# The averaged null energy condition on holographic evaporating black holes

Akihiro Ishibashi<EMAIL_ADDRESS>Department of Physics and Research
Institute for Science and Technology, Kindai University, Higashi-Osaka
577-8502, JAPAN  Kengo Maeda<EMAIL_ADDRESS>Faculty of
Engineering, Shibaura Institute of Technology, Saitama 330-8570, JAPAN

###### Abstract

We examine the averaged null energy condition (ANEC) for strongly coupled
fields, along the event horizon of an evaporating black hole by using the
AdS/CFT duality. First, we consider a holographic model of a $3$-dimensional
evaporating black hole with a perturbed 4-dimensional black droplet geometry
as the bulk dual, and investigate how negative energy flux going into the
boundary black hole horizon appears. We show that the ingoing negative energy
flux always appears at the boundary black hole horizon when the horizon area
decreases. Second, we test the ANEC in a holographic model whose boundary
geometry is a 4-dimensional asymptotically flat spacetime, describing the
formation and subsequent evaporation of a spherically symmetric black hole. By
applying the “bulk-no-shortcut principle”, we show that the ANEC is always
satisfied when the local null energy is averaged with a weight function along
the incomplete null geodesic on the event horizon from beginning of the
formation to the final instant of the black hole evaporation. Our results
indicate that the total ingoing negative energy flux is compensated by a large
amount of positive energy flux in the early stage of the black hole formation.

##  I Introduction

Black hole evaporation is one of the most mysterious phenomena in quantum
gravity. Hawking’s discovery [1] that a black hole evaporates by emitting
thermal radiation is based on quantum field theory in a fixed curved
background. There have been a number of calculations of the vacuum expectation
value of the stress-energy tensor for quantum fields on various black hole
backgrounds. In particular, negative energy flux going into the black hole
horizon was found in a two-dimensional model of gravitational collapse for the
Unruh vacuum state [2, 3]. This effect is consistent with the thermal
radiation from the black hole observed at spatial infinity, according to the
energy conservation. Because of the technical difficulties, however, most of
the calculations are restricted to free field theories with no interaction.

For interacting fields such as a strongly coupled field, the AdS/CFT
correspondence [4] provides a useful tool to investigate the Hawking radiation
on a fixed black hole background. So far, several models of AdS bulk solutions
associated with boundary black holes have been numerically constructed [5, 6,
7, 8, 9]. The black droplet is such an AdS bulk black hole solution in which
the boundary black hole horizon is disconnected from the bulk horizons [10].
When there are no non-extremal bulk horizons isolated from the AdS boundary,
the black droplet solution is regarded as the gravity dual to the Unruh vacuum
state in the boundary field theory, as the negative energy induced by a vacuum
polarisation effect decays to zero at the spatial infinity [5].

One peculiar behavior in strongly coupled fields in Unruh vacuum state is that
there is no negative energy flux through the horizon, except two dimensional
conformal field theory [11] 111On the non-equilibrium thermal state, heat flow
between two boundary black holes occurs for a non-stationary four-dimensional
bulk solution [12].. This would be explained by the formation of quantum
“hair” around the black hole [5]. Due to its strong attractive self-
interaction, the quantum field tends to collapse into the black hole, and it
is balanced to the radiation pressure from the black hole. Although this
picture is quite interesting, we need further investigations on a more general
non-stationary black droplet solution containing a time-dependent boundary
black hole, since most of the calculations so far made are based on the
assumption that the AdS bulk spacetime is time-independent.

In this paper, we consider evaporating black holes constructed on time-
dependent boundaries of some asymptotically AdS bulk spacetime, for which, as
shown in Sec. II, the null-null component of the boundary Ricci tensor
$R_{\mu\nu}l^{\mu}l^{\nu}$ is found to be negative, where $l^{\mu}$ is the
tangent vector to a null geodesic generator of the boundary black hole
horizon. This does not immediately imply that the null-null component of the
vacuum expectation value of the boundary stress-energy tensor
$\left\langle~{}T_{\mu\nu}~{}\right\rangle l^{\mu}l^{\nu}$ is necessarily
negative because the AdS/CFT correspondence itself does not provide any
dynamical equations of motion such as $R_{\mu\nu}l^{\mu}l^{\nu}=8\pi
G\left\langle~{}T_{\mu\nu}~{}\right\rangle l^{\mu}l^{\nu}$ on the boundary.
However, we can still expect that $\left\langle~{}T_{\mu\nu}~{}\right\rangle
l^{\mu}l^{\nu}$ is negative, or negative energy flux across the horizon
appears when the horizon area decreases from the perspective of the
conjectured generalized second law (GSL) as follows. The GSL states that the
sum of the gravitational and the matter entropy outside the black hole cannot
decrease even during the evaporation process [13, 1]. The GSL is widely
believed to be true and it was established for super-renormalizable quantum
field theories [14] (for a holographic setting, see also [15]). If the GSL is
always satisfied for any quantum field theories, ingoing negative energy flux
should appear on the horizon when the area decreases, in accord with the
energy conservation law. In this paper, we consider two such time-dependent
holographic models; one is a $3$-dimensional circular symmetric boundary black
hole and the other a $4$-dimensional spherically symmetric boundary black
hole. In the $3$-dimensional case, we will explicitly show that ingoing
negative energy flux always appears on the horizon when the area decreases, in
accord with the GSL picture.

For the $3$-dimensional time-dependent model, we start from the
$4$-dimensional static black droplet solution [16] as our background bulk
geometry, whose boundary metric is conformal to the $3$-dimensional Banados-
Teitelboim-Zanelli (BTZ) black hole [17]. Then, we consider metric
perturbations on this background so that the perturbed geometry describes a
regular bubble in the bulk and an evaporating black hole on the boundary. We
are concerned with what happens when we dynamically perturb the static BTZ
black hole with Unruh vacuum. As expected from the above observation, we show
that a negative energy flux going into the boundary black hole horizon always
appears when the horizon area decreases under the adiabatic approximation.
From the perspective of the AdS/CFT correspondence, the stress-energy tensor
on the boundary field theory depends on both of the boundary geometry and the
boundary condition deep inside the bulk. Since a regularity condition is
imposed on the bubble radius in our model as a natural boundary condition, the
stress-energy tensor is only determined by the boundary geometry. So, our
results indicate that the negative energy flux reflects the negative value of
$R_{\mu\nu}l^{\mu}l^{\nu}$ on the boundary black hole horizon.

Another key question regarding the negative energy flux is whether there is a
lower bound for the flux. Presumably, the absolute value of the total negative
energy flux would be bounded from above by the initial black hole mass by
energy conservation law. To evaluate the total amount of the flux, we consider
a holographic model in which the AdS boundary metric is conformal to a
$4$-dimensional asymptotically flat spherically symmetric metric which
describes an evaporating black hole formed by a gravitational collapse. The
null geodesic generator of the event horizon begins at a regular spacetime
point and ends at a zero mass naked singularity, where the evaporation is
completed. The behavior of the null geodesic congruence is very similar to the
one on the spatially compact $S^{3}$ universe in which the null geodesic
congruence expands from a point on the south pole and shrinks again to a point
on the north pole. In such spatially compact universes, it was shown that the
averaged null energy condition (ANEC) with a weight function is satisfied
along the null geodesic under the “no-bulk-shortcut condition”, stating that
no bulk causal curve can travel faster than the boundary achronal null
geodesics [18, 19]. When the null energy condition is satisfied in the bulk
spacetime, the no-bulk-shortcut condition is satisfied under the assumptions
that there are no pathological behavior such as a naked singularity formation
in the bulk and the boundary [20]. By applying the no-bulk-shortcut condition
to the holographic model of the evaporating black hole, we show that the ANEC
is satisfied.

The paper is organized as follows. In Section II, we consider the
$3$-dimensional evaporating black hole on the boundary of the perturbed
$4$-dimensional static black droplet solution and derive the negative energy
flux going into the boundary black hole horizon. In Section III, we derive the
ANEC with a weight function in a background of a $4$-dimensional evaporating
black hole in the context of the AdS/CFT correspondence. Section IV is devoted
to summary and discussions.

##  II $3$-dimensional evaporating black hole and negative energy flux

In this section, we construct a $4$-dimensional asymptotically locally AdS
spacetime with a $3$-dimensional time-dependent black hole on the AdS
boundary, describing the evaporation of a black hole. We assume that the
evaporating process of our boundary black hole proceeds very slowly so that
the boundary black hole can be viewed as almost static. Then, the
corresponding 4-dimensional bulk dual should also be well described by an
almost static or stationary black droplet geometry which has no non-extremal
bulk horizon isolated from the boundary black hole horizon. We construct such
a bulk geometry by perturbing the exact background solution of a static or
stationary black droplet.

###  II.1 A holographic model

We consider an analytic four-dimensional black droplet solution of the four-
dimensional vacuum Einstein equations with a negative cosmological constant
[16], in which there is a bubble of nothing deep inside the bulk, and the
boundary metric is conformal to the BTZ black hole. The metric of the
$4$-dimensional black droplet solution is written by

$\displaystyle
ds^{2}=\frac{d\rho^{2}}{f(\rho)}+\frac{\rho^{2}r_{0}^{2}}{r^{2}l^{2}}\left[-\frac{r^{2}-r_{0}^{2}}{l^{2}}dt^{2}+\frac{l^{2}dr^{2}}{r^{2}-r_{0}^{2}}+L^{2}\frac{r^{2}f(\rho)}{\rho^{2}}d\varphi^{2}\right],\qquad
f(\rho)=\frac{\rho^{2}}{L^{2}}+1-\dfrac{\mu_{0}}{\rho}\,,$ (1)

where $L$ and $l$ denote the AdS curvature length in the bulk and that in the
boundary, respectively, and where $\mu_{0}$ describes the mass parameter given
by

$\displaystyle{}\mu_{0}:=\rho_{0}\left(\dfrac{\rho_{0}^{2}}{L^{2}}+1\right)\,.$
(2)

Here, $\varphi$ is the periodic coordinate with period $2\pi$, and the circle
along $\varphi$ smoothly caps off at the bulk radius, $\rho=\rho_{0}$ when the
horizon radius $r_{0}$ of the boundary BTZ black hole is given by

$\displaystyle r_{0}=\frac{2\rho_{0}lL}{3\rho_{0}^{2}+L^{2}}\,.$ (3)

In this case, a bubble appears at the radius, $\rho=\rho_{0}$ instead of the
non-extremal bulk horizon.

Since the boundary metric of Eq. (1) at infinity $\rho\rightarrow\infty$ is
conformal to the static BTZ black hole, the solution describes a confining
vacuum in the dual field theory in the BTZ black hole background. Following
the procedure [21], the stress-energy tensor is calculated as

$\displaystyle
T_{tt}=-\dfrac{\mu_{0}r_{0}^{3}L\left(r^{2}-r_{0}^{2}\right)}{16\pi
G_{4}l^{4}r^{3}},\quad T_{rr}=\dfrac{L\mu_{0}r_{0}^{3}}{16\pi
G_{4}r^{3}(r^{2}-r_{0}^{2})}\,,\quad
T_{\varphi\varphi}=-\dfrac{\mu_{0}Lr_{0}^{3}}{8\pi G_{4}l^{2}r}\,,$ (4)

where $G_{4}$ is the 4-dimensional gravitational constant. It is clear that
there is no energy flux on the boundary black hole horizon, $r=r_{0}$, even
though the energy density outside the BTZ black hole is negative. In the
following subsections, we construct a time-dependent boundary evaporating
black hole geometry by considering perturbations of the static droplet
solution (1).

