# A Note on Islands in Schwarzschild
Black Holes

Irina Aref’eva and Igor Volovich

###### Abstract

We consider evaporation of the Schwarzschild black hole and note that island
configurations do not provide a bounded entanglement entropy. The same remark
is valid also for some other static black holes. A proposal for improving the
situation is discussed.

##  1 Introduction

The entropy of Hawking radiation of black holes grows up to infinity during
evaporation and it is a manifestation of the information paradox [1, 2]. This
increase contrasts with Page’s hypothetical behavior, in which entropy
decreases after the so-called Page time [3, 4, 5] and which ensures the
unitarity of quantum mechanics.

An approach to treating the black hole information problem was suggested in
[6, 7, 8, 9, 10], where the “island formula” for the entanglement entropy of
Hawking radiation based on quantum extremal surfaces [11] has been proposed
and it has been argued that the entanglement entropy is limited during the
evaporation of black holes. According to the prescription of quantum extremal
surfaces the entanglement entropy is given, after renormalization111See [12,
13] for renormalization of the entanglement entropy in four-dimensional case.,
by the formula

$\quad S(R)=\min\left\\{\mathop{\mathrm{ext}}_{{\cal
I}}\left[\frac{\mathrm{Area}(\partial{{\cal I}})}{4G}+S_{\rm
matter}(R\cup{{\cal I}})\right]\right\\}\,.$ (1.1)

Here ${\cal I}$ is a region, called the ”island”, whose boundary area is
denoted by $\mathrm{Area}(\partial{\cal I})$ and $S_{\rm matter}$ is the von
Neumann entropy $S_{\rm vN}(R\cup I)$ of union of the island and the region
$R$, $G$ is Newton’s constant. An extremization on any possible island and
then taking of the configuration with the minimum entropy is assumed. As the
system evolves, the form of island can change dynamically and different
extreme surfaces can dominate at different times. In several models this
change of dominance provides the bound on $S(R)$ as time evolves [6, 7, 10, 9,
21, 14, 15, 20, 16, 19, 22, 17, 18].

Although the prescription of the quantum extremal surface was proposed in the
framework of holography [23, 24], the island rule is applicable to black holes
in more general theories. The formula was confirmed for some two dimensional
models [8, 9]. For two dimensional gravity the island rule has been derived by
making use of replica trick [25, 26, 27] and the island contribution has been
associated with replica wormholes [20, 21]. The Page curve for evaporating
black holes in JT gravity has also been studied in [28]. The wormhole
configurations in JT gravity have been discussed in [29]. For a further
development in low dimensional case see [30, 31, 32, 33, 34, 36, 35, 37], and
references therein, in particular, for asymptotically flat two-dimensional
spacetime see [38, 39, 40]. The black holes in four and higher dimensional
asymptotically flat spacetime have been considered in [14, 41, 42, 43, 44].

Using the island formula (1.1) for asymptotically flat eternal Schwarzschild
black holes in four spacetime dimension the saturation at late time of the
entanglement entropy to the value

$S_{{\cal I}}=\frac{2\pi
r_{h}^{2}}{G}+\frac{c}{6}\frac{b-r_{h}}{r_{h}}+\frac{c}{6}\log\frac{16r_{h}^{3}(b-r_{h})^{2}}{G^{2}b}$
(1.2)

has been found in [41]. Here $r_{h}$ is the gravitational radius of the
Schwarzschild black hole, $r_{h}=2GM$, $M$ is the black hole mass, $c$ is the
number of massless matter fields, $b$ defines the boundaries of the
entanglement regions in the right and the left wedges of the Schwarzschild
geometry. The following inequalities are assumed: $\sqrt{G}\ll r_{h}\ll b$.
The entropy corresponding to non-island configuration dominates at small time
and increases with time linearly

$S_{n{\cal I}}\simeq\frac{c}{6}\frac{t}{r_{h}}.$ (1.3)

Equalizing the entropy without island with the entropy with island under
condition of dominating the first term in (1.2), one estimates the Page time
[41]

$t_{Page}\sim\frac{6\pi r_{h}^{3}}{c\,G}.$ (1.4)

The above considerations, as noted in [41], do not resort neither to
holographic correspondences, i.e. embedding into higher-dimensional AdS
spacetime, nor to coupling with an auxiliary system to absorb the radiation.

