11institutetext: Fehmi Ekmekçi, Lale Çelik, H. Volkan Şenavcı 22institutetext:
Ankara University, Faculty of Science, Department of Astronomy and Space
Sciences, 06100, Tandoğan, Ankara, Turkey, 22email:
<EMAIL_ADDRESS>
# RR Lyrae type stars, ST Boo and RR Leo: 2007 Observations and the
preliminary results of the frequency analaysis

Fehmi EKMEKÇİ  Lale ÇELİK  H. Volkan ŞENAVCI

###### Abstract

We present BVR light curves of pulsating stars, ST Boo and RR Leo, obtained
between March and September 2007 at the Ankara University Observatory (AUG)
and the TÜBİTAK National Observatory (TUG). Although these observational data
are insufficient to obtain the reliable results for a frequency analysis of ST
Boo and RR Leo stars, in this study, we tried to investigate the pulsation
phenomena of these two stars, as an overview, using the Period04 software
package. As preliminary results, we present the possible frequencies for ST
Boo and RR Leo.

##  1 Observations and Results

CCD observations of ST Boo and RR Leo were carried out by using an Apogee ALTA
$U47+CCD$ camera ($1024\times 1024$ pixels) with BVR filters mounted on both
40 cm Schmidt-Cassegrain telescopes of the Ankara University Observatory (AUG)
and the TÜBİTAK National Observatory (TUG) between March and September 2007.
BVR light curves of both ST Boo and RR Leo were normalized to maximum light
level to construct the data set for simultaneous multiple-frequency analysis
using Period04 (V 1.0) (Lenz05 ) which has a Fourier analysis definition of

$f(t)=Z+\sum_{i}A_{i}sin(2\pi(\Omega_{i}t+\phi_{i})).$ (1)

The results of multi-frequency solutions, with their errors calculated based
on Monte Carlo Simulation, for ST Boo and RR Leo are given in Table 1. Fig. 1
shows some of the light curve data with the fit curve of multi-frequency
solutions for ST Boo and RR Leo. Clearly, it must be included more and more
photometric data in the frequency analysis to have more definite and reliable
results for both of these pulsating stars.

Table 1: The results of multiple-frequency analysis of ST Boo and RR Leo

|  ST Boo |  |  |  |
---|---|---|---|---|---
f($cd^{-1}$) |  Amp.(mag.) |  S/N |  f($cd^{-1}$) |  Amp.(mag.) |  S/N
$4.201\pm 0.079$ |  $0.553\pm 0.664$ |  2.84 |  $18.925\pm 0.663$ |  $0.012\pm 0.022$ |  32.02
$4.221\pm 0.668$ |  $0.171\pm 0.754$ |  617.14 |  $28.315\pm 0.845$ |  $0.011\pm 0.012$ |  19.93
$0.187\pm 0.005$ |  $0.127\pm 0.209$ |  455.01 |  $16.236\pm 0.342$ |  $0.010\pm 0.033$ |  19.81
$7.311\pm 0.034$ |  $0.119\pm 0.072$ |  538.73 |  $20.333\pm 0.695$ |  $0.008\pm 0.026$ |  25.20
$6.337\pm 0.027$ |  $0.097\pm 0.071$ |  419.63 |  $33.717\pm 0.370$ |  $0.008\pm 0.004$ |  9.53
$10.247\pm 2.389$ |  $0.088\pm 0.143$ |  398.35 |  $28.724\pm 0.176$ |  $0.007\pm 0.009$ |  12.22
$10.649\pm 0.504$ |  $0.085\pm 0.178$ |  365.89 |  $23.532\pm 0.418$ |  $0.006\pm 0.014$ |  10.29
$7.500\pm 0.082$ |  $0.076\pm 0.114$ |  348.95 |  $602.962\pm 0.856$ |  $0.005\pm 0.002$ |  5.96
$17.073\pm 0.093$ |  $0.043\pm 0.055$ |  87.02 |  $35.863\pm 0.108$ |  $0.005\pm 0.006$ |  5.00
$0.227\pm 0.060$ |  $0.037\pm 0.560$ |  133.81 |  $604.857\pm 0.239$ |  $0.004\pm 0.002$ |  5.22
$10.875\pm 1.406$ |  $0.033\pm 0.139$ |  138.95 |  $510.716\pm 0.157$ |  $0.004\pm 0.002$ |  5.53
$3.113\pm 0.540$ |  $0.028\pm 0.099$ |  96.61 |  $30.484\pm 1.174$ |  $0.003\pm 0.006$ |  6.18
$21.491\pm 0.105$ |  $0.024\pm 0.020$ |  65.53 |  $568.959\pm 7.439$ |  $0.003\pm 0.002$ |  4.91
$11.980\pm 0.313$ |  $0.024\pm 0.043$ |  83.07 |  $599.575\pm 0.112$ |  $0.003\pm 0.003$ |  4.21
$14.403\pm 0.093$ |  $0.024\pm 0.043$ |  49.74 |  $566.738\pm 0.198$ |  $0.003\pm 0.002$ |  5.00
$16.399\pm 3.804$ |  $0.024\pm 0.055$ |  45.08 |  $559.057\pm 0.150$ |  $0.003\pm 0.002$ |  4.03
$6.465\pm 0.255$ |  $0.021\pm 0.135$ |  90.25 |  $47.909\pm 0.366$ |  $0.003\pm 0.002$ |  4.28
$24.493\pm 2.307$ |  $0.016\pm 0.020$ |  23.59 |  $562.309\pm 7.929$ |  $0.002\pm 0.002$ |  4.29
$21.501\pm 0.250$ |  $0.012\pm 0.016$ |  31.06 |  - |  - |  -
|  RR Leo |  |  |  |
f($cd^{-1}$) |  Amp.(mag.) |  S/N |  f($cd^{-1}$) |  Amp.(mag.) |  S/N
$3.418\pm 0.001$ |  $0.672\pm 0.041$ |  2.88 |  $10.788\pm 0.003$ |  $0.055\pm 0.016$ |  10.81
$8.634\pm 0.001$ |  $0.229\pm 0.026$ |  35.15 |  $19.478\pm 0.001$ |  $0.049\pm 0.010$ |  14.42
$4.541\pm 0.206$ |  $0.166\pm 0.060$ |  17.06 |  $16.500\pm 0.239$ |  $0.031\pm 0.017$ |  7.18
$13.469\pm 0.001$ |  $0.132\pm 0.022$ |  28.57 |  $21.518\pm 0.037$ |  $0.031\pm 0.006$ |  9.97
$6.962\pm 0.057$ |  $0.075\pm 0.038$ |  10.08 |  $24.141\pm 0.001$ |  $0.030\pm 0.005$ |  9.44
$2.259\pm 0.246$ |  $0.062\pm 0.049$ |  4.04 |  $27.494\pm 0.159$ |  $0.016\pm 0.004$ |  4.48

