aainstitutetext: Department of Physics, Faculty of Basic Sciences, University
of Mazandaran P. O. Box 47416-95447, Babolsar, Iran

# The correlation of WGC and Hydrodynamics bound with $R^{4}$ correction in
the charged AdSd+2 black brane

E.Naghd Mezerji a J.Sadeghi<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we focus on the possible correlation between conjectures KSS
bound and weak gravity conjecture (WGC). The hydrodynamic values KSS bound and
weak gravity conjecture constraint the low-energy effective field theory.
These conjectures identify UV complete theories. We give four, six and eight
order derivative corrections to corresponding action and employ the
hyperscaling violating charged AdSd+2 black brane solution. These corrections
lead us to find correlation between conjectures KSS bound and weak gravity
conjecture. We see that, with increasing perturbation correction, this
correlation is more likely to appear. We consider dynamical constant $z=1$,
$d=5$ and obtain the range of hyperscaling violation exponent
$d+z-2\leq\theta\leq d+z-1$ for the above mentioned black brane. Here, we show
that higher derivative corrections reduce the ratio of $\frac{M}{Q}$ to
extremal black holes. Likewise, we also obtain the universal relaxation bound
$\tau\geq\frac{1}{\pi T}$ and KSS bound $\frac{\eta}{s}\geq\frac{1}{4\pi}$ for
our model. The results indicate that there is a possibility of a relationship
between the two conjectures. Our studies also show the consistency of the WGC
and the KSS bound conjectures for all corrections (except curvature-cubed,
$\beta_{2}$) in the extremal and near-extremal condition.

Keywords: Black brane, Gravitational corrections, Weak gravity conjecture, KSS
bound, Universal relaxation bound

##  1 Introduction

As we know string theory proves that there is some extra dimension and the
physical properties of (3+1)-dimension phenomena depend on the choice of a
small compact space. The existence of allowed small space give us many
effective field theories in (3+1) dimension with consistence and inconsistence
quantum gravity . The effective field theory (EFT) that is consistent with
quantum gravity is called the landscape. However, many effective theories are
inconsistent and do not derive from string theory, they belong to the
swampland. Such theory is an EFT but is not consistent with quantum gravity.
How can we determine the difference between landscape and swampland and than
say that such a theory is perfectly compatible with quantum gravity? There are
some fundamental solution of EFT or some universal test to distinguish between
the two theory as landscape and swampland. One of the criteria for the
distinguishes of two theories is weak gravity conjecture (WGC). Here it means
that gravity is always the weakest force 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ;
10 . Here we first try to explain WGC conjecture with the present of charged
particle and black hole. As we know, the electric force between the elementary
charge states is stronger than the gravitational attraction between them11 ;
12 ; 13 ; 14 ; 15 ; 16 ; 17

$(F_{e}=\frac{q^{2}}{r^{2}})\geq(F_{g}=\frac{m^{2}}{r^{2}})\longrightarrow
q\geq m$ (1)

On the other hand, in the black hole, WGC is established in extremal
conditions $M=Q$ and $T=0$. When a black hole does Hawking-radiation, the
particles will out from it with mass $m$ and charge $q$. This means $Q-q\leq
M-m$; as a result, we have $M=Q\longrightarrow q\geq m$. The mode of $m=q$
only occurs in the case of supersymmetry (the PBS state) and global symmetry.
On the other hand, there are no such symmetries in the QG theory, so quantum
corrections must remove the parameters of the black hole from their classical
extremal values. In that case, even to extremal limit the corresponding
corrections allow black holes to disappear. There are two possible predictions
for quantum correction of black holes, $\frac{M}{Q}\leq 1$ and
$\frac{M}{Q}\geq 1$. Calculations for various corrections show that the
condition of $\frac{M}{Q}\leq 1$ is satisfied 18 ; 19 ; 20 ; 21 ; 22 ; 23 ; 24
; 25 ; 97 .

Here, to investigate the corresponding effective field theory and find the
distinguishing of landscape and swampland we give some higher order derivative
correction to the corresponding action. We know that Einstein’s theory of
gravity works well in the field of weak gravity, but when the scales become
very small, the curvature becomes significant. At such scales, the Euclidean
topological structure is improbable. The gravitational quantum fluctuations
near the Planck length are so severe that they may impose a dynamical variable
on the topological structure of the universe. The power of quantum effects on
gravitational interactions in this range can be understood. As a result, we
need to use generalized Lagrangian instead of Einstein-Hilbert Lagrangian.
Here we note that higher-order terms are unconstrained terms and also
independent of the original theory25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ; 33 ;
34 ; 35 ; 36 ; 37 ; 38 ; 39 ; 40 ; 41 .

In this paper, we use two special correction terms. Our first correction terms
are Gauss-Bonnet, with two corrections related to charge
${\alpha}_{1}+{\alpha}_{3}$,

${\alpha}_{1}{R}^{2}+{\alpha}_{2}({R}^{2}_{abcd}+4{R}^{2}_{ab}+{R}^{2})+{\alpha}_{3}{R}^{2}_{ab}.$
(2)

and the second terms is curvature-cubed. These two corrections do not change
under field invariant and do not break the supersymmetry. They are related to
non-supersymmetry CFTs,

${\beta}_{1}({{R}^{ab}}_{mn}{{R}^{mn}}_{pc}{{R}^{pc}}_{ab})+{\beta}_{2}({{R}^{ab}}_{cd}{{R}^{cq}}_{pq}{{R}^{dp}}_{qb}).$
(3)

The last terms here is an eighth-order derivative that describes the expansion
of general relativity with the string theory in sub-scale energies,

${\gamma}_{1}{({R}_{bcdm}{R}^{bcdm})}^{2}+{\gamma}_{2}{({R}_{bcdm}{\bar{R}}^{bcdm})}^{2}+{\gamma}_{3}({R}_{bcdm}{\bar{R}}^{bcdm}{R}_{bcdm}{R}^{bcdm}).$
(4)

For simplicity, we skipped the Maxwell field corrections in the corresponding
model and postponed it to work later. All the above corrections at Planck
scale are essential for the describing of full QG . However, a small area can
be considered where these reforms prevail. By calculating the various
constraints, we take a step towards the quantum mechanical regime. In that
case we take advantage from Abbot-Deser-Tekin method (ATD)and obtain the
gravitational mass of these corrections. This method is fully described in
subsection 3.1.

Another case for distinguishing landscape from swampland is to investigate
some hydrodynamic values in the asymptotically AdS space-time. These values
are presented as a conjecture for the holographic field theory of dual. As we
know, AdS/CFT conjecture is coming from holography and black hole objects play
an important role in such conjecture. On the other hand, due to Hawking-
radiation black hole objects follow statistical mechanics, so they have
temperature and entropy. Quantum mechanics can be said to be a diffusion and
attenuate theory at the fundamental level. It can be said that the
asymptotically AdS space-time with event horizon is interpreted as thermal
states in the field theory of dual, this means that the slight perturbations
of a black hole or black brane correspond to minor deviations from the
thermodynamic equilibrium in the field theory of dual. Our corrections lead to
perturbation in the behavior of the black hole and black brane. Also here we
say that the disturbance causes a similar behavior to hydrodynamics 42 ; 43 ;
44 ; 45 ; 46 ; 47 ; 48 ; 49 ; 50 ; 51 ; 52 ; 53 . As we know, hydrodynamics is
an effective theory that describes the system’s dynamics at large distances
and time scale. According to the general second law of thermodynamics (GSL),
the universal relaxation bound $\tau$ is related to the mean free path
$\ell_{mtp}$, which is given by,

$\ell_{mtp}\sim\tau\sim\frac{1}{T},$ (5)

$\ell_{mtp}$ limits the description of hydrodynamics. These holographic
arguments help to link quantum field theory to gravity. As a result, GSL is a
powerful law that connects quantum theory, gravity and thermodynamics with
respect to each other. Another factor that characterizes the universal
relaxation properties of perturbation fluids is the lower limit of
$\frac{\eta}{s}\geq\frac{1}{4\pi}$ or KSS bound (Kovtun-Starinets-Son)54 ; 55
; 56 ; 57 ; 58 ; 59 ; 60 ; 61 ; 62 ; 63 ; 64 ; 65 ; 66 . Fluids respond to
perturbations in two ways: the first leads to the sound state, which is caused
by longitudinal oscillations, and the second leads to the shear viscosity
state, which arises from transverse oscillations. The shear viscosity state
$\frac{\eta}{s}$ indicates how close a fluid is to perfection. This value is
equal to $\frac{\hbar}{4\pi k_{B}}$ for a set of the quantum field theory with
strong interactions whose dual description involves black holes in AdS space-
time . This boundary is saturated for the theory of boundary fields in
finite’t Hooft coupling and the number of colors $N_{c}$. Such theories are
Einstein’s gravity dual; these corrections show that it increases
$\frac{\eta}{s}$.2 . Another proof of conjecture is obtained from the explicit
calculations of the $\alpha^{\prime}$ correction leading to type IIB string
theory, which is compacted into 5-dimensions2 . Also, the above corrections
show that $\frac{\eta}{s}$ can be violated2 . The first time that
$\frac{\eta}{s}$ does not disappear is the 6-derivative correction. From
thermal field theory point of view , the calculation of shear viscosity
through Kubo’s formula, the poles of the stress-energy tensor correlation
function and the equation of state $D=\frac{\eta}{sT}$ have the same results.
We also compute the diffusion equation for our model. We relate the fluid
behavior of the boundary theory to the fluid of the bulk 2 . It seems that our
knowledge of the relationship between hydrodynamics and the physics of black
holes is still incomplete, and this article improves our understanding of this
issue67 ; 68 ; 69 ; 70 ; 71 ; 72 ; 73 ; 74 ; 75 ; 76 ; 77 ; 78 ; 79 ; 80 ; 81
.

Another perturbation that causes the species’ hydrodynamic behavior in black
holes is the entry of Dilaton into it. Another application of the AdS/CFT is
the study of coupled systems near critical points. At these points, the system
has scale symmetry and is described by a CFT theory. In many physical systems,
critical points are represented by a dynamic scale, under which the space-time
scale varies,82 ; 83 ; 84 ; 85 ; 86 ; 87 .

${ds}^{2}={r}^{\frac{-2\theta}{d}}(-{r}^{2z}{dt}^{2}+\frac{{dr}^{2}}{{r}^{2}}+{r}^{2}{d\vec{x}}^{2}),$
(6)

$d$, $z$ and $\theta$ are the dimension of space-time, the dynamic constant,
and hyperscaling violations parameter, respectively. We consider $d=5$, $z=1$
and $d-2+z\leq\theta\leq d-1+z$ in our background. This metric is not
inherently relativity, so that it can be considered for a toy model. In the
geometries of non-relativistic hyperscaling, the violation can also be
considered an effective holographic description that lives in a finite branch
$r$. The QFT states show its boundary in UV cutoff. For such theories,
Einstein’s gravity must be include the Abelian gauge field. In this article,
we investigate Einstein-Hilbert-Maxwell-Dilaton’s theory (EHMD). This is a
reasonable extension of the scalar-tensor theory, which is based on the low-
energy approximation of string theory and its space based on the geometries of
Lifshitz-like black brane88 ; 89 ; 90 ; 91 ; 92 ; 93 ; 94 ; 95 ; 96 .

All above mentioned information give us motivation to organize paper as
follow. In section 2, we describe the action and solution of the equations
corresponding to of matter Lagrangian $\epsilon\epsilon$ . In section 3, we
first obtain the gravitational mass from the ADT method for the above
mentioned corrections, then we calculate the mass-to-charge ratio for all
terms of action. In section 4, we calculate the hydrodynamic values, including
the ratio of shear viscosity to entropy density, etc. In section 5, we specify
the common constraint of WGC and KSS bound, for all corrections. In the last
section 6, we describe the results of our work.

##  2 The Einstein-Hilbert-Maxwell-Dilaton with corrections

In the beginning, to make exciting action for our model we use higher-order
curvature corrections . Using various references such as 95 ; 96 , we tried to
get solution of the corresponding corrected action.