In this paper we note that while applying the above estimates to the
evaporation of a black hole, the second term $c\,b/6r_{h}$ in (1.2) plays an
important role and becomes dominant for small $r_{h}$. Thus, the entropy
starts to increase with decreasing mass of the black hole. Or in other words,
although the inclusion of the island and the extremization of it’s location
leads to the time-independent entropy of the eternal black hole at late times,
when the black hole evaporates to the Schwarzschild radius
$r_{h}\ll(c\,G\,b/24\pi)^{1/3}$ entropy grows explosively over time.

Let us discuss the origin for the appearance of the term $c\,b/6r_{h}$ in
(1.2) which is singular for small $r_{h}=2GM$. We will see that the origin of
this term lies in using of the Kruskal coordinates which are singular for
small $G$. The island formula (1.1) has been derived in [9, 10] by using the
replica method and expanding the quantum gravity path integral over
gravitational constant $G$. The semi-classical expansion of the quantum
gravity path integral over $G$ is different from the standard semi-classical
expansion over the Planck constant because in the case of the $G$-expansion
the background metric itself depends on $G$. Moreover, the Schwarzschild
solution in the Kruskal coordinates is singular for small $G$ (or small $M$),
see a discussion of this question in [48]. As a result, the term $S_{\rm
matter}(R\cup{{\cal I}})$ is of the order $1/G$ for small $G$. Before
extremization the first term is of the same order and it is given by $2\pi
a^{2}/G$, here $a$ is a location of the island. However the extremization
procedure makes $a\approx r_{h}$ and $\,2\pi a^{2}/G=8\pi GM^{2}$, i.e. this
term becomes subdominant for small $G$ yielding leadership to the term
$c\,b/(12GM)$ 222In more detail, the entanglement entropy before extrimization
as a function of $a$ has the form  $S(a)=\frac{2\pi
a^{2}}{G}-\frac{c}{6}\left(\frac{a}{r_{h}}+\log\frac{a}{r_{h}}\right)+const.$
One can see that $S(a)$ is an increasing function for $a\geq r_{h}$ if
$r_{h}^{2}>cG/12\pi$. Therefore, the function has its minimum at $a=r_{h}$..

This is just an opposite behavior as compared with the estimation based on
taking only the first term $2\pi r_{h}^{2}/G$ into account in (1.2) and
leading to decreasing of entropy to zero with decreasing of the black hole
mass after the Page time $t_{Page}$ given by (1.4). By taking these two terms
into account different scenarios depending on the initial parameters of the
evaporating black hole can be realized.

  * •

In first scenario, before the Page time $t_{Page}$ the entropy increases. Then
the entropy decreases for some time, but then at the moment $t_{expl}$ an
explosive growth of the entropy begins. One can say that time evolution
follows the anti-Page curve. This anti-Page part of evolution ends with a blow
up at the point $t_{blow}$, where the mass of the black hole is completely
lost, see schematic plot in Fig.1.A.

  * •

For another scenario, the period of decreasing entropy disappears, see
Fig.1.B. In this case the initial increase of the entropy, inherent in the
configuration without an island, at moment $t_{expl}$ is replaced by explosive
behavior up to $t_{blow}$, typical for an island configuration with a low
black hole mass. This blow up continues until complete evaporation at the
moment $t_{blow}$. In this case, the Page point, where the increase of the
entropy changes to decrease disappears, altogether, and the anti-Page point,
where the rate of growth of entropy changes, appears.

  * •

For more special case, Fig.1.C. the no-island configuration contribution
dominates all the time till total evaporation.

We also consider the question when the effect of an explosive growth of the
entropy is behind the Planck scale. A regularization of equation (1.2) at
$r_{h}\to 0$ is discussed.

$\Large{t_{Page}}$$\Large{t_{expl}}$$\Large{t_{blow}}$$\Large{t_{Anti-
Page}}$$\Large{t_{blow}}$$\Large{t_{blow}}$

A) B) C)

Figure 1: The entropy with the island increases in the end of evaporation and
becomes infinite at $t_{blow}$. A) Increasing starts after some time of
decreasing. B) There is no decreasing period. The entropy increases after the
intersection point $t_{Anti-Page}$. C) All the time $S_{n{\cal I}}<S_{{\cal
I}}$.

The paper is organized as follows. In the setup Section 2 we remind the
formula for the entanglement entropy for configurations without and with an
island for two sided eternal 4-dimensional Schwarzschild black hole obtained
in [41]. In Section 3 we study the exchange of dominance between different
configurations with an island and without an island for the evaporating black
hole in details, especially in the end of evaporation. Special attention is
paid to the localization of the occurrence of quantum effects in this process.
In Section 4 we study a regularization of previous calculations that permits
to consider the total evaporation of the black hole.

