# Geometrical Approximation to the AdS/CFT Correspondence

M. A. Martin Contreras, J. M. R. Roldan Giraldo
High Energy Group, Department of Physics, University of los Andes
ma.martin41<EMAIL_ADDRESS>
###### Abstract

In this paper an analysis of the geometrical construction of the AdS/CFT
Correspondence is made. A geometrical definition of the configuration manifold
and the boundary manifold in terms of the conformal compactification scheme is
given. As a conclusion, it was obtained that the usual definition of the
correspondence [2] is strongly dependent of the unicity of the conformal class
of metrics on the boundary. Finally, a summary of some of the geometrical
issues of the correspondence is made, along with a possible way to avoid them.

##  1 Introduction

Gravity/Gauge duality is maybe one of the most important developments of the
latest times in String Theory. From its very begining, dual models have been
applied in many areas different from High Energy Physics or Black Hole
Physics. Any branch of Physics that exhibits phase transitions can be modeled
using dual models[1].

The central idea of Gravity/Gauge duality is the geometrical connection
existing between any Gravity Theory (Superstrings, for example) in $d+1$
dimensions to a QFT living in $d$ dimensions. In fact, it can be said that _we
can extract information about QFT from spacetime_ , and viceversa. This is
just a conjecture, and it still needs a proof. Once the connection between
bulk and boundary is stablished, the next step is to write of a proper
holographic dictionary, allowing to switch between gravity and QFT.

AdS/CFT Correspondence [2] is the most relevant realization of the
Gravity/Gauge duality, but is not the only successful one. Some examples of
this kind of duality are the Klebanov–Strassler duality [3] or the
NS5–branes/LST [4]. In all the three cases mentioned above, the bulk is a
non–compact manifold endowed with gravity, such that the dual gauge theory is
encoded in its asymptotic behavior.

##  2 AdS/CFT Correspondence in a Nutshell

The most representative holographic duality is the AdS/CFT Correspondence
(Maldacena 1998). In this duality we link gravity in a weakly curved
AdS${}_{5}\,\times\,S^{5}$ with a CFT in $3+1$ dimensions, which is in the
conformal boundary of AdS. AdS/CFT Correspondence has strong/weak duality too,
which relates SUGRA backgrounds at strong coupling with CFT at weak coupling.
Thanks to this, it has been possible to construct toy models for thermal (non
perturbative) QCD, as for example, the dual models of QGP using Dp/Dq branes
as gravitational background.

The idea behind the AdS/CFT Correspondence is the geometrical connection
between the isomeries of AdS and the conformal group. To be more precise,
since AdS is a maximally symmetric space, its isomeries are holomorphic to the
Poincare Group. This implies that, at inner level, AdS and any CFT are
_essentially_ the same thing. The statement of the correspondence is

$Z_{\text{String}}\left[\phi,\mathcal{M}\right]=Z_{CFT}\left[\phi_{0},\mathcal{O};\partial\,\mathcal{M},\eta\right],$
(1)

where $\mathcal{M}$ is the manifold where gravity lies, $\phi$ is a bulk field
with $\phi_{0}$ as the value at the conformal boundary $\partial\mathcal{M}$.
The conformal boundary carries a metric in a fixed conformal class $[\eta]$.
The conjecture stablishes that $\phi_{0}$ acts as a Schwinger source for any
CFT operator $\mathcal{O}$ living on $\partial\,\mathcal{M}$. This is the
essence of the correspondence.

Some remarks. The conjecture in principle can be made with any background
$\mathcal{M}$ that satisfies string equations of motion and has a the pair
$\left(\partial\,\mathcal{M}\,,\eta\right)$. Since the solution is not unique,
i.e., the charts over $M$ are not trivial, the correlation functions are
dependent from the choice of coordinates. As a conclusion, it is possible to
obtain different holographies according to the choice of chart. For example,
in the Maldacena’s original proposal, the Anti de Sitter space is covered
partially with a Poincare chart $AdS_{5}$ that picks up one of the two folds
of the hyperbolic space, fixing a conformal boundary at the origin of the
radial coordinate of $AdS_{5}$. This conformal boundary has a topology of
$\mathbb{R}^{1,3}$. Choices of different charts on Anti de Sitter space lead
to boundaries as $\mathbb{R}^{4}$, $S^{1}\times S^{3}$,
$S^{1}\times\mathbb{R}^{3}$ or $S^{1}\times\mathbb{H}^{3}$ [6]. All of these
topologies are diffeomorphical between each other. This has a deeper
implication in the foundations of the correspondence, because different charts
could lead to different dualities.

The utility of the correspondence comes in the calculation procedure, encoded
in the holographic dictionary, which is the relation between the bulk and the
boundary physics. Since the Anti de Sitter radius and the string lenght are
free parameters, it is possible to take a low energy limit in (1) in order to
reduce the string generating function to a supergravity one,

$W_{CFT}\left[\phi_{0},\mathcal{O};\partial\,\mathcal{M},\eta\right]=-\text{ln}\,Z_{CFT}\left[\phi_{0},\mathcal{O};\partial\,\mathcal{M},\eta\right]=\sum_{i}Z_{\text{SUGRA}}\left[\phi,\mathcal{M}_{i}\right]+O\left(\frac{1}{N}\right)+O\left(\frac{1}{\sqrt{\lambda}}\right),$
(2)

where the sum in the supergravity action appears to take into account the
chart dependence. The supergravity description is valid only for the large $N$
and large ’tHooft coupling $\lambda$. Note that the supergravity action can
carry divergences due to infinite volume or IR behaviour. These divergences
must be renormalized [7] and could lead to anomalies.

The dictionary is obtained following the saddle point approximation and the
functional standard techniques from the supergravity on-shell action:

$\langle\mathcal{O}\left(x_{1}\right)\,\mathcal{O}\left(x_{2}\right)...\mathcal{O}\left(x_{n}\right)\rangle_{CFT}=\left.\frac{\delta^{n}S_{\text{SUGRA}}^{\text{on-
shell}}\left[\phi_{0},...\right]}{\delta\phi_{0}\left(x_{1}\right)...\phi_{0}\left(x_{n}\right)}\right|_{\text{Sources}=0}.$
(3)

Expresion (3) tells how to connect fields in both sides. For example, the
dilaton is related with the string coupling. For each possible supergravity
action a dictionary can be constructed. This is the path followed, for
example, in AdS/QCD models [8].

##  3 Geometrical Approximation to the Correspondence

###  3.1 Formal Aspects

Geometrically speaking, the correspondence is build up using the complex
geometry language. Consider a open $n+1$-dimensional manifold
$\left(M,g\right)$. This manifold $M$ will be the configuration space for the
possible physical states on the bulk. Along with this manifold, we define a
closed $n+1$-dimensional manifold $(\tilde{M},\tilde{g})$ with no empty
$n$-dimensional boundary $\partial\tilde{M}$, such that $M\subset\tilde{M}$. A
complete Riemmann metric $g$ on $M$ is called _conformally compact_
111Conformally compact is equivalent to Penrose compact. if a function
$f\in\,\Omega_{0}(\tilde{M})$ on $\tilde{M}$ exist such that

$\tilde{g}=f^{2}\,g,$ (4)

with $f^{-1}\left(0\right)=\partial\tilde{M}$ and $df$ is not zero on
$\partial\tilde{M}$. Such a function is called a _defining_ function [9]. The
metric $\tilde{g}$ is called _compactification_ of the metric $g$. The
compactification defines an induced metric
$\eta=\tilde{g}|_{\partial\tilde{M}}$ on $\partial\tilde{M}$.

There are many defining functions, and hence many conformal compactifications
of a given metric $g$, then the choice of $\eta$ is not unique. This problem
can be avoided using the conformal class $[\eta]$ (called conformal infinity)
of $\eta$ on $\partial\tilde{M}$ defined by conformal transformations of
$\eta$. Recall that $[\eta]$ is uniquely determined by the pair $(M,g)$.
Physically, the choice of $[\eta]$ implies that the causal structure of
spacetime is conserved under conformal transformations. The pair
$(\partial\tilde{M},\eta)$ with $\eta\in[\eta]$, defines the _conformal
boundary_ , where the CFT operators are constructed.

Since the symmetries of $M$ and $\partial\tilde{M}$ must be the same222Both
manifolds must have the same causal structure., the moduli space of
$\partial\tilde{M}$, $\mathcal{M}_{\partial\tilde{M}}$ is defined by
$\mathcal{M}_{\partial\tilde{M}}=\text{Teich}(M)/\text{MCG}(M)$, since both
manifolds must have the same conformal stucture because they are
diffeomorphic. Following the discussions above, the entire moduli space of
$M$, $\mathcal{M}$, is restricted by the choice of a conformal class $[\eta]$,
thus not all the metrics $g$ on $M$ will contribute to the partition function
on the bulk. The restricted moduli space of $M$,
$\mathcal{M}_{(\partial\tilde{M},[\eta])}$, is defined as the set of all the
conformally compact metrics $g$ on $M$ [10].

Under these ideas, the AdS/CFT Correspondence can be summarized saying that
given any bulk data $(M,g)$, it is possible to construct (or obtain) a
boundary $(\partial\tilde{M},[\eta])$ by means of the conformal
compactification scheme (4), i.e,

$\overbrace{Z\left(\partial\tilde{M},[\eta]\right)}^{\text{Boundary}}=\overbrace{\sum_{g\in\mathcal{M}_{(\partial\tilde{M},[\eta])}}Z\left(g,M\right).}^{\text{Bulk}}$
(5)

Following physical arguments from AdS/CFT Correspondence, $M$ must be
10-dimensional.Thus, in order to have a conformal boundary as
$\mathbb{R}^{(1,3)}$, $M$ has to be decomposed into $M=\mathbb{H}^{5}\times
X^{5}$, with $X^{5}$ some compact space, such that in the compactification
limit $M\backsim\mathbb{R}^{2,4}\subset\mathbb{H}^{5}$, as in the AdS/CFT
Correspondence, in which $M$ is factorized as $AdS_{5}\times S^{5}$. All the
metrics $g$ that satisfies these conditions are the so called _asymptotically
hyperbolic Einstein metrics_.

###  3.2 Geodesic Compactifications

Any compactification (4) with a defining function given by
$f_{g}=\text{Dist}_{g}\left(x,M\right)$ is called _geodesic_ [9, 10]. These
compactifications are useful for computational purposes, and because given a
conformal infinity $[\eta]$ of $(M,g)$ exists a unique geodesic defining
function $f_{[\eta]}$ that has $\eta\in[\eta]$ as a boundary metric.

Following the Gauss lemma, the compactificacion $\tilde{g}$ can be expanded
into

$\tilde{g}=dr^{2}+g_{f},$ (6)

where $g_{f}$ is a family of metrics on $\partial\tilde{M}$. The
Fefferman–Graham expansion [11] of $g$ is a truncated Taylor-type expansion of
the family of metrics $g_{f}$, that depends on the dimensionality $n$ of $M$.
The exact form of the series depends on whether $n$ is even or odd. In a
general case, the series can be written as

$g_{f}=g_{\left(0\right)}+r\,g_{\left(1\right)}+r^{2}\,g_{\left(2\right)}+\ldots+r^{n}\,g_{\left(n\right)}+\text{terms
depending of even or odd }n,$ (7)

where $\eta:=g_{\left(0\right)}$ and the coefficients $g_{\left(k\right)}$
with $1<k<n$ are locally fixed by the curvature of $\eta$ and its covariant
derivatives. The extra terms depending on the even–odd character of $n$ are
calculated from the Einstein equations for $\eta$.

The $g_{\left(n\right)}$ term is a little more complex. For even dimensions,
$g_{\left(n\right)}$ is transverse traceless, but is determined by global
properties of $M$. In odd dimensions, $g_{\left(n\right)}$ is not traceless
but is still determined by global aspects of $M$. The $g_{\left(n\right)}$
factor corresponds to the stress–energy tensor of the CFT living in
$\partial\tilde{M}$.