Figure 1: The Penrose diagram of the two-dimensional de Sitter spacetime. The
dot-dashed curve represents an imaginary surface of the matter of the
gravitationally collapse.

###  II.2 Metric perturbations in the 4-dimensional bulk

Under the coordinate transformation
$(r,t,\varphi)\rightarrow(\bar{r},\bar{t},\bar{\varphi})$,

$\displaystyle r=\frac{r_{0}}{\bar{r}},\quad
t=\frac{l^{2}\bar{t}}{r_{0}},\quad\varphi=\frac{l}{Lr_{0}}\bar{\varphi}\,,$
(5)

the metric (1) becomes the following warped product type metric that includes,
as a part, the $2$-dimensional de Sitter metric,

$\displaystyle ds^{2}$
$\displaystyle=f(\rho)d\bar{\varphi}^{2}+\dfrac{d\rho^{2}}{f(\rho)}+\rho^{2}\left[-(1-\bar{r}^{2})d\bar{t}^{2}+\dfrac{d\bar{r}^{2}}{1-\bar{r}^{2}}\right]\,.$
(6)

Note that the de Sitter horizon at $\bar{r}=1$ corresponds to the BTZ black
hole horizon at $r=r_{0}$ on the boundary, as can be seen in (1).

In order to consider perturbations on this geometry, it is more convenient to
take the double wick rotation:

$\displaystyle\bar{\varphi}=i\tau,\quad\bar{t}=i\phi,\quad\bar{r}=\cos\theta\,.$
(7)

Then the black droplet metric takes the standard form of the $4$-dimensional
Schwarzschild AdS metric:

$\displaystyle
ds^{2}=g_{ab}(y)dy^{2}dy^{b}+\rho^{2}\gamma_{ij}(z)dz^{i}dz^{j}=-f(\rho)d\tau^{2}+\dfrac{d\rho^{2}}{f(\rho)}+\rho^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\,,$ (8)

where here and hereafter we use the coordinate notation $y^{a}=(\tau,\rho)$
and $z^{i}=(\theta,\phi)$. On this background we perform metric perturbations
by applying the gauge-invariant formalism of [22]. For our purpose, it is
sufficient to consider the scalar(polar)-type metric perturbations, which can
be expanded in terms of the scalar harmonic functions ${\mathbb{S}}{}_{\bf k}$
on the unit $2$-sphere,

$\displaystyle\hat{D}^{i}\hat{D}_{i}{\mathbb{S}}{}_{\bf
k}+k^{2}{\mathbb{S}}{}_{\bf k}=0\,,$ (9)

where $\hat{D}_{i}$ denotes the derivative operator associated with the unit
$2$-sphere metric $\gamma_{ij}dz^{i}dz^{j}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}$. We can express the components $(\delta g_{ab},\delta g_{ai},\delta
g_{ij})$ of the scalar-type metric perturbations as

$\displaystyle\delta g_{ab}=\epsilon f_{ab{\bf k}}{\mathbb{S}}_{\bf
k},\quad\delta g_{aj}=\epsilon\rho f_{a{\bf k}}\hat{D}_{j}{\mathbb{S}}_{\bf
k}\,,\quad\delta g_{ij}=2\epsilon\rho^{2}(H_{L{\bf
k}}\gamma_{ij}{\mathbb{S}}_{\bf k}+H_{T{\bf
k}}{\mathbb{S}}_{\bf(k)}{}_{ij})\,,$ (10)

where $\epsilon$ is an infinitesimally small parameter and where
${\mathbb{S}}_{\bf k}{}_{ij}$ is defined by

$\displaystyle{\mathbb{S}}_{\bf(k)}{}_{ij}=\dfrac{1}{k^{2}}\hat{D}_{i}\hat{D}_{j}{\mathbb{S}}_{\bf
k}+\dfrac{1}{2}\gamma_{ij}{\mathbb{S}}_{\bf k}\,.$ (11)

Hereafter we omit the mode index ${\bf k}$, for the notational simplicity. We
introduce the vector field $X_{a}$ on the $2$-dimensional spacetime spanned by
$y^{a}=(\tau,\rho)$ by

$\displaystyle
X_{a}=\dfrac{\rho}{k}\left(f_{a}+\dfrac{\rho}{k}D_{a}H_{T}\right)\,,$ (12)

where $D_{a}$ is the derivative operator associated with the $2$-dimensional
metric $g_{ab}$. We find the following combinations are gauge invariant;

$\displaystyle
F=H_{L}+\dfrac{1}{2}H_{T}+\dfrac{1}{\rho}D^{a}\rho\,X_{a}\,,\quad
F_{ab}=f_{ab}+D_{a}X_{b}+D_{b}X_{a}\,.$ (13)

For convenience, we introduce

$\displaystyle X:={F_{\tau}}^{\tau}-2F,\quad Y={F_{\rho}}^{\rho}-2F\,.$ (14)

Furthermore, we assume that $\partial_{\tau}$ is a Killing vector on the
perturbed spacetime. Thus, we consider the static perturbations in the
background (8). This, in turn, implies that we are concerned with dynamical
but circular perturbations along $\varphi$ in the original black droplet
background via the double wick rotation (7) and the rescaling (5). In this
case, $X$, $F$, and ${F_{a}}^{b}$ are determined by $Y$ as

$\displaystyle
X=Y+2\dfrac{f}{f^{\prime}}Y\,,\quad{F_{\tau}}^{\tau}=-{F_{\rho}}^{\rho}=\dfrac{f}{f^{\prime}}Y^{\prime},\quad
F=-\dfrac{1}{2}\left(Y+\dfrac{f}{f^{\prime}}Y^{\prime}\right)\,,$ (15)

where the prime denotes the derivative with respect to $\rho$.

For later convenience, we choose the gauge

$\displaystyle f_{\rho}=f_{\rho\rho}=f_{\tau}=0\,.$ (16)

Then, $X_{\tau}=0$ for the static perturbation. By Eqs. (13), (15), and (16),
$X_{\rho}$, $H_{T}$, and $H_{L}$ are determined by $Y$ as

$\displaystyle
X_{\rho}=-\dfrac{1}{2\sqrt{f}}\displaystyle\int\sqrt{f}\,\dfrac{Y^{\prime}}{f^{\prime}}d\rho,\quad
H_{T}=k^{2}\int\dfrac{X_{\rho}}{\rho^{2}}d\rho\,,\quad
H_{L}=-\dfrac{1}{2}\left(Y+\dfrac{fY^{\prime}}{f^{\prime}}\right)-\dfrac{H_{T}}{2}-\dfrac{f}{\rho}X_{\rho}\,.$
(17)

The master variable $Y$ satisfies

$\displaystyle Y^{\prime\prime}+\alpha Y^{\prime}+\beta Y=0,$ (18)

where

$\displaystyle\rho^{2}ff^{\prime}\alpha=4x-\dfrac{4\rho^{2}}{L^{2}}-2x^{2}+\dfrac{16}{L^{2}}x\rho^{2}+\dfrac{4\rho^{4}}{L^{4}}\,,\quad\rho^{2}f\beta=-k^{2}+2\,,$
(19)

with $x:={\mu_{0}}/{\rho}$.

###  II.3 The boundary conditions in the bulk

Let us consider appropriate boundary conditions for our perturbations. We
first consider boundary conditions from the bulk view point, which we need to
impose at the bubble radius, $\rho=\rho_{0}$, and at the AdS infinity,
$\rho=\infty$. The former condition comes from the regularity condition
[23]222 The right-hand side of (20) comes from the fact that $\bar{\varphi}$
has the period $2\pi Lr_{0}/l$ via the rescaling (5). If $\xi^{2}$ is chosen
to be the norm squared of $\partial/\partial\varphi$, the r.h.s should be unit
since the $\varphi$ circle has period $2\pi$. ,

$\displaystyle\lim_{\rho\to\rho_{0}}\dfrac{\nabla_{M}(\xi^{2})\nabla^{M}(\xi^{2})}{4\xi^{2}}=\left(\dfrac{l}{Lr_{0}}\right)^{2}\,,$
(20)

where $\xi^{2}$ denotes the norm squared of the circular Killing vector
$\partial/\partial\bar{\varphi}$. Note that $\bar{\varphi}=i\tau$, and
therefore that the r.h.s is equivalent to square of the surface gravity
$\kappa=l/Lr_{0}$ of the Killing horizon with respect to
$\partial/\partial\tau$ in the Schwarzschild-AdS black hole (8). Up to
$O(\epsilon)$, this regularity condition reduces to

$\displaystyle f_{\tau\tau}=\alpha_{\tau\tau}(\rho-\rho_{0})^{2}+\cdots.$ (21)

This determines the boundary condition for $f_{\tau\tau}$ and it is derived
from Eqs. (13) and (17) as

$\displaystyle
f_{\tau\tau}=F_{\tau\tau}+ff^{\prime}X_{\rho}=-\dfrac{f^{2}}{f^{\prime}}Y^{\prime}-\dfrac{\sqrt{f}}{2}f^{\prime}\int^{\rho}_{\rho_{0}}\sqrt{f}\dfrac{Y^{\prime}}{f^{\prime}}d\rho\,,$
(22)

which is equivalent to the condition

$\displaystyle X_{\rho}(\rho_{0})=0\,.$ (23)

Note that there is another integration constant which remains undetermined for
$H_{T}$ in Eq. (17), while the integration constant of $X_{\rho}$ is
determined by the regularity condition, $X_{\rho}(\rho_{0})=0$. To determine
the integration constant for $H_{T}$, we impose the following condition at the
AdS boundary:

$\displaystyle\lim_{\rho\to\infty}\dfrac{H_{T}}{\rho^{2}}=0.$ (24)

This condition for $H_{T}$, together with our gauge condition (16) and the
coordinate transformation (5), implies that our perturbations deform the
boundary BTZ metric to the following form;

$\displaystyle
ds^{2}=e^{A(t,r)}\left(-\dfrac{r^{2}-r_{0}^{2}}{l^{2}}dt^{2}+\dfrac{l^{2}dr^{2}}{r^{2}-r_{0}^{2}}\right)+e^{B(t,r)}r^{2}d\varphi^{2}\,.$
(25)

Namely, the perturbed boundary black hole is time-dependent and circularly
symmetric as intended, but the coordinate location of the event horizon is
fixed at $r=r_{0}$ under our gauge.