Figure 1: An example of the light curve measurements of ST Boo (left panel)
and RR Leo (right panel) together with the fit curve of multi-frequency
solution. The axis of the time is in unit of HJD(2450000+…) and observed BVR
is in normalized values to maximum level of the light curve

## References

  * (1) Lenz, P., Breger, M.: Period04 User Guide. Comm. in Asteroseismol. 146, 53–136 (2005)


Page Curve and Island in EGB gravity

Ankit Anand 111E-mail -<EMAIL_ADDRESS>

1Department of Physics,
Indian Institute of Technology Madras, Chennai 600 036, India

ABSTRACT

In this paper, the information paradox on lower-dimensional Gauss–Bonnet
gravity, known as a three-dimensional EGB black hole, is studied using the
quantum extremal island approach. For that, we connect an auxiliary flat bath
system to this timelike singularity spacetime and estimate the entropy of
hawking radiation in its asymptotic regions when gravity is weak. The addition
of island areas to the entanglement wedge of radiation causes its entropy to
obey the Page curve, as shown. The quantum extremal surface(QES), or island
boundary, is located outside the horizon. This yields a time-independent
equation for Hawking radiation fine-grained entropy that is compatible with
the appropriate Page curve. We also discover the modifications to this entropy
and Page time.

##  1 Introduction

From the quantum gravity theory point of view, the investigation of the
information loss paradox has been one of the most important challenges [1].
Hawking radiation has the same properties as thermal radiation, implying that
the entanglement entropy(EE) outside the black hole(BH) is increasing
monotonically. Quantum mechanics, on the other hand, mandates that the EE at
the end of the evaporation be zero because it must be the pure state. The EE
changes with time and results in the generation of the so-called Page curve
[2, 3]. As a result, the information loss paradox is transformed into getting
the Page curve for Hawking radiation’s EE. By incorporating a scale for the
time measurement known as the Page time $t_{p}$, the Page curve effectively
helps to answer whether the problem of the information loss paradox can be
resolved or not. In terms of the Page curve, the information loss paradox can
be explained as follows. The quantum field’s von Neumann entropy on the
radiation region $R$ outside the BH gives the fine-grained entropy of Hawking
radiation. We assume that the entire Cauchy slice is in the pure state, the
fine-grained entropy of radiation $S(R)=S(R_{c})$, where $S(R_{c})$ is the BH
subsystem’s fine-grained entropy. Because the whole system is the BH Plus
radiation region, the region $R_{c}$ should correspond to the BH’s fine-
grained entropy, and the region $R$ corresponds to the radiation’s fine-
grained entropy. This means that there is always a condition that $S(R)$ has
to satisfy i.e., $S(R)\leq S_{BH}$, where $S_{BH}$ is the BH subsystem’s
coarse-grained entropy. However, it has been discovered that $S(R)>S_{BH}$
immediately after the Page time $t_{p}$, resulting in the contradiction.

The Page curve has recently been postulated to develop from the action of
islands [4, 5, 6, 7]. The density matrix of R is generally determined by
taking the partial trace across the states in Rc, which is the complementary
area of $R$, when considering the state of the Hawking radiation as that in a
region R outside the BH. Recent research [8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 22, 23, 21, 24, 25, 26, 27] has revealed that specific
supplementary regions known as islands contribute to the entropy of Hawking
radiation, with their boundaries defined as surfaces known as QES. This
demonstrates the existence of a non-trivial QES in spacetime at Page time,
which cancels out the time-dependency of $S(R)$ and leads to a saturated
Hawking radiation’s fine-grained entropy. By inclusion of island
contributions, Hawking’s fine-grained entropy is [5]

$S(R)=\text{min}\;\;\;\;\;\text{ext}\left[\frac{Area(\partial
I)}{4G_{N}}+S_{\text{matter}}(I\cup R)\right]\ .$ (1)

The island rule was first postulated due to a conjectured QES prescription,
and it was recently deduced using the replica approach for the gravitational
path integral. When the replica trick [28, 29, 30] is used in gravitational
theories, only the boundary conditions of the replica geometries can be fixed,
and new saddles, where bulk wormholes join distinct copies of spacetime, must
be considered. Replica wormholes, as these new saddles are known, lead to
islands [31, 8]. When using replicas and in the semi-classical gravity limit,
there is of the partition function of the geometry is the one with the lowest
entanglement entropy dominant. The replica trick for gravitational theories
yield the same formula (1) as the QES prescription in this fashion.