Mathematically, these expansions can be obtained by compactifying the Einstein
equations and taking Lie derivatives of $\tilde{g}$ with $f_{g}=0$:

$g_{\left(k\right)}=\frac{1}{k!}\,\mathcal{L}_{\tilde{\nabla}f_{g}}^{\left(k\right)}\tilde{g}.$
(8)

If the metric is Hoelder, all the expansions hold up to order $m+\alpha$, with
$\alpha$ the Hoelder exponent.

As a conclusion, knowing $g_{\left(0\right)}$ and $g_{\left(n\right)}$ allow
to construct the bulk metric field $g$ from the expansion (7). The real
problem here is to know the convergence of the series and how its inclusion
may introduce anomalies [12].

###  3.3 General Decomposition of $M$

Until now, all of the approach to the conjecture was classical, i.e, real
manifolds only. A quantum approach (thinking on strings instead of
supergravity) requires a more general factorization $M=X\times Y$, where
$X\in\mathbb{H}^{5}$ and $Y$ is a 5-dimensional Calabi–Yau manifold. The
Calabi-Yau manifold can be justified on the grounds that classical mechanics
requires a simplectic structure while quantum mechanics requires complex
structure to implement unitarity. The main problem with these structures lies
on the construction of Calabi-Yau metrics. This problem can be partially
avoided by considering $Y$ as a 5-dimensional Sasaki-Einstein manifold [13].

##  4 Geometrical Issues

As it was said above, the central idea for the construction of the conjecture
is the existence of a conformal infinty $[\eta]$ that fixes a conformal
boundary $\partial\tilde{M}$. This process is highly depending on the
convergence of the Fefferman-Graham expansion (7), which could introduce
undesirable anomalies due to the holographic renormalization. But this is not
the only problem.

In a realistic approximation, Gravity/Gauge duality suggests that any quantum
field theory must have a string dual. The large $N$ and large $\lambda$ limits
restrict the possible dual models to the AdS/CFT Correspondence, that is no
realistic. Leaving aside the limits, to obtain a non-conformal holography
would imply the naive idea of taking a different background from Type IIB
supergravity.

Advances in this scenario were given by Skenderis and Taylor with their
precision holography [14]. The idea is to categorize all the possible $X$
manifolds in the decomposition $M=X\times Y$ into spaces that are
_asymptotically AdS_ and those which are not. Asymptotically AdS spaces are
related to the usual 10-dimensional $AdS_{n+2}\times S^{8-n}$ through a Weyl
transformation. This transformation redefines the coupling constant of the QFT
on the conformal boundary including an energy scale with no trivial running.
As a conclusion from [14], only on asymptotically AdS spaces it is possible to
do non conformal holography. This implies that holographic extension only can
be made on AdS-like spaces.

Another issue arises when the index theory comes into play. Following [9, 13],
the conjecture is build up in conformal boundaries, where the index of any
pseudodifferential operator is well defined. When closed and compact manifolds
are considered, the index theorem fails. This problem leads to the
consideration of the definition and the role of the boundary in AdS/CFT
Correspondence [10].

##  5 Conclusions

AdS/CFT Correspondence is strongly related to the concept of conformal
boundary. The construction of this boundary is dependent on the chosen charts,
thus the holographic dictionary (3) is not univocally. The usual chart used to
do holography is the Poincare chart. Non-conformal extensions are made
relaxing the conformal symmetry of $AdS_{5}\times S^{5}$.

The choice of a Calabi-Yau (or a Sasaki–Einstein) manifold as the compact
space in the factorization $AdS_{5}\times Y^{5}$, besides the relaxation of
the large $N$ and large $\lambda$ limits could lead to string/QFT duality.

## References

  * [1] E. Papantonopoulos, Editor. _From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence_. Springer. 2011.
  * [2] J. M. Maldacena._The large N limit of superconformal field theories and supergravity_ Adv. Theor. Math. Phys. 2, 231 (1998). [arXiv:hep-th/9711200].
  * [3] I. R. Klebanov, M. J. Strassler, _Supergravity and confining gauge theory: Duality Cascades and $\chi$SB-resolution of Naked Singularities_, JHEP 0008:052 (2000) [arXiv:hep-th/0007191].
  * [4] O. Aharony, M. Berkooz, D. Kustasov, N. Seiberg. _Linear Dilatons, NS5-branes and holography_ , JHEP 9810:004 (1998) [arXiv:hep-th/9808149]
  * [5] E. Witten, _Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories_. arXiv:hep-th/9803131v2. 1998.
  * [6] C. A. Ballon Bayona, N. R. F. Braga, _Anti-de Sitter boundary in Poincare coordinates_. Gen. Rel. Grav. 39: 1367-1379,(2007).
  * [7] I. Papadimitriou, K. Skenderis. _AdS/CFT Correspondence and Geometry_. IRMA Lectures in Mathematics and Theoretical Physics: AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, (2005).
  * [8] U. Gursoy, E. Kiritsis. _Exploring improved holographic theories for QCD Part 1_. JHEP 0802, 019 (2008).
  * [9] M. T. Anderson. _Geometrical Aspects of the AdS/CFT Correspondence_. IRMA Lectures in Mathematics and Theoretical Physics: AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, (2005).
  * [10] M. Sanchez. _Causal boundaries and holography on wave type spacetimes_. Nonlinear Analysis 71, e1744e1764 (2009).
  * [11] C. Fefferman, C. Graham. _Conformal invariants_. Ellie Cartan et les mathematiques d’aujourd’hui (1984), Asterisque (1985), 95 - 116.
  * [12] C. R. Graham, E. Witten. _Conformal Anomaly Of Submanifold Observables In AdS/CFT Correspondence_. Nucl. Phys. B 546: 52-64 (1999).
  * [13] J. P. Gaunttlet, D. Martelli, J. Sparks, D. Waldram. _Supersymmetric AdS backgrounds in String and M-theory_. IRMA Lectures in Mathematics and Theoretical Physics: AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, (2005).
  * [14] I. Kanitscheider, K. Skenderis, M. Taylor. _Precision Holography for non-conformal branes_. JHEP 09, 094 (2008).


††institutetext: Center for Joint Quantum Studies and Department of Physics,
School of Science, Tianjin University, Yaguan Road 135, Jinnan District,
300350 Tianjin, P. R. China

# Upper bounds of holographic entanglement entropy growth rate for thermofield
double states

Ze Li and Run-Qiu Yang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We studied the upper bounds of the holographic entanglement entropy growth
rate for thermofield double (TFD) states. By comparing the cases of vacuum AdS
and charged AdS black holes, we conjecture: for all static planar or
spherically symmetric asymptotically Schwarzschild-AdS black holes of same
mass density or entropy density, the vacuum AdS black hole gives the maximum
entanglement entropy growth rate. We gave proofs by assuming dominant energy
condition. We also considered the AdS black hole spacetime with real scalar
fields case, where the scalar fields violate the dominant energy condition and
the bulk geometry is not asymptotically Schwarzschild-AdS. Numerical results
show that this case vacuum black hole still has maximal growth rate if we
fixed entropy. However, in the case of fixed energy, vacuum case has maximal
growth rate of entanglement entropy only under standard quantization scheme.

##  1 Introduction

In recent years, exciting works based on the correspondence between the
gravity of asymptotically anti-de Sitter spacetime and the conformal field
theory, i.e. AdS/CFT correspondence, emerge one after another. Roughly
speaking, the AdS/CFT correspondence says that there is an equivalent
relationship between $d+1$-dimensional gravity theory in asymptotically AdSd+1
spacetime and $d$-dimensional conformal field theory on the boundary of the
AdSd+1 spacetime. Since it was first proposed in Maldacena:1997re , there have
been more and more evidences to support it in various different models. In
these models people have concluded a few of universal dual relationship
between the quantities in gravity theory and boundary field theory.

The entanglement entropy is an important quantity in the studies of both
gravity itself and the dual boundary field theory. In quantum field theory and
quantum mechanical systems it is usually computed by the von Neumann entropy
formula

$S_{vN}=-\rm Tr\rho\rm ln\rho,$ (1)

where $\rho$ is density matrix of a quantum system. In some situations, the
von Neumann entropy is not easy to be calculated directly. However, we can
calculate it by the AdS/CFT correspondence. The Ryu-Takayanagi (RT)
prescription tells us that the entanglement entropy of boundary CFT can be
calculated by the R-T surface $\gamma_{A}$ in the bulk as Ryu:2006bv

$S_{A}=\frac{\rm{Area}(\gamma_{A})}{4G_{N}^{(d+1)}},$ (2)

here $\gamma_{A}$ is the codimension-2 bulk minimal surface homologous to
subsystem $A$ of dual CFT and $G_{N}^{(d+1)}$ is the $(d+1)$-dimensional
Newton’s constant. The original proposal (2) was given on a particular
spacelike slice and is not covariant. As for covariant entanglement entropy,
we have the Hubeny-Rangamani-Takayanagi (HRT) surface correspondingly
Hubeny:2007xt . Furthermore, if we take the contribution of bulk matters into
account, it will require the extremal surface to minimize the generalized
entropy

$S_{\rm{EE}}=\rm{min}\left\\{\text{ext}\left[\frac{\text{Area}(\gamma_{A})}{4G_{N}^{(d+1)}}+S_{\rm
bulk}(\Sigma_{\gamma_{A}})\right]\right\\},$ (3)

where $\Sigma_{\gamma_{A}}$ is the region surrounded by $\gamma_{A}$ and
boundary subsystem $A$. The $\gamma_{A}$ that makes $S_{EE}$ to the minimum
value is called the quantum extremal surface (QES) Faulkner:2013ana ;
Engelhardt:2014gca .

The AdS black hole spacetime of maximum analytical continuation is dual to two
CFTs with finite temperature on its two boundaries respectively Israel:1976ur
; Maldacena:2001kr . The states in this model are called the thermofield
double(TFD) states. Usually, the extremal surface will not penetrate the
horizon Ryu:2006bv ; Hubeny:2012ry . However, for the two-sided AdS black hole
spacetime of maximum analytical continuation, the extremal surface has
possibility to penetrate the horizon if its two "endpoints" are anchored on
the two side boundaries respectively Hubeny:2007xt ; Hartman:2013qma ; Abajo-
Arrastia:2010ajo ; Aparicio:2011zy ; Albash:2010mv ; Balasubramanian:2010ce ;
Balasubramanian:2011ur . Such surfaces are also called Hartman-Maldacena(H-M)
surfaces. The H-M surface is an important tool for computing the holographic
entanglement entropy. It has some special properties, such like it increases
over time and the growth rate will approach to constant when time goes to
infinite Hartman:2013qma .

It is physical interesting to study the possible maximal growth rate of
entanglement entropy in many situations. For example, in quantum computation
theory, the quantum entanglement is regarded as a computing resource, which
provides the computational speed-up Ekert:1997wh ; Jozsa:1997hc . Studying the
growth rate of entanglement entropy can help us understand how to reach the
maximum entanglement state quickly. In addition, entanglement entropy plays an
important role in the black hole information paradox. To interpret the Page
Curve is recognized as an important task of solving black hole information
paradox Page:1993wv ; Page:2013dx . Some recent studies show that we can
explain the Page Curve from the perspective of holography, which says that a
region called "island" will appear in the later period of black hole
evaporation Almheiri:2020cfm ; Almheiri:2019qdq ; Penington:2019kki ;
Chen:2020hmv . According to holographic interpretation Almheiri:2019hni ;
Chen:2020jvn , the holographic entanglement entropy is calculated by H-M
surface before the "island" appears Almheiri:2019psy ; Ling:2020laa . When the
holographic entanglement entropy increases to the Beckenstein-Hawking entropy,
the "island" appears Almheiri:2019hni ; Chen:2019uhq ; Chen:2020hmv . By
studying the maximum growth rate, we can give the fastest situation that the
islands emerge.