##  2 Setup

###  2.1 Two-sided black hole

$\Large{R_{-}}$$\Large{R_{+}}$$\Large{{\cal
I}}$$\Large{b_{+}}$$\Large{a_{+}}$$\Large{a_{-}}$$\Large{b_{-}}$

Figure 2: The island configuration for two-sided black hole considered in
[41].

One of the simple examples explicitly demonstrated how an island can help to
make the bounded entanglement entropy of the Hawking radiation is the two-
sided black hole [41]. The island formula for the generalized entropy consists
of two parts

$S_{gen}=S_{gr}+S_{vN}.$ (2.1)

the gravity part $S_{gr}$ that is associated with a nontrivial quantum
extremal surface, island, and the matter (radiation) von Neumann entropy
$S_{vN}$.

One supposes that the radiation is located at the union of two regions $R_{+}$
and $R_{-}$, which are located in the right and left wedges in the Penrose
diagram (see Fig.2) near the null infinities, where the gravity is negligible.

The states that one considers are maximally symmetric, so the location of a
possible island is fixed by its position in $(t,r)$ coordinates and
effectively one deals with two-dimensional models with $(t,r)$ coordinates (or
deals only with s-modes) and computes the corresponding entanglement entropy
between several entangling regions using two dimensional answers [27].

It is convenient to work within the Kruskal coordinates [45] related with the
Schwarzschild coordinates $t,r$ as

$\displaystyle
U=-\sqrt{\frac{r-r_{h}}{r_{h}}}\,e^{-\frac{t-(r-r_{h})}{2r_{h}}},\qquad
V=\sqrt{\frac{r-r_{h}}{r_{h}}}\hskip 1.42262pte^{\frac{t+(r-r_{h})}{2r_{h}}}$
(2.2)

and by which the corresponding two-dimensional part of the Schwarzschild
metric is

$\displaystyle ds_{2-dim\,part\,Schw}^{2}=-W^{-2}dUdV,\qquad
W=\sqrt{\frac{r}{4r_{h}^{3}}}\,e^{\frac{r-r_{h}}{2r_{h}}}\,.$ (2.3)

For the configuration without islands the entanglement entropy $S_{n{\cal I}}$
of the Hawking radiation is identified with that in the region $R=R_{+}\cup
R_{-}$ and

$S_{n{\cal I}}=\frac{c}{6}\log d(\ell_{1},\ell_{2}),$ (2.4)

$d(\ell_{1},\ell_{2})$ is the geodesic distance between points $\ell_{1}$ and
$\ell_{2}$ given by

$\displaystyle
d(\ell_{1},\ell_{2})=\sqrt{\frac{\big{(}U(\ell_{2})-U(\ell_{1})\big{)}\big{(}V(\ell_{1})-V(\ell_{2})\big{)}}{W(\ell_{1})W(\ell_{2})}}.$
(2.5)

Here points $\ell_{1}$ and $\ell_{2}$ are located at $(t_{b},b_{+})$ and
$(-t_{b}+i2\pi r_{h},b_{-})$. The total entanglement entropy for this
configuration is given by

$S_{n{\cal
I}}=\frac{c}{6}\log\left[\frac{16r_{h}^{2}(b-r_{h})}{b}\cosh^{2}\frac{t_{b}}{2r_{h}}\right].$
(2.6)

At $t_{b}\gg b\,(>r_{h})$, the above result is approximated as

$S_{n{\cal I}}\simeq\frac{c}{6}\frac{t_{b}}{r_{h}}\,\qquad r_{h}=2GM$ (2.7)

and grows linearly in time. At the late times $t\gg\frac{r_{h}^{3}}{cG}$ this
entropy becomes much larger than the black hole entropy, and this contradicts
with the finiteness of the von Neumann entropy for a finite-dimensional black
hole system. In such a case an island is expected to emerge [41].

