# Hawking-Page transition in holographic QCD at finite density

Nelson R. F. Braga<EMAIL_ADDRESS>UFRJ — Universidade Federal do Rio de
Janeiro, Instituto de Física,
Caixa Postal 68528, Rio de Janeiro, Brasil  Octavio C. Junqueira
<EMAIL_ADDRESS>UFRJ — Universidade Federal do Rio de Janeiro,
Instituto de Física,
Caixa Postal 68528, Rio de Janeiro, Brasil

###### Abstract

We study the confinement/deconfinement transition of QCD matter for a quark-
gluon plasma at finite density using AdS/QCD duality. In order to determine
the critical temperature and its dependence on the quark chemical potential,
the semi-classical Hawking-Page approach is considered for a charged AdS black
hole. The result obtained is consistent with the QCD phase diagram, with the
critical temperature decreasing as the chemical potential increases. Using the
soft wall holographic model, the value of quark density in which the phase
transition occurs at zero temperature is estimated. For higher densities the
QCD matter is deconfined, independent of the temperature.

##  1 Introduction

Heavy-ion collisions, produced in particle accelerators such as the
Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) led
to the discovery of a new state of matter: the quark gluon plasma (QGP).
Understanding the factors that affect the confinement/deconfinement transition
between the hadronic and quark-gluon plasma (QGP) phases of strongly
interacting matter is one of the major goals of high energy physics. In the
hadronic phase, the Quantum Chromodynamics (QCD) matter is confined, while in
the QGP the quarks and gluons interact strongly but are not confined [1]. The
QGP cannot be observed directly. All the information about this special state
of matter, that behaves like a perfect fluid, comes from the particles
detected after hadronization, see [2, 3, 4].

Gauge/gravity duality [5, 6, 7] motivated the development of many interesting
tools for describing some aspects of systems governed by strong interactions,
as can be seen, for example, in[8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In
particular, it is possible to find holographic representations for a plasma
with finite density, that corresponds in the QGP to a finite quark chemical
potential [18, 19, 20, 21, 22, 14, 23, 24, 25, 26, 27, 28, 29]. The
gravitational dual representation of QCD matter at finite density is given by
a charged anti de-Sitter (AdS) black hole (BH), where the BH charge is related
to the quark density in the gauge theory side of the duality. From this type
of models, it is possible to investigate the confinement/deconfinement phase
transition at large quark density.

In a recent work [26], the confinement/deconfinement transition in the
Einstein-Maxwell-dilaton model was studied, using the dilaton that
characterizes the soft wall model[12]. The critical temperatures at different
values of the chemical potential was obtained from the analysis of Polyakov
loops. The results obtained are in agreement with the QCD phase diagram, in
the sense that the critical temperature decreases as the quark density
increases. In reference [13], the critical temperature for a plasma at zero
quark density was determined by applying the Hawking-Page transition approach,
which is based on the analysis of the regularized black hole free energy. Our
purpose here is to study the confinement/deconfinement phase transition in the
soft wall model at finite quark density, following the Hawking-Page approach,
and compare our results with the ones obtained from the Polyakov loop approach
in [26].

This work is organized as follows: in Section 2, we present the charged AdS
black hole geometry and discuss how the Hawking temperature is affected by the
BH charge. In Section 3, we construct the regularized BH action in the soft
wall AdS/QCD model at finite density and temperature, in the presence of the
heavy vector mesons. The critical temperatures of confinement/deconfinement
transition at different quark chemical potentials are then studied in Section
4, where the results are obtained from numerical analysis. The Hawking-Page
curve of the system is obtained. Section 5 contains our final considerations.

##  2 Charged AdS black hole geometry

The geometry that corresponds to a dual description of the quark-gluon plasma
with finite density $\mu$ was studied in refs. [18, 19, 20], and is given by a
5-D AdS charged black hole space given by the metric

$ds^{2}=\frac{L^{2}}{z^{2}}\left(-f(z)dt^{2}+d\overrightarrow{x}^{2}+\frac{dz^{2}}{f(z)}\right)\;,$
(2.1)

with

$f(z)=1-\frac{z^{4}}{z_{h}^{4}}-q^{2}zh^{2}z^{4}+q^{2}z^{6}\;,$ (2.2)

where $z_{h}$ is the location of the horizon, such that $f(z_{h})=0$, and $q$
is a parameter that is proportional to the BH charge. This metric is
equivalent to the Reissner-Nordstrom (RN) AdS BH, assuming that the BH charge
($Q$) is related to $q$ through, see [21],

$Q^{2}=\frac{3g_{5}^{2}L^{2}}{2\kappa^{2}}q^{2}\;.$ (2.3)

The hadronic (confined) matter phase, in turn, is described by the thermal AdS
space without a black hole, which is given by Eq. (2.1) taking $f(z)=1$, i.e.,
by the line element

$ds^{2}=\frac{L^{2}}{z^{2}}\left(-dt^{2}+d\overrightarrow{x}^{2}+dz^{2}\right)\;.$
(2.4)

Performing the Wick rotation, $t\rightarrow i\tau$, the compactified time
coordinate ($\tau$) is periodic with the BH period given by $\beta=1/T$, being
$T$ the BH temperature, which is associated to the charge $q$ and the horizon
position of the black hole according to the Hawking equation

$T=\frac{|f^{\prime}(z)|_{(z=z_{h})}}{4\pi}=\frac{1}{\pi
z_{h}}\left(1-\frac{q^{2}z_{h}^{6}}{2}\right)\;,$ (2.5)

that comes from the condition of absence of conical singularity at the horizon
[30]. By requiring that both geometries, the thermal and BH AdS ones, possess
the same shape at the asymptotic ultraviolet limit, $z=\epsilon$ with
$\epsilon\rightarrow 0$, one finds that the period of the thermal AdS space in
the Euclidean signature is $\beta^{\prime}=\beta\sqrt{f(\epsilon)}$, see [13].
With the periods defined in this way, we are able to apply the Hawking-Page
approach, and then analyze the confinement/deconfinement phase transition via
holography.

##  3 Soft wall AdS/QCD model at finite density

In the holographic soft wall AdS/QCD model, one introduces an energy parameter
that breaks the conformal symmetry of the action. This is done trough the
introduction of a dilaton background field, $\Phi(z)=c^{2}z^{2}$, that
contains the infrared (IR) energy parameter ($\sqrt{c}$). At finite
temperature, the gravity part of the action of the model is written in the
general form

$I_{G}=-\frac{1}{2\kappa^{2}}\int_{0}^{z_{f}}dz\int
d^{4}x\sqrt{-g}e^{-\Phi}\left(R-\Lambda\right)\;,$ (3.1)

where $\kappa$ is the gravitational coupling associated with the Newton
constant, and $\Lambda$, the cosmological constant. The time coordinate runs
from $0$ to $\beta$ (or $\beta^{\prime}$ for the thermal AdS case). The upper
limit of the integration over the holographic coordinate ($z_{f}$) depends on
the geometry, such that $z_{f}=z_{h}$ for the BH space, while
$z_{f}\rightarrow\infty$ for the thermal AdS one that does not posses an
horizon. In AdS spaces, $\Lambda$ and the curvature are related to the AdS
radius by

$\Lambda=\frac{3}{5}R=-\frac{12}{L^{2}}\;.$ (3.2)

In order to represent the quark density in this holographic approach, one
introduces an U(1) vector field $V_{M}=(V_{\mu},V_{z})$ with
$\mu=\\{0,1,2,3\\}$ and action

$I_{VF}=-\frac{1}{4g_{5}^{2}}\int_{0}^{z_{f}}dz\int
d^{4}x\sqrt{-g}e^{-\Phi}F_{MN}F^{MN}\;,$ (3.3)

where $F_{MN}=\partial_{M}V_{N}-\partial_{N}V_{M}$, and $g^{2}_{5}$ is a five-
dimensional gauge coupling constant.

The total action is then

$I=I_{G}+I_{VF}\;,$ (3.4)

and one chooses a solution for the vector field in the RN AdS black hole
background of the form[18, 21]

$\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle
A_{0}(z)\;=i(C-Qz^{2})\;,$ $\displaystyle V_{i}$ $\displaystyle=$
$\displaystyle V_{z}=0\quad(i=1,2,3)\;,$ (3.5)

where $C$ is a constant, $\kappa_{0}$ is a dimensionless parameter, and $Q$,
the black hole charge. The $i$-factor appears due to Wick rotation.

Associating the boundary value of the $A_{0}$ field with the quark chemical
potential $A_{0}(0)=i\mu$, one finds $C=\mu$. Thus

$A_{0}(z)=i(\mu-\eta qz^{2})\;,$ (3.6)

where $\eta=\sqrt{\frac{3g^{2}_{5}L^{2}}{2\kappa^{2}}}$, according to Eq.
(2.3).