Similar to the holographic entanglement entropy which is calculated by the
area of codimension-2 extremal surface, the holographic complexity in
“complexity-volume” (CV)-conjecture is calculated by the volume of
codimension-1 extremal surface. The quantum computation theory gives an upper
bound of the growth rate of complexity when the total energy of the system is
fixed, which is known as the Lloyd’s bound Lloyd2000-jm . There is also a
bound proved to be applicable to holographic complexity in CV-conjecture
Yang:2019alh . This inspires us that there may also be an upper bound for the
growth rate of entanglement entropy if the energy of the system is fixed.

In order to find the maximum growth rate of entanglement entropy for TFD
states in holography, we consider the entanglement between two half planes
which locate at two AdS boundaries symmetrically. This paper first calculates
the growth rate of holographic entanglemant entropy for the cases of vacuum
spherically/planar/hyperbolically symmetric black hole spacetime and
corresponding charged black hole spacetime with fixed mass, entropy density
and temperature. We find the that the entanglement growth rate of vacuum black
holes are always faster than the corresponding charged black hole with fixed
mass density and entropy density. Then we will give a conjecture: for static
asymptotically planar/spherically symmetric Schwarzschild-AdS black holes, the
vacuum black hole gives the upper bound of the growth rate under the
conditions of fixed mass density and entropy density. Then we will give proofs
under dominant energy condition. Furthermore, we perform numerical calculation
for the case that the spacetime with real scalar fields. Here the dominant
energy condition is not satisfied and the bulk spacetime is asymptotically AdS
but not asymptotically Schwarzschild-AdS. We find that the previous conclusion
is still valid under one of the two quantization schemes Marolf:2006nd .

This paper is organized as follows: We will introduce basics of TFD states and
H-M surfaces in section 2. Then we study the growth rate of holographic
entanglement entropy in Schwarzchild-AdS black hole spacetime case and RN-AdS
black hole spacetime cases as examples in section 3. We next give a general
proof for the static spacetime with planar and spherical symmetries in section
4 and assuming dominant energy condition. In scetion 5, we will consider
spacetime with matter fields that do not obey the dominant energy condition
and the bulk is asymptotically AdS but not Schwarzschild-AdS.

##  2 Area of extremal surface and its growth rate

In this section, we first introduce our holographic model of black hole
spacetime dual to boundary thermal CFTs. Then we will calculate the area of
the extremal surface. We will also give a method to compute the growth rate of
the area.

###  2.1 The holographic model

We consider the $(d+1)$-dimensional static AdS black hole spacetime with
spherical/planar/hyperbolic symmetries as the bulk gravity theories, of which
metric reads

$\text{d}s^{2}=\frac{1}{z^{2}}\left[-f(z)\mathrm{e}^{-\chi(z)}\text{d}t^{2}+\frac{\text{d}z^{2}}{f(z)}+\text{d}\Sigma_{k,d-1}^{2}\right].$
(4)

The $\text{d}\Sigma_{k,d-1}^{2}$ is $(d-1)$-dimensional angular direction line
element, which is given by

$\text{d}\Sigma_{k,d-1}^{2}=\begin{cases}\text{d}\Omega_{d-1}^{2}=\text{d}\theta^{2}+\sin^{2}\theta\text{d}\Omega_{d-2}^{2}&\text{
for }k=+1\\\
\text{d}\mathbf{x}_{d-1}^{2}=\sum_{i=1}^{d-1}\text{d}x_{i}^{2}/\ell_{\rm
AdS}^{2}&\text{ for }k=0\\\
\text{d}\Xi_{d-1}^{2}=\text{d}\theta^{2}+\sinh^{2}\theta\text{d}\Omega_{d-2}^{2}&\text{
for }k=-1\end{cases}.$ (5)

Where $k=\\{1,0,-1\\}$ represent spherical, planar and hyperbolic symmetries
of $(d-1)$-dimensional spatial directions respectively. The
$\text{d}\Omega_{d-2}^{2}$ is $(d-2)$-dimensional spherical coordinates
metric. The $\ell_{\mathrm{AdS}}$ is the AdS radius. The maximum analytical
continuation of the black hole spacetime described by metric (4) is a two-
sided black hole spacetime. It is dual to two copies of finite temperature
CFTs defined on the left and right side boundaries. The temperature of the
CFTs is just equal to the black hole temperature.

We first take a time slice
$\mathcal{W}(t_{B})=\mathcal{W}_{R}(t_{B})\cup\mathcal{W}_{L}(t_{B})$ of the
two-sided CFT as the total system. Here $t_{B}$ is the time coordinate on the
boundaries. Then we choose the subsystem
$\mathcal{D}(t_{B})=\mathcal{D}_{R}(t_{B})\cup\mathcal{D}_{L}(t_{B})$, where
$\mathcal{D}_{R}(t_{B})$ and $\mathcal{D}_{L}(t_{B})$ are subsystems of the
right and left side CFT time slice $\mathcal{W}_{R}(t_{B})$ and
$\mathcal{W}_{L}(t_{B})$ respectively. We set $\mathcal{D}_{R}(t_{B})$ and
$\mathcal{D}_{L}(t_{B})$ to satisfy

$\displaystyle\begin{cases}\mathcal{D}_{R}(t_{B})&=\\{x^{1}\textgreater
0|\mathcal{W}_{R}(t_{B})\\}\\\ \mathcal{D}_{L}(t_{B})&=\\{x^{1}\textgreater
0|\mathcal{W}_{L}(t_{B})\\}\end{cases}\qquad\text{ for }k=0,$ (6)
$\displaystyle\begin{cases}\mathcal{D}_{R}(t_{B})&=\\{0\textless\theta\textless\pi|\mathcal{W}_{R}(t_{B})\\}\\\
\mathcal{D}_{L}(t_{B})&=\\{0\textless\theta\textless\pi|\mathcal{W}_{L}(t_{B})\\}\end{cases}\qquad\text{
for }k=\pm 1.$

That is to say, we separate the $(d-1)-$dimensional space $\mathcal{W}$ into
two half spaces. The time evolution of the boundary subsystem
$\mathcal{D}(t_{B})$ is non-trivial, since it is generated by two different
time-like Killing vectors $\xi^{a}$ and $-\xi^{a}$ on the right and left
boundaries respectively(see Figure 1), where $\xi^{a}$ is the bulk Killing
vector which stands for the static symmetry. The entanglement entropy will
also change over time, we can compute it by computing the area of extremal
surface that anchored on the two boundaries $\partial\mathcal{D}_{L}$ and
$\partial\mathcal{D}_{R}$ of subsystem $\mathcal{D}$. Such surface is known as
the Hartman-Maldacena(H-M) surface Hartman:2013qma .

Figure 1: The sketch of boundary CFT time slice and extremal surface. The two
boundary slices $\mathcal{W}_{R}$ and $\mathcal{W}_{L}$ combine into the total
system $\mathcal{W}$. $\mathcal{D}$ is the subsystem of $\mathcal{W}$, which
marked in green. The blue parts are the complement subsystem of $\mathcal{D}$.
The extremal surface $\Gamma_{0}$ anchors on the subsystem boundaries
$\partial\mathcal{D}$, which marked in red. $\xi^{a}$ and $-\xi^{a}$ are
Killing vectors on the right and left boundaries.

###  2.2 Area of extremal surface and its Growth rate

Since we can always let $t_{R}=-t_{L}=t_{B}\geqslant 0$ by Lorentz boost, the
extremal surface could be symmetric in the left and right parts. Due to the
symmetry, this extremal surface can be parameterized locally by

$z=z(\lambda),t=t(\lambda)\,.$ (7)

with the boundary condition
$z(-\infty)=z_{L}=0,z(\infty)=z_{R}=0,t(-\infty)=t_{L}=-t_{B}$ and
$t(\infty)=t_{R}=t_{B}$. Also due to the symmetry, we can see that there is
“point” (Strictly speaking, is codimensional-2 sub-manifold) where $z_{A}$ is
local maximal and $t_{A}=0$. See Figure 2. This means that $\partial
z/\partial t|_{A}=0$. Let us denote the area of extremal surface $\Gamma$ to
be $A(\Gamma)$. Using the “point” A we can seperate the extremal surface
$\Gamma$ into two symmetric parts. The area $\mathcal{A}(\Gamma)$ can be
computed by the right half part multiplied by two. For convenience, we
eliminate the parameter $\lambda$ and rewrite the $z$-coordinate as the
function of $t$ locally. At the right part, since the $t$-coordinates first
runs from zero to infinity and then runs from infinity to $t_{B}$, the
function $z=z(t)$ should separated into two parts. The area of the extremal
surface $\Gamma$ reads

$\mathcal{A}(\Gamma)=2V_{k,d-2}\left(\int_{0}^{+\infty}\text{d}t\frac{1}{z^{d-1}}\sqrt{-f(z)\mathrm{e}^{-\chi(z)}+\frac{\dot{z}^{2}}{f(z)}}+\int_{+\infty}^{t_{B}}\text{d}t\frac{1}{z^{d-1}}\sqrt{-f(z)\mathrm{e}^{-\chi(z)}+\frac{\dot{z}^{2}}{f(z)}}\right).$
(8)

Figure 2: Set $t_{R}=-t_{L}=t_{B}$. A is the middle point with $t=0$.

Where $V_{k,d-2}=\int\text{d}\sigma_{k,d-2}$ is the unit volume of the
$(d-2)-$dimensional spatial directions, here $\text{d}\sigma_{k,d-2}^{2}$ is
the $(d-2)-$dimensional line element produced by $\text{d}\Sigma_{k,d-1}^{2}$
fixing one coordinate $x_{1}$ or $\theta$. The $V_{k,d-2}$ will be divergent
with $k=0,-1$, but we can set it as a large constant by taking a cut off, this
does not change our final results111We can also define a area density by
dividing the area by $V_{k,d-2}$ to eliminate divergence.. The integration of
Eq. (8) will be divergent since it goes to the bulk boundary, but we will see
later that its derivative with respect to time is finite.

In order to get the growth rate of area, we can regard the area $\mathcal{A}$
as an action, and the integrand function of (8) as a Lagrangian, which is a
functional of generalized coordinate and generalized velocity $z,\dot{z}$

$\mathcal{A}=2V_{k,d-2}\int\mathcal{L}(z,\dot{z})\mathrm{d}t,~{}~{}\mathcal{L}(z,\dot{z})=\frac{1}{z^{d-1}}\sqrt{-f(z)\mathrm{e}^{-\chi(z)}+\frac{\dot{z}^{2}}{f(z)}}.$
(9)

According to the Hamilton-Jacobi equation of classical mechanics, the partial
derivative of on-shell action $S$ over time $\partial S/\partial t$ is
negative Hamiltonian, so the partial derivative of area over time is
$\partial\mathcal{A}/\partial t=-2V_{k,d-2}\mathcal{H}$. Where $\mathcal{H}$
is the "Hamiltonian" of Lagrangian (9). At the boundary we have $z=0$, so that
$t=t_{B}$, we get the rate of area growth over the boundary time

$\frac{\text{d}\mathcal{A}}{\text{d}t_{B}}=\left.-2V_{k,d-2}\mathcal{H}\right|_{t=t_{B},z=0}.$
(10)

Since the Lagrangian $\mathcal{L}$ does not contain generalized coordinate
$t$, we can get a conserved quantity $\mathcal{H}$ on the extremal surface

$\mathcal{H}=\frac{\partial\mathcal{L}}{\partial\dot{z}}\dot{z}-\mathcal{L}=\frac{f(z)\mathrm{e}^{-\chi(z)}}{z^{d-1}\sqrt{-f(z)\mathrm{e}^{-\chi(z)}+\frac{\dot{z}^{2}}{f(z)}}}.$
(11)

This “Hamiltonian” is constant along the extreme surface. In order to figure
out the conserved quantity $\mathcal{H}$, we focus on the middle point A with
$z=z_{A}$, where we have $\dot{z}=0$. Then we can obtain

