\beta}{r_{0}}^{3}N^{1/2}{f_{x^{10}x^{10}}}({r_{0}})}{11{g_{s}}^{3/2}M{N_{f}}^{5/3}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}\log^{\frac{2}{3}}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}\beta.$ (3.130)

In a manner analogous to that of the ${\cal O}(\beta^{0})$ scenario (3.125),
comparing on-shell action density results of ${\cal O}(\beta)$ terms occurring
in (3.4.3) at the ultraviolet (UV) cut-off, i.e.,

$\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm UV}^{4}}\right)\frac{S_{D=11,\ {\rm
on-shell\ UV-finite}}^{\beta,\rm BH}}{{\cal V}_{4}}=\frac{S_{D=11,\ {\rm on-
shell\ UV-finite}}^{\beta,\rm thermal}}{{\cal V}_{4}}.$ (3.131)

We got the following results using the equations (3.4.3) and (3.131):

$\displaystyle\hskip-36.135ptf_{x^{10}x^{10}}({r_{0}})=-\frac{b^{2}{g_{s}}^{3}M^{2}{N_{f}}^{14/3}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)^{4}\log^{{3}}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)\log\left(\frac{{r_{h}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\log\left(1-\frac{{r_{h}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)}{{\kappa_{\rm EH,th}^{\beta,\ \rm
IR}}{N}\left(\frac{r_{0}}{{\cal R}_{D5/\overline{D5}}^{\rm th}}\right)^{3}}$
$\displaystyle\hskip 21.68121pt\times\left(-11{\cal
C}_{\theta_{1}x}{\kappa_{\sqrt{G^{(1)}}R^{(0)}}^{\rm
IR}}\log^{3}(N)-10{\kappa_{\rm EH,bh}^{\beta,\ \rm IR}}\left(-{\cal
C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm bh}-3{\cal C}_{\theta_{1}x}^{\rm
bh}\right)\right).$ (3.132)

The “UV-IR” mixing that is described in the equation (3.4.3) is almost
identical to the UV-IR mixing that is described in [3] and discussed earlier
in this chapter. The combination of the constants of integration such as
$\left(-{\cal C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm bh}-3{\cal
C}_{\theta_{1}x}^{\rm bh}\right)$ possesses the memory of flavor $D7$-branes
of type IIB string dual referred to as the “Flavor Memory” effect, and the
identical effect also seen in ${\cal O}(\beta)$ correction of the Hawking
temperature (3.4.1). Because the previously mentioned group of constants of
integration is zero, there is no ${\cal O}(\beta)$ correction to the Hawking
temperature (3.121), and there is also no ${\cal O}(\beta)$ correction to the
deconfinement temperature $(T_{c})$ due to gauge-gravity duality. This
suggests that $T_{c}$ does not undergo renormalization at ${\cal O}(\beta)$.
The holographic renormalization of this part is discussed in appendix B.5.

###  3.5 Summary

We obtained the following results in this chapter.

  * •

UV-IR Mixing and Flavor Memory: During the computation of $T_{c}$ from the
semiclassical method [4], we obtained “UV-IR” mixing described as the
connection between the ${\cal O}(\beta)$ corrections to the ${\cal M}$-theory
metric along the ${\cal M}$-theory circle in thermal background and the ${\cal
O}(\beta)$ correction to a specific combination of the ${\cal M}$-theory
metric components along the compact part of the four-cycle “wrapped” by the
flavor $D7$-branes of the parent type IIB (warped resolved deformed) conifold
geometry - the latter referred to as “Flavor Memory” in the ${\cal M}$-theory
uplift on equating the ${\cal O}(\beta)$ portion of the on-shell actions of
the black hole and thermal backgrounds in the absence and presence of rotation
both.

  * •

Non-Renormalization of $T_{c}$:

    * –

Semiclassical computation: We have shown that the leading order result for
$T_{c}$ is still valid after incorporating the cal ${\cal O}(R^{4})$ terms in
supergravity action. In the large-$N$ limit, the $t_{8}t_{8}R^{4}$ terms
dominate the ${\cal O}(R^{4})$ contributions (together with the sub-dominant
term, $\epsilon_{11}\epsilon_{11}R^{4}$), which, from a type IIB viewpoint in
the zero-instanton sector, translate to a tree-level contribution at ${\cal
O}\left((\alpha^{\prime})^{3}\right)$ and one-loop contribution to the
amplitude of four-graviton scattering which is obtained from integration of
fermionic zero modes. The $SL(2,\mathbb{Z})$ completeness of such $R^{4}$
terms [73], indicates that they are not perturbatively renormalized beyond one
loop of the zero-instanton sector, indicates that they are not renormalized
from a type IIB point of view. This shows that $T_{c}$ is not renormalized at
all the loops in ${\cal M}$-theory at ${\cal O}(R^{4})$.

    * –

$T_{c}$ from Entanglement entropy: We computed the entanglement entropy
between two parts by dividing one of the spatial coordinates of the thermal
${\cal M}$-theory background into a segment of length $\it l$ and its
complement. This generalization of [106] to ${\cal M}$-theory was made
possible by an obvious extension of [106] to ${\cal M}$-theory. In the same
vein as [106], there are two different RT surfaces: connected and unconnected.
If one is located below the critical value $itl_{c}rit$, then the connected
surface is the one that dominates the entanglement entropy, and if one is
located above the critical value ${\it l_{crit}}$, then the disconnected
surface is the one that dominates the entanglement entropy; In large $N_{c}$
gauge theories, the ${\it l}<{\it l_{crit}}$ region belongs to the confining
phase, whereas the ${\it l}>{\it l_{crit}}$ region belongs to the deconfining
phase of the same theory. In high $N_{c}$ gauge theories, this phenomenon is
regarded as a phase transition between confinement and deconfinement phases.

As a result of a precise and delicate cancellation between the ${\cal
O}(\beta)$ correction from a subset of the aforementioned metric components,
it is remarkable to observe that, $T_{c}$ receives no ${\cal O}(\beta)$
correction which is consistent with the conjecture discussed above from
semiclassical computation.

    * –

Hawking temperature of rotating cylinder black hole background is proportional
to $\left(-{\cal C}_{zz}^{\rm BH}+2{\cal C}_{\theta_{1}z}^{\rm BH}-3{\cal
C}_{\theta_{1}x}^{\rm BH}\right)$, as we have obtained in (3.4.1). To make the
correction to the three-form potential zero, it was proved in [3] that ${\cal
C}_{zz}^{\rm BH}=2{\cal C}_{\theta_{1}z}^{\rm BH}$ and $|{\cal
C}_{\theta_{1}x}^{\rm BH}|\ll 1$. This suggests that even in the presence of
rotating Quark Gluon Plasma, there is no ${\cal O}(R^{4})$ correction to the
deconfinement temperature, and hence the “Non-Renormalization of $T_{c}$” in
the rotating QGP as well.

  * •

Deriving ${\cal M}\chi$PT-Phenomenology compatibility: As demonstrated in
chapter 2, in order to match the phenomenological value corresponding to the
one-loop renormalized low energy constant associated with the ${\cal
O}(p^{4})$ $SU(3)\ \chi$PT Lagrangian term
“$\left(\nabla_{\mu}U^{\dagger}\nabla^{\mu}U\right)^{2}$” with the value found
from the type IIA string dual of thermal QCD-like theories including the
${\cal O}(R^{4})$ corrections, we need to impose a constraint, negative sign
of the combination of integration constants mentioned above.
$\mathbb{R}^{2}\times S^{1}(\frac{1}{M_{\rm KK}})$ is the thermal supergravity
background counterpart of type IIB (solitonic) $D3$-branes at lower
temperatures. In order to restore a boundary four-dimensional QCD-like theory
after compactifying on the base of a $G_{2}$-structure cone, one must take the
limit of$M_{\rm KK}\rightarrow 0$, and in this limit we derived the values of
the aforementioned integration constants explicitly and found that this
combination has negative sign.

  * •

Wald Entropy at ${\cal O}(R^{4})$: In the high-temperature ${\cal M}$-theory
dual, matching of ${\cal O}(R^{4})$ computation of Wald entropy with the black
hole entropy puts a linear constraint on the same linear combination of the
aforementioned integration constants.

  * •

Holographic Dual of Rotating QGP: By applying Lorentz transformations (3.117)
along $(t,\phi)$ coordinates on the gravity dual side, we are able to derive
the rotating QGP on the gauge theory side via gauge-gravity duality.
Therefore, for $T>T_{c}$ and $T<T_{c}$ on the gauge theory side (where $T_{c}$
serves as the deconfinement temperature of thermal QCD-like theories), the
gravity dual includes a rotating cylinder black hole and thermal backgrounds.

  * •

$T_{c}$ in Rotating QGP: We found that the deconfinement temperature of
rotating QGP (3.129) is inversely proportional to the Lorentz factor,
$\gamma$, by equating the ${\cal O}(\beta^{0})$ terms for the rotating
cylindrical black hole and the cylindrical thermal backgrounds at the UV cut-
off (3.125). This means that when the angular velocity of rotating QGP
increases, the deconfinement temperature of thermal QCD-like theories
decreases and vice-versa. Experimental results from the noncentral
Relativistic Heavy Ion Collider (RHIC) have shown that the Quark Gluon Plasma
created in these collisions has an angular velocity, and that the
deconfinement temperature decreases with increasing rotation [123]. To ensure
that the deconfinement temperature of rotating QGP computed using a top-down
technique is in agreement with actual results, our study serves as a good
check for the top-down duals of [14] and [1].

##  Chapter 4 Conclusion and Future Outlook

We explored the intermediate coupling regime of thermal QCD-like theories from
a top-down model constructed in [1]. The model is obtained as the extension of
[15] by including higher derivative terms (${\cal O}(R^{4})$) in the eleven
dimensional supergravity action. We studied the following issues in the
context of holographic thermal QCD at intermediate coupling.

  * •

$SU(3)$ Chiral Perturbation Theory from IIA String Theory: We obtained the low
energy coupling constants (LECs) appearing in the $SU(3)$ chiral perturbation
theory Lagrangian in the chiral limit from the type IIA string dual inclusive
of ${\cal O}(R^{4})$ corrections [2]. We matched our results with the
phenomenological values of the LECs available in the literature, and in this
process we found that resolution parameter of the conifold geometry receives
an ${\cal O}\left(\frac{l_{p}^{3}}{N}\right)$ correction. We also found the
connection between higher derivative terms and $\frac{1}{N}$ correction.

  * •

Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate
Coupling: We discussed the deconfinement phase transition in thermal QCD from
the semi-classical method and entanglement entropy point of view. The semi-
classical method is based on the Hawking Page phase transition between thermal
and black hole background on the gravity dual side. The procedure is to obtain
the on-shell action densities of thermal and black hole backgrounds and
equates these two at the UV cut-off. We obtained the deconfinement temperature
of thermal QCD by equating the leading order terms. We observed the “UV-IR”
mixing, “Flavor Memory” effect, and “non-renormalization of $T_{c}$”. Let us
explain these effects in some detail.
UV-IR Mixing and Flavor Memory Effect: When we equated the on-shell action
densities at the UV cut-off, then we obtained a relationship between the
integration constants appearing in the black hole background along the compact
part of the non-compact four cycle, which is part of the flavor $D7$-branes
worldvolume and higher derivative correction to the thermal background along
the ${\cal M}$-theory circle. This is called “UV-IR mixing” in our setup.
Since the aforementioned integration constants are associated with the flavor
$D7$-branes of parent type IIB string dual and hence they have the information
about the flavor branes in the ${\cal M}$-theory uplift, which has no branes.
This is interpreted as the “Flavor Memory” effect in our setup.

Non-Renormalization of $T_{c}$: The deconfinement phase transition in thermal
QCD corresponds to phase transitions between the entanglement entropies of
connected and disconnected surfaces in the language of gauge-gravity duality.
We showed that the deconfinement phase transition occurs at the critical value
of the length of the interval chosen to compute the entanglement entropy. In
this process, we found that entanglement entropies of connected and
disconnected surfaces do not receive the ${\cal O}(R^{4})$ corrections, and
hence $T_{c}$ does not receive the same. This is known as the non-
renormalization of the deconfinement temperature of thermal QCD-like theories.

${\cal M}\chi$PT-$T_{c}$ Compatibility: While matching one of the one-loop
renormalized LECs with the phenomenological data in [2], we imposed the
constraint on the combination of integration constants mentioned earlier but
now for the thermal background. The constraint was that one would get the
exact matching with phenomenological data provided the combination, as
mentioned above, should have a negative sign. In this paper, we explicitly
calculated the expressions of integration constants for the thermal background
and showed that the constraint is satisfied, and hence we showed the
compatibility between our chiral perturbation theory study and deconfinement
temperature computation.

  * •

Deconfinement Phase Transition in Rotating QGP at Intermediate Coupling: We
constructed the holographic dual of rotating QGP by making one of the spatial
coordinates in the ${\cal M}$-theory dual periodic and performing the Lorentz
transformation along the temporal and the periodic coordinate mentioned
earlier. The gravity dual will be a rotating cylindrical black hole and
thermal backgrounds when $T>T_{c}$ and $T<T_{c}$ on the thermal QCD side.
Since the cylindrical black hole has the angular velocity, this maps to the
angular velocity of rotation of QGP in thermal QCD via gauge-gravity duality.
We followed the semi-classical method to study the deconfinement phase
transition in rotating QGP from a top-down model. We found that the
deconfinement temperature of thermal QCD is inversely proportional to the
Lorentz factor, implying that as QGP rotation increases, $T_{c}$ decreases and
vice-versa. When we analyze the effect of higher derivative terms similar to
[3], we found that one again observed “UV-IR mixing”, “Flavor Memory effect”,
and “non-renormalization of $T_{c}$” even in the rotating QGP. In this paper,
we showed the normalization of $T_{c}$ by calculating the higher derivative
correction to the Hawking temperature too. Since, Hawking temperature maps to
$T_{c}$ on the QCD side via Hawking-Page phase transition in the context of
gauge-gravity duality. We found that Hawking temperature receives no higher
derivative corrections, guaranteeing that $T_{c}$ is non-renormalized.

Future Physics outlook:

  * •

The LEC $H_{1}$ of (2.2) and, in general, the LECs of the $\chi$PT Lagrangian
at ${\cal O}(p^{6})$ [124] may be calculated using the given values of the
parameters of our ${\cal M}$-theory dual of thermal QCD-like theories.

  * •

No prior work has applied gauge-gravity duality approaches to the study of the
perturbative regime of thermal QCD-like theories in order to explain, for
example, the low-frequency peaks expected to appear in spectral functions
related to transport coefficients, from ${\cal M}$-theory, in the context of
intermediate ’t Hooft coupling top-down holography. We would like to explore
more application of [1], e.g., we can obtain the spectral/correlation
functions using the techniques available from the gauge-gravity duality with
the gravity dual as ${\cal M}$ theory uplift of [15] including the ${\cal
O}(R^{4})$ corrections. This will allow us to get more insight about the
experimental results of the QCD at Relativistic Heavy Ion Collider(RHIC) and
test the QCD/${\cal M}$-theory duality used in part-I of this thesis. In
addition, the dependence on temperature of the speed of sound, attenuation
constant, and the bulk viscosity can all be obtained from its solution, along
with the ${\cal O}(l_{p}^{6})$ and the non-conformal corrections to them from
the same holographic dual. One could attempt to reproduce the well-known weak-
coupling result from ${\cal M}$-theory, which states that the ratio of the
bulk viscosity to the shear viscosity has a lower bound that is proportional
to the square of the deviation of the square of the speed of sound from its
conformal value (last reference of [60]). In a general sense, the dissipative
quasi-normal modes of the spectral functions at low frequencies could be
studied to analyze the occurrence of peaks at low frequencies in transport
coefficients, providing connections between perturbative QCD results along
with QCD plasma RHIC experiments.

Part-II
(HD) Gravity Islands, and Multiverse

“God does not play dice.” - Albert Einstein
“God not only plays dice, he sometimes throws the dice where they cannot be
seen.” - Stephen Hawking
“If God throws dice where they cannot be seen, they cannot affect us.” - Don
Page

##  Chapter 5 Introduction

In this chapter, we present the introduction of the materials needed to
understand the information paradox and its resolution from holography. We
start with the discussion on entanglement entropy in section 5.1, we discuss
the information paradox and the Page curve in section 5.2 and finally we
discuss the resolution of the information paradox in 5.3 from the island
proposal, doubly holographic setup and wedge holography in 5.3.1, 5.3.2 and
5.3.3 respectively.

###  5.1 Holographic Entanglement Entropy: Ryu-Takayanagi and Dong’s Proposals

Entanglement Entropy in Quantum Mechanics (QM): Let us first discuss the
entanglement entropy in quantum mechanical system. Let us consider a system
whose state is denoted by $|\psi\rangle$. The density matrix of the system is
defined as:

$\rho_{AB}={|\psi\rangle}_{AB}{\langle\psi|}_{AB}.$ (5.1)

Entanglement entropy is measured by the von-Neumann entropy. For this, first
we have to partition the system into two subsystems $A$ and $B$. States in the
subsystems $A$ and $B$ are denoted by ${|\psi\rangle}_{A}$ and
${|\psi\rangle}_{B}$ such that
$|\psi\rangle={|\psi\rangle}_{AB}={|\psi\rangle}_{A}\otimes{|\psi\rangle}_{B}$.
Reduced density matrix of the subsystem $A$ is obtained by tracing over the
degrees of freedom of the subsystem $B$ and vice-versa.

$\rho_{A}={\rm Tr}_{B}\left(\rho_{AB}\right)={\rm
Tr}_{B}\left({|\psi\rangle}_{AB}{\langle\psi|}_{AB}\right)={|\psi\rangle}_{A}{\langle\psi|}_{A}.$
(5.2)

Now, the von-Neumann entropy is defined as:

$\displaystyle S_{\rm EE}=-{\rm Tr}\left(\rho_{A}{\rm ln}\rho_{A}\right).$
(5.3)

Entanglement Entropy in Quantum Field Theory (QFT): It is not easy to compute
the entanglement entropy in quantum field theories (QFTs) by factoring the
system into subsystems because factorization is not always possible in QFTs.
Entanglement entropy in QFT is calculated using the replica trick. First, let
us define the Renyi entropy:

$\displaystyle S_{A}^{(n)}=\frac{1}{1-n}\log\left({\rm
Tr}\rho_{A}^{n}\right),$ (5.4)

from the Renyi entropy, von-Neumann entropy is defined as: $S_{\rm EE}={\rm
Limit}_{n\rightarrow 1}S_{A}^{(n)}$. In QFTs, usually we calculate the Renyi
entropy and then take the limit $n\rightarrow 1$. $\rho_{A}^{n}$ is obtained
from taking the $n$ copies of the same subsystem $A$, tracing in (5.4)
corresponds to the gluing of the $n$-sheeted surfaces where the subsystem $A$
is defined. It is found that:

$\displaystyle{\rm Tr}\rho_{A}^{n}=\frac{Z_{n}(A)}{Z_{1}^{n}},$ (5.5)

where $Z_{n}(A)$ is the partition function of the $n$-sheeted surface.
Therefore Renyi entropy is defined as:

$\displaystyle
S_{A}^{(n)}=\frac{1}{1-n}\log\left(\frac{Z_{n}(A)}{Z_{1}^{n}}\right).$ (5.6)

Hence, if we can obtain the von-Neumann entropy when $n\rightarrow 1$ in
(5.6).

