\right)=\frac{4\pi}{3}r_{c}^{3}\left(1-x^{3}\right),$
(11)

if the energy $M$, charge $Q$ and volume $V$ in space-time are taken as the
state parameters of a thermodynamic system, the first law of thermodynamics
should be satisfied

$dM=T_{eff}dS-P_{eff}dV+\phi_{eff}dQ,$ (12)

Considering the interaction between the black hole horizon and the
cosmological horizon, we set the entropy of the system (Huaifan Li et al.,
2017; Li-Chun Zhang et al., 2016; Yu-Bo Ma et al., 2018)

$S=\pi r_{c}^{2}\left[1+x^{2}+f(x)\right],$ (13)

here the undefined function $f(x)$ represents the extra contribution from the
correlations of the two horizons. When the radiation temperature of black hole
horizon and cosmological horizon is equal, the effective temperature of space-
time should also be equal to the radiation temperature of two horizons

$T_{eff}=T=T_{c}=T_{+},$ (14)

From equations (12), (13) and (14), the effective temperature $T_{eff}$,
pressure $P_{eff}$ and potential $\phi_{eff}$ of the system are obtained

$T_{eff}=\frac{B(x,q)\left(1-x^{3}\right)}{4\pi
r_{c}x\left(1+x^{4}\right)},\quad
P_{eff}=\frac{D(x,q)\left(1-x^{3}\right)}{16\pi
r_{c}^{2}x\left(1+x^{4}\right)},\quad\phi_{eff}=\left(\frac{\partial
M}{\partial
Q}\right)_{S,V}=\frac{4\pi^{2}Q(1+x)\left(1+x^{2}\right)}{v_{2}^{2}r_{c}x\left(1+x+x^{2}\right)}.$
(15)

where

$\displaystyle B(x,q)=$
$\displaystyle\left(k+m^{2}c_{2}\right)\frac{\left(1+x-2x^{2}+x^{3}+x^{4}\right)}{\left(1+x+x^{2}\right)}$
(16)
$\displaystyle-\frac{q^{2}}{4r_{c}^{2}}\frac{\left(1+x+x^{2}-2x^{3}+x^{4}+x^{5}+x^{6}\right)}{x^{2}\left(1+x+x^{2}\right)}+r_{c}m^{2}c_{1}x\frac{(2+x)(1-x)+2x^{3}}{2\left(1+x+x^{2}\right)},$

$\displaystyle D(x,q)=$
$\displaystyle\frac{\left(k+m^{2}c_{2}\right)}{\left(1+x+x^{2}\right)^{2}}\left\\{(1+2x)\left[1+x^{2}+f(x)\right]-x(1+x)\left(1+x+x^{2}\right)[2x+f^{\prime}(x)]\right\\}$
(17)
$\displaystyle-\frac{q^{2}}{4r_{c}^{2}x^{2}\left(1+x+x^{2}\right)^{2}}\left\\{2\left(1+2x+3x^{2}\right)\left[1+x^{2}+f(x)\right]-x(1+x)\left(1+x^{2}\right)\left(1+x+x^{2}\right)[2x+f(x)]\right\\}$
$\displaystyle+\frac{r_{c}m^{2}c_{1}x}{\left(1+x+x^{2}\right)^{2}}\left\\{2(2+x)\left[1+x^{2}+f(x)\right]-x\left(1+x+x^{2}\right)\left[2x+f^{\prime}(x)\right]\right\\}.$

Considering the initial conditions $f(0)=0$, we can get

$f(x)=-\frac{2\left(4-5x^{3}-x^{5}\right)}{5\left(1-x^{3}\right)}+\frac{8}{5}\left(1-x^{3}\right)^{2/3}$
(18)

we set the initial parameters to $m=2.12$, $c_{1}=2$,$c_{2}=3.18$,$q=1.7$,
$r_{c}=1$,$k=1$, and then take different values for $k$, $q$, $m$, $c_{1}$ and
$c_{2}$ respectively, to get the curve $P_{eff}-x$ and $T_{eff}-x$ curve (take
$r_{c}=1$) as shown in FIG.1 and FIG.2

Figure 1: $P_{eff}-x$ diagrams when the parameters change respectively.

Figure 2: $T_{eff}-x$ diagrams when the parameters change respectively.

##  III Critical phenomena

Heat capacity of the system at constant volume:

$\displaystyle C_{V}$ $\displaystyle=T_{eff}\left(\frac{\partial S}{\partial
T_{eff}}\right)_{V}=T_{eff}\frac{\left(\frac{\partial S}{\partial
r_{c}}\right)_{x}\left(\frac{\partial V}{\partial
x}\right)_{r_{c}}-\left(\frac{\partial S}{\partial
x}\right)_{r_{c}}\left(\frac{\partial V}{\partial
r_{c}}\right)_{x}}{\left(\frac{\partial V}{\partial
x}\right)_{r_{c}}\left(\frac{\partial
T_{eff}}{\partial_{r_{c}}}\right)_{x}-\left(\frac{\partial V}{\partial
r_{c}}\right)_{x}\left(\frac{\partial T_{eff}}{\partial x}\right)_{r_{c}}}$
(19) $\displaystyle=\frac{2\pi
r_{c}^{2}B(x,q)x\left(1+x^{4}\right)}{\left(1-x^{3}\right)\left[B^{\prime}(x,q)(1-x^{3})x^{2}\bar{B}(x,q)-\frac{B(x,q)(1+2x^{3}+5x^{4}-2x^{7})}{x\left(1+x^{4}\right)}\right]}$

where

$\bar{B}(x,q)=\left(k+m^{2}c_{2}\right)\frac{\left(1+x-2x^{2}+x^{3}+x^{4}\right)}{\left(1+x+x^{2}\right)}-\frac{3}{4}\mu_{c}^{2}\frac{\left(1+x+x^{2}-2x^{3}+x^{4}+x^{5}+x^{6}\right)}{x^{2}\left(1+x+x^{2}\right)}$
(20)

$B^{\prime}(x,q)=\frac{dB(x,q)}{dx}$, and $B(x,q)$ is presented by Eq.(16).

Figure 3:  $C_{V}-x$ diagrams when the parameters change respectively and the
initial parameters $m=2.12$, $c_{1}=2$,$c_{2}=3.18$,$q=1.7$, $r_{c}=1$,$k=1$.

From FIG.3, we know that the heat capacity $C_{V}$ of the system in DSBHMG is
not zero, which is different from the result for AdS black holes where
$C_{V}=0$ .

The heat capacity at constant pressure is

$C_{P_{eff}}=T_{eff}\left(\frac{\partial S}{\partial
T_{eff}}\right)_{P_{eff}}=T_{eff}\frac{\left(\frac{\partial S}{\partial
r_{c}}\right)_{x}\left(\frac{\partial P_{eff}}{\partial
x}\right)_{r_{c}}-\left(\frac{\partial S}{\partial
x}\right)_{r_{c}}\left(\frac{\partial P_{eff}}{\partial
r_{c}}\right)_{x}}{\left(\frac{\partial T_{eff}}{\partial
r_{c}}\right)_{x}-\left(\frac{\partial P_{eff}}{\partial
r_{c}}\right)_{x}\left(\frac{\partial T_{eff}}{\partial
x}\right)_{r_{c}}}=\frac{2\pi r_{c}^{2}B(x,q)E(x,q)}{F(x,q)}$ (21)

Here

$E(x,q)=\left[1+x^{2}+f(x)\right]\left[D^{\prime}(x,q)\left(1-x^{3}\right)-\frac{D(x,q)\left(1+2x^{3}+5x^{4}-2x^{7}\right)}{x\left(1+x^{4}\right)}\right]+\left[2x+f^{\prime}(x)\right]\frac{\bar{D}(x,q)\left(1-x^{3}\right)}{2x},$
(22)

$\displaystyle\bar{D}(x,q)=$
$\displaystyle\frac{2\left(k+m^{2}c_{2}\right)}{\left(1+x+x^{2}\right)^{2}}\left\\{(1+2x)\left[1+x^{2}+f(x)\right]-x(1+x)\left(1+x+x^{2}\right)\left[2x+f^{\prime}(x)\right]\right\\}$
(23)
$\displaystyle-\frac{\mu_{c}^{2}}{x^{2}\left(1+x+x^{2}\right)^{2}}\left\\{2\left(1+2x+3x^{2}\right)\left[1+x^{2}+f(x)\right]-x(1+x)\left(1+x^{2}\right)\left(1+x+x^{2}\right)\left[2x+f^{\prime}(x)\right]\right\\}$
$\displaystyle{+\frac{r_{c}m^{2}c_{1}x}{\left(1+x+x^{2}\right)^{2}}\left\\{2(2+x)\left[1+x^{2}+f(x)\right]-x\left(1+x+x^{2}\right)\left[2x+f^{\prime}(x)\right]\right\\}},$

$\displaystyle F(x,q)=$
$\displaystyle\bar{B}(x,q)\left[\frac{D(x,q)\left(1+2x^{3}+5x^{4}-2x^{7}\right)}{x\left(1+x^{4}\right)}-D^{\prime}(x,q)\left(1-x^{3}\right)\right]$
(24)
$\displaystyle+\bar{D}(x,q)\left[B^{\prime}(x,q)\left(1-x^{3}\right)-\frac{B(x,q)\left(1+2x^{3}+5x^{4}-2x^{7}\right)}{x\left(1+x^{4}\right)}\right].$

$D^{\prime}(x,q)=\frac{xD(x,q)}{dx}$, and $D(x,q)$ is given by Eq.(17).From
Eq.(21), the curves are depicted in FIG.4.

Figure 4: $C_{P_{eff}}-x$ diagrams when the parameters change respectively and
the initial parameters $m=2.12$, $c_{1}=2$,$c_{2}=3.18$,$q=1.7$,
$r_{c}=1$,$k=1$.

The expansion coefficient is

$\displaystyle\alpha$ $\displaystyle=\frac{1}{V}\left(\frac{\partial
V}{\partial T_{eff}}\right)_{P_{eff}}$ (25)
$\displaystyle=\frac{1}{V}\frac{\left(\frac{\partial V}{\partial
r_{c}}\right)_{x}\left(\frac{\partial P_{eff}}{\partial
x}\right)_{r_{c}}-\left(\frac{\partial V}{\partial
x}\right)_{r_{c}}\left(\frac{\partial P_{eff}}{\partial
r_{c}}\right)_{x}}{\left(\frac{\partial P_{eff}}{\partial
x}\right)_{r_{c}}\left(\frac{\partial T_{eff}}{\partial
r_{c}}\right)_{x}-\left(\frac{\partial P_{eff}}{\partial
r_{c}}\right)_{x}\left(\frac{\partial T_{eff}}{\partial x}\right)_{r_{c}}}$
$\displaystyle=\frac{12r_{c}x\left(1+x^{4}\right)}{\left(1-x^{3}\right)F(x,q)}\left[D^{\prime}(x,q)\left(1-x^{3}\right)-\frac{D(x,q)\left(1+2x^{3}+5x^{4}-2x^{7}\right)}{x\left(1+x^{4}\right)}-\bar{D}(x,q)x^{2}\right]$

The $\alpha-x$ curve is shown in FIG.5.

Figure 5: $\alpha-x$ diagrams when the parameters change respectively and the
initial parameters $m=2.12$, $c_{1}=2$,$c_{2}=3.18$,$q=1.7$, $r_{c}=1$,$k=1$.