###  II.4 Boundary conditions on the boundary

Let us turn to boundary conditions from the boundary view point, which are to
be imposed at the horizon of the 3-dimensional boundary black hole. For this
purpose, we take the double wick rotation (7) again and go back to the
background geometry (6). By doing so, we can now view that our metric
perturbations considered in the previous subsection II.2 are expanded
by—instead of the harmonic functions on the $2$-sphere defined in (9)—the mode
functions ${\mathbb{S}}_{\bf k}$ on the $2$-dimensional de Sitter spacetime
spanned by $(\bar{t},\bar{r})$, which now satisfy the wave equation below,

$\displaystyle\left[-\dfrac{\partial^{2}}{\partial{\bar{t}}^{2}}+(1-\bar{r}^{2})\left\\{\dfrac{\partial}{\partial\bar{r}}(1-\bar{r}^{2})\dfrac{\partial}{\partial\bar{r}}\right\\}+(1-\bar{r}^{2})k^{2}\right]{\mathbb{S}}_{\bf
k}=0\,.$ (26)

We set ${\mathbb{S}}_{\bf k}=e^{-i\omega\bar{t}}S_{\bf k}(\bar{r})$ so that
$S_{\bf k}(\bar{r})$ satisfies

$\displaystyle\left[(1-\bar{r}^{2})^{2}\dfrac{d^{2}}{d\bar{r}^{2}}-2\bar{r}(1-\bar{r}^{2})\dfrac{d}{d\bar{r}}+k^{2}(1-\bar{r}^{2})+\omega^{2}\right]S_{\bf
k}=0\,,$ (27)

and find that the general solution of $S_{\bf k}(\bar{r})$ is represented by
the associated Legendre functions, $P^{m}_{n}$ and $Q^{m}_{n}$ as

$\displaystyle S_{\bf k}=a\,P^{m}_{n}(\bar{r})+b\,Q^{m}_{n}(\bar{r}),\quad
m:=i\omega,\,\,n:=\frac{1}{2}(-1+\sqrt{1+4k^{2}})\,,$ (28)

where

$\displaystyle P^{m}_{n}(x)=\dfrac{e^{m\pi
i}}{\Gamma(1-m)}\left(\dfrac{1+x}{1-x}\right)^{m/2}F\left(-n,n+1,1-m;\dfrac{1-x}{2}\right),$
$\displaystyle Q^{m}_{n}(x)=\dfrac{\pi}{2\sin m\pi}\left[\cos
m\pi\,P^{m}_{n}(x)-\dfrac{\Gamma(n+m+1)}{\Gamma(n-m+1)}P^{-m}_{n}(x)\right].$
(29)

Here, note that we assume that $|m|$ is very small, since we consider the case
that the time evolution of the bulk geometry is very slow.

Since we are interested in the future horizon, $\bar{r}=1$ at the right-hand
side (r.h.s.) in Fig. 1, the ingoing boundary condition should be imposed as a
regularity condition. In terms of a retarded null coordinate $u$, $S_{\bf k}$
behaves near the horizon as

$\displaystyle{\mathbb{S}}_{\bf k}\sim e^{-i\omega u},\quad
u:=\bar{t}-\int\dfrac{d\bar{r}}{1-\bar{r}^{2}}\simeq\bar{t}+\dfrac{1}{2}\ln(1-\bar{r}).$
(30)

The ingoing mode comes from the left past horizon, $\bar{r}=-1$ in Fig. 1, as
an outgoing wave. So, we shall impose that $S_{\bf k}$ behaves near the past
horizon, $\bar{r}=-1$ at the left-hand side (l.h.s.) of Fig. 1 as

$\displaystyle{\mathbb{S}}_{\bf k}\sim e^{-i\omega u},\quad
u:=\bar{t}-\int\dfrac{d\bar{r}}{1-\bar{r}^{2}}\simeq\bar{t}-\dfrac{1}{2}\ln(1+\bar{r}).$
(31)

From the boundary condition (30) and the form of the associated Legendre
function (28), $S_{\bf k}\sim P^{m}_{n}(x)$ near the future horizon,
$\bar{r}=1$, and then, we obtain

$\displaystyle b=0.$ (32)

Using the linear transformation formulae for hypergeometric functions, $S_{\bf
k}$ behaves near the past horizon, $\bar{r}=-1$ as

$\displaystyle S_{\bf k}\sim P^{m}_{n}(\bar{r})\simeq e^{m\pi
i}\left(\dfrac{1+\bar{r}}{2}\right)^{\frac{m}{2}}\Biggl{[}\dfrac{\Gamma(m)}{\Gamma(-n)\Gamma(n+1)}\left(\dfrac{1+\bar{r}}{2}\right)^{-m}F\left(1+n-m,-n-m,1-m;\dfrac{1+\bar{r}}{2}\right)$
$\displaystyle+\dfrac{\Gamma(-m)}{\Gamma(1-m+n)\Gamma(-n-m)}F\left(-n,n+1,m+1;\dfrac{1+\bar{r}}{2}\right)\Biggr{]}\,.$
(33)

By imposing the boundary condition (31), we find that the first term should be
zero, and thus,

$\displaystyle n=N,\quad k^{2}=N(N+1)$ (34)

for an arbitrary positive integer $N$. Note that here, we do not consider a
particular class of the mode function, $k=0$ in which $H_{T}=0$ in Eq. (10).
In Sec. III we will consider a $4$-dimensional time-dependent boundary
geometry that describes a black hole formation and evaporation via Hawking
radiation, in which the energy flux comes in from past null infinity. Our
boundary conditions (32) and (34) may be viewed as boundary conditions that
correspond to such a black hole formation and evaporation induced via some
incoming flux from past infinity.

###  II.5 Negative energy flux on the evaporating black hole

In this section we derive the stress-energy tensor of the boundary field
theory by taking the Fefferman-Graham coordinate system and using the AdS/CFT
dictionary [21], and then we evaluate it on the horizon. Thanks to the gauge
choice (16), the Fefferman-Graham coordinates system

$\displaystyle
g_{MN}dx^{M}dx^{N}=\frac{L^{2}}{z^{2}}\left(dz^{2}+g_{mn}(x,z)dx^{m}dx^{n}\right),$
$\displaystyle g_{mn}(z,x)=g_{(0)mn}(x)+z^{2}g_{(2)mn}(x)+\cdots$ (35)

is derived from

$\displaystyle\rho(z)=\dfrac{L}{z}\left(1-\dfrac{1}{4}z^{2}+\dfrac{\mu}{6L}z^{3}+\cdots\right)\,.$
(36)

The stress-energy tensor $T_{mn}$ can be read off from the coefficient
$g_{(3)mn}(x)$ as

$\displaystyle T_{mn}=\dfrac{3L^{2}}{16\pi G_{4}}g_{(3)mn},$ (37)

where $G_{4}$ is the bulk gravitational constant. To derive the coefficient
$g_{(3)mn}(x)$, let us expand the master variable $Y$ near the infinity as

$\displaystyle
Y=a_{0}+\dfrac{a_{1}}{\rho}+\dfrac{a_{2}}{\rho^{2}}+\dfrac{a_{3}}{\rho^{3}}+\cdots.$
(38)

From Eq. (18), each coefficient $a_{n}~{}(n\geq 2)$ is determined by $a_{0}$
and $a_{1}$ as

$\displaystyle a_{2}=\dfrac{1}{2}a_{0}(k^{2}-2)L^{2},\qquad
a_{3}=\dfrac{1}{6}a_{1}(k^{2}-6)L^{2},$ $\displaystyle
a_{4}=\dfrac{1}{24}\\{a_{0}(24-14k^{2}+k^{4})L^{4}+18a_{1}\rho_{0}(L^{2}+\rho_{0}^{2})\\}\,,\cdots.$
(39)

Under the boundary conditions (23) and (24), $H_{T}$, $H_{L}$, and
$f_{\bar{\varphi}\bar{\varphi}}$ are expanded as a series of $1/\rho$ from
Eqs. (17) and (22) as

$\displaystyle
H_{T}=\dfrac{c_{1}k^{2}L}{4\rho^{2}}+\dfrac{a_{1}k^{2}L^{2}}{12\rho^{3}}+\cdots\,,$
(40) $\displaystyle H_{L}$ $\displaystyle=$
$\displaystyle\dfrac{c_{1}-a_{0}L}{2L}+\dfrac{L(k^{2}-2)(a_{0}L-c_{1})}{8\rho^{2}}+\dfrac{a_{1}(k^{2}-2)L^{3}-6c_{1}\rho_{0}(L^{2}+\rho_{0}^{2})}{24L\rho^{3}}+\cdots\,,$
$\displaystyle f_{\bar{\varphi}\bar{\varphi}}$ $\displaystyle=$
$\displaystyle\dfrac{c_{1}\rho^{2}}{L^{3}}+\dfrac{1}{4}\left\\{a_{0}(k^{2}-2)-\dfrac{2c_{1}}{L}\right\\}-\dfrac{a_{1}(k^{2}-2)}{6\rho}+\cdots\,,$
(41)

where the coefficient $c_{1}$ is defined as

$\displaystyle
c_{1}:=\int^{\infty}_{\rho_{0}}\sqrt{f}\,\dfrac{Y^{\prime}}{f^{\prime}}d\rho\,.$
(42)

Substituting (II.5) into (10) and expanding the metric as a series in $z$ with
the help of Eq. (36), we obtain the boundary metric,

$\displaystyle g_{(0)mn}dx^{m}dx^{n}$ $\displaystyle=$
$\displaystyle\dfrac{r_{0}^{2}}{l^{2}r^{2}}\left[r^{2}\left(1+\epsilon\dfrac{c_{1}{\mathbb{S}}_{\bf
k}}{L}\right)d\varphi^{2}+\left(1+\epsilon\dfrac{c_{1}-a_{0}L}{L}{\mathbb{S}}_{\bf
k}\right)\left(2dvdr-\dfrac{r^{2}-r_{0}^{2}}{l^{2}}dv^{2}\right)\right]$ (43)
$\displaystyle=:$ $\displaystyle\left(\dfrac{r_{0}}{r}\right)^{2}ds_{\rm
dBTZ}^{2}\,,$

where

$\displaystyle v:=t+\int\dfrac{l^{2}}{r^{2}-r_{0}^{2}}dr\,.$ (44)

Note that the stress-energy tensor for the boundary metric $ds_{\rm
dBTZ}^{2}$—which stands for the deformed BTZ metric—cannot be proportional to
the coefficient $g_{(3)mn}$, as $g_{(0)mn}$ is different from the metric
$ds_{\rm dBTZ}^{2}$ by a conformal factor, $(r_{0}/r)^{2}$. In general, for
the conformal transformation

$\displaystyle\hat{g}_{mn}dx^{m}dx^{n}=\Omega^{2}g_{mn}dx^{m}dx^{n}\,,$ (45)

the stress-energy tensor transforms as

$\displaystyle\hat{T}_{mn}=\frac{1}{\Omega}T_{mn}\,,$ (46)

for the 3-dimensional boundary spacetime. Since we are interested in the
stress-energy tensor on the horizon, $r=r_{0}$, the conformal factor is equal
to $1$, and then, the null-null component of the stress energy tensor for the
metric $ds_{\rm dBTZ}^{2}$ can be read off from the formula (37) as

$\displaystyle T_{vv}(v,r_{0})=\dfrac{\epsilon a_{1}L}{32\pi
G_{4}}\left(\partial_{v}^{2}{{\mathbb{S}}_{\bf
k}}-\dfrac{r_{0}}{l^{2}}\partial_{v}{{\mathbb{S}}_{\bf k}}\right)\,.$ (47)

The Ricci tensor for the deformed BTZ metric $ds_{\rm dBTZ}^{2}$ on the
horizon is also calculated as

$\displaystyle
R_{vv}(v,r_{0})=\epsilon\dfrac{c_{1}}{2L}\left(\dfrac{r_{0}}{l^{2}}\partial_{v}{{\mathbb{S}}_{\bf
k}}-\partial_{v}^{2}{{\mathbb{S}}_{\bf k}}\right)\,.$ (48)

So, the ratio between $T_{vv}(v,r_{0})$ and $R_{vv}(v,r_{0})$ is determined by
the dimensionless coefficient $\zeta$,

$\displaystyle\zeta:=\frac{-a_{1}}{c_{1}}\,.$ (49)

As shown in the proposition below, this coefficient $\zeta$ is always
positive, except the trivial case, $a_{1}=c_{1}=0$.

