# Quasinormal Modes of Near-Extremal
Electric and Magnetic Black Branes

Swapnil Nitin Shah
Electronic address<EMAIL_ADDRESS>
(Department of Physics, University of Washington)

###### Contents

  1. 1 Introduction
    1. 1.1 Quark Gluon Plasma
    2. 1.2 The AdS5/CFT4 Correspondence
    3. 1.3 Quasinormal Modes of Black Branes
  2. 2 Homogenous Isotropization in $\mathcal{N}$= 4 SYM Plasma
    1. 2.1 Gravitational Description
    2. 2.2 Numerics
    3. 2.3 Results
  3. 3 Reissner-Nordstrom Black Branes
    1. 3.1 Numerics
    2. 3.2 Results and Analysis
  4. 4 Magnetic Black Branes
    1. 4.1 Numerics
    2. 4.2 Results and Analysis
  5. 5 Conclusion

##  1 Introduction

###  1.1 Quark Gluon Plasma

Quark gluon plasma (QGP) is a state of matter in which quarks and gluons
comprising hadrons become free from their bound states at high energy
densities. Present understanding of cosmology suggests that all hadronic
matter from 10-12 to 10-6 seconds after the Big Bang existed in the form of
QGP. This state of matter was first observed in heavy ion collision
experiments conducted at Brookhaven National Laboratory (BNL) in 2005. Quantum
Chromodynamics (QCD) is the fundamental theory of strong interactions between
quarks and gluons. Asymptotic freedom in QCD states that the interaction
between quarks and gluons gets increasingly weak at extremely high energies
where perturbative treatment is applicable. This approach, however, fails in
describing dynamics of QGP where the coupling between quarks and gluons is
strong. Various attempts using non-perturbative techniques such as lattice QCD
have been made to study properties of QGP. One of the important predictions of
lattice QCD is the crossover temperature $\sim$200MeV from the confined
hadronic phase to the deconfined QGP phase as illustrated in the hypothesized
QCD phase diagram (Fig. 1)[1]. These methods, however, run into problems with
computing real time correlation functions at strong coupling and non-zero
temperature, which are important for describing non-equilibrium processes.

Figure 1: QCD phase diagram. Source: [1]

In this deconfined QGP phase, the colored partons can propagate over large
distances which accounts for non-trivial collective behavior. Consider a non-
central collision of two Lorentz contracted nuclei in the plane normal to the
collision direction (Fig. 2) [1]. The two nuclei only interact in the ’almond’
shaped region and the parts outside do not interact. If the observed hadron
jets emanating from the collision were assumed to form from individual
nucleon-nucleon collisions in the almond shaped region, the resulting hadron
jets would be uniformly distributed over the azimuthal angle in the collision
plane, independent of the shape of the interaction region. This is in conflict
with experimental data obtained at Relativistic Heavy Ion Collider (RHIC) and
Large Hadron Collider (LHC). Only strong interactions between partons in a
fluid medium with low shear viscosity can prevent the loss of asymmetry
information from the collision which has been experimentally verified. This
yields a straightforward treatment of the QGP medium near thermal equilibrium
in relativistic hydrodynamics.

Figure 2: Collision of two heavy nuclei as seen along the collision axis.
Pressure gradients along $x$-axis are shown to be greater than along $y$-axis.
Source: [1]

####  1.1.1 Relativistic Hydrodynamics

L. Landau pioneered the use of ultrarelativistic hydrodynamics in describing
multiparticle production from collisions of nuclei. The model describes QGP
medium as a perfect fluid which was further simplified by Bjorken for the case
of boost invariant flow. In Bjorken’s simplistic model, the system undergoes
one dimensional expansion along the collision axis and has boost invariance
along this axis. Bjorken’s model provides a further generalization of the
perfect fluid case by incorporating shear/bulk viscosities and thermal
conductivity for dissipative flows [2]. In the late time regime, when the
microscopic degrees of freedom have relaxed, such a hydrodynamic approximation
can be readily applied to QGP dynamics. In Bjorken’s model, the stress-energy-
momentum tensor $T^{\mu\nu}$ takes the familiar hydrodynamic form

$T^{\mu\nu}=(\varepsilon+P)u^{\mu}u^{\nu}+Pg^{\mu\nu}+\Pi^{\mu\nu}$ (1)

where $\varepsilon$ is the local energy density, $P$ is pressure and $u^{\mu}$
is the local 4-velocity. $\Pi^{\mu\nu}$ is the sum of dissipative
contributions to the energy-momentum tensor. Up to first order in gradients it
is given by