The isothermal compressibility is

$\displaystyle\kappa_{T_{eff}}$
$\displaystyle=-\frac{1}{V}\left(\frac{\partial V}{\partial
P_{eff}}\right)_{T_{eff}}$ (26)
$\displaystyle=\frac{1}{V}\frac{\left(\frac{\partial V}{\partial
r_{c}}\right)_{x}\left(\frac{\partial T_{eff}}{\partial
x}\right)_{r_{c}}-\left(\frac{\partial V}{\partial
x}\right)_{r_{c}}\left(\frac{\partial T_{eff}}{\partial
r_{c}}\right)_{x}}{\left(\frac{\partial P_{eff}}{\partial
x}\right)_{r_{c}}\left(\frac{\partial T_{eff}}{\partial
r_{c}}\right)_{x}-\left(\frac{\partial P_{eff}}{\partial
r_{c}}\right)_{x}\left(\frac{\partial T_{eff}}{\partial x}\right)_{r_{c}}}$
$\displaystyle=\frac{48\pi
r_{c}^{2}x\left(1+x^{4}\right)}{\left(1-x^{3}\right)F(x,q)}\left[B^{\prime}(x,q)\left(1-x^{3}\right)-\frac{B(x,q)\left(1+2x^{3}+5x^{4}-2x^{7}\right)}{x\left(1+x^{4}\right)}-\bar{B}(x,q)x^{2}\right].$

According to Eq.(26), we plot $\kappa_{T_{eff}}-x$ curve in FIG.6.

Figure 6: $\kappa_{T_{eff}}-x$ diagrams when the parameters change
respectively and the initial parameters $m=2.12$,
$c_{1}=2$,$c_{2}=3.18$,$q=1.7$, $r_{c}=1$,$k=1$.

It can be seen from the curve $C_{P_{eff}}-x$, $\alpha-x$,
$\kappa_{T_{eff}}-x$ , $C_{P_{eff}}$, $\alpha$ and $\kappa_{T_{eff}}$ are
divergence at $x=x_{c}$. The entropy and volume of the system are continuous.
According to Ehrenfest’s classification of phase transition, $x_{c}$ is the
second-order phase transition point of the system.

Gibbs function is given by

$\displaystyle G$ $\displaystyle=M-T_{eff}S$ (27)
$\displaystyle=\frac{\left(k+m^{2}c_{2}\right)r_{c}x(1+x)}{2\left(1+x+x^{2}\right)}+\frac{q^{2}(1+x)\left(1+x^{2}\right)}{8r_{c}x\left(1+x+x^{2}\right)}+\frac{r_{c}^{2}m^{2}c_{1}x^{2}}{4\left(1+x+x^{2}\right)}-\frac{r_{c}B(x,q)\left(1-x^{3}\right)}{4x\left(1+x^{4}\right)}\left[1+x^{2}+f(x)\right].$

the $G-T_{eff}$ curve is following

Figure 7: $G-T_{eff}$ diagrams when the parameters change respectively and the
initial parameters $m=2.12$, $c_{1}=2$,$c_{2}=3.18$,$q=1.7$, $r_{c}=1$,$k=1$.

From Figures $C_{P_{eff}}-x$, $\alpha-x$, $\kappa_{T_{eff}}$ and $G-T_{eff}$,
DSBHMG satisfies Ehrenfest’s second-order phase transition conditions for
phase transition classification at $x=x_{c}$, so secondary phase transition
occurs in DSBHMG space-time at $x=x_{c}$.

From the expressions (11) and (13) of the total entropy and volume of space-
time, it is known that in the range $0<x<1$ the entropy and volume of DSBHMG
are continuous. According to the curve $G-T_{eff}$, DSBHMG does not have a
first-order phase transition, which is different from the first-order phase
transition of AdS black holes.

##  IV The entropic force of interaction between two horizons

For a general thermodynamic system, when two systems are independent of each
other, the corresponding entropy of system A and system B are $S_{A}$ and
$S_{B}$. When two systems have interactions, the entropy of the total system
is

$S_{total}=S_{A}+S_{B}+S_{AB}$ (28)

In the formula, the $S_{AB}$ is an extra entropy caused by the interaction of
two systems. From formula (13), we know that the total entropy of the
effective thermodynamic system in DSBHMG is divided into two parts, one is the
entropy corresponding to the two horizons, and the other term is the increase
of the system entropy after considering the interaction of the two horizons as
a thermodynamic system, so the entropy is

$S_{f}=S_{AB}=\pi r_{c}^{2}f(x)=\pi
r_{c}^{2}\left[\frac{8}{5}\left(1-x^{3}\right)^{2/3}-\frac{2\left(4-5x^{3}-x^{5}\right)}{5\left(1-x^{3}\right)}\right]$
(29)

Entropic force in thermodynamic system is (Erik P. Verlinde et al., 2011;
Plastino et al., 2018, 2018; Li-Chun Zhang et al., 2019; Panos et al., 2019;
Dmitri E. Kharzeev et al., 2014)

$F=-T\frac{\partial S}{\partial r}$ (30)

which $T$ is the temperature of the system, $r$ is the location of the
boundary surface.

The entropic force in DSBHMG is

$F=T_{eff}\left(\frac{\partial S_{f}}{\partial r}\right)_{T_{eff}}$ (31)

Where $T_{eff}$ is the equivalent temperature of the system,
$r=r_{c}-r_{+}=r_{c}(1-x)$, from Eq.(29), we can obtain Eq.(32),

$\displaystyle F(x)$ $\displaystyle=T_{eff}\frac{\left(\frac{\partial
S_{f}}{\partial r_{c}}\right)_{x}\left(\frac{\partial T_{eff}}{\partial
x}\right)_{r_{c}}-\left(\frac{\partial S_{f}}{\partial
x}\right)_{r_{c}}\left(\frac{\partial T_{eff}}{\partial
r_{c}}\right)}{(1-x)\left(\frac{\partial T_{eff}}{\partial
x}\right)_{r_{c}}+r_{c}\left(\frac{\partial T_{eff}}{\partial
r_{c}}\right)_{x}}$ (32)
$\displaystyle=\frac{B(x,q)\left(1-x^{3}\right)}{4x\left(1+x^{4}\right)}\frac{\left[2B^{\prime}(x,q)\left(1-x^{3}\right)f(x)-\frac{2B(x,q)\left(1+2x^{3}+5x^{4}-2x^{7}\right)}{x\left(1+x^{4}\right)}f(x)+\bar{B}(x,q)\left(1-x^{3}\right)f^{\prime}(x)\right]}{\left[B^{\prime}(x,q)\left(1-x^{3}\right)(1-x)-\frac{B(x,q)\left(1+2x^{3}+5x^{4}-2x^{7}\right)}{x\left(1+x^{4}\right)}(1-x)-\bar{B}(x,q)\left(1-x^{3}\right)\right]}.$

The $F(x)-x$ curve is shown in FIG.8.

Figure 8: $F-x$ diagrams when the parameters change respectively and the
initial parameters $m=2.12$, $c_{1}=2$,$c_{2}=3.18$,$q=1.7$, $r_{c}=1$,$k=1$.

For different parameters, the intersection position of the $F(x)-x$ curve and
the axis is $x_{0}$. The entropic force between the two horizons is
positive($F>0$) when $1>x>x_{0}$, which means that the two horizons are
mutually exclusive; and the entropic force between the two horizons is
negative ($F<0$) when $x_{0}>x>0$, indicating that the two horizons are
mutually attractive. Therefore, the position ratio of the two horizons $x$ is
different, and the entropic force between them is different. The two horizons
are accelerated to separate under the action of entropic force, and the
cosmological horizon expands faster than the black hole horizon when
$1>x>x_{0}$, and they are decelerating to be separated by the entropic force,
and the cosmological horizon undergoes a decelerating expansion relative to
the black hole horizon when $x_{0}>x>0$. So we get that the different values
$S_{1}-s_{2}$ between the area $S_{1}$ of the curve and the axis in interval
$1>x>x_{0}$ and the area $S_{2}$ of the curve and the axis in interval
$x_{0}>x>0$ determines whether the cosmological horizon accelerates or
oscillates with respect to the black hole horizon. The cosmological horizon
changes from accelerating expansion to decelerating expansion when
$S_{1}-S_{2}>0$. When it is equal to or less than zero, the cosmological
horizon is changing from accelerating expansion to decelerating expansion, and
then changing from accelerating contraction to decelerating contraction. One
cycle ends and the next cycle begins. The cosmological horizon is oscillating
relative to the horizon of the black hole.

Interactions between neutral molecules or atoms with a center of mass
separation $r$ are often approximated by the so-called Lennard-Jones potential
energy $\phi_{L,J}$ which is given by (David C. Johnston et al., 2; Plastino
et al., 2018, 2018; Yan-Gang Miao et al., 2018)

$\phi_{L,J}=4\phi_{\min}\left[\left(\frac{r_{0}}{r}\right)^{12}-\left(\frac{r_{0}}{r}\right)^{6}\right],$
(33)

where the first term is a short-range repulsive interaction and the second
term is a longer-range attractive interaction. A plot of
$\frac{\phi_{L,J}}{\phi_{min}}$ versus $\frac{r}{r_{0}}$ is shown in FIG.3.
The value $r=r_{0}$ corresponds to $\phi_{L,J}=0$, and the minimum value of
$\phi_{L,J}$ is $\frac{\phi_{L,J}}{\phi_{min}}=-1$ at

$\frac{r_{min}}{r_{0}}=2^{\frac{1}{6}}\approx 1.122$ (34)

By the definition of potential energy, the force between a molecule and a
neighbor in the radial direction from the first molecule is
$F_{r}=\frac{-d\phi_{L,J}}{dr}$ , which is positive (repulsive) for
$r<r_{min}$ and negative (attractive) for $r>r_{min}$. When the center of the
first particle coincides with the coordinate center, let the radius of the
particle be $\frac{r_{0}}{2}$. When the center of the second particle is at
$r$, and the boundary of the second particle is at $r_{2}$, then
$r=r_{2}+\frac{r_{0}}{2}$. Let’s take $y=\frac{r_{0}}{2r_{2}}$, $0<y\leq 1$.
Eq.(33) can be expressed as

$\phi_{L,J}(y)=4\phi_{\min}\left[\left(\frac{r_{0}}{r}\right)^{12}-\left(\frac{r_{0}}{r}\right)^{6}\right]=4\phi_{min}2^{6}\left[2^{6}\left(\frac{y}{1+y}\right)^{12}-\left(\frac{y}{1+y}\right)^{6}\right].$
(35)

The interaction between the two particles is

$F(y)=-\frac{d\phi_{L,J}}{dr}=4\phi_{\min}\frac{6}{r}\left[2\left(\frac{r_{0}}{r}\right)^{12}-\left(\frac{r_{0}}{r}\right)^{6}\right]=\frac{3\phi_{min}2^{10}}{r_{0}}\left[2^{7}\left(\frac{y}{1+y}\right)^{13}-\left(\frac{y}{1+y}\right)^{7}\right].$
(36)

Figure 9: $\phi_{L,J}(y)-y$ diagram when $y$ from $0$ to $1$.

Figure 10: $F(y)-y$ diagram when $y$ from $0$ to $1$.

From FIG.8 and FIG.10, we can see the relation curve of entropic force between
two horizons in DSBHMG with the ratio of the position of the two horizons is
very similar to that of the Lenard-Jones interaction force with the position
ratio of the two particle boundaries, which has the same change rule.

In order to show the similarity of the two curves more clearly, we show the
two curves in the same coordinate as is shown in FIG.11.