In that case, we show that the solution of action correspond to hyperscaling
violation charged $AdS_{d+2}$ black brane. On the other hand, such action can
identified by the constraints of commonalities of the two theories as WGC and
KSS bound. So, the above mentioned action with corresponding constraints lead
us to obtain two parameters of hyperscaling violation metric as $\theta$ and
dynamical scaling $z$. Here, we face by two part in action as geometric with
higher derivatives ${\ell}_{G}$ and matter part of Lagrangian as ${\ell}_{m}$.
So, the corresponding action in $(d+2)$ dimension is given by,

$\begin{split}S=&\frac{1}{16\pi{G}_{d+2}}{\int}{d}^{d+2}x\sqrt{-g}(R-2\Lambda+{\ell}_{G}+{\ell}_{m}),\\\
&{\ell}_{G}={\alpha}_{1}{R}^{2}+{\alpha}_{2}({R}^{2}_{abcd}+4{R}^{2}_{ab}+{R}^{2})+{\alpha}_{3}{R}^{2}_{ab}+{\beta}_{1}({{R}^{ab}}_{mn}{{R}^{mn}}_{pc}{{R}^{pc}}_{ab})+{\beta}_{2}({{R}^{ab}}_{cd}\\\
&{{R}^{cq}}_{pq}{{R}^{dp}}_{qb})+{\gamma}_{1}{({R}_{bcdm}{R}^{bcdm})}^{2}+{\gamma}_{2}{({R}_{bcdm}{\bar{R}}^{bcdm})}^{2}+{\gamma}_{3}({R}_{bcdm}{\bar{R}}^{bcdm}{R}_{bcdm}{R}^{bcdm}),\\\
&{\ell}_{m}=-\frac{1}{4}\sum_{i=1}^{2}{e}^{{\lambda}_{i}\phi}{F}^{2}_{i}-\frac{1}{2}{(\partial\phi)}^{2}+{\upsilon}_{0}{e}^{\upsilon\phi}\,,\qquad{\bar{R}}^{bcdm}={{\varepsilon}^{bc}}_{an}{R}^{andm}\end{split}$
(7)

${\upsilon}_{0}$, ${\upsilon}$, ${\lambda}_{1,2}$ and $\Lambda$ are free
parameters and cosmological constant respectively. In order to have quantum
gravity in the Planck scale we essential to consider higher derivative
correction terms. We note here the couplings${\alpha}_{1,3}$, ${\alpha}_{2}$,
${\beta}_{1,2}$ and ${\gamma}_{1,2,3}$ are charge, the Gauss-Bonnet, the
curvature-cubed and curvature-quartile respectively. In the matter part of
lagrangian, we need to find gravitational theories in the framework of the
$AdS/CFT$ that describe the gravity of critical points (Like Lifshitz fixed
points). Therefore, in this section, we consider the EHMD model. This model
consists of a scalar field $\phi$ and two gauge fields $U(1)$, one for charge
and the other along with scalar field generated an anisotropic scaling. We
also use a typical exponential potential of string theory for the dilaton
field. This potential is crucial to achieving a solution for this action and
can be found near the horizons of different branes. We employ the equation of
motion corresponding to matter Lagrangian ${\ell}_{M}$ 95 .

$\begin{split}&\tau_{matter}=\frac{1}{2}\sum_{i=1}^{2}e^{\lambda_{i}\phi}({F^{c}_{i}}_{a}{F_{i}}_{cb}-{F}^{2}_{i}\frac{g_{ab}}{2d})+\frac{1}{2}{\partial_{a}\phi}{\partial_{b}\phi}-\upsilon_{0}e^{\upsilon\phi}\frac{g_{ab}}{d},\\\
&\nabla^{2}\phi=-\upsilon_{0}e^{\upsilon\phi}\upsilon+\frac{1}{4}\sum_{i=1}^{2}\lambda_{i}e^{\lambda_{i}\phi}F^{2}_{i}\,,\qquad{\nabla}_{a}(\sqrt{-g}{e}^{{\lambda}_{i}\phi}{F}^{ab}_{i})=0,\end{split}$
(8)

The solution of matter sector without the higher derivatives terms is given
by,

$\begin{split}&{ds}^{2}=\frac{-r^{z}(r^{d+z}-Mr^{\theta}+Q^{2}r^{2(\theta+1)})}{r^{d+\frac{2\theta}{d}}}{dt}^{2}+\frac{r^{z+d-2-\frac{2\theta}{d}}}{(r^{z+d}-Mr^{\theta}+Q^{2}r^{2(\theta+1)})}{dr}^{2}+r^{2}{d\vec{x}}^{2},\\\
&F_{2}=Qr^{(\theta+1-d-z)}e^{\sqrt{\frac{1-z+\frac{\theta}{d}}{2(\theta-d)}}\varphi_{0}}\sqrt{2(\theta-d)(\theta+2-d-z)},\\\
&e^{\phi}=e^{{\varphi}_{0}}r^{\sqrt{2(\theta-d)(1-z+\frac{\theta}{d})}}\,,\qquad\lambda_{2}=\sqrt{\frac{2(1-z+\frac{\theta}{d})}{\theta-d}},\end{split}$
(9)

where $K=0$ is flat space-time and $l=1$ is AdS radius. Here, $M$ and $Q$ are
the mass and charge of the black hole, respectively. In the next section to
include the terms of higher-derivatives ${\ell}_{G}$ in the solution, first
one can obtained the gravitational mass by the ADT method.

##  3 The weak gravity conjecture

In this section, we are going to calculate the extremality condition of WGC.
First, we use ADT method and obtain energy ${\ell}_{G}$, then we couple
corresponding theory to the gauge and dilaton fields, and finally we calculate
the ratio of $M/Q.$

###  3.1 The gravitational mass with the ADT method

Now we are to going briefly describe the ADT method. Our way is similar to the
Landau-Lifshitz method, which is asymptotically flat space-time. In this
method, we obtain the equations of motion corresponding to the action. Here,
the gravitational theory is given by,

${\Psi}_{ab}(g,R,{R}^{2},{R}^{3},{R}^{4},...)=\omega{\tau}_{ab},$ (10)

where $\omega$ and $\tau$ are gravitational coupling and stress-energy tensor
of matter. For linearization, we decompose the metric into
${g}_{ab}={\bar{g}}_{ab}+{h}_{ab}$. Metric $\bar{g}_{ab}$ is to the equations
(7) with ${\tau}_{ab}=0$, in this mode we have $E=0$ in this mode. Also here,
the deviation ${h}_{ab}$ is for the linear state that relates to our
gravitational terms and it disappears infinitely. The effective stress-tensor
can be obtained by linearization,

${\Theta(\bar{g})}_{abcd}{h}^{cd}={\psi}_{ab}=\omega{T}_{ab},$ (11)

where $\Theta(\bar{g})$ is the hermitian operator that depends on the
background metric. The equation (8) is obtained by the generalized Bianchi
identity ${\bar{\nabla}}^{a}{\psi}_{ab}=0$, the covariant conservation of the
stress tensor ${\bar{\nabla}}^{a}{T}_{ab}=0$ and by placing ${g}_{ab}$. Then,
to calculate a conserved charge, we use the tensor ${T}_{ab}$. In this method,
the gravitational mass (corresponding energy) can be obtained by the following
integral,

$E={\int}_{\varsigma}{d}^{d-1}x\sqrt{\bar{g}\varsigma}{n}_{a}{T}^{ab}{\bar{\xi}}_{b},$
(12)

According to the above equation, using time-like Killing vectors
${\bar{\xi}}_{b}$ and a constant-time hypersurface ${\varsigma}$ with normal
unit vector ${n}_{a}$, one can write energy as integral of bulk. We can now
express the conserved current ${T}^{ab}{\bar{\xi}}_{b}$ as a complete
derivative of potential ${\Gamma}^{ab}$, it means that
${T}^{ab}{\bar{\xi}}_{b}={\bar{\nabla}}_{a}{\Gamma}^{ab}$. The upper bulk
integral can be written as a boundary integral, and ${r}_{b}$ is the boundary
unit.

$\textbf{E}={\int}_{\partial\varsigma}{d}^{d-2}x\sqrt{\bar{g}\partial\varsigma}{n}_{a}{r}_{b}{\Gamma}^{ab},$
(13)

Now, we are going to calculate the corresponding energy (gravitational mass).
As we know such energy is calculated for the six-derivative and the four-
derivative theories by 96 . Now, we take advantage from 96  and obtain the
above mentioned energy for the eight-derivative theory. The equation motion
correspond to action (7) is obtained by the following expression,

$-\frac{16\pi{G}_{d+2}}{\sqrt{-g}}\frac{\delta{S}_{m}}{\delta{g}^{ab}}={R}_{ab}-\frac{1}{2}{g}_{ab}R+\Lambda{g}_{ab}+{\psi}^{(i)}_{ab}=16\pi{G}_{d+2}{\tau}_{ab},$
(14)

where ${\psi}^{(i)}_{ab}$ corresponds to the equation of motion for the i=4,
6, and 8-derivative corrections, which are given by,

$\begin{split}\psi^{4}_{ab}=&2\alpha_{1}R(R_{ab}-\frac{1}{4}Rg_{ab})+(2\alpha_{1}+\alpha_{3})(-\nabla_{b}\nabla_{a}+g_{ab}g^{ab}\nabla_{b}\nabla_{a})R+\alpha_{3}g^{ab}\nabla_{b}\nabla_{a}(R_{ab}\\\
&-\frac{1}{2}Rg_{ab})+2\alpha_{2}[RR_{ab}-2R_{adbc}R^{dc}+R_{adcp}{R_{b}}^{dcp}-2R_{ad}{R^{d}}_{b}-\frac{1}{4}(R^{2}_{pfcd}-4R^{2}_{cd}\\\
&+R^{2})g_{ab}]+2\alpha_{3}(R_{adbc}-\frac{1}{4}R_{dc}g_{ab})R^{dc},\end{split}$
(15)
$\begin{split}\psi^{6}_{ab}=&\frac{1}{2}[-6\beta_{1}\nabla_{f}\nabla_{c}({R_{a}}^{cpq}{{R_{pq}}^{f}}_{b})+3\beta_{2}\nabla_{f}\nabla_{c}(R_{aqpb}R^{cqpf}-{R_{a}}^{qpf}{R^{c}}_{qpb})+\beta_{1}(3R_{dapc}\\\
&R^{pcmn}{R^{d}}_{bmn}-\frac{1}{2}{R^{de}}_{pc}R^{pcmn}R_{mnde}g_{ab})+\beta_{2}(3{R^{n}}_{acd}{R^{cq}}_{pn}{{{R^{d}}_{q}}^{p}}_{b}-\frac{1}{2}{R^{ne}}_{cd}\\\
&{R^{cq}}_{pn}{{{R^{d}}_{q}}^{p}}_{e}g_{ab})]+\frac{1}{2}[a\longleftrightarrow
b],\end{split}$ (16)
$\begin{split}\psi^{8}_{ab}=&\gamma_{1}(8R_{anbc}\nabla^{n}\nabla^{c}R_{bcdm}R^{bcdm}+\frac{g_{ab}}{2}(R_{bcdm}R^{bcdm})^{2})+\gamma_{2}(8R_{apbn}\nabla^{p}\nabla^{n}R_{bcdm}{\bar{R}}^{bcdm}\\\
&+(R_{bcdm}{\bar{R}}^{bcdm})^{2}\frac{g_{ab}}{2})+\gamma_{3}(4{\bar{R}}_{apbn}\nabla^{p}\nabla^{n}R_{bcdm}R^{bcdm}+4R_{apbn}\nabla^{p}\nabla^{n}R_{bcdm}{\bar{R}}^{bcdm}\\\
&+\frac{g_{ab}}{2}R_{bcdm}{\bar{R}}^{bcdm}R_{bcdm}R^{bcdm}),\end{split}$ (17)

Then, by placing the Riemann tensor, Ricci tensor and the scalar curvature in
$d+2$-dimensions in the above equations, we obtain AdS solution with the
effective cosmological constant 47,

$\begin{split}\frac{32{(d+2)}^{2}(\gamma_{3}+\gamma_{2}+\gamma_{1})}{d^{4}{(d+1)}^{2}}\Lambda^{4}_{eff}+\frac{4(d-4)(d\beta_{2}+4\beta_{1})}{d^{3}{(d+1)}^{2}}\Lambda^{3}_{eff}\\\
+\frac{2(d-2)(\alpha_{3}+\frac{d(d-1)}{(d+1)}\alpha_{2}+(d+2)\alpha_{1})}{d^{2}}\Lambda^{2}_{eff}+{\Lambda}_{eff}=\Lambda\end{split}$
(18)

Therefore, the perturbation solution of cosmological constant
${\Lambda}_{eff}$ is given by,

$\begin{split}{\Lambda}_{eff}=&\Lambda-\frac{2(2-d)(-\alpha_{3}-(2+d)\alpha_{1})}{d^{2}}\Lambda^{2}-\frac{2(2-d)(1-d)\alpha_{2}}{d(2+d)}\Lambda^{2}+\frac{16(4-d)\beta_{1}}{d^{3}(1+d)^{2}}\Lambda^{3}\\\
&+\frac{4(4-d)\beta_{2}}{d^{2}(1+d)^{2}}\Lambda^{3}-\frac{32(2+d)^{2}(\gamma_{3}+\gamma_{2}+\gamma_{1})}{d^{4}(1+d)^{2}}\Lambda^{4}+\frac{8(-2+d)^{2}(\alpha_{3}+(2+d)\alpha_{1})^{2}}{d^{4}}\\\
&\Lambda^{3}+\frac{8(-1+d)^{2}(-2+d)^{2}\alpha^{2}_{2}}{d^{2}(1+d)^{2}}\Lambda^{3}+\frac{16(-1+d)(-2+d)^{2}(\alpha_{3}+(2+d)\alpha_{1})\alpha_{2}}{d^{3}(1+d)}\Lambda^{3}\\\
&+\frac{32(-4+d)^{2}(4\beta_{1}(\beta_{2}d^{2}+4\beta_{1})+d^{2}\beta^{2}_{2})}{d^{6}(1+d)^{4}}\Lambda^{4}+...,\end{split}$
(19)