Proposition
For the mode functions ${\mathbb{S}}_{\bf k}$ satisfying Eq. (26) with (34),
$\zeta>0$ for all integers $N>1$, and $a_{1}=c_{1}=0$ for $N=1$.
proof)
In terms of the new variable, $u=\rho_{0}/\rho$, the master Eq. (18) can be
rewritten as

$\displaystyle(1-u)\\{\hat{\rho}_{0}^{2}(1+u)+(1+\hat{\rho}_{0}^{2})u^{2}\\}\\{2\hat{\rho}_{0}^{2}+(1+\hat{\rho}_{0}^{2})u^{3}\\}\ddot{Y}$
$\displaystyle\qquad-2u\\{-4\hat{\rho}_{0}^{2}+9\hat{\rho}_{0}^{2}(1+\hat{\rho}_{0}^{2})u+(1+\hat{\rho}_{0}^{2})u^{3}\\}\dot{Y}+(2-k^{2})\\{2\hat{\rho}_{0}^{2}+(1+\hat{\rho}_{0}^{2})u^{3}\\}Y=0\,,$
$\displaystyle\dot{Y}(1)=\dfrac{1-\dfrac{k^{2}}{2}}{1+3\hat{\rho}_{0}^{2}}Y(1)\,,$
(50)

where $\hat{\rho}_{0}=\rho_{0}/L$, and the dot denotes the derivative with
respect to $u$. The second equation comes from the regularity on the bubble
radius, $\rho=\rho_{0}$. Since the differential equation is linear, we can set
$Y(1)=1$ without loss of generality. First, consider the case $N>1$. Then,
$k^{2}>2$ and hence $\dot{Y}(1)<0$. Now suppose $Y$ would have a maximum at
$u=u_{m}~{}(0\leq u_{m}\leq 1)$. In this case, $Y(u_{m})>1$,
$\dot{Y}(u_{m})=0$, and $\ddot{Y}(u_{m})\leq 0$. This is impossible from Eq.
(II.5). So, $\dot{Y}<0$ for $0\leq u\leq 1$. In terms of $u$, the coefficient
$c_{1}$ in Eq. (42) is rewritten by

$\displaystyle\frac{c_{1}}{L}=-\int^{1}_{0}\dfrac{\sqrt{g(u)}\,\dot{Y}}{2+\left(1+\dfrac{1}{\hat{\rho}_{0}^{2}}\right)u^{3}}du\,,\qquad
g(u):=1+\dfrac{u^{2}}{\hat{\rho}_{0}^{2}}-\left(1+\dfrac{1}{\hat{\rho}_{0}^{2}}\right)u^{3}\,.$
(51)

From this, $g(u)\geq 0$ and $c_{1}>0$, while $a_{1}<0$ from the expansion
(38). Thus, we have shown $\zeta>0$. As for the $N=1$ case, the $Y$ term in
Eq. (II.5) vanishes, as $k^{2}=2$. This is analytically solved and the only
regular solution is $Y=1$. In this case, $a_{1}=c_{1}=0$.

$\Box$.

Now let us consider the time dependence of our boundary black hole. By Eq.
(43), the area on the horizon is given by

$\displaystyle{\cal A}=2\pi r_{0}\sqrt{1+\epsilon\dfrac{c_{1}{\mathbb{S}}_{\bf
k}}{L}}\sim r_{0}\left(1+\dfrac{\epsilon c_{1}}{2L}{\mathbb{S}}_{\bf
k}\right)\,,$ (52)

where ${\mathbb{S}}_{\bf k}$ is evaluated at $u=1$. By setting $0<m=i\omega\ll
1$ and choosing the sign of $\epsilon$ so that $\epsilon
c_{1}{\mathbb{S}}_{\bf k}>0$, the area decreases very slowly, thus describing
the evaporating process. In this case, $\partial_{v}^{2}{\mathbb{S}}_{{\bf
k}}$ is negligible compared with $\partial_{v}{\mathbb{S}}_{{\bf k}}$, and
hence, $R_{vv}(v,r_{0})$ in Eq. (48) is negative on the horizon. The above
proposition means that the null-null component of the stress-energy tensor is
always negative during the evaporating phase. By the energy conservation law,
this implies that the energy outside the evaporating black hole increases.
This picture agrees with the Hawking radiation, and thus, the entropy outside
the horizon would increase, supporting the GSL conjecture.

At first glance, the existence of the negative energy flux would violate the
averaged null energy condition (ANEC) on the horizon. In the next section, we
examine the ANEC in a background of $4$-dimensional evaporating black hole in
the context of the AdS/CFT correspondence.

##  III $4$-dimensional evaporating black hole and ANEC

Figure 2: The Penrose diagram of a four-dimensional spherically symmetric
evaporating black hole which is formed by a collapsing null dust shell. The
(red) dotted line and the (green) dashed-dotted line represent, respectively,
the event horizon and the apparent horizon. The (blue) solid line represents
the collapsing null dust shell, which is characterized in terms of the
advanced null coordinate $v$ as $v=v_{0}$.

In this section, to examine the ANEC, we construct the 5-dimensional bulk
geometry dual to a $4$-dimensional asymptotically flat, spherically symmetric
time-dependent boundary black hole, describing the formation by gravitational
collapse and subsequent evaporation. Such a bulk spacetime can be constructed,
at least locally near the boundary, by taking the Vaidya metric as our
boundary geometry and performing the Fefferman-Graham expansion into
$5$-dimensional bulk. As such, although our construction of the holographic
time-dependent black hole is limited, it is sufficient for the purpose of
examining the ANEC on our time-dependent boundary black hole by applying the
method of Refs. [18, 19], based on the no-bulk shortcut principle.

###  III.1 An asymptotically flat evaporating black hole

We consider an asymptotically flat spherically symmetric evaporating black
hole formed by injecting a null dust shell. Such a model of evaporating black
hole is well described by the Vaidya model [24] as,

$\displaystyle
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=2dvdr-\left(1-\frac{2m(v)}{r}\right)dv^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\,,$ (53)

where $v$ denotes the advanced null coordinate and $m(v)$ the mass of the
black hole. Note that here and hereafter, we use the coordinate index
$x^{\mu}$ to denote tensor components on the $4$-dimensional boundary
spacetime. As shown in Fig. 2, the null dust shell is injected at $v=v_{0}$ to
form the black hole, of which the event horizon (EH) begins at the center of
spherical symmetry [point $p$ at ($r=0,v=v_{i}$)] and terminates at a zero
mass naked singularity at point $q$ ($r=0,v=v_{f}$), where the evaporation is
completed. The apparent horizon (AH) is located outside the EH during the
evaporating process, as the area decreases. We assume that the spacetime is
flat, i.e., $m(v)=0,v<v_{0}$ before the collapse of the null shell.

As is often discussed in the consideration of backreaction effects of the
Hawking radiation, for a free massless field, the negative energy flux ${\cal
F}$ flowing into the black hole is determined by the dimensional analysis as

$\displaystyle{\cal F}\simeq\frac{A}{m(v)^{2}}\,,$ (54)

where $A$ is of order unity in Planck units [25]. Assuming that the mass of
the black hole is lost by the negative energy flux, ${dm}/{dv}\simeq-A/m^{2}$,
one obtains

$\displaystyle m(v)\sim(v-v_{f})^{1/3}\,.$ (55)

Generalizing Eq. (55), we shall consider a holographic model in which the AdS
boundary is conformal to the Vaidya metric (53) with $m(v)$ satisfying

$\displaystyle
m(v)=m_{0}\left(1-\frac{v_{0}}{v_{f}}\right)^{-\alpha}\left(1-\frac{v}{v_{f}}\right)^{\alpha}\,,\qquad
m_{0}>0\,,\quad 0<\alpha<1\,,$ (56)

where $m_{0}$ is the mass of the black hole just after the collapse of the
null dust shell. The radial null geodesic on the EH obeys the equation

$\displaystyle\frac{dr}{dv}=\frac{1}{2}\left(1-\frac{2m(v)}{r}\right)\,.$ (57)

Note that the AH is located at the radius $r_{\rm AH}$ where the r.h.s. of the
above equation vanishes, i.e., $r_{\rm AH}=2m(v)=2m_{0}(1-v/v_{f})^{\alpha}$.
In the evaporation process $v_{0}<v<v_{f}$, the EH is located inside the AH as
depicted in Fig. 2, which implies that the EH radius $r_{\rm EH}$ is bounded
from above by $r_{\rm AH}=2m(v)$. This can be the case since the quantum field
stress-energy will not satisfy the weak energy condition under the evaporation
process. In particular, at the last stage of the evaporation, the geometry
rapidly changes and it is plausible that the EH radius could be much smaller
than that of the AH, close to the final point of the evaporation $v\simeq
v_{f}$. In such a case, by assuming $r_{\rm EH}\ll 2m(v)$, we can find the
approximated solution of (57) near $v=v_{f}$ as,

$\displaystyle r_{\rm
EH}(v)\simeq\sqrt{\frac{2m_{0}v_{f}}{\alpha+1}}\left(1-\frac{v_{0}}{v_{f}}\right)^{-\frac{\alpha}{2}}\left(1-\frac{v}{v_{f}}\right)^{(\alpha+1)/2}\,.$
(58)

Along the radial outgoing null vector

$\displaystyle
k^{\mu}=(\partial_{v})^{\mu}+\dfrac{1}{2}\left(1-\dfrac{2m(v)}{r}\right)(\partial_{r})^{\mu}\,,$
(59)

on the EH, the affine parametrized null geodesic generator $\gamma$ is given
by

$\displaystyle K^{\mu}:=h(v)k^{\mu}\,,$ (60)

where the factor $h$ obeys the following equation,

$\displaystyle\frac{dh}{dv}=-{h}\frac{m}{r^{2}}\,.$ (61)

Near the naked singularity, $v=v_{f}$, $r=0$, the solution is approximately
given by the functions (56) and (58) as

$\displaystyle\frac{dv}{dV}=h\propto\left(1-\frac{v}{v_{f}}\right)^{(\alpha+1)/2}\,,$
(62)

where $V$ is the affine parameter of the null geodesic $\gamma$ with the
tangent $K^{\mu}$.