The requirement of a regular norm of the vector field is satisfied imposing
the Dirichlet boundary condition $A_{0}(z_{h})=0$, see [22, 14, 31, 28], which
yields the following relation between the charge $q$ and the quark chemical
potential:

$\displaystyle\mu=\eta qz_{h}^{2}=Qz_{h}^{2}\;.$ (3.7)

This relation will be fundamental to obtain the critical temperatures at
different densities of the medium, and to determine the Hawking-Page curve.

In order to obtain the on-shell action, one must replace Eq. (3.2) into Eq.
(3.1), and Equations (3.5) and (3.6) into Eq. (3.3). Having done this, one
gets the following total on-shell action for the AdS black hole,

$\displaystyle
I_{BH}(\epsilon)=V_{3D}\,\beta\int_{\epsilon}^{z_{h}}dz\frac{e^{-cz^{2}}}{z^{5}}\left(\frac{4L^{3}}{\kappa^{2}}+\frac{2L}{g^{2}_{5}}Q^{2}z^{6}\right)\;,$
(3.8)

and, for the thermal AdS space,

$\displaystyle
I_{AdS}(\epsilon)=V_{3D}\,\beta\sqrt{1-\frac{\epsilon^{4}}{z^{4}}-q^{2}z_{h}^{2}\epsilon^{4}+q^{2}\epsilon^{6}}\int_{\epsilon}^{\infty}dz\frac{e^{-cz^{2}}}{z^{5}}\left(\frac{4L^{3}}{\kappa^{2}}+\frac{2L}{g^{2}_{5}}Q^{2}z^{6}\right)\;,$
(3.9)

where $V_{3D}$ is the trivial spatial volume corresponding to coordinates
$\vec{x}$, and $\epsilon$ is the ultraviolet (Uv) regulator. We need to
introduce this regulator because both actions, $I_{BH}$ and $I_{AdS}$, are
infinite in the UV limit $\epsilon\rightarrow 0$.

###  3.1 Regularized black hole action density

In order to eliminate the trivial transverse volume, one defines the action
density by $\mathcal{E}=I/V_{3D}$, such that

$\displaystyle\mathcal{E}_{BH}(\epsilon)=I_{BH}(\epsilon)/V_{3D}\quad\text{and}\quad\mathcal{E}_{AdS}(\epsilon)=I_{AdS}(\epsilon)/V_{3D}\;,$
(3.10)

according to Equations (3.8) and (3.9), respectively. Then, to obtain a finite
physical quantity, we apply the Hawking-Page approach and define the
regularized black hole action density ($\bigtriangleup\mathcal{E}$), given by
the difference between the black hole and thermal AdS action densities,
namely,

$\displaystyle\bigtriangleup\mathcal{E}=\lim_{\epsilon\rightarrow
0}\left[\mathcal{E}_{BH}(\epsilon)-\mathcal{E}_{AdS}(\epsilon)\right]\;.$
(3.11)

This regularized density does not possess UV divergences, which allows us to
take the limit $\epsilon\rightarrow 0$. After integrating over the holographic
coordinate and using Eq. (2.3), one finds the expression

$\displaystyle\bigtriangleup\mathcal{E}(\bar{\beta},\bar{q},\bar{z_{h}})=\frac{e^{-\bar{z_{h}}^{2}}\bar{\beta}c^{3/2}L^{3}}{\kappa^{2}\bar{z_{h}}^{4}}\left[(-1+\bar{z_{h}}^{2})-\frac{3}{2}\bar{q}^{2}\bar{z_{h}}^{4}+e^{\bar{z_{h}}^{2}}\bar{z_{h}^{4}}\text{Ei}(-\bar{z_{h}}^{2})\right]\;,$
(3.12)

wherein we have used the definition of the dimensionless variables:

$\displaystyle\bar{z_{h}}$ $\displaystyle=$ $\displaystyle z_{h}\sqrt{c}\;,$
(3.13) $\displaystyle\bar{\beta}$ $\displaystyle=$
$\displaystyle\beta\sqrt{c}\;,$ (3.14) $\displaystyle\bar{q}$ $\displaystyle=$
$\displaystyle q/c^{3/2}\;,$ (3.15)

and $\text{Ei}(x)=-\int_{-x}^{\infty}e^{-t}/t\,dt$. If we take the limit of
null chemical potential, $\mu\rightarrow 0$ (or $q\rightarrow 0$), we recover
the Herzog expression for the regularized BH action density in the soft wall
for the case with zero quark density [13], eliminating the contribution that
comes from the vector field action.

From the definition of the dimensionless variables (3.13)-(3.15), we can
obtain results that do not depend on the value of the IR energy parameter
$\sqrt{c}$. The only condition is that $c$ must be positive, $c>0$, otherwise,
the expression of Eq. (3.12) is not valid. The stability of the system is
determined by the sign of $\bigtriangleup\mathcal{E}$. When
$\bigtriangleup\mathcal{E}$ is positive, the free energy associated to the
black hole will be greater than the thermal AdS one, meaning that the thermal
AdS configuration is stable. On the other hand, if $\bigtriangleup\mathcal{E}$
is negative, the black hole is stable. In the gauge/gravity duality, the AdS
space with a black role corresponds to the quark-gluon plasma phase, while the
thermal AdS, to the confined hadronic one. The confinement/deconfinement phase
transition occurs when $\bigtriangleup\mathcal{E}=0$. This equation, however,
does not possess analytical solution, what compels us to apply numerical
methods.

##  4 Critical temperatures and Hawking-Page curve

The critical temperature depends on the critical horizon position
(${z_{h}}_{c}$) in which the regularized BH action density vanishes. As we
will see, this critical horizon is sensitive to the value of the BH charge (or
quark chemical potential). In order to write $\bigtriangleup\mathcal{E}$ as a
function of the dimensionless quark chemical potential

$\bar{\mu}=\mu/\sqrt{c}=\eta\bar{q}\bar{z_{h}}^{2}$ (4.1)

and $\bar{z_{h}}$, one must replace

$\displaystyle\bar{\beta}=\frac{\pi\bar{z_{h}}}{1-\frac{\bar{\mu}^{2}\bar{z_{h}}^{2}}{2}}\;,$
(4.2)

into Eq. (3.12). In this case, one can define

$\displaystyle\bar{\bigtriangleup\mathcal{E}}(\bar{\mu},\bar{z_{h}})=\frac{\pi\bar{z_{h}}}{1-\frac{\bar{\mu}^{2}\bar{z_{h}}^{2}}{2}}H(\bar{\mu},\bar{z_{h}})\;,$
(4.3)

where
$\bar{\bigtriangleup\mathcal{E}}=\kappa^{2}/(c^{3/2}L^{3})\bigtriangleup\mathcal{E}$,
together with

$\displaystyle
H(\bar{\mu},\bar{z_{h}})=e^{-\bar{z_{h}}^{2}}\left[\frac{(-1+\bar{z_{h}}^{2})}{\bar{z_{h}}^{4}}-\frac{3\bar{\mu}^{2}}{2\eta^{2}\bar{z_{h}}^{4}}+e^{\bar{z_{h}}^{2}}\text{Ei}(-\bar{z_{h}}^{2})\right]\;.$
(4.4)

From now on, we will assume that $\eta=1$. Naturally, the phase transition
occurs when

$\displaystyle
H(\bar{\mu},\bar{z_{h}})=0\quad\text{at}\quad\bar{z_{h}}=\bar{z_{h}}_{c}\;,$
(4.5)

being this equation only determined by the BH horizon position and the quark
chemical potential. In Figure 1, we plot $H$ as a function of $\bar{z_{h}}$ at
different fixed values of the chemical potential. We observe that the critical
horizon positions, the roots of $H(\bar{\mu},\bar{z_{h}})$, increases as
$\bar{\mu}$ increases.

Figure 1: $H(\bar{\mu},\bar{z_{h}})$ as a function of $\bar{z_{h}}$ at
different fixed values of the quark chemical potential. The
confinement/deconfinement transition occurs at $H(\bar{\mu},\bar{z_{h}})=0$,
i.e., the roots of the respective functions.

The values of the critical horizons can be determined by applying numerical
methods, and their values are displayed in Table 1 below. From each pair
$(\bar{\mu},\bar{z_{c}})$, we determine a critical $\bar{\beta}_{c}$ using Eq.
(4.2), and the corresponding dimensionless critical temperature is then
computed through the relation

$\bar{T}_{c}=T_{c}/\sqrt{c}=1/\bar{\beta}_{c}\;.$ (4.6)

This result shows that $T_{c}$ decreases as the quark chemical potential
increases. Such a description is in agreement with the QCD phase diagram. As
it is well known, as the density of the medium increases, the temperature
where the transition between the hadronic and plasma phases occurs decreases.

In Figure 2, we plot the points that describe the behaviour of the Hawking-
Page curve, using the data from Table 1. The region below the curve
corresponds to the confined phase of QCD matter, while the region above
represents the quark-gluon plasma.