According to ${g}_{ab}={\bar{g}}_{ab}+{h}_{ab}$ and appendix A, we linearize
the equation of motion and obtain the stress tensor ${T}_{ab}$,

$\begin{split}16G\pi{T}_{ab}=&\textbf{\bigg{[}}1+\frac{4((d+2)\alpha_{1}+\alpha_{3})}{d}\Lambda_{eff}+\frac{4(1-d)(2-d)\alpha_{2}}{d(1+d)}\Lambda_{eff}-\frac{48(2d+1)\beta_{1}}{d^{2}(d+1)^{2}}\\\
&\Lambda^{2}_{eff}+\frac{12(3d-2)\beta_{2}}{d(d+1)^{2}}\Lambda^{2}_{eff}+\frac{32(d+2)(\gamma_{1}-\gamma_{2}+\gamma_{3})}{d^{3}(d+1)^{2}}\Lambda^{3}_{eff}\textbf{\bigg{]}}(R^{L}_{ab}-\frac{1}{2}{\bar{g}}_{ab}R_{L}\\\
&-\frac{2\Lambda_{eff}}{d}h_{ab})+\textbf{\bigg{[}}2\alpha_{1}+\alpha_{3}+\frac{24\beta_{1}}{d(d+1)}\Lambda_{eff}+\frac{128(\gamma_{1}+\gamma_{2}+\gamma_{3})}{d^{2}(d+1)^{2}}\Lambda^{2}_{eff}\textbf{\bigg{]}}({\bar{g}}_{ab}{\bar{g}}^{ab}\\\
&{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}-{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}+\frac{2\Lambda_{eff}}{d}{\bar{g}}_{ab})R_{L}+\textbf{\bigg{[}}\alpha_{3}+\frac{6(8\beta_{1}-\beta_{2})}{d(d+1)}\Lambda_{eff}\textbf{\bigg{]}}({\bar{g}}^{ab}{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}(R^{L}_{ab}\\\
&-\frac{1}{2}{\bar{g}}_{ab}R_{L}-\frac{2\Lambda_{eff}}{d}h_{ab})-\frac{2\Lambda_{eff}}{d}{\bar{g}}_{ab}R_{L}),\end{split}$
(20)

${R}^{L}_{ab}$ and ${R}_{L}$ are the linearized Ricci tensor and scalar
curvature. As we have stated, ${\bar{\nabla}}^{a}{T}_{ab}=0$; therefore,
${T}^{ab}{\bar{\xi}}_{b}={\bar{\nabla}}_{a}{\Gamma}^{ab}$, we can obtain
potential ${\Gamma}_{ab}$. Hence, the second and third terms of equation (20)
will not affect on the energy. And also, in our black brane model (7) will
fall off at infinity. It means that,

$\begin{split}&{\bar{\nabla}}^{a}({\bar{g}}_{ab}{\bar{g}}^{ab}{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}-{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}+\frac{2{\Lambda}_{eff}}{d}{\bar{g}}_{ab}){R}_{L}=0\\\
&{\bar{\nabla}}^{a}({\bar{g}}^{ab}{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}({R}^{L}_{ab}-\frac{1}{2}{\bar{g}}_{ab}{R}_{L}-\frac{2{\Lambda}_{eff}}{d}{h}_{ab})-\frac{2{\Lambda}_{eff}}{d}{\bar{g}}_{ab}{R}_{L})=0,\end{split}$
(21)

So for the first term of equation (20),one can obtain the potential
${\Gamma}_{ab}$ as,

$\begin{split}16G\pi\Gamma^{ab}=&\textbf{\bigg{[}}1+\frac{4((d+2)\alpha_{1}+\alpha_{3})}{d}\Lambda_{eff}+\frac{4(d-1)(d-2)\alpha_{2}}{d(d+1)}\Lambda_{eff}-\frac{48(2d+1)\beta_{1}}{d^{2}(d+1)^{2}}\Lambda^{2}_{eff}\\\
+&\frac{12(3d-2)\beta_{2}}{d(d+1)^{2}}\Lambda^{2}_{eff}+\frac{32(d+2)(\gamma_{1}-\gamma_{2}+\gamma_{3})}{d^{3}(d+1)^{2}}\Lambda^{3}_{eff}\textbf{\bigg{]}}({\bar{\xi}_{p}}\bar{\nabla^{a}}h^{bp}-{\bar{\xi}_{p}}\bar{\nabla^{b}}h^{ap}\\\
-&{\bar{\xi}^{b}}\bar{{\nabla}^{a}}h+{\bar{\xi}^{b}}\bar{\nabla_{p}}h^{ap}+{\bar{\xi}^{a}}\bar{{\nabla}^{b}}h-{\bar{\xi}^{a}}\bar{{\nabla}_{p}}h^{bp}+h^{ap}\bar{{\nabla}^{b}}\bar{{\xi}_{p}}-h^{bp}\bar{{\nabla}^{a}}\bar{\xi_{p}}+h\bar{\nabla^{a}}\bar{\xi^{b}}),\end{split}$
(22)

We note here that the energy is located in the couplings. Finally, using
equations (13), (19) and (22), we can obtain the energy of perturbation case,
which is given by.

$\begin{split}E=&\frac{md}{32G\pi}(1+\frac{4((d+2)\alpha_{1}+\alpha_{3})}{d}\Lambda_{eff}+\frac{4(1-d)(2-d)\alpha_{2}}{d(1+d)}\Lambda_{eff}-\frac{48(2d+1)\beta_{1}}{d^{2}(d+1)^{2}}\Lambda^{2}_{eff}\\\
+&\frac{12(3d-2)\beta_{2}}{d(d+1)^{2}}\Lambda^{2}_{eff}+\frac{32(d+2)(\gamma_{1}-\gamma_{2}+\gamma_{3})}{d^{3}(d+1)^{2}}\Lambda^{3}_{eff}),\end{split}$
(23)

where ${E}_{0}=\frac{md}{32G\pi}$ is the case of non-correction of action. Due
to the perturbation of equation (20) , our energy is also corrected by the
corresponding terms.

###  3.2 The Ratio $M/Q$

To obtain a ratio of $M$ to $Q$, we consider the following general ansatz,

${ds}^{2}=g_{tt}{dt}^{2}+g_{rr}{dr}^{2}+g_{xx}{d\vec{x}}^{2}$ (24)

The solution of equation (7) without corrections is given by,

$\begin{split}g_{tt}=&r^{4+2z-\theta}(-1+Mr^{2(\theta-\frac{d}{2}-\frac{\theta}{d})}-Q^{2}r^{2(1-d-z+3\frac{\theta}{2}-\frac{\theta}{d})}),\\\
g_{rr}=&{r}^{\frac{-2\theta}{d}}({r}^{6}-M{r}^{(6-d-z+{\theta})}+{Q}^{2}{r}^{2(4-d-z+{\theta})})^{-1},\\\
g_{xx}=&r^{2(1-\frac{\theta}{d})},\end{split}$ (25)

The corrections solution is fully described in appendix B. For the non-
correction mode, we take $r_{h}$ as the radius of the outer horizon and put it
in $f(r)$ (9) to get $M(r_{h})=r^{1/2}(1+Q^{2}r_{h})$. Then from
$f(r_{h},r)=0$, we calculate
${Q_{1}}^{2}=\frac{-r^{d+z-\theta}+{r_{h}}^{d+z-\theta}}{r^{2-d-z+\theta}-{r_{h}}^{2-d-z+\theta}}$.
Hawking’s temperature can be obtained by general formula of black hole, which
is given by,

$T=-\frac{\partial_{r}g_{tt}}{4\pi\sqrt{-g_{tt}g_{rr}}}{\bigg{|}}_{r=r_{h}}$
(26)

In that case the corresponding temperature for non-correction mode will be as,

$T_{(0)}=\frac{r^{4+z}}{2\pi}(2+z-\frac{2\theta}{d}+M\frac{r^{\theta-
d-z}}{2}(d-z-4+\frac{2\theta}{d}-\theta)-Q^{2}r^{2(\theta+1-d-z)}(d-3+\frac{\theta}{d}-\theta))$
(27)

By placing $M(r_{h})$ obtained from (9) and examining the extremal condition
$T=0$,

${Q_{2}}^{2}=\frac{-r^{d+z-\theta}(\theta-2d-dz)(d^{2}+\theta-d(3+\theta))+{r_{h}}^{d-\theta+z}(d^{2}+2\theta-d(4+\theta+z))}{2r^{2-d-z+\theta}(d^{2}+\theta-d(3+\theta))-{r_{h}}^{2-d+\theta-z}(d^{2}+2\theta-d(4+\theta+z))}$
(28)

In the extremal case, by setting ${Q_{1}}^{2}={Q_{2}}^{2}$, the $\theta$ range
is obtained $d-2+z\leq\theta\leq d-1+z$. This range is where the outer and
inner horizons coincide. Now again we write the non-correction temperature for
$d=5$, $z=1$ and on average of $\theta=4.5$, also we place $r=r_{h}$ and
$M=r^{1/2}(1+Q^{2}r)$.

$T_{0}(r,Q)=\frac{r^{4}}{8\pi}(Q^{2}+3r)$ (29)

The extremal condition is setting to $Q^{2}=-3r$. We also calculated the
Hawking temperature of the corrected solution, and we have given it in
appendix C.

On the other hand, we know that in the dual CFT, the temperature and time must
be set to $T_{CFT}\equiv l_{eff}$ and $t\rightarrow t/l_{eff}$. Also, the
energy density of the field theory is obtained by $M\equiv l_{eff}E$. To
compare the corrected solutions with the non-corrected ones, the temperature
must not change at higher order-correction. It means that $\frac{M}{Q}$ is
written in terms of $T_{CFT}$ instead of $r$. To do this, we fix the new
parameter $\tilde{r}$ in $T_{CFT}$.

$T_{CFT}(\tilde{r},Q)=\frac{\tilde{r}^{4}}{8\pi}(Q^{2}+3\tilde{r})$ (30)

We get $r$ based on $\tilde{r}$ to remove $r$ in all terms in favor of
$\tilde{r}$ by solving equation $T_{CFT}\equiv l_{eff}T$. Then we make sure
that the corrected and non-corrected solutions have the same temperature. The
non-corrected mode is $r=\tilde{r}$, and for the rest of the higher order-
derivative terms, it is calculated similarly in the extermal solution,
$Q^{2}=-3\tilde{r}$.

Now we calculate ${(\frac{M}{Q})}_{0}$ for the non-correction state. Mass is
obtained through the metric (25) and perturbation energy $E_{0}$ (23),
$M_{0}=\frac{5\tilde{r}^{1/2}}{32G\pi}(Q^{2}+\tilde{r})$. The charge is
obtained for ${\Gamma}^{ab}=\sqrt{-g}{e}^{{\lambda}_{i}\phi}{F}^{ab}_{i}$ and
(9) by solving integral (13),
$Q_{0}=\frac{i}{\sqrt{2}}Q\tilde{r}^{(\frac{3i}{\sqrt{10}}-\frac{1}{2})}\exp(\frac{9i\varphi_{0}}{\sqrt{10}})$.
As a result,

${(\frac{M}{Q})}_{0}=\frac{5\sqrt{2}\exp(\frac{9}{\sqrt{-10}}\varphi_{0})}{32G\pi
i}(Q^{2}+\tilde{r})\frac{\tilde{r}^{\frac{3}{\sqrt{-10}}+1}}{Q}$ (31)

Hence, all correction terms are calculated by appendix B with placing
$(\tilde{r}=1)$.