Figure 11: $F(x)-x$ and $F(y)-y$ diagrams when abscissa $x$ and $y$ from $0$
to $1$.

From FIG.11, we know that the trend of change in curve $F(x)-x$ and $F(y)-y$
is exactly the same, only the intersection position of the two curves and the
abscissa is slightly different. It shows that the entropic force between the
horizon of black hole and the cosmological horizon is very similar to the
Lenard Jones interaction force between two particles in RN dssq space-time,
which provides a new way to explore the interaction force between particles in
black hole.

##  V Conclusion and discussion

When the black hole horizon and the cosmological horizon are viewed separately
as independent thermodynamic systems without considering the correlation
between them, the space-time does not meet the requirements of thermodynamic
equilibrium stability because the radiation temperature of the two horizons
are different. Thus the space-time is unstable. When the correlation of two
horizons is considered, the effective temperature $T_{eff}$, effective
pressure $P_{eff}$ and effective potential $\phi_{eff}$ reflecting the thermal
properties of DSBHMG are given by Eq.(15). From curve $C_{p_{eff}}-x$,
$\alpha-x$ and $\kappa_{T_{eff}}-x$, when the ratio of the positions of the
two horizons in DSBHMG is $x=x_{c}$, phase transition occurs in the system.
According to Ehrenfest’s classification of phase transition, the phase
transition of the system at this point is of second-order, which is similar to
that of AdS black hole. In the study of the thermodynamic properties of
spherically symmetric AdS black holes, people study the critical phenomena of
AdS black holes by comparing the thermodynamic quantities of AdS black holes
with the thermodynamic quantities of Van der Waals equation and obtain the
critical exponents and the heat capacity of AdS black holes, which provides a
basis for further study of the thermal effect of black holes and experimental
observation. However, it is difficult to accept the fact that the heat
capacity at constant volume of AdS black holes is zero. It is shown that the
effective thermodynamic quantity of DSBHMG also has the phase transition
characteristics similar to that of van der Waals system. From formula (19),
the heat capacity of DSBHMG at constant volume is not zero, which is
consistent with the result that this heat capacity of van der Waals system is
not zero. From the curve of $C_{P_{eff}}-x$, we know that the stability of
DSBHMG depends on the value of $x$ . When $x<x_{c}$, space-time meets the
requirements of thermodynamic stability. When $1>x>x_{c}$, space-time does not
meet the requirements of thermodynamic equilibrium stability, and space-time
is unstable. Therefore, there is no DSBHMG steady-state space-time satisfying
the position ratio $1>x>x_{c}$ of two horizons in the universe, which provides
a theoretical basis to find black holes. The influence of parameters in DSBHMG
on space-time stability is shown in $C_{P_{eff}}-x$ curve seeing FIG.4.

In the framework of general relativity, the entropic force of interaction
between the horizon of black hole and the cosmological horizon given by theory
is very similar to the Lennard-Jones force between two particles confirmed by
experiments. Since the space-time metric is derived from relativity, the
thermodynamics of space-time is based on general relativity, and the thermal
effect of black hole is obtained from quantum mechanics. The physical
quantities obtained meet the first law of thermodynamics. Therefore, the
conclusion we have given reveals the internal relationship among general
relativity, quantum mechanics and thermodynamics, and provide a new way for us
to study the interaction between particles in black holes and the micro state
of particles in black holes, and the relationship between Lennard-Jones
potential and micro state of particles in ordinary thermodynamic systems.

Through the analysis of the fourth part, it is known that under the action of
entropic force, when the ratio of the positions of the two horizons satisfies
$x_{0}<x<1$, the entropic force between the two horizons is mutually
exclusive, and the two horizons in this region are accelerating expansion
under the action of entropic force; when $0<x<x_{0}$, the entropic force
between the two horizons is mutually attractive, and the two horizons are
decelerating expansion; until $x\rightarrow{0}$, the entropic force between
the two horizons is zero, and the two horizons tend to be relatively static.
Therefore, in the $0<x<1$ range, the cosmological horizon accelerates to
expand relative to the horizon of the black hole under the action of the
entropic force. Our universe is a quasi de Sitter space-time, so our
discussion provides a new way to explore the internal cause of the expansion
of the universe.

## Acknowledgments

We thank Prof. Z. H. Zhu for useful discussions.

This work was supported by the Scientific and Technological Innovation
Programs of Higher Education Institutions of Shanxi Province, China (Grant No.
2020L0471, No. 2020L0472, No. 2019L0743) and the National Natural Science
Foundation of China (Grant Nos. 12075143, 11847123, 11475108, 11705106,
11705107, 11605107).