To obtain the physical values of $T_{c}$ and $\mu$, we must determine the
value of $\sqrt{c}$ using QCD phenomenology. For the soft wall model
describing light mesons, this has already been done in [13]. The IR energy
parameter can be fixed at zero density, from the mass of the lightest
$\rho$-meson, that yields $\sqrt{c}=338\,\text{MeV}$. The physical
temperatures are then evaluated multiplying $\bar{T}_{c}$ by this value. At
zero density, one finds $T_{c}(\mu=0)=0.491728\sqrt{c}=191\,\text{MeV}$, see
[13]. This prediction is in accordance with lattice predictions [32].

$\bar{\mu}$ | $\bar{z_{h}}_{c}$ | $\bar{T}_{c}$
---|---|---
$\,\,\,0.000000\,\,\,$ | $\,\,\,0.647329\,\,\,$ | $\,\,\,0.491728\,\,\,$
$\,\,\,0.081649\,\,\,$ | $\,\,\,0.651867\,\,\,$ | $\,\,\,0.487613\,\,\,$
$\,\,\,0.163299\,\,\,$ | $\,\,\,0.664767\,\,\,$ | $\,\,\,0.476008\,\,\,$
$\,\,\,0.244948\,\,\,$ | $\,\,\,0.684109\,\,\,$ | $0.458759$
$\,\,\,0.326598\,\,\,$ | $\,\,\,0.707323\,\,\,$ | $0.438013$
$\,\,\,0.408248\,\,\,$ | $\,\,\,0.731865\,\,\,$ | $0.415517$
$\,\,\,0.489897\,\,\,$ | $\,\,\,0.755725\,\,\,$ | $0.392332$
$\,\,\,0.571547\,\,\,$ | $\,\,\,0.777637\,\,\,$ | $0.368900$
$\,\,\,0.653197\,\,\,$ | $\,\,\,0.797013\,\,\,$ | $0.345257$
$\,\,\,0.734846\,\,\,$ | $\,\,\,0.813743\,\,\,$ | $0.321232$
$\,\,\,0.816497\,\,\,$ | $\,\,\,0.827991\,\,\,$ | $0.296584$
$\,\,\,0.898146\,\,\,$ | $\,\,\,0.840045\,\,\,$ | $0.271071$
$\,\,\,0.979795\,\,\,$ | $\,\,\,0.850223\,\,\,$ | $0.244480$
$\,\,\,1.061445\,\,\,$ | $\,\,\,0.858825\,\,\,$ | $0.216634$
$\,\,\,1.143095\,\,\,$ | $\,\,\,0.866117\,\,\,$ | $0.187394$
$\,\,\,1.224744\,\,\,$ | $\,\,\,0.872325\,\,\,$ | $0.156646$
$\,\,\,1.306394\,\,\,$ | $\,\,\,0.877634\,\,\,$ | $0.124304$
$\,\,\,1.388044\,\,\,$ | $\,\,\,0.878122\,\,\,$ | $0.102522$
$\,\,\,1.469693\,\,\,$ | $\,\,\,0.870546\,\,\,$ | $0.066372$
$\,\,\,1.551343\,\,\,$ | $\,\,\,0.875152\,\,\,$ | $0.028507$
$\,\,\,1.610000\,\,\,$ | $\,\,\,0.878122\,\,\,$ | $0.000224$
Table 1: Quark chemical potentials, critical horizon positions, and the
corresponding critical temperatures of confinement/deconfinement phase
transition: dimensionless variables.

An interesting point to investigate is the prediction for phase transitions
that occur at high densities and very low temperatures. Performing a numerical
analysis one obtains that the phase transition at zero temperature occurs
between $\bar{\mu}=1.6102$ and $\bar{\mu}=1.6104$. Within this precision, we
estimate that the transition at zero temperature occurs at

$\displaystyle\bar{\mu}_{0}\equiv\bar{\mu}(T=0)=1.610\;,$ (4.7)

that is,

$\displaystyle\mu_{0}=1.610\sqrt{c}=544.18\,\text{MeV}\;.$ (4.8)

This is the holographic prediction, based on the Hawking-Page method, for the
quark density in which the transition between the confined hadronic and
deconfined plasma phases occurs at approximately zero temperature. For higher
values of $\mu$ the QCD matter is a quark-gluon plasma.

The relation between the quark and baryon densities is $\mu=\mu_{B}/3$, see
[18]. Therefore, we estimate that the transition at zero temperature occurs at
baryon density $\mu_{B}\approx 1632,54\,\text{MeV}$. From Polyakov loops in
the $2+1$ flavor model [26], it is predicted that the transition without
plasma rotation would occur approximately at $\mu_{B}\approx
1200\,\text{MeV}$, see Figure 6 of [26]. Our prediction is valid assuming that
$\eta=1$ in the vector field solution (3.6). Other predictions could be
obtained by varying the parameter $\eta$, that changes the relation between
$q$ and the chemical potential. The fact that $\eta$ affects the
confinement/deconfinement transition was observed in [18]. The predictions
would have agreed perfectly if we had assumed $\eta\approx 0.7351$. We must
observe that the action used in [26] is not the same as ours, since they have
considered an AdS/QCD model for a Einstein-Maxwell-Dilaton gravitational
system, see Eq. (2.1) of [26]. The interesting part of the result is that both
holographic models predict a baryon density at the same energy scale.

Figure 2: Hawking-Page transition: Critical temperature versus quark density
in the soft wall AdS/QCD model.

##  5 Final considerations

Holographic models for confinement/deconfinement phase transition in high-
energy physics at finite density were studied in [20, 21, 22, 25, 26, 23]. The
authors of [20] analyzed the scalar glueball and vector meson spectral
functions in the soft wall model. The confinement/deconfinement transition in
the hard wall model was investigated in [21, 22], by introducing a IR cutoff
in the holographic coordinate direction. In both cases, the authors considered
the RN/AdS metric solution, that was also used here, see Equations
(3.5)-(3.6). In [23], the geometry of a charged black hole is not considered,
and a prediction of the quark density at low critical temperatures is not
presented. The chiral phase transition in an improved soft wall was
investigated in [25], for a model with $2+1$ flavors, in which the authors
also studied the confinement/deconfinement transition, but not considering the
Hawking Page transition between the black hole geometry and the thermal AdS. A
similar analysis was performed in [26], in which the phase transition was
investigated through Polyakov loops, taking into account rotation effects.
Again, the critical temperatures at different densities showed to be in
harmony with QCD phase diagram, with $T_{c}$ decreasing as $\mu$ increases.

In the present work, we have applied the the Hawking-Page approach to the soft
wall model in the case of finite quark density, in a similar way as it was was
done for the zero density case in [13]. We construct, a regularized black hole
action density, from which the UV divergencies are eliminated, allowing us to
apply numerical methods to study the plasma thermodynamics at very low
critical temperatures and large densities. Such a construction is possible in
AdS/QCD setup through the geometry of a charged AdS black hole in the
gravitation side of the duality, see Eq. (2.1), being the BH charge associated
to the quark chemical potential, according to the relation (3.7). Differently
of [13], where the author assumed that $\mu=0$, we have considered the
contribution of the vector mesons to the total action, as discussed in Section
3.

The total regularized action density is given by Eq. (3.12), that defines the
critical temperatures at different densities, when the regularized free energy
density of the charged BH vanishes. As the quark density increases, we observe
that the horizon positions at the phase transition also increases, see Table
1. By this reason, the critical temperatures decrease as $\mu$ increases,
being this result in harmony with QCD phase diagram, see Hawking-Page curve of
Figure 2. This shows the power of the Hawking-Page method, based on the
holographic description of dense matter, in agreement with the results
obtained in [26] from the analysis of Polyakov loops.

Throughout the paper, we have worked with dimensionless variables, as defined
by Equations (3.13)-(3.15), (4.1) and (4.6). In this way, our results are
independently of the IR parameter introduced by the dilaton background field,
under only the condition that it must be positive. This parameter can be fixed
using QCD phenomenology. For the soft wall model, it is estimated that
$\sqrt{c}=338\,\text{MeV}$, thus determined by the mass of the lightest
$\rho$-meson, see [13]. This value leads to a critical temperature at $\mu=0$
in complete agreement with lattice QCD data, see [13, 32]. This matching
brings confidence for the new holographic predictions of critical temperatures
of confinement/deconfinement transition at finite quark density, displayed on
Table 1, where the same approach of [13] was applied. As the quark density
increases, there is an interesting point where the phase transition occurs at
zero temperature. From a numerical analysis in the soft wall model, we
estimate that it happens at quark density $\mu_{0}=544.18\,\text{MeV}$. Our
result was obtained assuming that $\eta$ (the free parameter that appears in
the RN solution for the vector meson field) is equal to one, see Eq. (3.6).
Then, for $\mu\geq\mu_{0}$, the QCD matter will be described by a quark-gluon
plasma, even at very low temperatures. The present holographic prediction of
$\mu_{0}$ is at the same energy scale of the one obtained via Polyakov loops
in [26], where the authors have used an EMD gravitational model.