$\mathcal{H}=-\frac{\sqrt{-f(z_{A})\mathrm{e}^{-\chi(z_{A})}}}{z^{d-1}_{A}}.$
(12)

Since (11) and (12) are both conserved quantity on the extremal surface, we
can get an equation

$-\frac{\sqrt{-f(z_{A})\mathrm{e}^{-\chi(z_{A})}}}{z^{d-1}_{A}}=\frac{f(z)\mathrm{e}^{-\chi(z)}}{z^{d-1}\sqrt{-f(z)\mathrm{e}^{-\chi(z)}+\frac{\dot{z}^{2}}{f(z)}}}.$
(13)

Solve this equation, we can get

$\frac{\text{d}t}{\text{d}z}=\frac{\sqrt{-f(z_{A})\mathrm{e}^{-\chi(z_{A})}}}{z^{d-1}_{A}}\frac{1}{f(z)\mathrm{e}^{-\frac{\chi(z)}{2}}\sqrt{\frac{-f(z_{A})\mathrm{e}^{-\chi(z_{A})}}{z^{2(d-1)}_{A}}+\frac{f(z)\mathrm{e}^{-\chi(z)}}{z^{2(d-1)}}}}.$
(14)

For convenience, we define a function $G(z)$ (which is also very important in
the following) as

$G(z)\equiv\frac{\sqrt{-f(z)\mathrm{e}^{-\chi(z)}}}{z^{d-1}}.$ (15)

Integrate (14) over $z$, we get the boundary time $t_{B}$, which is a function
of $z_{A}$

$\displaystyle t_{B}(z_{A})$
$\displaystyle=\int_{z_{A}}^{0}\frac{G(z_{A})}{f(z)\mathrm{e}^{-\frac{\chi(z)}{2}}\sqrt{G(z_{A})^{2}+\frac{f(z)}{z^{2(d-1)}}}}\text{d}z.$
(16)
$\displaystyle=\int_{z_{A}}^{z_{h}+\epsilon}\frac{G(z_{A})}{f(z)\mathrm{e}^{-\frac{\chi(z)}{2}}\sqrt{G(z_{A})^{2}+\frac{f(z)}{z^{2(d-1)}}}}\text{d}z$
$\displaystyle-\int_{0}^{z_{h}-\epsilon}\frac{G(z_{A})}{f(z)\mathrm{e}^{-\frac{\chi(z)}{2}}\sqrt{G(z_{A})^{2}+\frac{f(z)}{z^{2(d-1)}}}}\text{d}z.$

Here we introduce a cut off $\epsilon$, since the integral diverges at the
horizon. The divergences at two sides just cancel with each other, so the
integral obtains a finite contribution when it pass through horizon. Figure 3
is obtained by numerical calculating $t_{B}$ by (16), which shows that
boundary time $t_{B}$ goes to infinite with $z_{A}$ limit to a finite value.
At the same time, the growth rate increases monotonically to its maximum. The
(14) tells us $\dot{z}\rightarrow 0$ with $z\rightarrow z_{A}$, which means
the extremal surface will be infinitely close to the surface $z=z_{A}$ inside
the horizon(see Figure 4).

Figure 3: Left: The image of $z_{A}$ and $t_{B}$. Right: The image of
$G(z_{A})$ and $t_{B}$. Here we set $d=3$, $\ell_{\rm AdS}=100$, and
$f(z)=h(z)$ as 20 with $k=0$.

Consider the conserved quantity $\mathcal{H}$ in (12) and the function defined
in (15), the growth rate finally is written as

$\frac{\text{d}\mathcal{A}}{\text{d}t_{B}}=2V_{k,d-2}G(z_{A}).$ (17)

We can also use a single parameter $\lambda$ to parameterize the extremal
surface and obtain the a same result, which is shown in appendix A. The growth
rate can be determined by $z$ coordinate of the middle point $A$. The $z_{A}$
will be a constant at late time as we have discussed before. This means the
area of the extremal surface will growth linearly.

To find maximal growth rate of entanglement entropy, one needs to find the
extremum of $G(z)$(at this time $z=z_{\rm ext}$). The derivative of $G(z)$
reads

$G^{\prime}(z)=(1-d)z^{-d}\sqrt{-\mathrm{e}^{-\chi(z)}f(z)}+\frac{z^{1-d}\mathrm{e}^{-\chi(z)}\left[f(z)\chi^{\prime}(z)-f^{\prime}(z)\right]}{2\sqrt{-\mathrm{e}^{-\chi(z)}f(z)}}=0.$
(18)

then we get

$2(d-1)f(z_{\rm ext})-z_{\rm ext}f^{\prime}(z_{\rm ext})+z_{\rm ext}f(z_{\rm
ext})\chi^{\prime}(z_{\rm ext})=0.$ (19)

For a given $f(z)$, we can figure out a corresponding $z_{\rm ext}$. It is
clear that $z_{h}\textless z_{\rm ext}\textless+\infty$, i.e., the extreme
point is inside the horizon but does not touch the singularity.

Figure 4: The red curve is the extremal surface, the blue curve is the
extremal surface with $z_{A}=z_{\rm ext}$. With the time $t_{B}$ goes to
infinite, the extremal surface will be infinitely approach to the blue
surface.

##  3 Examples of vacuum black holes and charged black holes

We have given the growth rate of extreme surface area in the previous section,
now we consider some examples to calculate and compare the growth rates. We
will study the cases of vacuum and charged black holes with different
symmetries and fixed mass density, entropy density and temperature
respectively.

###  3.1 Schwarzchild black holes

For Schwarzchild black hole, the metric is given by (4). And the $f(z)$ and
$\chi(z)$ is given by

$f(z)=kz^{2}+\frac{1}{\ell_{\rm AdS}^{2}}-f_{0}z^{d},\qquad\chi(z)=0.$ (20)

Here $f_{0}$ is a parameters related to black hole mass which will be shown in
the later formula (22). For $k=0,1$, the $f_{0}$ is required to be positive,
but for $k=-1$, $f_{0}$ could be negative, since we require the metrics are
black hole solutions. Now we let $f(z_{h})=0$, which can give us $f_{0}$ as

$f_{0}=\frac{1}{z_{h}^{d-2}}\left(\frac{1}{z_{h}^{2}\ell_{\rm
AdS}^{2}}+k\right),$ (21)

here $z_{h}$ is the $z$ coordinate of the horizon. From the metric, we can
obtain the mass $M$, Bekenstein-Hawking entropy $S$, and temperature $T$ of
the black hole

$\displaystyle M$ $\displaystyle=\frac{(d-1)V_{k,d-1}}{16\pi
G_{N}}f_{0}=\frac{(d-1)V_{k,d-1}}{16\pi
G_{N}}\frac{1}{z_{h}^{d-2}}\left(\frac{1}{z_{h}^{2}\ell_{\rm
AdS}^{2}}+k\right),$ (22) $\displaystyle S$
$\displaystyle=\frac{V_{k,d-1}}{4G_{N}}z_{h}^{1-d},$ $\displaystyle T$
$\displaystyle=-\frac{1}{4\pi}\frac{\partial f(z)}{\partial
z}\bigg{|}_{z=z_{h}}=\frac{1}{4\pi}\left((d-2)kz_{h}+\frac{d}{z_{h}\ell_{\rm
AdS}^{2}}\right).$

Where $V_{k,d-1}=\int\text{d}\Sigma_{k,d-1}$ is the unit volume of the $d-1$
dimensional spatial directions.

The function $G(z)$ defined in (15) will be $G(z)=\sqrt{-f(z)}/z^{d-1}$, and
the maximum of $G(z)$ satisfies

$2(d-1)f(z_{\rm ext})-z_{\rm ext}f^{\prime}(z_{\rm ext})=0.$ (23)

Substitute $f(z)$ into (23), we obtain

$2kz_{\rm ext}^{2}+\frac{2(d-1)}{(d-2)}\frac{1}{\ell_{\rm
AdS}^{2}}-f_{0}z_{\rm ext}^{d}=0.$ (24)

So the maximum growth rate $G(z_{\rm ext})$ can be written as

$G(z_{\rm ext})=\frac{\sqrt{kz_{\rm ext}^{2}+\frac{d}{d-2}\frac{1}{\ell_{\rm
AdS}^{2}}}}{z_{\rm ext}^{d-1}}.$ (25)

The mass, Bekenstein-Hawking entropy, and temperature are given according to
Eq. (22), we can fix each of them respectively to study the maximal growth
rate with different topologies.

####  3.1.1 Fixed mass density

We will calculate and compare the growth rate of the cases with different $k$
with fixed mass density in this subsection. The first line of (22) shows that,
the mass $M$ is divergent in the cases of $k=0$ and $k=-1$. For different $k$,
only the volume factor $V_{k,d-1}$ is different. We define the "mass density"
$\mathcal{M}=M/V_{k,d-1}$. We keep $\mathcal{M}$ fixed, i.e., $f_{0}$ is
fixed, and then compare the growth rates of the cases with different $k$.

When $k=0$, we have $f(z)=\frac{1}{\ell_{\rm AdS}^{2}}-f_{0}z^{d}$, then we
can get the solution of (24)

$z_{\rm ext}=z_{p\rm ext}:=\left[\frac{2(d-1)}{(d-2)\ell_{\rm
AdS}^{2}f_{0}}\right]^{\frac{1}{d}}=\left[\frac{2(d-1)}{(d-2)}\right]^{\frac{1}{d}}z_{h}.$
(26)

For convenience, we define a coefficient $\gamma$ as $\gamma=2(d-1)/(d-2)$.
Substitute (26) into (24), we can obtain

$G_{p}(z_{\rm ext})=\sqrt{\frac{d}{d-2}}\gamma^{\frac{1}{d}-1}\ell_{\rm
AdS}^{1-\frac{2}{d}}f_{0}^{1-\frac{1}{d}}.$ (27)

Here we use a footmark "$p$" to mark the growth rate of the vacuum planar
symmetry case. The growth rate can be written as

$\frac{\text{d}\mathcal{A}}{\text{d}t_{B}}=2V_{k,d-2}G_{p}(z_{\rm
ext})=2V_{k,d-2}\sqrt{\frac{d}{d-2}}\gamma^{\frac{1}{d}-1}\ell_{\rm
AdS}^{1-\frac{2}{d}}f_{0}^{1-\frac{1}{d}},$ (28)

which only depends on $V_{k,d-2}$, $d$, $\ell_{\mathrm{AdS}}$, and $f_{0}$.

For the spherically or hyperbolically symmetric Schwarzschild-AdS black hole,
the function $f(z)$ is $f(z)=kz^{2}+\frac{1}{\ell_{\rm AdS}^{2}}-f_{0}z^{d}$,
with $k=1$ or $k=-1$. Then we find the solution of (24) satisfies

$z_{\rm ext}^{d}=\frac{2kz_{\rm ext}^{2}}{f_{0}}+\frac{2(d-1)}{(d-2)\ell_{\rm
AdS}^{2}f_{0}}.$ (29)

Noting Eq. (26), we find that the (29) can be written as

$z_{\rm ext}^{d}=\frac{2kz_{\rm ext}^{2}}{f_{0}}+z_{p\rm ext}^{d}.$ (30)

Substituting (30) into the expression of $G(z_{\rm ext})$, we can eliminate
the $k$ and obtain the growth rate of the spherically or hyperbolically
symmetric case

$G(z_{\rm ext})=\frac{\sqrt{-f(z_{\rm ext})}}{z_{\rm
ext}^{d-1}}=\frac{\sqrt{\frac{f_{0}}{2}(z_{\rm ext}^{d}+z_{p\rm
ext}^{d})-\frac{1}{\ell_{\rm AdS}^{2}}}}{z_{\rm ext}^{d-1}}.$ (31)

We first consider the case of $k=1$. At this case, we have $z_{\rm
ext}\textgreater z_{p\rm ext}$, then we get

$G(z_{\rm ext})\textless\frac{\sqrt{f_{0}z_{\rm ext}^{d}-\frac{1}{\ell_{\rm
AdS}^{2}}}}{z_{\rm ext}^{d-1}}.$ (32)

For the planar case, we have already known that $G_{p}(z_{p\rm
ext})=\sqrt{f_{0}z_{p\rm ext}^{d}-1/\ell_{\rm AdS}^{2}}/z_{p\rm ext}^{d-1}$.
It’s easy for us to know that function $G(x)=\sqrt{f_{0}x^{d}-1/\ell_{\rm
AdS}^{2}}/x^{d-1}$ with $f_{0}x^{d}-1/\ell_{\rm AdS}^{2}\textgreater
0$(Corresponding to the $f(z)$ inside the horizon) and $d\geqslant 3$ is a
monotone decreasing function. We compare $G(z_{\rm ext})$ and $G_{p}(z_{p\rm
ext})$, combine the fact $z_{\rm ext}\textgreater z_{p\rm ext}$, and finally
obtain

$G(z_{\rm ext})\textless\frac{\sqrt{f_{0}z_{\rm ext}^{d}-\frac{1}{\ell_{\rm
AdS}^{2}}}}{z_{\rm ext}^{d-1}}\textless\frac{\sqrt{f_{0}z_{p\rm
ext}^{d}-\frac{1}{\ell_{\rm AdS}^{2}}}}{z_{p\rm ext}^{d-1}}=G_{p}(z_{p\rm
ext}).$ (33)

That is to say, the growth rate of spherical symmetry case is less than planar
symmetry case with the same $f_{0}$.