Holographic Entanglement Entropy (HEE): As we saw in the earlier discussion
that once we know the partition function of the $n$-sheeted surface then we
are able to compute the entanglement entropy in QFT but calculation of the
partition function itself is a cumbersome process. The AdS/CFT correspondence
make this job easier because of the Ryu-Takayanagi proposal to compute the
entanglement entropy in dual CFT from the gravity side [107]. Suppose, we are
dealing with $AdS_{d+1}/CFT_{d}$ duality and the bulk is denoted by ${\cal
M}_{d+1}$, subsystem on the boundary CFT is denoted by $A$ with boundary
$\partial A$ then the proposal states that

  * •

We need to find out a co-dimension two surface $(\epsilon_{A})$ in the bulk
${\cal M}_{d+1}$ which is anchored on $\partial A$.

  * •

There is possibility of many surfaces but we have to consider the one which
satisfy the homology constraint, i.e., $\epsilon_{A}$ is smoothly retractable
to the boundary region $A$.

  * •

Out of those surfaces which satisfy homology constraint, we need to pick the
one with the minimal area then the entanglement entropy is defined as:

$\displaystyle S_{A}=\frac{{\rm min}\left({\rm
Area}\epsilon_{A}\right)}{4G_{N}^{(d+1)}}.$ (5.7)

The Ryu-Takayangi formula has certain limitation, it is applicable to the the
time-independent backgrounds. For the time dependent background, one is
required to use the HRT formula [125] where HRT stands for Hubney, Rangamani
and Takayanagi. Quantum corrections to all order in $\hbar$ to the Ryu-
Takayanagi formula was incorporated in [126] where one is required to
extremize the generalised entropy. Surfaces which extremize the generalised
entropy are known as quantum extremal surfaces(QES). If there are more than
one quantum extremal surfaces then we need to consider the one with minimal
area. In [6], authors generalised the QES prescription to island surfaces
where we are required to extremize the generalised entropy like functional
which includes contribution from the island surfaces. In this case, extremal
surfaces are known as quantum extremal islands. Since, in this thesis, we are
confining ourselves to the time independent backgrounds and therefore we will
not discuss the HRT formula.

Dong’s Proposal to Compute HEE in Higher Derivative Theories of Gravity: One
can obtain the HEE in in general higher derivative theories of gravity by
using the formula given by X. Dong in [127]. In the context of
$AdS_{d+1}/CFT_{d}$ correspondence the formula is written below:

$S_{EE}=\int d^{d-1}y\sqrt{-g}\Biggl{[}\frac{\partial{\cal L}}{\partial
R_{z\bar{z}z\bar{z}}}+\sum_{\alpha}\left(\frac{\partial^{2}{\cal L}}{\partial
R_{zizj}\partial
R_{\bar{z}m\bar{z}l}}\right)_{\alpha}\frac{8K_{zij}K_{\bar{z}ml}}{(q_{\alpha}+1)}\Biggr{]},$
(5.8)

where $(a,b)$ and $(i,j,k,l)$ denote the normal and tangential directions, $g$
is the induced metric’s determinant on the co-dimension two surface in the
bulk, $K_{zij}=\frac{1}{2}\partial_{z}G_{ij}$ with trace
$K_{z}=K_{zij}G^{ij}$. (5.8) consists of two terms: Wald entropy like term and
the second term can be calculated by following these steps:

  * •

Let us label the each term as $\alpha$th term after getting the final
expression of the second term which is obtained from the differentiation of
Lagrangian with respect to Riemann tensor twice.

  * •

We need to perform the following transformations on certain components of the
Riemann tensors:

$\displaystyle
R_{abij}=r_{abij}+g^{ml}\left(K_{ajm}K_{bil}-K_{aim}K_{bjl}\right),$ (5.9)
$\displaystyle\hskip-231.26378ptR_{aibj}=r_{aibj}+g^{ml}K_{ajm}K_{bil}-Q_{abij},$
$\displaystyle\hskip-231.26378ptR_{ijml}=r_{ijml}+g^{ab}\left(K_{ail}K_{bjm}-K_{aij}K_{bml}\right),$

where $Q_{abij}=\partial_{a}K_{bij}$.

  * •

In the $\alpha$th term, the number of $Q_{aaij}$ and $Q_{bbij}$ be denoted by
$\gamma$ and the number of $K_{aij}$, $R_{abci}$ and $R_{aijk}$ will be
denoted $\delta$. Then $q_{\alpha}$ corresponding to the $\alpha$th term is
defined as:

$q_{\alpha}=\gamma+\frac{\delta}{2}.$ (5.10)

  * •

From (5.9), we can obtain $r_{abij},r_{aibj},r_{ijml}$ in terms of
$R_{abij},R_{aibj},R_{ijml}$, substitutions of the same in (5.8) results the
entanglement entropy in terms of original Riemann tensors i.e. in terms of
$R_{abij},R_{aibj}$ etc.

The reason for discussing these proposals is that when we compute the Page
curve of black holes in doubly holographic setup and wedge holography then
these proposals will be useful. The holographic entanglement entropy has also
been computed from the holographic stress tensor and surface terms in [128]
and [129] respectively.

###  5.2 Hawking’s Information Paradox and Page Curve

Hawking’s black hole information paradox is a long-time puzzle that started
with his papers [130, 131]. When matter collapses to form a black hole, the
whole matter is stored in the singularity. The horizon of the black hole
covers the black hole singularity. Initially, the system is in a pure state.
Hawking studied the creation of particles in pairs with negative and positive
energy in the presence of quantum effects, and he found that a particle with
negative energy gets trapped inside the black hole, whereas the particle with
positive energy scattered off to infinity is what we receive in Hawking
radiation. We can get the radiation from the black hole due to quantum
mechanics, which allows the possibility of quantum tunneling through a
potential barrier. In the case of a black hole, the horizon acts as the
potential barrier. Hawking calculated the spectrum of the particles coming out
of the black hole and found that the spectrum behaves as the spectrum of
thermal radiation with a temperature known as Hawking temperature, which
implies a mixed state. This means that the black hole evolves from the pure
state to the mixed state, and hence unitary evolution of quantum mechanics is
not preserved. This leads to the famous “information paradox”.

Page suggested that when we include the quantum effects, then the black hole
must follow the unitary evolution [132]. If we consider the black hole and
radiation region as a single system, then one should get the Page curve to
resolve the paradox. For the evaporating black hole, entanglement entropy of
the Hawking radiation first increases linearly with time up to the Page time
and then falls back to zero [132]. We are interested in eternal black holes,
and for these black hoes, instead of falling to zero of the entanglement
entropy, one obtains the constant entanglement entropy after the Page time,
and this constant value is equal to the twice the thermal entropy of the black
holes.

In this part of the thesis, we focus on getting the Page curve of eternal
black holes using the recent proposals given in the literature, e.g., island
proposal, doubly holographic setup, and wedge holography. Apart from getting
the Page curve, we have also obtained other exciting results, which are
discussed in the upcoming chapters.

###  5.3 Resolution of Information Paradox from Holography

Following three proposals are available in the literature which started with
the idea of holography to resolve the black hole information paradox.

####  5.3.1 Island Proposal and its Extension to HD Gravity

Authors in [6] proposed a method to resolve the information paradox which is
equivalent to getting the Page curve. Idea is that at early times we get only
contribution from the radiation region which gives the divergent part of
entanglement entropy at late times because the entanglement entropy of Hawking
radiation turns out to be proportional to the time. According to [6] at early
times situation remains the same whereas at late times interior of the black
holes becomes part of the entanglement wedge and hence at late times
entanglement entropy receives the contributions from the radiation as well as
interior of the black holes. The part of the interior the black holes which
contributes to the entanglement entropy is known as “island”.

Island rule was proposed from a setup where we couple the evaporating
JT(Jackiw Teitelboim) black hole plus conformal matter on the Planck brane
with the two-dimensional CFT bath [6]. The black hole is contained on the
Planck brane, and the Hawking radiation is collected in the 2D conformal bath.
This setup has the following three descriptions.

  * •

2D-Gravity: The Planck brane is coupled to the external CFT bath, which acts
as the sink for the Hawking radiation.

  * •

3D-Gravity: Two-dimensional conformal field theory has the three-dimensional
gravity dual with metric $AdS_{3}$ via AdS/CFT correspondence.

  * •

QM: The boundary of the external CFT bath is one dimensional where quantum
mechanics (QM) is present.

The island formula was derived from the gravitational path integral using the
replica trick for special JT black holes in [133, 134]. The authors obtained
the Page curve from the disconnected and connected saddles. One obtains the
linear time dependence in the Page curve from the disconnected saddles,
whereas connected saddles produce the finite part of the Page curve. The
discussion of [133] holds for the replica wormholes with $n$ boundary as well.
The generalised entropy in the presence of island surface is written as
follows:

$\displaystyle S_{\rm gen}(r)={\rm Min}_{\cal I}\Biggl{[}{\rm Ext}_{\cal
I}\Biggl{(}\frac{Area(\partial{\cal I})}{4G_{N}}+S_{\rm matter}({\cal
R}\cup{\cal I})\Biggr{)}\Biggr{]},$ (5.11)

where ${\cal R}$, $G_{N}$ and ${\cal I}$ are representing the radiation
region, Newton constant and the island surface. Equation (5.11) contains two
terms: island surface’s area and the matter contribution from the radiation
and island regions. From (5.11), we can easily see that when island surface is
absent then $S_{\rm gen}(r)=S_{\rm matter}({\cal R})$. It has been shown in
the literature that island surface emerges at late times and hence initially
one obtains the linear time dependence in the Page curve and at late times,
when island surface’s contribution dominates then one obtains the fall of the
entanglement entropy for the evaporating black holes where as constant part
(twice of their thermal entropies) for the eternal black holes. Hence, when we
include these contributions, we obtain the Page curve. If there are more than
one island surfaces then we have to consider the one with the minimal area. We
have followed this proposal to obtain the Page curve of Schwarzschild de-
Sitter black hole in [12] and discussed in detail in the chapter 8 of this
thesis. See [135, 136, 137] for the application of island proposal in the
context of JT gravity and other issues [138, 139, 140].

Island proposal was extended for higher derivative gravity in [141]. The
proposal is exactly similar to [6] but we have to replace the first term of
(5.11) by the term which can give the information about the entanglement
entropy of higher derivative gravity and the formula for the same was proposed
by X. Dong in [127] and hence the island proposal in the presence of higher
derivative terms in the gravitational action is written as [141]

$\displaystyle S_{\rm total}=S_{\rm gravity}+S_{\rm matter},$ (5.12)

where $S_{\rm matter}$ is the same as the $S_{\rm matter}({\cal R}\cup{\cal
I})$ of (5.11) and $S_{\rm gravity}$ will be calculated using the Dong’s
formula [127]. For the $AdS_{d+1}/CFT_{d}$ correspondence the Dong’s formulas
is given below111We have already written the formula in (5.1), here we are
writing the covariant form of (5.1). In this formula, $a$ and $i,j$ represents
the tangential and normal directions..

$\displaystyle\hskip-21.68121ptS_{\rm gravity}=\frac{1}{4G_{N}}\int
d^{d-1}y\sqrt{h}\Biggl{[}-\frac{\partial{\cal L}}{\partial
R_{{\mu_{1}}{\nu_{1}}{\rho_{1}}{\sigma_{1}}}}\epsilon_{{\mu_{1}}{\nu_{1}}}\epsilon_{{\rho_{1}}{\sigma_{1}}}+\sum_{\alpha}\left(\frac{\partial^{2}{\cal
L}}{\partial R_{{\mu_{3}}{\nu_{3}}{\rho_{3}}{\sigma_{3}}}\partial
R_{{\mu_{1}}{\nu_{1}}{\rho_{1}}{\sigma_{1}}}}\right)_{\alpha}$
$\displaystyle\frac{2K_{{\lambda_{3}}{\rho_{3}}{\sigma_{3}}}K_{{\lambda_{1}}{\rho_{1}}{\sigma_{1}}}}{q_{\alpha}+1}[\left(n_{{\mu_{3}}{\mu_{1}}}n_{{\nu_{3}}{\nu_{1}}}-\epsilon_{{\mu_{3}}{\mu_{1}}}\epsilon_{{\nu_{3}}{\nu_{1}}}\right)n^{{\lambda_{3}}{\lambda_{1}}}+\left(n_{{\mu_{3}}{\mu_{1}}}\epsilon_{{\nu_{3}}{\nu_{1}}}+\epsilon_{{\mu_{3}}{\mu_{1}}}n_{{\nu_{3}}{\nu_{1}}}\right)\epsilon^{{\lambda_{3}}{\lambda_{1}}}]\Biggr{]},$
(5.13)

where

$\displaystyle n_{\mu\nu}=n_{\mu}^{(i)}n_{\nu}^{(j)}g_{ij},$
$\displaystyle\epsilon_{\mu\nu}=n_{\mu}^{(i)}n_{\nu}^{(i)}\epsilon_{ij},$
$\displaystyle\epsilon_{\mu\nu}\epsilon_{\rho\sigma}=n_{\mu\rho}n_{\nu\sigma}-n_{\mu\sigma}n_{\nu\rho},$
$\displaystyle
K_{\lambda\mu\nu}=n_{\lambda}^{(i)}m_{\mu}^{(a)}n_{\nu}^{(b)}K_{iab},$ (5.14)

where $m_{\mu}^{(a)}$ and $n_{\mu}^{(i)}$ are the unit normal vectors along
the tangential ($y^{a}$) and normal directions, , $h$ is the determinant of
the induced metric on co-dimension two surface in the bulk. $q_{\alpha}$ can
be obtained similar to (5.10).

####  5.3.2 Doubly Holographic Setup

The doubly holographic setup is a nice setup to calculate the Page curve of
black holes. As its name suggests, it is the double copy of the usual
holography proposed by J. Maldacena. First, we need to take the bulk and
truncate the geometry along one of the spatial coordinates [142, 143]. By
doing so, one generates $d$-dimensional geometry embedded in the
($d+1$)-dimensional bulk. The $d$-dimensional geometry is known as end-of-the-
world brane or Karch-Randall brane in the literature, and this holography is
called “braneworld holography”. The doubly holographic setup is obtained by
joining the two copies of the Karch-Randall model. The setup consists of an
eternal black hole living on the brane and two baths where we can collect the
Hawking radiation. These two baths behave as thermofield double states because
these are like two copies of the boundary conformal field theory (BCFT). Let
us discuss the double holography in the context of $AdS_{d+1}/BCFT_{d}$
correspondence using a bottom-up approach, and the setup is shown in figure
5.1.

Figure 5.1: Description of doubly holographic setup. Blue curves are the
island surfaces and red curve is the Hartman-Maldacena surface. $\delta M$ is
the conformal boundary, $z_{*}$ and $z_{T}$ are the turning points of Hartman-
Maldacena and island surfaces.

The doubly holographic setup has the three descriptions summarised below.

  * •

Boundary description: $d$-dimensional BCFT at the conformal boundary of the
bulk $AdS_{d+1}$. The boundary of ${BCFT}_{d}$ is the ($d-1$) dimensional
defect.

  * •

Intermediate description: Gravity on the end-of-the-world brane is coupled to
BCFT via transparent boundary condition at the defect.

  * •

Bulk description: The holographic dual of ${BCFT}_{d}$ is $AdS_{d+1}$
spacetime.

The intermediate description is very crucial to resolving the information
paradox. Because in this description black hole living on the end-of-the-world
brane directly couples with the external CFT bath. Define $S({\cal R})$ as the
von Neumann entropy of the subregion ${\cal R}$ on a constant time slice in
description 1. One can obtain the $S({\cal R})$ in second description from the
island rule [6]:

$S({\cal R})={\rm min}_{\cal I}~{}{\rm ext}_{\cal I}~{}S_{gen}({\cal
R}\cup{\cal I}),$ (5.15)

where generalised entropy functional $(S_{gen}({\cal R}\cup{\cal I}))$ is
[126]:

$S_{gen}({\cal R}\cup{\cal I})=\frac{A(\partial{\cal
I})}{4G_{N}}+S_{matter}({\cal R}\cup{\cal I}),$ (5.16)

A doubly holographic setup is advantageous in the sense that we can obtain
$S({\cal R})$ very easily using the classical Ryu-Takayanagi formula [107].
When bulk is $(d+1)$ dimensional then [107]:

$S_{gen}({\cal R}\cup{\cal I})=\frac{A(\gamma)}{4G_{N}^{(d+1)}},$ (5.17)

where $\gamma$ is the minimal co-dimension of two surface in bulk.

In figure 5.1, there are two BCFTs on the conformal boundary of the bulk. The
vertical line is the end-of-the-world brane which contains the black hole. The
CFT bath collects the Hawking radiation emitted by the black hole. This setup
has two possible extremal surfaces: Hartman-Maldacena [144] and island
surfaces. The Hartman-Maldacena surface connects the two BCFTs; it starts at
the CFT bath, crosses the horizons, reaches up to the turning point, and then
meets the thermofield double partner of BCFT. The entanglement entropy is
divergent at late times for the Hartman-Maldacena surface, which implies
Hawking’s information paradox. The island surface starts at the external CFT
bath and lands on the end-of-the-world brane. The island surface’s
entanglement entropy turns out to be a constant value (twice of thermal
entropy of the black hole). Therefore, one recovers the Page curve by
combining the contributions of the entanglement entropies of both these
extremal surfaces. See [7, 145, 146, 147, 151, 152, 153, 154, 155, 156, 157,
158, 159, 148, 149, 150, 160, 161] for the extensive literature on the doubly
holographic setup.

Some authors found that gravity is massive on the end-of-the-world brane [163,
164, 162, 165] when we couple the brane to the external CFT bath. In some
papers, it was shown by the authors that we could construct the doubly
holographic setup with massless gravity on the brane [166, 154, 11, 167]. We
constructed the doubly holographic setup from a top-down approach in [11] and
details are given in chapter 7. We have a non-conformal bath ($QCD_{2+1}$) and
the holographic dual is ${\cal M}$ theory inclusive of ${\cal O}(R^{4})$
corrections [1]. The reason for the existence of massless graviton in our
setup is that we required the wave function of the graviton to be normalized,
the second reason is due to the Dirichlet boundary condition on the wave
function of the graviton, and the third reason is that end-of-the-world brane
had non-zero tension and hence localization of graviton is possible on the
brane in a “volcano”-like potential. We obtained the Page curve with massless
gravity in our setup, which was impossible in other doubly holographic setups
without the DGP term. One alternate method to deduce the massless gravity on
the brane is to include the Dvali-Gabadadze-Porrati (DGP) term [168] on the
brane [166].

####  5.3.3 Wedge Holography

In the doubly holographic setup, the external bath is a fixed CFT bath. In
some of the papers, it was found that gravity is massive on the end-of-the-
world brane and island prescription is not valid in the masssless gravity.
Some of the authors considered the bath as gravitating too [8, 9, 169, 162].
This setup is known as wedge holography in the literature. It was also argued
that in wedge holography, Hartman-Maldacena surface does not exist and hence
no Page curve in wedge holography. In [13], we showed that entanglement
entropy of Hartman-Maldacena surface is non-zero for the AdS and Schwarzschild
black hole and it is zero for the de-Sitter black hole. This implies that one
could get the Page curve for the AdS and Schwarzschild black hole but not for
the de-Sitter space using wedge holography. See Fig. 5.2 for the pictorial
description of the wedge holography.