Acknowledgments: The authors are supported by FAPERJ — Fundação Carlos Chagas
Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, CNPq - Conselho
Nacional de Desenvolvimento Científico e Tecnológico. This work received also
support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brasil (CAPES) - Finance Code 001.

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# Black holes in Einstein-Gauss-Bonnet gravity: dynamical and 4-dimensional
novel stationary black hole

Bhramar Chatterjee<EMAIL_ADDRESS>
###### Abstract

We consider black hole in Einstein-Gauss-Bonnet(EGB) gravity in a dynamical
setup and also the stationary black hole in recently discoveredGlavan
$4$-dimensional novel Einstein-Gauss-Bonnet gravity. For dynamical EGB black
hole in completely local settings we calculate the temperature associated with
the horizon. For the stationary black hole in $4$-dimensional novel EGB theory
we explore the thermodynamic properties including calculation of temperature
and entropy.

Gauss-Bonnet gravity, trapping horizon; horizon temperature.

###### pacs:

04.50.Kd ; 04.70.Dy

##  I Introduction

Alternative theories of gravity have been a prevalent subject of current
research in gravitational physics. The motivation behind this is to look for
answers to some fundamental problems which can not be resolved in Einstein’s
general relativity (GR), for example the singularity. Since inclusion of
higher order curvature terms in the action as corrections seems natural
progression from Einstein’s GR, many such theories exist, and Einstein-Gauss-
Bonnet gravity(EGB) which comprises of the Gauss-Bonnet term (quadratic in
curvature) added to the Lagrangian of Einstein-Hilbert action is one of the
most promising of such theories Boulware ; Zumino . This particular
combination of quadratic curvature terms appears as the second order
correction in the action of Lanzcos-Lovelock gravityLovelock ; Lovelock1 ;
Lanzcos ,the zeroth and first order terms being the cosmological constant and
Einstein-Hilbert action respectively. The advantage of Lanzcos-Lovelock
gravity is that the equations of motion contain only up to second derivative
of metric even if higher curvature terms are present in the action, which
makes the theory ghost free. Also, Gauss-Bonnet term appears in the low energy
limit of heterotic string theoryZwiebach ; Gross ; Gross1 ; Metsaev ; Metsaev1
which is another reason to consider this theory as an alternative to
Einstein’s general relativity.
Now, the effect of these higher order curvature terms is expected to become
apparent in extremely curved domain of spacetime. So study of black holes in
Einstein-Gauss-Bonnet gravity can reveal different features from Einstein’s
gravity. Recent availability of high precision observational data in the
strong gravity regime around black holes made it possible to study different
phenomena related to black holes more thoroughly and preciselyBerti ; Barack .
So this is a good time to test alternative theories of gravity in the
background of black holes compared to standard general relativity.
Stationary black holes has been studied exhaustively over the years,
particularly for their thermodynamic properties. But black holes in reality
rarely reach equilibrium. A physical black hole may grow by absorbing nearby
matter or coalesce with another black hole or emit Hawking radiation, all of
which are highly dynamical processes. Moreover the notion of black hole event
horizon, being a global one, is not at all suitable for practical purposes. To
locate the event horizon of a black hole one has to know the entire structure
of the spacetime beforehand, which is not possible always. For all of these
reasons some local definitions of black hole horizons which incorporate the
dynamical nature of the system has been proposed over the last few years with
varying degrees of success. Examples include trapping horizonHayward1 ;
Hayward2 , slowly evolving horizonBooth , dynamical horizonAshtekar1 ;
Krishnan  and isolated horizonAshtekar2 . All these horizons are locally
defined, i.e unlike event horizon only local properties of the spacetime are
sufficient to locate the horizon. Also these horizons are endowed with nice
properties i.e they obey the laws of black hole mechanics and a temperature
and entropy can be associated with them.

In the first part of this work we will consider a dynamical black hole in
Einstein-Gauss-Bonnet gravity in the context of Hayward’s trapping horizon.
For a $4$-dimensional spacetime with spherical symmetry the trapping horizon
is defined in a $2+2$ framework using only the local geometry without any
assumptions on the global developments of the spacetime in which the horizon
is embedded. For Einstein-Gauss-Bonnet gravity, trapping horizon and it’s
properties has been studied in detail in Nozawa . There are two branches of
solutions in EGB gravity : the GR branch which has an asymptotic general
relativistic limit, and the non-GR branch which has no such limit. It is found
that for the GR branch, for a $n$-dimensional spacetime which has symmetries
corresponding to a $(n-2)$-dimensional maximally symmetric space with constant
curvature, the trapping horizon has very similar behavior to that of in
general relativity under the null energy condition, whereas the non-GR branch
solution behaves completely differently. Since the spacetime is dynamical,
there is no timelike Killing vector, but because of the maximally symmetric
nature of the spacetime one can define a Kodama vector Kodama , in a similar
manner to that of spherically symmetric $4$-dimensional dynamical spacetime,
which plays an equivalent role of Killing vector. We shall calculate the
temperature of the trapping horizon in EGB gravity by considering radiation
from the horizon in the form of positive frequency field modes of the Kodama
vector. This method has previously been employed to calculate temperature of
horizons in different $4$-dimensional spacetimes Chatterjee1 ; Chatterjee2 ;
Chatterjee3 . The set up is essentially local and dynamical. The nature of the
trapping horizon is characterized by the energy conditions, without specifying
the energy momentum tensor of the matter field.

For the second part of this paper we will consider the stationary black hole
solution in the recently discovered $4$-dimensional novel Einstein-Gauss-
Bonnet gravity Glavan . Following Lovelock’s theorem Lovelock ; Lovelock1 ,
Einstein’s GR with the cosmological constant is the unique theory of gravity
in $4$-dimension if we assume diffeomorphism invariance, metricity and second
order equations of motion. The Gauss-Bonnet term, which is the next higher
order term in curvature, becomes a total derivative in $4$-dimensions, hence
does not appear in the Einstein’s equations. However, recently Glavan and Lin
Glavan  has proposed a modified theory of gravity which has Gauss-Bonnet term
contributing in the $4$-dimensional gravitational dynamics. According to
Glavan , this theory is formulated in $D>4$ dimensions and the action consists
of Einstein-Hilbert term plus the cosmological constant and the Gauss-Bonnet
term multiplied by a factor of $1/(D-4)$. The $4$-dimensional theory is
recovered as the limit $D\rightarrow 4$ at the level of equations of motion
and there is significant contribution from the Gauss-Bonnet term. The choice
of action is ad hoc at the best but this theory has some very interesting
features. Particularly the vacuum black hole solution which behaves like the
Schwarzschild black hole at large distances is very different from
Schwarzschild solution near the horizon. This black hole solution admits two
horizons, outer and inner, despite being static and without any charges and
the gravitational potential inside the inner horizon becomes repulsive, which
suggests a white hole spacetime inside the black hole. There has been a lot of
activities concerning the black hole in this $4$-dimensional novel Einstein-
Gauss-Bonnet gravity Fernandes ; Ai ; Mukhoyama ; Konoplya ; Ghosh ; Ghosh1 ;
Dadhich . In this paper we shall construct double null coordinates for this
black hole and calculate the temperature, entropy and other thermodynamic
properties associated with the black hole horizon.

The plan of the paper is as follows : in the first section we will discuss
basics of Einstein-Gauss-Bonnet gravity briefly starting from the action. The
dynamical black hole will be discussed in the next section. After introducing
the concept of future outer trapping horizon (FOTH, which characterizes a
black hole in dynamical scenario) for higher dimensional spacetime and Kodama
vector, we shall construct the field modes of Kodama vector and calculate the
probability current which will give the temperature of the horizon. In the
last section we will discuss $4$-dimensional novel EGB black hole and it’s
thermodynamic properties and conclude. We have chosen $c=\hbar=1$.