Higher-curvature gravity theories continue to pique attention, partly because
the majority of quantum gravity theories indicate that such corrections modify
the Einstein–Hilbert(EH) action and because it provides a unique setting to
understand classical gravity in strong Gravitational fields. Lovelock theories
[32], which are the broadest theories formed from the Riemann curvature tensor
and preserve metric’s second-order motion equations, are the most promising
and most investigated options. For $D>5$, although they have the feature that
any additional contributions they make to the action are topological or zero.
The cosmological and Einstein–Hilbert factors are combined in this theory,
which includes new corrections for each odd spacetime dimension above four.
The Gauss-Bonnet term is the first new correction of this type

$\mathcal{G}=R_{\alpha\beta\gamma\tau}R^{\alpha\beta\gamma\tau}-4R_{\alpha\beta}R^{\alpha\beta}+R^{2}\
,$ (2)

has five or more dimensions of activity. Significantly, for D $<$ 5, the new
terms that come from this are either zero, or it is topological, establishing
Einstein’s theory as the most general second-order metric theory of gravity in
four dimensions. Recently, a novel approach [33] for overcoming this
constraint for Gauss-Bonnet gravity (the most fundamental of the Lovelock
theories) has attracted attention. By considering the theory’s parameter as
the spacetime dimension and scaling the Gauss-Bonnet coupling ($\alpha$), it
is possible to build $D=4$ and $D=3$ variations of this theory. Even though
the initial approach involved taking limits of solutions to the field
equations, which raised several consistency concerns [34, 35, 36], maintaining
consistency in $D<5$ can be obtained unbiased of any real solutions or
additional dimensions [37, 38]. This approach is a modification of one
previously in use to reach a D $\rightarrow 2$ limit of GR [39], and it is
congruent with a newly proposed dimensional reduction technique [40, 41], the
inside space must be flat. The resulting scalar-tensor theory of gravity is as
follows

$S=\int
d^{D}x\sqrt{-g}\left[R-2\Lambda+\alpha\left(\phi\mathcal{G}+4G^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi-4(\partial\phi)^{2}\Box\phi+2((\nabla\phi)^{2})^{2}\right)\right]\
,$ (3)

where $\mathcal{G}$ is Gauss-Bonet term given in equation (2) and
$G_{\alpha\beta}$ is the Einstein tensor. The Gauss–Bonnet term vanishes
identically for the case of $D<4$ dimensions. For $D=3$, no scalar degree of
freedom is propagating [42]. The repercussions of lower-dimensional
Gauss–Bonnet gravity, including BHs, solutions that radiate and collapse, BH
thermodynamics, and additional physical effects, have been studied
extensively. Till now, only the spherically symmetric BH has been studied.

In general, the entropy formula for higher derivative gravity may be written
as

$S_{\text{total}}=S_{\text{gravity}}+S_{\text{matter}}\ ,$ (4)

where $S_{\text{gravity}}$ and $S_{\text{matter}}$ are the gravitational and
matter contributions to total entanglement entropy, respectively. In higher
derivative gravity theories, the first component in equation (4) may be
determined using the Dong formula [43] while the $S_{\text{matter}}$ term can
be calculated using the Cardy formula [44, 45]. Then we must extremize the
overall entanglement entropy with respect to the location of the island
surfaces. If there are several surfaces, we must select the surface with the
smallest area from among those surfaces. The authors of [19]constructed Page
curves of Schwarzschild black holes in the presence of higher derivative
components that are $\order{R^{2}}$. Following [19], [21] calculates the Page
curves of an eternal Reissner-Nordström black hole in four dimensions in the
presence of $\order{R^{2}}$ terms as discussed in [19] and in Einstein-Gauss-
Bonnet gravity [46]. Throughout the paper we set $l=1$.

This paper is organized as follows. In Section 2, we set the stage for further
discussions (see the reference[20]). In Section 3, we discuss the EGB gravity.
In section 4 we go over how to compute the entanglement entropy without and
with the island in EGB gravity. Section 5 displays the Page curve and
discusses the Page time and scrambling time. Section 6 is devoted to
conclusions.

##  2 Preliminaries

###  2.1 BTZ Black Hole

Let’s start with the metric of BTZ BH

$dS^{2}=-\left(r^{2}-r_{+}^{2}\right)dt^{2}+\left(\frac{1}{r^{2}-r_{+}^{2}}\right)dr^{2}+r^{2}d\phi^{2}\
,$ (5)

where $r_{+}$ represents the location of the horizon. The thermodynamics of
BTZ well studied in literature and can be written as

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{m}{8}=\frac{r_{+}^{2}}{8}\ ,\hskip
14.22636ptT=\frac{f^{\prime}}{4\pi}=\frac{r_{+}}{2\pi}\ ,\hskip
14.22636pt\Omega=0\ ,$ $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi r_{+}}{2}\ ,\hskip 14.22636ptP=\frac{1}{8\pi}\ ,\hskip
14.22636ptV=\pi r_{+}^{2}\ .$ (6)

The Kruskal extension of BTZ spacetime with identification as

Right Wedge : $U=-e^{-k(t-r_{*})}$ and $V=e^{k(t-r_{*})}$

Left Wedge : $U=e^{-k(t-r_{*})}$ and $V=-e^{k(t-r_{*})}$

where the surface gravity is defined as $k=\frac{2\pi}{\beta}$, and the
tortoise coordinate is indicated by $r_{*}$ and given as

$r_{*}=\int\frac{dr}{f(r)}=\frac{1}{2r_{+}}\left[\text{log}\left(\frac{|r-r_{+}|}{r+r_{+}}\right)\right]\
.$ (7)

Using the linking auxiliary baths [50][22], we can now make the BTZ spacetime
boundary transparent. We can treat the bath as a flat Minkowski spacetime
devoid of gravitational effects and suppose it is in thermal equilibrium with
the black hole. In terms of Kruskal coordinate, the metric can be written as

$dS^{2}=\Omega^{-2}dUdV+r^{2}d\phi^{2},$ (8)

with $\Omega=\frac{r_{+}}{r+r_{+}}$ and $\phi$=constant represents the two
dimensional space. By ignoring the island’s contribution, eq.(1) takes the
conventional form $S(R)=S_{gen}(R)$, meaning that the von Neumann entropy of
quantum fields must be computed on the union of $R_{+}$ and $R_{-}$.