$\begin{split}\frac{M}{Q}=&{(\frac{M}{Q})}_{0}(1-\alpha_{1}\big{[}\frac{42(291.6+459.7Q^{2}+177Q^{4}+8.4Q^{6})}{291.01+566.4Q^{2}+295.2Q^{4}+19.8Q^{6}}\big{]}-\alpha_{2}\big{[}\frac{48(132+139Q^{2}+7Q^{4})}{264+287Q^{2}+23Q^{4}}\big{]}\\\
-&\alpha_{3}\big{[}\frac{192(1+1.5Q^{2}+0.6Q^{4}+0.05Q^{6})}{15.6+32.1Q^{2}+19Q^{4}+2.5Q^{6}}\big{]}-\beta_{1}\big{[}\frac{5376(1+1.3Q^{2}+0.07Q^{4}-0.1Q^{6})}{70.1+87.2Q^{2}-23.7Q^{4}-40.8Q^{6}}\big{]}\\\
+&\beta_{2}\big{[}\frac{4160(1+1.5Q^{2}+0.5Q^{4}+0.02Q^{6})}{32+61.7Q^{2}+31.3Q^{4}+1.6Q^{6}}\big{]}+\gamma_{2}\big{[}\frac{168(24.2+38.9Q^{2}+9.9Q^{4}-1.5Q^{6})}{27.5+43.8Q^{2}+10.08Q^{4}-6.1Q^{6}}\big{]}\\\
-&(\gamma_{1}+\gamma_{3})\big{[}\frac{168(12.9+20.4Q^{2}+7.9Q^{4}+0.4Q^{6})}{291.01+566.4Q^{2}+295.2Q^{4}+19.8Q^{6}}\big{]}+.)\end{split}$
(32)

At extremality case as $Q^{2}=-3$, one can write following expression,

$\frac{M}{Q}=\frac{4.08}{16G\pi}e^{-2.84i\varphi_{0}}(1-16.3\alpha_{1}-27.3\alpha_{2}-6.6\alpha_{3}-20.06\beta_{1}+48.8\beta_{2}-68.4(\gamma_{1}+\gamma_{3})+42.1\gamma_{2}+.)$
(33)

The perturbation of black holes in the extremal and near-extremal states is
satisfied by the $\frac{M}{Q}\leq 1$ condition . Our results show that the
establishment of this condition depends on the sign of the couplings. In
section 5, we examine both $\alpha_{1}>0$ and $\alpha_{1}<0$ modes for all
couplings.

##  4 The Hydrodynamics

As a mentioned before another approach of distinguishing a landscape from
swampland is KSS bound which is coming from hydrodynamic side. For this reason
in this section we describe hydrodynamics and its values.

###  4.1 Limits of thermodynamic

As we know, the second law of thermodynamic is the bridge between quantum
theory, gravity and thermodynamics. According to it, the entropy of
$S_{matter}+S_{hole}$ never decreases in the interaction between matter and
black hole. It predicts the quantity of entropy generally will be universal.
We note here, this corresponding law creates two quantum constraints. I. The
minimum length scale:
The first constraint coming from minimum length scale and there is a specific
length range to the thermodynamic system. It is $\ell_{min}=\frac{1}{2\pi T}$
for a fluid with zero chemical potential. So, our corrections lead us to have
following expression,

$\ell_{min}=(9\times
10^{15}-342784.2\alpha_{1}-0.3\alpha_{3}-0.09\beta_{1}-159.08i\beta_{2}+1467.5i(\gamma_{1}+\gamma_{3})+311229i\gamma_{2})$
(34)

The temperature for term $\alpha_{2}$ is zero, which means that we have
infinite length. Generally, one can say that here hydrodynamics is an
effective theory for perturbation of Gauss-Bonnet at all lengths. Other above
values also indicate the effectiveness of the hydrodynamics theory for
perturbations of hyperscaling violating charged AdSd+2 black brane.

#### II. Universal relaxation bound:

Another constraint coming from $\tau\geq\frac{1}{\pi T}$ which is the time
limit of a perturbed thermodynamic system to achieve universal relaxation.
This constraint for our corrections,

$\tau=(1.8\times
10^{16}-685568.4\alpha_{1}-0.6\alpha_{3}-0.18\beta_{1}-318.1i\beta_{2}+2935.08i(\gamma_{1}+\gamma_{3})+622458.1i\gamma_{2}+.)$
(35)

So, this above limits of length give us the validity of the effective
hydrodynamic description. On the other hand, each fluid has two types of
physical responses to different perturbations, shear viscosity and sound. The
most extended turbulence state has wave-length as $\lambda_{max}=2\pi\ell$ and
wave-number as $\kappa_{min}=\frac{2\pi}{\lambda}=2\pi T$. In the following we
have two parts, first we calculate the shear mode $(\frac{\eta}{s}$ for the
correction case, and then second part we obtain its diffusion relation.

###  4.2 The KSS Bounds

This section we calculate the ratio of shear viscosity to entropy density for
hyperscaling violation charged AdSd+2 black brane with higher-derivative
corrections. The shear viscosity is given by Kubo’s formula with two-point
correlation function of the stress-energy tensor will be as,

$\eta=\lim_{\omega\longrightarrow\infty}\int d^{4}\vec{x}\frac{e^{i\omega
t}}{2\omega}\langle[T_{xy}(x),T_{xy}(0)]\rangle$ (36)

We express the two-point function as a retarded Green’s function of $T_{xy}$,

$G_{tx,tx}(\omega,q)=-i\int\Theta(t)d^{4}\vec{x}e^{i(\omega
t-qz)}{\langle[T_{tx}(x),T_{tx}(0)]\rangle}\propto(-Dq^{2}+i\omega)^{-1}$ (37)

It has a pole at $\omega=-iDq^{2}$. The shear diffusion constant $D$ is
related to the plasma entropy density of the gauge theory, which is given by,

$D=\frac{\eta}{sT}$ (38)

In the thermal field theory, the calculation of shear viscosity via all three
of the above equations has the same results. According to the temperature
calculation in subsection 3.2, we obtain the shear viscosity from equation
(38). We have given the temperature (26) for all correction terms in appendix
C. At present, with the corrected solutions B, we can acquire the diffusion
constant for all corrective terms,

$D=\sqrt{\frac{g(r_{h})}{g_{tt}(r_{h})g_{rr}(r_{h})}}\int_{r_{h}}^{\infty}dr\frac{-g_{tt}g_{rr}}{g_{xx}\sqrt{-g}}$
(39)

We can now examine the universality of the shear viscosity ratio to the
entropy density $\frac{\eta}{s}=TD$ for our action . In case of non-corrective
state of the hyperscaling violation charged AdSd+2 black brane, this ratio is,

$\frac{\eta}{s}=\frac{1}{4\pi}(-2.45Mr^{5/2}+2.9Q^{2}r^{3}+3.81r^{4})$ (40)

For $r=r_{h}$, $d=5$ and $z=1$, we get an average of $\theta=4.5$. Using
equations (39), appendix B and C for extremal conditions, we obtain the ratio
$\frac{\eta}{s}$. In appendix D, we also provide $\frac{\eta}{s}$ for all
above mentioned corrections to the system .

$\begin{split}\frac{\eta}{s}=&\frac{1}{4\pi}(0.014+0.0001\alpha_{1}-0.0000008\alpha_{3}+0.000001\beta_{1}-6893.63\beta_{2}\\\
+&0.002\gamma_{1}-0.00005\gamma_{2}+0.002\gamma_{3}+..)\simeq\frac{1}{4\pi}(-6893.63\beta_{2})\end{split}$
(41)

The above results indicate the violation of condition KSS bound in different
couplings except for $\beta_{2}$, for this reason we consider only
$\beta_{2}$. We thoroughly compare corrections to the KSS bound and the WGC
for all coupling marks in section 5.

###  4.3 The Dispersion Law

Shear viscosity is the inherent ability of a turbulent fluid to relax toward
equilibrium, so there is a possibility of a relationship between KSS bound and
a low thermodynamic limit of the universal relaxation bound. We express this
relationship with the help of the diffusion equation.

$\omega=i\frac{\eta}{sT}\kappa^{2}+i\tau(\frac{\eta}{sT})^{2}\kappa^{4}+...\cong
0$ (42)

According to equations (34), (38) and wave-number, one can be obtain for their
corrections $\omega\cong 0$. This value of diffusion determines the inherent
ability of a fluid to eliminate perturbation and approach thermal equilibrium.

##  5 The connection WGC and KSS

We now examine the results of weak gravity conjecture and hydrodynamics for
the corresponding corrections. Also here we show that under what conditions
they have relationship to each other. To better perception the common
constraint, we categorized the perturbations into three sections as follows:

###  5.1 Four-Derivatives

We first consider the case where the perturbations for the six and eight
order-derivatives are off, i.e.$\beta_{1,2}=\gamma_{1,2,3}=0$. The terms of
this part is satisfied by the supersymmetry. Modes $\alpha_{1}$, $\alpha_{2}$,
and $\alpha_{3}$ are created by the energy of effective action correspond to
the heterotic string. According to (33) and (41), for the corresponding above
conditions in the extremal condition, we have

$\frac{M}{Q}=(\frac{M}{Q})_{0}(1-16.38\alpha_{1}-27.32\alpha_{2}-6.6\alpha_{3})\,,\qquad\frac{\eta}{s}=\frac{1}{4\pi}$
(43)

These results show that, the common range between two conjectures as WGC and
KSS bound in the extremal conditions is equal to $\alpha_{1}=0.061$,
$\alpha_{2}=0.036$ and $\alpha_{3}=0.151$.

$Coupling$ | $Charge-range$ | $Common-range$
---|---|---
$\alpha_{1}>0$ | $-1.877\leq Q^{2}\leq-1.003$ | $-1.877\leq Q^{2}\leq-1.003$
$\alpha_{1}<0$ | $-1.934\leq Q^{2}\leq-1.003$ |
$\alpha_{2}>0$ | $Q^{2}=-18.38$ | $-19.39\leq Q^{2}\leq-18.38$
$\alpha_{2}<0$ | $Q^{2}=-19.39$ |
$\alpha_{3}>0$ | $-1.68\leq Q^{2}\leq-1.083$ | $-1.68\leq Q^{2}\leq-1.085$
$\alpha_{3}<0$ | $-1.82\leq Q^{2}\leq-1.085$ |
Table 1: Charge range for 4-derivative couplings in near-extermal condition.

The range of KSS bound, near-extremal condition (32) covers everywhere due to
the low value of the ratio shear viscosity to entropy density (41); therefore,
the WGC space specifies the joint range. According to the sign of couplings,
we have different charging ranges in table 1. The joint range of their charge
gives us the common constraint the WGC and KSS bound. In that case, we have
drawn figure 1.

We do not consider the values $-18.5\leq Q^{2}\leq-17.86$ for $\alpha_{1}$ and
$-12.5\leq Q^{2}\leq-9.4$ for $\alpha_{3}$. For $\alpha_{2}$, we also saw the
results of $-2\leq Q^{2}\leq 2$, were not in our answers. The third figure 1
shows that even in this case, we have a joint range. Also, we considered the
coupling range to be $-1\leq\alpha_{1,2,3}\leq+1$. It should be noted that
mode $\alpha_{2}$ is the Gauss-Bonnet term, which we have shown for this mode;
both conditions are satisfied within specific charging ranges.

Figure 1: Plot of $(\alpha_{i}-Q^{2})$. The solar colors’ region represent
where both the KSS bound and the WGC are satisfied, for different couplings of
4-derivative perturbations.

###  5.2 Six-Derivatives

In the second part, we examine theories considered toy models for non-
supersymmetry string theory compactifications. Those theories may be dual to
CFTs. These theories are the same corrections as the six order-derivatives
with $\beta_{1}$ and $\beta_{2}$. We consider the state
$\alpha_{1,2,3}=\gamma_{1,2,3}=0$. First, we examine $\beta_{1}$, so we have
equations (33) and (41) for the extremal conditions.

$\frac{M}{Q}=(\frac{M}{Q})_{0}(1-20.06\beta_{1})\,,\qquad\frac{\eta}{s}=\frac{1}{4\pi}$
(44)

In extremal condition, two modes of the KSS bound and WGC are satisfied by the
point of $\beta_{1}=0.049)$.

| $Coupling$ | $Charge-range$ | $Common-range$
---|---|---|---
$WGC$ | $\beta_{1}>0$ | $-1.96\leq Q^{2}\leq 3.176$ | $-1.96\leq Q^{2}\leq 3.3$
| $\beta_{1}<0$ | $-2.003\leq Q^{2}\leq 3.3$ |
$WGC$ | $\beta_{2}>0$ | $-1.887\leq Q^{2}\leq-0.999$ | $-1.869\leq Q^{2}\leq-1.00$
| $\beta_{2}<0$ | $-1.869\leq Q^{2}\leq-1.00$ |
$KSS$ | $\beta_{2}>0$ | $-2.88\leq Q^{2}\leq 2.88$ | $-2.88\leq Q^{2}\leq 2.88$
| $\beta_{2}<0$ | $-2.88\leq Q^{2}\leq 2.88$ |
Table 2: Charge range for 6-derivative couplings in near-extermal condition.