$\Pi^{\mu\nu}=-\eta\left(\Delta^{\mu\lambda}\nabla_{\lambda}u^{\nu}+\Delta^{\nu\lambda}\nabla_{\lambda}u^{\mu}-\frac{2}{3}\Delta^{\mu\nu}\nabla_{\lambda}u^{\lambda}\right)-\zeta\Delta^{\mu\nu}\nabla_{\lambda}u^{\lambda},\qquad\qquad\Delta^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$
(2)

where $\eta$ is the shear viscosity and $\zeta$ is the bulk viscosity.
$T^{\mu\nu}$ is diagonal in the local rest frame given by coordinate
transformation
$\\{t,x_{1},x_{2},x_{3}\\}\equiv\\{\tau\cosh(y),\tau\sinh(y),x2,x3\\}\rightarrow\\{\tau,y=0,x2,x3\\}$
for collision along the $x_{1}$ axis. Here $\tau$ is the proper time and $y$
is the rapidity. In this frame,

$T^{\mu\nu}=\begin{pmatrix}\varepsilon&0&0&0\\\
0&\frac{1}{\tau^{2}}\left(P-\frac{4\eta}{3\tau}-\frac{\zeta}{\tau}\right)&0&0\\\
0&0&P+\frac{2\eta}{3\tau}-\frac{\zeta}{\tau}&0\\\
0&0&0&P+\frac{2\eta}{3\tau}-\frac{\zeta}{\tau}\end{pmatrix}$ (3)

For a 4-dimensional Conformal Field Theory (CFT), energy-momentum conservation
$\nabla_{\mu}T^{\mu\nu}=0$ yields

$T^{\mu\nu}=\begin{pmatrix}\varepsilon&0&0&0\\\
0&\frac{1}{\tau^{2}}\left(-\varepsilon-\tau\frac{\partial\varepsilon}{\partial\tau}\right)&0&0\\\
0&0&\varepsilon+\frac{1}{2}\tau\frac{\partial\varepsilon}{\partial\tau}&0\\\
0&0&0&\varepsilon+\frac{1}{2}\tau\frac{\partial\varepsilon}{\partial\tau}\end{pmatrix}$
(4)

From (3) and (4), one finds the equation of state to be $\varepsilon=3P$ with
$\varepsilon\sim\tau^{-4/3}$ as $\tau\rightarrow\infty$. Although this
approach is quite successful in explaining late time QGP dynamics, it cannot
be used to study QGP formation from nuclei collisions as such a system is far
away from equilibrium. Also, the hydrodynamic equations are highly susceptible
to fluctuations of initial data making it highly non-trivial to compute
transport coefficients. A radically different yet highly successful approach
to understanding QGP dynamics derives from the AdS5/CFT4 correspondence.

###  1.2 The AdS5/CFT4 Correspondence

AdS5/CFT4 establishes a correspondence between parameters in a strongly
coupled conformal field theory on the 4-d Minkowski boundary of a 5-d AdS5
(anti-deSitter) space with those in a dual weakly interacting gravitational
theory in its 5-d bulk [3]. It has been highly successful in describing QGP
formation, its properties in thermal equilibrium as well as its transport
coefficients close to equilibrium. This approach has also been employed
recently to make predictions about far from equilibrium QGP by solving the
dual gravitational problem in the bulk. Using the duality, the formation of
QGP and thermalization can be described as being dual to black brane formation
in the AdS5 space with temperature given by the Hawking temperature. A known
example of such correspondence is the $\mathcal{N}=4$ supersymmetric Yang-
Mills theory, with matter fields in the adjoint representation of the
SU($N_{c}$) gauge group, that is dual to the type IIB superstring theory in
AdS5$\times$S5 space [3]. Before delving in the details, it is worthwhile
mentioning the limitations of this approach.

One of the hurdles in generalizing this approach, especially to non-abelian
plasma far away from equilibrium is that the dual Einstein Field Equations
(EFEs) turn out to be highly coupled and do not always yield closed form
analytical solutions [4]. In a special class of problems, symmetries of the
system are exploited to yield insight into the hydrodynamic behavior of QGP
close to equilibrium. These are based on evaluating the gravity dual to
Bjorken’s boost invariant flow in relativistic hydrodynamics described in the
previous section.