###  2.2  Island and the Information Recovery

We start by calculating the radiation’s EE in the absence of islands. The area
term does not affect the gravitational fine-grained entropy (1) in this case.
Only the right and left wedges of the Penrose diagram have Hawking radiation
zones, and the borders of these entanglement regions are marked by $b_{+}$ and
$b_{-}$, respectively

Figure 1: Penrose diagrams of the BTZ BH (left) and one with an island $I$
(right)

As a result, the generalized EE is made up entirely considering the von
Neumann entropy of the constrained quantum matter proposed by

$S_{\text{gen}}=\frac{c}{3}\text{Log}[\ell(b_{+},b_{-})]\ ,$ (9)

where $\ell(b_{+},b_{-})$ denotes the distance between $b_{+}$ and $b_{-}$.
The expression for the distances is already calculated in [20]. So we have

$S_{\text{gen}}=\frac{c}{6}\text{Log}\left[\frac{4(b^{2}-r_{+}^{2})}{r_{+}^{2}}\text{Cosh}^{2}(2r_{+}t_{b})\right]\
,$ (10)

$c$ is the central charge. At late time i.e., $t\rightarrow\infty$ then
$\text{Cosh}(r_{+}t)\sim e^{r_{+}t}$, with this approximation the above
expression reduces to

$S(R)\sim\frac{c}{3}r_{+}t\ ,$ (11)

with the identification of $t_{b}$ as $t$. We can see from equation (11), that
in the absence of an island surface, the EE of the Hawking radiation increases
linearly with time and becomes infinite at late times, resulting in BTZ BH’s
information paradox.

Now we will show that at the late time( i.e., after page time), an island
appears and result in the constant value of EE of the Hawking radiation.

To check whether the island is present or not in the BTZ+Bath system, for the
s-wave approximation to be effective, we presume that the right wedge is far
apart from the left wedge. Moreover, the von Neumann entropy of quantum matter
is adequate for unifying the radiation and island over the whole region.

$S(R\cup
I)=\frac{c}{6}\text{log}\left[\frac{\ell(a_{+},a_{-})\ell(b_{+},b_{-})\ell(a_{+},b_{+})\ell(a_{-},b_{-})}{\ell(a_{+},b_{-})\ell(a_{-},b_{+}}\right]\
,$ (12)

where $a_{\pm}=(\pm t_{a},a)$. Because large distances between two wedges are
assumed, it follows that

$\ell(a_{+},a_{-})\simeq\ell(b_{+},b_{-})\simeq\ell(a_{\pm},b_{\mp})\gg\ell(a_{\pm},b_{\pm})\
.$ (13)

So, the EE of the whole system can be written as the sum of the $R\cup I$ in
left as well as right wedges.

The expression of the generalized entropy will be

$\displaystyle S_{\text{gen}}$ $\displaystyle=$ $\displaystyle\frac{\pi
a}{G_{N}}+\frac{c}{3}\text{log}[\ell(a_{+},b_{+})]\ ,$ (14) $\displaystyle=$
$\displaystyle\frac{\pi
a}{G_{N}}+\frac{c}{6}\text{log}\Bigg{[}\frac{(a+r_{+})(b+r_{+})}{r_{+}}\Bigg{[}\left(\frac{a-r_{+}}{a+r_{+}}\right)+\left(\frac{b-r_{+}}{b+r_{+}}\right)$
$\displaystyle-2\Bigg{(}\sqrt{\left(\frac{b-r_{+}}{b+r_{+}}\right)}\sqrt{\left(\frac{a-r_{+}}{a+r_{+}}\right)}\text{Cosh}(r_{+}(t_{a}-t_{b}))\Bigg{]}\
.$

Now, Extremizing the $S_{\text{gen}}$ (14) with respect to $t_{a}$ and
equating to zero results in $t_{a}=t_{b}$. Substituting this value to the
equation (14) and then by extremizing w.r.t $a$ i.e., $\frac{\partial
S_{\text{gen}}}{\partial a}=0$ reveals that

$a=r_{+}+\frac{1}{2}\left(\frac{cG_{N}}{6\pi}\right)^{2}+\dots\ .$ (15)

Substituting this value in $S_{\text{gen}}$ we get

$\displaystyle S_{\text{gen}}$ $\displaystyle=$ $\displaystyle\frac{\pi
r_{+}}{G_{N}}+\frac{c^{2}}{72}\frac{G_{N}}{\pi}$ (16) $\displaystyle=$
$\displaystyle 2S_{BH}+\cdots\ .$

The above phrase contains a few distinguishing features. The statement
includes universal corrections to the BH’s Hawking entropy, in addition to the
leading item $S_{BH}$. Because these adjustments are quantum gravity signals,
this is a pleasant surprise. It is astounding how the mutual information
interpretation explains how the aforementioned conclusion yields the right
Page curve, and so resolves the information loss paradox.