For the $k=-1$ case, we find that we cannot analytically calculate and compare
the $G(z_{\rm ext})$ with other case. We numerically calculated the growth
rate and obtained Figure 5. From Figure 5 we can see, with the same $f_{0}$,
$G(z_{\rm ext})$ with $k=-1$ is larger than $G(z_{\rm ext})$ with $k=0$, and
$G(z_{\rm ext})$ with $k=0$ larger than $G(z_{\rm ext})$ with $k=1$.

(a) d=3 (b) d=4 (c) d=5 (d) d=6

Figure 5: For $\ell_{\rm AdS}=1$, the vertical axis represents $G(z_{\rm
ext})/G_{p}(z_{\rm ext})$ and the horizontal axis represents $f_{0}$.

####  3.1.2 Fixed entropy density

We will calculate and compare the growth rate of different $k$ with fixed
entropy density in this subsection. The second line of (22) shows that, the
entropy $S$ is also divergent in the cases of $k=0$ and $k=-1$. We can also
define the "entropy density" $\mathcal{S}=S/V_{k,d-1}$. The $\mathcal{S}$ only
depends on $z_{h}^{1-d}$. So keeping the entropy density $\mathcal{S}$ means
to keep $z_{h}$ constant. The function $f_{0}$ will be determined by (21).

For the planar case($k=0$), the maximum of $G(z)$ is also given by (27)

$G(z_{\rm ext})=\sqrt{\frac{d}{d-2}}\gamma^{\frac{1}{d}-1}\ell_{\rm
AdS}^{-1}z_{h}^{1-d}=4G_{N}\gamma^{\frac{1}{d}-1}z_{1}\mathcal{S}$ (34)

where

$z_{1}=\frac{1}{\ell_{\rm AdS}}\sqrt{\frac{d}{d-2}}.$ (35)

When $k\neq 0$, there are no analytical expressions of $z_{\rm ext}$ and
$G(z_{\rm ext})$. To compare $G(z_{\rm ext})$ with different $k$, we can treat
$k$ as continuous real number, and then calculate the derivative of $G(z_{\rm
ext})$ with respect to $k$ in the interval $k\in[-1,1]$

$\displaystyle\frac{\text{d}G(z_{\rm ext})}{\text{d}k}$
$\displaystyle=\frac{1}{2G(z_{\rm ext})}\frac{\text{d}G^{2}(z_{\rm
ext})}{\text{d}k}$ (36) $\displaystyle=\frac{1}{2G(z_{\rm
ext})}\left[\frac{\partial G^{2}(z_{\rm ext})}{\partial k}+2G(z)\frac{\partial
G(z)}{\partial z}\frac{\text{d}z_{\rm ext}}{\text{d}k}\Big{|}_{z=z_{\rm
ext}}\right].$

Since we have $\frac{\partial G(z)}{\partial z}\Big{|}_{z=z_{\rm ext}}=0$,
(36) can be simplified to

$\frac{\text{d}G(z_{\rm ext})}{\text{d}k}=\frac{1}{2G(z_{\rm
ext})}\frac{\partial G^{2}(z_{\rm ext})}{\partial k}=\frac{1}{2G(z_{\rm
ext})}\left(\frac{\text{d}f_{0}}{\text{d}k}z_{\rm ext}^{2-d}-z_{\rm
ext}^{4-2d}\right).$ (37)

Since we have $f(z_{h})=kz_{h}^{2}+\frac{1}{\ell_{\rm
AdS}^{2}}-f_{0}z_{h}^{d}=0$, and $z_{h}$ is a constant. So we can obtain
$\frac{\text{d}f_{0}}{\text{d}k}=z_{h}^{2-d}$. So the (37) becomes to

$\frac{\text{d}G(z_{\rm ext})}{\text{d}k}=\frac{z_{\rm ext}^{2-d}}{2G(z_{\rm
ext})}\left(z_{h}^{2-d}-z_{\rm ext}^{2-d}\right).$ (38)

Since we always have $z_{h}\textless z_{\rm ext}$ and $d\textgreater 2$, the
RHS of (38) is obviously greater than zero, which means $G(z_{\rm ext})$
increases as $k$ increases. That is to say, when we fix entropy density
$\mathcal{S}$, the spherical symmetry case($k=1$) gives the upper bound of
growth rate, which is larger than
$4G_{N}\gamma^{\frac{1}{d}-1}z_{1}\mathcal{S}$ given by the planar symmetry
case($k=0$). We also verify this conclusion by numerical calculation, which is
shown in Figure 6.

(a) d=3 (b) d=4 (c) d=5 (d) d=6

Figure 6: For $\ell_{\rm AdS}=1$, the vertical axis represents $G(z_{\rm
ext})/Gp(z_{\rm ext})$ and the horizontal axis represents $\mathcal{S}$.

####  3.1.3 Fixed temperature

We will calculate and compare the growth rate of different $k$ with fixed
temperature in this subsection. The third line of (22) implies that fixing the
temperature $T$ is much more complicated since both $z_{h}$ and $f_{0}$ will
be different with $k=1,0,-1$. We can rewrite the temperature $T$ as

$T=\frac{d-2}{4\pi}\left(kz_{h}+\frac{z_{1}^{2}}{z_{h}}\right)\,,$ (39)

Figure 7: The sketch of $T(z_{h})$ with different $k$. The $T_{min}$ is the
lower bound of the $f_{0}$.

where $z_{1}$ is defined in (35). The sketch of $T(z_{h})$ is shown as Figure
7. We can see that to keep the three cases have the same temperature $T$, The
temperature has a minimum $T_{\text{min}}$ and the relationship of $z_{h}$ is
$z_{h}(k=-1)\textless z_{h}(k=0)\textless z_{h}(k=1)$. There is an upper bound
of spherical horizon $z_{h}(k=1)\leqslant z_{1}$. Similarly, there are also
upper bounds $z_{1}/2$ and $(\sqrt{2}-1)z_{1}$ for the planar horizon and
hyperbolic horizon respectively.

For the planar case($k=0$), the maximum of $G(z)$ is again given by (27)

$G(z_{\rm ext})=\sqrt{\frac{d}{d-2}}\gamma^{\frac{1}{d}-1}\ell_{\rm
AdS}^{-1}z_{h}^{1-d}=\left(\frac{d-2}{4\pi}\right)^{1-d}\gamma^{\frac{1}{d}-1}z_{1}^{3-2d}T^{d-1}.$
(40)

We can see that the growth rate of entanglement entropy is proportional to the
$T^{d-1}$. When $k\neq 0$, there are also no analytical expressions of $z_{\rm
ext}$ and $G(z_{\rm ext})$. We adopt the prescription that calculating the
derivative of $G(z_{\rm ext})$ with respect to a continuous real number $k$ in
the interval $k\in[-1,1]$ again to compare $G(z_{\rm ext})$ with different
$k$. We will first obtain (37), but the $\text{d}f_{0}/\text{d}k$ will be
different. Since $z_{h}$ is no longer a constant, but a function of $k$.
Combine the two conditions that $f(z_{h})=0$ and $\text{d}T=0$, we finally
obtain

$\frac{\text{d}f_{0}}{\text{d}k}=z_{h}^{2-d}+(2-d)z_{h}^{2-d}\frac{z_{1}^{2}+kz_{h}^{2}}{z_{1}^{2}-kz_{h}^{2}}.$
(41)

So the derivative of $G(z_{\rm ext})$ with respect to $k$ is

$\frac{\text{d}G(z_{\rm ext})}{\text{d}k}=\frac{z_{\rm ext}^{2-d}}{2G(z_{\rm
ext})}\left[z_{h}^{2-d}\left(1+(2-d)\frac{z_{1}^{2}+kz_{h}^{2}}{z_{1}^{2}-kz_{h}^{2}}\right)-z_{\rm
ext}^{2-d}\right].$ (42)

To compare the (42) and 0, we define a new function $\Theta(d,k,z_{h})$ as

$\Theta(d,k,z_{h})=z_{h}^{2-d}\left(1+(2-d)\frac{z_{1}^{2}+kz_{h}^{2}}{z_{1}^{2}-kz_{h}^{2}}\right)-z_{\rm
ext}^{2-d}.$ (43)

Where $d\geqslant 3$, $-1\leqslant k\leqslant 1$, and $0\textless
z_{h}\leqslant(\sqrt{2}-1)z_{1}$. The $z_{\rm ext}$ is determined by

$kz_{h}^{2-d}+\frac{1}{\ell_{\rm AdS}^{2}z_{h}^{d}}=2kz_{\rm
ext}^{2-d}+\frac{\gamma}{\ell_{\rm AdS}^{2}z_{\rm ext}^{d}}=f_{0}.$ (44)

When $k\geqslant 0$, there is obviously $\Theta(d,k,z_{h})\textless 0$. For
the $k\textless 0$ case, we have to calculate $\Theta(d,k,z_{h})$ numerically.
The result shows that $\Theta(d,k,z_{h})\textless 0$ always holds, which means
$G(z_{\rm ext})$ is the decreasing function of $k$. That is to say, when we
fix the temperature $T$, the hyperbolic symmetry case($k=-1$) gives the upper
bound of growth rate, which is larger than
$\left(\frac{d-2}{4\pi}\right)^{1-d}\gamma^{\frac{1}{d}-1}z_{1}^{3-2d}T^{d-1}$
given by the planar symmetry case($k=0$). We also verify this conclusion by
numerical calculation, which is shown in Figure 8.

(a) d=3 (b) d=4 (c) d=5 (d) d=6

Figure 8: For $\ell_{\rm AdS}=1$, the vertical axis represents $G(z_{\rm
ext})/Gp(z_{\rm ext})$ and the horizontal axis represents $T$.

###  3.2 Black holes with charge

We have studied the maximum growth rates of holographic entanglement entropy
for the vacuum black hole spacetime with different topologies. Now we will
study the case that black holes with charge. We consider the simplest charged
case, the RN-AdS black hole spacetime, which is also described by the metric
(4), and the function $f(z)$ and $\chi(z)$ is given by

$f(z)=kz^{2}+\frac{1}{\ell_{\rm
AdS}^{2}}-f_{0}z^{d}+\widetilde{q}z^{2d-2},\qquad\chi(z)=0$ (45)

Where the $f_{0}$ is the mass parameter, and $\widetilde{q}=q^{2}\geqslant 0$
is the charge parameter. Substitute (45) into (23), we obtain a similar
equation as $(\ref{eq:405045})$

$2kz_{\rm ext}^{2}+\frac{2(d-1)}{(d-2)}\frac{1}{\ell_{\rm
AdS}^{2}}-f_{0}z_{\rm ext}^{d}=0,$ (46)

The growth rate of charged case

$G_{q}(z_{\rm ext})=\frac{\sqrt{-f(z_{\rm ext})}}{z_{\rm
ext}^{d-1}}=\frac{\sqrt{-\left(kz^{2}+\frac{1}{\ell_{\rm
AdS}^{2}}-f_{0}z^{d}+\widetilde{q}z^{2d-2}\right)}}{z_{\rm ext}^{d-1}}.$ (47)

Here we used subscript $q$ to implies that here we consider the charged cases.