A more fundamental problem with this approach is that AdS5/CFT4 correspondence
is exact only between a supersymmetric conformal field theory, i.e.
$\mathcal{N}=4$ Supersymmetric Yang Mills theory on the AdS5 boundary and
classical Supergravity in the bulk for group rank $N_{c}\to\infty$ and t’Hooft
coupling $\lambda\to\infty$. QCD is neither super-symmetric nor conformal in
general, i.e., its coupling constant changes with the energy scale in
consideration. Also, quarks in QCD transform under the fundamental
representation of color gauge group SU(3) whereas charges in $\mathcal{N}=4$
SYM transform under the adjoint representation of the group SU($N_{c}$).
However, lattice gauge theory calculations suggest that in the deconfined QGP
regime, QCD is quasi conformal from the standpoint of thermodynamics. On the
other hand, it has been shown how supersymmetry in $\mathcal{N}=4$ SYM is
absent for collective excitations at finite non-zero temperature using Keldysh
diagram techniques [1, 4]. Owing to these observations and experimental data
from heavy ion collisions, $\mathcal{N}=4$ SYM serves as a good model for
describing QGP dynamics despite the significant differences between the two
theories.

####  1.2.1 AdS5/CFT4 Dictionary

The metric of AdS5 space is a solution of vacuum Einstein equations (5) in 5
spacetime dimensions with a constant negative curvature and negative
cosmological constant [3]

$R_{mn}-\frac{1}{2}g_{mn}R-6g_{mn}=0$ (5)

where $R_{mn}$ is the 5-d Ricci tensor and $g_{mn}$ the metric
($m,n=t,x,y,z,r$). Its line element has the form:

$ds^{2}=\frac{r^{2}}{L^{2}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{L^{2}}{r^{2}}dr^{2}$
(6)

where $r$ is the radial coordinate orthogonal to the 4-d spacetime directions,
$L$ is the curvature radius of AdS5 and $\eta_{\mu\nu}$ is the 4-d Minkowski
metric ($\mu,\nu=t,x,y,z$). In the low energy limit, AdS5/CFT4 correspondence
relates $\mathcal{N}=4$ SYM theory, with gauge group SU($N_{c}$), and Type IIB
(closed) string theory in AdS5$\times$S5. The parameters characterizing the
gauge theory are rank of the gauge group $N_{c}$ and the t’Hooft coupling
$\lambda=g^{2}_{YM}N_{c}$. On the other side of the duality, parameters
characterizing Type IIB string theory are the string coupling $g_{s}$ and the
string length $l_{s}$. AdS5/CFT4 relates these parameters as under

$g_{s}\sim\frac{g^{2}_{YM}}{4\pi}\qquad\qquad\frac{L}{l_{s}}\sim\lambda^{1/4}\qquad\qquad\frac{G_{10}}{L^{8}}\sim\frac{\pi^{4}}{2N^{2}_{c}}$
(7)

where $G_{10}$ is the 10-d gravitational constant and $g_{YM}$ is the Yang-
Mills coupling in $\mathcal{N}=4$ SYM.

In the limit of $\lambda\rightarrow\infty$, $N_{c}\rightarrow\infty$, the
boundary gauge theory is strongly coupled while closed strings in dual Type
IIB theory are weakly coupled (from (7)). In this regime, Type IIB string
theory simplifies to Type IIB supergravity which on compactification over
$S_{5}$ reproduces classical Einstein field equations coupled to matter
fields. In absence of matter fields, the solution to compactified supergravity
equations is simply the AdS5 metric (6) as expected in classical general
relativity.

A $p$-form bulk field $\Phi(r,x)$ dual to an operator $O(x)$ of the boundary
gauge theory deforms the gauge theory action [4] as

$S\rightarrow
S+\int{d^{4}x\,\phi(x)O(x)},\qquad\qquad\phi(x)=\lim_{r\to\infty}r^{4-p-\Delta}\,\Phi(r,x)$
(8)

where $\phi(x)$ is like a source field in the gauge theory Lagrangian.
$\Delta$ is the conformal dimension of its dual gauge theory operator $O(x)$
given by the larger root of

$m^{2}L^{2}=(\Delta-p)(\Delta+p-4)$ (9)

where $m$ is the mass of the gauge theory field. Defining coordinate
$z=\frac{L^{2}}{r}$, the near boundary expansion of the source field
$\Phi(z,x)$ is