## References

  * David Kubiznak et al. (2012) D. Kubiznak and R. B. Mann, _P-V criticality of charged AdS black holes, JHEP_ 07 (2012) 033.
  * Robie A. Hennigar et al. (2017) R. A. Hennigar, E. Tjoa and R. B. Mann, _Thermodynamics of hairy black holes in Lovelock gravity, JHEP_ 02 (2017) 70.
  * Aruna Rajagopal et al. (2014) A. Rajagopal, D. Kubiznak and R. B. Mann, _Van der Waals black hole, Phys. Lett. B_ 737 (2014) 277.
  * Hendi et al. (2017) S. H. Hendi, R. B. Mann, S. Panahiyan and B. Eslam Panah, _van der Waals like behaviour of topological AdS black holes in massive gravity, Phys. Rev. D_ 95 (2017) 021501(R).
  * Hendi et al. (2017) S. H. Hendi, G. Q. Li, J. X. Mo, S. Panahiyan and B. Eslam Panah, _New perspective for black hole thermodynamics in Gauss-Bonnet-Born-Infeld massive gravity, Eur. Phys. J. C_ 76 (2016) 571.
  * Hendi et al. (2015) S. H. Hendi, B. Eslam Panah and S. Panahiyan, _Einstein-Born-Infeld-massive gravity: AdS black hole solutions and their thermodynamical properties, JHEP_ 11 (2015) 157.
  * Panahiyan et al. (2019) S. Panahiyan, S. H. hendi and N. Riazi, _$AdS_{4}$ dyonic black holes in gravity’s rainbow, Nucl. Phys. B_ 938 (2019) 388.
  * Rong-Gen Cai et al. (2013) R. G. Cai, L. M. Cao, L. Li and R. Q. Yang, _P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space, JHEP_ 09(2013) 005.
  * Rong-Gen Cai et al. (2016) R. G. Cai, S. M. Ruan, S. J. Wang, R. Q. Yang and R. H. Peng, _Complexity Growth for AdS Black Holes, JHEP_ 09 (2016) 161.
  * Jia-Lin Zhang et al. (2015) J. L. Zhang, R. G. Cai and H. W. Yu, _Phase transition and Thermodynamical geometry of Reissner-Nordström-AdS Black Holes in Extended Phase Space, Phys.Rev. D_ 91 (2015) 044028.
  * Jia-Lin Zhang et al. (2015) J. L. Zhang, R. G. Cai and H. W. Yu, _Phase transition and thermodynamical geometry for Schwarzschild AdS black hole in $AdS_{5}\times S^{5}$ spacetime, JHEP_ 02 (2015) 143.
  * Hendi et al. (2016) S. H. Hendi, B. Eslam Panah and S. Panahiyan, _Massive charged BTZ black holes in asymptotically (a)dS spacetimes, JHEP_ 05 (2016) 029.
  * Wei Xu et al. (2014) W. Xu, H. Xu and L. Zhao, _Gauss-Bonnet coupling constant as a free thermodynamical variable and the associated criticality, Eur. Phys. J. C_ 74 (2014) 2970.
  * Wen et al. (2014) S. W. Wen and Y. X. Liu, _Triple points and phase diagrams in the extended phase space of charged Gauss-Bonnet black holes in AdS space, Phys. Rev. D_ 90 (2014) 044057.
  * Wilson G. Brenna et al. (2015) W. G. Brenna, R. B. Mann and M. Park, _Mass and Thermodynamic Volume in Lifshitz Spacetimes, Phys. Rev. D_ 92 (2015) 044015.
  * Rabin Banerjee et al. (2017) R. Banerjee, B. R. Majhi and S. Samanta, _Thermogeometric phase transition in a unified framework, Phys. Lett. B_ 767 (2017) 25.
  * Rabin Banerjee et al. (2011) R. Banerjee and D. Roychowdhury, _Thermodynamics of phase transition in higher dimensional AdS black holes, JHEP_ 11 (2011) 004.
  * Rabin Banerjee et al. (2012) R. Banerjee and D. Roychowdhury, _Critical behavior of Born Infeld AdS black holes in higher dimensions, Phys. Rev. D_ 85 (2012) 104043.
  * Meng-Sen Ma et al. (2017) M. S. Ma, R. Zhao and Y. S. Liu, _Phase transition and thermodynamic stability of topological black holes in Ho $\check{r}$ava-Lifshitz gravity, Class. Quant. Grav._ 34 (2017) 165009.
  * Meng-Sen Ma et al. (2017) M. S. Ma and R. H. Wang, _Peculiar P-V criticality of topological Ho $\check{r}$ava-Lifshitz black holes, Phys. Rev. D_ 96 (2017) 024052.
  * Hendi et al. (2017) S. H. Hendi, B. Eslam Panah, S. Panahiyan and M. S. Talezadeh, _Geometrical thermodynamics and P-V criticality of the black holes with power-law Maxwell field, Eur. Phys. J. C_ 77 (2017) 133.
  * Dayyani et al. (2017) Z. Dayyani, A. Sheykhi and M. H. Dehghani, _Counterterm method in Einstein dilaton gravity and the critical behavior of dilaton black hole with a power-Maxwell field, Phys. Rev. D_ 95 (2017) 84004.
  * De-Cheng Zou et al. (2017) D. C. Zou, Y. Q. Liu and R. H. Yue, _Behavior of quasinormal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity, Eur. Phys. J. C_ 77 (2017) 365.
  * Peng Cheng et al. (2016) P. Cheng, S. W. Wei and Y. X. Liu, _Critical phenomena in the extended phase space of Kerr-Newman-AdS black holes, Phys. Rev. D_ 94 (2016) 024025.
  * Mozhgan Mir et al. (2017) M. Mir and R. B. Mann, _Charged Rotating AdS Black Holes with Chern-Simons coupling, Phys. Rev. D_ 95 (2017) 024005.
  * Zixu Zhao et al. (2014) Z. X. Zhao and J. L. Jing, _Ehrenfest scheme for complex thermodynamic systems in full phase space, JHEP_ 11 (2014) 037.
  * Ren Zhao et al. (2013) R. Zhao, H. H. Zhao, M. S. Ma and L. C. Zhang, _On the critical phenomena and thermodynamics of charged topological dilaton AdS black holes, Eur. Phys. J. C_ 73 (2013) 2645.
  * Zeinab et al. (2018) Zeinab, Dayyani and A. Sheykhi, _Critical behavior of Lifshitz dilaton black holes, Phys. Rev. D_ 98 (2018) 104026.
  * Run Zhou et al. (2019) R. Zhou and S. W. Wei, _Novel equal area law and analytical charge-electric potential criticality for charged Anti-de Sitterblack holes, Phys. Lett. B_ 792 (2019) 406.
  * Toledo et al. (2019) J. M. Toledo and V. b. Bezerra, _Some remarks on the thermodynamic of charged AdS black holes with cloud of strings and quintessence, Eur. Phys. J. C_ 79 (2019) 110.
  * Saoussen Mbarek et al. (2019) S. Mbarek and R. B. Mann, _Reverse Hawking-Page Phase Transition in de Sitter Black Holes, JHEP_ 02 (2019) 103.
  * Fil Simovic et al. (2019) F. Simovic and R. B. Mann, _Critical Phenomena of Charged de Sitter Black Holes in Cavities, Class. Quant. Grav._ 36 (2019) 014002.
  * Brian P. Dolan et al. (2013) B. P. Dolan, D. Kastor, D. Kubiznak, R. B. Mann and J. Traschen, _Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes, Phys. Rev. D_ 87 (2013) 104017.
  * Hendi et al. (2017) S. H. Hendi, A. Dehghani and Mir Faizal, _Black hole thermodynamics in Lovelock gravity’s rainbow with (A)dS asymptote, Nucl. Phys. B_ 914 (2017) 117.
  * Sekiwa et al. (2006) Y. Sekiwa, _Thermodynamics of de Sitter black holes: Thermal cosmological constant, Phys. Rev. D_ 73 (2006) 084009.
  * David Kubiznak et al. (2016) D. Kubiznak, _Thermodynamics of horizons: de Sitter black holes and reentrant phase transitions, Class. Quant. Grav._ 33 (2016) 24.
  * James McInerney et al. (2016) J. McInerney, G. Satishchandran and J. Traschen, _Cosmography of KNdS Black Holes and Isentropic Phase Transitions, Class. Quant. Grav._ 33 (2016) 10.
  * Miho Urano et al. (2009) M. Urano and A. Tomimatsu, _Mechanical First Law of Black Hole Spacetimes with Cosmological Constant and Its Application to Schwarzschild-de Sitter Spacetime, Class. Quant. Grav._ 26 (2009) 105010.
  * Sourav Bhattacharya et al. (2013) S. Bhattacharya and A. Lahiri, _Mass function and particle creation in Schwarzschild-de Sitter spacetime, Eur. Phys. J. C_ 73 (2013) 2673.
  * Cai et al. (2002) R. G. Cai, _Cardy-Verlinde formula and asymptotically de Sitter spaces, Phys. Lett. B_ 525 (2002) 331 .
  * Cai et al. (2002) R. G. Cai, _Cardy-Verlinde formula and thermodynamics of black holes in de Sitter spaces, Nucl. Phys. B_ 628 (2002) 375.
  * Yu-Bo Ma et al. (2018) Y. B. Ma, S. X. Zhang, Y. Wu, L. Ma and S. Cao, _Thermodynamics of de Sitter Black Hole in Massive Gravity, Commun. Theor. Phys._ 69 (2018) 544-550.
  * Azreg-Aiou et al. (2015) M. Azreg-Aiou, _Charged de Sitter-like black holes: quintessence-dependent enthalpy and new extreme solutions, Eur. Phys. J. C_ 75 (2015) 34.
  * Zhang et al. (2016) L. C. Zhang and R. Zhao, _The critical phenomena of Schwarzschild-de Sitter Black hole, EPL_ 113 (2016) 10008.
  * Li-Chun Zhang et al. (2014) X. Y. Guo, H. F. Li, L. C. Zhang and R. Zhao, _Continuous phase transition and microstructure of charged AdS black hole with quintessence, Eur. Phys. J. C_ 80 (2020) 168.
  * Hui-Hua Zhao et al. (2014) H. H. Zhao, L. C. Zhang, M. S. Ma and R. Zhao, _P-V criticality of higher dimensional charged topological dilaton de Sitter black holes, Phys. Rev. D_ 90 (2014) 064018.
  * Huaifan Li et al. (2017) Y. B. Ma, Y. Zhang, L. C. Zhang, L. Wu, Y. M. Huang and Y. Pan, _Thermodynamic properties of higher-dimensional dS black holes in dRGT massive gravity, Eur. Phys. J. C_ 80 (2020) 213.
  * Li-Chun Zhang et al. (2016) L. C. Zhang, R. Zhao and M. S. Ma, _Entropy of Reissner-Nordström de Sitter black hole, Phys. Lett. B_ 761 (2016) 74.
  * Yu-Bo Ma et al. (2018) Y. B. Ma, L. C. Zhang, T. Peng, Y. Pan and S. Cao, _Entropy of the electrically charged hairy black holes, Eur. Phys. J. C_ 78 (2018) 763 .
  * Hinterbichler et al. (2012) K. Hinterbichler, _Theoretical aspects of massive gravity, Rev. Mod. Phys._ 84 (2012) 671.
  * Fierz et al. (1939) M. Fierz, _On the relativistic theory of force-free particles with any spin, Helv. Phys. Acta_ 12 (1939) 3.
  * Fierz et al. (1939) M. Fierz and W. Pauli, _On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc.Roy.Soc.Lond. A_ 173 (1939) 211.
  * van Dam et al. (1970) H. van Dam and M. J. G. Veltman, _Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B_ 22 (1970) 397.
  * Zakharov et al. (1970) V. I. Zakharov, _Linearized gravitation theory and the graviton mass, JETP Lett._ 12 (1970) 312.
  * Boulware et al. (1972) D. G. Boulware and S. Deser, _Can gravitation have a finite range, Phys. Rev. D_ 6 (1972) 3368.
  * Boulware et al. (1972) D. G. Boulware and S. Deser, _Inconsistency of finite range gravitation, Phys. Lett. B_ 40 (1972) 227.
  * de Rham et al. (2010) C. de Rham, G. Gabadadze, _Generalization of the Fierz-Pauli action, Phys. Rev. D_ 82 (2010) 044020.
  * de Rham et al. (2011) C. de Rham, G. Gabadadze and A. J. Tolley, _Resummation of Massive Gravity,Phys. Rev. Lett._ 106 (2011) 231101.
  * B. Eslam Panah et al. (2019) B. Eslam Panah and S. H. Hendi, _Black hole solutions correspondence between conformal and massive theories of gravity, EPL_ 125 (2019) 60006.
  * B. Eslam Panah et al. (2020) B. Eslam Panah, S. H. Hendi and Y. C. Ong, _Black Hole Remnant in Massive Gravity, Phys. Dark Universe_ 27 (2020) 100452.
  * S. H. Hendi et al. (2018) S. H. Hendi, G. H. Bordbar, B. Eslam Panah and S. Panahiyan, _Neutron stars structure in the context of massive gravity, JCAP_ 07 (2017) 004.
  * B. Eslam Panah et al. (2019) B. Eslam Panah and H. L. Liu, _White dwarfs in de Rham-Gabadadze-Tolley like massive gravity, Phys. Rev. D_ 99 (2019) 104074.
  * Hendi et al. (2017) S. H. Hendi, S. Panahiyan, S. Upadhyay and B. Eslam Panah, _Charged BTZ black holes in the context of massive gravity’s rainbow, Phys. Rev. D_ 95 (2017) 084036.
  * Majhi et al. (2017) B. R. Majhi and S. Samanta, _P-V criticality of AdS black holes in a general framework, Phys. Lett. B_ 773 (2017) 203.
  * Cai et al. (2015) R. G. Cai, Y. P. Hu, Q. Y. Pan and Y. L. Zhang, _Thermodynamics of Black Holes in Massive Gravity, Phys. Rev. D_ 91 (2015) 024032 .
  * Jianfei Xu et al. (2015) J. F. Xu, L. M. Cao and Y. P. Hu, _P-V criticality in the extended phase space of black holes in massive gravity, Phys. Rev. D_ 91 (2015) 124033.
  * Upadhyay et al. (1) S. Upadhyay, B. Pourhassan and H. Farahani, _P-V criticality of first-order entropy corrected AdS black holes in massive gravity, Phys. Rev. D_ 95 (2017) 106014.
  * David C. Johnston et al. (2) D. C. Johnston, _Thermodynamic Properties of the van der Waals Fluid, Online ISBN:_ 978-1-627-05532-1, arXiv:1402.1205.
  * Hassan et al. (2011) S. F. Hassan, R. A. Rosen, _On Non-Linear Actions for Massive Gravity, JHEP_ 07 (2011) 009.
  * Allan Adams et al. (2015) A. Adams, D. A. Roberts and O. Saremi, _Hawking-Page transition in holographic massive gravity, Phys. Rev. D_ 91 (2015) 046003.
  * De-Cheng Zou et al. (2017) D. C. Zou, R. H. Yue and M. Zhang, _Reentrant pha AdS black holes in dRGT massive gravity, Eur. Phys. J. C_ 77 (2017) 256.
  * Petarpa Boonserm et al. (2018) P. Boonserm and T. Ngampitipan, _Greybody factor for black holes in dRGT massive gravity, Eur. Phys. J. C_ 78 (2018) 492.
  * Erik P. Verlinde et al. (2011) E. P. Verlinde, _On the origin of gravity and the laws of Newton, JHEP_ 04 (2011) 029.
  * Panos et al. (2019) C. P. Panos and Ch. C. Moustakidis, _A simple link of information entropy of quantum and classical systems with Newtonian r-2 dependence of Verlinde’s entropic force, Physica A_ 518 (2019) 384.
  * Dmitri E. Kharzeev et al. (2014) D. E. Kharzeev, _Deconfinement as an entropic self-destruction: A solution for the quarkonium suppression puzzle, Phys. Rev. D_ 90 (2014) 074007.
  * Li-Chun Zhang et al. (2019) L. C. Zhang and R. Zhao, _The entropic force in Reissner-Nordström-de Sitter spacetime, Phys. Lett. B_ 797 (2019) 134798.
  * Plastino et al. (2018) A. Plastino, M. C. Rocca and G. L. Ferri, _Quantum treatment of Verlinde entropic force conjecture, Physica A_ 511 (2018) 139.
  * Plastino et al. (2018) A. Plastino and M. C. Rocca, _On the entropic derivation of the r-2 Newtonian gravity force, Physica A_ 505 (2018) 190.
  * Yan-Gang Miao et al. (2018) Y. G. Miao and Z. M. Xu, _On thermal molecular potential among micromolecules in charged AdS black holes, Phys. Rev. D_ 98 (2018) 044001.


# Black hole with confining electric potential in scalar-tensor description of
regularized 4-dimensional Einstein-Gauss-Bonnet gravity

A. Övgün<EMAIL_ADDRESS>Physics Department, Eastern Mediterranean
University, Famagusta, 99628 North Cyprus via Mersin 10, Turkey.

###### Abstract

In this paper, we analytically present an exact confining charged black hole
solution to the scalar-tensor description of regularized 4-dimensional
Einstein-Gauss-Bonnet gravity (4DEGBG) coupled to non-linear gauge theory
containing $\sqrt{-F^{2}}$ term. Part of the importance of using a non-trivial
$4D$ theory of Gauss-Bonnet gravity is that it is proven the absence of the
Ostrogradsky instability was formulated in [D. Glavan and C. Lin, Phys. Rev.
Lett. 124, 081301 (2020) Glavan:2019inb ] and then regularized 4DEGBG theory
was obtained in [Hennigar et al. JHEP 07, 027 (2020) Hennigar:2020lsl ] and
[Fernandes et al. Phys. Rev. D 102, no.2, 024025 (2020) Fernandes:2020nbq ] as
well as this theory gains many attention because of its important finding for
black hole physics. We therefore also study some properties of this solution
such as temperature, specific heat,shadow and quasinormal modes. Our results
here show that a confining charge $f$ gives a significant contribution to the
shadow of the black hole as well as quasinormal modes frequencies.