For the configuration with the island, presented in Fig.2, the entanglement
entropy for the conformal matter is given by

$\displaystyle S_{\text{matter}}$ $\displaystyle=$
$\displaystyle\frac{c}{3}\log\frac{d(a_{+},a_{-})d(b_{+},b_{-})d(a_{+},b_{+})d(a_{-},b_{-})}{d(a_{+},b_{-})d(a_{-},b_{+})}.$
(2.8)

Locations of $a_{\pm}$ and $b_{\pm}$ are indicated in Fig.2 and
$d(\ell_{1},\ell_{2})$ is given by (2.5).Supposing that $\sqrt{G}<r_{h}\ll b$
and the extremizing of the total entropy about the location of the island, one
gets [41] a unique island location, that is $t_{a}=t_{b}$ and
$a=r_{h}+r_{h}X^{2}(b,G,c)$, where $X^{2}<<1$. At this configuration the total
entanglement entropy is given by (1.2). The main claim of [41], see also [43],
is that although at early times one has a linear growth, the island comes to
rescue the unitarity at late times in agreement with the Page curve.
Equalizing the entropy without island $S_{n{\cal I}}$ with the entropy
$S_{{\cal I}}$ with island one estimates the Page time given by (1.4). The
mutual information between the island and the region of collecting the Hawking
radiation has been considered in [46] and the subsystem volume complexity in
this context has been considered in [47].

Note that the consideration in [41] and presented above is relied on the
assumptions, that Hawking radiation has no gravitational interaction and the
main contribution to the matter von Neumann entropy comes from the
entanglement between s-wave modes of the quantum field, so one can reduce the
theory into two dimensional conformal field theory.

###  2.2 Time dependence of the mass of the evaporating black hole

In four dimensions due to radiation the mass $M$ of the black hole is reduced
as [4]

$M(t)=\frac{r_{0}}{2G}\left(1-\frac{24\alpha\,c\,Gt}{r_{0}^{3}}\right)^{1/3},$
(2.9)

where $\alpha$ is a constant dependent on the spin of the radiating particle,
$c$ is the number of massless matter fields and $r_{0}$ is the Schwarzschild
radius at $t=0$. The semiclassical estimate of the black hole lifetime is

$t_{evaporate}=\frac{r_{0}^{3}}{24c\,\alpha G}.$ (2.10)

##  3 Time dependence of the entanglement entropy of the evaporating black
hole

In this section using equation (1.2) we analyze what happens, when the black
hole is losing its mass. We consider this process adiabatically just
supposing, that the dependence of the entanglement entropy on time is defined
by dependence of mass of black hole on time $M(t)$ considered in [41]. From
formula (1.2) one sees, that for small $r_{h}$, the term $cb/6r_{h}$ in (1.2)
dominates and in this case the entropy increases when mass of the black hole
goes to zero. We will show that just this increasing leads to so called anti-
Page time dependence of entropy of the system.

Figure 3: Dependence of $S_{Island}$ on $M$ (green lines). The dashed lines
show $S^{\prime}_{Island,M}$. The red dashed line shows the location of the
Planck mass.

The typical dependence of the entanglement entropy (1.2) on mass is presented
in Fig.3. We see that this dependence has a minimum located for large $b$,
$b>r_{h}$ at

$M_{min}=\frac{1}{4}\left(\frac{bc}{3\pi G^{2}}\right)^{1/3}.$ (3.1)

This minimum can be realized outside of the Planck domain, i.e.

$M_{min}>M_{Planck}\simeq 1/\sqrt{G},$ (3.2)

that corresponds in the used approximation to

$\displaystyle\sqrt{G}$ $\displaystyle<$
$\displaystyle\frac{1}{64}\frac{bc}{3\pi}.$ (3.3)

In all our plots $c=3$, and therefore should be $\sqrt{G}<\frac{b}{64\pi}.$

One can compare the entropy with an island and the entropy for a configuration
without an island, see Fig.4. We see, that for a given mass of the black hole,
after some time, depending on the mass of the black hole, the entropy without
island (red lines with increasing thickness for increasing time) reaches the
generalized entropy for the configuration with the island (the green line).
This is true for both increasing and decreasing branches of the entanglement
entropy with the island. This observation gives the Page time, where two
entropies are equalized. In the cases of cyan points indicated in Fig.4 the
time dependence of the entropy for the eternal black has a desirable form
[41]. In these cases dependence of the Page time on mass of the black hole is
given by

$t_{Page}=\frac{48\pi M^{3}}{c\,G^{2}}.$ (3.4)

However, if the red line intersects the green line at $M$, that is less then
corresponding $M_{cr}$ (two red points at Fig.4), the increasing is preserved,
only the slopes of the increase change. We call this point the anti-Page one,
see the schematic plot presented in Fig.1.B.