###  III.2 The no bulk-short cut principle in the dual bulk spacetime

We briefly recall the statement of the no-bulk-short-cut principle proposed in
[26, 18, 19]. Suppose that a pair of boundary points $p$ and $q$ are connected
by an achronal null geodesic $\gamma$ on the boundary spacetime. Then, the no-
bulk-short-cut principle states that there is no bulk timelike curve $\lambda$
which connects $p$ to $q$. In other words, $\gamma$ is the fastest causal
curve among all causal curves from $p$ to $q$, including the bulk causal
curves. In a class of spacetime, this principle was shown in the Gao-Wald
theorem [20]. By applying this principle to boundary Minkowski spacetime in a
holographic setting, the ANEC was proved in [26]. The principle is also
applied to the spatially compact universes and the CANEC or the ANEC with a
weight function were derived [18, 19]. In this regard, see also [27] for the
ANEC in de Sitter and AdS spacetime. Here, we shall apply this principle to
the holographic model with the boundary geometry descrbing the evaporating
black hole (53) discussed above, whose event horizon is generated by achronal
null geodesics $\gamma$ from $v=v_{i}$ to $v=v_{f}$. We wish to derive the
lower bound of the null-null component of the boundary stress-energy tensor
along boundary achronal null geodesics $\gamma$. Our strategy is very similar
to the one performed in Refs. [18, 19], which is as follows. Given such a
boundary achronal null geodesic $\gamma$, we consider all the bulk causal
curves connected with $\gamma$ via Jacobi fields, thus considered to be in the
neighborhood of $\gamma$ covered by Fefferman-Graham coordinates. Then, by
imposing the no-bulk-short-cut principle that $\gamma$ be the fastest causal
curve among all the causal curves in the neighborhood, we will obtain the
lower bound.

The $5$-dimensional bulk spacetime in the neighbourhood of the boundary metric
(53) including the boundary null geodesics is represented by the following
Fefferman-Graham coordinates

$\displaystyle
g_{ab}dx^{a}dx^{b}=\frac{L^{2}}{z^{2}}\left(dz^{2}+g_{\mu\nu}(x,z)dx^{\mu}dx^{\nu}\right):=\frac{L^{2}}{z^{2}}\hat{g}_{ab}dx^{a}dx^{b}\,,$
$\displaystyle
g_{\mu\nu}(z,x)=g_{(0)\mu\nu}(x)+z^{2}g_{(2)\mu\nu}(x)+\cdots+z^{4}g_{(4)\mu\nu}(x)+h_{(4)\mu\nu}\,z^{4}\ln
z^{2}+\cdots\,,$ (63)

where $g_{(0)\mu\nu}(x)$ corresponds to the boundary metric (53), and
$\hat{g}_{ab}$ is the unphysical metric conformal to $g_{ab}$. The subleading
coefficients $g_{(2)\mu\nu}$, $g_{(4)\mu\nu}(x)$, and $h_{(4)\mu\nu}$ are
given by [21],

$\displaystyle
g_{(2)\mu\nu}=-\frac{1}{2}\left(R_{\mu\nu}-\frac{R}{6}g_{(0)\mu\nu}\right)\,,$
$\displaystyle
g_{(4)\mu\nu}=t_{\mu\nu}+\frac{1}{8}g_{(0)\mu\nu}\left[(\mbox{Tr}g_{(2)})^{2}-\mbox{Tr}(g^{2}_{(2)})\right]+\frac{1}{2}g_{(2)\mu\alpha}{g_{(0)}}^{\alpha\beta}g_{(2)\beta\nu}-\frac{1}{4}g_{(2)\mu\nu}\mbox{Tr}(g_{(2)})\,,$
$\displaystyle
h_{(4)\mu\nu}=\frac{1}{2}g_{(2)\mu\alpha}{g_{(0)}}^{\alpha\beta}g_{(2)\beta\nu}-\frac{1}{8}g_{(0)\mu\nu}\mbox{Tr}(g_{(2)}^{2})\,,$
(64)

where $R_{\mu\nu}$, $R$ are the Ricci tensor, Ricci scalar of the metric
$g_{(0)\mu\nu}$, and the indices are raised and lowered by the metric
$g_{(0)\mu\nu}$. According to the AdS/CFT dictionary [21], the expectation
value of the stress-energy tensor $\left\langle~{}T_{\mu\nu}~{}\right\rangle$
is determined by $t_{\mu\nu}$ as

$\displaystyle\left\langle~{}T_{\mu\nu}~{}\right\rangle=\frac{1}{4\pi
G_{5}}t_{\mu\nu}\,.$ (65)

It is convenient to use the following double null coordinates

$\displaystyle
g_{(0)\mu\nu}dx^{\mu}dx^{\nu}=-e^{f(U,V)}dUdV+r^{2}(U,V)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\,,$ (66)

instead of the metric (53). On the EH, we can set

$\displaystyle U=0\,,\quad f(0,V)=0\,,$ (67)

wthout loss of generality. In this case, $V$ is the affine parameter of the
boundary null geodesic $\gamma$ with the tangent vector $K^{\mu}$ defined by
Eq. (60). We denote $V_{i}$ and $V_{f}$ as the values of the point $p$
($v=v_{i}$) and the point $q$ ($v=v_{f}$), respectively.

Now consider a bulk causal curve $\lambda$ with the tangent vector $K^{a}$
near the boundary null geodesic $\gamma$, which connects the boundary point
$p$ to the boundary point $q$ on $\gamma$. From spherical symmetry, it is
sufficient to consider the case $K^{\theta}=K^{\phi}=0$, and $K^{a}$ can be
expanded as a series of small parameter $\epsilon$ as

$\displaystyle
K^{a}=(K^{z},K^{U},K^{V},K^{\theta},K^{\phi})=\left(\frac{dz}{dV},\frac{dU}{dV},1,0,0\right)\,,$
$\displaystyle z=\epsilon z_{1}+\epsilon^{2}z_{2}+\cdots\,,$
$\displaystyle\frac{dU}{dV}=\epsilon^{2}\frac{dU_{2}}{dV}+\epsilon^{3}\frac{dU_{3}}{dV}+\epsilon^{4}\frac{dU_{4}}{dV}+\cdots\,.$
(68)

Substituting this into Eq. (III.2), the constraint of
$\hat{g}_{ab}K^{a}K^{b}\leq 0$ is expressed as

$\displaystyle\hat{g}_{ab}K^{a}K^{b}$
$\displaystyle=\epsilon^{2}\left(-\frac{dU_{2}}{dV}+\dot{z}_{1}^{2}+z_{1}^{2}\,g_{(2)VV}(0,V)\right)$
$\displaystyle\quad+\epsilon^{3}\left(2\dot{z}_{1}\dot{z}_{2}+2z_{1}z_{2}\,g_{(2)VV}(0,V)-\frac{dU_{3}}{dV}\right)$
$\displaystyle\quad+\epsilon^{4}\Biggl{(}\dot{z}_{2}^{2}+z_{2}^{2}\,g_{(2)VV}(0,V)+2\dot{z}_{1}\dot{z}_{3}$
$\displaystyle\qquad\qquad+2z_{1}z_{3}\,g_{(2)VV}(0,V)+2z_{1}^{4}\ln
z_{1}\,h_{VV}(0,V)$
$\displaystyle\qquad\qquad+2z_{1}^{2}\,g_{(2)UV}(0,V)\frac{dU_{2}}{dV}-\frac{dU_{4}}{dV}+z_{1}^{4}\,g_{(4)VV}(0,V)$
$\displaystyle\qquad\qquad+z_{1}^{2}\partial_{U}(g_{(2)VV})(0,V)U_{2}-(\partial_{U}f)(0,V)U_{2}\frac{dU_{2}}{dV}\Biggr{)}$
$\displaystyle\quad+2\epsilon^{4}\ln(\epsilon)z_{1}^{4}\,h_{VV}(0,V)+\cdots$
$\displaystyle\quad\leq 0\,,$ (69)

where the dot denotes the derivative with respect to $V$. At
$O(\epsilon^{2})$, we obtain the inequality of the variation of $U$ from
$V_{i}$ to $V_{f}$ as

$\displaystyle\Delta
U_{2}=\int_{V_{i}}^{V_{f}}\dfrac{dU_{2}}{dV}dV\geq\int^{V_{f}}_{V_{i}}(\dot{z}_{1}^{2}+z_{1}^{2}\,g_{(2)VV}(0,V))dV\,,$
(70)

where the equality is satisfied for the null curve. The r.h.s. of Eq. (70) is
minimized for $z_{1}$ obeying

$\displaystyle\ddot{z}_{1}=g_{(2)VV}(0,V)z_{1}=\frac{\ddot{r}(0,V)}{r(0,V)}z_{1}\,,$
(71)

where we have used that on the EH, $R_{VV}(0,V)=-2\ddot{r}/r$, under our gauge
condition $f(0,V)=0$ and the holographic formula (III.2) for $g_{(2)VV}$.
Therefore the solution satisfying $z_{1}(p)=z_{1}(q)=0$ is simply given by

$\displaystyle z_{1}=cr(0,V)\,,$ (72)

for some positive constant $c$. Then, the integrand of the r.h.s. of (70) is
expressed simply as $(z_{1}\dot{z}_{1})\dot{}$ and thus, integrating the
r.h.s. of Eq. (70) results in the boundary values of $z_{1}\dot{z}_{1}$. As
shown in Ref. [19], $z_{1}$ is identified as the magnitude of a Jacobi field
for a null geodesic congruence on EH. Then, $z_{1}$ must vanish at
$(U=0,V=V_{i})$ [the point $p$ in Fig. 2)], where the EH is formed. Also,
since the EH completely evaporates at $(U=0,V=V_{f})$ [the point $q$ in Fig.
2)], the Jacobi field $z_{1}$ is expected to vanish there too;

$\displaystyle z_{1}(p)=z_{1}(q)=0\,.$ (73)

This guarantees that the bulk null curve $\lambda$ remains in the
neighbourhood of the boundary null geodesic $\gamma$ from the point $p$ to the
point $q$ when $\epsilon$ is small enough. In fact, one can directly check
that the boundary value $\dot{z}_{1}z_{1}(V)$ goes to zero in the limit $V\to
V_{f}$ by utilizing Eqs. (58) and (62), as

$\displaystyle\lim_{V\to V_{f}}\dot{z}_{1}(V)z_{1}(V)=\lim_{V\to
V_{f}}h(v)\frac{dz_{1}}{dv}z_{1}\sim\lim_{v\to
v_{f}}\left(1-\frac{v}{v_{f}}\right)^{(3\alpha+1)/2}\to 0\,.$ (74)

Thus, the r.h.s. of Eq. (70) must vanish, implying that there is no time delay
between $\gamma$ and $\lambda$ at $O(\epsilon^{2})$ when $\lambda$ is the null
curve, as the variation of $U$ between $\gamma$ and $\lambda$ is zero.
Actually, $U_{2}$ is analytically written by $z_{1}$ as

$\displaystyle U_{2}=z_{1}\dot{z}_{1}\,.$ (75)

Similarly, the variation of $U$ from $V_{i}$ to $V_{f}$ is minimized for
$z_{2}$ and $z_{3}$ obeying the same Jacobi equation (71), and $\Delta U_{3}$
is also zero at $O(\epsilon^{3})$.