Black hole; Exact solutions; Gauss-bonnet gravity; QCD-inspired confining
potential; Shadow

###### pacs:

95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.+h

##  I Introduction

Gravitational tests have been conducted in a variety of cosmic settings since
the the 1919 solar eclipse, which is the first evidence of General Relativity
(GR) Einstein:1916vd ; Dyson:1920cwa . Not long ago, gravitational tests have
been organized to explore gravity outside the solar system and even on a
cosmological scale. For instance, the Laser Interferometer Gravitational-Wave
Observatory (LIGO) detects the gravitational waves that propagate through the
fabric of spacetime as predicted by GR Abbott:2016blz . On the other hand, the
Event Horizon Telescope (EHT) collaboration, has recently imaged M87’s central
black hole, and provided a number of enlightening answers Akiyama:2019cqa .
Based on an analysis of the black hole’s shadow, the team showed a solitary
search of GR, boosting knowing about the properties of black holes (BHs) and
eliminating many modified gravity theories Psaltis:2020lvx ; Akiyama:2021qum .

GR stands among the most famous theories ever created, but it has some long-
standing weak points for instance, combining GR with quantum mechanics
(quantum gravity) or singularity problem Capozziello:2011et ; Demir:2021moq .
One of the important description of quantum theory for a linear confinement
phenomena using the actions of effective nonlinear gauge field (NGF) was
studied by ‘t Hooft tHooft:2002gdr . It was shown that the nonlinear terms in
NGF interpret as “infrared counterterms”. Along with the Maxwell term, one can
add a square root of the field strength squared, and construct a special type
of non-Maxwell nonlinear effective gauge field model as given by
Vasihoun:2014pha ; Guendelman:2003ib ; Gaete:2006xd ; Korover:2009zz ;
Guendelman:2011sm ; Guendelman:2012ve ; Guendelman:2018tzi :

$\begin{array}[]{c}S=\int d^{4}xL\left(F^{2}\right)\quad,\quad
L\left(F^{2}\right)=-\frac{1}{4}F^{2}-\frac{f}{2}\sqrt{-F^{2}}\\\ F^{2}\equiv
F_{\mu\nu}F^{\mu\nu}\quad,\quad
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\end{array}$ (1)

where $f$ being a positive coupling constant. Moreover, the square root of the
Maxwell term is appeared naturally as a conclusion of spontaneous scale
symmetry breaking of scale invariant theory of Maxwell with $f$. Here $f$ is
an integration constant and stands for the spontaneous breakdown. It is seen
from Eq. (1) that a confining effective potential which is Coulomb plus linear
one, in the form of well-known “Cornell” potential in quantum chromodynamics
(QCD) is arisen as $V(r)=-\frac{\alpha}{r}+\beta r$. Adding both the usual
Maxwell term with non analytic form of $F^{2}$ provide the form of nonlinear
electrodynamics, which is different from the $-\frac{f}{2}\sqrt{F^{2}}$
Lagranigian models using in string theory Nielsen:1973qs . Note that
$\sqrt{-F^{2}}$ is for “electrically” dominated models on the other hand
$\sqrt{F^{2}}$ is for “magnetically” dominated Nielsen-Olesen models
Nielsen:1973qs .

To solve these problems, modified gravity theories appear as alternative
theories, for example, adding higher curvature corrections to the Einstein
action, which is inspired by the low energy limit of string theory
Nojiri:2010wj . A vast range of modified theories and their applications to
black holes and wormholes now exist in the literature Jusufi:2017mav ;
Jusufi:2017lsl . Since the Lovelock gravity and its simplest case Einstein-
Gauss-bonnet (EGB) gravity are the most well-known theories of modified
gravity Lovelock:1972vz ; Boulware:1985wk , there were many studies on these
theories for $D>4$ black holes Wiltshire:1988uq , on the other hand for $D=4$,
EGB gravity, is topological invariant, has not any physical contribution to
dynamics Simon:1990ic . Based on non-Lagrangian approach Tomozawa:2011gp ;
Glavan:2019inb  by rescaling the Gauss–Bonnet dimensional coupling constant
$\alpha$ as $(D-4)\alpha\to\alpha$, it was shown that it is possible to
construct a new theory at limit $D\to 4$ of solutions without singularities,
which is free from Ostrogradsky instability as well as bypass the Lovelock’s
theorem. In addition, the GB term contributes to GR as a quantum correction.
The properties of the EGB in $4D$ and its solutions have been studied in
various papers Fernandes:2020rpa ; Kruglov:2021stm ; Kumar:2020owy ;
Guo:2020zmf ; Konoplya:2020bxa ; Donmez:2020rnf ; Donmez:2021fbk ;
Konoplya:2020qqh ; Ghosh:2020syx ; Singh:2020mty . On the other hand, EGB in
4D was debated in literature Arrechea:2020evj ; Gurses:2020rxb ;
Gurses:2020ofy ; Arrechea:2020gjw ; Cao:2021nng ; Aoki:2020iwm ; Aoki:2020lig
; Aoki:2020ila ; Hennigar:2020lsl ; Lu:2020iav ; Fernandes:2020nbq ;
Fernandes:2021dsb . Gürses et al. claim that the EGB theory lacks of the
continuity and covariant description at 4D Gurses:2020rxb , similarly Arrechea
et al. show the divergent contributions to the field equations in 4D
Arrechea:2020gjw . However, other authors claim that the problems disappear in
certain circumstances and the theory is (locally) conformally flat, but the
theory in 4D has preferred spacetimes, and it is not diffeomorphism invariant
Cao:2021nng . Some authors recognize the problems but also argue that the
theory can be formulated in a consistent way to cosmology and gravitational
waves Aoki:2020iwm , and show the theory either breaks the diffeomorphism
invariance or has additional degrees of freedom in line with the Lovelock
theorem. Moreover, the dimensional regularization of Glavan:2019inb  is made
righteous in the sight of some class of spherically symmetric spacetime metric
that typically present a new scalar degree of freedom Aoki:2020lig ;
Aoki:2020ila ; Hennigar:2020lsl ; Lu:2020iav . Moreover, it is shown that the
field equation for the EGB is intimately connected with generalized conformal
properties of the scalar field Fernandes:2020nbq ; Fernandes:2021dsb . Aoki et
al. study the gravitational waves (GW) in EGB and find a bound for the EGB
parameter as $\tilde{\alpha}\lesssim\mathcal{O}(1)\mathrm{eV}^{-2}$
Aoki:2020iwm , on the other hand using the velocity propagation of GW, Clifton
el al. show the bound at $\alpha\approx 10^{49}\mathrm{eV}^{-2}$
Clifton:2020xhc . Hence, the spherically symmetric black hole solutions in the
4DEGBG are still valid in Glavan:2019inb  and therefore, it is worth studying
more features of the black hole solutions in regularized 4DEGBG
Fernandes:2020nbq ; Hennigar:2020lsl .

Main aim of the study is to find a possible new effects by coupling the
confining potential generating nonlinear gauge field system in the scalar-
tensor description of regularized 4DEGBG black hole. Here, we mainly attend to
a regularized 4DEGB theory which is obtained in Ref. Fernandes:2020nbq ;
Hennigar:2020lsl , that present a counter-term to eliminate the divergent part
of the theory. In 1992 Mann also used same procedure in 2 dimensions
Mann:1992ar , which construct a well defined theory, costing only adding new
scalar degree of freedom.

This paper is organized as follows. Section 2 contains the brief review of
scalar-tensor description of regularized 4DEGBG, and then we solve the theory
and present an exact black hole solution for the scalar-tensor description of
regularized 4DEGBG with confining electric potential. In Section 3, we study
some basic thermodynamics properties of the black hole, discuss their
properties, and give numerical analyses. In Section 4, shadow of the black
hole, and in Section 5, quasinormal modes frequencies are discussed. Finally,
a conclusion is presented in Sect. 6.

##  II Black hole with Confining Electric Potential in regularized 4DEGBG

The action of the scalar-tensor formulation of regularized 4DEGBG coupled to
non-linear gauge theory containing $\sqrt{-F^{2}}$ term is written as follows
Hennigar:2020lsl ; Lu:2020iav ; Kobayashi:2020wqy ; Mann:1992ar ;
Fernandes:2020nbq ; Fernandes:2021dsb ; Fernandes:2021ysi :

$S=\frac{1}{16\pi}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\Big{[}R+\alpha\big{(}4G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-\phi\mathcal{G}+4\square\phi(\nabla\phi)^{2}+2(\nabla\phi)^{4}\big{)}+\mathcal{L}_{NGF}\Big{]}\,.$
(2)

Note that $g$ is the determinant of the metric tensor $g_{ab}$, $\alpha$ is a
dimensionless coupling constant, and $\mathcal{L}_{NGF}$ is the Lagrangian for
the nonlinear gauge field. On the other hand, $\mathcal{G}$ stands for the
Gauss-Bonnet invariant and $\phi$ is a scalar field. The action in Eq. 2 can
be derived from the Lovelock theory
$\mathcal{L}=\sqrt{-g}\left(-2\Lambda+R+\alpha\mathcal{G}+\dots\right)\,,$
with $\mathcal{G}\equiv
R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}\,,$
using the by the addition of a counter-term that consists of the Gauss-Bonnet
invariant of a conformally transformed geometry
$\tilde{g}_{\mu\nu}=e^{2\phi}g_{\mu\nu}$. Then it gives Fernandes:2020nbq ;
Hennigar:2020lsl

$\lim_{D\to
4}\frac{\int_{\mathcal{M}}d^{D}x\sqrt{-g}\mathcal{G}-\int_{\mathcal{M}}d^{D}x\sqrt{-\tilde{g}}\tilde{\mathcal{G}}}{D-4}=\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(4G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-\phi\mathcal{G}+4\square\phi(\nabla\phi)^{2}+2(\nabla\phi)^{4}\right)\,,$
(3)

where the factor of $D-4$ is appeared due to regularization of
$\alpha\rightarrow\alpha/(D-4)$ for observational constraints $\alpha>0$
Clifton:2020xhc  which is different than the method of Glavan & Lin. Here it
is shown that the counter-term has tildes, to remove divergences in the 4
dimensional limit. Hence, there is a well-defined scalar-tensor theory of
gravity in (2).

We vary the action Eq. (2) with respect to metric, and obtain the following
equations of motions (EOMs):

$G_{\mu\nu}+\alpha\mathcal{H}_{\mu\nu}=8\pi\,T_{\mu\nu}\,,$ (4)

with the energy-momentum tensor $T_{\mu\nu}$ and

$\displaystyle\mathcal{H}_{\mu\nu}=$ $\displaystyle
2G_{\mu\nu}\left(\nabla\phi\right)^{2}+4P_{\mu\alpha\nu\beta}\left(\nabla^{\beta}\nabla^{\alpha}\phi-\nabla^{\alpha}\phi\nabla^{\beta}\phi\right)+4\left(\nabla_{\alpha}\phi\nabla_{\mu}\phi-\nabla_{\alpha}\nabla_{\mu}\phi\right)\left(\nabla^{\alpha}\phi\nabla_{\nu}\phi-\nabla^{\alpha}\nabla_{\nu}\phi\right)$
(5)
$\displaystyle+4\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\nabla_{\nu}\nabla_{\mu}\phi\right)\Box\phi+g_{\mu\nu}\Big{(}2\left(\Box\phi\right)^{2}-\left(\nabla\phi\right)^{4}+2\nabla_{\beta}\nabla_{\alpha}\phi\left(2\nabla^{\alpha}\phi\nabla^{\beta}\phi-\nabla^{\beta}\nabla^{\alpha}\phi\right)\Big{)}\,,$

where the double dual of the Riemann tensor is

$P_{\alpha\beta\mu\nu}\equiv\frac{1}{4}\epsilon_{\alpha\beta\gamma\delta}R^{\rho\sigma\gamma\delta}\epsilon_{\rho\sigma\mu\nu}=2\,g_{\alpha[\mu}G_{\nu]\beta}+2\,g_{\beta[\nu}R_{\mu]\alpha}-R_{\alpha\beta\mu\nu}.$
(6)

Note the square brackets stands for anti-symmetrization.