Figure 5.2: Description of wedge holography. Two $d$-dimensional Karch-Randall
branes joined at the $(d-1)$ dimensional defect, Karch-Randall branes are
embedded in $(d+1)$-dimensional bulk.

One can get the Page curve or not in wedge holography is a debatable topic.
Some progress in this direction has been made in [166]. It was shown by the
author that we can get the Page curve with massless gravity localize on the
Karch-Randall brane provided we have to include the DGP term on the Karch-
Randall brane, see [170, 171] for the detailed analysis with examples.

Take into consideration the following action, [8, 9, 169], to describe the
mathematical description of wedge holography:

$\displaystyle S=-\frac{1}{16\pi G_{N}^{(d+1)}}\int d^{d+1}x\sqrt{-g_{\rm
bulk}}\left(R[g_{\rm bulk}]-2\Lambda\right)$ $\displaystyle-\frac{1}{8\pi
G_{N}^{(d+1)}}\int_{\alpha=1,2}d^{d}x\sqrt{-h_{\alpha}}\left({\cal
K}_{\alpha}-T_{\alpha}\right),$ (5.18)

where the first term serves as the Einstein-Hilbert term with the cosmological
constant $\left(\Lambda=-\frac{d(d-1)}{2}\right)$ and the second term being
the boundary terms on the Karch-Randall branes with tensions $T_{\alpha=1,2}$.
The bulk action’s Einstein equation (5.3.3) is as follows:

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{d(d-1)}{2}g_{\mu\nu}.$ (5.19)

The above equation has the following solution [9]:

$ds_{(d+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+\cosh^{2}(r)h_{ij}^{\alpha}dy^{i}dy^{j},$
(5.20)

where $h_{ij}^{\alpha}$ denote the induced metric on Karch-Randall branes. The
Neumann boundary condition (NBC) obtained from the variation of the bulk
action (5.3.3) with respect to the metric on Karch-Randall branes
$h_{ij}^{\alpha}$ is as follows:

${\cal K}_{ij}^{\alpha}-({\cal K}^{\alpha}-T^{\alpha})h_{ij}^{\alpha}=0.$
(5.21)

The wedge holography exist if the metric (5.20) is the solution of the
Einstein equation (5.19) and the bulk metric (5.20) satisfies the NBC (5.21)
on the branes, i.e., at $r=\pm\rho$. For $-\rho_{1}\leq r\leq\rho_{2}$ with
$\rho_{1}\neq\rho_{2}$ [9], the branes will have different tensions [9]. It is
also required that the induced metric $h_{ij}^{\alpha}$ on the branes should
be the solution of Einstein equation with a negative cosmological constant to
have the anti de-Sitter branes in wedge holography.

$\displaystyle
R_{ij}^{\alpha}-\frac{1}{2}h_{ij}^{\alpha}R[h_{ij}]^{\alpha}=\frac{(d-1)(d-2)}{2}h_{ij}^{\alpha},$
(5.22)

Similar to the double holography, wedge holography has also the three
descriptions:

  * •

Boundary description: ${BCFT}_{d}$ on the conformal boundary of the bulk
$AdS_{d+1}$ with the $(d-1)$ dimensional defect.

  * •

Intermediate description: two gravitating systems are connected with each
other via the transparent boundary condition at the defect.

  * •

Bulk description: the holographic dual of ${BCFT}_{d}$ is classical gravity
$AdS_{d+1}$ spacetime.

The wedge holographic dictionary for the $(d+1)$\- dimensional bulk is stated
as: holographic dual of the $(d-1)$-dimensional defect conformal field theory
is the classical gravity in $(d+1)$\- dimensions. Therefore it is a co-
dimension two holography. Now let us understand the how this duality exists.

Classical gravity in $(d+1)$-dimensional bulk
$\equiv$ (Quantum) gravity on two $d$-dimensional Karch-Randall branes with
metric $AdS_{d}$
$\equiv$ CFT living on $(d-1)$-dimensional defect.

Braneworld holography [142, 143] relates first and second line whereas AdS/CFT
correspondence [17] between the dynamical gravity on the Karch-Randall brane
and defect CFT connects second and third line. Therefore, classical gravity in
$(d+1)$ bulk is dual to $CFT_{d-1}$ at the defect. Wedge holography helps us
in getting the Page curve of the black holes similar to doubly holographic
setup discussed in 5.3.2. One is required to compute the entanglement
entropies of Hartman-Maldacena and island surfaces and plot of these entropies
with time will give the Page curve.

##  Chapter 6 Page Curves of Reissner-Nordström Black Hole in HD Gravity

###  6.1 Introduction and Motivation

Applying the island proposal described in 5.3.1, we are able to compute the
Page curves of eternal black holes [6]. In the papers [172, 173, 174, 175],
the Page curves of the Reissner-Nordström black hole, charged dilaton black
hole, Schwarzschild black hole, and hyperscaling violating black branes were
found. In the research referred to as [176], the page curve in the charged
linear dilaton model for both the non-extremal black hole and the extremal
black hole was analyzed. Islands in Kerr-de Sitter spacetime and generalized
dilaton theories were the subject of research in the article [177, 178]. In
the research referred to as [141], the effect that mutual knowledge between
subsystems on the Page curve was analyzed. Calculations of the page curve used
in higher derivative gravity theories could be found in the papers [141, 11].
Page curves of Schwarzschild black holes were obtained by the authors of the
paper mentioned above [141] when higher derivative terms, known as ${\cal
O}(R^{2})$ and Gauss-Bonnet terms, were present in the gravitational action.

In the papers [185, 179, 180, 181, 182, 183, 184] researchers looked at
charged black holes in Einstein-Gauss-Bonnet gravity in four dimensions. See
[186] for a review of the Einstein-Gauss-Bonnet theory of gravity when applied
to four dimensions. Since we were dealing with higher derivative theories of
gravity, we employed [127] to compute the entanglement entropy for those cases
where it was necessary. The non-extremal black holes are what we’ll be
discussing in this chapter. For extremal black holes, see [187, 176, 188]. It
has come to our attention that research on the effect of the Gauss-Bonnet
coupling on the Page curve of a charged Einstein-Gauss-Bonnet black hole had
not been done. As a result, it would be quite fascinating to investigate how
the Page curves of the Reissner-Nordström black hole will change when the
Gauss-Bonnet term is present. As a result of this inspiration, we have
computed the Page curves of charged black holes in higher derivative gravity
with ${\cal O}(R^{2})$ terms and for charged Einstein-Gauss-Bonnet black holes
in this work.

###  6.2 Review

This section has been broken up into two different subsections. We will
provide a concise review of the charged black hole in Einstein-Guass-Bonnet
gravity in the vanishing cosmological constant limit [189] and the Page curve
calculation of the Reissner-Nordström black hole [172] in four dimensions in
6.2.1 and 6.2.2.

####  6.2.1 Brief Review of Charged Black Hole in Einstein-Gauss-Bonnet
Gravity

It has been demonstrated by the authors of the work [190] that the Gauss-
Bonnet term in four dimensions can be made dynamical by rescaling the Gauss-
Bonnet coupling to the form $\alpha\rightarrow\frac{\alpha}{D-4}$. The authors
in [191] produced a consistent theory of Gauss-Bonnet gravity in
$(d+1)$-dimensions with all of the necessary degrees of freedom. The action of
Aoki, Gorji, and Mukhohyama (AGM) theory can be written as follows:

$\displaystyle S=\frac{1}{2\kappa^{2}}\int
d^{(d+1)}x\sqrt{-g}\left(R[g]+\alpha{\cal L}_{\rm GB}\right),$ (6.1)

where ${\cal L}_{\rm
GB}=R_{\mu\nu\rho\sigma}[g]R^{\mu\nu\rho\sigma}[g]-4R_{\mu\nu}[g]R^{\mu\nu}[g]+R[g]^{2}$
represents Gauss-Bonnet term. The limit $d\rightarrow 3$ of (6.1) implies
consistent theory of four dimensional neutral Gauss-Bonnet black hole. By
scaling the Gauss-Bonnet coupling, the author of [189] produced a charged
Einstein-Gauss-Bonnet black hole solution in four dimensions. In the vanishing
cosmological constant limit, we discuss about this solution. The Maxwell term,
along with Einstein-Gauss-Bonnet gravity, has the following action:

$\displaystyle S=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}[R[g]+\alpha{\cal
L}_{\rm GB}-F_{\mu\nu}F^{\mu\nu}],$ (6.2)

where $R[g]$ represents the Ricci scalar, $\alpha$ denotes the Gauss-Bonnet
coupling, ${\cal L}_{\rm GB}$ represents the Gauss-Bonnet term, and
$F_{\mu\nu}$ denotes the field strength tensor associated with $A_{\mu}$ gauge
field, which are described as follows:

$\displaystyle{\cal L}_{\rm
GB}=R_{\mu\nu\rho\sigma}[g]R^{\mu\nu\rho\sigma}[g]-4R_{\mu\nu}[g]R^{\mu\nu}[g]+R[g]^{2},$
$\displaystyle F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$ (6.3)

The gravitational action (6.2)has the following black hole solution:

$\displaystyle ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{F(r)}+r^{2}d\Omega^{2},$ (6.4)

where the black hole function $F(r)$ is defined as:

$F(r)=1+\frac{r^{2}}{2\alpha}\left(1\pm\sqrt{1+4\alpha\left(\frac{2M}{r^{3}}-\frac{Q^{2}}{r^{4}}\right)}\right).$
(6.5)

If we do a small expansion in $\alpha$ with a negative sign chosen in (6.5),
then we get the following result:

$F(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{\left(Q^{2}-2Mr\right)^{2}}{r^{6}}\alpha.$
(6.6)

Solving $F(r)=0$ with negative sign of (6.5) yields the black hole’s horizons,
which are presented in the following form:

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}-\alpha},$ (6.7)

where the physical horizon of the charged black hole is denoted by the symbol
$r_{+}$ and the Cauchy horizon is denoted by the symbol $r_{-}$.

Let us now examine the relationship between the Gauss-Bonnet coupling $\alpha$
and the black hole horizon $r_{+}$. We have depicted it in figure 6.1 for
$M=1$ and for a variety of different values for the black hole charges. When
the Gauss-Bonnet coupling ($\alpha$) is zero, as shown by the figure 6.1 and
the equation (6.7), we get the typical Reissner-Nordström black hole. The
decrease of Gauss-Bonnet coupling results in the increase of the black hole
horizon and vice-versa.

Figure 6.1: Plot of $r_{+}$ versus $\alpha$

When $M=1$, the following values for the Gauss-Bonnet coupling $\alpha$ have
been specified by the authors of [179, 182]:

  * •

A:$Q^{2}-4-2\sqrt{4-2Q^{2}}<\alpha<1-Q^{2},\ \ {\rm when}\ \
0<Q<\frac{\sqrt{3}}{2}$.

  * •

B:$Q^{2}-4-2\sqrt{4-2Q^{2}}<\alpha<Q^{2}-4+2\sqrt{4-2Q^{2}},\ \ {\rm when}\ \
0<Q<\sqrt{2}$.

In contrast to the region B, which only contains the physical horizon
($r_{+}$), the region A contains both horizons ($r_{\pm}$). Since we’re
looking for the non-extremal black hole, the region A is of interest to us.
Using $Q=0.8$, we found that for the region A, $\alpha\in(-6.66,0.36)$.

####  6.2.2 Review of Page Curve of Reissner-Nordström Black Hole

Here, we take a look back at [172]’s computation of the Reisnner-Nordström
black hole’s Page curve. The action that describes the Einstein-Maxwell
gravity is given as:

$\displaystyle I=\frac{1}{16\pi G_{N}}\int_{\cal
M}d^{4}x\sqrt{-g}\left(R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right)+I_{\rm
matter},$ (6.8)

where $R$ represents the Ricci scalar, $F_{\mu\nu}$ denotes the field strength
tensor associated with the $A_{\mu}$ gauge field, and $I_{\rm matter}$
represents the matter part of the action. The Reissner-Nordstörm black hole
has the following metric:

$\displaystyle ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{F(r)}+r^{2}d\Omega^{2},$ (6.9)

where, $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ and black hole
function:

$F(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.$ (6.10)

The following are the black hole horizons that result from solving $F(r)=0$:

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}.$ (6.11)

Below is a version of metric (6.9) written in Kruskal coordinates:

$\displaystyle
ds^{2}=-\frac{r_{+}r_{-}}{r^{2}\kappa_{+}^{2}}\left(\frac{r_{-}}{r-r_{-}}\right)^{\frac{\kappa_{+}}{\kappa_{-}}-1}e^{-2\kappa_{+}r}dUdV+r^{2}d\Omega^{2},$
(6.12)

where,

$U=-e^{-\kappa_{+}(t-r_{*})},V=e^{\kappa_{+}(t+r_{*})},$ (6.13)

and the tortoise coordinate $r_{*}$ is given by:

$\displaystyle
r_{*}=r+\frac{r_{+}^{2}}{r_{+}-r_{-}}\log|r-r_{+}|-\frac{r_{-}^{2}}{r_{+}-r_{-}}\log|r-r_{-}|.$
(6.14)

For both $r_{+}$ and $r_{-}$, we have the following expressions for the
surface gravities $\kappa_{\pm}$:

$\displaystyle\kappa_{\pm}=\frac{r_{\pm}-r_{\mp}}{2r_{\pm}^{2}}.$ (6.15)

The following are definitions of the charged black hole’s Hawking temperature
and Bekenstein-Hawking entropy:

$\displaystyle T_{\rm RN}=\frac{\kappa_{+}}{2\pi},$ $\displaystyle S_{\rm
BH}^{(0)}=\frac{\pi r_{+}^{2}}{G_{N}}.$ (6.16)

The conformal factor in the metric, expressed in Kruskal coordinates, is
defined as follows:

$\displaystyle
g^{2}(r)=\frac{r_{+}r_{-}}{r^{2}\kappa_{+}^{2}}\left(\frac{r_{-}}{r-r_{-}}\right)^{\frac{\kappa_{+}}{\kappa_{-}}-1}e^{-2\kappa_{+}r}.$
(6.17)

As a result, the metric can be expressed as follows:

$\displaystyle ds^{2}=-g^{2}(r)dUdV+r^{2}d\Omega^{2}.$ (6.18)

The Penrose diagrams of an eternal Reissner-Nordström black hole are shown in
figures 6.3 and 6.3. In figures 6.3 and 6.3, the $R_{-}$ and $R_{+}$ denote
the left and right wedges of radiation regions, the $b_{-}$ and $b_{+}$ denote
the boundaries of $R_{-}$ and $R_{+}$, and the $a_{-}$ and $a_{+}$ denote the
boundaries of island surface in the left and right wedges, respectively.

Figure 6.2: Penrose diagram of an eternal
Reissner-Nordström black hole [172] in the
absence of island surface.

Figure 6.3: Penrose diagram of an eternal
Reissner-Nordström black hole [172] in the
presence of island surface.

#####  6.2.2.1 Entanglement Entropy without Island

Since there is no island surface to begin with, the entanglement entropy of
the Hawking radiation will be obtained by applying the formula shown below
[192, 193, 173]:

$\displaystyle S=\frac{c}{6}\log[d^{2}(b_{+},b_{-})],$ (6.19)

where $b_{+}(t_{b},b)$ and $b_{-}(-t_{b}+\iota\frac{\beta}{2},b)$ represent
the boundaries of the radiation regions in the right and left wedges of the
Reissner-Nordstrom black hole, respectively. The value of $\beta$, which is
defined as $\beta=2\pi/\kappa_{+}$, represents the inverse of the Hawking
temperature. Equation (6.19) with no island is applicable to two-dimensional
vacuum CFT. Higher-dimensional spacetime’s entanglement entropy formula is
unknown. However, by assuming that the observer/cut-off surface is located at
a large distance from the black hole horizon ($r_{+}$) ($b_{\pm}\gg r_{+}$),
we can use the (6.19) for higher-dimensional spacetime in the s-wave
assumption. Therefore, the angular component of the metric may be ignored in
the s-wave approximation, leaving us with an asymptotically flat black hole
for which we can utilize the (6.19). The formula for calculating the geodesic
distance among two points $(l_{1},l_{2})$ is as follows:

$d(l_{1},l_{2})=\sqrt{g(l_{1})g(l_{2})(U(l_{2})-U(l_{1}))(V(l_{1})-V(l_{2}))}.$
(6.20)

Now, utilizing equations (6.13), (6.14), (6.17), (6.19) and (6.20), the
entanglement entropy of Hawking radiation in the absence of an island surface
has been simplified to [172]:

$\displaystyle
S=\frac{c}{6}\log[4g(b)^{2}e^{2\kappa_{+}r_{*}(b)}\cosh^{2}\kappa_{+}t_{b}],$
(6.21)

If $t_{b}\rightarrow\infty$, i.e., in late time approximation, we can write
$\cosh\kappa_{+}t_{b}\sim e^{\kappa_{+}t_{b}}$. The time-dependent component
of the equation (6.21) can be simplified as:

$\displaystyle S_{\rm EE}^{\rm WI}\sim\frac{c}{3}\kappa_{+}t_{b}.$ (6.22)

Therefore, we can observe that the entanglement entropy of the Hawking
radiation in without an island surface is increasing linearly with time and
approaches infinite at later times via the equation (6.22). This is what
causes the information paradox for the Reissner-Nordström black hole.