In near-extremal condition, the KSS bound is satisfied to all charges, so
again, we only need to find the WGC range. According to equation (32), we
calculate the charge range for $\beta_{1}$ different sign and obtain the joint
range, see table 2. Then, to satisfy KSS bound and WGC for the coupling
$\beta_{1}$, we draw figure 2. According to equations (33) and (41), we see
the affect of coupling $\beta_{2}.$

$\frac{M}{Q}=(\frac{M}{Q})_{0}(1-48.84\beta_{2})\,,\qquad\frac{\eta}{s}=\frac{1}{4\pi}(-6893.63\beta_{2})$
(45)

The above results show that the extremal condition of KSS bound and WGC are
not satisfied together. Here, checking the near-extremal condition can yield
exciting results. We calculate the charge range separately from equations (32)
and (60) and obtain their joint range equal to $-1.869\leq Q^{2}\leq-1.00$,
table 2. Then we draw them separately, figure 2. According to this figure,
there is no satisfying between the KSS and the WGC in condition near-extremal.
Note that we have checked mode $\theta=4.5$, $d=5$, $z=1$ at hyperscaling
violating charged AdSd+2 black brane. As we calculated in section 3, $\theta$
has a range of $d-2+z\leq\theta\leq d-1+z$. It indicates that the possibility
of compatibility of these conjectures with other values has not yet been
eliminated.

Figure 2: Plot of $(\beta_{i}-Q^{2})$. For $\beta_{1}$ the solar colors’
region represent where both the KSS bound and the WGC are satisfied, but for
$\beta_{2}$ show these two conjectures have no satisfy region.

###  5.3 Eight-Derivatives

The latest corrections reviewed are the eight order-derivatives. These
corrections describe the development of general relativity with string theory
at sub-scale energies. In this case, too, we first consider the extremal
condition for all three terms of equations (33) and (41) with
$\alpha_{1,2,3}=\beta_{1,2}=0$.

$\frac{M}{Q}=(\frac{M}{Q})_{0}(1-68.43\gamma_{1}+42.18\gamma_{2}-68.43\gamma_{3})\,,\qquad\frac{\eta}{s}=\frac{1}{4\pi}$
(46)

Conjectures of the KSS bound and the WGC are consistent in points of
$\gamma_{1,3}=0.14$ and $\gamma_{2}=-0.023$.

$Coupling$ | $Charge-range$ | $Common-range$
---|---|---
$\gamma_{1}>0$ | $-1.883\leq Q^{2}\leq-1.017$ | $-1.883\leq Q^{2}\leq-1.017$
$\gamma_{1}<0$ | $-1.897\leq Q^{2}\leq-1.017$ |
$\gamma_{2}>0$ | $-2.083\leq Q^{2}\leq 9.480$ | $-2.064\leq Q^{2}\leq 9.232$
$\gamma_{2}<0$ | $-2.064\leq Q^{2}\leq 9.232$ |
$\gamma_{3}>0$ | $-1.883\leq Q^{2}\leq-1.017$ | $-1.883\leq Q^{2}\leq-1.017$
$\gamma_{3}<0$ | $-1.897\leq Q^{2}\leq-1.017$ |
Table 3: Charge range for 8-derivative couplings in near-extermal condition.

For the near-extremal condition, we calculate the charge range of all three
couplings from equation (32), Table 3. We do not consider $-15.73\leq
Q^{2}\leq-15.56$ answers for $\gamma_{1}$ and $\gamma_{3}$. According to the
common charge limits for the near-extremal condition of two conjectures, the
WGC and the KSS bound, we have drawn the consistent region for all couplings,
we see figure 3.

Figure 3: Plot of $(\gamma_{i}-Q^{2})$. The solar colors’ region represent
where the KSS bound and the WGC are satisfied.

##  6 Conclusions

In this paper, we first calculated the gravitational mass for the higher
derivative corrections (toy-models (7)) from the covariant ADT method in the
AdS background. In this method, gravitational mass (energy) can be obtained by
linearizing the equations of motion without the need for counter-terms. Also,
the effective stress tensor (20) has maintained its state in all terms and
only its coefficients have changed. These results are equal to 96 . Given
that, this effective stress tensor has nothing to do with the matter part of
the action. It can be examined whether this effective stress tensor can be a
general measure for different modes in higher corrections or not? The results
of our study prove this for the first-order corrections in the $4$, $6$, and
$8$-derivatives. We then obtained a ratio of $\frac{M}{Q}$ and show that
higher derivative corrections reduce the mass to charge ratio of the extremal
black brane (WGC). We also examined the hydrodynamic values of the black hole
in section 4. Likewise, we obtained the universal relaxation bound
$(\tau\geq\frac{1}{\pi T})$ by Hawking temperature, shear viscosity to entropy
ratio $(\frac{\eta}{s}\geq\frac{1}{4\pi})$ by shear diffusion equation and
dispersion equation $\omega$ of these hydrodynamic values. Our result shown
that there is a possibility of violating these values in higher corrections.
It also seems that probability there is a relationship between the universal
relaxation bound and KSS bound. Such constraints helped us to distinguish
which theories can be UV completed. Finally, we analyzed the results of the
KSS bound and the WGC in section 5 and our correction terms shown that the
joint range between the two conjectures for the hyperscaling violation charged
AdSd+2 black brane in the state $z=1,d=5,\theta=4.5$. The quantities must be
measured at one temperature for a meaningful comparison because these values
are in different theories. We rewrite these values by putting $\tilde{r}$
(expressed in (30)) in them. One of our most important results was that the
correction signs in KSS bound (41) and WGC (33) are opposite (except term
$\beta_{2}$). It means that the constraints are exclusive, and these theories
may not be UV completed, i.e., four-derivative, Gauss-Bonnet, and eight-
derivative. (Notice: this result is obtained for our particular black brane in
extremal conditions, it is interesting to study these quantities for such a
particular black brane. So there is an opportunity for possible changes in
other conditions.) On the other hand, for the six-derivative coefficient
$\beta_{2}$, both conjectures need the same sign. As a result, in the sense
KSS bound and WGC, $\beta_{2}$ shown good behavior, which can mean that the
ratio of $\frac{\eta}{s}$ does not vanish in extremal conditions. These
results are consistent with the results of other black branes 57 ; 96 , which
could mean the universal behavior of corrections in these conjectures. We also
considered first-order $\alpha^{\prime}$ corrections in our study. It is
possible that at higher-levels these two conjectures can be satisfied more
beautifully and more widely. In our future work, it may be interesting to
consider Maxwell field with other higher order derivative corrections. These
can serve as probes for the relationship between conjectures of the KSS bound
and the WGC. Also, we will rewrite our results in terms of parameters CFT,
thereby expanding our analysis of these conjectures.

## Appendix

##  Appendix A The Linearization

The Riemann tensor, Ricci tensor, and the scalar curve in $d+2$-dimensions,
used in equations (18) and (19).

${\bar{R}}_{acbd}=\frac{2{\Lambda}_{eff}}{d(d+1)}(\bar{g}_{ab}\bar{g}_{cd}-\bar{g}_{ad}\bar{g}_{cb})\,,\qquad{\bar{R}}_{ab}=\frac{2{\Lambda}_{eff}}{d}\bar{g}_{ab}\,,\qquad\bar{R}=\frac{2(d+2){\Lambda}_{eff}}{d},$
(47)

We present the linear equation of motion used in section 3.1. These
calculations are in $d+2$-dimensions.

$\begin{split}&{R}^{2}_{ab}{\big{|}}_{L}=\frac{4\Lambda}{d}{R}_{L},\hskip
154.0ptR{R}_{ab}{\big{|}}_{L}=\frac{2\Lambda}{d}((d+2){R}^{L}_{ab}+{\bar{g}}_{ab}{R}_{L}),\\\
&{R}^{L}_{acbd}{\bar{g}}^{cd}{\big{|}}_{L}=\frac{2\Lambda}{d(d+1)}({\bar{g}}_{ab}h-{h}_{ab})+{R}^{L}_{ab},\hskip
34.0pt{{R}^{d}}_{a}{R}_{bd}{\big{|}}_{L}=\frac{4\Lambda}{d^{2}}(d{R}^{L}_{ab}-\Lambda{h}_{ab}),\\\
&{R}_{acbd}{R}^{cd}{\big{|}}_{L}=\frac{2\Lambda}{d+1}({R}^{L}_{ab}+\frac{{\bar{g}}_{ab}}{d}{R}_{L}+\frac{2\Lambda}{d^{2}}{h}_{ab}),\hskip
18.0pt{R}_{abcd}{R}^{abcd}{\big{|}}_{L}=\frac{8\Lambda}{d(d+1)}{R}_{L},\\\
&{R}_{aecd}{{R}_{b}}^{ecd}{\big{|}}_{L}=\frac{8\Lambda}{d(d+1)}({R}^{L}_{ab}-\frac{\Lambda}{d}{h}_{ab}),\hskip
46.0pt{R}^{2}_{abcd}-4{R}^{2}_{cd}+{R}^{2}{\big{|}}_{L}=\frac{4\Lambda(d-1)}{(d+1)}{R}_{L},\end{split}$
(48)
$\begin{split}\nabla_{f}\nabla_{c}({{R}_{a}}^{qpf}{{R}^{c}}_{qpb}){\big{|}}_{L}=&\frac{2\Lambda}{d(d+1)}{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}{R}_{L}+\frac{4\Lambda^{2}(d+2)}{d^{2}(d+1)^{2}}(({R}^{L}_{ab}-{R}_{L}\frac{{\bar{g}}_{ab}}{(d+2)})-\frac{2\Lambda}{d}{h}_{ab}),\\\
\nabla_{f}\nabla_{c}({{R}_{a}}^{cpq}{{{R}_{pq}}^{f}}_{b}){\big{|}}_{L}=&\frac{-8\Lambda}{d(d+1)}(\bar{\Box}{R}^{L}_{ab}+\frac{1}{2}{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}{R}_{L})+\frac{16\Lambda^{2}}{d^{2}(d+1)^{2}}((d+1)\bar{\Box}{h}_{ab}-{R}_{L}{\bar{g}}_{ab}\\\
&-(d+2){R}^{L}_{ab})-\frac{16(d+2)\Lambda^{3}}{d^{3}(d+1)^{2}}{h}_{ab},\\\
\nabla_{f}\nabla_{c}({R}_{aqpb}{R}^{cqpf}){\big{|}}_{L}=&\frac{-2\Lambda}{d(1+d)}({\bar{\nabla}}_{a}{\bar{\nabla}}_{b}{R}_{L}-\bar{\Box}({R}^{L}_{ab}-\frac{{\bar{g}}_{ab}}{2}{R}_{L}))+\frac{8\Lambda^{2}}{d^{2}(1+d)^{2}}((2+d){R}^{L}_{ab}\\\
&+\frac{(1+d)}{2}\bar{\Box}{h}_{ab}-{R}_{L}{\bar{g}}_{ab})-\frac{16(2+d)\Lambda^{3}}{d^{3}(1+d)^{2}}{h}_{ab},\\\
\end{split}$ (49)
$\begin{split}{R}_{dapc}{R}^{pcmn}{{R}^{d}}_{bmn}{\big{|}}_{L}=&\frac{24\Lambda^{2}}{d^{2}(d+1)^{2}}({R}^{L}_{ab}-\frac{4\Lambda}{3d}{h}_{ab}),\\\
{{R}^{n}}_{acd}{{R}^{cq}}_{pn}{{{{R}^{d}}_{q}}^{p}}_{b}{\big{|}}_{L}=&\frac{8\Lambda^{2}}{d^{2}(d+1)^{2}}((d-1){R}^{L}_{ab}+\frac{{\bar{g}}_{ab}}{d}{R}_{L}-\frac{(d-2)\Lambda}{d}{h}_{ab}),\\\
\end{split}$ (50)

##  Appendix B The Ansatz Solutions

The solution is calculated by placing the gravitational mass (23) in principal
action (7). In fact, according to the solution non-corrections (25) and the
cosmological constant (19), we compute the perturbation equations of motion up
to the first order for each action term. This does, corrections the mass in
the metric.