###  2.3 Page Curve

EE increases linearly with time in the absence of an island surface, according
to equation (11), and is constant at late times, according to equation (16).
The Page time island surface forms, saturating the linear expansion of
entanglement entropy and reaching a constant value double the Bekenstein-
Hawking entropy of the BTZ BH, and we have the Page curve of the BTZ BH.

  * •

Page time $(t_{p})$ : Page time is defined as the point at which the Hawking
radiation’s EE begins to decline to zero for an evaporating black hole and
achieves a constant value for an eternal BH,

$\displaystyle t_{p}$ $\displaystyle=$
$\displaystyle\frac{6S_{BH}}{cr_{+}}+\cdots$ (17) $\displaystyle=$
$\displaystyle\left(\frac{6\beta}{c\pi}\right)S_{BH}-\cdots$

So, the leading component in the above equation represents the page
time($t_{p}$).

  * •

Scrambling time: $(t_{\text{scr}})$ : The time interval during which we
retrieve the information sent into the BH in the form of Hawking radiation is
known as scrambling time. Furthermore, it is the amount of time taken for the
information to reach the island’s surface. If we wish to transfer data from
the cutoff surface $r=b$ to the BH, the time it takes to get to the island
surface $r=a$ is:

$\displaystyle t_{\text{scr}}$ $\displaystyle\equiv$
$\displaystyle|r_{*}(b)-r_{*}(a)|$ $\displaystyle=$
$\displaystyle\frac{1}{2r_{+}}\text{log}\left(\frac{b-r_{+}}{b+r_{+}}\frac{a+r_{+}}{a-r_{+}}\right)$
$\displaystyle\simeq$
$\displaystyle\frac{\beta}{2\pi}\text{log}\left(\frac{r_{+}}{cG_{N}}\right)+\cdots\
.$

In the above calculation, we have set the order of $b$ as same as that of
$r_{+}$, fix the island location as in equation (15), and by assuming the
$S_{\text{BH}}\simeq\frac{r_{+}}{G_{N}}$. We have an expression for Scrambling
time as

$t_{\text{scr}}\simeq\frac{\beta}{2\pi}\text{log}\left(S_{\text{BH}}\right)$
(18)

##  3 EGB Gravity in Lower dimensions

Let’s start with the action that describes the Gauss-Bonnet (GB) gravity in
the lower dimension from eq.(3) and is given by

$S=\int
d^{3}x\sqrt{-g}\left[R-2\Lambda+\alpha\left(\phi\mathcal{G}+4G^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi-4(\partial\phi)^{2}\Box\phi+2((\nabla\phi)^{2})^{2}\right)\right]\
,$ (19)

with
$\mathcal{G}=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}-R_{\alpha\beta}R^{\alpha\beta}+R^{2}$,
in addition having extra scalar field $\phi$ and the GB coupling $\alpha$.
Following [48, 38], some extra term has to be added in action to reduce the
complications, and the term is

$S_{\lambda}=-2\int d^{3}x\sqrt{-g}\left[\lambda
e^{-2\phi}(R+6(\partial\phi)^{2})+3\lambda^{2}e^{-4\phi}\right]\ ,$ (20)

where the interior space’s curvature is represented by the $\lambda$. We
investigate the equations of motion resulting from the action (1) along with
the extra term (2) and the line element for deriving BH solutions to the GB
gravity; we start with the line element

$ds^{2}=-f(r)dt^{2}+\frac{1}{h(r)f(r)}dr^{2}+r^{2}\left(d\phi^{2}-\frac{J^{2}}{2r^{2}}dt\right)^{2}\
,$ (21)

where $J$ is some constant. We have three radial coordinate dependence
functions $f(r),h(r)$ and the scalar field $\phi(r)$. So, varying the action
(eq. (19) along with eq.(20)), we have three different equations of motion. As
we are interested in BTZ-like solutions, we set $h(r)=1$. From [48, 38],

  1. 1.

For the case “$\phi$=costant” and “$\lambda=0$” :

This choice results in

$f(r)=-m+\frac{r^{2}}{l^{2}}+\frac{J^{2}}{4r^{2}}\;\;\;\;\;\;\Lambda=-\frac{1}{l^{2}}\
,$ (22)

where $m$ is BH mass and $J$ is the angular momentum of the BH. This situation
is nothing but the BTZ case.

  2. 2.

For the case “$\phi(r)=\text{ln}\left(\frac{r}{l}\right)$” and “$\lambda=0$” :

This situation results in

$f_{\pm}=-\frac{r^{2}}{2\alpha}\left(1\pm\sqrt{1+\frac{4\alpha}{r^{2}}\left[\frac{r^{2}}{l^{2}}-m\right]}\right)\
,$ (23)

As the GB coupling constant approaches $0$, the $f(r)_{+}$ of solution (23)
lacks a well-defined limit, i.e., in the limit $\alpha\rightarrow 0$,
$f(r)_{+}$ blows up to infinity. So, only the $f(r)_{-}$ has a well-defined
limit as $\alpha\rightarrow 0$, and it gives the BTZ metric. The $f(r)_{-}$ in
this limit is

$f(r)_{-}=\frac{r^{2}}{l^{2}}-m-\frac{\alpha}{r^{2}}\left(\frac{r^{2}}{l^{2}}-m\right)^{2}+\order{\alpha^{2}}\
.$ (24)

Now, the authors of [38] discussed the entropy expression based on the Iyer-
Wald method [51][52], expression for entropy is

$S=\frac{\pi r_{+}}{2}\left\\{1-2\lambda\alpha e^{-2\phi}\right\\}\ .$ (25)