We vary the action Eq. (2) with respect to $\phi$, and obtain the following
equations of motions:

$\displaystyle
R^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi-\Box\phi\left(\nabla\phi\right)^{2}+(\nabla_{\mu}\nabla_{\nu}\phi)^{2}-(\Box\phi)^{2}-2\nabla_{\mu}\phi\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi=\frac{1}{8}\mathcal{G}\,.$
(7)

This equation is conformal invariance under the transformation $g_{\mu\nu}\to
g_{\mu\nu}e^{2\sigma}$ and $\phi\to\phi-\sigma$ Fernandes:2021dsb . Moreover,
using Eq. (7), with (4), it becomes Fernandes:2020nbq

$R+\frac{\alpha}{2}\mathcal{G}=-8\pi\,T.$ (8)

Hence Fernandes shows that this purely geometric relation is a direct
consequence of the conformal invariance of the scalar field equation
Fernandes:2021dsb . Then one can calculate a Noether current with vanishing
divergence Saravani:2019xwx :

$j^{\mu}=\frac{1}{\sqrt{-g}}\frac{\delta
S}{\delta(\partial_{\mu}\phi)},\qquad{\rm
such\,\,that}\qquad\nabla_{\mu}j^{\mu}=0\,.$ (9)

Note that one can vanish divergence $\nabla_{\mu}j^{\mu}=0$ to show that
$\partial_{\mu}\left(\sqrt{-g}j^{\mu}\right)=0$, and then find the equation of
motion (7).

Using the field equations (4) and (7), we will construct the black hole
solution with confining electric potential in the regularized 4DEGBG gravity.
To construct the black hole, we consider the $4$ dimensional static and
spherically symmetric metric:

$ds^{2}=-A(r)dt^{2}+\frac{1}{A(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (10)

On the other hand, Lagrangian for the nonlinear gauge field is given by
Vasihoun:2014pha :

$L_{NGF}=-\frac{1}{4}F^{2}-\frac{f}{2}\sqrt{-F^{2}},$ (11)

where $F^{2}\equiv F_{\kappa\lambda}F_{\mu\nu}g^{\kappa\mu}g^{\lambda\nu}.$
This model produces a confining effective potential which is
$V=-\frac{a}{r}+\beta r$ of the form of the well-known Cornell potential in
quantum chromodynamics (QCD) Korover:2009zz ; Gaete:2006xd . It is crucial to
stress that the Lagrangian contains both the usual Maxwell term as well as a
non-analytic function of $F^{2}$ and thus it is a non-standard form of
nonlinear electrodynamics. The energy-momentum tensor of the nonlinear gauge
field is Vasihoun:2014pha ; Guendelman:2003ib ; Gaete:2006xd ; Korover:2009zz
; Guendelman:2011sm ; Guendelman:2012ve ; Guendelman:2018tzi :

$T_{\mu\nu}^{(NGF)}=\left(1-\frac{f}{\sqrt{-F^{2}}}\right)F_{\mu\kappa}F_{\nu\lambda}g^{\kappa\lambda}-\frac{1}{4}\left(F^{2}+2f\sqrt{-F^{2}}\right)g_{\mu\nu},$
(12)

and

$\partial_{\nu}\left(\sqrt{-g}\left(1-\frac{f}{\sqrt{-F^{2}}}\right)F_{\kappa\lambda}g^{\mu\kappa}g^{\nu\lambda}\right)=0.$
(13)

In this case the gauge field equations of motion (13) become:

$\partial_{r}\left(r^{2}\left(F_{0r}-\frac{\varepsilon_{F}f}{\sqrt{2}}\right)\right)=0\quad,\quad\varepsilon_{F}\equiv
sign\left(F_{0r}\right)$ (14)

in which

$F_{\mu\nu}=0\text{ for }(\mu,\nu)\neq(0,r)\quad,\quad F_{0r}=F_{0r}(r)$ (15)

and its solution reads:

$F_{0r}=\frac{\varepsilon_{F}f}{\sqrt{2}}+\frac{Q}{\sqrt{4\pi}r^{2}}\quad,\quad\varepsilon_{F}=sign(Q).$
(16)

Again, as in the flat space-time case (1), the electric field contain a radial
constant piece ${\varepsilon_{F}f}\sqrt{2}$ alongside with the Coulomb term.
The form of a QCD-like (Cornell -type) confining-type potential (provided
$\varepsilon_{F}f<0$ )with $\varepsilon_{F}$ . Then the energy momentum tensor
for the nonlinear gauge field is

$T_{0}^{0}=T_{\eta}^{(F)}{\eta}=-\frac{1}{2}F_{0\eta}^{2}\quad,\quad
T_{ij}=g_{ij}\left(\frac{1}{2}F_{0\eta}^{2}-\frac{f}{\sqrt{2}}\left|F_{0\eta}\right|\right).$
(17)

After we solve the field equations Eq. 17 within Eq. 4:

$\displaystyle\frac{\left[2\alpha(1-A(r))+r^{2}\right]A^{\prime}(r)}{r^{3}}-\frac{[1-A(r)]\left(r^{2}-\alpha(1-A(r))\right)}{r^{4}}+4\pi\left(\frac{f}{\sqrt{2}}-\frac{Q}{2\sqrt{\pi}r^{2}}\right)^{2}=0.$
(18)

with $M$ the ADM mass, we get

$A(r)=1+\frac{r^{2}}{2\alpha}\left(1\pm\sqrt{1+4\alpha\left(\frac{2\pi
f^{2}}{3}-\frac{2\sqrt{2\pi}fQ}{r^{2}}+\frac{2M}{r^{3}}-\frac{Q^{2}}{r^{4}}\right)}\right),$
(19)

with the scalar field profile for this solution is

$\phi^{\prime}(r)=\frac{1-\sqrt{A(r)}}{r\sqrt{A(r)}}.$ (20)

Note that the prime stands for a derivative with respect to $r$. The $\pm$
sign in front of the square root term in Eq. (19), corresponds to two
different branches of solutions. It is shown that the negative branch of
solution is for a charged 4D black hole, on the other hand, the positive
branch gives instabilities of the graviton, because of the positive sign in
the mass Singh:2020mty . Thus, since only the negative branch leads to a
physically meaningful solution, we will limit our analysis to this branch of
the solution. Note that the metric function reduces to the scalar-tensor
description of regularized 4DEGBG solution at the limit of $f=Q=0$ founded by
Fernandes et al. Fernandes:2020nbq . Moreover, when a confining-type charge
term equals to zero $f=0,$ it reduces to the charged EGBG black hole solution
founded by Fernandes Fernandes:2020rpa .

The asymptotic of the metric function $A(r)$ when $r\to\infty$ is

$A(r)=1-\,{\frac{2M}{r}}+{\frac{{Q}^{2}}{{r}^{2}}}+\frac{2}{3}{r}^{2}\pi\,{f}^{2}+2\sqrt{2\pi}fQ+\mathcal{O}\left(M^{2},Q^{3},f^{3}\right).$
(21)

The event horizon of the black holes $r_{+}$ is the larger root of the
equation Eq. (19), $A(r)=0$. As shown in Fig. (1), the number of horizons
depends on the parameters $f$.

Figure 1: The lapse function $A(r)$ as a function of $r$ for $M=2$, $Q=1$,
$\alpha=+0.2$ and for the different values of $f$.

##  III Thermodynamics and physical properties

In this section, we study some thermodynamic properties of the black hole.
First, the black hole mass can be calculated by
$\left.g_{00}\right|_{r=r_{+}}=0$

$M(r_{+})=\frac{3\alpha-2\pi
f^{2}r_{+}^{4}+6\sqrt{2\pi}fQr_{+}^{2}+3Q^{2}+3r_{+}^{2}}{6r_{+}}.$ (22)

Note that one can take a limit of vanishing Gauss-bonnet coupling parameter,
and charges $\alpha=Q=f=0$, then the mass of the black hole reduces to mass of
Schwarzschild black holes: $M_{+}=\frac{r_{+}}{2}$. Second, the Hawking
temperature associated with horizon radius $r_{+}$, can be obtained through:

$T_{+}=\frac{\kappa}{2\pi}=\frac{\frac{r_{+}\left(1-\sqrt{\frac{4\alpha\left(2\pi
f^{2}r_{+}^{4}-6\sqrt{2\pi}fQr_{+}^{2}+6Mr_{+}-3Q^{2}\right)}{3r_{+}^{4}}+1}\right)}{\alpha}-\frac{2\left(2\sqrt{2\pi}fQr_{+}^{2}-3Mr_{+}+2Q^{2}\right)}{r_{+}^{3}\sqrt{\frac{8}{3}\pi\alpha
f^{2}-\frac{8\sqrt{2\pi}\alpha fQ}{r_{+}^{2}}+\frac{8\alpha
M}{r_{+}^{3}}-\frac{4\alpha Q^{2}}{r_{+}^{4}}+1}}}{4\pi},$ (23)

where $\kappa$ is the surface gravity given by

$\kappa=\left.\sqrt{-\frac{1}{2}\nabla_{\mu}\chi_{\nu}\nabla^{\mu}\chi^{\nu}}\equiv\frac{1}{2}\frac{\partial
A(r)}{\partial r}\right|_{r=r_{+}},$ (24)

and the Hawking temperature is plotted in Fig. (2). Furthermore, we can take a
limit of vanishing Gauss-bonnet coupling parameter, and charges
$\alpha=Q=f=0$, then the Hawking temperature of the black hole reduces to
temperature of Schwarzschild black holes: $T_{r_{+}}=\frac{1}{8M\pi}$.

Figure 2: Hawking temperature $T$ versus $r$ for $M=2$, $Q=1$, $\alpha=+0.2$
and for the different values of $f$.

Moreover, to check the thermodynamics stability of black holes, the behaviour
of specific heat $(C_{+})$ of the black holes is calculated. The positive
(negative) specific heat signifies the local thermodynamics stability
(instability) of the black holes. By using the relation $C_{+}=\frac{\partial
M_{+}}{\partial r_{+}}/\frac{\partial T_{+}}{\partial r_{+}}$, the specific
heat is plotted in Fig. (3). It is shown that the regularized 4DEGBG black
hole with confining electric potential is thermodynamically stable, where the
Hawking temperature in Fig. (2) and the specific heat in Fig. (3) are
positive, in some range of confining charge parameters and event horizon
radii.

Figure 3: Specific heat $C_{+}$ versus horizon $r_{+}$ for $M=2$, $Q=1$,
$\alpha=+0.2$ and for the different values of $f$.