Then the (46) will be identical to (24), so will be the $z_{\rm ext}$. We have

$\frac{\sqrt{-\left(kz^{2}+\frac{1}{\ell_{\rm
AdS}^{2}}-f_{0}z^{d}+\widetilde{q}z^{2d-2}\right)}}{z_{\rm
ext}^{d-1}}\leqslant\frac{\sqrt{-\left(kz^{2}+\frac{1}{\ell_{\rm
AdS}^{2}}-f_{0}z^{d}\right)}}{z_{\rm ext}^{d-1}},$ (48)

since $\widetilde{q}\geqslant 0$, so we obtain

$G_{q}(z_{\rm ext})\leqslant G_{0}(z_{\rm ext}).$ (49)

So the maximum growth rate of holographic entanglement entropy in the charged
cases is less than that of the cases without charge when we fixed the mass
parameter.

(a) d=3 (b) d=4 (c) d=5 (d) d=6

Figure 9: For the case of $\ell_{\rm AdS}=1$. The vertical axis represents
$G_{q}(z_{\rm ext})/G_{0}(z_{\rm ext})$, where $G_{0}(z_{\rm ext})$ is the
maximum growth rate of vacuum case. The horizontal axis represents the charge
parameter $q^{2}$.

We next consider the cases that the entropy density $\mathcal{S}$ remains the
same as that of vacuum black holes. According to the second line of (22), the
same $\mathcal{S}$ means the same $z_{h}$. For the charged cases, we choose
the outermost horizon to calculate the entropy density. The numerical results
are shown in the figure 9. We can see that the maximum growth rate will
decrease as the charge parameter $\tilde{q}$ increases. So the growth rate in
the charged cases is less than that of the cases without charge when we fixed
the entropy density $\mathcal{S}$.

The cases with fixed temperature $T$ is different from other cases. The
temperature is calculated by the third line of (22), and we also choose the
outermost horizon to calculate the charged cases. The numerical results are
shown in the figure 10. We can see that the maximum growth rate will increase
as the charge parameter $\tilde{q}$ increases. So the growth rate in the
charged cases is faster than that of the cases without charge when we fixed
the temperature $T$. This conclusion is different from the previous case. The
vacuum cases do not give the fastest growth rate when we fixed the
temperature.

(a) d=3 (b) d=4 (c) d=5 (d) d=6

Figure 10: For $\ell_{\rm AdS}=1$, the vertical axis represents $G_{q}(z_{\rm
ext})/G_{0}(z_{\rm ext})$, where $G_{0}(z_{\rm ext})$ is the maximum growth
rate of vacuum case. The horizontal axis represents the charge parameter
$q^{2}$.

When we fix the mass and entropy density respectively, the vacuum cases give
the faster growth rate. Then we are going to ask, would the presence of other
matter fields reduce the growth rate under the planar and spherical symmetries
with fixed mass or entropy density? We conjecture: for all static planar or
spherically symmetric asymptotically Schwarzschild-AdS black holes of same
mass density or entropy density, the vacuum AdS black hole gives the maximum
entanglement entropy growth rate. We will consider more general case and give
proofs in next section.

##  4 A proof about planar and spherically symmetric spacetime with matter
fields

The results of the previous section suggest that, the maximum growth rate of
giving energy and entropy density will decrease when there is a Maxwell field.
In this section, we will show that such property is universal at least for
spherically symmetric and planar symmetric asymptotically Schwarzschild-AdS
black holes if matters satisfy the dominant energy condition. We here
emphasize that the dominant energy condition is just one sufficient condition.
In next section, we will give an example to show that such property is still
true even if the dominant energy condition is broken.

###  4.1 Planar and spherical symmetry cases with fixed mass density

In this subsection, we will keep the mass density to be fixed. And we use the
metric (50) to describe our spacetime with planar or spherical symmetry, which
reads

$\text{d}s^{2}=\frac{1}{z^{2}}\left[-f\mathrm{e}^{-\chi}\text{d}t^{2}+\frac{\text{d}z^{2}}{f}+\mathfrak{s}_{ij}\text{d}x^{i}\text{d}x^{j}\right],$
(50)

where $f$ and $\chi$ are only the functions of $z$, their expression is
undetermined now. $\mathfrak{s}_{ij}$ is $d-2$ dimensional angular direction
metric. The derivative of area with respect to timeis given by (17), and now
we take its square

$\dot{\mathcal{A}}^{2}=-4V_{k,d-2}^{2}z^{2(1-d)}f\mathrm{e}^{-\chi}.$ (51)

For convenience, we define $F(z):=-z^{2(1-d)}f$, then we rewrite (51) as
$\dot{\mathcal{A}}^{2}=4V_{k,d-2}^{2}F(z)\mathrm{e}^{-\chi}$.

Assume that $T_{\mu\nu}$ is the energy momentum tensor of matters. According
to the Einstein’s equation, we can get Madler:2016xju

$\partial_{z}\chi=\frac{8\pi z}{d-1}T_{zz}.$ (52)

and

$-(d-1)z^{d-1}\partial_{z}\left(z^{-d}f\right)=\mathrm{e}^{-\chi}\mathfrak{R}+\frac{d(d-1)\mathrm{e}^{-\chi}}{z^{2}\ell_{\mathrm{AdS}}^{2}}-8\pi
z^{-2}\mathrm{e}^{-\chi}(\rho-P).$ (53)

Where $T_{zz}$ is the $zz$ component of $T_{\mu\nu}$.
$\rho=T_{\mu\nu}n^{\mu}n^{\nu}$, $P=T_{\mu\nu}m^{\mu}m^{\nu}$, where $n^{\mu}$
and $m^{\mu}$ are orthogonal timelike and spacelike normal vectors of subspace
spanned by $\\{x^{i}\\}$. The dominant energy condition implies
$T_{zz}\geqslant 0$ and $\rho-P\geqslant 0$. $\mathfrak{R}$ is the scalar
curvature of the $d-2$ dimensional spatial directions. For the planar or
spherical symmetry cases, we can write the $\mathfrak{R}$ as

$\mathfrak{R}=(d-1)(d-2)k,$ (54)

where $k=0$ or $k=1$. Since we have $T_{zz}\geqslant 0$, the (52) gives us

$\partial_{z}\chi\geqslant 0.$ (55)

Since we have the boundary condition $\chi|_{z=0}=0$, we get $\chi\geqslant
0$, which leads to $\mathrm{e}^{-\chi}\leqslant 1$, combining the (51) we
obtain

$\dot{\mathcal{A}}^{2}\leqslant 4V_{k,d-2}^{2}F(z).$ (56)

We we can rewrite (53) as

$\frac{\text{d}}{\text{d}z}\left(z^{d-2}F\right)=\frac{d\mathrm{e}^{-\chi}}{z^{d+1}\ell_{\mathrm{AdS}}^{2}}+\frac{\mathrm{e}^{-\chi}\mathfrak{R}}{(d-1)z^{d-1}}-\frac{\mathrm{e}^{-\chi}Q^{2}}{(d-1)z^{d-1}}$
(57)

where $Q=\sqrt{8\pi(\rho-P)}/z\geq 0$. Near the boundary $z\rightarrow 0$,
$\chi$ and $Q$ are required to decay fast enough, so the spacetime is
asymptotically Schwarzschild-AdS and

$F\rightarrow-\frac{k}{z^{d-2}}-\frac{1}{z^{d}\ell_{\mathrm{AdS}}^{2}}+f_{0},$
(58)

where $f_{0}$ is the integration constant. We can rewrite (58) as

$F(z)\rightarrow-z^{-2(1-d)}\left[kz^{2}+1/\ell_{\mathrm{AdS}}^{2}-f_{0}z^{d}\right]=\left[G_{0}(z)\right]^{2}$
(59)

We find that $f_{0}$ can be just right regarded as the mass parameter which is
fixed. And $G_{0}(z)$ is the growth rate function of the vacuum cases.
According to (57), for finite $z$, and consider that $\chi\geqslant 0$ and
$k\geqslant 0$, we have

$\frac{\text{d}}{\text{d}z}\left(z^{d-2}F\right)\leqslant\frac{(d-2)k}{z^{d-1}}+\frac{d}{z^{d+1}\ell_{\mathrm{AdS}}^{2}}$
(60)

Integrating it and noting the asymptotically behavior (59), we find

$F(z)\leqslant\left[G_{0}(z)\right]^{2}$ (61)

Consider (62),we finally get

$\dot{\mathcal{A}}\leqslant 2V_{k,d-2}G_{0}(z)\leqslant 2V_{k,d-2}\max G_{0}.$
(62)

This result shows that, when the mass density is fixed, in the case of planar
or spherically symmetric asymptotically Schwarzschild-AdS black holes with
matter field that following the dominant energy condition, the growth rate of
holographic entanglement entropy will less than the vacuum case.

###  4.2 Planar and spherical symmetry cases with fixed entropy density

The previous subsections considered the cases with fixed mass density. Now we
consider the cases with fixed entropy density $\mathcal{S}$. Integrate the
right side over $z$, we can write (60) as

$\frac{\text{d}}{\text{d}z}\left(z^{d-2}F\right)\leqslant\frac{\text{d}}{\text{d}z}\left(-\frac{k}{z^{d-2}}-\frac{1}{z^{d}\ell_{\mathrm{AdS}}^{2}}+\tilde{f}_{0}\right)=\frac{\text{d}}{\text{d}z}\left(-\frac{f_{\text{sch}}(z)}{z^{d}}\right),$
(63)

where $\tilde{f}_{0}$ is the integration constant (independent of $z$) that
satisfies

$kz_{h}^{2}+1/\ell_{\mathrm{AdS}}^{2}-\tilde{f}_{0}z_{h}^{d}=0\,$ (64)

and we define
$f_{\text{sch}}(z):=kz^{2}+1/\ell_{\mathrm{AdS}}^{2}-\tilde{f}_{0}z^{d}$.
According to the second line of (22), the fixed $\mathcal{S}$ means the fixed
horizon coordinate $z_{h}$. Since the metric is (64), we have
$f_{\text{sch}}(z_{h})=F(z_{h})=0$. This shows that

$z_{h}^{d-2}F(z_{h})=-\frac{f_{\text{sch}}(z_{h})}{z_{h}^{d}}=0.$ (65)

When $z>z_{h}$, combine (63) and (65),we have

$z^{d-2}F(z)\leqslant-\frac{f_{\text{sch}}(z)}{z^{d}}.$ (66)

So we obtain

$F(z)\leqslant-\frac{f_{\text{sch}}(z)}{z^{2d-2}}=\left[G_{0}(z)\right]^{2},$
(67)

where $G_{0}(z)$ is the growth rate function of the vacuum black hole cases.
We finally get

$\dot{\mathcal{A}}\leqslant 2V_{k,d-2}G_{0}(z)\leqslant 2V_{k,d-2}\max G_{0}.$
(68)

This result shows that, when the entropy density is fixed, in the case of
planar or spherically symmetric asymptotically Schwarzschild-AdS black holes
with matter field that following the dominant energy condition, the growth
rate of holographic entanglement entropy will less than the vacuum case.