#####  6.2.2.2 Entanglement Entropy with Island

Here, we will go through the computation of the entanglement entropy of the
Hawking radiation when an island surface is present. We are able to compute it
by utilizing the formula [192, 193, 173], which is as follows:

$\displaystyle S({\cal R}\cup{\cal
I})=\frac{c}{3}\log\left(\frac{d(a_{+},a_{-})d(b_{+},b_{-})d(a_{+},b_{+})d(a_{-},b_{-})}{d(a_{+},b_{-})d(a_{-},b_{+})}\right),$
(6.23)

where $a_{+}(t_{a},a)$ and $a_{-}(-t_{a}+\iota\frac{\beta}{2},a)$ represent
the boundaries of the island surface in the right and left wedges of Reissner-
Nordström geometry, respectively. In the s-wave approximation ($b_{\pm}\gg
r_{+}$), the entanglement entropy formula (6.23) can be applied to higher-
dimensional spacetime similar to (6.19). In late time assumption
($t_{a},t_{b}\gg b>r_{+}$) utilizing (6.13), (6.14), (6.17) and (6.20); (6.23)
simplified to the following form [172]:

$\displaystyle S({\cal R}\cup{\cal I})^{\rm late\
times}=\frac{c}{6}\log[g^{2}(a)g^{2}(b)]+\frac{2c}{3}\kappa_{+}r_{*}(b)$
$\displaystyle+\frac{c}{3}\Biggl{[}-2e^{-\kappa_{+}(b-a)}\Biggl{|}\frac{a-r_{+}}{b-r_{+}}\Biggr{|}^{1/2}\Biggl{|}\frac{a-r_{-}}{b-r_{-}}\Biggr{|}^{-(r_{-}^{2}/r_{+}^{2})}-e^{\kappa_{+}(r_{*}(b)-r_{*}(a)-2t_{b})}\Biggr{]}.$
(6.24)

Therefore, generalized entropy with inclusion of an island surface at late
times is as follows [172]:

$\displaystyle S_{gen}^{\rm late\ times}(a)=\frac{2\pi
a^{2}}{G_{N}}+\frac{c}{6}\log[g^{2}(a)g^{2}(b)]+\frac{2c}{3}\kappa_{+}r_{*}(b)$
$\displaystyle+\frac{c}{3}\Biggl{[}-2e^{-\kappa_{+}(b-a)}\Biggl{|}\frac{a-r_{+}}{b-r_{+}}\Biggr{|}^{1/2}\Biggl{|}\frac{a-r_{-}}{b-r_{-}}\Biggr{|}^{-(r_{-}^{2}/r_{+}^{2})}-e^{\kappa_{+}(r_{*}(b)-r_{*}(a)-2t_{b})}\Biggr{]}.$
(6.25)

By taking the variation of (6.2.2.2) with respect to $a$, i.e.,
$\frac{\partial S_{gen}^{\rm late\ times}(a)}{\partial a}=0$, we have to find
the solution of the equation given below:

$\displaystyle\frac{4\pi
r_{+}}{G_{N}}+\frac{c\left(\frac{r_{+}-r_{-}}{b-r_{-}}\right){}^{-\frac{r_{-}^{2}}{r_{+}^{2}}}e^{\kappa_{+}\left(r_{+}-b\right)}}{3\left(r_{+}-b\right)\sqrt{\frac{a-r_{+}}{b-r_{+}}}}=0,$
(6.26)

which has solution:

$\displaystyle a\approx
r_{+}+\Biggl{[}\frac{c^{2}\left(\frac{r_{+}-r_{-}}{b-r_{-}}\right){}^{-\frac{2r_{-}^{2}}{r_{+}^{2}}}G_{N}^{2}e^{2\kappa_{+}\left(r_{+}-b\right)}}{144\pi^{2}r_{+}^{2}\left(b-r_{+}\right)}\Biggr{]},$
(6.27)

Substitution of $a$ using (6.27) in (6.2.2.2), results in Hawking radiation’s
total entanglement entropy at late times:

$\displaystyle S_{\rm total}^{(0)}=\frac{2\pi
r_{+}^{2}}{G_{N}}+\frac{c}{3}\log\left(\frac{r_{-}e^{2\kappa_{+}\left(b-r_{+}\right)}\left(\frac{r_{-}^{2}}{\left(r_{+}-r_{-}\right)\left(b-r_{-}\right)}\right){}^{\frac{1}{2}\left(\frac{\kappa_{+}}{\kappa_{-}}-1\right)}}{b\kappa_{+}^{2}}\right)+small,$
$\displaystyle S_{\rm total}^{(0)}=2S_{\rm
BH}^{\rm(0)}+\frac{c}{3}\log\left(\frac{r_{-}e^{2\kappa_{+}\left(b-r_{+}\right)}\left(\frac{r_{-}^{2}}{\left(r_{+}-r_{-}\right)\left(b-r_{-}\right)}\right){}^{\frac{1}{2}\left(\frac{\kappa_{+}}{\kappa_{-}}-1\right)}}{b\kappa_{+}^{2}}\right)+small.$
(6.28)

We may deduce from the preceding equation that the total entanglement entropy
remains the same, which means that it takes precedence after the Page time,
and as a result, we have the Page curve of an eternal Reissner-Nordström black
hole.

#####  6.2.2.3 Page Curve

We showed that the entanglement entropy without an island surface increases
linearly with time by using (6.22), and we found that the entanglement entropy
remains constant at late times by using (6.2.2.2). Island surface appears
after the Page time, which saturates the linear growth of entanglement entropy
and attains a fix value, which equals twice of the Bekenstein-Hawking entropy
of the Reissner-Nordström black hole, we are able to get the Page curve of the
Reissner-Nordström black hole.

We used only the leading order term in the $G_{N}$ in equation (6.2.2.2) when
we drew the Page curve of the Reissner-Nordström black hole (shown in figure
6.4) with the value $M=1,Q=0.8,G_{N}=1$. The blue line in figure 6.4 refers to
the linear time growth of the entanglement entropy equation (6.22)), and the
orange line refers to the entanglement entropy with island surface (equation
(6.2.2.2)), which takes over after the Page time and results in the Page
curve.

Figure 6.4: Page curve of an eternal Reissner-Nordström black hole by
considering only leading order term in $G_{N}$.

Page time: Page time is described as the point in time at which the
entanglement entropy of the Hawking radiation begins to approach zero for a
black hole that is evaporating and reaches a value that is constant for a
black hole that is eternal. From matching the equations (6.22) and (6.2.2.2),
we are able to obtain the Page time, which is represented by the following:

$\displaystyle t_{Page}^{(0)}\sim\frac{6\pi
r_{+}^{2}}{c\kappa_{+}G_{N}}=\frac{12\pi r_{+}^{4}}{c\
G_{N}(r_{+}-r_{-})}=\frac{3S_{\rm BH}^{(0)}}{2\pi cT_{\rm RN}}.$ (6.29)

Scrambling time: The term “scrambling time” refers to the time when we are
able to retrieve the information that was lost when it was sucked into the
black hole in the form of Hawking radiation [194, 195]. Information from a
black hole that has evaporated halfway can be quickly retrieved as Hawking
radiation, as detailed in [194]. If the black hole hasn’t evaporated halfway,
we’ll have to wait until it does so that we can swiftly retrieve the
information we’ve lost. Scrambling time, as used in the terminology of
entanglement wedge reconstruction [196], is the amount of time it takes the
information from the cut-off surface ($r=b_{+}$) to arrive at the island
surface’s boundary ($r=a_{+}$). Once the information thrown into the black
hole has traveled to the boundary of the island surface ($r=a_{+}$), the
entanglement entropy of Hawking radiation begins to include contribution from
the island surface, as the degrees of freedom of the island become part of the
entanglement wedge of the Hawking radiation. Time it takes the information to
travel from the cut-off surface ($r=b$) to the boundary of the island surface
($r=a$) when sent into a black hole is provided by:

$\displaystyle
t_{scr}^{(0)}=r_{*}(b)-r_{*}(a)=b-a+\frac{r_{-}^{2}}{r_{+}-r_{-}}\log\left(\frac{a-r_{-}}{b-r_{-}}\right)-\frac{r_{+}^{2}}{r_{+}-r_{-}}\log\left(\frac{a-r_{+}}{b-r_{+}}\right).$
(6.30)

Scrambling time of the Reissner-Nordström black hole is found to be when the
value of $a$ from equation (6.27) is substituted into the previous equation
and the obtained expression is written as:

$\displaystyle
t_{scr}^{(0)}\sim\frac{2r_{+}^{2}}{(r_{+}-r_{-})}\log\left(\frac{\pi
r_{+}^{2}}{G_{N}}\right)+small=\frac{1}{2\pi T_{\rm RN}}\log\left(S_{\rm
BH}^{(0)}\right)+small.$ (6.31)

This result is comparable to [195], which suggests that black holes are the
most efficient scramblers. We can observe this by looking at the equation
(6.31), which shows that the scrambling time is logarithmic of the thermal
entropy of charged black hole analogous to [195].

###  6.3 Page Curves of Charged Black Hole in HD Gravity up to ${\cal
O}(R^{2})$

We studied generic higher derivative terms of ${\cal O}(R^{2})$ for the
gravitational action as that of considered in [141], and we obtained the Page
curves of an eternal Reissner-Nordström black hole. The gravitational action
of the Einstein-Maxwell theory with inclusion of ${\cal O}(R^{2})$ is given as
below:

$\displaystyle S=\frac{1}{16\pi G_{N}}\int
d^{4}x\sqrt{-g}\left(R[g]+\lambda_{1}R^{2}[g]+\lambda_{2}R_{\mu\nu}[g]R^{\mu\nu}[g]+\alpha{\cal
L}_{GB}[g]-F_{\mu\nu}F^{\mu\nu}\right),$ (6.32)

where $R[g]$ denotes the Ricci scalar, $R_{\mu\nu}[g]$ represents the Ricci
tensor, and ${\cal L}_{GB}$ denotes the Gauss-Bonnet term defined in (6.2.1).
The formula to calculate the Wald entropy in higher derivative theories of
gravity is given as [102, 103]:

$\displaystyle S_{\rm
Wald}=\frac{1}{4G_{N}}\oint\sqrt{h}\left(\frac{\partial{\cal L}}{\partial
R_{\mu\nu\rho\sigma}}\right)\epsilon_{\mu\nu}\epsilon_{\rho\sigma}d^{2}x,$
(6.33)

where $h$ denotes the determinant of the induced metric of the co-dimension
two surface. The metric (6.9) is rewritten as follows:

$ds^{2}=-\left(\frac{(r-r_{+})(r-r_{-})}{r^{2}}\right)dt^{2}+\frac{dr^{2}}{\left(\frac{(r-r_{+})(r-r_{-})}{r^{2}}\right)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (6.34)

We found the following for the action (6.32):

$\displaystyle\frac{\partial{\cal L}}{\partial
R_{\mu\nu\rho\sigma}}=\Biggr{[}g^{\nu\sigma}g^{\mu\rho}(1+2\alpha
R[g])+2\alpha R^{\mu\nu\rho\sigma}[g]-8\alpha
g^{\mu\rho}R^{\nu\sigma}[g]\Biggl{]}$
$\displaystyle+2\lambda_{1}R[g]g^{\nu\sigma}g^{\mu\rho}+2\lambda_{2}g^{\mu\rho}R^{\nu\sigma}[g],$
(6.35)

For the metric (referred to as (6.34)), the Wald entropy that corresponds to
the action (referred to as (6.32)) can be calculated utilizing the equation
(referred to as (6.3)) as follows:

$\displaystyle S_{\rm Wald}^{(R^{2})}=S_{\rm
BH}^{(\alpha)}=\frac{1}{4G_{N}}\oint\sqrt{h}\left(\frac{\partial{\cal
L}}{\partial R_{trtr}}\right)d\theta d\phi$
$\displaystyle=\frac{\pi}{G_{N}}\left(r_{+}^{2}+\frac{4\alpha(r_{+}+6r_{-})}{r_{+}}-2\lambda_{2}\left(\frac{r_{-}r_{+}}{r_{+}^{2}}\right)\right).$
(6.36)

Given that $r_{+}\gg r_{-}$, the preceding equation can be simplified to read
as follows:

$\displaystyle S_{\rm Wald}^{(R^{2})}=S_{\rm
BH}^{(\alpha)}\sim\frac{\pi}{G_{N}}\left(r_{+}^{2}+4\alpha\right).$ (6.37)

The formula which was given in [127] and is described in chapter 5, can be
utilized to determine the holographic entanglement entropy that is associated
with higher derivative gravity theories. According to what was discussed in
chapter 5, we are able to write the total entanglement entropy in the presence
of higher derivative terms as follows [141]:

$\displaystyle S_{\rm total}=S_{\rm gravity}+S_{\rm matter}(R\cup{\cal I}).$
(6.38)

The result of calculating gravity’s contribution to the entanglement entropy
corresponding to the action (6.32) is as follows [141]:

$\displaystyle S_{\rm gravity}=\frac{A[\partial{\cal I}]}{4G_{N,\rm
ren}}+\frac{1}{4G_{N,\rm ren}}\int_{\partial{\cal I}}\left(2\lambda_{1,\rm
ren}R[g]+\lambda_{2,\rm
ren}\sum_{i=1}^{2}[R_{\mu\nu}[g]n^{\mu}_{i}n^{\nu}_{i}-\frac{1}{2}K_{i}K_{i}]+2\alpha_{\rm
ren}R[\partial{\cal I}]\right),$

where $i$ refers to normal directions, $K_{i}$ representing the trace of
extrinsic curvature, $K_{i,\mu\nu}=-h^{\alpha}_{\mu}h_{\nu\beta}$, and
$h_{\mu\nu}$ being the induced metric on island surface’s boundary. In
addition, $G_{N,\rm ren},\lambda_{1,\rm ren},\lambda_{2,\rm ren}$ and
$\alpha_{\rm ren}$ are representing the renormalized Newton constant and the
coupling constants that appear in higher derivative gravity action (6.32).
These are what are utilized in the process of absorbing the UV divergences of
the von Neumann entropy of matter field that was covered in [141]. We can
calculate the matter’s contribution to the entanglement entropy of the Hawking
radiation in the presence of the island surface by utilizing the equations
(6.13), (6.14), (6.17), (6.20) and (6.23), which is written as [172]111The
contribution of matter to the entanglement entropy, denoted by the notation
$S_{\rm matter}(R\cup{\cal I})$, is comparable, at least to some extent, to
the result given in [172]. Because of this, we currently wrote the result
until that point.:

$\displaystyle S_{\rm matter}(R\cup{\cal
I})=\frac{c}{6}\log[2^{4}g^{2}(a)g^{2}(b)\cosh^{2}(\kappa_{+}t_{a})\cosh^{2}(\kappa_{+}t_{b})]+\frac{c}{3}\kappa_{+}(r_{*}(a)+r_{*}(b))$
$\displaystyle+\frac{c}{3}\Biggl{[}\frac{\cosh\left(\kappa_{+}(r_{*}(a)-r_{*}(b))\right)-\cosh\left(\kappa_{+}(t_{a}-t_{b})\right)}{\cosh\left(\kappa_{+}(r_{*}(a)-r_{*}(b))\right)+\cosh\left(\kappa_{+}(t_{a}-t_{b})\right)}\Biggr{]}.$
(6.40)

In late time approximation, when $t_{a},t_{b}\gg b>r_{+}$, the preceding
equation can be simplified into the following simpler version:

$\displaystyle S_{\rm matter}^{\rm late\ time}(R\cup{\cal
I})=\frac{c}{6}\log[g^{2}(a)g^{2}(b)]+\frac{2c}{3}\kappa_{+}r_{*}(b)$
$\displaystyle+\frac{c}{3}\Biggl{[}-2e^{-\kappa_{+}(b-a)}\Biggl{|}\frac{a-r_{+}}{b-r_{+}}\Biggr{|}^{1/2}\Biggl{|}\frac{a-r_{-}}{b-r_{-}}\Biggr{|}^{-(r_{-}^{2}/r_{+}^{2})}-e^{\kappa_{+}\left(r_{*}(b)-r_{*}(a)-2t_{b}\right)}\Biggr{]}.$
(6.41)

Because of this, the generalized entropy of the action (6.32) is as follows
[141]:

$\displaystyle S_{gen}(r)={\rm Min}_{\cal I}\Biggl{[}{\rm Ext}_{\cal
I}\Biggl{(}\frac{A[\partial{\cal I}]}{4G_{N,\rm ren}}+\frac{1}{4G_{N,\rm
ren}}\int_{\partial{\cal I}}\Biggl{(}2\lambda_{1,\rm ren}R[g]+\lambda_{2,\rm
ren}\sum_{i=1}^{2}[R_{\mu\nu}[g]n^{\mu}_{i}n^{\nu}_{i}-\frac{1}{2}K_{i}K_{i}]$
$\displaystyle\hskip 72.26999pt+2\alpha_{\rm ren}R[\partial{\cal
I}]\Biggr{)}+S_{\rm matter}(R\cup{\cal I})\Biggr{)}\Biggr{]}.$ (6.42)

As we have the metric (6.9), we obtained: $R[g]=0,R[\partial{\cal
I}]=\frac{2}{a^{2}},n_{1}^{t}=\frac{1}{\sqrt{F(r)}},n_{2}^{r}=\sqrt{F(r)}$,$K_{1}=0$
and $K_{2}=-\frac{2}{r}\sqrt{F(r)}$, implying

$\displaystyle\sum_{i=1}^{2}R_{\mu\nu}[g]n^{\mu}_{i}n^{\nu}_{i}=0,$
$\displaystyle K_{1}K_{1}=0,$ $\displaystyle
K_{2}K_{2}=\frac{4}{a^{2}}\left(1-\frac{2M}{a}+\frac{Q^{2}}{a^{2}}\right).$
(6.43)

The gravitational component of the entanglement entropy of the Hawking
radiation (6.3) can therefore be simplified as follows:

$\displaystyle S_{\rm gravity}(a)=\frac{2\pi}{G_{N,\rm
ren}}\Biggl{[}a^{2}-2\lambda_{2,\rm
ren}\left(1-\frac{2M}{a}+\frac{Q^{2}}{a^{2}}\right)+4\alpha_{\rm
ren}\Biggr{]},$ (6.44)

Thus, the total generalized entropy is going to be calculated as the sum of
the contributions to the entanglement entropy from the gravity component
(6.44) and the matter part (6.3), which is given as follows:

$\displaystyle S_{\rm gen}(a)=S_{\rm gravity}(a)+S_{matter}^{\rm late\
times}(R\cup{\cal I})(a)$ (6.45) $\displaystyle S_{\rm
gen}(a)\sim\frac{2\pi}{G_{N,\rm ren}}\Biggl{[}a^{2}-2\lambda_{2,\rm
ren}\left(1-\frac{2M}{a}+\frac{Q^{2}}{a^{2}}\right)+4\alpha_{\rm
ren}\Biggr{]}+\frac{c}{6}\log[g^{2}(a)g^{2}(b)]+\frac{2c}{3}\kappa_{+}r_{*}(b)$
$\displaystyle\hskip
54.2025pt+\frac{c}{3}\Biggl{[}-2e^{-\kappa_{+}(b-a)}\Biggl{|}\frac{a-r_{+}}{b-r_{+}}\Biggr{|}^{1/2}\Biggl{|}\frac{a-r_{-}}{b-r_{-}}\Biggr{|}^{-(r_{-}^{2}/r_{+}^{2})}\Biggr{]},$

where $r_{*}(b)$, $g(a)$, and $g(b)$ can be replaced from the equations (6.14)
and (6.17) in the equation (6.45). The small $t_{b}$ dependent contribution is
ignorable in (6.3). The position of the island surface can be determined by
extremizing the equation shown above with respect to the variable $a$, which
can be written as:

$\displaystyle\frac{\partial S_{\rm gen}(a)}{\partial
a}\sim\frac{4\pi\left(-2Mr_{+}\lambda_{2,\rm ren}+2Q^{2}\lambda_{2,\rm
ren}+r_{+}^{4}\right)}{r_{+}^{3}G_{N}}+\frac{c\left(\frac{r_{+}-r_{-}}{b-r_{-}}\right){}^{-\frac{r_{-}^{2}}{r_{+}^{2}}}e^{\kappa_{+}\left(r_{+}-b\right)}}{3\left(r_{+}-b\right)\sqrt{\frac{a-r_{+}}{b-r_{+}}}}=0,$
(6.46)

solving above equation, we obtained:

$\displaystyle a\approx
r_{+}+\frac{c^{2}r_{+}^{6}\left(\frac{r_{+}-r_{-}}{b-r_{-}}\right){}^{-\frac{2r_{-}^{2}}{r_{+}^{2}}}G_{N}^{2}e^{2\kappa_{+}\left(r_{+}-b\right)}}{144\pi^{2}\left(b-r_{+}\right)\left(2\lambda_{2,\rm
ren}\left(Q^{2}-Mr_{+}\right)+r_{+}^{4}\right){}^{2}}.$ (6.47)

Because the island is located beyond the event horizon of the black hole, this
creates a causality paradox. It was demonstrated in [197] that this conclusion
occurs in all two-sided eternal black holes or black holes in the Hartle-
Hawking state, and it is possible to restore causality using quantum focusing
conjecture (QFC) [198]. When we separate the black hole from the bath, a
finite quantity of energy flux is generated. This energy flux pulls the black
hole horizon outwards, and as a result, the island always stays behind the
horizon. It was pointed out in [199] that a limited quantity of energy is also
generated even when we couple the black hole to bath, so such energy flux
drives the horizon outwards, that suggests that island lies behind the horizon
similar to the way that the decoupling process works, and that we are able to
get rid of the causality paradox.