##  IV Shadow of the black hole with confining electric potential in
regularized 4DEGBG

Studying the shadow of the black hole is recently gained more attention, (see,
for example, Falcke:1999pj ; Cunha:2018acu ; Atamurotov:2013sca ; Bisnovatyi-
Kogan:2018vxl ; Cunha:2016wzk ; Vagnozzi:2019apd ; Shaikh:2018lcc ;
Cunha:2017eoe ; Tsukamoto:2014tja ; Ovgun:2020gjz ; Ovgun:2018tua  and
references therein). Therefore, now we analyse the shadow of the black hole
and study the effect of the confining charge on the shadow cast. The Hamilton-
Jacobi approach for a photon in the equatorial plane $\theta=\frac{\pi}{2}$
can be written in the form of

$H=\frac{1}{2}g^{\mu
v}p_{\mu}p_{v}=\frac{1}{2}\left(\frac{L^{2}}{r^{2}}-\frac{E^{2}}{A(r)}+\frac{\dot{r}^{2}}{A(r)}\right)=0.$
(25)

Note that $p_{\mu}$is the photon momenta, $\dot{r}=\partial H/\partial p_{r}$
, $E=-p_{t}$ is the energy and $L=p_{\phi}$ is the angular momentum. Using the
(25), a complete description of the dynamics with effective potential $V$ is
given by

$V+\dot{r}^{2}=0,\quad
V=A(r)\left(\frac{L^{2}}{r^{2}}-\frac{E^{2}}{A(r)}\right).$ (26)

The stability condition of the circular null geodesics provides that
$V(r)=V^{\prime}(r)=0$ and $V^{\prime\prime}(r)>0$ Ovgun:2020gjz . For the
circular photon orbits, unstability associated to the maximum value of the
effective potential so that

$\left.V(r)\right|_{r=r_{p}}=0,\left.\quad V^{\prime}(r)\right|_{r=r_{p}}=0,$
(27)

where the impact parameter
$b\equiv\frac{L}{E}=\frac{r_{p}}{\sqrt{A\left(r_{p}\right)}}$ and $r_{p}$ is
the radius of the photon sphere, which can be calculated by finding the
largest root of the this relation:

$\frac{A^{\prime}(r_{p})}{A(r_{p})}=\frac{2}{r_{p}},$ (28)

The above equation 28 is complicated to solve analytically, just so, the
numerical methods are used to obtain the photon sphere radius $r_{p}$ which
are presented in Table I. It is shown that increasing values of the confining
charge parameter tend to increase the photon sphere.

The radius of the black hole shadow as observed by a static observer at the
position $r_{0}$ is

$R_{s}=r_{p}\sqrt{\frac{A\left(r_{0}\right)}{A\left(r_{p}\right)}},$ (29)

and for large distant observer ($A(r_{0})=1$) and

$R_{s}^{2}=\frac{r_{p}^{2}}{A(r_{p})}.$ (30)

$f$ |  $r_{p}$ |  $R_{s}$
---|---|---
0.20 |  2.52725 |  9.65169
0.30 |  1.89395 |  8.88559
0.40 |  1.4573 |  4.68954
0.50 |  1.1166 |  2.62366
Table 1: Effects of the confining charge on the BH shadow for fixed $M=2$,
$Q=1$ and $\alpha=0.2$. Figure 4: Shadow of the black hole for different
values of $f$ for $M=2$, $Q=1$, $\alpha=+0.2$.

The apparent shape of a shadow is plotted by a stereographic projection in
terms of the celestial coordinates $X$ and $Y$ in Fig. (4), that the radius of
the shadow increases with the decreasing value of $f$. Fig. (4) shows the
evidence that confining charge $f$ has a stronger effect on the shadow size of
the 4D EGBG black hole.

The relation between the high energy absorption cross section and the shadow
for the observer located at infinity is

$\frac{d^{2}E(\omega)}{dtd\omega}=\frac{2\pi^{3}\omega^{3}R_{s}^{2}}{\exp\left(\omega/T_{+}\right)-1},$
(31)

where $\omega$ stands for the emission frequency. In Fig. (5), indicates that
the there is a peak of the energy emission rate and increasing the value of
$f$, the peak of the energy emission rate increases.

Figure 5: Energy emission rate of the black hole for different values of $f$
for $M=2$, $Q=1$, $\alpha=+0.2$.

##  V Quasinormal modes

Since the discovery of the LIGO experiment, gravitational waves which is
emitted by perturbed black holes become popular, that are dominated by
‘quasinormal ringing’. Quasinormal modes are damped oscillations at single
frequencies which are characteristic of the underlying system and can be
calculated using the scalar perturbation of massless field around black hole:

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu
v}\partial_{\nu}\Phi\right)=0,$ (32)

and separate the above equation using the

$\Phi=e^{-i\omega t}\frac{R(r)}{r}e^{im\phi}Y_{lm}(\vartheta),$ (33)

where separation of variables for radial function $R(r)$ and spherical
harmonics $Y_{lm}$. Using the tortoise coordinates $r_{*}=\int dr/f(r)$, the
Eq. 32 for the radial field becomes:

$\frac{d^{2}R\left(r_{*}\right)}{dr_{*}^{2}}+\left(\omega^{2}-V\left(r_{*}\right)\right)R\left(r_{*}\right)=0,$
(34)

with the effective potential ($l=1,2,3$ is multipole number, and
$\omega=\omega_{R}-i\omega_{I}$ is a complex quasinormal mode frequency)

$V(r)=A(r)\left(\frac{A^{\prime}(r)}{r}+\frac{l(l+1)}{r^{2}}\right).$ (35)

which is plotted in Fig. (6).

Figure 6: Effective potential for different values of $f$ for $M=2$, $Q=1$,
$l=1$, and $\alpha=+0.2$.

Here we use the WKB approximation method to find quasinormal mode frequency
and the unstable circular null geodesic method Cardoso:2008bp ;
Konoplya:2017wot . The imaginary part of the quasinormal mode frequency (Im
$\omega$=-$\omega_{I}$) which is responsible for the temporal, exponential
decay can be calculated , in the large-$l$ limit as follows:

$\omega_{l\gg
1}=l\Omega_{p}-i\left(n+\frac{1}{2}\right)\left|\lambda_{\mathrm{L}}\right|,$
(36)

with the angular velocity $\Omega_{p}$:

$\Omega_{p}=\frac{\sqrt{A\left(r_{p}\right)}}{r_{p}},$ (37)

and Lyapunov exponent $\lambda_{\mathrm{L}}$:

$\lambda_{\mathrm{L}}=\sqrt{\frac{A\left(r_{p}\right)\left[2A\left(r_{p}\right)-r_{p}^{2}A^{\prime\prime}\left(r_{p}\right)\right]}{2r_{p}^{2}}},$
(38)

where $n$ is the overtone number and take values $n=0,1,2,...$. Table 2 shows
the real part as well as imaginary part of the quasinormal modes increasing
with the increasing $f$. Hence, the modes are stable because in the Table 2
the the imaginary parts of the quasinormal modes frequencies (Im
$\omega$=-$\omega_{I}$) are negative, furthermore, the real part stands for
the frequency of oscillations. Explained differently, increasing the confining
charge parameter $f$, the scalar perturbations oscillate with greater
frequency $\omega$ which means that oscillates decay faster.

$f$ |  $\omega_{R}$ |  $\omega_{I}$
---|---|---
0.20 |  0.103609 |  0.0641446
0.30 |  0.112542 |  0.0723644
0.40 |  0.213241 |  0.134316
0.50 |  0.381146 |  0.209336
Table 2: Effects of the confining charge on the quasinormal modes frequencies
for fixed $M=2$, $Q=1$, $n=0$, $l=1$ and $\alpha=0.2$.

##  VI Conclusion

In this paper, we obtain the exact BH solution in the scalar-tensor
description of regularized 4DEGBG including confining electric charge. EGB
gravity theory is known as nontrivial for in $D>4$, notwithstanding, till one
shows that dimensional reduction by re-scaling of the coupling constant
$\alpha$ as $(D-4)\alpha\to\alpha$, open on to the novel regularization method
to build the effective solutions in 4D Glavan:2019inb . On the other hand, we
show new effects of regularized 4DEGBG by coupling the confining potential
which generates nonlinear gauge field system. This is also clear way to
generate dynamically the cosmological constant through the non-Maxwell gauge
fields Vasihoun:2014pha ; Guendelman:2003ib ; Gaete:2006xd ; Korover:2009zz ;
Guendelman:2011sm ; Guendelman:2012ve . The metric function for the derived
black hole solution in Eq. (19), reduces to the regularized 4DEGBG solution at
the limit of $f=Q=0$ founded by Fernandes et al. Fernandes:2020nbq  as well as
it reduces to the charged 4DEGBG black hole solution founded by Fernandes
Fernandes:2020rpa  when a confining-type charge term equals to zero $f=0$. In
addition, at the limit of $\alpha\to 0$ and $Q=0$, the solution reduces to
Schwarzschild-de-Sitter black hole with effective cosmological constant:
$\Lambda_{eff}=2\pi f^{2}$.

Next, we study thermodynamics of the regularized 4DEGBG black hole with
confining electric potential, and analyze its thermodynamics stability using
the black hole mass, Hawking temperature, and specific heat of the black hole.
We show that the regularized 4DEGBG with confining electric potential is
thermodynamically stable, where the Hawking temperature in Fig. (2) and the
specific heat in Fig. (3) are positive, in some range of confining charge
parameters and event horizon radii. Moreover, the shadow size, the energy
emission rate, and at last, quasinormal modes of the black hole are
investigated to see the effect of confining charge on the regularized 4DEGBG
black hole. The possible range of the confining charge parameter is
constrained and the results indicate that increasing the parameter of
confining charge shrinks the shadow’s radius in Fig. (4) and Table 1. We show
that there is a peak of the energy emission rate and increasing the value of
$f$, the peak of the energy emission rate increases in Fig. (5). Furthermore,
using the correspondence between null geodesics and quasinormal modes in the
eikonal regime for test fields, we also observe that the increasing the
parameter of confining charge, increase the real part and also imaginary part
of the quasinormal modes frequencies as well as the modes are stable because
in the Table 2 the the imaginary parts of the quasinormal modes frequencies
(Im $\omega$=-$\omega_{I}$) are negative. It is worth to mention that
increasing the confining charge parameter $f$, the scalar perturbations
oscillate with greater frequency $\omega$ which means that oscillates decay
faster. Thus, we conclude that these results showing the contribution of the
NGF, mainly with confining charge, on the black holes in regularized 4DEGBG
tells us clearly that, this is an important analytical solution to the
regularized 4DEGBG. In future, we plan to extend this model to a case without
spherical symmetry.

###### Acknowledgements.

The author is grateful for helpful discussions with Eduardo I. Guendelman.