$\Large{M_{cr,2}}$$\Large{M_{cr,1}}$$\Large{M_{Pl}}$

Figure 4: Dependence of the entropy for the configuration with the island,
$S_{{\cal I}}$ (green line), and the configuration without the island,
$S_{n{\cal I}}$ (red lines), on $M$. Red lines of different thickness
correspond to different $t_{b}$ for the case of $S_{n{\cal I}}$. Green lines
of different thickness correspond to different $b$ (here $b=150$ and $500$)
for the case of $S_{{\cal I}}$. The gray lines show the derivative of
corresponding $S_{{\cal I}}$ with respect to $M$. These derivatives are equal
to zero at $M_{cr,1}$ and $M_{cr,2}$ for $b=150$ and $500$, respectively. We
see that $M_{cr,2}>M_{cr,2}$. The points show the intersections of the green
lines with the red ones: darker cyan points show that at these points the
increasing of entropy with time changes to decreasing, while in the red points
the increasing is preserved, only the slopes of the increase change. The
magenta dot shows the intersection at masses less than the Planck mass.

Now we would like to consider the modification of the Page curve for the black
hole evaporation following equation (2.9). In plots Fig.5 and Fig.6 the
competition between two entropies is shown. The position of the Page time is
indicated by the cyan line. A new feature arises at the end of evaporation. As
was mentioned in the Introduction, due to the presence of the term inverse of
$r_{h}$ the island entropy starts to increase and blows up in the end of
evaporation. This period of evolution may obscure behind the Planck scale or
may not, depending on the parameters of the theory.

$\Large{t_{Page}}$$\Large{t_{blow}}$$\Large{t_{expl}}$$\Large{t_{Pl}}$$\Large{t_{blow}}$

A B

Figure 5:  A) The green line shows the evolution of the configuration with the
island, and the red one – the configuration without island. The cyan line
indicates the Page time. B) Plot near the end of evaporation. The darker red
line shows the decreasing of mass of evaporating BH, the darker magenta line
shows the Planck mass, corresponding to $G=0.01$, i.e.
$M_{Plank}=1/\sqrt{G}=10$, $t_{Pl}$ shows the time when $M(t)=M_{Pl}$, and
$t_{expl}$ and $t_{blow}$ show the times when the entropy starts to increase
and increases to infinity, respectively.

It may happen, that the growth of the entropy of the configuration with the
island begins earlier than the two curves intersect, see Fig.6 and the
schematic picture presented at Fig.1. B.

$\Large{t_{Anti-Page}}$$\Large{t_{blow}}$

Figure 6: The green line shows the evolution of the configuration with the
island and the red one shows evolution of the configuration without island.
The thick red line indicates the anti-Page time. The darker red line shows the
decreasing of mass of evaporating BH, the darker magenta line shows the Planck
mass, $M_{Planck}=1/\sqrt{G}$. Here $G=0.1$ and $M_{Planck}=3.16$.

Moreover, the equalizing of two types of entropy may not appear before the
complete evaporation of the black hole, see Fig.7 and the schematic picture
presented at Fig.1. C. We see at Fig.7 that entropy for the island
configuration begins to increase at some time and continues to exceed the
entropy of the non-island configuration for all the time until the end of
evaporation.

$\Large{t_{Pl}}$$\Large{t_{blow}}$

Figure 7:  The green line shows the evolution of the configuration with the
island and the red one configuration without island. We see that there is no
Page time at all, since the entropy for the configuration without island lies
below the entropy of the configuration with the island in all periods of
evaporation. The darker red line shows the decrease of mass of the evaporating
black hole, the magenta line shows the Planck mass, $M_{Planck}=1/\sqrt{G}$.
The red line lies below the green one. Here $G=0.1$ and $M_{Planck}=3.16$.

Summarising this considerations, we get the plots schematically presented in
Introduction, Fig.1. Note that it is interesting to know if explosion regime
is behind the Planck scale. For this purpose we indicate the Planck scale on
all our plots. There are several cases:

  * •

by the time $t_{expl}$ the black hole has not yet evaporated to the Planck
mass, see Fig.5. B;

  * •

by the time $t_{expl}$ the black hole has already evaporated to a mass less
than the Planck mass, see Fig.6.

In the first case we see the increasing region and therefore the island does
not solve the information problem. In the second case one can say, that in
this increasing entropy region one has to modify the semi-classical
approximation and include the quantum gravity corrections, since the point of
the increasing $t_{expl}$ is out of the semiclassical approximation. The same
concerns also to the case where the period of decreasing is absent. In this
case we have found the situation presented in Fig.8. C and D, i.e. the
increasing point is obscured by quantum effects.