When the value of “a” from the equation (6.47) is substituted into the
equation (6.45), the generalized entropy equation is simplified to the
following form:

$\displaystyle S_{\rm total}^{{\cal O}(R^{2})}=S_{\rm gravity}+S_{\rm
matter}({\cal R}\cup{\cal I}),$ $\displaystyle S_{\rm total}^{{\cal
O}(R^{2})}\sim\frac{2\pi\left(r_{+}^{2}+4\alpha_{\rm
ren}\right)}{G_{N}}+\frac{c}{3}\log\left(\frac{r_{-}e^{-\kappa_{+}\left(b+r_{+}\right)}\left(\frac{r_{-}^{2}}{\left(r_{+}-r_{-}\right)\left(b-r_{-}\right)}\right){}^{\frac{1}{2}\left(\frac{\kappa_{+}}{\kappa_{-}}-1\right)}}{b\kappa_{+}^{2}}\right)+{\cal
O}(G_{N}),$ $\displaystyle S_{\rm total}^{{\cal O}(R^{2})}=2S_{\rm
BH}^{(\alpha)}+\frac{c}{3}\log\left(\frac{r_{-}e^{-\kappa_{+}\left(b+r_{+}\right)}\left(\frac{r_{-}^{2}}{\left(r_{+}-r_{-}\right)\left(b-r_{-}\right)}\right){}^{\frac{1}{2}\left(\frac{\kappa_{+}}{\kappa_{-}}-1\right)}}{b\kappa_{+}^{2}}\right)+{\cal
O}(G_{N}).$ (6.48)

If we only look at the leading order term in $G_{N}$, we are able to see that
the total entanglement entropy of an eternal Reissner-Nordström black hole
with inclusion of ${\cal O}(R^{2})$ terms in the gravitational action
approaches a constant value that is compatible with the literature. In the
case where $\alpha\rightarrow 0$, the equation (6.3) can be simplified to:

$S_{\rm total}^{(0)}=\frac{2\pi
r_{+}^{2}}{G_{N}}+\frac{c}{3}\log\left(\frac{r_{-}e^{-\kappa_{+}\left(b+r_{+}\right)}\left(\frac{r_{-}^{2}}{\left(r_{+}-r_{-}\right)\left(b-r_{-}\right)}\right){}^{\frac{1}{2}\left(\frac{\kappa_{+}}{\kappa_{-}}-1\right)}}{b\kappa_{+}^{2}}\right).$
(6.49)

It is nearly identical to what is seen in [172]. The Page curves are obtained
by combining 6.22) and (6.3), respectively. Taking into account only the
leading order term in (6.3), and then replacing $r_{+}=M+\sqrt{M^{2}-Q^{2}}$,
the following is what we get:

$\displaystyle S_{\rm
total}=\frac{2\pi\left(\left(M+\sqrt{M^{2}-Q^{2}}\right)^{2}+4\alpha\right)}{G_{N}},$
(6.50)

where $\alpha$ is $\alpha_{\rm ren}$ in (6.50) and in this chapter, where
we’ve just used $\alpha$ for the purpose of simplicity. In figure 6.5, we
displayed the Page curves of the charged black hole in with the presence of
${\cal O}(R^{2})$ terms for the parameters $M=1,Q=0.8,G_{N}=1$ for a range of
different values of the Gauss-Bonnet coupling ($\alpha$). The diagonal green
line indicates (6.22) and the constant red, green and blue horizontal lines
are representing the twice of thermal entropy of the black hole when
$\alpha=0.2,0,-0.2$ with Page times $t_{1_{\rm Page}},t_{2_{\rm Page}}$ and
$t_{3_{\rm Page}}$. Figure 6.5 represents the shifting of Page curves with
Gauss-Bonnet coupling. When $\alpha$ rises then Page curves shift towards
later times and vice-versa.

Figure 6.5: Page curves of an eternal Reissner-Nordström black hole using a
number of different Gauss-Bonnet couplings.

Page Time: As at $t=t_{\rm Page}^{\rm HD}$, $S_{\rm EE}^{\rm WI}=S_{\rm
total}$, thus (6.22) and (6.3) implying the following Page time:

$\displaystyle t_{\rm Page}^{\rm
HD}\sim\frac{6\pi\left(r_{+}^{2}+4\alpha\right)}{c\kappa_{+}G_{N}}=\frac{3S_{\rm
BH}^{(0)}}{2\pi cT_{\rm RN}}+\frac{12\alpha}{c\ G_{N}T_{\rm RN}}=t_{\rm
Page}^{(0)}+\frac{12\alpha}{c\ G_{N}T_{\rm RN}}.$ (6.51)

The preceding equation makes it abundantly evident that in the case when
$\alpha\rightarrow 0$ is being considered, the Page time of the Reissner-
Nordström black hole with the presence of ${\cal O}(R^{2})$ terms simplifies
to the Page time of the Reissner-Nordström black hole (6.29) when higher
derivative terms are removed.

$\displaystyle t_{\rm Page}^{(0)}\sim\frac{6\pi r_{+}^{2}}{c\kappa_{+}G_{N}}.$
(6.52)

As a result of the fact that the Gauss-Bonnet coupling ($\alpha$) appears with
a positive sign in (6.3) and (6.51), we receive the Page curves at a later or
earlier times when $\alpha$ increases or decreases.
Scrambling time: The scrambling time of the charged black hole with the
presence of ${\cal O}(R^{2})$ terms can be found by putting $a$ from (6.47)
into (6.30). The resulting equation is as follows:

$\displaystyle t_{scr}^{\rm
HD}\sim\frac{2r_{+}^{2}}{r_{+}-r_{-}}{\log\left(\frac{\pi\left(2\lambda_{2,\rm
ren}\left(Q^{2}-Mr_{+}\right)+r_{+}^{4}\right)}{r_{+}^{2}G_{N}}\right)}+small.$
(6.53)

In the previous equation, if we consider the limit of $\lambda_{2,\rm
ren}\rightarrow 0$, we will get the scrambling time for the Reissner-Nordström
black hole (6.31) when higher derivative terms are not taken into account. The
small $\lambda_{2,\rm ren}$ expansion of (6.53) results in:

$\displaystyle t_{\rm scr}^{\rm
HD}\sim\frac{2r_{+}^{2}}{r_{+}-r_{-}}{\log\left(\frac{\pi
r_{+}^{2}}{G_{N}}\right)}+\frac{2\lambda_{2,\rm
ren}\left(Q^{2}-Mr_{+}\right)}{\left(r_{+}-r_{-}\right)r_{+}^{2}},$
$\displaystyle=\frac{1}{2\pi T_{\rm RN}}{\log\left(S_{\rm
BH}^{(0)}\right)}+\frac{2\lambda_{2,\rm
ren}\left(Q^{2}-Mr_{+}\right)}{\left(r_{+}-r_{-}\right)r_{+}^{2}}=t_{\rm
scr}^{(0)}+\frac{2\lambda_{2,\rm
ren}\left(Q^{2}-Mr_{+}\right)}{\left(r_{+}-r_{-}\right)r_{+}^{2}}.$ (6.54)

Hence, scrambling time of the charged black hole with inclusion of ${\cal
O}(R^{2})$ terms may increase or decrease based on the sign of second term of
(6.3).

###  6.4 Page Curves of Charged Black Hole in Einstein-Gauss-Bonnet Gravity

We now obtain the Page curves of charged black hole in Einstein-Gauss-Bonnet
gravity. The gravitational action of Einstein-Maxwell-Gauss-Bonnet gravity
without cosmological constant has the following form [189]:

$\displaystyle S=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}[R[g]+\alpha{\cal
L}_{\rm GB}-F_{\mu\nu}F^{\mu\nu}].$ (6.55)

The Wald entropy of the charged black hole in Einstein-Gauss-Bonnet gravity
has been obtained utilizing (6.33),(6.34),(6.3) and (6.55) and the obtained
results is written below:

$\displaystyle S_{\rm Wald}^{(\rm EGB)}=S_{\rm
BH}^{(\alpha)}=\frac{\pi}{G_{N}}\left(r_{+}^{2}+\frac{4\alpha(r_{+}+6r_{-})}{r_{+}}\right)\sim\frac{\pi}{G_{N}}\left(r_{+}^{2}+4\alpha\right).$
(6.56)

The following is a result of the holographic entanglement entropy of the
gravitational action (6.55) that was done in [200, 127, 201]:

$\displaystyle S_{\rm gravity}=\frac{1}{4G_{N}}\int
d^{2}y\sqrt{h}\Biggl{[}1+\alpha\Biggl{(}2R[g]-4\left(R_{\mu\nu}[g]n^{\nu
i}n^{\mu}_{i}-\frac{1}{2}K^{i}K_{i}\right)+2\left(R_{\mu\nu\rho\sigma}[g]n^{\mu
i}n^{\nu
j}n^{\rho}_{i}n^{\sigma}_{j}-K^{i}_{ab}K_{i}^{ab}\right)\Biggr{)}\Biggr{]},$

where $h$ denotes the determinant of a given induced metric on a surface with
constant coordinates ($t,r=a$):

$\displaystyle ds^{2}=h_{ab}dx^{a}dx^{b}=a^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (6.58)

Now equation (6.4) was simplified by the use of Gauss-Codazzi equation [141,
201]:

$\displaystyle R[g]=R[\partial{\cal I}]-R_{\mu\nu\rho\sigma}[g]n^{\mu i}n^{\nu
j}n^{\rho}_{i}n^{\sigma}_{j}+2R_{\mu\nu}[g]n^{\nu
i}n^{\mu}_{i}-K^{i}K_{i}+K^{i}_{ab}K_{i}^{ab}.$ (6.59)

Holographic entanglement entropy in Einstein-Gauss-Bonnet gravity can be
simplified to the following form through the use of the equations (6.4) and
(6.59), which represent the gravitational contribution to the generalized
entropy:

$\displaystyle S_{\rm gravity}=\frac{1}{4G_{N}}\int
d^{2}y\sqrt{h}\left(1+2\alpha R[\partial{\cal I}]\right),$ (6.60)

and contribution of matter to the entanglement entropy is same as given in
6.3. For the induced metric (6.58), we found that: $R[\partial{\cal
I}]=\frac{2}{a^{2}}$. Hence, (6.60) has the following form for the metric
(6.58):

$\displaystyle S_{\rm gravity}=\frac{2\pi
a^{2}}{G_{N}}\left(1+\frac{4\alpha}{a^{2}}\right),$ (6.61)

Hence, similar to 6.3, the generalised entropy for this case is obtained as:

$\displaystyle S_{\rm gen}(a)\sim\frac{2\pi
a^{2}}{G_{N}}\left(1+\frac{4\alpha}{a^{2}}\right)+\frac{c}{6}\log[g^{2}(a)g^{2}(b)]+\frac{2c}{3}\kappa_{+}r_{*}(b)$
$\displaystyle+\frac{c}{3}\Biggl{[}-2e^{-\kappa_{+}(b-a)}\Biggl{|}\frac{a-r_{+}}{b-r_{+}}\Biggr{|}^{1/2}\Biggl{|}\frac{a-r_{-}}{b-r_{-}}\Biggr{|}^{-(r_{-}^{2}/r_{+}^{2})}\Biggr{]}.$
(6.62)

The variation of (6.4) with respect to $a$ leads to the following equation:

$\displaystyle\frac{\partial S_{\rm gen}(a)}{\partial a}\sim\frac{4\pi
r_{+}}{G_{N}}+\frac{c\left(\frac{r_{+}-r_{-}}{b-r_{-}}\right){}^{-\frac{r_{-}^{2}}{r_{+}^{2}}}e^{\kappa_{+}\left(r_{+}-b\right)}}{3\left(r_{+}-b\right)\sqrt{\frac{a-r_{+}}{b-r_{+}}}}=0,$
(6.63)

from the previous equation, we found the location of the island surface as
written below:

$\displaystyle a\approx
r_{+}+\frac{c^{2}\left(\frac{r_{+}-r_{-}}{b-r_{-}}\right){}^{-\frac{2r_{-}^{2}}{r_{+}^{2}}}G_{N}^{2}e^{2\kappa_{+}\left(r_{+}-b\right)}}{144\pi^{2}r_{+}^{2}\left(b-r_{+}\right)}.$
(6.64)

Using (6.64), the total entanglement entropy of the charged Einstein-Gauss-
Bonnet black hole (6.4) in late time approximation is obtained as:

$\displaystyle S_{\rm total}^{\rm
EGB}\sim\frac{2\pi\left(4\alpha+r_{+}^{2}\right)}{G_{N}}+\frac{c}{3}\log\left(\frac{r_{-}e^{-\kappa_{+}\left(b+r_{+}\right)}\left(\frac{r_{-}^{2}}{\left(r_{+}-r_{-}\right)\left(b-r_{-}\right)}\right){}^{\frac{1}{2}\left(\frac{\kappa_{+}}{\kappa_{-}}-1\right)}}{b\kappa_{+}^{2}}\right),$
$\displaystyle S_{\rm total}^{\rm
EGB}=2S_{BH}^{(\alpha)}+\frac{c}{3}\log\left(\frac{r_{-}e^{-\kappa_{+}\left(b+r_{+}\right)}\left(\frac{r_{-}^{2}}{\left(r_{+}-r_{-}\right)\left(b-r_{-}\right)}\right){}^{\frac{1}{2}\left(\frac{\kappa_{+}}{\kappa_{-}}-1\right)}}{b\kappa_{+}^{2}}\right).$
(6.65)

Since the total entanglement entropy of the charged black hole in Einstein-
Gauss-Bonnet gravity at late times (6.4) is the same as the total entanglement
entropy of the charged black hole in higher derivative gravity with the ${\cal
O}(R^{2})$ terms specified in the equation (6.3), it may be concluded that the
two quantities are identical. Because of this, the Page curves and Page times
of the charged black hole in Einstein-Gauss-Bonnet gravity will turn out to be
the same as the Page curves and Page times of the charged black hole in higher
derivative gravity with ${\cal O}(R^{2})$ terms , figure 6.5.
Scrambling time: In this particular instance, the scrambling time is not
influenced by the higher derivative terms and is identical to the scrambling
time of the Reissner-Nordström black hole [172]:

$\displaystyle\hskip-14.45377ptt_{scr}^{\rm
EGB}\sim\frac{2r_{+}^{2}}{(r_{+}-r_{-})}\log\left(\frac{\pi
r_{+}^{2}}{G_{N}}\right)+small=\frac{1}{2\pi T_{\rm RN}}\log\left(S_{\rm
BH}^{(0)}\right)+small=t_{\rm scr}^{(0)}+small.$ (6.66)

###  6.5 Conclusion and Discussion

We obtained the Page curves of an eternal Reissner-Nordstrom black hole in
this chapter. We concentrated on the non-holographic model and made the
assumption that the observer is located a large distance from the black hole.
As a result, we were able to employ the s-wave approximation to compute the
entanglement entropy of Hawking radiation utilizing the formula associated
with two-dimensional conformal field theory. The affect of higher derivative
terms on the Page curve was studied in [141] for the neutral black hole. The
researchers in that paper took into account the eternal black hole and the
evaporating black hole both. They were solely concerned with the more generic
${\cal O}(R^{2})$ terms. We considered the general ${\cal O}(R^{2})$ terms as
well as Gauss-Bonnet term both as the higher derivative terms for the non-
extremal eternal Reissner-Nordström black hole and found that Page time,
scrambling time, and the Page curves of eternal black holes are all affected
when higher derivative terms are present in the gravitational action. The key
results that have been found are summarized below.

  1. 1.

In the first scenario, we took into account the general terms (${\cal
O}(R^{2})$) in the gravitational action described in [141] and calculated the
Page curves. We came to the conclusion that, because there is initially no
island surface, the entanglement entropy of the Hawking radiation will
continue to increase in a linear fashion with the passage of time
indefinitely. As a result, we ended up at an information paradox for the
charged black hole. At later times, an island will appear, and the
entanglement entropy of the Hawking radiation will attain a constant value.
This value will be equal to double the Bekenstein-Hawking entropy of the black
hole, and it will allow us to derive the Page curves for fixed values of the
Gauss-Bonnet coupling ($\alpha$). The higher derivative terms shifts the Page
curve and hence Page time. Page curves move towards later times as the Gauss-
Bonnet coupling ($\alpha$) grows, and Page curves shift towards earlier times
as the Gauss-Bonnet coupling ($\alpha$) drops.

  2. 2.

In the next scenario, we solely included the Gauss-Bonnet term as the higher
derivative term because it is crucial for the investigation of the charged
black hole in Einstein-Gauss-Bonnet gravity [189]. In a manner analogous to
the first scenario, we observe linear time growth of the entanglement entropy
of the Hawking radiation at the beginning. This is followed by the emergence
of an island at later times, that dominates the linear time growth of the
Hawking radiation. As a result, the entanglement entropy of the Hawking
radiation eventually reaches exactly two times the Bekenstein-Hawking entropy
of the black hole, and we obtained the Page curve exactly similar to the first
case. It is interesting that we found that the Page curves with Gauss-Bonnet
coupling behave in a manner that is comparable to that which was explained
earlier in the first case.

  3. 3.

In the first scenario, the higher derivative terms have an effect on how long
it takes the charged black hole to scramble its information. If the correction
term is positive, it will make it so that it is larger, but if it is negative,
it will make it so that it is smaller. In addition, if we take the coupling
that appears in scrambling time and set it to zero, then we are able to get
back the scrambling time of the Reissner-Nordström black hole. Scrambling time
is unaffected by the higher derivative term in the second scenario, which is
when we look at solely the Gauss-Bonnet term as the higher derivative term. In
this scenario, scrambling time remains identical, just like the scrambling
time associated with the Reissner-Nordström black hole.

  4. 4.

When the Gauss-Bonnet coupling goes to zero, we are able to recover the Page
curve of the Reissner-Nordsrtöm black hole, which was obtained in [172].
Further, all of our results reduces to that of [172] in the same limit.

##  Chapter 7 Entanglement Entropy and Page Curve from the ${\cal M}$-Theory
Dual of Thermal QCD Above $T_{c}$ at Intermediate Coupling

###  7.1 Introduction and Motivation

In doubly holographic approaches, as described in 5.3.2 of chapter 5, a
gravitational dual black hole is connected with an external CFT bath [202]. As
an illustration, gravity in $d$-dimensions is connected with the external bath
in $d$-dimensions, wherein $d$-dimensional exterior bath possesses a
corresponding holographic dual in $(d+1)$-dimensions. In such models, we take
into consideration two variants of the previously mentioned configuration. We
look at two distinct types of extremal surfaces in these models. The very
first one corresponds to the Hartman-Maldacena-like surface [144], that begins
from the position at which gravity interacts with with an external bath, i.e.,
at the defect, and traverses the black hole horizon to get to the defect of
the thermofield double associated with doubly holographic setups, and the
entanglement entropy involvement via the aforementioned surface possesses
linear time dependency, leading to information paradox at later times. The
other type of surface includes the island surface, that begins at the exterior
bath and ends on the Karch-Randall brane [142, 143]. The contribution of
entanglement entropy coming from the island surface becomes time independent
and takes over beyond the Page time. The Page curve is obtained through the
combination of the entanglement entropy contributions obtained from both the
extremal surfaces.