In the case of RN black hole we have found that the Maxwell field always
decreases the maximal growth rate of entanglement entropy. However, the proofs
here only cover the spherically and planar symmetric cases. The reason is that
we require $\mathfrak{R}\geqslant 0$ to obtain inequality (60) from Eq. (55).
It is not clear that whether dominant energy condition can insure the same
conclusion or not for hyperbolically symmetric black hole. We hope we could
address this issue in the future.

##  5 Black holes with scalar hair

We have proved that the vacuum black holes with $k\geqslant 0$ have the
maximum growth rate by the dominant energy condition and the Einstein
equation. In this section, we consider the planar symmetric AdS black holes
with real scalar hair. We will choose the negative ”mass-square” for the
scalar field so that the dominant energy condition can be broken. In addition,
due to the presence of external source, the bulk spacetime is asymptotically
AdS but not asymptotically Schwarzschild-AdS.

The model is Einstein gravity with a negative cosmological minimally coupled
to a real scalar field $\phi$, whose action reads

$S=\frac{1}{16\pi}\int\mathrm{d}^{d+1}x\sqrt{-g}\left(R-2\Lambda-\frac{1}{2}\nabla^{\mu}\phi\nabla_{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}\right),$
(69)

where $\Lambda=-\frac{d(d-1)}{2\ell_{\rm AdS}^{2}}$, and $m$ is the mass
parameter of the scalar field $\phi$. Performing variation on the action (69)
with respect to the metric $g_{\mu\nu}$, and the scalar field $\phi$, we
obtain the equation of motion

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}$
$\displaystyle=\frac{1}{2}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\mu}\phi\nabla^{\mu}\phi-\frac{1}{2}g_{\mu\nu}m^{2}\phi^{2}\right),$
(70) $\displaystyle\nabla_{\mu}\nabla^{\mu}\phi-m^{2}\phi$ $\displaystyle=0.$

Consider the planar symmetry ansatz given by

$\displaystyle ds^{2}$
$\displaystyle=-f(r)e^{-\chi(r)}\text{d}t^{2}+\frac{\text{d}r^{2}}{f(r)}+r^{2}\text{d}\mathbf{x}_{d-1}^{2},$
(71) $\displaystyle\phi$ $\displaystyle=\phi(r).$

where $f,\chi,\phi$ are only functions of coordinate $r$, and
$\text{d}\mathbf{x}_{d-1}^{2}=\text{d}x_{1}^{2}+\text{d}x_{2}^{2}+\cdots+\text{d}x_{d-1}^{2}$
is the $d-1$ dimensional spatial directional line element.

With the ansatz, the equation of motion (70) reduce to

$\displaystyle\frac{\chi^{\prime}}{r}+\frac{1}{d-1}\phi^{\prime 2}$
$\displaystyle=0,$ (72)
$\displaystyle\frac{2}{r}\frac{f^{\prime}}{f}-\frac{\chi^{\prime}}{r}+\frac{1}{d-1}\frac{m^{2}\phi^{2}}{f}+\frac{2(d-2)}{r^{2}}$
$\displaystyle=0,$
$\displaystyle\frac{f^{\prime\prime}}{f}-\chi^{\prime\prime}+\frac{1}{2}\chi^{\prime
2}+\frac{(d-2)\chi^{\prime}}{r}+\left(\frac{d-3}{r}-\frac{3}{2}\chi^{\prime}\right)\frac{f^{\prime}}{f}-\frac{2(d-2)}{r^{2}}$
$\displaystyle=0,$
$\displaystyle\phi^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{\chi^{\prime}}{2}+\frac{d-1}{r}\right)\phi^{\prime}-\frac{m^{2}\phi}{f}$
$\displaystyle=0.$

Since there are only three independent functions of coordinate $r$, only three
equations of (72) are independent.222One can verify that by transforming the
first three equations to get the fourth equation. For convenience, we can
choose the first, second and fourth equations for later calculation.

The entropy density and temperature are given by

$\mathcal{S}=\frac{1}{4}r_{h}^{d-1},\qquad
T=\frac{1}{4\pi}f^{\prime}(r_{h}){\rm e}^{-\frac{\chi(r_{h})}{2}}.$ (73)

We assume that the solution is an asymptotically AdS spacetime, which means
the functions $f,\chi,\phi$ have the following behavior at the AdS boundary

$\lim_{r\to\infty}f(r)=\frac{r^{2}}{\ell_{\rm
AdS}^{2}},\qquad\lim_{r\to\infty}\chi(r)=0,\qquad\lim_{r\to\infty}\phi(r)=0.$
(74)

The scalar field $\phi(r)$ can be expanded at the AdS
boundary($r\rightarrow\infty$) as Skenderis:2002wp ; Marolf:2006nd

$\phi(r)=\frac{\phi_{\alpha}}{r^{d-\Delta}}\left(1+\cdots\right)+\frac{\phi_{\beta}}{r^{\Delta}}\left(1+\cdots\right),$
(75)

where $\phi_{\alpha}$ and $\phi_{\beta}$ are non-trivial parameters. According
to the holographic renormalization of the massive scalar field
Skenderis:2002wp ; Marolf:2006nd , the parameter $\Delta$, which is called the
scaling dimension of the boundary field, is given by

$\Delta=\frac{1}{2}\left(d+\sqrt{d^{2}+4m^{2}\ell_{\rm AdS}^{2}}\right),$ (76)

where $m$ is the mass of the bulk scalar field. The surd in (76) implies that
$m^{2}$ can be negative, and there is a lower bound of $m^{2}$ which is called
the Breitenlohner-Freedman (BF) bound $m^{2}_{BF}=-\frac{d^{2}}{4\ell_{\rm
AdS}^{2}}$ Breitenlohner:1982jf ; Breitenlohner:1982bm . Hence, the scalar
field may not obey the dominant energy conditions.

Now we consider a specific case to calculate the growth rate of holographic
entanglement entropy. Without loss of generality, we will set $\ell_{\rm
AdS}=1$ in this section. We take four-dimensional space-time$(d=3)$, and
choose the mass parameter $m^{2}=-2$.333Note that it is satisfied the
Breitenlohner-Freedman bound, $m^{2}=-2\textgreater m^{2}_{BF}=-2.25$. And we
can get the scaling dimension $\Delta=2$ by (76). Substitute it into (75), we
obtain asymptotic expansion of $\phi(r)$ on the AdS boundary

$\phi(r)=\frac{\phi_{\alpha}}{r}+\frac{\phi_{\beta}}{r^{2}}+\frac{\phi_{3}}{r^{3}}+\mathcal{O}(\frac{1}{r^{4}}),$
(77)

where $\phi_{3}$ is undetermined coefficient that can be represented by
$\phi_{\alpha}$, $\phi_{\beta}$. Since here we choose negative mass-square,
the dominant energy condition may be broken somewhere in the bulk. We can also
write the expansion of $f(r)$ and $\chi(r)$ on the AdS boundary according to
(74)

$\displaystyle f(r)$
$\displaystyle=r^{2}\left[1+\frac{f_{2}}{r^{2}}+\frac{f_{3}}{r^{3}}+\mathcal{O}(\frac{1}{r^{4}})\right],$
(78) $\displaystyle\chi(r)$
$\displaystyle=\frac{\chi_{1}}{r}+\frac{\chi_{2}}{r^{2}}+\frac{\chi_{3}}{r^{3}}+\mathcal{O}(\frac{1}{r^{4}}),$

with undetermined parameters $\\{f_{2},f_{3},\chi_{1},\chi_{2},\chi_{3}\\}$.
Substitute (77) and (78) into the equations (72) at $r\rightarrow\infty$, we
obtain Li:2020spf

$\displaystyle\phi(r)$
$\displaystyle=\frac{\phi_{\alpha}}{r}+\frac{\phi_{\beta}}{r^{2}}-\frac{\phi_{\alpha}^{3}}{8r^{3}}+\mathcal{O}(\frac{1}{r^{4}}),$
(79) $\displaystyle f(r)$
$\displaystyle=r^{2}\left[1+\frac{\phi_{\alpha}^{2}}{4r^{2}}+\frac{f_{3}}{r^{3}}+\mathcal{O}(\frac{1}{r^{4}})\right],$
$\displaystyle\chi(r)$
$\displaystyle=\frac{\phi_{\alpha}^{2}}{4r^{2}}+\frac{2\phi_{\alpha}\phi_{\beta}}{3r^{3}}+\mathcal{O}(\frac{1}{r^{4}}),$

there are only three free parameters $\phi_{\alpha}$, $\phi_{\beta}$ and
$f_{3}$, which will give us the energy(or the mass) of space-time. In fact,
according to the boundary conditions of the equation (72), we can find that
$\phi_{\alpha}$ and $\phi_{\beta}$ are related, that is, there are only two
free parameters. One can see that the function $f(r)$ is not asymptotically
Schwarzschild if the coefficient $\phi_{\alpha,\beta}\neq 0$.

In order to obtain the total energy, we need the prescription of holographic
renormalization. Since there are two parameters $\phi_{\alpha}$ and
$\phi_{\beta}$ in the (75), we can fix them respectively at the boundary. This
will give us two different quantization schemes Marolf:2006nd , and we will
discuss them reseparately.

We first consider standard quantization scheme which takes the leading order
$\phi_{\alpha}$ as the source, the boundary action will given by Marolf:2006nd

$S_{\partial}^{(\alpha)}=\frac{1}{16\pi}\int_{r\rightarrow\infty}\text{d}x^{3}\sqrt{-h}\left[2K-4-\frac{1}{2}\phi^{2}\right]$
(80)

where $h=\rm{det}$$(h_{ij})$, $h_{ij}$ is the induced metric at the AdS
boundary, and $K$ is the extrinsic curvature scalar of the boundary manifold.
The first term of (80) is the Gibbons-Hawking-York term. The second and third
terms are counter terms for removing divergences.444In most cases, there is a
finite counter term (the contact term) $-c\phi^{3}$ after the counter terms in
(80) Skenderis:2002wp ; Li:2020spf . Here we choose the minimum coupling case
with $c=0$.

Then we can obtain the holographic stress tensor by performing variation on
the boundary action (80)

$T_{\mu\nu}^{(\alpha)}=\frac{1}{16\pi}\lim_{r\rightarrow\infty}r\left[2\left(Kh_{\mu\nu}-K_{\mu\nu}-2h_{\mu\nu}\right)-\frac{1}{2}h_{\mu\nu}\phi^{2}\right].$
(81)

Substituting the expansions (79) into the stress tensor (81), we obtain the
$T_{tt}$ component reads

$16\pi
T_{tt}^{(\alpha)}=16\pi\mathcal{E}^{(\alpha)}=-2f_{3}+\phi_{\alpha}\phi_{\beta},$
(82)

where $\mathcal{E}^{(\alpha)}$ is the energy(or mass) density. When the
parameter $\phi_{\alpha}=0$, the scalar field will vanish, the metric will
degenerate to vacuum black hole solution.

Now we solve the equation (72) numerically. The asymptotic infinity of the
numerical solution will give the parameters $\phi_{\alpha}$, $\phi_{\beta}$
and $f_{3}$, and then we can compute the the energy $\mathcal{E}^{(\alpha)}$
by (82). We are interested in different cases that with the same energy. So we
vary the parameters $\phi_{\alpha}$and $f_{3}$ while keeping the
$\mathcal{E}^{(\alpha)}$ a constant.

We can compute the growth rate of holographic entanglement entropy by (15).
Figure 11(a) shows the maximum value of growth rate function with different
$\phi_{\alpha}$. When the parameter $\phi_{\alpha}=0$, we have the vacuum
solution, which have the maximum growth rate. When we increase the parameter
$\phi_{\alpha}$, which means a stronger scalar field in spacetime, the maximum
growth rate will decrease. That is to say, vacuum black holes will have a
greater growth rate of holographic entanglement entropy.

(a) $\mathcal{E}^{(\alpha)}=1/16\pi$ (b) $\mathcal{S}=0.25$

Figure 11: For $\ell_{\rm AdS}=1$, relationship between growth rate and
$\phi_{\alpha}$. Fixed the $\mathcal{E}=1/16\pi$, $\mathcal{S}=0.25$
respectively, the larger the parameter $\phi_{\alpha}$, the smaller the
maximum value of the growth rate function $G(r)$.