##  II Einstein Gauss-Bonnet gravity

We begin with a brief discussion of Einstein-Gauss-Bonnet(EGB) gravity. In the
presence of a cosmological constant the action is given by,

$\displaystyle S=\int d^{n}x\sqrt{-g}\Big{[}\frac{1}{16\pi
G_{n}}(R-2\Lambda+\alpha L_{GB})\Big{]}+S_{matter},$ (1)

where $G_{n}$,$R$ and $\Lambda$ are the $n$-dimensional gravitational
constant, Ricci scalar and the cosmological constant respectively. The Gauss-
Bonnet term $L_{GB}$ ,which is quadratic in curvature, is a combination of
Ricci scalar, Ricci tensor $R_{\mu\nu}$ and Riemann tensor
$R_{\mu\nu\alpha\beta}$ as

$\displaystyle
L_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}.$
(2)

$\alpha(\geq 0)$ is the Gauss-Bonnet coupling constant and $S_{matter}$ is the
contribution from matter field. In $4$-dimensions $L_{GB}$ does not contribute
to the field equations since it becomes a total derivative. For $D>4$, the
field equation derived from the action (1) is

$\displaystyle G^{\mu}_{\nu}+\alpha
H^{\mu}_{\nu}+\Lambda\delta^{\mu}_{\nu}=8\pi G_{n}T^{\mu}_{\nu},$ (3)

where,

$\displaystyle G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R,$ (4)
$\displaystyle
H_{\mu\nu}=2[RR_{\mu\nu}-2R_{\mu\alpha}R^{\alpha}_{\nu}-2R^{\alpha\beta}R_{\mu\alpha\nu\beta}+R_{\mu}^{\alpha\beta\gamma}R_{\nu\alpha\beta\gamma}]-\frac{1}{2}g_{\mu\nu}L_{GB}.$
(5)

The field equations (3) contain only up to linear second derivative of the
metric. $T_{\mu\nu}$ is the energy-momentum tensor of the matter fields. Now
following Nozawa , if we assume the $n$-dimensional spacetime
$(M^{n},g_{\mu\nu})$ is a warped product of a $(n-2)$-dimensional constant
curvature space $(K^{n-2},\gamma_{ij})$ and a $2$-dimensional orbit spacetime
$(M^{2},g_{ab})$ under the isometry of $(K^{n-2},\gamma_{ij})$ , the metric
can be written as,

$\displaystyle
g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{ab}(y)dy^{a}dy^{b}+r^{2}(y)\gamma_{ij}dz^{i}dz^{j},$
(6)

where $a,b=0,1;i,j=2,...,n-1$, $\gamma_{ij}$ is the unit metric on
$(K^{n-2},\gamma_{ij})$ with sectional curvature $k=-1,0,+1$, $r$ is a scalar
on $(M^{2},g_{ab})$ with the boundary at $r=0$. If there are no matter fields,
we will get the vacuum solution which is given by the generalized Boulware-
Deser-Wheeler solutionBoulware ; Wheeler ; Cai ; Soh

$\displaystyle
ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{F(r)}+r^{2}\gamma_{ij}dz^{i}dz^{j},$ (7)

with

$\displaystyle
F(r)=k+\frac{r^{2}}{2\tilde{\alpha}}\Big{[}1\mp\sqrt{1+\frac{64\pi
G_{n}\tilde{\alpha}m}{(n-2)V^{k}_{n-2}r^{n-1}}+4\tilde{\alpha}\tilde{\Lambda}}\Big{]},$
(8)

where
$\tilde{\alpha}=(n-3)(n-4)\alpha,\tilde{\Lambda}=\frac{2\Lambda}{(n-1)(n-2)}$.
$V^{k}_{n-2}$ is the area of the unit $(n-2)$-dimensional space with constant
curvature and $m$ is the gravitational mass of the solution. The upper sign in
$F(r)$ refers to the GR branch because in the asymptotic limit this gives a
Schwarzschild-like solution. The lower sign is the non-GR branch.
When matter fields are present, the setup becomes dynamic. There are no longer
an event horizon or a Killing vector. But if the spacetime is maximally
symmetric, as we have assumed here, one can define a trapping horizon which
characterizes a black hole using only local geometry. This we will discuss in
the next section.

##  III Dynamical EGB black hole

To describe the trapping horizon of a dynamical black hole in EGB gravity it
is convenient to write the metric in double null coordinates as,

$\displaystyle ds^{2}=-2e^{-f(u,v)}dudv+r^{2}(u,v)\gamma_{ij}dz^{i}dz^{j}.$
(9)

The null vectors $\partial/\partial u$ and $\partial/\partial v$ are taken to
be future directed with $\partial/\partial u(\partial/\partial v)$ in the
inward(outward) direction. The expansions of the two radial null geodesics are
given by,

$\displaystyle\theta_{+}=(n-2)\frac{\partial_{v}r}{r},$ (10)
$\displaystyle\theta_{-}=(n-2)\frac{\partial_{u}r}{r}$ (11)

Now a marginal surface in this spacetime is a $(n-2)$-surface with
$\theta_{+}\theta_{-}=0$ and without any loss of generality we can choose
$\theta_{+}=0$. A future outer trapping horizon (FOTH) which corresponds to a
black hole horizon in this spacetime is a $(n-1)$\- hypersurface foliated by
marginal surfaces such that on each $(n-2)$-marginal surfaces $\theta_{-}<0$
and $\partial_{u}\theta_{+}<0$. This implies the ingoing null rays are
converging $(\theta_{-}<0)$, the outgoing null rays are converging inside the
FOTH $(\partial_{u}\theta_{+}<0)$, instantaneously parallel to the horizon
$(\theta_{+}=0)$ and diverging just outside the horizon , which indicates a
black hole spacetime.
The active gravitational mass of the trapping horizon is the generalized
Misner-Sharp mass $m$. Since the equations of motion contain quadratic
curvature terms, $m$ is quadratic in $e^{f}\theta_{+}\theta_{-}$ and is given
by Nozawa ,

$\displaystyle\frac{2}{(n-2)^{2}}r^{2}e^{f}\theta_{+}\theta_{-}=-k-\frac{r^{2}}{2\tilde{\alpha}}\Big{[}1\mp\sqrt{1+\frac{64\pi
G_{n}\tilde{\alpha}m}{(n-2)V^{k}_{n-2}r^{n-1}}+4\tilde{\alpha}\tilde{\Lambda}}\Big{]}$
(12)

The ”$-$” sign in front of the square root corresponds to the GR branch
solution which has a Schwarzschild like limit as $\alpha\rightarrow 0$. The
”$+$” sign gives the non-GR branch which behaves rather pathologically. The
FOTH in the GR branch has very similar properties as that of in general
relativity under the null energy condition (for details see Nozawa ) and in
the next subsection we will calculate temperature of this horizon.

###  III.1 Kodama vector and surface gravity

For a dynamical spacetime there is no Killing vector, so definition of time
becomes ambiguous. But in $4$-dimension if the spacetime is spherically
symmetric, Kodama vector $(K)$ generates a preferred flow of time Kodama .
Using Einstein’s equations it can be shown that the vector field
$J^{a}=G^{ab}K_{b}$ is divergence free where $G^{ab}$ is the Einstein tensor
Kodama ; Visser . This implies that even though the spacetime is completely
dynamical, $J^{a}$ is the locally conserved energy flux. So the Kodama vector
provides a preferred timelike direction. Also, it is shown in Hayward4 , that
the Noether charge associated with the Kodama current $J^{a}$ leads to the
Misner-Sharp mass Misner  of the spacetime. For $(n\geq 5)$-dimension, if the
spacetime is maximally symmetric as we have assumed here, we can define a
generalized Kodama vector by,

$\displaystyle K^{a}=\epsilon^{ab}D_{b}r,$ (13)

where $\epsilon^{ab}$ is a volume element of $(M^{2},g_{ab})$ and $D_{a}$ is a
linear connection compatible with the metric in $(M^{2},g_{ab})$. This vector
lies entirely on the transverse $(u,v)$ plane and using the metric (9) can be
written as,

$\displaystyle K=e^{f}(\partial_{v}r\partial_{u}-\partial_{u}r\partial_{v}).$
(14)

The dynamical surface gravity of the trapping horizon $(\kappa_{TH})$ is
defined as,

$\displaystyle K^{b}D_{[b}K_{a]}=\kappa_{TH}K_{a}.$ (15)

So, using (9) we have,

$\displaystyle\kappa_{TH}=\frac{1}{2}D^{2}r=-e^{f}\partial_{u}\partial_{v}r.$
(16)

The FOTH condition $(\partial_{u}\theta_{+}<0)$ makes sure that $\kappa_{TH}$
is positive on the FOTH. We will show that $\kappa_{TH}$ is indeed the
temperature of the FOTH using field modes of Kodama vector $(K)$ in the
following subsection.