As we are working in $\lambda=0$ and
$\phi(r)=\text{ln}\left(\frac{r}{l}\right)$. the thermodynamics of such BHs
can be written as

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{m}{8}=\frac{r_{+}^{2}}{8l^{2}}\ ,\hskip
14.22636ptT=\frac{f^{\prime}}{4\pi}=\frac{r_{+}}{2\pi l^{2}}\ ,\hskip
14.22636pt\Phi=0\ ,$ $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi r_{+}}{2}\ ,\hskip 14.22636ptP=\frac{1}{8\pi l^{2}}\
,\hskip 14.22636ptV=\pi r_{+}^{2}\ .$ (26)

It is the same as the well-known BTZ black hole in Einstein’s gravity. The
first law of BH thermodynamics states that $\delta M=T\delta S+V\delta
P+\Phi_{\alpha}\delta$, it is to verify that it is verified with above
parameters. Also the Smarr formula $0=TS-2PV+2\Phi_{\alpha}\alpha$, is also
verified. This is interesting because, regardless of the value of $\alpha$,
the thermodynamic parameters remain the same even if the curvature is not
constant.

The metric is

$dS^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\phi^{2}\ ,$ (27)

where
$f(r)=\frac{r^{2}}{l^{2}}-m-\frac{\alpha}{r^{2}}\left(\frac{r^{2}}{l^{2}}-m\right)^{2}+\order{\alpha^{2}}=\frac{r^{2}-r_{+}^{2}}{l^{2}}-\frac{\alpha}{r^{2}}\left(\frac{r^{2}-r_{+}^{2}}{l^{2}}\right)^{2}+\order{\alpha^{2}}$.
In the limit of small $\alpha$, it reproduces the typical BTZ solution.
$r_{+}$ is the location of the horizon, and it matches with the BTZ horizon of
the same mass.

Working with the two-dimensional surface with $\phi$ = constant, the metric
can be written as

$dS^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}\ .$ (28)

Furthermore, the Kruskal coordinate can be defined to achieve the maximal BTZ
spacetime extension.

Right Wedge : $U=-e^{-k(t-r_{*})}$ and $V=e^{k(t-r_{*})}$

Left Wedge : $U=e^{-k(t-r_{*})}$ and $V=-e^{k(t-r_{*})}$

with the expression of surface gravity is $k=\frac{2\pi}{\beta}$ and $r_{*}$
is the tortoise coordinate and given by

$r_{*}=\int\frac{dr}{f(r)}=\frac{1}{2r_{+}}\left[\text{log}\left(\frac{|r-r_{+}|}{r+r_{+}}\right)-\gamma\hskip
2.84544pt\text{log}\left(\frac{|r-\gamma r_{+}|}{r+\gamma
r_{+}}\right)\right]\ ,$ (29)

where $\gamma=\sqrt{\frac{\alpha}{1-\alpha}}$. As we discussed above, the
procedure of linking flat space. In terms of Kruskal coordinate, the metric
can be written as

$dS^{2}=\Omega^{-2}dUdV\ ,$ (30)

with $\Omega=\frac{r+r_{+}}{rr_{+}}\sqrt{\left(\left[\frac{|r-\gamma
r_{+}|}{r+\gamma r_{+}}\right]^{\gamma}[r^{2}+(r^{2}-1)\gamma^{2}]\right)}$ .
$R_{-}$ and $R_{+}$ are the left and right wedges of the radiation regions,
$b_{-}$ and $b_{+}$ are the boundaries of the $R_{-}$ and $R_{+}$, and $a_{-}$
and $a_{+}$ are the boundaries of the island surface in the left and right
wedges for an eternal EGB black hole.

##  4 Island and the Information Recovery

###  4.1 Without Island

In this subsection, we discuss the calculation of EE without an island.

Figure 2:  The Penrose diagram of an EGB BH when there is no island.

Because there is no island surface initially, Hawking radiation’s EE can be
estimated using the formula below

$S=\frac{c}{3}\ell(b_{+},b_{-})\ ,$ (31)

where the points $b_{\pm}=(\pm t_{b},b)$ are the radiation region limits in
the right and left wedges of the EGB black hole. Defining $\beta$ as inverse
Hawking temperature as $\beta=\frac{2\pi}{k}$. Between the boundary points
$b_{+}$ and $b_{-}$, the geodesics distance is

$\ell(b_{+},b_{-})=\sqrt{\frac{4(b^{2}-r_{+}^{2})[b^{2}+(b^{2}-1)\gamma^{2}]}{(1+\gamma^{2})b^{2}r_{+}^{2}}\text{Cosh}^{2}(r_{+}t_{b}))}\
,$ (32)

putting eq.(32) in eq.(31) we get

$S(R)=\left(\frac{c}{6}\right)\text{log}\left[\frac{4(b^{2}-r_{+}^{2})[b^{2}+(b^{2}-1)\gamma^{2}]}{(1+\gamma^{2})b^{2}r_{+}^{2}}\text{Cosh}^{2}(r_{+}t))\right]\
,$ (33)

$c$ is the central charge. At late time i.e., $t\rightarrow\infty$ then
$\text{Cosh}(r_{+}t)\sim e^{r_{+}t}$, the above equation can be approximated
as

$S(R)\sim\frac{c}{3}r_{+}t\ .$ (34)

As a result of equation (1), we can see that the EE of the Hawking radiation
in the absence of an island surface increases linearly with time and becomes
infinite at late times, resulting in the EGB black hole’s information paradox.
Next, We’ll show that at late periods, an island appears, and the Hawking
radiation’s EE in the presence of an island surface remains constant and
dominates after the Page time. The Page curve is obtained by combining the two
contributions.