We next consider black holes with the same entropy density $\mathcal{S}$ but
different scalar fields, which means we have a fixed horizon radius $r_{h}$.
We change the parameter of the scalar field, which will give us the growth
rate function curves(see Figure 11(b)). It still follows the law that vacuum
black holes have the greater growth rate, and increasing the scalar field
reduces the growth rate.

We next consider alternative quantization scheme which takes the sub-leading
order $\phi_{\beta}$ as the source555Noting that $\phi_{\alpha}$ and
$\phi_{\beta}$ are related. If $\phi_{\beta}$ is positive, $\phi_{\alpha}$
will be negative. Since $\phi_{\alpha}$ and $\phi_{\beta}$ always appear in
the energy expression in the form of $\phi_{\alpha}\phi_{\beta}$, there is a
symmetry of the $\phi_{\beta}$ and $-\phi_{\beta}$. Here we set $\phi_{\beta}$
is positive., the boundary action is given by Marolf:2006nd

$S_{\partial}^{(\beta)}=\frac{1}{16\pi}\int_{r\rightarrow\infty}\text{d}x^{3}\sqrt{-h}\left[2K-4+\phi
n^{a}\partial_{a}\phi+\frac{1}{2}\phi^{2}\right],$ (83)

where the $n^{a}$ is the outward unit normal vector of the AdS boundary. Then
we can obtain the holographic stress tensor as

$T_{\mu\nu}^{(\beta)}=\frac{1}{16\pi}\lim_{r\rightarrow\infty}r\left[2\left(Kh_{\mu\nu}-K_{\mu\nu}-2h_{\mu\nu}\right)+h_{\mu\nu}(\phi
n^{a}\partial_{a}\phi+\frac{1}{2}\phi^{2})\right].$ (84)

The total energy is given by

$16\pi\mathcal{E}^{(\beta)}=16\pi
T_{tt}^{(\beta)}=-2f_{3}+2\phi_{\alpha}\phi_{\beta}.$ (85)

We perform similar numerical calculations as before, and the results are shown
in the Figure 12. The Figure 12(a) shows that the growth rate will increase
when the $\phi_{\beta}$ increase while keeping the $\mathcal{E}^{(\beta)}$ a
constant. This means that the vacuum case has the minimum growth rate with
fixed mass. The Figure 12(b) shows that the growth rate will decrease when the
$\phi_{\beta}$ increase with a fixed entropy density $\mathcal{S}$. That is to
say the vacuum case gives the maximum growth rate with fixed entropy density.

(a) $\mathcal{E}^{(\alpha)}=1/16\pi$ (b) $\mathcal{S}=0.25$

Figure 12: For $\ell_{\rm AdS}=1$, relationship between growth rate and
$\phi_{\beta}$. Fixed the $\mathcal{E}^{(\beta)}=1/16\pi$, $\mathcal{S}=0.25$
respectively, the larger the parameter $\phi_{\beta}$, the smaller the maximum
value of the growth rate function $G(r)$.

From these numerical results we obtain following two conclusions for the case
when the dominant energy condition is broken and the bulk is not
asymptotically Schwarzschild. Firstly, if we fix entropy density, the vacuum
planar symmetric black hole will still have maximal entanglement growth rate.
This indicates that the dominant energy condition and asymptotically
Schwarzschild bulk geometry are not necessary conditions for our conclusion in
the case of fixing entropy density. Further research needs to be done in the
future to find the necessary and sufficient conditions. When we fix the energy
density, the case become a little complicated. If we use standard
quantization, we find the vacuum planar symmetric black hole will still have
maximal entanglement growth rate. However, if we used alternative
quantization, numerical results show that the matter will increase the maximal
growth rate of entanglement. It is interesting to study if there is any deep
physics resulting such difference in the future.

##  6 Summary and Discussion

In this paper, we have studied the upper bounds of the holographic
entanglement entropy growth rate which calculated by the H-M surface. We first
study the Schwarzschild-AdS black hole spacetime with planar, spherical and
hyperbolic symmetries and find that there are three different maximums of the
growth rate with given mass density, entropy density and temperature
respectively. Then we study the RN-AdS black hole cases in the same way and
find that the growth rates are always less than that in the Schwarzschild-AdS
black hole cases with the same symmetries and given mass density, entropy
density respectively. That is to say, the existence of Maxwell electromagnetic
fields slows down the growth rate of holographic entanglement entropy if the
black hole has spherical or planar symmetry.

The example of RN black hole inspires us to give a conjecture: For all static
planar and spherically symmetric asymptotically Schwarzschild-AdS black holes
of same mass density or entropy density, the vacuum AdS black hole gives the
maximum entanglement entropy growth rate. We prove this conjecture by under
dominant energy condition and Einstein’ equation. Furthermore, we considered
the black hole spacetime with real scalar fields and take a negative “mass-
square”, which does not obey the dominant energy condition and the bulk
geometry is not asymptotically Schwarzschild. We perform numerical
calculations to give the relationships between the maximum growth rate and the
parameters $\phi_{\alpha}$ and $\phi_{\beta}$. When we fix the total energy,
under the standard quantization scheme, the vacuum black hole case will give
the upper bound of the growth rate, and the maximum growth rate will decrease
with the increase of parameter $\phi_{\alpha}$. However, when we choose
alternative quantization scheme, the result will be opposite. The maximum
growth rate will increase with the increase of parameter $\phi_{\beta}$, and
the growth rate given by vacuum black hole is the smallest. When we fix the
entropy density, in both quantization schemes, the vacuum black hole case
gives the upper bound of the growth rate, and the maximum growth rate will
decrease with the increase of parameter $\phi_{\alpha}$ or $\phi_{\beta}$.

We have gotten the conjecture that the vacuum black hole spacetime gives the
upper bound of the entanglement entropy growth rate with a given mass density
or entropy. But with a given temperature, we have not discussed which case
gives the upper bound. Our proofs in section 4 have not discussed the cases
with hyperbolic symmetry, but some examples we have shown in this paper
suggest that the hyperbolic cases also follow the same conclusion. Our proofs
are based on the dominant energy condition, but the result of the cases with
real scalar field indicates that the dominant energy condition is not a
necessary condition for our conclusion. For the cases that do not obey the
dominant energy condition, we only consider the example of the AdS black hole
with minimum coupled real scalar field to study the non vacuum case. We have
not covered more complicated situations, there may be further discussed in our
future work. Further research needs to be done in the future to find the
necessary and sufficient conditions.

###### Acknowledgements.

The work is supported by the NSFC(Natural Science Foundation of China) under
Grant No. 12005155\.

##  Appendix A The area of extremal surface parameterized by continuous
parameter

We have calculated the area of extremal surface by using the $t$ coordinate as
the integral variable in the section 2. The extremal surface is continuous at
the horizon, but the $t$ coordinate is divergent. In this appendix, we will
parameterize the extremal surface with a continuous real number $\lambda$, and
get the the same result as that in section 2. The codimension-2 extremal
surface $\Gamma$ can be parameterize by $t=t(\lambda,x^{i})$ and
$z=z(\lambda,x^{i})$(see Figure 13). We set $\lambda\rightarrow-\infty$ and
$\lambda\rightarrow+\infty$ at the left and right boundaries respectively, and
$\lambda=0$ at the "middle point" A(Where we have $t=0$.).

Since we can always let $t_{R}=t_{L}$ by Lorentz boost, the extremal surface
will be symmetric in the left and right parts. At the middle point
$A$($\lambda_{A}=0$), we will have $\partial z/\partial\lambda=0$ because of
the symmetry and extremum constraints. The area $A(\Gamma)$ also can be
computed by the right half part($0\leqslant\lambda\leqslant+\infty$)
multiplied by two. We use a dot to represent the derivative with respect to
$\lambda$. The area of the extremal surface $\Gamma$ reads

$\mathcal{A}(\Gamma)=2V_{k,d-2}\int_{0}^{+\infty}\text{d}\lambda\frac{1}{z^{d-1}}\sqrt{-f\mathrm{e}^{-\chi}\dot{t}^{2}+f^{-1}\dot{z}^{2}}.$
(86)

Figure 13: Set $t_{R}=t_{L}=t_{B}$. A is the middle point with $\lambda=0$.

Where $V_{k,d-2}=\int\text{d}\sigma_{k,d-2}$ is the unit volume of the $d-2$
dimensional spatial directions, here $\text{d}\sigma_{k,d-2}^{2}$ is the
induced metric of the constant of $x_{1}$ in $\text{d}\Sigma_{k,d-1}^{2}$.

In order to get the growth rate of area, we can regard the area $\mathcal{A}$
as an action, and the integrand function of (86) as a Lagrangian, which is a
functional of generalized coordinate and generalized velocity
$z,\dot{z},\dot{t}$

$\mathcal{L}(z,\dot{z},\dot{t})=\frac{1}{z^{d-1}}\sqrt{-f\mathrm{e}^{-\chi}\dot{t}^{2}+f^{-1}\dot{z}^{2}}.$
(87)

According to classical mechanics, the partial derivative of action over
generalized coordinate $\partial S/\partial q$ is generalized momentum, so the
partial derivative of area over $t$(as a generalized coordinate) is
$\partial\mathcal{A}/\partial t=2V_{k,d-2}\mathcal{P}$. Where $\mathcal{P}$ is
the "generalized momentum" of the extremal surface. At the boundary we have
$z=0$, so that $t=t_{B}$, we get the rate of area growth over the boundary
time

$\frac{\text{d}\mathcal{A}}{\text{d}t_{B}}=2V_{k,d-2}\mathcal{P}|_{t=t_{B},z=0}.$
(88)

Since the Lagrangian $\mathcal{L}$ does not depend explicitly on $t$, we can
get a conserved quantity on the extremal surface

$\mathcal{P}=\frac{\partial\mathcal{L}}{\partial\dot{t}}=\frac{-f\mathrm{e}^{-\chi}\dot{t}+f^{-1}\dot{z}}{z^{d-1}\sqrt{-f\mathrm{e}^{-\chi}\dot{t}^{2}+f^{-1}\dot{z}^{2}}}.$
(89)

Since the parameter $\lambda$ can be chosen freely, we take $\lambda$ as the
length parameter, which satisfies $\mathcal{A}=2V_{k,d-2}\int\text{d}\lambda$,
thus

$\frac{1}{z^{d-1}}\sqrt{-f\mathrm{e}^{-\chi}\dot{t}^{2}+f^{-1}\dot{z}^{2}}=1.$
(90)

Combining (89) and (90), eliminate $\dot{t}$, we can get

$\mathcal{P}=\frac{-f\mathrm{e}^{-\chi}\sqrt{f^{-2}\mathrm{e}^{\chi}\dot{z}^{2}-f^{-1}\mathrm{e}^{\chi}z^{2(d-1)}}+f^{-1}\dot{z}}{z^{2(d-1)}}.$
(91)

In order to figure out the conserved quantity $\mathcal{P}$, we focue on the
middle point A with $z=z_{A}$, there is an extra conditions $\dot{z}=0$, then
we get

$\mathcal{P}=\frac{\sqrt{-f(z_{A})\mathrm{e}^{-\chi(z_{A})}}}{z^{d-1}_{A}}.$
(92)

So we obtain the growth rate of the area

$\frac{\text{d}\mathcal{A}}{\text{d}t_{B}}=2V_{k,d-2}\mathcal{P}=2V_{k,d-2}\frac{\sqrt{-f(z_{A})\mathrm{e}^{-\chi(z_{A})}}}{z^{d-1}_{A}}.$
(93)

The result is the same as (17). The extremal surface is smooth at the horizon,
where there is only coordinate singularity of $t$. However, it is easier for
us to do the following calculations with the $t$ coordinate as the integral
variable in the section 2.

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