As we discussed in 5.3.2, in some cases, the gravity becomes massive on the
Karch-Randall branes, and one was unable to get the Page curve with massless
gravity localized on the Karch-Randall branes. In this chapter, we have looked
at this major issue of doubly holographic setup from ${\cal M}$-theory
perspective in the presence of ${\cal O}(R^{4})$ terms in the supergravity
action. We have a different situation from the literature that we have a non-
conformal bath (QCD bath), and the doubly holographic setup is being
constructed from a top-down approach111See [148, 149, 150], for some papers
from the top-down approach where the authors have worked with type IIB string
theory. with the inclusion of higher derivative terms in the supergravity
action. Interestingly, the issue of the massive graviton is absent in our
setup, and one can get the Page curve with massless gravity on the end-of-the-
world(ETW) brane.

We cannot utilize the island approach for non-conformal backgrounds because
currently there is no recognized Cardy-like formula for the non-conformal
theories. As a result, we used the previously mentioned doubly holographic
frameworks technique to arrive at the Page curve. In the context of our
scenario, on the gravity dual side, there is a ${\cal M}$-theory dual
consisting of higher derivative terms, whereas on the gauge theory side, we
find non-conformal thermal QCD-like theories at the intermediate coupling, the
model is discussed in 1.4 of chapter 1. In the present chapter, we created a
doubly holographic setup analogous to [7, 203], where the researchers utilize
a conformal theory on their gauge theory side. There are two types of
potential surfaces to think about: Hartman-Maldacena-like and island surfaces.
The Hartman-Maldacena-like surface has been accountable for the gradual linear
temporal growth of Hawking radiation’s entanglement entropy. Above the Page
time, the entanglement entropy component via the island surface takes over and
this is not dependent upon time, so we extract the Page curve using the ${\cal
M}$-theory dual, which includes ${\cal O}(R^{4})$ corrections. As a remark, we
used the formula of [127] to compute the entanglement entropy with the
inclusion of higher derivative terms and when higher derivative terms are
absent then we used the Ryu-Takayanagi formula to compute the entanglement
entropy [107].

###  7.2 Doubly Holographic Setup in ${\cal M}$ Theory

We generalized these types of setups to ${\cal M}$-theory background after
being inspired by the double holographic setup presented in 5.3.2. After a
double Wick-rotation, we have ${\rm QCD}_{2+1}$ at r=0, along $x^{1,2}$ and
Wick-rotated $x^{3}$. One can think of a fluxed ETW-hypersurface or “brane"
$M_{10}=\mathbb{R}^{2}(x^{2,3})\times_{w}M_{8}$ where $M_{8}$ is the (non-
compact) $SU(4)/Spin(7)$-structure ([1], Table 2) eight-fold
$M_{8}^{SU(4)/Spin(7)}(r,t,\theta_{1,2},\phi_{1,2},\psi,x^{10})$ at $x^{1}=0$
which has a black hole. The ETW-hypersurface can be interpreted as
$\mathbb{R}^{2}(x^{2,3})\times_{w}M_{8}^{SU(4)/Spin(7)}(t,r,\theta_{1,2},\phi_{1,2},\psi,x^{10})$
containing black $M5$-brane at $x^{1}=0$ with world volume
$\Sigma^{(6)}=S^{1}(t)\times_{w}\mathbb{R}_{\geq 0}(r)\times\Sigma^{(4)}$
where
$\Sigma^{(4)}=n_{1}S^{3}(\theta_{1},\phi_{1},\psi)\times[0,1]_{\theta_{2}}+n_{2}S^{2}(\theta_{1},\phi_{1})\times
S^{2}(\theta_{2},x^{10})$ with $n_{1}$ determined by
$\int_{S^{3}\times[0,1]}G_{4}$ and $n_{2}$ by $\int_{S^{2}\times S^{2}}G_{4}$;
the ${\rm QCD}_{2+1}$ can be thought of as living on $M2$-branes with world
volume $\Sigma^{(3)}(x^{1,2,3})$. ${\rm QCD}_{2+1}$ at $r=0$ would interact
gravitationally via the pull-back of the ambient
$M_{11}=\mathbb{R}^{1,2,3}\times_{w}M_{8}=\mathbb{R}(x^{1})\times_{w}M_{10}$
metric used to contract the non-abelian field strength in the gauge kinetic
term obtained as part of the $M2$-brane($x^{1,2,3}$) world-volume action. This
has some similarity with points 2 and 3 of the doubly holographic setup as
discussed in 5.3.2. The doubly holographic setup constructed by us is shown in
Fig. 7.1. Let us be more specific and briefly describe the three equivalent
descriptions alluded to towards the beginning of this sub-section. The doubly
holographic setup constructed from the bottom-up approach (usually followed in
the literature) as described at the beginning of this sub-section, has the
following ${\cal M}$-theory description (the one we follow) of top-down double
holography with $QCD_{2+1}$ bath with the numbering matches the one used in
the aforementioned three equivalent description:

  1. 1.

Boundary-like Description: $QCD_{2+1}$ (could be thought of as supported on an
$M2$-brane with world-volume $\Sigma^{(3)}(x^{1,2,3})$, and) is living at the
tip ($r=0$) of a non-compact seven-fold of $G_{2}$ structure which is a cone
over a warped non-Kähler resolved conifold. The two-dimensional “defect”
$\Sigma^{(2)}\cong\Sigma^{(3)}(x^{2,3};x^{1}=0)\cong\mathbb{R}^{2}(x^{2,3})$.

  2. 2.

Non-Conformal Bath-ETW Interaction Description: Fluxed End-of-The-World (ETW)
hypersurface
$M_{10}\cong\mathbb{R}^{2}(x^{2,3})\times_{w}M_{8}^{SU(4)/Spin(7)}(t,r,\theta_{1,2},\phi_{1,2},\psi,x^{10})$
containing black $M5$-brane at $x^{1}=0$ coupled to $QCD_{2+1}$ bath living on
$M2$ brane with world volume $\Sigma^{(3)}(x^{1,2,3})$ along the defect
$\left.\Sigma^{(2)}\cong M_{10}\cap\Sigma^{(3)}(x^{1,2,3})\right|_{x^{1}=0}$
via exchange of massless graviton.

  3. 3.

Bulk Description: $QCD_{2+1}(x^{1,2,3})$ has holographic dual which is eleven
dimensional ($M_{11}\cong\mathbb{R}(x^{1})\times_{w}M_{10}\cong
S^{1}(t)\times_{w}\mathbb{R}^{3}(x^{1,2,3})\times_{w}M_{7}^{G_{2}}(r,\theta_{1,2},\phi_{1,2},\psi,x^{10})$)
${\cal M}$-theory background (compactified on a seven-fold with $G_{2}$
structure) containing the fluxed ETW-hypersurface $M_{10}(x^{1}=0)$.

Figure 7.1: Doubly holographic setup in ${\cal M}$-theory dual. ETW-black
“brane” is coupled the thermal bath (at the tip of seven-fold of
$G_{2}$-structure which is a cone over a warped non-Kähler resolved conifold),
where the Hawking radiation is collected, which effectively is thermal
$QCD_{2+1}$ along $x^{1,2,3}$ after Wick-rotation along $x^{3}$ at $r=0$. Blue
curves correspond to the island surfaces and red curve corresponds to the
Hartman-Maldacena-like surface.

The pictorial representation of the aforementioned ETW-brane(containing a
black $M5$-brane)/$M2$-brane(supporting the non-conformal “bath” -
$QCD_{2+1}$)-setup focusing only on the $x^{1}-r$-plane, along with the
Hartmann-Maldacena-like surface and island surface, is given in Fig. 7.1.
Unlike [6], [202] and [162], the (non-conformal) bath is at $r=0$ instead of
the UV cutoff $r=r_{\rm UV}$222The cut-off, unlike most references in the
literature is not at infinity but is such that $r_{\rm
UV}\stackrel{{\scriptstyle<}}{{\sim}}L\equiv\left(4\pi g_{s}N\right)^{1/4}$
thereby justifying dropping the “1” in the 10-dimensional warp factor $h$ in
[14], [15], [1] [that appears in (3.1)], which otherwise would have been
$1+\frac{L^{4}}{r^{4}}\left[1+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right]$.. In the high temperature
($T>T_{c}$) ${\cal M}$-theory dual (3.1), evidently $r\geq r_{h}$ (the metric
component $g_{tt}^{\cal M}$ being proportional to a warp factor $e^{B}$ where
$B\sim\log\left(1-\frac{r_{h}^{4}}{r^{4}}+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right)$ from the solutions to the
supergravity EOM implies that for $B\in\mathbb{R},r\geq r_{h}$ in the MQGP
limit (1.39)).333Further, as will be shown later, the area/entanglement
entropy (inclusive of ${\cal O}(R^{4})$ contributions) of the HM surface (see
(7.4.1.1), (7.39), (7.5.1), (7.70) and (7.105)) as well as the IS (see
(7.4.2.1), (7.4.2.1), (7.5.3) and (7.108)) are proportional to a positive
power of $M$ \- the number of fractional $D3$-branes in the parent type IIB
dual [14]. Now, the contribution to $r$ integral from $r\in$ UV, i.e.,
$r>\sqrt{3}a\sim\biggl{[}1+{\cal O}\left(\frac{\sqrt{\beta}}{N}\right)+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\biggr{]}r_{h}\sim\left(1+\epsilon\right)r_{h},|\epsilon|\ll
1$ [2] will involve replacing $M$ by $M^{\rm UV}\equiv M_{\rm eff}(r\in{\rm
UV}\Leftrightarrow r>\sqrt{3}a)$ \- the UV-valued effective number of
fractional $D3$-branes in the aforementioned parent type IIB dual - which is
vanishing small ensuring UV conformality. Therefore all integrals
$\int^{r_{\rm UV}}_{r_{*}\ {\rm or}\ r_{T}}$ of integrands relevant to the
HM/IS areas or entanglement entropies, will vanish as $r_{*,T}>r_{h}$ though
being nearer to $r_{h}$ than $\sqrt{3}a$.

From the work of [127], we could compute the entanglement entropy associated
with holographic dual theories with the inclusion of higher derivative terms.
We could express the analogue of the generalised entropy functional described
in equation (5.17) using [127] for ${\cal M}$-theory dual in the presence of
${\cal O}(R^{4})$ terms as follows:

$S_{gen}({\cal R}\cup{\cal I})=S_{\rm gravity},$ (7.1)

where

$S_{\rm gravity}=\frac{S_{\rm EE}}{4G_{N}^{(11)}},$ (7.2)

where $S_{\rm EE}$ is defined in (5.8). The Page curve is obtained from (7.1)
and (5.8) for the Hartman-Maldacena-like and island surfaces in 7.5.

###  7.3 ETW-“brane” Embedding

End of the World (ETW)-“brane” embedding is a crucial issue associated with
the presence of islands in the doubly-holographic approach. The situation is
widely recognized at the stage of Einstein gravity at LO (i.e. Einstein-
Hilbert(EH) and the boundary Gibbons-Hawking-York surface terms), but little
is understood for higher derivative gravity, e.g. $D=11$ supergravity at
${\cal O}(R^{4})$; in top-down frameworks like the one taken into account by
us, the ETW-“brane” is a fluxed hypersurface ${\cal W}$. We tackled the
situation in two subsections within this section: first we obtained the ETW
embedding exactly the stage of EH+GHY action, and the subsequent one
demonstrated how the aforementioned LO embedding obtains no modifications in
the MQGP limit (1.39). The ETW-“brane” ${\cal
W}:AdS_{4}^{\infty}\times_{w}M_{6}$ with $G_{4}$ fluxes threading a homologous
sum of four-cycles $S^{3}\times[0,1]$ and $S^{2}\times S^{2}$ in
$M_{6}=M_{5}(\theta_{1,2},\phi_{1,2},\psi)\times S^{1}(x^{10})\rightarrow
M^{SU(4)/Spin(7)}_{8}(t,r,\theta_{1,2},\phi_{1,2},\psi,x^{10})$ (in the
large-$N$ MQGP limit) implies: $\partial M_{11}=\left\\{r=r_{\rm
UV}\right\\}\cup{\cal W}$. Considering the end-of-the-world (ETW)-“Brane”
embedding as,

$x^{1}=x^{1}(r),$ (7.3)

and inserting it into (7.20) yields:

$ds^{2}=\alpha(r)\left[-g(r)dt^{2}+\left(\sigma(r)+\left(\frac{dx^{1}}{dr}\right)^{2}\right)dr^{2}+\sum_{\mu=2}^{3}dx_{\mu}dx^{\mu}\right]+g_{mn}dx^{m}dx^{n}.$
(7.4)

####  7.3.1 At ${\cal O}(\beta^{0})$

If $T$ is the tension associated with the ETW-“brane”, and if ${\cal K}$ and
${\cal K}_{\tilde{m}\tilde{n}},\tilde{m},\tilde{n}=r,t,\mu,m$ are the
extrinsic curvature scalar and tensor associated with the end-of-the-world
(ETW)-“brane”, then [204],

${\cal K}_{mn}-{\cal K}h_{mn}=-9Th_{mn},$ (7.5)

where $h_{mn}$ represents the induced metric on the end-of-the-world brane.
Focus on the $rr$-component of (7.5) in the IR. We could argue that in the IR,
the embedding function $x^{1}(r)=x(r)$ always appears as
$x^{\prime}(r),x^{\prime\prime}(r)$ in (7.5), and writing $a=\left(b+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right)r_{h}$ [60], the terms LO and NLO in
$N$ and

$\displaystyle{\cal K}\left(N,g_{s};r_{h}\right)$ $\displaystyle\sim$
$\displaystyle-\kappa_{K}\frac{\left(3b^{2}-1\right)\sqrt{{r_{h}}}(\log
N-9\log({r_{h}}))}{\sqrt[4]{{g_{s}}}\sqrt[4]{N}\sqrt{r-{r_{h}}}\left(\left(3b^{2}-1\right)\log
N+9\log({r_{h}})\right)\sqrt[3]{{N_{f}}(\log N-3\log({r_{h}}))}}$
$\displaystyle\hskip-7.22743pt{\cal
K}_{rr}\left(N,M,N_{f},g_{s};r_{h};x^{\prime}(r);x^{\prime\prime}(r)\right)$
$\displaystyle\sim$
$\displaystyle-\kappa_{K_{rr}}^{(1)}\frac{\sqrt{{g_{s}}N}\Sigma(r_{h};N,N_{f})}{r_{h}(r-{r_{h}})^{2}}$
$\displaystyle+\kappa_{K_{rr}}^{(2)}\frac{{g_{s}}^{3}M^{2}{N_{f}}\log({r_{h}})(\log
N-12\log({r_{h}}))\Sigma(r_{h};N,N_{f})}{{r_{h}}^{2}\sqrt{{g_{s}}N}(r-{r_{h}})^{2}}$
$\displaystyle+\kappa_{K_{rr}}^{(3)}\frac{{r_{h}}x^{\prime}(r)\Sigma(r_{h};N,N_{f})}{\sqrt{{g_{s}}N}}+\kappa_{K_{rr}}^{(4)}\frac{{r_{h}}^{2}x^{\prime\prime}(r)\Sigma(r_{h};N,N_{f})}{\sqrt{{g_{s}}N}},$
$\displaystyle h_{rr}\left(N,M,N_{f},g_{s};r_{h};x^{\prime}(r)\right)$
$\displaystyle\sim$
$\displaystyle-2\kappa_{K_{rr}}^{(1)}\frac{\sqrt{g_{s}N}\Sigma(r_{h};N,N_{f})}{r_{h}(r-{r_{h}})}-\kappa_{K_{rr}}^{(3)}\frac{{r_{h}}^{2}x^{\prime}(r)\Sigma(r_{h};N,N_{f})}{\sqrt{{g_{s}}N}}$
(7.6)
$\displaystyle-2\kappa_{K_{rr}}^{(2)}\frac{3{g_{s}}^{3}M^{2}{N_{f}}\log({r_{h}})(\log
N-12\log({r_{h}}))\Sigma(r_{h};N,N_{f})}{{r_{h}}\sqrt{{g_{s}}N}(r-{r_{h}})},$

wherein $\Sigma(r_{h};N,N_{f})\equiv\left(6\log
N{N_{f}}-3{N_{f}}\log\left(9a^{2}{r_{h}}^{4}+{r_{h}}^{6}\right)\right)^{2/3}$,
numerical pre-factors appear in $\kappa_{K},\kappa_{K_{rr}^{(i=1,2,3,4)}}$.
Since $b=\frac{1}{\sqrt{3}}+\epsilon$, (e.g., in the
$\psi=2n\pi,n=0,1,2$-coordinate patches, $|\epsilon|\sim r_{h}^{2}\left(|\log
r_{h}|\right)^{9/2}N^{-9/10-\alpha_{b}},\alpha_{b}>0$ [1]), (7.5) may be
approximated with $K_{mn}\sim-9Th_{mn}$. Consider $m=n=r$ component of the
same, we found that unlike ${\cal
K}_{rr}\left(N,M,N_{f},g_{s};r_{h};x^{\prime}(r);x^{\prime\prime}(r)\right)$,
there is no $x^{\prime\prime}(r)$ term in
$h_{rr}\left(N,M,N_{f},g_{s};r_{h};x^{\prime}(r)\right)$. Hence,

$x^{\prime\prime}(r)=0.$ (7.7)

urthermore, the LO-in-$N$ terms in the IR ($r={\cal O}(1)r_{h}$ of) ${\cal
K}_{rr}\left(N,M,N_{f},g_{s};r_{h};x^{\prime}(r);x^{\prime\prime}(r)\right)$
are proportional to $h_{rr}\left(N,M,N_{f},g_{s};r_{h};x^{\prime}(r)\right)$
with a proportionality constant “-2”. This implies that ETW-“brane” has the
following tension:

$T\sim\frac{1}{r_{h}}.$ (7.8)

However, at NLO in $N$, the coefficients are proportional to one another,
although having a constant for proportionality of “-1”. This is balanced by
$x^{\prime}(r)=0$, i.e.,

$x^{1}(r)={\rm constant},$ (7.9)

and we assume the constant is zero.