###  4.2  Entanglement Entropy with Island

The assumption for the verification of the presence of the island in the EGB
BH-Bath System are :

Figure 3:  The Penrose diagram of EGB BH when there is an island $I$ indicated
in orange with endpoints $a_{-}$ and $a_{+}$.

  * •

The right wedge is far apart from the left wedge such that the s-wave
approximation can be applied.

  * •

The quantum matter’s von Neumann entropy throughout the entire region for the
$R\cup I$ is sufficiently large.

Using eq.(38), one may determine the von Neumann entropy, one can get the
expression for $S_{\text{generalized}}$. Extremizing the
$S_{\text{generalized}}$ with respect to $t_{a}$ yields i.e., $\frac{\partial
S_{\text{generalized}}}{\partial t_{a}}=0$ results in $t_{a}=t_{b}$. By
executing this result, one can easily conclude that

$\displaystyle S_{\text{generalized}}$ $\displaystyle=$
$\displaystyle\frac{\pi
a}{G_{N}}+\frac{c}{6}\text{log}\Bigg{[}\frac{1}{r_{+}^{2}}\sqrt{\frac{(b^{2}-r_{+}^{2})[b^{2}+(b^{2}-1)\gamma^{2}]}{b^{2}(1+\gamma^{2})}}\sqrt{\frac{(a^{2}-r_{+}^{2})[a^{2}+(a^{2}-1)\gamma^{2}]}{a^{2}(1+\gamma^{2})}}$
$\displaystyle\Bigg{(}\frac{1}{(b-r_{+})(f-r_{+})}\Bigg{\\{}\sqrt{\frac{(b-r_{+})}{(b+r_{+})}\left[\frac{(b+\gamma
r_{+})}{(b-\gamma
r_{+})}\right]^{\gamma}}\sqrt{\frac{(a-r_{+})}{(a+r_{+})}\left[\frac{(a+\gamma
r_{+})}{(a-\gamma r_{+})}\right]^{\gamma}}$
$\displaystyle\Bigg{(}(f-r_{+})(b+r_{+})\left[\frac{b-\gamma r_{+}}{b+\gamma
r_{+}}\right]^{\gamma}+(b-r_{+})(f+r_{+})\left[\frac{f-\gamma r_{+}}{f+\gamma
r_{+}}\right]^{\gamma}\Bigg{)}-2\Bigg{\\}}\Bigg{]}\ .$

The location of QES can be obtained by extremizing the above equation w.r.t
$a$

$a=r_{+}+\frac{1}{2r_{+}}\left(\frac{cG}{6\pi}\right)^{2}\left(\frac{b+r_{+}}{b-r_{+}}\right)\left(\frac{1+\gamma}{1-\gamma}\right)^{\gamma}\left(\frac{b-\gamma
r_{+}}{b+\gamma r_{+}}\right)^{\gamma}\ .$ (35)

Substituting this value in $S_{generalized}$, we have

$\displaystyle S_{generalized}$ $\displaystyle\simeq$ $\displaystyle\frac{\pi
r_{+}}{G_{N}}+\frac{c^{2}G_{N}}{72\pi^{2}}\left[\frac{1+\gamma}{1-\gamma}\right]^{\gamma}+\order{G_{N}^{2}}$
(36) $\displaystyle\simeq$ $\displaystyle
2S_{BH}+\frac{c^{2}G_{N}}{72\pi^{2}}\left[\frac{1+\gamma}{1-\gamma}\right]^{\gamma}+\order{G_{N}^{2}}\
.$

The above expression has some notable characteristics. Aside from the first
item $S_{BH}$, the phrase includes universal adjustments to the Hawking
entropy of a BH. So, these corrections can be thought of as quantum gravity
signals, this is a surprise. The information loss paradox is solved by the
above conclusion yielding the proper Page curve and has a lovely mutual
information interpretation.

##  5 Page Curve

Figure 4: Page curves of an eternal BTZ black hole and for EGB BH for various
values of Gauss-Bonnet coupling($\alpha$).

  * •

Page time($t_{\text{page}}$) : Page time is the point at which Hawking
radiation’s EE reaches its maximum. The entropy of an eternal black hole does
not alter after it. By comparing (34) and (36)

$\displaystyle t_{\text{page}}$ $\displaystyle=$
$\displaystyle\frac{6}{cr_{+}}S^{BH}+\frac{3}{cr_{+}}\frac{c^{2}G_{N}}{72\pi^{2}}\left[\frac{1+\gamma}{1-\gamma}\right]^{\gamma}$
(37) $\displaystyle=$
$\displaystyle\frac{3\beta}{2c\pi^{2}}S_{BH}+\frac{(cG_{N})}{96\pi^{4}}\left[\frac{1+\gamma}{1-\gamma}\right]^{\gamma}\
.$

In the above expression we have used the expression
$\beta=\frac{2\pi}{T}=\frac{4\pi^{2}}{r_{+}}$.

  * •

Scrambling time ($t_{\text{scr}}$): the Hayden-Preskill experiment [53]
establishes the scrambling time is the smallest time during which information
can be retrieved from Hawking radiation. Due to connections between the
island’s degree of freedom and radiation regions, the scrambling time is also
given by the time for information to enter the island in the entanglement
wedge reconstruction proposal [4]. One can compute the Scrambling time by

$\displaystyle t_{\text{scr}}$ $\displaystyle\equiv$
$\displaystyle|r_{*}(b)-r_{*}(a)|$ $\displaystyle=$
$\displaystyle\frac{1}{2r_{+}}\text{log}\left[\left(\frac{b-r_{+}}{b+r_{+}}\frac{a+r_{+}}{a-r_{+}}\right)\left(\frac{\Gamma
b+r_{+}}{\Gamma b-r_{+}}\frac{\Gamma a+r_{+}}{\Gamma
a+r_{+}}\right)^{1/\Gamma}\right]$ $\displaystyle\simeq$
$\displaystyle\frac{\beta}{2\pi}\text{log}\left(\frac{r_{+}}{cG_{N}}\right)+\cdots\
.$

The higher derivative terms do not affect scrambling time, similar to the BTZ
BH’s scrambling time.