$\begin{split}g_{tt}=&\big{[}-r^{4+2z-\theta}+Mr^{2-2z+\theta+\frac{\theta}{d}-d}-Q^{2}r^{2(3+\theta-\frac{\theta}{d}-d)}\big{]}+\alpha_{1}\big{[}2A_{j}-(d^{2}+2+3d)A_{i}\big{]}-\alpha_{2}\\\
&\big{[}(d^{2}-3d+2)A_{i}\big{]}+\alpha_{3}\big{[}A_{k}+A_{j}-(1+d)A_{i}\big{]}-12\beta_{1}\big{[}2A_{k}+A_{j}+(d-\frac{3}{2})A_{i}\big{]}+3\beta_{2}\\\
&\big{[}A_{k}+(\frac{3d^{2}}{2}-d)A_{i}\big{]}+(\gamma_{1}+2\gamma_{3})\big{[}16A_{j}-(2+3d+d^{2})A_{i}\big{]}+2\gamma_{2}\big{[}16A_{j}+(2+d)A_{i}\big{]}\end{split}$
(51)
$\begin{split}{g}_{rr}=&\big{[}\frac{{r}^{-2(\frac{\theta}{d}+3)}}{1-M{r}^{{\theta}-d-z}+{Q}^{2}{r}^{2({\theta}+1-d-z)}}\big{]}+{\alpha}_{1}\big{[}2{B}_{j}-(d^{2}+3d+2){B}_{i}\big{]}-{\alpha}_{2}\big{[}(d^{2}-3d+2)\\\
&{B}_{i}\big{]}+{\alpha}_{3}\big{[}{B}_{k}+{B}_{j}-(d+1){B}_{i}\big{]}-12{\beta}_{1}\big{[}2{B}_{k}+{B}_{j}+(d-\frac{3}{2}){B}_{i}\big{]}+3{\beta}_{2}\big{[}(\frac{3d^{2}}{2}-d)\\\
&{B}_{i}+{B}_{k}\big{]}+2({\gamma}_{1}+{\gamma}_{3})\big{[}16{B}_{j}-(d^{2}+3d+2){B}_{i}\big{]}+2{\gamma}_{2}\big{[}16{B}_{j}+(2+d){B}_{i}\big{]}\end{split}$
(52)

and,

$\begin{split}g_{xx}=&r^{2(1-\frac{\theta}{d})}+\alpha_{1}\big{[}2C_{j}-(d^{2}+3d+2)C_{i}\big{]}-\alpha_{2}\big{[}(d^{2}-3d+2)C_{i}\big{]}+\alpha_{3}\big{[}C_{k}+C_{j}-(1+d)\\\
&C_{i}\big{]}-12\beta_{1}\big{[}2C_{k}+C_{j}+(d-\frac{3}{2})C_{i}\big{]}+3\beta_{2}\big{[}C_{k}+d(\frac{3d}{2}-1)C_{i}\big{]}+2(\gamma_{1}+2\gamma_{3})\big{[}16C_{j}-\\\
&(d^{2}+3d+2)C_{i}\big{]}+2\gamma_{2}\big{[}16C_{j}+(2+d)C_{i}\big{]}\end{split}$
(53)

We have summarized the above equations and is written by.

$\begin{split}&R_{xx}=r^{2}(1-\frac{\theta}{d})\big{[}\theta-z-\frac{1}{d}+Q^{2}\frac{r^{2(\theta+1)}}{r^{2(z+d)}}(z-\theta-d)\big{]},\\\
&R_{tt}=\big{[}r^{2z}-M\frac{r^{z+\theta}}{r^{d}}+Q^{2}\frac{r^{2(1+\theta)}}{r^{2d}}\big{]}\big{[}(d-\theta+z)(z-\frac{\theta}{d})+Q^{2}\frac{r^{2(1+\theta+z)}}{r^{2d}}(z-\theta+d-2)\\\
&(d-1-\theta+\frac{\theta}{d})\big{]},\\\
&R_{rr}=\frac{(\theta-d)r^{z+d-2}\big{[}1+\frac{z}{d}+z\frac{(z-1)}{(d-\theta)}-\frac{Mr^{\theta}}{r^{z+d}}(\frac{\theta}{d}-z+1)+Q^{2}\frac{r^{2(\theta+1)}}{r^{2(z+d)}}(\frac{(2\theta-z+2)}{d}-\theta+d-2)\big{]}}{r^{-z-d}-Mr^{\theta}+Q^{2}r^{2\theta}},\\\
&R=r^{\frac{2\theta}{d}}\big{[}(2z-\theta)\frac{\theta}{d}-d(2z+d+1)-2z^{2}+2\theta(z-\theta+d+1)\big{]}-M\frac{r^{\theta+\frac{2\theta}{d}}}{r^{z+d}}\big{[}d-z\theta+\frac{\theta^{3}}{d}\\\
&+dz\big{]}+Q^{2}\frac{r^{2(1+\theta+\frac{\theta}{d})}}{r^{2(z+d)}}\big{[}2z-\theta(6+\theta)-4+d(3+2\theta-d)+(4+3\theta-2z)\frac{\theta}{d}\big{]},\end{split}$
(54)
$\begin{split}\Upsilon_{i}=&R_{\iota\iota}+g_{\iota\iota}(d+1-\frac{R}{2}),\hskip
6.0pt\Upsilon_{j}=(d+1)(g_{\iota\iota}R+\frac{R^{\prime\prime}}{g_{\iota\iota}^{\prime\prime}}),\hskip
6.0pt\Upsilon_{k}=g_{\iota\iota}((d+1)R+\frac{\Upsilon_{i}^{\prime\prime}}{g_{\iota\iota}^{\prime\prime}}),\\\
&\Upsilon(g_{\iota\iota})\rightarrow
A(g_{tt}),B(g_{rr}),C(g_{xx}),\end{split}$ (55)

##  Appendix C The Hawking Temperatur

By placing correction solutions B in the Hawking temperature equation (26), we
obtain the temperature of the hyperscaling charged AdSd+2 black brane. We
consider $d=5$, $z=1$ and we get an average of $\theta=4.5$ from equation
$d+z-2\leq\theta\leq d+z-1$. We also consider $r=r_{h}$ and
$M=r^{1/2}(1+Q^{2}r)$ by inserting the given values,

$\begin{split}T=&\frac{1}{4\pi}(1.5r^{5}+0.5Q^{2}r^{4}+{\alpha}_{1}A_{1}+{\alpha}_{2}A_{2}+{\alpha}_{3}A_{3}+{\beta}_{1}B_{1}+{\beta}_{2}B_{2}+({\gamma}_{1}+{\gamma}_{3})C_{1}+{\gamma}_{2}C_{2}),\end{split}$
(56)
$\begin{split}A_{1}=&\big{[}0.1Q^{4}r^{\frac{-12}{5}}-2.3Q^{8}+2.9Q^{2}r^{\frac{-7}{5}}+12.5r^{\frac{-2}{5}}-21.8Q^{6}r-62.4Q^{4}r^{2}-31.9Q^{2}r^{3}-\\\
&73Q^{6}r^{\frac{16}{5}}+69.7r^{4}+0.1Q^{8}r^{4}-750.5Q^{4}r^{\frac{21}{5}}+1.4Q^{6}r^{5}-2550.2Q^{2}r^{\frac{26}{5}}-35.1Q^{4}r^{6}-\\\
&2873.2r^{\frac{31}{5}}-203.5Q^{2}r^{7}-272.4r^{8}\big{]}\big{/}\big{[}(0.1Q^{4}r+0.7Q^{2}r^{2}+1.2r^{3})((-0.05Q^{2}-2.6r\\\
&-1.1Q^{4}r^{\frac{17}{5}}-1.1Q^{6}r^{\frac{32}{5}}-1.1Q^{2}r^{\frac{33}{5}}-1.08Q^{4}r^{\frac{37}{5}}-1.1Q^{2}r^{\frac{42}{5}})10^{16}+47.2Q^{2}r^{\frac{22}{5}}+\\\
&63r^{\frac{27}{5}}+552r^{\frac{47}{5}})^{\frac{1}{2}}\big{]},\\\
A_{2}=&0,\end{split}$ (57)
$\begin{split}A_{3}=&r^{\frac{41}{10}}\big{[}0.07Q^{8}-0.05Q^{4}r^{\frac{-12}{5}}-1.7Q^{2}r^{\frac{-7}{5}}-7.5r^{\frac{-2}{5}}+0.6Q^{6}r+1.9Q^{4}r^{2}+1.01r^{3}Q^{2}+\\\
&1.7Q^{6}r^{\frac{16}{5}}-1.9r^{4}-0.1Q^{8}r^{4}+17.8Q^{4}r^{\frac{21}{5}}-0.4Q^{6}r^{5}+60.5Q^{2}r^{\frac{26}{5}}+4.3Q^{4}r^{6}+24.8\\\
&Q^{2}r^{7}+68.2r^{\frac{31}{5}}+33.6r^{8}\big{]}\big{/}\big{[}(0.2Q^{4}+2.05Q^{2}r+3.7r^{2})i(1+48.4rQ^{-2})^{\frac{1}{2}}\big{]},\end{split}$
(58)
$\begin{split}B_{1}=&r^{\frac{51}{10}}\big{[}16.5r^{4}-0.6Q^{4}r^{\frac{-12}{5}}-21.6Q^{2}r^{\frac{-7}{5}}-90.8r^{\frac{-2}{5}}-3.9Q^{6}r-11.4Q^{4}r^{2}-3.9r^{3}Q^{2}-\\\
&23.1Q^{6}r^{\frac{16}{5}}-0.4Q^{8}-1.6Q^{8}r^{4}-237.03Q^{4}r^{\frac{21}{5}}-6.2Q^{6}r^{5}-804.9Q^{2}r^{\frac{26}{5}}+51.2r^{6}Q^{4}-\\\
&907.5r^{\frac{31}{5}}+304.1Q^{2}r^{7}+416.4r^{8}\big{]}\big{/}\big{[}(Q^{4}r+7.2Q^{2}r^{2}+13.04r^{3})i(1+48.4rQ^{-2})^{\frac{1}{2}}\big{]},\end{split}$
(59)
$\begin{split}B_{2}=&Qr^{\frac{12}{5}}\big{[}4.8Q^{8}+45.8Q^{6}r+130.9Q^{4}r^{2}+62.5Q^{2}r^{3}+150.7Q^{6}r^{\frac{16}{5}}-144.2r^{4}-1.1r^{4}Q^{8}+\\\
&1541.3Q^{4}r^{\frac{21}{5}}-3.3Q^{6}r^{5}+5234.3Q^{2}r^{\frac{26}{5}}+48.8Q^{4}r^{6}+5901.09r^{\frac{31}{5}}+259.1Q^{2}r^{7}+r^{8}\\\
&344.1\big{]}\big{/}\big{[}(\sqrt{2}Q^{4}+10.2Q^{2}r+18.4r^{2})(3Q^{4}r^{4}-2Q^{4}+64Q^{2}r^{\frac{16}{5}}+256r^{\frac{21}{5}}-32r^{6})^{\frac{1}{2}}\big{]},\end{split}$
(60)
$\begin{split}C_{1}=&r^{\frac{46}{10}}\big{[}0.03Q^{4}r^{\frac{-12}{5}}-0.1Q^{8}+1.1Q^{2}r^{\frac{-7}{5}}+4.7r^{\frac{-2}{5}}-1.03Q^{6}r-2.9Q^{4}r^{2}-1.5Q^{2}r^{3}-3.4\\\
&Q^{6}r^{\frac{16}{5}}+3.3r^{4}+0.03Q^{8}r^{4}-35.6Q^{4}r^{\frac{21}{5}}+0.19Q^{6}r^{5}-121.02Q^{2}r^{\frac{26}{5}}-2.6Q^{4}r^{6}-136.3\\\
&r^{\frac{31}{5}}-15.6Q^{2}r^{7}-20.9r^{8}\big{]}\big{/}\big{[}(0.1Q^{2}r+0.7r^{2}+1.2r^{3}Q^{-2})(0.05Q^{2}r^{\frac{22}{5}}-4.8\times
10^{12}Q^{2}\\\ &-2.3\times 10^{14}r-1.27\times
10^{13}Q^{4}r^{\frac{17}{5}}+0.07r^{\frac{27}{5}}-1.27\times
10^{13}Q^{6}r^{\frac{32}{5}}-1.27\times 10^{13}r^{\frac{33}{5}}Q^{2}\\\
&-7.9\times 10^{12}Q^{4}r^{\frac{37}{5}}-1.27\times
10^{13}Q^{2}r^{\frac{42}{5}}+r^{\frac{47}{5}})^{\frac{1}{2}}\big{]},\end{split}$
(61)
$\begin{split}C_{2}=&r^{\frac{46}{10}}\big{[}0.03Q^{8}+0.06Q^{4}r^{\frac{-12}{5}}+1.9Q^{2}r^{\frac{-7}{5}}+8.19r^{\frac{-2}{5}}+0.29Q^{6}r+0.84Q^{4}r^{2}+0.4Q^{2}r^{3}+\\\
&0.99Q^{6}r^{\frac{16}{5}}-0.9r^{4}+0.06Q^{8}r^{4}+10.2Q^{4}r^{\frac{21}{5}}+0.2Q^{6}r^{5}+34.7Q^{2}r^{\frac{26}{5}}-1.5Q^{4}r^{6}+r^{\frac{31}{5}}\\\
&39.09-9.19Q^{2}r^{7}-12.3r^{8}\big{]}\big{/}\big{[}(-0.1Q^{2}r-0.7r^{2}-1.2r^{3}Q^{-2})(-1.4\times
10^{13}Q^{2}-7.06\\\ &\times 10^{14}r+6.3\times
10^{12}Q^{4}r^{\frac{17}{5}}-0.02Q^{2}r^{\frac{22}{5}}-0.03r^{\frac{27}{5}}+6.3\times
10^{12}Q^{6}r^{\frac{32}{5}}+6.3\times 10^{12}\\\
&r^{\frac{33}{5}}Q^{2}+2.07\times 10^{13}Q^{4}r^{\frac{37}{5}}+6.3\times
10^{12}Q^{2}r^{\frac{42}{5}}+r^{\frac{47}{5}})^{\frac{1}{2}}\big{]},\end{split}$
(62)

##  Appendix D The Ratio $\eta/s$

We obtain the ratio $\frac{\eta}{s}=TD$ using equations appendix B, C, and
(39) for $Q^{2}=-3r$ and $M=r^{1/2}(1+Q^{2}r)$, $r=r_{h}$.