$\Phi(z,x)\equiv\Phi(z,x)_{\mu_{0}\dots\mu_{p}}\approx\alpha(x)_{\mu_{0}\dots\mu_{p}}\,z^{4-p-\Delta}+\beta(x)_{\mu_{0}\dots\mu_{p}}\,z^{\Delta-p}$
(10)

Based on the duality prescription, it can be shown that the expectation value
of operator $O(x)$ in the presence of source field $\phi(x)=\alpha(x)$ is
given by (indices not shown for brevity)

$\langle O(x)\rangle_{\phi}=\lim_{z\to 0}z^{4-p-\Delta}\frac{\delta
S^{\mathrm{(ren)}}[\Phi_{c}]}{\delta\Phi_{c}(z,x)}=2v\beta(x),\qquad\qquad
v=\sqrt{(2-p)^{2}+m^{2}L^{2}}$ (11)

where $S^{\mathrm{(ren)}}$ is the renormalized bulk action and $\Phi_{c}(z,x)$
is the solution of the classical equations of motion $\delta S[\Phi_{c}]=0$.
To illustrate, consider a small perturbation $h_{\mu\nu}$ of the Minkowski
boundary metric $\eta_{\mu\nu}$. Using (11), one can compute the boundary
stress-energy tensor

$\langle T_{\mu\nu}(x)\rangle_{h}=\lim_{z\to 0}\frac{1}{4!}\cdot\frac{1}{4\pi
G_{5}}\frac{\partial^{4}g_{\mu\nu}(z,x)}{\partial z^{4}}=\lim_{z\to
0}\frac{1}{4!}\cdot\frac{N^{2}_{c}}{2\pi^{2}L^{3}}\frac{\partial^{4}g_{\mu\nu}(z,x)}{\partial
z^{4}},\qquad\qquad g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ (12)

####  1.2.2 Equilibrium QGP in AdS5/CFT4

The geometry dual to $\mathcal{N}=4$ SYM plasma at finite non-zero temperature
is the AdS5-Schwarschild black brane [4] with the following metric (line
element)

$z^{2}\,ds^{2}=-\left(1-\frac{z^{4}}{z^{4}_{0}}\right)dt^{2}+d\mathrm{\bf{x}}^{2}+\frac{dz^{2}}{\left(1-\frac{z^{4}}{z^{4}_{0}}\right)}$
(13)

The corresponding Hawking temperature is $T_{H}=\frac{1}{\pi z_{0}}$. With a
coordinate transformation
$z\to\tilde{z}\left(1+\tilde{z}^{4}/\tilde{z}^{4}_{0}\right)^{-1/2}$, one can
write (13) in the Fefferman-Graham form as

$\tilde{z}^{2}\,ds^{2}=-\frac{\left(1-\frac{\tilde{z}^{4}}{\tilde{z}^{4}_{0}}\right)^{2}}{\left(1+\frac{\tilde{z}^{4}}{\tilde{z}^{4}_{0}}\right)}dt^{2}+\left(1+\frac{\tilde{z}^{4}}{\tilde{z}^{4}_{0}}\right)d\mathrm{\bf{x}}^{2}+d\tilde{z}^{2}$
(14)

From (12) and (14), the stress-energy-momentum tensor for QGP in equilibrium
corresponds to that of a perfect fluid in hydrodynamics ($\varepsilon=3P$)

$T_{\mu\nu}=\frac{N^{2}_{c}}{2\pi^{2}}\mathrm{\bf{diag}}\left\\{\frac{3}{\tilde{z}^{4}_{0}},\frac{1}{\tilde{z}^{4}_{0}},\frac{1}{\tilde{z}^{4}_{0}},\frac{1}{\tilde{z}^{4}_{0}}\right\\}$
(15)

###  1.3 Quasinormal Modes of Black Branes

Quasinormal modes (QNMs) are analogues of normal modes (characteristic
oscillations) for perturbed physical systems in the presence of dissipation
[5]. More formally, they are the eigenstates of linearized equations of motion
for dissipative dynamical systems under perturbation. Gravitational
backgrounds such as black holes/branes are intrinsically dissipative due to
the presence of an event horizon. Such a system is not time-symmetric and the
associated boundary value problem is not Hermitian. In general, their QNMs
have complex frequencies with the imaginary part being associated with the
decay timescale of the perturbation.