$\Large{t_{Page}}$$\Large{t_{Pl}}$$\Large{t_{expl}}$$\Large{t_{blow}}$$\Large{t_{Page}}$$\Large{t_{expl}}$$\Large{t_{Pl}}$$\Large{t_{blow}}$$\Large{t_{Pl}}$$\Large{t_{Anti-
Page}}$$\Large{t_{blow}}$$\Large{t_{Anti-
Page}}$$\Large{t_{Pl}}$$\Large{t_{blow}}$

A B

C D

Figure 8:  The entropy of configurations with the island increasing in the end
of evaporation. A) Increasing starts after some decreasing period and by the
time $t_{expl}$ the black hole has evaporated to a mass less than the Planck
mass. B) Increasing starts after some decreasing period and at the moment
$t_{expl}$ the black hole is still more that the Planck mass. C) There is no
decreasing period and new increasing period starts at $t_{Anti-Page}$ and the
mass of the BH in this time is less than the Planck mass. D) There is no
decreasing period and new increasing period starts before the black hole
becomes of the Planck mass. In all of these plots, $t_{Pl}$ indicates the time
when the mass of the black hole becomes equal to the Planck mass.

##  4 Time dependence of the entanglement entropy of the evaporating black
hole in thermal coordinates

As we have seen in the previous section, time dependence of the entanglement
entropy of the evaporating black hole in the end of evaporation has singular
behaviour. This behaviour is related with singular behaviour of the Kruskal
coordinates in the limit of the black hole mass $M\to 0$, see discussion of
this problem in [48]. One can remove this singularity using thermal
coordinates [48], that provide a regularization of the Kruskal coordinates
near $M=0$. These coordinates are introduced as

$\displaystyle{\mathscr{U}}$ $\displaystyle=$
$\displaystyle-e^{-\frac{t-(r-r_{h})}{B}}\left(\frac{r-r_{h}}{r_{h}}\right)^{\frac{r_{h}}{B}},$
(4.1) $\displaystyle{\mathscr{V}}$ $\displaystyle=$ $\displaystyle
e^{\frac{t+(r-r_{h})}{B}}\left(\frac{r-r_{h}}{r_{h}}\right)^{\frac{r_{h}}{B}},$
(4.2)

where

$B=(4M+\mu)G,$ (4.3)

and $\mu$ is a positive constant. In these coordinates the standard
Schwarzschild solution has the temperature $T=1/2\pi B$. Two dimensional part
of the Schwarzschild metric is

$\displaystyle ds_{2-dim\,part\,Schw}^{2}$ $\displaystyle=$
$\displaystyle\frac{\,d{\mathscr{U}}d{\mathscr{V}}}{{\mathscr{W}}^{2}},\qquad{\mathscr{W}}=\frac{1}{B}\left(\frac{{\mathscr{U}}{\mathscr{V}}}{(1-\frac{2GM}{r})}\right)^{1/2},$
(4.4)

compare with (2.3). For the configuration without islands, by analogy with
(4.5), the entanglement entropy of the Hawking radiation is identified with
that in the region $R=R_{+}\cup R_{-}$ and it is

$S_{n{\cal I}}=\frac{c}{6}\log{\cal D}(\ell_{1},\ell_{2}),$ (4.5)

where $d(\ell_{1},\ell_{2})$ is the geodesic distance between points
$\ell_{1}$ and $\ell_{2}$ given by

$\displaystyle{\cal D}(\ell_{1},\ell_{2})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\left({\mathscr{U}}(\ell_{2})-{\mathscr{U}}(\ell_{1})\right)\left({\mathscr{V}}(\ell_{1})-{\mathscr{V}}(\ell_{2})\right)}{{\mathscr{W}}(\ell_{1}){\mathscr{W}}(\ell_{2})}}.$
(4.6)

Here points $\ell_{1}$ and $\ell_{2}$ are located at $(t_{b},b_{+})$ and
$(-t_{b}+i\pi B,b_{-})$. The regularized entanglement entropy for the
configuration presented in Fig.2 is given by

$S_{{\cal I},reg}=\frac{2\pi a^{2}}{G}+\frac{c}{3}\log{\cal
L}_{reg}(a,b,t_{a},t_{,}b,r_{h},\mu),$ (4.7)

where

$\displaystyle{\cal L}_{reg}(a,b,t_{a},t_{,}b,r_{h},\mu)^{2}={\cal
R}_{reg}(a,b,t_{a},t_{,}b,r_{h},\mu)=\frac{{\cal D}(a_{+},a_{-}){\cal
D}(b_{+},b_{-}){\cal D}(a_{+},b_{+}){\cal D}(a_{-},b_{-})}{{\cal
D}(a_{+},b_{-}){\cal D}(a_{-},b_{+})}.$ $\displaystyle{\cal
L}_{reg}=\frac{{\cal D}(a_{+},a_{-}){\cal D}(b_{+},b_{-}){\cal
D}(a_{+},b_{+}){\cal D}(a_{-},b_{-})}{{\cal D}(a_{+},b_{-}){\cal
D}(a_{-},b_{+})}.$