As for $O(\epsilon^{4})$, by noting the fact that for the Vaidya metric (53),
only non-vanishing component among $g_{(2)\mu\nu}$ in Eqs. (III.2) is
$g_{(2)VV}=-R_{VV}/2$. Thus, $h_{VV}=0$, and we obtain

$\displaystyle\Delta U_{4}$ $\displaystyle\geq$
$\displaystyle\int^{V_{f}}_{V_{s}}z_{1}^{4}\,t_{VV}(0,V)dV+{\cal I}\,,$
$\displaystyle{\cal I}$ $\displaystyle:=$
$\displaystyle\int^{V_{f}}_{V_{s}}\left(z_{1}^{2}\partial_{U}(g_{(2)VV})(0,V)U_{2}-(\partial_{U}f)(0,V)U_{2}\frac{dU_{2}}{dV}\right)dV$
(76) $\displaystyle=$
$\displaystyle\frac{c^{4}}{16}\int^{V_{f}}_{V_{i}}r^{2}\left[r^{2}\mu\,\theta_{+}^{2}-2R_{VV}\right]dV\,,$

where $\theta_{\pm}$ and $\mu$ are outgoing and ingoing expansions, and the
mass density defined by

$\displaystyle\theta_{+}=\frac{2\dot{r}}{r}\,,\quad\theta_{-}=\frac{2r^{\prime}}{r}\,,\quad\mu:=\frac{1}{r^{2}}+\theta_{+}\theta_{-}\,,$
(77)

where the dash expresses the derivative with respect to $U$. The second
equality in Eq. (III.2) is shown in the Appendix. Applying the no-bulk-short-
cut principle to our case, $\Delta U_{4}$ should be non-negative, and we
finally obtain the following inequality with the weight function $r^{4}$,

$\displaystyle\int^{V_{f}}_{V_{s}}r^{4}t_{VV}dV\geq\frac{1}{16}\int^{V_{f}}_{V_{s}}r^{2}\left[2R_{VV}-r^{2}\mu\,\theta_{+}^{2}\right]dV\,.$
(78)

###  III.3 The ANEC in the evaporating BH

Now, we can derive the ANEC from Eq. (78). Using the fact that $f(0,V)=0$ and
$\partial_{U}=-1/(2h)\partial_{r}$ along the EH, the expansions $\theta_{\pm}$
and the mass density $\mu$ in Eqs. (77) are rewritten in terms of the advanced
null coordinate, $v$ and $r$ as

$\displaystyle\theta_{+}=\left(1-\frac{2m(v)}{r}\right)\frac{h}{r},\qquad\theta_{-}=-\frac{1}{hr},\qquad\mu=\frac{2m}{r^{3}}.$
(79)

Substituting Eq. (79) into the inequality (78), we find

$\displaystyle\int^{V_{f}}_{V_{s}}r^{4}t_{VV}dV\geq\frac{1}{4}\int^{v_{f}}_{v_{s}}h\left[\frac{dm}{dv}-\left(1-\frac{2m}{r(v)}\right)^{2}\frac{m}{2r(v)}\right]dv.$
(80)

In order to inspect the above inequality closely, let us devide the whole
region $v_{i}\leq v\leq v_{f}$ into four phases:

Phase (I) $v_{i}\leq v\leq v_{0}-\delta_{1}$: the geometry is the Minkowski
spacetime,

Phase (II) $v_{0}-\delta_{1}\leq v\leq v_{0}+\delta_{1}$: the null shell is
collapsing,

Phase (III) $v_{0}+\delta_{1}\leq v\leq v_{f}-\delta_{2}$: the BH is
evaporating,

Phase (IV) $v_{f}-\delta_{2}\leq v\leq v_{f}$: the final explosion.

Here, we assume that the time evolution of the evaporation is sufficiently
slow ($v_{f}\gg m_{0}$), and the time durations $\delta_{1}$, $\delta_{2}$ are
very small, i.e.,

$\displaystyle\delta_{1},\,\delta_{2}\ll|v_{f}-v_{0}|\,.$ (81)

In Phase (I), the r.h.s. of Eq. (80) vanishes, as $m=0$. In the collapsing
phase (II), the mass rapidly grows during the short time period $\delta_{1}$
by the condition (81), and the second term is negligible compared with the
first term. In Phase (III), the time evolution of the mass is very slow, and
$r\simeq 2m$ in this phase. Let us set the location of the EH as
$r=2m(1-\eta)$ with $\eta$ assumed to be small so that $|\eta|\simeq|dm/dv|\ll
1$ in Phase (III). Then, linearizing Eq. (57) with respect to $\eta$, one
obtains the approximate solution,

$\displaystyle\eta\simeq-2\dfrac{d}{dv}[2m(1-\eta)]\simeq-4\dfrac{dm}{dv}\,.$
(82)

This implies that the second term in the r.h.s. of Eq. (80) is small compared
with the first term, as

$\displaystyle\left(1-\frac{2m}{r}\right)^{2}\frac{m}{2r}\simeq\frac{\eta^{2}}{4}\ll\Bigl{|}\frac{dm}{dv}\Bigr{|}\simeq\Bigl{|}\frac{\eta}{4}\Bigr{|}\,,$
(83)

under the approximation (82).

In the final phase (IV), one might expect that the second term in the r.h.s.
of Eq. (80) could give a large contribution. However, in the case that the
time evolution of the mass is very slow, i.e., $v_{f}\gg m_{0}$, it turns out
that the contribution from the second term is still small compared with that
from the first term in Phase (IV). This can be seen from Eq. (58) that the
ratio of the second term to the first term is at least of
$O(\sqrt{m_{0}/v_{f}})$. Thus, it is sufficient for our purpose to evaluate
the first term of the r.h.s. of Eq. (80) in each phase. In Phase (I), the
spacetime is flat, and then $h=1$. In Phase (II), by integrating Eq. (61), we
obtain

$\displaystyle\int^{v_{0}+\delta_{1}}_{v_{0}-\delta_{1}}\frac{dh}{h}=\ln
h(v_{0}+\delta_{1})=-\int^{v_{0}+\delta_{1}}_{v_{0}-\delta_{1}}\frac{m}{r^{2}}dv=O(\delta_{1})\,,$
(84)

which implies $h\sim 1$ during Phase (II). Thus, we obtain

$\displaystyle\int^{v_{f}}_{v_{i}}h\frac{dm}{dv}dv$ $\displaystyle=$
$\displaystyle\int^{v_{0}+\delta_{1}}_{v_{0}-\delta_{1}}h\frac{dm}{dv}dv+\int^{v_{f}}_{v_{0}+\delta_{1}}h\frac{dm}{dv}dv$
(85) $\displaystyle\simeq$
$\displaystyle\int^{v_{0}+\delta_{1}}_{v_{0}-\delta_{1}}\frac{dm}{dv}dv+\int^{v_{f}}_{v_{0}+\delta_{1}}h\frac{dm}{dv}dv$
$\displaystyle=$ $\displaystyle
m_{0}+\int^{v_{f}}_{v_{0}+\delta_{1}}h\frac{dm}{dv}dv\,.$

By Eq. (61), $dh/dv<0$ during the evaporating phases (III) and (IV),

$\displaystyle 0\leq h\leq 1\quad\mbox{for}\quad v\geq v_{0}+\delta_{1}\,.$
(86)

Since $dm/dv<0$ during the evaporating phases (III) and (IV), we thus obtain

$\displaystyle\int^{V_{f}}_{V_{s}}r^{4}t_{VV}dV$ $\displaystyle\geq$
$\displaystyle\frac{1}{4}\left[m_{0}+\int^{v_{f}}_{v_{0}+\delta_{1}}h\frac{dm}{dv}dv\right]$
(87) $\displaystyle\geq$
$\displaystyle\frac{1}{4}\left[m_{0}+\int^{v_{f}}_{v_{0}+\delta_{1}}\frac{dm}{dv}dv\right]$
$\displaystyle\geq$
$\displaystyle\frac{1}{4}m_{0}+\frac{1}{4}[m(v)]^{v_{f}}_{v_{0}+\delta_{1}}$
$\displaystyle\geq$ $\displaystyle 0\,.$

This shows that the ANEC with the weight function $r^{4}$ is satisfied for the
evaporating black hole.

##  IV Summary and discussions

We have investigated two holographic models of evaporating black holes. First,
by perturbing the 4-dimensional black droplet solution associated with BTZ
black holes on the AdS boundary [16], we have constructed the bulk geometry
dual to the boundary field theory in a time-dependent BTZ type black hole with
the horizon area decreasing. We found that negative energy flux going into the
time-dependent horizon always appears by calculating the boundary stress-
energy tensor. This calculation agrees with Hawking’s picture of evaporating
black holes [1], where the horizon area decreases by absorbing negative
quantum energy flux. According to the energy conservation, the total energy
and the matter entropy outside the evaporating black hole should increase by
the Hawking radiation. Our result supports the generalized second law (GSL)
[1, 13], which states that the total of the gravitational entropy and the
matter entropy outside the horizon should not decrease.

The existence of the negative energy flux shown in this paper implies a local
violation of the null energy condition (NEC). As first shown in a class of
holographic theories [26], ANEC is satisfied along a complete achronal null
geodesic in flat spacetimes. In curved spacetime, however, we cannot expect
that the ANEC is generically satisfied, except some particular class of curved
spacetimes [18, 19]. It was shown for some semi-analytic black droplet
solutions [28, 29] that along the incomplete null geodesics, which terminate
at a singularity inside the boundary black hole, the ANEC is violated due to
the negative divergence of the null energy toward the singularity. In our
present holographic model, the null geodesic generators along the event
horizon begin from the formation of the boundary black hole, and terminate at
the final instant of the black hole evaporation, where a naked singularity
appears, as shown in Fig. 2. Although this situation is very similar to the
incomplete null geodesics observed in Refs. [28, 29], we can expect the ANEC
to hold in our case, as the singularity at the very final moment of the
complete black hole evaporation is a weak singularity in the sense that the
mass $m(v)$ becomes zero at the final singularity. In addition, the null
geodesic congruence along the horizon expands from the point at the formation
of the black hole, and shrinks to zero again at the final zero mass
singularity. This behavior of the null geodesic congruence is very similar to
that in a spatially compact $S^{3}$ universe, in which a congruence expands
from the south pole and then shrinks to zero at the north pole. This peculiar
feature enables us to prove the ANEC along the null geodesic generator of the
event horizon, just like the case considered in spatially compact universes
[18, 19]. This indicates that the total negative energy flux is bounded from
below, and is compensated by a positive energy flux in the early stage of the
black hole formation.

## Acknowledgements

We would like to thank Takashi Okamura for useful discussions. This work was
supported in part by JSPS KAKENHI Grant Numbers 17K05451, 20K03975 (KM), and
15K05092, 20K03938, 21H05182 (AI), and 21H05186 (KM, AI).

##  Appendix A The derivation of Eq. (78)

In the Vaidya metric (53), $R_{\mu\nu}=0$ except $R_{vv}$. In the double null
coordinates (66), $R_{UV}=R_{\theta\theta}=0$ yields

$\displaystyle\dot{f^{\prime}}=-\dfrac{2\dot{r^{\prime}}}{r},\qquad\dot{r^{\prime}}=-\dfrac{1}{r}\left(\dot{r}r^{\prime}+\frac{1}{4}e^{f}\right).$
(88)

By the fact that $f=0$ and $\partial_{U}\sim h^{-1}\partial_{r}$ along the EH
where $U=0$, the asymptotic behavior of $\dot{r^{\prime}}$ and $f^{\prime}$ in
Eq. (88) becomes

$\displaystyle\dot{r^{\prime}}\sim\left(1-\dfrac{v_{f}}{v}\right)^{-1},\quad
f^{\prime}\sim\left(1-\dfrac{v_{f}}{v}\right)^{-\alpha-1}$ (89)

from Eqs. (56), (58), and (62).