In the context of gauge-gravity duality, the quasinormal spectra of dual
gravitational backgrounds provide the locations of poles of the retarded
correlators in the boundary gauge theory [3, 5]. They provide important
information about the gauge theory’s quasiparticle spectra and transport
coefficients. QNMs are therefore a powerful tool in studying the near-
equilibrium behavior of strongly coupled non-abelian plasmas with a dual
gravitational description.

In general relativity, linearized Einstein field equations for perturbed
gravitational backgrounds naturally give rise to QNMs when transformed to the
Fourier space. The perturbations obey linear second-order partial differential
equations (PDEs) which may be reduced to linear ordinary differential
equations (ODEs) by exploiting symmetries of the gravitational background
under consideration. Under appropriate boundary conditions, eigenmodes of the
system of ODEs can be computed which are QNMs of the gravitational background
[5].

####  1.3.1 Computation of QNMs

Asymptotically AdS5 black brane metrics with SO(3) rotational, translational
and boost symmetry have a line element of the form [5, 6]

$ds^{2}_{5}=\frac{r^{2}}{L^{2}}\left(-f(r)dt^{2}+d\textbf{x}^{2}\right)+\frac{L^{2}}{r^{2}}f(r)^{-1}dr^{2}$
(16)

where the metric function $f(r)$ encodes the bulk geometry and horizon
information. For the AdS5-Schwarschild black brane
$f(r)=\left(1-r^{4}_{0}/r^{4}\right)$, where $r_{0}$ is the horizon location
($f(r_{0})=0$). The metric (16) becomes singular at the horizon owing to the
above choice of coordinates [5]. One can circumvent this issue by defining the
tortoise coordinate $r^{*}$ related to the radial coordinate $r$ as
$\frac{dr^{*}}{dr}=\frac{L^{2}}{r^{2}f(r)}$.

The general prescription to compute the QNMs of (16) is as follows [5, 7]

  * •

Consider a plane wave perturbation of the metric $h_{mn}(r)\sim
e^{-i\omega(t+r^{*})+i\vec{k}\cdot\vec{x}}$. Using the tortoise coordinate
$r^{*}$ ensures only infalling modes are $\mathcal{C}^{\infty}$-smooth at the
horizon ($r\to r_{0}$)

  * •

For an SO(3) symmetric metric, it is convenient to set $\vec{x}=\vec{z}$

  * •

At the AdS5 boundary ($r\to\infty$), gauge-gravity duality dictates that the
perturbation vanish. Thus $\lim_{r\to\infty}h_{mn}(r)\sim
r^{-\gamma}\,\,(\gamma>0)$

  * •

Substitute the perturbed metric ansatz ($g_{mn}\to g_{mn}+h_{mn}$) in Einstein
field equations and linearize them w.r.t $h_{mn}(r)$

  * •

Solve the resultant eigenvalue equations for $h_{mn}(r)$ after fixing the
diffeomorphism gauge and momentum $k$. The eigenvalues obtained are the
complex QNM frequencies.

The metric perturbations are grouped into three sectors depending on the
symmetry channels they transform under, viz. the scalar, vector and tensor
perturbations [5]. Each sector is governed by an independent closed group of
differential equations. An easy choice for fixing the diffeomorphism gauge is
setting $h_{rm}=0,\,m=r,t,x,y,z$ [7]. For non-vanishing momentum, the scalar
sector comprises of perturbations that transform as a scalar under SO(2)
(rotations in the $x\leftrightarrow y$ plane) -
$h_{tt},\,h_{xx}+h_{yy},\,h_{tz},\,h_{zz}$. Similarly, the vectorial sector
comprises of those that transform as a vector under SO(2) -
$h_{tx},\,h_{ty},\,h_{zx},\,h_{zy}$ and tensorial sector comprises of those
that transform as a rank-2 tensor under SO(2) - $h_{xx}-h_{yy},\,h_{xy}$.

####  1.3.2 QNMs of AdS5-Schwarschild Black Branes

In this section, we will briefly illustrate the process of computing QNMs for
perturbations belonging to the tensorial sector using the example of
AdS5-Schwarschild black branes. A similar process can be followed to compute
the vectorial and scalar sector QNMs. As discussed in the previous section,
computation of QNMs can be most easily done by working in the infalling null
coordinates $(v=t+r^{*},x,y,z,r)$, also known as the generalized Eddington-
Finkelstein coordinates. Their advantage over Fefferman-Graham coordinates is
that the QNM boundary conditions only require solutions to be regular at the
horizon and vanish at the AdS5 boundary [5]. In these coordinates, the
AdS5-Schwarschild black brane metric takes the form