The explicit expression for ${\cal R}_{reg}$ is given by

$\displaystyle{\cal
R}_{reg}=\frac{16B^{4}(a-r_{h})(b-r_{h})\cosh^{2}\left(\frac{t_{a}}{B}\right)\cosh^{2}\left(\frac{t_{b}}{B}\right)\left(e^{\frac{a-b}{B}}\left(\frac{a-r_{h}}{b-r_{h}}\right)^{\frac{r_{h}}{B}}+e^{\frac{b-a}{B}}\left(\frac{b-r_{h}}{a-r_{h}}\right)^{\frac{r_{h}}{B}}-2\cosh\left(\frac{t_{a}-t_{b}}{B}\right)\right)^{2}}{abG^{2}\left(e^{\frac{a-b}{B}}\left(\frac{a-r_{h}}{b-r_{h}}\right)^{\frac{r_{h}}{B}}+e^{\frac{b-a}{B}}\left(\frac{b-r_{h}}{a-r_{h}}\right)^{\frac{r_{h}}{B}}+2\cosh\left(\frac{t_{a}+t_{b}}{B}\right)\right)^{2}}{\color[rgb]{1,0,0}.}$
(4.8)

For $\mu=0$ this expression reproduces the corresponding expression from [41]
and as we have seen in the previous Section 3 does not admit the finite limit
for $r_{h}=0$.

To get the limit $r_{h}\to 0$ of the regularized entanglement entropy
$S_{{\cal I},reg}$, we take the limit $r_{h}\to 0$ in the expression (4.8),
and then find the regularized entanglement entropy for zero mass by finding
extremum of the expression $S_{reg,0}$ varying over $a$ and $t_{a}$.
$S_{reg,0}$ is given as

$S_{reg,0}=\lim_{r_{h}\to 0}S_{reg,rh}=\frac{2\pi
a^{2}}{G}+\frac{c}{6}\log{\cal R}_{reg,0},$ (4.9)

where

$\displaystyle{\cal R}_{reg,0}$ $\displaystyle\equiv$ $\displaystyle{\cal
R}_{reg}\Big{|}_{r_{h}\to
0}=\frac{16B_{0}^{4}\cosh^{2}\left(\frac{t_{a}}{B_{0}}\right)\cosh^{2}\left(\frac{t_{b}}{B_{0}}\right)\left(e^{\frac{a-b}{B_{0}}}+e^{\frac{b-a}{B_{0}}}-2\cosh\left(\frac{t_{a}-t_{b}}{B_{0}}\right)\right)^{2}}{G^{2}\left(e^{\frac{a-b}{B_{0}}}+e^{\frac{b-a}{B_{0}}}+2\cosh\left(\frac{t_{a}+t_{b}}{B_{0}}\right)\right)^{2}},$
$\displaystyle B_{0}$ $\displaystyle=$ $\displaystyle\mu\,G.$ (4.10)

Extremizing on $t_{a}$ at large $t_{b},t_{b}\to\infty$ we get that
$t_{a}=t_{b}$ and at large $t_{b},t_{b}$

$\displaystyle{\cal
R}_{reg,0}\approx\frac{4B_{0}^{4}}{G^{2}}\left(\cosh\left(\frac{a-b}{B_{0}}\right)-1\right)^{2}.$

Assuming $a,b>B_{0}$ we have

$\displaystyle S_{0,reg,apr}$ $\displaystyle\simeq$ $\displaystyle\frac{2\pi
a^{2}}{G}+\frac{c}{6}\log\left[4G^{2}\mu^{4}\cosh\left(\frac{a-b}{B_{0}}-1\right)^{2}\right]{\color[rgb]{1,0,0}.}$
(4.11) $\displaystyle S_{{\cal I},reg}\Big{|}_{M=0}$ $\displaystyle\simeq$
$\displaystyle\frac{2\pi
a^{2}}{G}+\frac{c}{6}\log\left[4G^{2}\mu^{4}\cosh\left(\frac{a-b}{\mu\,G}-1\right)^{2}\right]{\color[rgb]{1,0,0}.}$
(4.12)

Extremizing (4.12) on $a$ we get

$\frac{c\sinh\left(\frac{a-b}{G\mu}\right)}{3\mu\left(\cosh\left(\frac{a-b}{\mu}\right)-1\right)}+4\pi
a=0.$ (4.13)

Finding solution of this equation numerically we get the dependence of
$S_{{\cal I},reg,r_{h}=0}$ on regularization parameter $\mu$, Fig.9 We see, as
should be, this expression is singular for $\mu\to 0$.