###  III.2 Field modes and probability current

From the expression of Kodama vector (14), it is easy to see that any smooth
function of $r$ is a zero mode of $K$. Now we can evaluate other positive
frequency eigenmodes of $K$ using,

$\displaystyle iKZ_{\omega}=\omega Z_{\omega}.$ (17)

Here, $Z_{\omega}$ are the eigenfunctions corresponding to the positive
frequency $\omega$. The method of finding eigenmodes of Kodama vector has been
discussed in detail in Chatterjee2 ; Chatterjee3 . Following these one can
find

$Z_{\omega}=C(r)\begin{cases}\theta_{+}^{-\frac{i\omega}{\kappa_{TH}}}&\text{for}\;\theta_{+}>0\\\
(|\theta_{+}|)^{-\frac{i\omega}{\kappa_{TH}}}&{\rm
for}\;\theta_{+}<0.\end{cases}$ (18)

where the $(n-2)$-surfaces are not trapped ‘outside the trapping horizon’
($\theta_{+}>0$) and fully trapped ‘inside’ ($\theta_{+}<0$). The only
assumption made in this calculation is that $\kappa_{TH}$ is a slowly varying
function in some small neighbourhood of the FOTH. These are precisely the
modes which are defined outside and inside the dynamical horizon respectively
but not on the horizon. Now we have to keep in mind that the modes (18) are
not ordinary functions, but are distribution-valued. And as distributions (or
generalized functions) these are perfectly well behaved on the horizon. Using
the standard results Gel'fand ,the appropriate distributions for the
$Z_{\omega}$ are,

$\displaystyle Z_{\omega}=\lim_{\epsilon\to
0}C(r)(\theta_{+}+i\epsilon)^{-\frac{i\omega}{\kappa_{TH}}}=\left\\{\begin{array}[]{ll}C(r)\left(\theta_{+}\right)^{-\frac{i\omega}{\kappa_{TH}}}&\mbox{
for $\theta_{+}>0$,}\\\
C(r)|\theta_{+}|^{-\frac{i\omega}{\kappa_{TH}}}e^{\frac{\pi\omega}{\kappa_{TH}}}&\mbox{
for $\theta_{+}<0$. }\end{array}\right.$ (21)

The distribution (21) is well-defined for all values of $\theta_{+}$ and
$\kappa_{TH}$, and it is differentiable to all orders. The modes
$Z^{*}_{\omega}$ are given by the complex conjugate distribution

$\displaystyle Z_{\omega}^{*}=\lim_{\epsilon\to
0}{C(r)}^{*}(\theta_{+}-i\epsilon)^{\frac{i\omega}{\kappa_{TH}}}=\left\\{\begin{array}[]{ll}{C(r)}^{*}\left(\theta_{+}\right)^{\frac{i\omega}{\kappa_{TH}}}&\mbox{
for $\theta_{+}>0$,}\\\
{C(r)}^{*}|\theta_{-}|^{\frac{i\omega}{\kappa_{TH}}}e^{\frac{\pi\omega}{\kappa_{TH}}}&\mbox{
for $\theta_{+}<0$. }\end{array}\right.$ (24)

Let us now calculate the probability density in a single particle Hilbert
space for positive frequency solutions across the dynamical horizon along the
direction of the Kodama vector $K$,

$\varrho(\omega)=-\frac{i}{2}\Big{[}Z^{*}_{\omega}KZ_{\omega}-KZ^{*}_{\omega}Z_{\omega}\Big{]}=\omega
Z_{\omega}^{*}Z_{\omega}.$ (25)

A straightforward calculation gives, apart from a positive function of $r$,

$\displaystyle\varrho(\omega)$ $\displaystyle=$
$\displaystyle\omega(\theta_{+}+i\epsilon)^{-\frac{i\omega}{\kappa_{TH}}}(\theta_{+}-i\epsilon)^{\frac{i\omega}{\kappa_{TH}}}.$
$\displaystyle=$ $\displaystyle\begin{cases}\omega&\text{for}\;\theta_{+}>0\\\
\omega
e^{\frac{2\pi\omega}{\kappa_{TH}}}&\text{for}\;\theta_{+}<0.\end{cases}$

The conditional probability that a particle emits when it is incident on the
horizon from inside is,

$\displaystyle
P_{(emission|incident)}=\frac{P_{(emission{\cap}incident)}}{P_{(incident)}}=e^{-\frac{2\pi\omega}{\kappa_{TH}}}$
(27)

comparing this with the Boltzmann factor $e^{-\beta\omega}$ gives the
temperature of this dynamical horizon $T_{H}=\kappa_{TH}/2\pi$, which is the
desired result. This temperature for the Gauss-Bonnet trapping horizon is
obtained for the GR branch solution only, which is very similar to the
temperature of trapping horizon in $4$-dimension Chatterjee2 . The effect of
the Gauss-Bonnet term is embedded in $\kappa_{TH}$ which is indeed different
from it’s $4$-dimensional counterpart since the spacetime metrics are
different. But otherwise the trapping horizon for the GR branch in EGB gravity
behaves very similarly to the trapping horizon in $4$-dimensional GR. For the
non-GR branch the trapping horizon behaves quite differently under the null
energy condition, and the results are not easy to interpret.

##  IV The novel 4D-EGB black hole

According to Glavan , the $4$-dimensional novel EGB theory is formulated in
$n>4$ dimensions with the action ,

$\displaystyle S=\int d^{n}x\sqrt{-g}\Big{[}\frac{1}{16\pi
G_{n}}(R-2\Lambda+\frac{\tilde{\alpha}}{n-4}L_{GB})\Big{]},$ (28)

and the $4$-D theory is recovered in the limit $n\rightarrow 4$ at the level
of equations of motion. For a static spherically symmetric $4$-D spacetime
with a vanishing cosmological constant this leads to the solution,

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (29)

where

$\displaystyle f(r)=1+\frac{r^{2}}{2\alpha}\Big{(}1\mp\sqrt{1+\frac{8\alpha
M}{r^{3}}}\Big{)}.$ (30)

Here $d\Omega^{2}$ is the usual metric on unit 2-sphere, $\alpha$ is rescaled
GB coupling constant $(\alpha=16\pi\tilde{\alpha})$ and $M$ is mass of
isolated body. We have also chosen $G=1$. In the asymptotic limit
$(r\rightarrow\infty)$ the $"+"$ sign in $f(r)$ gives Schwarzschild-de Sitter
black hole with negative gravitational mass and the $"-"$ sign gives
Schwarzschild black hole with positive gravitational mass. So we choose $"-"$
sign in $f(r)$ for attractive gravity. This $f(r)$ has roots at

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-\alpha}.$ (31)

For $\alpha>0$, the roots are real for $M^{2}>\alpha$, while $M^{2}=\alpha$
gives the extremal case. So unlike the Schwarzschild solution, in this case
the black hole has two horizons : outer$(r_{+})$ and inner$(r_{-})$, just like
the Reissner-Nordström case where the electric charge $Q^{2}$ is replaced by
GB coupling constant $\alpha$ which is quite exceptional for a static
uncharged black hole. As was shown in Glavan , the gravitational potential has
a minimum and is attractive to the right and repulsive to the left of this
minimum. For $M^{2}>\alpha$, where two horizon exist this means the outer
horizon is the event horizon of a black hole while the inner horizon envelopes
a white hole spacetime inside. In the following subsections we will explore
the thermodynamic properties of the black hole particularly the temperature
and entropy associated with the outer horizon. To calculate temperature we
will use the formalism adopted for calculating temperature of dynamical EGB
black hole in the previous section.For this we need the metric in the double
null Kruskal coordinates so that the coordinate singularity can be removed.

###  IV.1 Double-null coordinates

Because of the presence of coordinate singularities at $r_{\pm}$, Kruskal
coordinates are required to extend the metric across these surfaces. However
one fact has to be mentioned clearly i.e a single Kruskal patch will not cover
the entire manifold in this case due to the presence of two horizons. The
Kruskal coordinates we construct to remove the singularity at the outer
horizon $(r_{+})$ fails to be regular at the inner horizon $(r_{-})$ and we
need a different set of coordinate transformation for the inner horizon. This
is similar to Reissner-Nordström black hole again. Let us construct the
Kruskal coordinates across the outer horizon (the black hole event horizon).
Near $r=r_{+}$ Poisson ,

$\displaystyle f(r)\approx
f^{\prime}(r)\mid_{r_{+}}(r-r_{+})=2\kappa_{+}(r-r_{+})$ (32)

where $\kappa_{+}\equiv\frac{1}{2}f^{\prime}(r_{+}).$ Then we have,

$\displaystyle
r_{*}=\int\frac{dr}{f(r)}=\int\frac{dr}{2\kappa_{+}(r-r_{+})}=\frac{1}{2\kappa_{+}}\ln|\kappa_{+}(r-r_{+})|.$
(33)

Now we can define null coordinates $u=t-r_{*}$, and $v=t+r_{*}$, and the
Kruskal coordinates are defined as $U_{+}=\mp e^{-\kappa_{+}u}$ and
$V_{+}=e^{\kappa_{+}v}$. The upper sign in $U_{+}$ is for $r>r_{+}$, and the
lower sign is for $r<r_{+}$. In these coordinates $f(r)\simeq-2U_{+}V_{+}$
near $r_{+}$ and the metric becomes,

$\displaystyle
ds^{2}=\frac{f(r)}{\kappa_{+}^{2}U_{+}V_{+}}dU_{+}dV_{+}+r^{2}d\Omega^{2}\simeq-\frac{2}{\kappa_{+}^{2}}dU_{+}dV_{+}+r^{2}d\Omega^{2}.$
(34)

The future outer horizon is defined as $U_{+}=0,V_{+}>0$. This metric is
regular at the outer horizon. After crossing the outer horizon as one
approaches the inner horizon $r_{-}$, $r_{*}\rightarrow\infty$, which implies
$v-u\rightarrow\infty,$ i.e $U_{+}V_{+}\rightarrow\infty.$ So these Kruskal
coordinates are singular at inner horizon and should be used only in an
interval $r_{1}<r<\infty$ where $r_{1}>r_{-}$ is some cutoff radius.