$ds^{2}_{5}=\frac{r^{2}}{L^{2}}\left(-f(r)dv^{2}+d\textbf{x}^{2}\right)+2dvdr$
(17)

Without loss of generality, one can set the horizon $r_{0}$ as the AdS5
curvature radius $L$. On adding the tensorial metric perturbation
$h_{xy}\equiv\frac{r^{2}}{L^{2}}g(r)e^{-i\omega v+ikz}$ to (17), and
linearizing the corresponding (vacuum) Einstein field equations (5), we get a
QNM second order ordinary differential equation for $g(r)$

$L^{2}r(L^{2}k^{2}+3ir\omega)g(r)+(L^{4}-5r^{4}+2iL^{2}r^{3}\omega)g^{\prime}(r)+r(L^{4}-r^{4})g^{\prime\prime}(r)=0$
(18)

For every choice of momentum $k$, solutions to (18) only exist for a discrete
set of values of $\omega$. It is therefore an eigenvalue equation where the
solutions are the eigenvectors and corresponding $\omega$’s are the
eigenvalues. Solving (18) using pseudospectral methods will be covered in the
next few sections. For the case of vanishing momentum $k=0$ and $L=1$, the
lowest few QNM frequencies ($\omega$) for the AdS5-Schwarschild black brane
(17) are tabulated in Table. 1. All QNM frequencies have a negative imaginary
part which indicates oscillations damped in time.

n | Re($\omega_{n}$) | Im($\omega_{n}$)
---|---|---
1 | $\pm 3.119452$ | $-2.746676$
2 | $\pm 5.169521$ | $-4.763570$
3 | $\pm 7.187931$ | $-6.769565$
4 | $\pm 9.197199$ | $-8.772481$
5 | $\pm 11.202676$ | $-10.774162$
6 | $\pm 13.206247$ | $-12.775239$
Table 1: Lowest QNM frequencies (tensorial sector) for AdS5-Schwarschild black
brane ($k=0$)

##  2 Homogenous Isotropization in $\mathcal{N}$= 4 SYM Plasma

It is computationally challenging to study real time dynamics of strongly
coupled QGP using lattice methods. In order to study QGP formation from heavy
ion collisions and its dynamics far from equilibrium, one can study the
behavior of $\mathcal{N}=4$ SYM plasma as a model of QGP for aforementioned
reasons. A high degree of momentum anisotropy would seem to exist in partons
formed out of such collisions, which would subsequently form the QGP state
that behaves like a relativistic fluid. When this QGP cools below the
crossover temperature, hadrons form that fly outwards toward the detectors.
Although the near-equilibrium behavior of QGP is explained well by
hydrodynamics, the same cannot be said for the rapid isotropization and
mechanisms underlying QGP formation far from equilibrium. This process can be
studied for the $\mathcal{N}=4$ SYM plasma far from equilibrium using its dual
gravitational description for a given set of initial/boundary conditions. The
next few sections provide an example application of pseudospectral methods for
solving dual Einstein field equations for the case of homogenous
isotropization of far from equilibrium QGP. We use the setup and methods
described in article [8]

###  2.1 Gravitational Description

The most general asymptotically AdS5 metric with translational and boost
invariance (in Eddington-Finkelstein coordinates) has line element [8]

$ds^{2}_{5}=g_{ij}(r,v)dx^{i}dx^{j}+2dv(dr-A(r,v)dv)$ (19)

where $i,j=x,y,z$. For a fluid expansion restricted to the $z$ direction, one
can set $g_{ij}\equiv\\{\Sigma^{2}e^{B},\Sigma^{2}e^{B},\Sigma^{2}e^{-2B}\\}$.
The corresponding Einstein field equations (5) are:

$\displaystyle\Sigma^{\prime\prime}+\frac{1}{2}\left(B^{\prime}\right)^{2}\Sigma=0$
(20) $\displaystyle
A^{\prime\prime}=6\left(\frac{\Sigma^{\prime}}{\Sigma^{2}}\right)d_{+}\Sigma-\frac{3}{2}B^{\prime}d_{+}B-2$
(21) $\displaystyle
d_{+}\Sigma^{\prime}+2\left(\frac{\Sigma^{\prime}}{\Sigma}\right)d_{+}\Sigma=2\Sigma$
(22)
$\displaystyle\left(d_{+}B\right)^{\prime}+\frac{3}{2}\left(\frac{\Sigma^{\prime}}{\Sigma}\right)d_{+}B=-\frac{3}{2}B^{\prime}\frac{\left(d_{+}\Sigma\right)}{\Sigma}$
(23)