By Eq. (III.2), $g_{(2)VV}$ is given by

$\displaystyle
g_{(2)VV}=-\dfrac{R_{VV}}{2}=\dfrac{\ddot{r}-\dot{f}\dot{r}}{r}.$ (90)

Substituting this into the first term of ${\cal I}$ in Eq. (III.2), we obtain

$\displaystyle\int z_{1}^{2}\partial_{U}(g_{(2)VV})U_{2}\,dV$ $\displaystyle=$
$\displaystyle\int
z_{1}^{3}\dot{z}_{1}\left[\dfrac{\ddot{r^{\prime}}}{r}-\frac{r^{\prime}}{r^{2}}\ddot{r}-\dfrac{2\dot{r}}{r^{3}}\left(\dot{r}r^{\prime}+\dfrac{1}{4}\right)\right]dV$
(91) $\displaystyle=$ $\displaystyle
c^{4}\int\dot{r}\left[r^{2}\ddot{r}^{\prime}-rr^{\prime}\ddot{r}-2\dot{r}\left(\dot{r}r^{\prime}+\dfrac{1}{4}\right)\right]dV$
$\displaystyle=$ $\displaystyle\frac{c^{4}}{4}\int r\ddot{r}dV.$

Here, in the second line, we used the condition $f(0,V)=0$ along the EH and
also Eqs. (75), (88), and (72). In the third line, we performed integration by
parts for the first term in the second line, and used the fact that the
boundary term $r^{2}\dot{r^{\prime}}\dot{r}$ vanishes in the limit $v\to
v_{f}$ by Eqs. (58), (62), and (89).

Similarly, we evaluate the second term of ${\cal I}$ in Eq. (III.2) as

$\displaystyle-\int
f^{\prime}U_{2}\dot{U}_{2}dV=-\int\dfrac{\dot{r^{\prime}}}{r}U_{2}^{2}dV=c^{4}\int\left(\dot{r}r^{\prime}+\dfrac{1}{4}\right)\dot{r}^{2}dV,$
(92)

where we performed integration by parts in the first equality and used Eq.
(88). Note that the boundary term $f^{\prime}U_{2}^{2}$ vanishes in the limit
$v\to v_{f}$ by Eqs. (58), (62), and (89). To derive the second equality, we
used Eqs. (89) and (72). Substituting Eqs. (91) and (92) into ${\cal I}$ in
Eq. (III.2) and using Eqs. (79) and (90), we finally obtain the inequality
(78).

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# A modified proximal contraction principle with applications to variational
inequality problems

Aftab Alam

Formerly at: Department of Mathematics, Aligarh Muslim University,
Aligarh-202002, India.
Email Id<EMAIL_ADDRESS>
Orcid Id: 0000-0002-0479-9118
*Corresponding author<EMAIL_ADDRESS>
Abstract. In this paper, we introduce the notions of proximally completeness,
proximally closedness and proximally continuity and utilize the same to prove
a result on existence and uniqueness of best proximity points in the setting
of metric space (not necessarily complete). Our newly proved result enriches,
sharpens, improves and modifies the proximal contraction principle of Basha
[J. Optim. Theory Appl. 2011:151 (2011), 210-216]. In order to illustrate the
effectiveness of our finding, we discuss the sufficient conditions ensuring
the existence of a unique solution of certain variational inequality problem.

Keywords: best approximation; best proximity points; proximal contractions;
variational inequality.

AMS Subject Classification: 41A65, 47H09, 30L99, 47J20.

##  1 Introduction

Consider the non-self-mapping $T:A\rightarrow B$, whereas $A$ and $B$ stand
for two nonempty subset a metric space $(X,d)$. We say that an element $x\in
A$ is a fixed point of the mapping if $T(x)=x$ provided $A\cap
T(A)\neq\emptyset$. On the other hand, in case of $A\cap T(A)=\emptyset$, $T$
has no fixed point so that for each $x\in A$, $d(x,Tx)>0$. In such a case, we
are interesting to compute an optimal approximate solution of functional
equation $T(x)=x$ such that $d(x,Tx)$ is closet to zero. In this perspective,
best approximation theory and best proximity point analysis have been
appeared.

In $1969$, Fan [1] proved the first best approximation theorem in the setting
of Hausdorff locally convex topological vector space. In the subsequent years,
the classical best approximation theorem of Fan has been generalized and
extended by many researchers, but we merely refer [2, 3, 4, 5] and references
therein. A comprehensive and unified approach to such best approximation
theorems has been furnished by Vetrivel et al. [6]. On the other hand, Eldred
and Veeramani [7] established some results on existence of best proximity
points of cyclic contractions in the context of metric space and utilized the
same to discuss the existence, uniqueness and convergence of best proximity
points in the framework of uniformly convex Banach spaces. In the same
continuation, Basha [8, 9, 10] extended the Banach contraction principle
employing the minimum distance between two subsets of the metric space and
utilized the same to prove the best proximity point results for non-self
proximal contractions.

Indeed, the best proximity point theorems offer an approximate solution that
is optimal. However, the best approximation theorems yield approximate
solutions that are not necessarily optimal. It is interesting to see that best
proximity point theorems generalize fixed point theorems in a natural way. In
fact, when the mapping under consideration is a self-mapping, a best proximity
point boils down to a fixed point. The theory of best proximity point has the
great importance in nonlinear analysis, approximation theory, optimization
theory, game theory, fixed point theory and variational inequalities.

The aim of this paper is to refine and to improve the proximal contraction
principle due to Basha [10]. In process, we introduce some new notions, such
as: $T$-proximal sequence, proximally completeness, proximally closedness and
proximally continuity. As an application of our result, we study the existence
of a unique solution of a variational inequality problem.

##  2 Preliminaries

Given a pair $(A,B)$ of nonempty subsets of a metric space $(X,d)$, the
following notations will be utilized in our subsequential discussion:

  1. $d(A,B):=\inf\\{d(x,y):x\in A,y\in B\\}$,

  2. $d(x,B):=\inf\\{d(x,y):y\in B\\}$, for $x\in A$,

  3. $A_{0}:=\\{x\in A:d(x,y)=d(A,B),~{}~{}\mbox{for some}~{}y\in B\\}$,

  4. $B_{0}:=\\{y\in A:d(x,y)=d(A,B),~{}~{}\mbox{for some}~{}x\in A\\}$.

It can be noted that for each $x\in A_{0}$, $\exists$ $y\in B_{0}$ such that
$d(x,y)=d(A,B)$ and conversely, for each $y\in B_{0}$, $\exists$ $x\in A_{0}$
such that $d(x,y)=d(A,B)$. Consequently, $A_{0}$ is nonempty if and only if
$B_{0}$ is nonempty. Also, it is evident that both $A_{0}$ and $B_{0}$ are
nonempty, whenever $A$ intersects $B$.

######  Definition 1.

[7] Let $(X,d)$ be a metric space and $(A,B)$ a pair of two nonempty subsets
of $X$. An element $\overline{x}\in A$ is called a best proximity point of
$T:A\rightarrow B$ if

$d(\overline{x},T\overline{x})=d(A,B):=\inf\\{d(x,y):x\in A,y\in B\\}.$

In other words, we say that $\overline{x}\in A$ is a best proximity point of
$T$ if at $\overline{x}$ the function $d(x,Tx)$ attains its global minimum
with the value $d(A,B)$.

######  Definition 2.

[10] Let $(X,d)$ be a metric space and $(A,B)$ a pair of nonempty subsets of
$X$. A mapping $T:A\rightarrow B$ is called proximal contraction if $\exists$
$k\in[0,1)$ such that for all $x,y,u,v\in A$,

$d(u,Tx)=d(v,Ty)=d(A,B)\Rightarrow d(u,v)\leq kd(x,y).$

######  Remark 1.

Any proximal contraction mapping is not necessarily continuous. Under the
restriction $A=B=X$, the notion of proximal contraction coincides with that of
usual contraction.

######  Definition 3.

[7] Let $(X,d)$ be a metric space and $(A,B)$ a pair of nonempty subsets of
$X$. The set $B$ is called approximatively compact with respect to $A$ if
every sequence $\\{y_{n}\\}\subset B$ satisfying the condition that
$d(x,y_{n})\stackrel{{\scriptstyle\mathbb{R}}}{{\to}}d(x,B)$ for some $x\in
A$, has a convergent subsequence.

######  Remark 2.

[10] Every set is approximatively compact with respect to itself. Also, every
compact set is approximatively compact.

######  Remark 3.

[7] The sets $A_{0}$ and $B_{0}$ are nonempty if $A$ is compact and $B$ is
approximatively compact with respect to $A$.

######  Theorem 1.

[10] Let $(X,d)$ be a complete metric space, $(A,B)$ a pair of nonempty
subsets of $X$ and $T:A\rightarrow B$ a mapping. Suppose that the following
conditions hold:

  1. (i)

$A_{0}\neq\emptyset$ and $B_{0}\neq\emptyset$,

  2. (ii)

$T(A_{0})\subseteq B_{0}$,

  3. (iii)

$T$ is proximal contraction,

  4. (iv)

$A$ and $B$ are closed subspaces of $X$,

  5. (v)

$B$ is approximatively compact with respect to $A$.

Then $T$ has a unique best proximity point. Further, for any fixed element
$x_{0}\in A_{0}$, the sequence $\\{x_{n}\\}$, defined by

$d(x_{n+1},Tx_{n})=d(A,B)$

converges to the unique best proximity point of $T$.

##  3 Main Results

In this section, we first introduce several metrical notions in our setting
and then establish the result on existence and uniqueness of best proximity
points.

######  Definition 4.

Let $(X,d)$ be a metric space, $(A,B)$ a pair of nonempty subsets of $X$ and
$T:A\rightarrow B$ a mapping. A sequence $\\{x_{n}\\}\subset A$ is called
$T$-proximal if

$d(x_{n+1},Tx_{n})=d(A,B).$

######  Remark 4.

In particular for $A=B=X$, the notion of $T$-proximal sequence coincides with
that of sequence of Picard iteration.

######  Definition 5.

Let $(X,d)$ be a metric space, $(A,B)$ a pair of nonempty subsets of $X$ and
$T:A\rightarrow B$ a mapping. We say that the subspace $(A,d)$ is proximally
complete if every $T$-proximal Cauchy sequence in $A$ converges in $A$.

Clearly, every complete subspace of a metric space is proximally complete.

######  Definition 6.

Let $(X,d)$ be a metric space, $(A,B)$ a pair of nonempty subsets of $X$,
$T:A\rightarrow B$ a mapping and $E\subseteq A$. We say that $E$ is proximally
closed subspace of $A$ if the limit of each $T$-proximal convergent sequence
in $E$ belongs to $E$.

Clearly, every closed subspace of a metric space is proximally closed.

######  Proposition 1.

Let $(X,d)$ be a metric space, $(A,B)$ a pair of nonempty subsets of $X$ and
$T:A\rightarrow B$ a mapping. If $B$ is approximatively compact with respect
to $A$, then $A_{0}$ is proximally closed.