###  IV.2 Surface gravity

Since the spacetime is static, $K=\partial_{t}$ is a Killing vector and the
event horizon is also a Killing horizon. Surface gravity $(\kappa)$ is defined
using the Killing vector $(K^{\mu})$,

$\displaystyle\left(-K^{\mu}K_{\mu}\right);_{\nu}=2\kappa K_{\nu}.$ (35)

And in this case of 4D EGB black hole the surface gravity associated with the
outer horizon is found to be $\kappa_{+}=\frac{1}{2}f^{\prime}(r_{+})$. It
also satisfies the extremal limit i.e $\kappa_{+}=0$ when $\alpha=M^{2}$.

###  IV.3 Horizon temperature

To determine the temperature of the outer horizon we shall adopt the formalism
discussed in Chatterjee1  for static spherically symmetric black holes.
Considering a scalar field, we shall construct the modes of the Killing vector
across the horizon. The positive frequency modes$(\omega>0)$ of the Killing
vector satisfy,

$\displaystyle i\partial_{t}\Phi_{\omega}=\omega\Phi_{\omega}.$ (36)

In Kruskal coordinates $\partial_{t}$ becomes,

$\displaystyle\partial_{t}=-\kappa_{+}U_{+}\partial_{U_{+}}+\kappa_{+}V_{+}\partial_{V_{+}},$
(37)

and the modes are separated,

$\displaystyle\Phi_{\omega}=f_{\omega}(U_{+})+g_{\omega}(V_{+}).$ (38)

Equation (36) can be solved immediately, and the solutions are,

$\displaystyle
f_{\omega}(U_{+})=N_{\omega}|U_{+}|^{\frac{i\omega}{\kappa_{+}}}$ (39)
$\displaystyle
g_{\omega}(V_{+})=N_{\omega}\left(V_{+}\right)^{-\frac{i\omega}{\kappa_{+}}}.$
(40)

The ingoing $V_{+}$ modes are well behaved and of no interest to us. The
outgoing $U_{+}$ modes have a logarithmic singularity at the horizon and we
need these modes only to calculate the outgoing probability current. To remove
the singularity at the horizon we have used the fact that the field modes are
essentially distributions, and the distributions are well behaved at the
horizon. The appropriate form of the distribution isGel'fand ,

$\displaystyle f_{\omega}=\lim_{\epsilon\to
0}N_{\omega}|U_{+}+i\epsilon|^{\frac{i\omega}{\kappa_{+}}}=\left\\{\begin{array}[]{ll}N_{\omega}\left(U_{+}\right)^{\frac{i\omega}{\kappa_{+}}}&\mbox{
for $U_{+}>0$,}\\\
N_{\omega}|U_{+}|^{\frac{i\omega}{\kappa_{+}}}e^{-\frac{\pi\omega}{\kappa_{+}}}&\mbox{
for $U_{+}<0$, }\end{array}\right.$ (43)

and the complex conjugate form is

$\displaystyle\overline{f_{\omega}}=\lim_{\epsilon\to
0}N_{\omega}^{*}\,|U_{+}-i\epsilon|^{-\frac{i\omega}{\kappa_{+}}}=\left\\{\begin{array}[]{ll}N_{\omega}^{*}\left(U_{+}\right)^{-\frac{i\omega}{\kappa_{+}}}&\mbox{
for $U_{+}>0$,}\\\
N_{\omega}^{*}\,|U_{+}|^{-\frac{i\omega}{\kappa_{+}}}e^{-\frac{\pi\omega}{\kappa_{+}}}&\mbox{
for $U_{+}<0$. }\end{array}\right.$ (46)

Now, the emission probability is given by the outgoing probability current,

$\displaystyle
j^{out}=-i\left[-\kappa_{+}U_{+}\left(\partial_{U_{+}}\overline{\Phi_{\omega}}\right)\Phi_{\omega}+\kappa_{+}U_{+}\overline{\Phi_{\omega}}\left(\partial_{U_{+}}\Phi_{\omega}\right)\right].$
(47)

A straightforward calculation gives,

$\displaystyle j^{out}=\omega U_{+}|N_{\omega}|^{2}\lim_{\epsilon\to
0}\left[\frac{1}{U_{+}-i\epsilon}+\frac{1}{U_{+}+i\epsilon}\right]\left(U_{+}-i\epsilon\right)^{-\frac{i\omega}{\kappa_{+}}}\left(U_{+}+i\epsilon\right)^{\frac{i\omega}{\kappa_{+}}},$
(48)

so that,

$\displaystyle j^{out}=\left\\{\begin{array}[]{ll}|N_{\omega}|^{2}&\mbox{ for
$U_{+}>0$,}\\\ |N_{\omega}|^{2}e^{-\frac{2\pi\omega}{\kappa_{+}}}&\mbox{ for
$U_{+}<0$. }\end{array}\right.$ (51)

Then we get the conditional probability that a particle emits from the horizon
when it is incident on the horizon from the inside,

$\displaystyle
P_{(emission|incident)}=\frac{P_{(emission{\cap}incident)}}{P_{(incident)}}=\frac{|N_{\omega}|^{2}e^{-\frac{2\pi\omega}{\kappa_{+}}}}{|N_{\omega}|^{2}}.$
(52)

Comparing this with the Boltzmann factor, $e^{-\beta\omega}$, we found the
temperature of the outer horizon is $\kappa_{+}/2\pi$. This temperature is not
similar to the temperature of a Schwarzschild black hole, rather it looks like
the temperature of the outer horizon in Reissner-Nordström black hole with the
Gauss-Bonnet coupling constant playing the role of electric charge. So due to
the presence of the Gauss-Bonnet term in $4$-dimension the black hole acquires
two horizons. We will discuss this in detail in the next section.

###  IV.4 Entropy

In General Relativity black hole entropy $(S)$ is given by the Hawking-
Bekenstein formula $S=A/4$, where $A$ is the area of the event horizon.
However one cannot naively apply this formula to higher order curvature
theories of gravity to obtain entropy. In that case it is most logical to
calculate entropy from the first law of black hole thermodynamics $dM=TdS$
which givesCai

$\displaystyle S=\int\frac{dM}{T(M)},$ (53)

So, we get

$\displaystyle S=\int\frac{1}{T(r_{+})}\frac{\partial M}{\partial
r_{+}}dr_{+}=S_{0}+2\pi\int_{0}^{r_{+}}(r_{+}+\frac{2\alpha}{r_{+}})dr_{+},$
(54)

where $S_{0}$ is some integration constant which can be fixed by arguing
entropy should be zero when the horizon of the black hole shrinks to zero.
Then we find,

$\displaystyle S=\pi
r_{+}^{2}+2\pi\alpha\ln{r_{+}^{2}}+\tilde{S}_{0}=\frac{A}{4}+2\pi\alpha\ln{\frac{A}{A_{0}}},$
(55)

where $A$ is the area of outer horizon and $A_{0}$ is some constant with
dimension of area. This result is very surprising because though the first
term gives standard Hawking-Bekenstein entropyBekenstein ; Hawking , nowhere
in general relativity the logarithmic term appears in the expression for
entropy. It only appears as the leading order correction to entropy in some
quantum theories of gravityMajumdar ; Cai1 ; Cai2 ; Cognola .

###  IV.5 Heat capacity

The heat capacity is given by,

$\displaystyle C=(\frac{\partial M}{\partial T})=(\frac{\partial M}{\partial
r_{+}})(\frac{\partial r_{+}}{\partial T}).$ (56)

In this case we have,

$\displaystyle\frac{\partial M}{\partial
r_{+}}=1-\frac{r_{+}^{2}+\alpha}{2r_{+}^{2}}=\frac{r_{+}^{2}-\alpha}{2r_{+}^{2}},$
(57)

and

$\displaystyle\frac{\partial T}{\partial
r_{+}}=\frac{1}{4\pi}\frac{2\alpha^{2}+5\alpha
r_{+}^{2}-r_{+}^{4}}{r_{+}^{2}(r_{+}^{2}+2\alpha)^{2}},$ (58)

which gives,

$\displaystyle
C=\frac{2\pi(r_{+}^{2}-\alpha)(r_{+}^{2}+2\alpha)^{2}}{2\alpha^{2}+5\alpha
r_{+}^{2}-r_{+}^{4}}.$ (59)

###  IV.6 Free energy

The free energy of black holes is defined as $F=M-TS$, and in this case is
given by,

$\displaystyle
F=\frac{r_{+}^{2}+\alpha}{2r_{+}}-\frac{r_{+}(r_{+}^{2}-\alpha)}{4(r_{+}^{2}+2\alpha)}-\frac{\alpha(r_{+}^{2}-\alpha)}{2r_{+}(r_{+}^{2}+2\alpha)}\ln\frac{A}{A_{0}}.$
(60)