where ${}^{\prime}\equiv\partial_{r}$ and
$d_{+}\equiv\partial_{v}+A(r_{*},v_{*})\partial_{r}$ is the directional
derivative along the outgoing null geodesic passing through $(r_{*},v_{*})$.
Before solving the system of equations ($\ref{eq:Eq20}\to\ref{eq:Eq23}$), one
needs to fix the residual gauge freedom. One computationally convenient way of
doing this, for reasons mentioned in the next section, is demanding that the
location of apparent horizon (outermost trapped null surface) occurs at a
fixed value of radial coordinate $r=r_{h}$ and that it does not change with
time [8]. This requirement leads to two additional constraints for metric
functions

$\displaystyle d_{+}\Sigma\big{|}_{r_{h}}=0$ (24) $\displaystyle
A(r_{h},v)=-\frac{1}{4}\left(d_{+}B\right)^{2}\big{|}_{r_{h}}$ (25)

The metric form (19) is manifestly invariant under a radial shift $r\to
r+\lambda(v)$. In effect, (24) fixes the initial value of radial shift
$\lambda(v_{0})$ at time $v_{0}$ and (25) determines $\partial_{v}\lambda$ to
keep the horizon stationary. The asymptotic near boundary expansions
($r\to\infty$) for the metric functions are found to be

$\displaystyle\Sigma\sim r+\lambda(v)+O(r^{-7})$ (26) $\displaystyle
A\sim\frac{1}{2}\left(r+\lambda(v)\right)^{2}-\partial_{v}\lambda(v)+a^{(4)}(v)r^{-2}+O(r^{-3})$
(27) $\displaystyle
d_{+}\Sigma\sim\frac{1}{2}\left(r+\lambda(v)\right)^{2}+a^{(4)}(v)r^{-2}+O(r^{-3})$
(28) $\displaystyle B\sim b^{(4)}(v)r^{-4}+O(r^{-5})$ (29) $\displaystyle
d_{+}B\sim-2b^{(4)}(v)r^{-3}+O(r^{-4})$ (30)

where $a^{(4)}(v)$ and $b^{(4)}(v)$ are related to the boundary stress-energy
tensor based on the gauge-gravity duality as (parantheses omitted for brevity)

$T^{vv}=-\frac{3N_{c}^{2}}{4\pi^{2}}a^{(4)},\qquad\qquad
T^{xx}=T^{yy}=\frac{N_{c}^{2}}{2\pi^{2}}\left(b^{(4)}-\frac{1}{2}a^{(4)}\right),\qquad\qquad
T^{zz}=\frac{N_{c}^{2}}{2\pi^{2}}\left(-2b^{(4)}-\frac{1}{2}a^{(4)}\right)$
(31)

It can be seen that (26), (27) and (28) diverge as $r\to\infty$. It is
computationally convenient to use the inverse radial coordinate $u=1/r$ and
define subtracted and rescaled functions (i.e. functions with leading order
$r$ divergences removed and subsequently rescaled) such that these quantities
approach a constant or vanish linearly as $u\to 0$ [8]. Following is an
example of such redefinition from article [8]

$\sigma=\Sigma-1/u,\qquad\qquad\dot{\sigma}=u^{-1}\left(d_{+}\Sigma-\frac{1}{2}\Sigma^{2}\right),\qquad\qquad
a=A-\frac{1}{2}\Sigma^{2}$ (32) $b=u^{-3}B,\qquad\qquad\dot{b}=u^{-3}d_{+}B$
(33)

where resulting $u\to 0$ boundary conditions to be imposed for these redefined
fields are

$\sigma\to\lambda,\qquad\dot{\sigma}\to ua^{(4)},\qquad b\to
0,\qquad\dot{b}\to-2b^{(4)}$ (34)

###  2.2 Numerics

The system of linear ordinary differential equations (ODEs)
($\ref{eq:Eq20}\to\ref{eq:Eq23}$) has a very convenient nested structure.
Given the anisotropy function $B(v_{0},r)$, boundary energy density
$T^{vv}(v_{0})$ and radial shift $\lambda(v_{0})$ at some initial time
$v_{0}$, one can solve the equations one after the other in order
$(\ref{eq:Eq20})\to(\ref{eq:Eq22})\to(\ref{eq:Eq23})\to(\ref{eq:Eq21})$ such
that constraints (24) and (25) are satisfied. Using the definition of
$d_{+}B$, the function $B$ can be evolved in time and the aforementioned
process is repeated to get the time evolution of the other metric functions.
As discussed in the previous section, we choose to work with the redefined
fields ($\ref{eq:Eq26}\to\ref{eq:Eq30}$) for computational reasons, with
boundary conditions (34).