## References

  * (1) A. Einstein, “The Foundation of the General Theory of Relativity,” Annalen Phys. 49, no.7, 769-822 (1916).
  * (2) F. W. Dyson, A. S. Eddington and C. Davidson, “A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919,” Phil. Trans. Roy. Soc. Lond. A 220, 291-333 (1920).
  * (3) B. P. Abbott et al. [LIGO Scientific and Virgo], “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, no.6, 061102 (2016).
  * (4) K. Akiyama et al. [Event Horizon Telescope], “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. Lett. 875, L1 (2019).
  * (5) D. Psaltis et al. [Event Horizon Telescope], “Gravitational Test Beyond the First Post-Newtonian Order with the Shadow of the M87 Black Hole,” Phys. Rev. Lett. 125, no.14, 141104 (2020).
  * (6) K. Akiyama et al. [Event Horizon Telescope], “First M87 Event Horizon Telescope Results. VII. Polarization of the Ring,” Astrophys. J. Lett. 910, no.1, L12 (2021).
  * (7) S. Capozziello and M. De Laurentis, “Extended Theories of Gravity,” Phys. Rept. 509, 167-321 (2011).
  * (8) D. Demir, “Emergent Gravity as the Eraser of Anomalous Gauge Boson Masses, and QFT-GR Concord,” Gen. Rel. Grav. 53, no.2, 22 (2021).
  * (9) G. ’t Hooft, “On peculiarities and pitfalls in path integrals,” Phys. Status Solidi B 237, 13 (2003).
  * (10) M. Vasihoun and E. Guendelman, “Gravitational and topological effects on $\sqrt{-F^{2}}$ Confinement Dynamics,” Int. J. Mod. Phys. A 29, no. 23, 1430042 (2014).
  * (11) E. I. Guendelman, “Scale symmetry breaking from the dynamics of maximal rank gauge field strengths,” Int. J. Mod. Phys. A 19, 3255 (2004).
  * (12) P. Gaete and E. Guendelman, “Confinement from spontaneous breaking of scale symmetry,” Phys. Lett. B 640, 201-204 (2006).
  * (13) I. Korover and E. I. Guendelman, “Confinement effect as a result of spontaneous breaking of scale invariance,” Int. J. Mod. Phys. A 24, 1443-1456 (2009).
  * (14) E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva, “Asymptotically de Sitter and anti-de Sitter Black Holes with Confining Electric Potential,” Phys. Lett. B 704, 230-233 (2011) [erratum: Phys. Lett. B 705, 545-545 (2011)]
  * (15) E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva, “Dynamical Couplings, Dynamical Vacuum Energy and Confinement/Deconfinement from $R^{2}-Gravity$,” Phys. Lett. B 718, 1099 (2013).
  * (16) E. Guendelman, E. Nissimov and S. Pacheva, “Four-Dimensonal Gauss-Bonnet Gravity Without Gauss-Bonnet Coupling to Matter - Spherically Symmetric Solutions, Domain Walls and Spacetime Singularities,” Bulg. J. Phys. 48, no.2, 087-116 (2021).
  * (17) H. B. Nielsen and P. Olesen, “Local field theory of the dual string,” Nucl. Phys. B 57, 367-380 (1973).
  * (18) S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,” Phys. Rept. 505, 59-144 (2011).
  * (19) K. Jusufi and A. Övgün, “Gravitational Lensing by Rotating Wormholes,” Phys. Rev. D 97, no.2, 024042 (2018)
  * (20) K. Jusufi, M. C. Werner, A. Banerjee and A. Övgün, “Light Deflection by a Rotating Global Monopole Spacetime,” Phys. Rev. D 95, no.10, 104012 (2017)
  * (21) D. Lovelock, “The four-dimensionality of space and the einstein tensor,” J. Math. Phys. 13, 874-876 (1972).
  * (22) D. G. Boulware and S. Deser, “String Generated Gravity Models,” Phys. Rev. Lett. 55, 2656 (1985).
  * (23) D. L. Wiltshire, “Black Holes in String Generated Gravity Models,” Phys. Rev. D 38, 2445 (1988).
  * (24) J. Z. Simon, “Higher Derivative Lagrangians, Nonlocality, Problems and Solutions,” Phys. Rev. D 41, 3720 (1990).
  * (25) Y. Tomozawa, “Quantum corrections to gravity,” [arXiv:1107.1424 [gr-qc]].
  * (26) D. Glavan and C. Lin, “Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime,” Phys. Rev. Lett. 124, no.8, 081301 (2020).
  * (27) P. G. S. Fernandes, “Charged Black Holes in AdS Spaces in $4D$ Einstein Gauss-Bonnet Gravity,” Phys. Lett. B 805, 135468 (2020).
  * (28) S. I. Kruglov, “Einstein $-$ Gauss $-$ Bonnet gravity with nonlinear electrodynamics,” Annals Phys. 428, 168449 (2021).
  * (29) R. Kumar and S. G. Ghosh, “Rotating black holes in $4D$ Einstein-Gauss-Bonnet gravity and its shadow,” JCAP 07, 053 (2020).
  * (30) R. A. Konoplya and A. Zhidenko, “Black holes in the four-dimensional Einstein-Lovelock gravity,” Phys. Rev. D 101, no.8, 084038 (2020).
  * (31) M. Guo and P. C. Li, “Innermost stable circular orbit and shadow of the $4D$ Einstein–Gauss–Bonnet black hole,” Eur. Phys. J. C 80, no.6, 588 (2020).
  * (32) R. A. Konoplya and A. F. Zinhailo, “Quasinormal modes, stability and shadows of a black hole in the 4D Einstein–Gauss–Bonnet gravity,” Eur. Phys. J. C 80, no.11, 1049 (2020).
  * (33) O. Donmez, “Bondi-Hoyle Accretion around the Non-rotating Black Hole in 4D Einstein-Gauss-Bonnet Gravity,” Eur. Phys. J. C 81, no.2, 113 (2021).
  * (34) O. Donmez, “Dynamical Evolution of the Shock Cone around $4D$ Einstein-Gauss Bonnet Rotating Black Hole,” [arXiv:2103.03160 [astro-ph.HE]].
  * (35) S. G. Ghosh and R. Kumar, “Generating black holes in $4D$ Einstein-Gauss-Bonnet gravity,” Class. Quant. Grav. 37, no.24, 245008 (2020).
  * (36) S. G. Ghosh, D. V. Singh, R. Kumar and S. D. Maharaj, “Phase transition of AdS black holes in 4D EGB gravity coupled to nonlinear electrodynamics,” Annals Phys. 424, 168347 (2021)
  * (37) J. Arrechea, A. Delhom and A. Jiménez-Cano, “Inconsistencies in four-dimensional Einstein-Gauss-Bonnet gravity,” Chin. Phys. C 45, no.1, 013107 (2021).
  * (38) M. Gurses, T. Ç. Şişman and B. Tekin, “Comment on ”Einstein-Gauss-Bonnet Gravity in 4-Dimensional Space-Time”,” Phys. Rev. Lett. 125, no.14, 149001 (2020).
  * (39) M. Gürses, T. Ç. Şişman and B. Tekin, “Is there a novel Einstein–Gauss–Bonnet theory in four dimensions?,” Eur. Phys. J. C 80, no.7, 647 (2020).
  * (40) J. Arrechea, A. Delhom and A. Jiménez-Cano, “Comment on “Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime”,” Phys. Rev. Lett. 125, no.14, 149002 (2020)
  * (41) L. M. Cao and L. B. Wu, “On the ”Einstein-Gauss-Bonnet Gravity in Four Dimension”,” [arXiv:2103.09612 [gr-qc]].
  * (42) K. Aoki, M. A. Gorji and S. Mukohyama, “Cosmology and gravitational waves in consistent $D\to 4$ Einstein-Gauss-Bonnet gravity,” JCAP 09, 014 (2020) [erratum: JCAP 05, E01 (2021)]
  * (43) K. Aoki, M. A. Gorji and S. Mukohyama, “A consistent theory of $D\to 4$ Einstein-Gauss-Bonnet gravity,” Phys. Lett. B 810, 135843 (2020)
  * (44) K. Aoki, M. A. Gorji, S. Mizuno and S. Mukohyama, “Inflationary gravitational waves in consistent $D\to 4$ Einstein-Gauss-Bonnet gravity,” JCAP 01, 054 (2021)
  * (45) R. A. Hennigar, D. Kubizňák, R. B. Mann and C. Pollack, “On taking the D → 4 limit of Gauss-Bonnet gravity: theory and solutions,” JHEP 07, 027 (2020)
  * (46) H. Lu and Y. Pang, “Horndeski gravity as $D\rightarrow 4$ limit of Gauss-Bonnet,” Phys. Lett. B 809, 135717 (2020).
  * (47) T. Kobayashi, “Effective scalar-tensor description of regularized Lovelock gravity in four dimensions,” JCAP 07, 013 (2020).
  * (48) R. B. Mann and S. F. Ross, “The D —$>$ 2 limit of general relativity,” Class. Quant. Grav. 10, 1405-1408 (1993).
  * (49) P. G. S. Fernandes, P. Carrilho, T. Clifton and D. J. Mulryne, “Derivation of Regularized Field Equations for the Einstein-Gauss-Bonnet Theory in Four Dimensions,” Phys. Rev. D 102, no.2, 024025 (2020)
  * (50) P. G. S. Fernandes, “Gravity with a generalized conformal scalar field: theory and solutions,” Phys. Rev. D 103, no.10, 104065 (2021)
  * (51) P. G. S. Fernandes, P. Carrilho, T. Clifton and D. J. Mulryne, “Black Holes in the Scalar-Tensor Formulation of 4D Einstein-Gauss-Bonnet Gravity: Uniqueness of Solutions, and a New Candidate for Dark Matter,” [arXiv:2107.00046 [gr-qc]].
  * (52) M. Saravani and T. P. Sotiriou, “Classification of shift-symmetric Horndeski theories and hairy black holes,” Phys. Rev. D 99, no.12, 124004 (2019)
  * (53) T. Clifton, P. Carrilho, P. G. S. Fernandes and D. J. Mulryne, “Observational Constraints on the Regularized 4D Einstein-Gauss-Bonnet Theory of Gravity,” Phys. Rev. D 102, no.8, 084005 (2020)
  * (54) H. Falcke, F. Melia and E. Agol, “Viewing the shadow of the black hole at the galactic center,” Astrophys. J. Lett. 528, L13 (2000).
  * (55) P. V. P. Cunha and C. A. R. Herdeiro, “Shadows and strong gravitational lensing: a brief review,” Gen. Rel. Grav. 50, no.4, 42 (2018).
  * (56) F. Atamurotov, A. Abdujabbarov and B. Ahmedov, “Shadow of rotating non-Kerr black hole,” Phys. Rev. D 88, no.6, 064004 (2013).
  * (57) G. S. Bisnovatyi-Kogan and O. Y. Tsupko, “Shadow of a black hole at cosmological distances,” Phys. Rev. D 98, no.8, 084020 (2018).
  * (58) P. V. P. Cunha, C. A. R. Herdeiro, B. Kleihaus, J. Kunz and E. Radu, “Shadows of Einstein–dilaton–Gauss–Bonnet black holes,” Phys. Lett. B 768, 373-379 (2017).
  * (59) S. Vagnozzi and L. Visinelli, “Hunting for extra dimensions in the shadow of M87*,” Phys. Rev. D 100, no.2, 024020 (2019).
  * (60) R. Shaikh, P. Kocherlakota, R. Narayan and P. S. Joshi, “Shadows of spherically symmetric black holes and naked singularities,” Mon. Not. Roy. Astron. Soc. 482, no.1, 52-64 (2019).
  * (61) P. Cunha, V.P., C. A. R. Herdeiro and E. Radu, “Fundamental photon orbits: black hole shadows and spacetime instabilities,” Phys. Rev. D 96, no.2, 024039 (2017).
  * (62) N. Tsukamoto, Z. Li and C. Bambi, “Constraining the spin and the deformation parameters from the black hole shadow,” JCAP 06, 043 (2014).
  * (63) A. Övgün and İ. Sakallı, “Testing generalized Einstein–Cartan–Kibble–Sciama gravity using weak deflection angle and shadow cast,” Class. Quant. Grav. 37, no.22, 225003 (2020).
  * (64) A. Övgün, İ. Sakallı and J. Saavedra, “Shadow cast and Deflection angle of Kerr-Newman-Kasuya spacetime,” JCAP 10, 041 (2018).
  * (65) V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, “Geodesic stability, Lyapunov exponents and quasinormal modes,” Phys. Rev. D 79, 064016 (2009).
  * (66) R. A. Konoplya and Z. Stuchlík, “Are eikonal quasinormal modes linked to the unstable circular null geodesics?,” Phys. Lett. B 771, 597-602 (2017).