###### Proof.

Let $\\{x_{n}\\}\subset A_{0}$ be a $T$-proximal sequence converging to $x\in
A$. We have to show that $x\in A_{0}$. As $\\{x_{n}\\}$ is $T$-proximal, we
have

$d(x_{n+1},Tx_{n})=d(A,B).$

Further since $T(x_{n})\in B$, therefore we have

$\displaystyle d(x,B)$ $\displaystyle\leq$ $\displaystyle d(x,Tx_{n})$
$\displaystyle\leq$ $\displaystyle d(x,x_{n+1})+d(x_{n+1},Tx_{n})$
$\displaystyle=$ $\displaystyle d(x,x_{n+1})+d(A,B)$ $\displaystyle\leq$
$\displaystyle d(x,x_{n+1})+d(x,B)$

so that

$d(x,B)\leq d(x,Tx_{n})\leq d(x,x_{n+1})+d(x,B).$ (1)

Letting $n\to\infty$, inequality (1) gives rise to

$d(x,Tx_{n})\stackrel{{\scriptstyle\mathbb{R}}}{{\to}}d(x,B).$

By approximatively compactness of $B$, $\\{Tx_{n}\\}$ has a subsequence
$\\{Tx_{n_{k}}\\}$ converging to $y\in B$. Thus, we have

$d(x,y)=d(A,B)$

yielding thereby $x\in A_{0}$. Hence, $A_{0}$ is proximally closed. ∎

######  Definition 7.

Let $(X,d)$ be a metric space and $(A,B)$ a pair of nonempty subsets of $X$. A
mapping $T:A\rightarrow B$ is called proximally continuous at a point $x\in A$
if for any $T$-proximal sequence $\\{x_{n}\\}\subset A$ such that
$x_{n}\stackrel{{\scriptstyle d}}{{\longrightarrow}}x$, we have
$T(x_{n})\stackrel{{\scriptstyle d}}{{\longrightarrow}}T(x)$. $T$ is called
proximally continuous if it is proximally continuous at each point of $X$.

Clearly, every continuous function is proximally continuous.

Now, we present a sharpened version of Theorem 1. Our result improves Theorem
1 in the following respects:

  1. $\bullet$

“Completeness of whole metric space $X$" is replaced by “proximally
completeness of subspace $A$". Moreover, closedness of $A$ and $B$ can be
relaxed.

  2. $\bullet$

“Approximatively compactness of $B$ (w.r.t. $A$)" is replaced by relatively
weaker notion, namely, “proximally closedness of $A_{0}$". Moreover, this
condition is not necessary as it can alternately be replaced by the
“proximally continuity of $T$".

######  Theorem 2.

Let $(X,d)$ be a metric space, $(A,B)$ a pair of nonempty subsets of $X$ and
$T:A\rightarrow B$ a mapping. Suppose that the following conditions hold:

  1. (i)

$A_{0}\neq\emptyset$

  2. (ii)

$T(A_{0})\subseteq B_{0}$,

  3. (iii)

$T$ is proximal contraction,

  4. (iv)

$A$ is proximally complete subspace of $X$,

  5. (v)

either $T$ is proximally continuous or $A_{0}$ is proximally closed subspace
of $A$.

Then $T$ has a unique best proximity point. Further, for any fixed element
$x_{0}\in A_{0}$, the $T$-proximal sequence $\\{x_{n}\\}$ based on initial
point $x_{0}$ converges to the unique best proximity point of $T$.

###### Proof.

In view of assumption (i), $\exists$ $x_{0}\in A_{0}$. As $T(x_{0})\in
T(A_{0})\subseteq B_{0}$, $\exists$ $x_{1}\in A_{0}$ such that

$d(x_{1},Tx_{0})=d(A,B).$

As $T(x_{1})\in T(A_{0})\subseteq B_{0}$, $\exists$ $x_{2}\in A_{0}$ such that

$d(x_{2},Tx_{1})=d(A,B).$

Continuing this process, by induction, we can construct a sequence
$\\{x_{n}\\}\subset A_{0}$ such that

$d(x_{n+1},Tx_{n})=d(A,B).$ (2)

Hence, $\\{x_{n}\\}$ is a $T$-proximal sequence. Applying the contractivity
condition (iii) to (2), we deduce, for all $n\in\mathbb{N}_{0}$ and for some
$k\in[0,1)$, that

$d(x_{n+1},x_{n+2})\leq kd(x_{n},x_{n+1}),$

which by induction yields that

$d(x_{n+1},x_{n+2})\leq k^{n+1}d(x_{0},x_{1}).$ (3)

For all $m,n\in\mathbb{N}$ with $m<n$, using (3) and triangular inequality, we
get

$\displaystyle d(x_{m},x_{n})$ $\displaystyle\leq$ $\displaystyle
d(x_{m},x_{m+1})+d(x_{m+1},x_{m+2})+\cdots+d(x_{n-1},x_{n})$
$\displaystyle\leq$
$\displaystyle(k^{m}+k^{m+1}+\cdots+k^{n-1})d(x_{0},x_{1})$ $\displaystyle=$
$\displaystyle k^{m}(1+k+k^{2}+\cdots+k^{n-m-1})d(x_{0},x_{1})$
$\displaystyle\leq$ $\displaystyle\frac{k^{m}}{1-k}d(x_{0},x_{1}),~{}~{}({\rm
where}~{}0\leq k<1)$ $\displaystyle\rightarrow$ $\displaystyle 0\;{\rm
as}\;m\;({\rm and~{}hence}~{}n)\rightarrow\infty,$

which implies that the sequence $\\{x_{n}\\}$ is Cauchy. Hence, $\\{x_{n}\\}$
is a $T$-proximal Cauchy sequence in $A$. By proximally completeness of $A$,
$\exists~{}\overline{x}\in A$ such that $x_{n}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\overline{x}$.

Now, we use assumption (v) to show that $x$ is a best proximity point of $T$.
Suppose that $T$ is proximally continuous. As $\\{x_{n}\\}$ is a $T$-proximal
sequence satisfying $x_{n}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\overline{x}$, proximally continuity of $T$ implies that
$T(x_{n})\stackrel{{\scriptstyle d}}{{\longrightarrow}}T(\overline{x})$. Using
continuity of $d$ and (2), we get

$d(\overline{x},T\overline{x})=d(\lim\limits_{n\rightarrow\infty}x_{n+1},\lim\limits_{n\rightarrow\infty}Tx_{n})=\lim\limits_{n\rightarrow\infty}d(x_{n+1},Tx_{n})=d(A,B)$

so that $\overline{x}$ is a best proximity point of $T$. Alternately, assume
that $A_{0}$ is proximally closed subspace of $A$. As $\\{x_{n}\\}\subset
A_{0}$ is a $T$-proximal sequence satisfying $x_{n}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\overline{x}\in A$, we have $\overline{x}\in A_{0}$.
Also by assumption (ii), we get $T(\overline{x})\in B_{0}$. Therefore,
$\exists~{}\omega\in A_{0}$ such that

$d(\omega,T\overline{x})=d(A,B).$ (4)

Combining (2) and (4), we get

$d(x_{n+1},Tx_{n})=d(\omega,T\overline{x})=d(A,B).$ (5)

Using assumption (iii) and $x_{n}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}x$, we obtain for some $0\leq k<1$ that

$\displaystyle d(x_{n+1},\omega)\leq kd(x_{n},\overline{x})\rightarrow 0\;{\rm
as}\;n\rightarrow\infty$

so that $x_{n}\stackrel{{\scriptstyle d}}{{\longrightarrow}}\omega$. Owing to
the uniqueness of limit, we obtain $\omega=\overline{x}$. Hence, (4) reduces
to

$d(\overline{x},T\overline{x})=d(A,B),$

so that $\overline{x}$ is a best proximity point of $T$.

Finally, we prove the uniqueness of best proximity point. Suppose that
$\overline{x}$ and $\overline{y}$ are two best proximity points of $T$. Then,
we have

$d(\overline{x},T\overline{x})=d(\overline{y},T\overline{y})=d(A,B).$ (6)

Applying proximal contractivity condition (iii) to (6), we get

$d(\overline{x},\overline{y})\leq kd(\overline{x},\overline{y}),$

for some $0\leq k<1$, yielding thereby $\overline{x}=\overline{y}$. Hence, $T$
has a unique best proximity point. ∎

######  Remark 5.

Under the restriction $A=B=X$, Theorem 2 reduces to classical Banach
contraction principle.

##  4 An Application to Variational Inequality Problem

Let $H$ we a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$
and induced norm $\|\cdot\|$. Let $K$ be a nonempty closed and convex subset
of $H$ and $S:H\to H$ an operator. Consider the following variational
inequality problem:

$\mbox{Find~{}}u\in K\mbox{such~{}that~{}}\langle Su,v-u\rangle\geq
0,\quad\forall~{}v\in K.$ (7)

To solve problem (7), we use the theoretical results concerning the metric
projection operator $P_{K}:H\to K$. Notice that that for each $u\in H$, there
exists a unique nearest point $P_{K}(u)$ such that

$\|u-P_{K}u\|\leq\|u-v\|,\quad\forall~{}v\in K.$

The theory of variational inequalities is motivated by equilibrium problems.
During last three decades, the theory of variational inequalities emerged as a
rapidly growing area of research due to its appearance in the fields of
nonlinear analysis, operations research, economics, game theory, mathematical
physics and calculus of variations associated with the minimization of
infinite-dimensional functionals. For details see [11, 12, 13] and the
references therein. In the recent past, many authors solved the variational
inequality problems employing fixed point theorems. The following known
results correlate the solvability of a variational inequality problem to the
solvability of certain fixed point problem.

######  Proposition 2.

[13] Let $z\in H$. Then $u\in K$ satisfies the inequality $\langle
u-z,y-u\rangle\geq 0$, for all $y\in K$ if and only if $u=P_{K}(z)$.

######  Lemma 1.

[13] Let $S:H\to H$ an operator. Then $u\in K$ solves the inequality $\langle
Su,v-u\rangle\geq 0$, for all $v\in K$ if and only if $u=P_{K}(u-\lambda Su)$,
where $\lambda>0$.

The main result of this section runs as follows:

######  Theorem 3.

Let $K$ be a nonempty closed and convex subset of a real Hilbert space $H$. If
$S:H\to H$ is an operator such that $P_{K}(I-\lambda S):K\to K$, with
$\lambda>0$, forms a proximal contraction, then the problem (7) has a unique
solution $u^{\ast}\in K$. Moreover, for each $u_{0}\in K$, the recursive
sequence $\\{u_{n}\\}\subset K$ defined by $u_{n+1}:=P_{K}(u_{n}-\lambda
Su_{n})$, for all $n\in\mathbb{N}$, converges to $u^{\ast}$.

###### Proof.

Define the operator $T:K\rightarrow K$ by $T(x)=P_{K}(x-\lambda Sx)$, for all
$x\in K$. $T$ satisfies all the hypotheses mentioned in Theorem 2 for $A=B=K$
in the context of real Hilbert space $H$ equipped with norm metric $d$.
Consequently, there exists a unique $u^{\ast}$ such that
$d(u^{\ast},Tu^{\ast})=d(K,K)=0$ and $u_{n}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}u^{\ast}$. This yields that
$u^{\ast}=T(u^{\ast})=P_{K}(u^{\ast}-\lambda Su^{\ast})$ and hence from Lemma
1, it follows that $u^{\ast}$ is a solution of the problem (7). This completes
the proof. ∎

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