##  6 Results and Discussion

In this paper, we have calculated the holographic EE to calculate the Page
curves of an evaporating EGB BH in higher derivative terms. In this paper, we
have considered $\order{R^{2}}$ terms in gravitational action and using that,
we have plotted the Page curves 4. As a result, when we consider the
$\order{R^{2}}$ terms, initially due to the absence of an island, we obtain
that there is linear dependence of time in the EE, resulting in an information
paradox for the EGB BH. Late in the game, an island appears, and the Hawking
radiation’s EE achieves a constant value equal to double the BH’s Bekenstein-
Hawking entropy, yielding Page curves for fixed values of the Gauss-Bonnet
coupling ($\alpha$). We have also plotted the EE $(S_{\text{EE}})$ with
respect to time $(t)$ with fixed parameter $(c=1,G_{N}=1,r_{+}=1)$ for BTZ
black hole and then $\alpha=0.1$ and $0.2$ for EGB BH. From the figure 4, it
is clear that the Page curves change towards later times when Gauss-Bonnet
coupling ($\alpha$) rises in this example. We have also studied the page time
and the scrambling time in BTZ and the EGB case. For consistency check, for
$\alpha\rightarrow 0$, we get a similar expression as in the BTZ case from the
EGB case.

As we have discussed in the section, we have considered some specific choices
for $\phi(r)$ and $\lambda=0$. It is interesting to investigate what is the
expression for the function $f(r)$ if we considered the case with $\lambda\neq
0$. We will come back to these issues later on.

## Acknwoledgment

I am indebted to Prasanta K. Tripathy for a careful manuscript reading.

##  Appendix A Expressions for distances

The Expressions for distances are

$\displaystyle\ell(a_{+},b_{+})$ $\displaystyle=$
$\displaystyle\Bigg{[}\frac{1}{r_{+}^{2}}\sqrt{\frac{(b^{2}-r_{+}^{2})[b^{2}+(b^{2}-1)\gamma^{2}]}{b^{2}(1+\gamma^{2})}}\sqrt{\frac{(a^{2}-r_{+}^{2})[a^{2}+(a^{2}-1)\gamma^{2}]}{a^{2}(1+\gamma^{2})}}\Bigg{(}\frac{1}{(b-r_{+})(f-r_{+})}$
(38)
$\displaystyle\Bigg{\\{}\sqrt{\frac{(b-r_{+})}{(b+r_{+})}\left[\frac{(b+\gamma
r_{+})}{(b-\gamma
r_{+})}\right]^{\gamma}}\sqrt{\frac{(a-r_{+})}{(a+r_{+})}\left[\frac{(a+\gamma
r_{+})}{(a-\gamma
r_{+})}\right]^{\gamma}}\Bigg{(}(f-r_{+})(b+r_{+})\left[\frac{b-\gamma
r_{+}}{b+\gamma r_{+}}\right]^{\gamma}$
$\displaystyle+(b-r_{+})(f+r_{+})\left[\frac{f-\gamma r_{+}}{f+\gamma
r_{+}}\right]^{\gamma}\Bigg{)}-2\text{Cosh}[r_{+}(t_{a}-t_{b})]\Bigg{\\}}\Bigg{]}^{\frac{1}{2}}\
,$ $\displaystyle\ell(a_{+},b_{-})$ $\displaystyle=$
$\displaystyle\Bigg{[}\frac{1}{r_{+}^{2}}\sqrt{\frac{(b^{2}-r_{+}^{2})[b^{2}+(b^{2}-1)\gamma^{2}]}{b^{2}(1+\gamma^{2})}}\sqrt{\frac{(a^{2}-r_{+}^{2})[a^{2}+(a^{2}-1)\gamma^{2}]}{a^{2}(1+\gamma^{2})}}\Bigg{(}\frac{1}{(b-r_{+})(f-r_{+})}$
(39)
$\displaystyle\Bigg{\\{}\sqrt{\frac{(b-r_{+})}{(b+r_{+})}\left[\frac{(b+\gamma
r_{+})}{(b-\gamma
r_{+})}\right]^{\gamma}}\sqrt{\frac{(a-r_{+})}{(a+r_{+})}\left[\frac{(a+\gamma
r_{+})}{(a-\gamma
r_{+})}\right]^{\gamma}}\Bigg{(}(f-r_{+})(b+r_{+})\left[\frac{b-\gamma
r_{+}}{b+\gamma r_{+}}\right]^{\gamma}$
$\displaystyle+(b-r_{+})(f+r_{+})\left[\frac{f-\gamma r_{+}}{f+\gamma
r_{+}}\right]^{\gamma}\Bigg{)}-2\text{Cosh}[r_{+}(t_{a}+t_{b})]\Bigg{\\}}\Bigg{]}^{\frac{1}{2}}=\ell(a_{-},b_{-})\
,$ $\displaystyle\ell(b_{+},b_{-})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{4(b^{2}-r_{+}^{2})[b^{2}+(b^{2}-1)\gamma^{2}]}{(1+\gamma^{2})b^{2}r_{+}^{2}}\text{Cosh}^{2}(r_{+}t_{b}))}\
.$ (40)

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