##  V Discussions

In this paper we have discussed two different types of black holes in the
context of Einstein-Gauss-Bonnet gravity. For both of these black holes, we
have concentrated solely on the GR branch solution of the EGB gravity since
this branch has a general relativistic limit and it’s easy to compare the
results with black holes in general relativity. Non-GR branch solution on the
other hand behaves quite differently and the results are not so easy to
interpret. Since our aim is to compare the effects of Gauss-Bonnet term with
general relativity, we have not considered the non-GR branch here.
For dynamical EGB black hole we have shown the surface gravity associated with
the trapping horizon is indeed it’s temperature through our calculation. There
are several different definitions of black hole horizon as well as surface
gravity available in the literature for a dynamical spacetime. Among all
these, the Hayward-Kodama formalism stands out for it’s coherent and
consistent description of a dynamical black hole spacetime. Since most of the
previous works are done in a $4$-dimensional spacetime, it is not clear a
priori how this formalism would work for gravity theories in higher
dimensions. And it is reassuring to find that the trapping horizon formalism
works nicely for GR branch solution in dynamical EGB gravity and produce the
expected results. In Nozawa , the first law of black hole dynamics was proved
assuming $\kappa_{TH}$ as the temperature of the trapping horizon which we
have shown explicitly in this paper. So this is another point in favor of
Hayward-Kodama formalism.
The method we have employed to calculate temperature of the dynamical horizon
has been used to calculate temperature for the stationary (both non-extremal
and extremal)Chatterjee1  as well as dynamical horizonsChatterjee2 ;
Chatterjee3 . For dynamical horizons in $4$-dimensions the symmetry of the
spacetime plays an important role. The Kodama vector and the Misner-Sharp
energy can be defined uniquely only if the spacetime is spherically symmetric
and this is essential for our calculations. For dynamical horizons in higher
dimensional GB gravity we have chosen the spacetime to be maximally symmetric
which ensures the existence of Kodama vector and generalized Misner-Sharp
energy. For a different geometry, a Kodama-like vector field is not known yet
and its not obvious how our formalism can be extended to these type of
spacetimes.
Another important point is that this method calculates the correct temperature
for a horizon without resorting to any approximations unlike the Hamilton-
Jacobi tunneling formalism Vanzo , which depends heavily on WKB
approximations. The method is semiclassical in the sense that we use quantum
field theory in a classical gravity background but the mode solutions are
exact and this method is applicable to both extremal and non-extremal
spacetimes Chatterjee1 .
For the $4$-D novel EGB black hole we have calculated the temperature,
entropy, heat capacity and free energy associated with the event horizon for
the GR branch solution. Though the calculations are straightforward and
produce standard result for temperature, the expression for entropy is
somewhat surprising. Even with inclusion of higher order curvature term in
action, there is no classical example of the existence of the logarithmic term
in entropy. It appears only as leading order correction to entropy in quantum
theories of gravity. This expression for entropy was found in Fernandes  for a
charged black hole in AdS spaces in $4$-D EGB gravity. Surprisingly both the
contribution from the electric charge and the cosmological constant canceled
out and the result is same with what we have found for the static spherically
symmetric black hole. How such terms are coming into play is not clear at this
moment and maybe this unusual theory is an indicator of how these corrections
appear in quantum gravity, which remains to be seen.
There are several issues related to the $4$-D EGB gravity theory, the most
important being the absence of a valid action and consistent theory in
$4$-dimensionsDadhich ; Ai . As was claimed in the original work Glavan , this
theory is formulated in $n>4$ dimensions with a rescaled Gauss-Bonnet coupling
constant $\alpha\rightarrow\alpha/(n-4)$ and the $4$-D theory is recovered
only by taking limit $\alpha\rightarrow 4$ at the level of equations of motion
after variation of action. First of all, this action does not follow from
Lovelock gravity which is expected naturally for higher curvature gravity
theories. Secondly there are some unnatural consequences which follow from
this ad hoc choice of rescaling the Gauss-Bonnet coupling constant. The black
hole spacetime in this theory admits two horizons just like a Reissner-
Nordström black hole despite the absence of any electrical charges. What is
more peculiar is that the coupling constant $\alpha$, which is a geometrical
quantity behaves like an electric charge, a physical quantity, so that the
whole causal structure of the spacetime has changed and the central
singularity is timelike instead of spacelike. This is not comfortable. On the
other hand there is the expression for entropy for this black hole, which
indicates though the spacetime looks like Reissner-Nordström apparently, it is
not exactly so. There are some subtleties in this theory which are not fully
understood yet. There has been some effort to make a consistent $n\rightarrow
4$ Einstein-Gauss-Bonnet gravityMukhoyama  where it was shown that the
regularization is possible only for some broad, but limited, class of metrics
and only by breaking (a part of) the diffeomorphism invariance or by adding an
extra degree of freedom, in agreement with the Lovelock theorem, a consistent
$n\rightarrow 4$ theory can be constructed which is different from general
relativity in $4$-dimension. Whether this unusual theory can genuinely become
an alternative to Einstein’s general relativity and it’s interesting features
can provide some insights into quantum theories of gravity remains to be seen.

##  VI Acknowledgements

I would like to thank Narayan Banerjee for useful discussions. This work is
supported by DST Project Grant No. SR/WOS-A/PM-78/2017 under the DST Women
Scientists Scheme A.

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violated.

See 35

###### Proof.

We modify our monitoring approach as indicated in the proof sketch, and
describe here only the missing details: Let $\alpha:=\alpha_{n}$ be the last
assignment of a given trace, as in procedure Monitor. Let $\mathcal{K}$ be the
finite set of integers that contains all constants occuring in $\psi$ and
$\alpha$, as well as $0$. We use the finite summary $(\text{GC}_{K},\sim_{K})$
described in (Felli, Montali, and Winkler 2022b) when computing constraint
graphs, where $K=max\\{k-k^{\prime}\mid k,k^{\prime}\in\mathcal{K}\\}$. When
computing CGs in line 5 of the procedure, we start from the formula
$\varphi_{\alpha}:=\bigwedge v=\alpha(v)$ instead of
$\varphi_{\mathit{init}}$. For the correctness proof, the notion of a history
constraints changes such that for the empty word $w$, $h(w)=\varphi_{\alpha}$.
Thus, history constraints are formulas with free variables $V$ (the variables
$V_{0}$ are no longer necessary). Lem. 25 changes to the following statement:
For $w\in\Theta^{*}$ of length $k$, $h(w)$ is satisfied by assignment $\nu$
with domain $V$ iff $\langle\emptyset\rangle\,{\cdot}\,w$ is consistent with a
trace $\alpha_{0},\dots,\alpha_{k}$ such that $\nu(v_{0})\,{=}\,\alpha(v)$ and
$\nu(v)\,{=}\,\alpha_{k}(v)$ for all $v\in V$. That is, the statement is as
before, but the initial assignment is fixed to $\alpha$. The notions of a
history set and a summary as well as Lem. 28, are as before, but with the
updated notion of history constraints. By reasoning as in (Felli, Montali, and
Winkler 2022b, Thm. 5.5) a finite summary exists, so the CG computation will
terminate. The statement of Thm. 29 changes as follows:
_For a DFA $\mathcal{D}_{\psi}$ for $\psi$ and a trace $\tau$, let
$w\in\Theta_{\mathit{cur}}\Theta^{*}$ be the word consistent with $\tau$ and
$P$ the $\mathcal{D}_{\psi}$ state s.t. $\\{q_{0}\\}\to_{w}^{*}P$. For
$G:=\textup{CG}_{\psi}(P,\sim_{K})$ the constraint graph from $P$,_

  * •

_if_ $P\,{\in}\,P_{F}$ then
$\tau\models\llbracket\psi{=}\textsc{cs}\rrbracket$ if $G$ has a non-empty
path to a non-final state, and
$\tau\models\llbracket\psi{=}\text{{ps}}\rrbracket$ otherwise,

  * •

_if_ $P\,{\not\in}\,P_{F}$ then
$\tau\models\llbracket\psi{=}\textsc{cv}\rrbracket$ if $G$ has a non-empty
path to a final state, and $\tau\models\llbracket\psi{=}\textsc{pv}\rrbracket$
otherwise.

The proof of the theorem is almost as before. E.g. for the first case, if
$P\,{\in}\,P_{F}$ and $G$ has a path to a non-final state
$(P^{\prime},\varphi)$, then by Lem. 28 $P\to_{u}^{+}P^{\prime}$ and
$\varphi\sim h(u)$ is satisfiable. By the modified Lem. 25 there is a trace
consistent with $u$ starting with $\alpha$. We can build the extension
$\tau^{\prime\prime}$ of $\tau$ as in the proof of Thm. 29, which is
consistent with $wu$ and leads to the non-accepting state $P^{\prime}$, so by
the results on automata $\tau^{\prime\prime}\not\models\psi$ and hence
$\tau\models\llbracket\psi{=}\textsc{cs}\rrbracket$. ∎