####  7.3.2 No Boundary Terms Generated at ${\cal O}(R^{4})$

The eleven-dimensional supergravity action, which includes ${\cal O}(R^{4})$
terms, is presented by the following:

$\displaystyle S=\frac{1}{2\kappa_{11}^{2}}\int_{M_{11}}\sqrt{-G^{\cal
M}}\left[{\cal R}*_{11}1-\frac{1}{2}G_{4}\wedge*_{11}G_{4}-\frac{1}{6}C\wedge
G\wedge G\right]+\frac{1}{\kappa_{11}^{2}}\int_{r=r_{\rm UV}}d^{10}x\sqrt{h}K$
$\displaystyle\hskip
14.45377pt+\frac{1}{(2\pi)^{4}3^{2}2^{13}}\left(\frac{2\pi^{2}}{\kappa_{11}^{2}}\right)^{\frac{1}{3}}\int_{M_{11}}d^{11}x\sqrt{-G^{\cal
M}}\left(J_{0}-\frac{1}{2}E_{8}\right)+\left(\frac{2\pi^{2}}{\kappa_{11}^{2}}\right)\int
C_{3}\wedge X_{8}$ $\displaystyle\hskip
14.45377pt+\frac{1}{\kappa_{11}^{2}}\int_{\cal W}d^{10}x\sqrt{\gamma}({\cal
K}-9T),$

where the terms $J_{0}$ and $E_{8}$ have been given in (1.4.4). Because of the
hierarchical structure, $|t_{8}^{2}G^{2}R^{3}|<|E_{8}|<|J_{0}|$ [1] in the
MQGP limit (1.39), we worked with $J_{0}$; $\gamma$ , ${\cal K}$ and $T$ in in
(7.3.2) have already been defined. From [1],

$\displaystyle\delta J_{0}\sim-\delta
g_{\tilde{M}\tilde{N}}\Biggl{[}g^{M\tilde{M}}R^{H\tilde{N}NK}+g^{N\tilde{N}}R^{HM\tilde{M}K}+g^{K\tilde{M}}R^{HMN\tilde{N}}\Biggr{]}\tilde{\chi}_{HMNK}$
(7.11)
$\displaystyle-\frac{1}{2}\Biggl{[}g^{H\tilde{H}}\left(D_{[N_{1}}\Biggl{(}D_{K_{1}]}\delta
g_{M_{1}\tilde{H}}+D_{M_{1}}\delta g_{|K_{1}]\tilde{H}}-D_{\tilde{H}}\delta
g_{|K_{1}]M_{1}}\Biggr{)}\right)\chi_{H}^{M_{1}N_{1}K_{1}}\Biggr{]},$

where,

$\displaystyle\tilde{\chi}_{HMNK}\equiv R_{PMNQ}R_{H}^{\ RSP}R^{Q}_{\
RSK}-\frac{1}{2}R_{PQKN}R_{H}^{\ RSP}R^{Q}_{\ RSM};$
$\displaystyle\chi_{H}^{M_{1}N_{1}K_{1}}\equiv R_{P\ \ \ \ Q}^{\
M_{1}N_{1}}R_{H}^{\ RSP}R^{Q\ \ \ K_{1}}_{\ RS}-\frac{1}{2}R_{PQ}^{\ \
M_{1}N_{1}}R_{H}^{\ RSP}R^{Q\ \ \ K_{1}}_{\ RS}.$ (7.12)

As a result, we are able to identify that a usual boundary term including
covariant derivatives of metric fluctuations that would have to be cancelled
out by a suitable boundary term (using Stokes theorem) is:

$\displaystyle\int_{{\cal W}}D_{[K_{1}|}\delta
g_{M_{1}\tilde{H}}\chi^{\tilde{H}M_{1}[N_{1}K_{1}]}d\Sigma_{|N_{1}]}=\int_{{\cal
W}}D_{[K_{1}|}\delta
g_{M_{1}\tilde{H}}\chi^{\tilde{H}M_{1}[N_{1}K_{1}]}\eta_{|N_{1}]}\sqrt{-h}d^{10}y.$
(7.13)

The (dual to the) unit normal vector to ${\cal W}$ provided by:

$\eta_{M}=\left(\eta_{r},\eta_{x},\eta_{M\neq
r,x}\right)=\frac{\left(-\frac{dx^{1}(r)}{dr}\sqrt{G^{rr}_{\cal
M}},\sqrt{G^{x^{1}x^{1}}_{\cal M}},{\bf 0}\right)}{\sqrt{\left(G^{xx}_{\cal
M}\right)^{2}+\left(G^{rr}_{\cal
M}\right)^{2}\left(\frac{dx^{1}(r)}{dr}\right)^{2}}}.$ (7.14)

Hence,

$\left.\eta_{M}\right|_{x^{1}(r)={\rm constant}}=\left(0,\sqrt{G^{xx}_{\cal
M}},{\bf 0}\right).$ (7.15)

According to (7.15), one will be needed to considera $\chi^{HM[x^{1}N]}$.
Using the fact that the only non-vanishing linearly independent Riemann
curvature tensor for the metric (7.20) with one index along $x^{1}$ is
$R_{x^{1}tx^{1}t}$, and (7.3.2), we obtained:

$\displaystyle\chi^{tx^{1}tx^{1}}=\frac{1}{2}\left(g_{x^{1}x^{1}}^{\cal
M}\right)^{2}\left(R_{t}^{\ x^{1}x^{1}t}\right)^{2}R^{tx^{1}x^{1}t}.$ (7.16)

Utilizing,

$\displaystyle
R_{t}^{x^{1}x^{1}t}\sim-\frac{\left(9a^{2}+{r_{h}}^{2}\right)}{{N_{f}}{r_{h}}^{2}\left(6a^{2}+{r_{h}}^{2}\right)(\log
N-3\log({r_{h}}))\sqrt[3]{\frac{12\pi}{{g_{s}}}+3\log
N{N_{f}}-9{N_{f}}\log({r_{h}})}};$ $\displaystyle
R^{tx^{1}tx^{1}}\sim\frac{\left(9a^{2}({g_{s}}\log
N{N_{f}}+4\pi)-3{g_{s}}{N_{f}}\left(9a^{2}+{r_{h}}^{2}\right)\log({r_{h}})+{g_{s}}\log
N{N_{f}}{r_{h}}^{2}\right)}{\sqrt{{g_{s}}}{N_{f}}^{3}{r_{h}}^{3}\left(6a^{2}+{r_{h}}^{2}\right)(r-{r_{h}})(\log
N-3\log({r_{h}}))^{3}}.$

We found that up to leading order in $N$ and $\log N,|\log r_{h}|$, and in the
IR (near $r=r_{h}$),

$\displaystyle\left.\beta\chi^{tx^{1}tx^{1}}\right|_{x^{1}={\rm
constant}}\sim\frac{\beta}{N}\frac{1}{\sqrt{g_{s}}N_{f}^{25/3}r_{h}^{3}(r-r_{h})\left(\log
N-3|\log r_{h}|\right)^{10/3}}.$

As a result, we can observe from
(7.13)($\chi^{ttx^{1}t}=\chi^{x^{1}x^{1}x^{1}t}=0$):

$\displaystyle\left.\int_{{\cal W}}D_{[K_{1}|}\delta g^{\cal
M}_{M_{1}\tilde{H}}\chi^{\tilde{H}M_{1}[N_{1}K_{1}]}d\Sigma_{|N_{1}]}\right|_{x^{1}={\rm
constant}}\sim\int_{{\cal W}}D_{t}\delta g^{\cal
M}_{x^{1}t}\chi^{tx^{1}[x^{1}t]}\eta_{x^{1}}\sqrt{-h}d^{10}y.$ (7.17)

Since, $g^{\cal M}_{x^{1}t}=0$. As a result, confined to $x^{1}=$constant,
there isn’t any surface term produced if $\delta g^{\cal M}_{x^{1}t}=0$ is
chosen.

###  7.4 Page Curve without Higher Derivative Terms

Here, we obtained the Page curve using the doubly holographic setup
constructed in 7.2. In this part, we do not consider the ${\cal O}(R^{4})$
terms in the supergravity action. Therefore, we used the Ryu-Takayanagi
formula to compute the entanglement entropies of Hartman-Maldacena-like and
Island surfaces in 7.4.1 and 7.4.2. The Page curve has been obtained in 7.4.3.
In the absence of ${\cal O}(R^{4})$ terms, the ${\cal N}=1,D=11$ supergravity
action has been provided as follows:

$\displaystyle S=\frac{1}{2\kappa_{11}^{2}}\int_{M_{11}}\sqrt{-G^{\cal
M}}\left[{\cal R}*_{11}\wedge
1-\frac{1}{2}G_{4}\wedge*_{11}G_{4}-\frac{1}{6}C\wedge G\wedge G\right]$
$\displaystyle+\frac{1}{\kappa_{11}^{2}}\int_{r=r_{\rm
UV}}d^{10}x\sqrt{h}K+\frac{1}{\kappa_{11}^{2}}\int_{\cal
W}d^{10}x\sqrt{\gamma}({\cal K}-9T).$ (7.18)

$\gamma$ , ${\cal K}$ and $T$ in (7.4) correspond to the induced metric, trace
of extrinsic curvature (${\cal K}_{mn}$), and tension of the ETW brane(${\cal
W}$: $\partial M_{11}=\left\\{r=r_{\rm UV}\right\\}\cup{\cal W}$). For
$T>T_{c}$ on the gauge theory side, the black hole metric associated with the
${\cal M}$-theory dual is provided via the following:

$\displaystyle\hskip-7.22743ptds_{11}^{2}$ $\displaystyle=$ $\displaystyle
e^{-\frac{2\phi^{\rm
IIA}}{3}}\Biggl{[}\frac{1}{\sqrt{h(r,\theta_{1,2})}}\left(-g(r)dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}+\left(dx^{3}\right)^{2}\right)$
$\displaystyle\hskip-7.22743pt+\sqrt{h(r,\theta_{1,2})}\left(\frac{dr^{2}}{g(r)}+ds^{2}_{\rm
IIA}(r,\theta_{1,2},\phi_{1,2},\psi)\right)\Biggr{]}+e^{\frac{4\phi^{\rm
IIA}}{3}}\left(dx^{11}+A_{\rm IIA}^{F_{1}^{\rm IIB}+F_{3}^{\rm IIB}+F_{5}^{\rm
IIB}}\right)^{2},$

where $A_{\rm IIA}^{F^{\rm IIB}_{i=1,3,5}}$ represent the type IIA RR one-
forms produced from the type IIB triple T/SYZ-dual. $F_{1,3,5}^{\rm IIB}$
fluxes in [14], and $g(r)=1-\frac{r_{h}^{4}}{r^{4}}$. We could write down the
metric ((7.4) near the $\psi=2n\pi,n=0,1,2$-coordinate patch in the following
form:

$ds^{2}=\alpha(r)[-g(r)dt^{2}+\sigma(r)dr^{2}+dx_{\mu}dx^{\mu}]+g_{mn}dx^{m}dx^{n},$
(7.20)

where $x_{\mu}(\mu=1,2,3)$ denotes spatial coordinates, $r$ denotes radial
coordinates, and $x^{m}(m=5,6,7,8,9,10)$ denotes six angular coordinates
$(\theta_{1,2},\phi_{1,2},\psi,x^{10})$ within the conifold geometry. We found
the following results through the comparison of equations (7.4) and (7.20):

$\displaystyle\alpha(r,\theta_{1,2})=\frac{e^{-\frac{2\phi^{\rm
IIA}}{3}}}{\sqrt{h(r,\theta_{1,2})}},$
$\displaystyle\sigma(r,\theta_{1,2})=\frac{h(r,\theta_{1,2})}{g(r)},$ (7.21)

where $\phi^{\rm IIA}$ represents the type IIA dilaton profile, which could be
described as $G_{x_{10}x_{10}}^{\cal M}=e^{\frac{4\phi^{\rm IIA}}{3}}$. The
following equation has been used to calculate the volume of the compact six-
fold:

${\cal V}_{\rm int}=\int\prod_{m}dx^{m}\sqrt{-g}.$ (7.22)

####  7.4.1 Entanglement Entropy Contribution from Hartman-Maldacena-like
Surface

To calculate the time-dependent entanglement entropy associated with a
Hartman-Maldacena-like surface, we first look at the induced metric on a
constant $x^{1}$ slice, which was calculated utilizing (7.20) as follows:

$ds^{2}|_{x^{1}=x_{R}}=\alpha(r)\Biggl{[}\left(-g(r)\dot{t}(r)^{2}+\sigma(r)\right)dr^{2}+(dx^{2})^{2}+(dx^{3})^{2}\Biggr{]}+g_{mn}dx^{m}dx^{n}.$
(7.23)

The area density functional for a Hartman-Maldacena-like surface could be
calculated using (7.23) as follows:

${\cal A}_{HM}(t_{b})=\frac{A}{{\cal V}_{2}}=\int
dt\sqrt{\left(-g(r)H(r)+\sigma(r)H(r)\dot{r}(t)^{2}\right)}\equiv\int dt{\cal
L},$ (7.24)

where $H(r)={\cal V}_{int}^{2}\alpha(r)^{3}$ and ${\cal V}_{2}=\int\int
dx^{2}dx^{3}$ . The following is the constant of motion E (which in turn is
the energy of the minimum surface) since there is no explicit $t$ dependency
in the Lagrangian:

$E=\frac{\partial{\cal L}}{\partial\dot{r}(t)}\dot{r}(t)-{\cal L},$ (7.25)

where $\dot{r}(t)=\frac{dr(t)}{dt}$. Using equation (7.24), we can reduce the
preceding equation as follows:

$E=\frac{g(r)H(r)}{\sqrt{\left(-g(r)H(r)+\sigma(r)H(r)\dot{r}(t)^{2}\right)}}.$
(7.26)

When we solved the preceding equation for $\dot{r(t)}$, we got:

$\dot{r}(t)=\pm\sqrt{\left(\frac{g(r)}{\sigma(r)}\left(1+\frac{g(r)H(r)}{E^{2}}\right)\right)}.$
(7.27)

At $r=r^{*}$, $\dot{r}(t)|_{r=r^{*}}=0$. As a result, equation (7.27) is going
to be rewritten as follows:

$\frac{g(r^{*})}{\sigma(r^{*})}\left(1+\frac{g(r^{*})H(r^{*})}{E^{2}}\right)=0.$
(7.28)

implying

$E^{2}=-g(r^{*})H(r^{*}),$ (7.29)

The highest value of $r$ for the a surface having energy $E$ is $r^{*}$. In
the entire geometry, a Hartman-Maldacena-like surface begins at $x^{1}=x_{\cal
R}$, rises to $r^{*}$, and then descends to its thermofield double partner.
The equation (7.24) has been simplified as follows:

${\cal A}_{HM}(t_{b})=\int
dr\sqrt{-\frac{g(r)H(r)}{\dot{r}(t)^{2}}+\sigma(r)H(r)}.$ (7.30)

Using (7.27) and (7.30), we obtained:

${\cal
A}_{HM}(t_{b})=2\int_{r_{h}}^{r^{*}}dr{\frac{\sqrt{\sigma(r)g(r)}H(r)}{E\sqrt{\left(1+\frac{g(r)H(r)}{E^{2}}\right)}}}.$
(7.31)

Now we identify time via the integral:

$\int_{r_{h}}^{r^{*}}\frac{dr}{\dot{r}(t)}=t_{*}-t_{b},$ (7.32)

where $t_{*}=t(r^{*})$ and $t_{b}$ represents the boundary time at $r=r_{h}$.
The boundary time as expressed in the form of energy could be calculated using
equations (7.27) and (7.32) as follows:

$t_{b}=-P\int_{r_{h}}^{r^{*}}\frac{dr}{\sqrt{\frac{g(r)}{\sigma(r)}\left(1+\frac{g(r)H(r)}{E^{2}}\right)}}.$
(7.33)

As a result, the Hartman-Maldacena-like surface’s entanglement entropy is:

${\cal S}_{HM}(t_{b})=\frac{{\cal A}_{HM}(t_{b})}{4G_{N}^{(11)}}.$ (7.34)

#####  7.4.1.1 Hartman-Maldacena-like Surface Analytics/Numerics

After integrating all of the angular coordinates and consequently including a
$(2\pi)^{4}$ emerging from integration w.r.t. $\phi_{1,2},\psi,x^{10}$, the
area corresponding to the Hartman-Maldacena-like surface is obtained as:

$\displaystyle\hskip-21.68121ptA_{\rm
HM}\sim(2\pi)^{4}\int_{r_{h}}^{r^{*}}dr\Biggl{(}{\pi^{9}EM^{2}\sqrt[10]{N}\log^{2}(2)(\log(64)-1)^{2}N_{f}^{4}g_{s}^{9/4}\log^{2}(r)(\log(N)-3\log(r))^{4}}$
$\displaystyle\times\left(N_{f}g_{s}\left(r^{2}(2\log(N)-18\log(r)+3)-2r_{h}^{2}(\log(N)-54r\log(r))\right)+8\pi
r_{h}^{2}\right){}^{2}\Biggr{)}.$ (7.35)

Consider $\left|\log r_{h}\right|\gg\log N$ [82] in (C.9), we obtained:

$A_{\rm HM}\sim{\cal O}(1)\times
10^{5}M^{2}\sqrt[10]{N}N_{f}^{6}g_{s}^{17/4}r_{h}^{4}\left(r_{*}-r_{h}\right)\log^{4}\left(r_{h}\right)\left(\log(N)-3\log\left(r_{h}\right)\right)^{4}.$
(7.36)

Now, the principal value of the following integral gives $t_{b}$:

$\displaystyle t_{b}\sim{\cal
P}\int_{r_{h}}^{r_{*}}dr\Biggl{(}\frac{E^{2}\sqrt{N}r^{2}\sqrt{g_{s}}\left(N_{f}g_{s}\left(r^{2}(2\log(N)-18\log(r)+3)-2r_{h}^{2}(\log(N)-54r\log(r))\right)+8\pi
r_{h}^{2}\right){}^{2}}{\left(r^{4}-r_{h}^{4}\right)\left(N_{f}g_{s}\left(\left(2r^{2}-6a^{2}\right)\log(N)+3r\left(108a^{2}\log(r)+r-6r\log(r)\right)\right)+24\pi
a^{2}\right)^{2}}\Biggr{)}$ $\displaystyle\hskip
14.45377pt\sim\lim_{\epsilon_{1}\rightarrow
0}\left[4\pi^{33/2}E^{2}\sqrt{{g_{s}}}\sqrt{N}\left(\frac{\log({r_{h}}-r)}{4{r_{h}}}-\frac{\log(r+{r_{h}})}{4{r_{h}}}+\frac{\tan^{-1}\left(\frac{r}{{r_{h}}}\right)}{2{r_{h}}}\right)\right]_{r=r_{h}+\epsilon_{1}}^{r=r_{*}}$
$\displaystyle\hskip
14.45377pt-\frac{E^{2}\sqrt{{g_{s}}}\sqrt{N}(\log(-\epsilon)-\log({r_{h}}-{r_{*}}))}{{r_{h}}}+\frac{\pi^{33/2}E^{2}\sqrt{{g_{s}}}\sqrt{N}({r_{*}}-{r_{h}})}{2{r_{h}}^{2}}+{\cal
O}\left(({r_{*}}-{r_{h}})^{2}\right).$ (7.37)

The Principal value necessitates: $r_{*}=r_{h}+\epsilon_{1}$. Let
$a=\left(\frac{1}{\sqrt{3}}+\epsilon\right)r_{h}$, upto the leading order in
$\epsilon$, we obtained:

$\displaystyle
t_{b}=\frac{E^{2}\pi^{33/2}\sqrt{N}\sqrt{g_{s}}\left(r_{*}-r_{h}\right)}{2r_{h}^{2}}.$
(7.38)

Considering $\epsilon_{1}=\tilde{\epsilon}_{1}r_{h}$, we found that for (i)
$\tilde{\epsilon}_{1}\sim 0.5$, we have to include the terms up to ${\cal
O}(\tilde{\epsilon}_{1}^{2})$ in $t_{b}$, (ii)
$\tilde{\epsilon}_{1}\sim\frac{1}{\sqrt{2}}$, we have to include the terms up
to ${\cal O}(\tilde{\epsilon}_{1}^{3})$ in $t_{b}$, (iii)
$\tilde{\epsilon}_{1}\sim 1$, we have to include the terms up to ${\cal
O}(\tilde{\epsilon}_{1}^{4})$ in $t_{b}$, and (iv)
$\tilde{\epsilon}_{1}\sim\sqrt{5}$, we have to include the terms up to ${\cal
O}(\tilde{\epsilon}_{1}^{5})$ in $t_{b}$. We found that all the way to the
Page time,$\tilde{\epsilon}_{1}\ll 1$, and consequently it is justified in
preserving just linear terms in $\tilde{\epsilon}_{1}$ as in (7.38). We
calculated the entanglement entropy associated with a Hartman-Maldacena-like
surface using the equations (7.36) and (7.38) as follows:

$\displaystyle S_{\rm HM}^{\beta^{0}}=\frac{A_{\rm
HM}}{4G_{N}^{(11)}}\sim\frac{{\cal O}(1)\times
10^{-4}M^{2}N_{f}^{6}g_{s}^{15/4}r_{h}^{6}\log^{4}\left(r_{h}\right)\left(\log(N)-3\log\left(r_{h}\right)\right){}^{4}}{E^{2}G_{N}^{(11)}N^{2/5}}t_{b}.$
(7.39)

As a result, the entanglement entropy related with a Hartman-Maldacena-like
surface grows linearly with time.