Figure 9: A) Dependence of the regularized entropy
$S_{Island}^{(B)},B=2r_{h}+\mu$ on $M$, here $\mu=0.01$. B) Dependence of the
zero mass regularized entropy $S_{Island}^{(B)}\Big{|}_{B=\mu}$ on
regularization parameter $\mu$.

##  5 Conclusion and discussion

We have considered the evaporation of the Schwarzschild black hole and note
that, generally speaking, the island configurations do not provide a bounded
entanglement entropy in the end of the black hole evaporation. Despite the
fact that including the island and the extremization about its location
results in saturation of the entropy for the eternal black hole, the
evaporating of black hole ends up by unbounded increasing of the entropy.

One can argue that the formula (1.2) applies for a black hole held in
equilibrium with radiation, and not for a freely evaporating black hole. The
second term $cb/6r_{h}$ can be interpreted as the entropy of infalling
radiation333We thank Henry Maxfield for providing this interpretation.

General arguments of possible inconsistency of islands in theories with long-
range gravity have been presented in [49].

In [48] it was suggested that the origin for such singular behavior lies in
the use of the Kruskal coordinates. The Kruskal coordinates are suitable to
the analytical continuation of the Schwarzschild metric but they are singular
in the limit of vanishing of the black hole mass $M\to 0$. Thus, the Kruskal
coordinates are not appropriate to describe small black holes. Depending of
the initial state parameters the unboundeness of the entanglement entropy can
occur inside or outside of the Planck length. Similar remarks about islands
and Page/anti-Page curves are applied to other static black holes, in
particular, to Reissner-Nordström black holes [50, 51], black holes in
modified gravity [43] as well as black holes in dS space-time [52].

We have considered the regularization of applying the quantum extremal surface
prescription to gravity theory with matter fields to the Schwarzschild black
hole metric in $r_{h}\to 0$ using thermal coordinates [48]. This consideration
permits to consider the entanglement entropy of island configurations in the
end of evaporation of the black hole, i.e. take the limit $r_{h}\to 0$. This
regularization can be applied also to other static black holes. It would be
also interesting to consider the modification of the Page curve in BCFT models
of black hole [53, 54, 55, 56] and holographic moving mirror in the end of
evaporation [57, 58].

The consideration of small black hole presented here can be generalized to the
Reissner-Nordström black holes [50, 51], charged dilaton black holes in non-
asymptotically flat cases [60, 59] as well as to black holes in (A)dS [52].
The latter are of particular interest in the context of possible
phenomenological applications in the condense matter physics [62, 61] and
quark gluon plasma [63]. In particular, the entropy for the dilaton charged
black hole contains the $\log b/r_{h}$ term444We thank Hyun-Sik Jeong for
taking our attention to this fact..

As a general remark, let us note that in [64] the black hole information
problem has been considered as a particular example of the fundamental
irreversibility problem in statistical physics. It was pointed out that
similar problem occurs when we study ordinary gas or the formation of the
ordinary black body and its thermal radiation. Actually, one has to give a
quantum mechanical explanation for the emergence of the second law of
thermodynamics in macroscopic systems. In this context one can say that the
Boltzmann equation for the one-particle distribution function in the kinetic
theory of gases is an analog of the extrimization equation for the
entanglement entropy with island.

It can be assumed that if we take into account more detailed information about
the dynamic properties of matter in the radiation region (not limited to the
$s$-mode only) and use a more structured model of islands, perhaps we will be
able to more accurately reconstruct the fine grained entropy of quantum
gravity and avoid the problem with entropy growth in the end of the black hole
evaporation.

## Acknowledgements

We would like to thank Dmitry Ageev, Michael Khramtsov, Kristina Rannu,
Timofey Rusalev and Pavel Slepov for useful discussions. We would also like to
thank Hyun-Sik Jeong, Nori Iizuka, Henry Maxfield and Suvrat Raju for comments
and correspondences.

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