$\begin{split}\frac{\eta}{s}=\frac{1}{4\pi}(-1.36-0.45Q^{2}+\alpha_{1}A_{1}+\alpha_{2}A_{2}+\alpha_{3}A_{3}+\beta_{1}B_{1}+\beta_{2}B_{2}+\gamma_{1}C_{1}+\gamma_{3}C_{1}+\gamma_{2}C_{2})\end{split}$
(63)
$\begin{split}A_{1}=&\big{[}(0.149-1.19Q^{2}+19.07Q^{4}+1.19Q^{32}-0.018Q^{36}+0.0005Q^{38}-0.000009Q^{40})\\\
&10^{-7}+0.00003Q^{8}-0.00003Q^{10}+0.0002Q^{16}-0.0001Q^{18}-0.0002Q^{20}-0.00001\\\
&Q^{26}\big{]}\big{/}\big{[}74.08+1069.3Q^{2}+7263Q^{4}+30894Q^{6}+92400.7Q^{8}+206870Q^{10}+360336\\\
&Q^{12}+500836Q^{14}+564680.6Q^{16}+521348Q^{18}+395338.9Q^{20}+245366.7Q^{22}+Q^{24}\\\
&123362+49346.06Q^{26}+15277.4Q^{28}+3511.9Q^{30}+561Q^{32}+55.3Q^{34}+2.5Q^{36}\big{]},\\\
A_{2}=&0,\end{split}$ (64)
$\begin{split}A_{3}=&\big{[}1.16+18.6Q^{2}+74.5Q^{4}-596.04Q^{8}-2384.1Q^{10}-4768.3Q^{12}-2384.1Q^{14}-4768\\\
&Q^{16}-2384.1Q^{18}-149.01Q^{26}-37.2Q^{28}+0.03Q^{34}+0.002Q^{36}+7\times
10^{-5}Q^{38}+2.2\\\ &\times
10^{-6}Q^{40}\big{]}10^{-10}\big{/}\big{[}38.4+521.9Q^{2}+3289.9Q^{4}+12764.5Q^{6}+34075.6Q^{8}+66253.8\\\
&Q^{10}+96848.1Q^{12}+108285.1Q^{14}+93275.7Q^{16}+61832Q^{18}+31250.3Q^{20}+11814.8\\\
&Q^{22}+3234.3Q^{24}+607.05Q^{26}+70.8Q^{28}+4.17Q^{30}+0.06Q^{32}-1.1\times
10^{-13}Q^{34}\big{]},\end{split}$ (65)
$\begin{split}B_{1}=&\big{[}10^{-8}(5.9Q^{2}-71.5Q^{4}-572.2Q^{8}+762.9Q^{10}-47.6Q^{26}-2.98Q^{28}+0.01Q^{36}-\\\
&0.001Q^{38})-0.00003Q^{12}+0.00003Q^{14}-0.00001Q^{16}+0.00003Q^{18}\big{]}\big{/}\big{[}461.4+Q^{2}\\\
&6263+39478Q^{4}+153174Q^{6}+408908Q^{8}+795045.9Q^{10}+1162177Q^{12}+1299422\\\
&Q^{14}+1119309Q^{16}+741984Q^{18}+375004.5Q^{20}+141778Q^{22}+38812.5Q^{24}+Q^{26}\\\
&7284.6+850.15Q^{28}+50.1Q^{30}+0.7Q^{32}+1.36\times 10^{-12}Q^{34}+7.1\times
10^{-14}Q^{36}\big{]},\end{split}$ (66)
$\begin{split}B_{2}=&\big{[}1862.6+14901Q^{2}+238418.5Q^{4}+357627.8Q^{6}+238418.5Q^{8}-476837Q^{10}-Q^{12}\\\
&953674.3+476837Q^{14}+476837Q^{16}+119209.2Q^{20}+7450.5Q^{22}+1862.6Q^{24}+\\\
&Q^{28}0.01+0.05Q^{30}\big{]}\big{/}\big{[}698.4Q^{6}-1.81-291.03Q^{4}+931.3Q^{8}+465.6Q^{12}-465.6\\\
&Q^{14}-232.8Q^{16}-58.2Q^{18}-29.1Q^{20}+3.6Q^{22}+0.02Q^{24}-0.007Q^{26}\big{]},\end{split}$
(67)
$\begin{split}C_{1}=&\big{[}(76.2Q^{2}-4.7+38.1Q^{30}-9.5Q^{32}-1.1Q^{34}-0.1Q^{36}-0.002Q^{38})10^{-7}+0.00003\\\
&Q^{4}-0.0002Q^{8}+0.0009Q^{12}+0.001Q^{16}-0.0009Q^{18}+0.0009Q^{20}-0.0001Q^{26}-\\\
&0.00003Q^{28}\big{]}\big{/}\big{[}1185.4+16220.06Q^{2}+103269.5Q^{4}+405972.2Q^{6}+778772.5Q^{22}+\\\
&318289.9Q^{24}+109098Q^{26}+30833.5Q^{28}+6834.7Q^{30}+1090.4Q^{32}+108.9Q^{34}+\\\
&5.04Q^{36}+(1.1Q^{8}+2.1Q^{10}+3.3Q^{12}+3.8Q^{14}+3.6Q^{16}+2.65Q^{18}+1.5Q^{20})10^{6}\big{]},\end{split}$
(68)
$\begin{split}C_{2}=&\big{[}((4.7Q^{2}+19Q^{4}+38Q^{26}+1.1Q^{30}-0.14Q^{32})10^{5}-174.6Q^{34}-58.2Q^{36}-3.6Q^{38}\\\
&)10^{-12}+0.00002Q^{6}+0.00003Q^{8}-0.0001Q^{10}-0.00006Q^{12}+0.0001Q^{14}+0.0001\\\
&Q^{16}-0.0001Q^{18}-0.0001Q^{20}-0.00001Q^{24}\big{]}\big{/}\big{[}-1185.4-16071.8Q^{2}-101112Q^{4}\\\
&-391249.8Q^{6}-798581.7Q^{20}-254256.5Q^{22}-42372Q^{24}+4307.6Q^{26}+4767.5Q^{28}\\\
&+1391.3Q^{30}+224.3Q^{32}+20.3Q^{34}+0.8Q^{36}+(-1.04Q^{8}-2.01Q^{10}-Q^{12}2.9-\\\
&3.2Q^{14}-2.6Q^{16}-1.7Q^{18})10^{6}\big{]},\end{split}$ (69)