####  7.4.2 Entanglement Entropy Contribution from Island Surface

We take a constant $t$ slice to calculate the entanglement entropy for an
island surface. As a result, utilizing equation (7.20), we expressed the
induced metric associated with the island surface as follows:

$ds^{2}|_{constt-
time}=\alpha(r)\Biggl{[}\left(\sigma(r)+\dot{x}(r)^{2}\right)dr^{2}+(dx^{2})^{2}+(dx^{3})^{2}\Biggr{]}+g_{mn}dx^{m}dx^{n},$
(7.40)

where $x^{1}(r)\equiv x(r)$, and $\dot{x}(r)\equiv\frac{dx(r)}{dr}$. We
computed the area density functional associated with the island surface as
shown below:

${\cal A}_{IS}=\frac{A}{{\cal V}_{2}}=\int
dr\sqrt{\left(H(r)\sigma(r)+H(r)\dot{x}(r)^{2}\right)}\equiv\int dr{\cal L},$
(7.41)

wherein

$H(r)={\cal V}_{int}^{2}\alpha(r)^{3}.$ (7.42)

Because $x(r)$ represents a cyclic coordinate, the conjugate momentum related
to $x(r)$ is $p_{x(r)}=\frac{\partial{\cal L}}{\partial\dot{x}(r)}$.The
constant of motion is written as follows:

$p_{x(r)}={\cal C},$ (7.43)

where ${\cal C}$ is the constant. We simplified the expression (7.43)
utilizing equation (7.41) as given below:

$\frac{H(r)\dot{x}(r)}{{\cal L}}={\cal C}.$ (7.44)

When we solved the preceding equation for $\dot{x}(r)$, we were provided with:

$\dot{x}(r)=\pm{\cal C}\sqrt{\frac{\sigma(r)}{H(r)-{\cal C}^{2}\sigma(r)}}$
(7.45)

At the turning point, $r=r_{T}$:

$\left(\frac{dr}{dx}\right)_{r=r_{T}}=0.$ (7.46)

Constant ${\cal C}$ could be calculated using the equations (7.45) and (7.46),
as shown below:

${\cal C}=\pm\sqrt{\frac{H(r_{T})}{\sigma(r_{T})}}.$ (7.47)

Taking the previously stated expression of ${\cal C}$, the equation (7.44)
reduces to the following structure:

$\frac{dx(r)}{dr}=\pm\sqrt{\frac{H(r_{T})\sigma(r_{T})}{H(r)\sigma(r_{T})-H(r_{T})\sigma(r)}}.$
(7.48)

Now, the area density functional associated with the island surface has been
simplified using the equations (7.41) and (7.48).

${\cal
A}_{IS}=2\int_{r_{h}}^{r_{T}}\sqrt{H(r)\sigma(r)+\frac{H(r)H(r_{T})\sigma(r)}{H(r)\sigma(r_{T})-H(r_{T})\sigma(r)}}.$
(7.49)

Hence, the entanglement entropy associated with the island surface is obtained
as follows:

${\cal S}_{IS}=\frac{{\cal A}_{IS}}{4G_{N}^{(11)}}$ (7.50)

Since, there is no time dependence in (7.50), and hence entanglement entropy
of the Hawking radiation for the island surface is constant.

#####  7.4.2.1 Island Surface Analytics

When (7.49) is evaluated, the entanglement entropy is as follows:

$\displaystyle{\cal A}_{\cal
IS}=\int_{r_{h}}^{r_{T}}\Biggl{(}\frac{2\sqrt{2}\pi
MN^{3/10}rN_{f}^{2}g_{s}^{11/8}\log(r)(\log(N)-3\log(r))^{2}}{\sqrt{r^{4}-r_{h}^{4}}}$
$\displaystyle\times\left(N_{f}g_{s}\left(r^{2}(2\log(N)-18\log(r)+3)-2r_{h}^{2}(\log(N)-54r\log(r))\right)+8\pi
r_{h}^{2}\right)\Biggr{)},$ (7.51)

implying

$\displaystyle{\cal
S}_{IS}\sim\frac{1}{4G_{N}^{(11)}}\Biggl{[}MN^{3/10}N_{f}^{3}g_{s}^{19/8}\log\left(r_{h}\right)\left(\log(N)-3\log\left(r_{h}\right)\right){}^{2}\Biggl{(}216r_{h}\tilde{r}_{T}\log\left(r_{h}\right)\,_{2}F_{1}\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{1}{\tilde{r}_{T}^{4}}\right)$
$\displaystyle-
r_{h}\left(\log(N)\left(2\log\left(\sqrt{1-\frac{1}{\tilde{r}_{T}^{4}}}+1\right)-\log\left(\frac{1}{\tilde{r}_{T}^{4}}\right)\right)+\sqrt{\tilde{r}_{T}^{4}-1}\left(18\log\left(r_{h}\right)-2\log(N)-3\right)+73\sqrt{\pi}r_{h}\log\left(r_{h}\right)\right)\Biggr{)}\Biggr{]},$

where $\tilde{r}_{T}\equiv\frac{r_{T}}{r_{h}}>1$; $r_{T}$ is calculated in
(C.2.2) - (C.12).

####  7.4.3 Page curve at LO from Areas of Hartman-Maldacena-like and Island
Surfaces

According to (7.39), for the numerical values of $g_{s},M,N_{f},N,r_{h}$
chosen in appendix C.2.2, $S_{\rm EE}^{\rm HM}=\frac{5\times
10^{-4}t_{b}}{E^{2}}$, and from equation (7.4.2.1), the entanglement entropy
portion of the island’s surface remains independent of time, and can be
calculated as 421. This contribution takes precedence after the Page time.
Hence, we obtained the Page curve as shown in Fig. 7.2. The Page time is
calculated as $t_{\rm Page}=8.5\times 10^{5}E^{2}$. To produce the same
$t_{\rm Page}$ with the insertion of ${\cal O}(\beta)$“anomaly terms” as in
7.5.4, $E=1.1$. Using this $t_{\rm Page}$, one can observe that
$\tilde{\epsilon}_{1}\sim 10^{-5}\ll 1$, as indicated below (7.38).

Figure 7.2: Page curve up to ${\cal O}(\beta^{0})$ of an eternal neutral black
hole in ${\cal M}$-theory dual via doubly holographic setup. The blue line in
the figure indicates the area of the Hartman-Maldacena-like surface, and the
orange line refers for the area of the island surface; $E$ in (7.39) is set to
$1.1$ to obtain the same order of the magnitude of $t_{\rm Page}$ as in Fig.
7.7.

###  7.5 Page Curve with Higher Derivative Terms

We obtained the Page curve corresponding to a neutral black hole in with the
inclusion of higher derivative terms that are quartic in the Riemann curvature
tensor in this section. This section has been broken into four subsections. We
calculated the entanglement entropy associated with the HM-like surface in
subsection 7.5.1, discussed the “Swiss-Cheese” structure of the identical
surface in subsection 7.5.2, computed the entanglement entropy of island
surface in 7.5.3, and at last obtained the Page curve corresponding to an
eternal black hole in subsection 7.5.4 utilizing the outcomes that were
obtained from previous subsections.

We conducted the same analysis as in 7.4. [127] was used to compute the
entanglement entropy associated with Hartman-Maldacena-like surface and island
surface in with the inclusion of higher derivative terms. In generic higher
derivative gravity theories, we are able to calculate the holographic
entanglement entropy (5.8) as follows:

  * •

When the holographic dual consists of a $(d+1)$ dimensional gravitational
background, then with the help of the embedding function, compute the induced
metric associated with the co-dimension two surface.

  * •

Using the above-mentioned induced metric, compute (5.8). This yields the
holographic entanglement entropy as an expression of the embedding function
and its derivatives.

  * •

Find the solution to the embedding function’s equation of motion.

  * •

In higher derivative gravity theories, putting the solution found in the
previous step inside the action yields the holographic entanglement entropy.

Contribution from $J_{0}$ term: In addition to the leading order term in the
gravitational action, we need to consider the $J_{0}$ term to compute the
holographic entanglement entropy. Let us see, how $J_{0}$ term contribute to
the holographic entanglement entropy.

We derived the four terms via the Wald entanglement entropy formula in (5.8),
which is $\frac{\partial J_{0}}{\partial R_{z\bar{z}z\bar{z}}}$ for the two
extremal surfaces. They are listed in C.5.1 pertaining to the Hartman-
Maldacena-like surface and C.5.2 for the island surface.

$\displaystyle\frac{\partial J_{0}}{\partial
R_{z\bar{z}z\bar{z}}}=-4x^{2}\left(\frac{\partial J_{0}}{\partial
R_{txtx}}\right).$ (7.53)

To determine the second part in the formula (5.8), we must first compute the
four types of derivatives for the Hartman-Maldacena-like and island surfaces,
which are listed below:

$\displaystyle\left(\frac{\partial^{2}{J_{0}}}{\partial R_{zizj}\partial
R_{\bar{z}m\bar{z}l}}\right)K_{zij}K_{\bar{z}ml}=\Biggl{(}\frac{\partial^{2}{J_{0}}}{\partial
R_{xixj}\partial R_{xmxl}}+x^{4}\frac{\partial^{2}{J_{0}}}{\partial
R_{titj}\partial R_{tmtl}}-2x^{2}\frac{\partial^{2}{J_{0}}}{\partial
R_{titj}\partial R_{xmxl}}$
$\displaystyle+4x^{2}\frac{\partial^{2}{J_{0}}}{\partial R_{tixj}\partial
R_{xmtl}}\Biggr{)}\left(\frac{1}{x^{2}}K_{tij}K_{tml}+K_{xij}K_{xml}-\frac{i}{x}K_{tij}K_{xml}+\frac{i}{x}K_{xij}K_{tml}\right),$
(7.54)

The numerator and denominator coefficients in the preceding equation are
$x=x_{R}$ for the Hartman-Maldacena-like surface and $x=x(r)$ for the island
surface. All four types of variations occurring in equation (7.5) for the
Hartman-Maldacena-like surface in appendix C.5.1 and island surface in
appendix C.5.2 have been computed and listed.

####  7.5.1 Entanglement Entropy Contribution from Hartman-Maldacena-like
Surface

Hartman-Maldacena-like surface in ${\cal M}$-theory dual corresponds to a co-
dimension two surface situated at $x^{1}=x_{R}$. Utilizing equation (5.8), we
are able to write the mathematical expression that describes the entanglement
entropy associated with a Hartman-Maldacena-like surface as follows:

$S_{EE}=\int
drdx^{2}dx^{3}d\theta_{1}d\theta_{2}dxdydzdx^{10}\sqrt{-g}\Biggl{[}\frac{\partial{\cal
L}}{\partial R_{z\bar{z}z\bar{z}}}+\sum_{\alpha}\left(\frac{\partial^{2}{\cal
L}}{\partial R_{zizj}\partial
R_{\bar{z}m\bar{z}l}}\right)_{\alpha}\frac{8K_{zij}K_{\bar{z}ml}}{(q_{\alpha}+1)}\Biggr{]},$
(7.55)

where $g$ denotes determinant of the induced metric defined in (7.23) for the
co-dimension two surface. For the metric (7.23), the entanglement entropy
associated with the Hartman-Maldacena-like surface using (7.55) and appendix
C.5.1 after the angular integrations is obtained as444The angular integrations
have been performed using (C.2.1) and (C.2.1). We have taken the most dominant
terms in the leading order in $N$ from C.5.1 for the ${\cal O}(\beta^{0})$ and
${\cal O}(\beta)$ contributions.:

$S_{\rm EE}^{\rm total,\ {\rm HM}}=\int dr{\cal L}_{\rm{Total}}^{\rm{HM}}=\int
dr\left({\cal L}_{0}^{\rm HM}+{\cal L}_{\cal W}^{\rm HM}+{\cal L}_{\cal
A}^{\rm HM}\right),$ (7.56)

where

$\displaystyle{\cal L}_{0}^{\rm
HM}=\left(x_{R}^{2}\lambda(r)\sqrt{\alpha(r)\left(\sigma(r)-\left(1-\frac{r_{h}^{4}}{r^{4}}\right)t^{\prime}(r)^{2}\right)}\right),$
$\displaystyle{\cal L}_{\cal W}^{\rm
HM}=\left(-4x_{R}^{2}(\lambda_{1}(r)+\lambda_{2}(r))\sqrt{\alpha(r)\left(\sigma(r)-\left(1-\frac{r_{h}^{4}}{r^{4}}\right)t^{\prime}(r)^{2}\right)}\right),$
$\displaystyle{\cal L}_{\cal A}^{\rm HM}=\frac{1}{x_{R}^{2}}\Biggl{(}Z(r){\cal
L}_{1}+x_{R}^{4}W(r){\cal L}_{2}-2x_{R}^{2}U(r){\cal
L}_{3}+4x_{R}^{2}\left(U(r)+2V(r)\right){\cal L}_{4}\Biggr{)},$ (7.57)

where $x_{R}$ being a constant, various $r$ dependent functions appearing in
(7.5.1) are listed in (• ‣ C.3), and appendix C.1, equations (• ‣ C.3), (C.15)
and (• ‣ C.3) have been used to calculate (7.56). Utilizing (C.17) - (• ‣
C.3), and writing $t(r)=t_{0}(r)+\beta t_{1}(r)$, the equation of motion for
the embedding $t(r)$ up to ${\cal O}(\beta^{0})$ is as follows:

$\displaystyle
N^{3/10}(r-r_{h})^{5/2}p_{1}\left(r_{h}\right){t_{0}}^{\prime\prime}(r)+\frac{5}{2}N^{3/10}(r-r_{h})^{3/2}p_{1}\left(r_{h}\right){t_{0}}^{\prime}(r)=0.$
(7.58)

The equation (7.58) has the following solution:

$\displaystyle{t_{0}}(r)=c_{2}-\frac{2c_{1}}{3\left(r-r_{h}\right){}^{3/2}}.$
(7.59)

substitution of (7.59) in (• ‣ C.3) yields:

$\displaystyle\hskip-21.68121pt{t_{1}}(r)=c_{3}+\frac{2r_{h}^{3/2}\sqrt{\kappa_{\sigma}}\left(11{r_{h}}^{7}\left(p_{8}^{\beta}\left(r_{h}\right)+p_{9}^{\beta}\left(r_{h}\right)\right)-3\left(r-r_{h}\right){}^{7}\left(p_{8}^{\beta}\left(r_{h}\right)+p_{9}^{\beta}\left(r_{h}\right)\right)-11c_{1}{}^{3}c_{2}p_{1}\left(r_{h}\right)\right)}{33c_{1}{}^{3}MN_{f}^{5/3}g_{s}^{7/3}\left(r-r_{h}\right){}^{3/2}\sqrt{\kappa_{\alpha}}\kappa_{\lambda}\log\left(r_{h}\right)\left(\log(N)-9\log\left(r_{h}\right)\right)\left(\log(N)-3\log\left(r_{h}\right)\right){}^{5/6}}.$
(7.60)

Keeping the term up to the leading order in $N$, we could write the embedding
function $t(r)$ as follows:

$\displaystyle
t(r)=c_{2}-\frac{2c_{1}}{3\left(r-r_{h}\right){}^{3/2}}+\beta\left(c_{3}-\frac{2c_{2}}{3\left(r-r_{h}\right){}^{3/2}}\right).$
(7.61)

If $t(r=r_{h})\equiv t_{b}$ is intended to be assessed as
$t\left(r_{h}\biggl{[}1+\frac{1}{N^{n_{t_{b}}}}\biggr{]}\right),n_{t_{b}}\equiv{\cal
O}(1)$, in the large-$N$ limit. Since (in ${\cal
R}_{D5/\overline{D5}}=1$-units) $\frac{c_{1}}{r_{h}^{3/2}}\sim c_{2}$, i.e.,
$c_{1}\sim r_{h}^{\alpha}c_{2}$ implying $\alpha=3/2$, i.e. $l_{p}^{6}\sim
r_{h}^{3/2}$ or $l_{p}\sim r_{h}^{1/4}$ i.e.,

$l_{p}\sim\left(g_{s}^{\frac{4}{3}}\alpha^{\prime}\
{}^{2}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}}\right)\right)^{\frac{1}{4}}.$ (7.62)

To identify the Hartman-Maldacena-like surface’s turning point, $r_{*}$, we
must apply the condition given below:

$\left(\frac{1}{t^{\prime}(r)}\right)_{r=r_{*}}=\frac{\left(r_{*}-r_{h}\right){}^{5/2}}{c_{1}}-\frac{\beta
c_{2}\left(r_{*}-r_{h}\right){}^{5/2}}{c_{1}{}^{2}}=0,$ (7.63)

As $\frac{c_{1}}{r_{h}^{3/2}}\sim c_{2}$. As a result, for the purpose of
determining $r_{*}$, (7.63) could be approximated by the equation that
follows:

$(r_{*}-r_{h})^{3/2}-\beta\kappa_{r_{*}}^{\beta}=0,$ (7.64)

which has the following solution,

$r_{*}=r_{h}+\beta^{2/3}{\left(\kappa_{r_{*}}^{\beta}\right)}^{2/3}$ (7.65)

The entanglement entropy of the Hawking radiation corresponding to the
Hartman-Maldacena-like surface is computed as given below:

$\displaystyle S_{\rm EE}^{\beta^{0},\rm
HM}=\int_{r_{h}}^{r_{*}}dr\Biggl{(}\lambda(r)\sqrt{\alpha(r)\left(\sigma(r)-\left(1-\frac{r_{h}^{4}}{r^{4}}\right)t_{0}^{\prime}(r){}^{2}\right)}\Biggr{)},$
(7.66)

utilizing (• ‣ C.3) and (7.59), (7.66) is expressed as:

$\displaystyle S_{\rm EE}^{\beta^{0},\rm
HM}\sim\int_{r_{h}}^{r_{*}}\left(\frac{\sqrt{\left(r-r_{h}\right)r_{h}}\log\left(r_{h}\right)\left(\log(N)-9\log\left(r_{h}\right)\right)\left(\log(N)-3\log\left(r_{h}\right)\right){}^{5/6}}{r_{h}^{3}\sqrt{r_{h}^{2}}}\right)$
$\displaystyle\times\Biggl{(}-\frac{2^{5/6}MN^{13/10}\log(2)(\log(64)-1)N_{f}^{5/3}g_{s}^{10/3}\sqrt{\kappa_{\alpha}}\kappa_{\lambda}\sqrt{\kappa_{\sigma}}}{81\
3^{2/3}\pi^{11/12}}\Biggr{)}$
$\displaystyle\sim\frac{2\left(r_{*}-r_{h}\right){}^{3/2}\log\left(r_{h}\right)\left(\log(N)-9\log\left(r_{h}\right)\right)\left(\log(N)-3\log\left(r_{h}\right)\right){}^{5/6}}{3r_{h}^{7/2}}$