## References

  * (1) E. Palti, _A Brief Introduction to the Weak Gravity Conjecture_ , LHEP 2020 (2020) 176.
  * (2) E. Palti, _The Swampland: Introduction and Review_ , Fortsch. Phys. 67 (2019) 1900037, arxiv:1903.06239.
  * (3) T. D. Brennan, F. Carta and C. Vafa, _The String Landscape, the Swampland, and the Missing Corner_ , PoS TASI2017 (2017) 015, arxiv:1711.00864.
  * (4) T. Banks and N. Seiberg, _Symmetries and Strings in Field Theory and Gravity_ , Phys. Rev. D 83 (2011) 084019, arxiv:1011.5120.
  * (5) H.-C. Kim, G. Shiu and C. Vafa, _Branes and the Swampland_ , Phys. Rev. D 100 (2019) 066006, arxiv:1905.08261.
  * (6) B. Heidenreich, M. Reece and T. Rudelius, _The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff_ , Eur. Phys. J. C 78 (2018) 337, arxiv:1712.01868.
  * (7) C. Vafa, _The String landscape and the swampland_ , arxiv:hep-th/0509212.
  * (8) C. Cheung, J. Liu and G. N. Remmen, _Proof of the Weak Gravity Conjecture from Black Hole Entropy_ , JHEP 10 (2018) 004, arxiv:1801.08546.
  * (9) N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, _The String landscape, black holes and gravity as the weakest force_ , JHEP 06 (2007) 060, arxiv:hep-th/0601001.
  * (10) S. Hod, _A proof of the weak gravity conjecture_ , Int. J. Mod. Phys. D 26 (2017) 1742004, arxiv:1705.06287.
  * (11) A. Urbano, _Towards a proof of the Weak Gravity Conjecture_ , arxiv:1810.05621.
  * (12) M. Montero, _A Holographic Derivation of the Weak Gravity Conjecture_ , JHEP 03 (2019) 157, arxiv:1812.03978.
  * (13) N. Arkani-Hamed, S. Dimopoulos and S. Kachru, _Predictive landscapes and new physics at a TeV_ , arxiv:hep-th/0501082.
  * (14) E. Palti, _The Weak Gravity Conjecture and Scalar Fields_ , JHEP 08 (2017) 034, arxiv:1705.04328.
  * (15) S.-J. Lee, W. Lerche and T. Weigand, _A Stringy Test of the Scalar Weak Gravity Conjecture_ , Nucl. Phys. B 938 (2019) 321–350, arxiv:1810.05169.
  * (16) P. Saraswat, _Weak gravity conjecture and effective field theory_ , Phys. Rev. D 95 (2017) 025013, arxiv:1608.06951.
  * (17) H. Ooguri and C. Vafa, _On the Geometry of the String Landscape and the Swampland_ , Nucl. Phys. B 766 (2007) 21–33, arxiv:hep-th/0605264.
  * (18) D. Harlow and H. Ooguri, _Constraints on Symmetries from Holography_ , Phys. Rev. Lett. 122 (2019) 191601, arxiv:1810.05337.
  * (19) E. Gonzalo and L. E. Ibanez, _A Strong Scalar Weak Gravity Conjecture and Some Implications_ , JHEP 08 (2019) 118, arxiv:1903.08878.
  * (20) D. Lust, E. Palti and C. Vafa, _AdS and the Swampland_ , Phys. Lett. B 797 (2019) 134867, arxiv:1906.05225.
  * (21) J. Sadeghi. E. Naghd Mezerji. and S. Noori Gashti, _Study of some cosmological parameters in logarithmic corrected f(R) gravitational model with swampland conjectures_ , Modern Physics Letters A 36.05 (2021): arxiv:2150027.
  * (22) J. Sadeghi and E. Naghd Mezerji and S. Noori Gashti, _Logarithmic corrected f(R) gravitational model and swampland conjectures_ , (2019) arxiv:1910.11676[hep-th].
  * (23) J. Sadeghi. E. Naghd Mezerji. and S. Noori Gashti, _Universal relations and weak gravity conjecture of AdS black holes surrounded by perfect fluid dark matter with small correction_ , arXiv preprint, Nov 29 (2020) arxiv:2011.14366.
  * (24) J. Sadeghi. S Noori Gashti. E Naghd Mezerji, _The investigation of universal relation between corrections to entropy and extremality bounds with verification WGC_ , Physics of teh Dark Universe, 30 ,(2020) 100626.
  * (25) J. Sadeghi.and S. Noori Gashti. E. Naghd Mezerji. B. Pourhassan, _Weak Gravity Conjecture of Charged-Rotating-AdS Black Hole Surrounded by Quintessence and String Cloud_ , arXiv preprint. Nov 10 (2020) arxiv:2011.05109.
  * (26) S. Cremonini, J. T. Liu and P. Szepietowski, _Higher Derivative Corrections to R-charged Black Holes: Boundary Counterterms and the Mass-Charge Relation_ , JHEP 1003, 042 (2010) arxiv:0910.5159[hep-th].
  * (27) A. Giveon and D. Gorbonos, _On black fundamental strings_ , JHEP 0610, 038 (2006) arxiv:hep-th/0606156.
  * (28) A. Giveon, D. Gorbonos and M. Stern, _Fundamental Strings and Higher Derivative Corrections to d-Dimensional Black Holes_ , JHEP 1002, 012 (2010) arxiv:0909.5264[hep-th].
  * (29) A. Ghodsi and M. Alishahiha, _Non-relativistic D3-brane in the presence of higher derivative corrections_ , Phys. Rev. D 80, 026004 (2009) arxiv:0901.3431[hep-th].
  * (30) J. de Boer, M. Kulaxizi and A. Parnachev, _AdS7/CFT6, Gauss-Bonnet Gravity, and Viscosity Bound_ , JHEP 1003, 087 (2010) arxiv:0910.5347[hep-th].
  * (31) X. O. Camanho and J. D. Edelstein, _Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity_ , JHEP 1004, 007 (2010) arxiv:0911.3160[hep-th].
  * (32) D. M. Hofman and J. Maldacena, _Conformal collider physics: Energy and charge correlations_ , JHEP 0805, 012 (2008) arxiv:0803.1467[hep-th].
  * (33) R. C. Myers and B. Robinson, _Black Holes in Quasi-topological Gravity_ , arxiv:1003.5357[gr-qc].
  * (34) R. R. Metsaev and A. A. Tseytlin, _Curvature Cubed Terms In String Theory Effective Actions_ , Phys. Lett. B 185, 52 (1987).
  * (35) S. Deser and B. Tekin, _Gravitational energy in quadratic curvature gravities_ , Phys. Rev. Lett. 89, 101101 (2002) arxiv:hep-th/0205318.
  * (36) L. F. Abbott and S. Deser, _Stability of gravity with a cosmological constant_ , Nucl. Phys. B 195, 76 (1982).
  * (37) Y. Kats, L. Motl and M. Padi, _Higher-order corrections to mass-charge relation of extremal black holes_ , JHEP 0712, 068 (2007) arxiv:hep-th/0606100.
  * (38) S. Deser and B. Tekin, _Energy in generic higher curvature gravity theories_ , Phys. Rev. D 67, 084009 (2003) arxiv:hep-th/0212292.
  * (39) Y. Kats, L. Motl and M. Padi, _Higher-order corrections to mass-charge relation of extremal black holes_ , JHEP 12 (2007) 068, arxiv:hep-th/0606100.
  * (40) D. M. Hofman, _Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT_ , Nucl. Phys. B 823, 174 (2009) arxiv:0907.1625[hep-th].
  * (41) G. Barnich and F. Brandt, _Covariant theory of asymptotic symmetries, conservation laws and central charges_ , Nucl. Phys. B 633, 3 (2002) arxiv:hep-th/0111246.
  * (42) G. Koutsoumbas, E. Papantonopoulos, G. Siopsis, _Shear Viscosity and Chern-Simons Diffusion Rate from Hyperbolic Horizons_ , Phys. Lett. B 677, 74 (2009), arxiv:0809.3388[hep-th].
  * (43) R. G. Cai, Z. Y. Nie and Y. W. Sun, _Shear Viscosity from Effective Couplings of Gravitons_ , Phys. Rev. D 78, 126007 (2008) arxiv:0811.1665[hep-th].
  * (44) R. G. Cai, Z. Y. Nie, N. Ohta and Y. W. Sun, _Shear Viscosity from Gauss-Bonnet Gravity with a Dilaton Coupling_ , Phys. Rev. D 79, 066004 (2009) arxiv:0901.1421[hep-th].
  * (45) A. Buchel, R. C. Myers and A. Sinha, _Beyond eta/s=1/4pi_ , JHEP 03, 084 (2009), arxiv:0812.2521[hep-th].
  * (46) D. Harlow, _TASI Lectures on the Emergence of Bulk Physics in AdS/CFT_ , PoS TASI2017 (2018) 002, arxiv:1802.01040.
  * (47) A. Sinha and R. C. Myers, _The viscosity bound in string theory_ , Nucl. Phys. A 830, 295C (2009) arxiv:0907.4798[hep-th].
  * (48) S. S. Pal, _eta/s at finite coupling_ , Phys. Rev. D 81, 045005 (2010), arxiv:0910.0101[hep-th].
  * (49) Y. Kats and P. Petrov, _Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory_ , JHEP 0901, 044 (2009) arxiv:0712.0743[hep-th].
  * (50) R. C. Myers, M. F. Paulos and A. Sinha, _Holographic Hydrodynamics with a Chemical Potential_ , JHEP 0906, 006 (2009) arxiv:0903.2834[hep-th].
  * (51) X. O. Camanho and J. D. Edelstein, _Causality in AdS/CFT and Lovelock theory_ , JHEP 1006, 099 (2010) arxiv:0912.1944[hep-th].
  * (52) R. Brustein and A. J. M. Medved, _Proof of a universal lower bound on the shear viscosity to entropy density ratio_ , arxiv:0908.1473[hep-th].
  * (53) M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, _The Viscosity Bound and Causality Violation_ , Phys. Rev. Lett. 100, 191601 (2008) arxiv:0802.3318[hep-th].
  * (54) N. Banerjee and S. Dutta, _Near-Horizon Analysis of eta/s_ , arxiv:0911.0557[hep-th].
  * (55) R. C. Myers, M. F. Paulos and A. Sinha, _Holographic studies of quasi-topological gravity_ , arxiv:1004.2055[hep-th].
  * (56) S. Cremonini, K. Hanaki, J. T. Liu and P. Szepietowski, _Higher derivative effects on eta/s at finite chemical potential_ , Phys. Rev. D 80, 025002 (2009) arxiv:0903.3244[hep-th].
  * (57) S. S. Pal, _Weak Gravity Conjecture, Central Charges and eta/s_ , arxiv:1003.0745[hep-th].
  * (58) M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, _Viscosity Bound Violation in Higher Derivative Gravity_ , Phys. Rev. D 77, 126006 (2008), arxiv:0712.0805[hep-th].
  * (59) P. Kovtun, D. T. Son and A. O. Starinets, _Viscosity in strongly interacting quantum field theories from black hole physics_ , Phys. Rev. Lett. 94, 111601 (2005) arxiv:hep-th/0405231.
  * (60) T. Banks, M. Johnson and A. Shomer, _A note on gauge theories coupled to gravity_ , JHEP 0609, 049 (2006) arxiv:hep-th/0606277].
  * (61) A. Buchel, J. T. Liu and A. O. Starinets, _Coupling constant dependence of the shear viscosity in N=4 supersymmetric Yang-Mills theory_ , Nucl. Phys. B 707, 56 (2005) arxiv:hep-th/0406264.
  * (62) A. Buchel, _Resolving disagreement for eta/s in a CFT plasma at finite coupling_ , Nucl. Phys. B 803, 166 (2008) arxiv:0805.2683[hep-th].
  * (63) R. C. Myers, M. F. Paulos and A. Sinha, _Quantum corrections to eta/s_ , Phys. Rev. D 79, 041901 (2009) arxiv:0806.2156[hep-th].
  * (64) M. Henningson and K. Skenderis, _The holographic Weyl anomaly_ , JHEP 9807, 023 (1998) arxiv:hep-th/9806087.
  * (65) M. Henneaux and C. Teitelboim, _Asymptotically Anti-De Sitter Spaces_ , Commun. Math. Phys. 98, 391 (1985). Cambridge (1979).
  * (66) X. H. Ge, Y. Matsuo, F. W. Shu, S. J. Sin and T. Tsukioka, _Viscosity Bound, Causality Violation and Instability with Stringy Correction and Charge_ , JHEP 10, 009 (2008), arxiv:0808.2354[hep-th].
  * (67) G. Barnich and G. Comp‘ere,_Surface charge algebra in gauge theories and thermodynamic integrability_ , J. Math. Phys. 49, 042901 (2008) arxiv:0708.2378[gr-qc].
  * (68) G. Comp‘ere, _Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions_ , arxiv:0708.3153[hep-th].
  * (69) V. Iyer and R. M. Wald, _Some properties of Noether charge and a proposal for dynamical black hole entropy_ , Phys. Rev. D 50, 846 (1994) arxiv:gr-qc/9403028.
  * (70) R. M. Wald and A. Zoupas, _A General Definition of Conserved Quantities in General Relativity and Other Theories of Gravity_ , Phys. Rev. D 61, 084027 (2000) arxiv:gr-qc/9911095.
  * (71) D. T. Son and A. O. Starinets, _Minkowski-space correlators in AdS/CFT correspondence: Recipe and applications_ , JHEP 0209, 042 (2002) arxiv:hep-th/0205051.
  * (72) G. Policastro, D. T. Son and A. O. Starinets, _From AdS/CFT correspondence to hydrodynamics_ , JHEP 0209, 043 (2002) arxiv:hep-th/0205052.
  * (73) N. Banerjee and S. Dutta, _Higher Derivative Corrections to Shear Viscosity from Graviton’s Effective Coupling_ , JHEP 0903, 116 (2009) arxiv:0901.3848 [hep-th].
  * (74) N. Iqbal and H. Liu, _Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm_ , Phys. Rev. D 79, 025023 (2009), arxiv:0809.3808 [hep-th].
  * (75) R. M. Wald, _Black hole entropy is the Noether charge_ , Phys. Rev. D 48, 3427 (1993) arxiv:gr-qc/9307038.
  * (76) T. Jacobson, G. Kang and R. C. Myers, _On Black Hole Entropy_ , Phys. Rev. D 49, 6587 (1994) arxiv:gr-qc/9312023.
  * (77) R. R. Metsaev and A. A. Tseytlin, _Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the Sigma Model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor_ , Nucl. Phys. B 293, 385 (1987).
  * (78) V. Balasubramanian and P. Kraus, _A stress tensor for anti-de Sitter gravity_ , Commun. Math. Phys. 208, 413 (1999) arxiv:hep-th/9902121.
  * (79) M. Alishahiha and H. Yavartanoo, _On Holography with Hyperscaling Violation_ , arxiv:1208.6197[hep-th].
  * (80) X. Dong, S. Harrison, S. Kachru, G. Torroba and H. Wang, _Aspects of holography for theories with hyperscaling violation_ , JHEP 1206, 041 (2012) arxiv:1201.1905[hep-th].
  * (81) K. Narayan, _On Lifshitz scaling and hyperscaling violation in string theory_ , Phys. Rev. D 85, 106006 (2012) arxiv:1202.5935[hep-th].
  * (82) P. Dey and S. Roy, _Lifshitz-like space-time from intersecting branes in string/M theory_ , arxiv:1203.5381[hep-th].
  * (83) Y. S. Myung and T. Moon, _Quasinormal frequencies and thermodynamic quantities for the Lifshitz black holes_ , Phys. Rev. D 86, 024006 (2012) arxiv:1204.2116[hep-th].
  * (84) J. Tarrio and S. Vandoren, _Black holes and black branes in Lifshitz spacetimes_ , JHEP 1109, 017 (2011) arxiv:1105.6335[hep-th].
  * (85) K. Goldstein, S. Kachru, S. Prakash and S. P. Trivedi, _Holography of Charged Dilaton Black Holes_ , JHEP 1008, 078 (2010) arxiv:0911.3586[hep-th].
  * (86) J. M. Maldacena, _The large N limit of superconformal field theories and supergravity_ , Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] arxiv:hep-th/9711200.
  * (87) S. Kachru, X. Liu and M. Mulligan, _Gravity duals of Lifshitz-like Fixed Points_ , Phys. Rev. D78 (2008) 106005, arxiv:0808.1725.
  * (88) K. Goldstein, N. Iizuka, S. Kachru, S. Prakash, S. P. Trivedi and A. Westphal, _Holography of Dyonic Dilaton Black Branes_ , JHEP 1010, 027 (2010) arxiv:1007.2490[hep-th].
  * (89) G. Bertoldi, B. A. Burrington and A. W. Peet, _Thermal behavior of charged dilatonic black branes in AdS and UV completions of Lifshitz-like geometries_ , Phys. Rev. D 82, 106013 (2010) arxiv:1007.1464[hep-th].
  * (90) G. Bertoldi, B. A. Burrington, A. W. Peet and I. G. Zadeh, _Lifshitz-like black brane thermodynamics in higher dimensions_ , Phys. Rev. D 83, 126006 (2011) arxiv:1101.1980[hep-th].
  * (91) M. Cadoni and P. Pani, _Holography of charged dilatonic black branes at finite temperature_ , JHEP 1104, 049 (2011) arxiv:1102.3820[hep-th].
  * (92) P. Berglund, J. Bhattacharyya and D. Mattingly, _Charged Dilatonic AdS Black Branes in Arbitrary Dimensions_ , arxiv:1107.3096[hep-th].
  * (93) M. Cadoni and S. Mignemi, _Phase transition and hyperscaling violation for scalar Black Branes_ , JHEP 1206, 056 (2012) arxiv:1205.0412[hep-th].
  * (94) L. D. Landau and E. M. Lifshitz, _The Classical Theory of Fields_ , Pergamon Press, Oxford (1975).
  * (95) M. Alishahiha, E. Colgáin, and H. Yavartanoo, _Charged black branes with hyperscaling violating factor_ , JHEP 11, (2012) 137 arxiv:1209.3946.
  * (96) Aaron J. Amsel and Dan. Gorbonos, _The weak gravity conjecture and the viscosity bound with six-derivative corrections_ , JHEP 033, (2010) 1011 arxiv:1005.4718v2.
  * (97) M. van Beest, J. Calderón Infante, D. Mirfendereski, I. Valenzuela, _Lectures on the Swampland Program in String Compactifications_ , arxiv:2102.01111v1[hep-th].