The initial value of radial shift $\lambda(v_{0})$ is not known apriori and
must be computed using a root finding technique such as Newton-Raphson method
to solve constraint (24) in conjunction with solving (20) and (21). The next
section describes how pseudospectral methods (esp. those employing Chebyshev
basis functions) can be used to solve linear boundary value problems (BVPs).

####  2.2.1 Pseudospectral ODE Solvers

Traditionally a system of ODEs is solved numerically using some variant of
Finite Element Method (FEM). It involves discretization of the domain space
($r$ in this case) into a grid. The functions are represented as piecewise
polynomial basis functions defined on the grid. This turns the BVPs into a
system of linear equations, one per each domain element/piece per function.
The derivatives of functions are approximated by the local derivatives of the
basis functions. These methods work quite well when there are no singular
points in the computational domain (including the domain boundary).

For the system described in the previous section, we can see that $u\to 0$ and
$u\to 1$ are singular points which are not well tolerated by FEM. One can
circumvent this issue by using spectral methods for solving ODEs as one can
directly apply spectral methods to BVPs with regular singular points as long
as the solution of interest is regular at the singular point [9, 8]. Also,
they show superior convergence as compared to FEM and their accuracy improves
exponentially with the number of basis functions. Unlike FEM, spectral methods
use a truncated linear combination of long range basis functions which are
defined over the entire computational domain of interest. As with FEM,
spectral methods also transform a BVP into a linear algebraic problem. For
periodic functions defined on an interval, the basis of complex exponentials
(Fourier basis) is a natural choice and the expansion is the truncated Fourier
series [8]. For aperiodic functions, the convenient basis is the Chebyshev
polynomials $T_{n}(z)\equiv\cos(n\cos^{-1}(z))$. For a computational domain
$0<u<1$, the (truncated) expansion of a function $f(u)$ is given by

$f(u)\approx\sum^{N}_{n=0}\alpha_{n}T_{n}(2u-1)$ (35)

Solving a BVP for $f(u)$ reduces to solving for the coefficients $\alpha_{n}$
after substituting (35) into the BVP. In pseudospectral or collocation
approaches, one makes such a substitution and demands that the residual of the
BVP vanishes exactly at a predetermined special set of points called the
collocation grid points [9, 8]. The number of such points equals the expansion
order $N$. In this approach, the knowledge of coefficients $\alpha_{n}$ is
equivalent to the value of function $f(u)$ at collocation grid points points
$u_{n}$. For the Chebyshev basis, appropriate collocation points in the domain
interval $[0,1]$ are

$u_{n}=\frac{1}{2}\left(1-\cos{\frac{n\pi}{N}}\right)$ (36)

####  2.2.2 Time Evolution

For each timestep, once the system of (radial) ODEs is solved using a
pseudospectral solver, the anisotropy function $B$ can be evolved in time by
solving the Initial Value Problem (IVP) $\partial_{v}B=d_{+}B-AB^{\prime}$
[8]. This can be solved using an explicit fourth order Runge-Kutta (RK4)
method which has a convergence error of $O(\Delta v)^{5}$ in the timestep
$\Delta v$. In RK4, for an IVP of the form

$\frac{\partial f}{\partial v}=g(v,f),\qquad\qquad f(v_{0})=f_{0}$ (37)

and a chosen timestep size of $\Delta v\equiv h>0$, the function $f(v+h)$ can
be approximated by

$f(v+h)\approx f(v)+\frac{h}{6}\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right)$ (38)

where,

$\displaystyle k_{1}=g(v,f)$ (39) $\displaystyle
k_{2}=g\left(v+\frac{h}{2},f+h\frac{k_{1}}{2}\right)$ (40) $\displaystyle
k_{3}=g\left(v+\frac{h}{2},f+h\frac{k_{2}}{2}\right)$ (41) $\displaystyle
k_{4}=g(v+h,f+hk_{3})$ (42)

###  2.3 Results

The following was chosen as the initial data and parameters for the example
simulation

$b(v_{0},u)=5ue^{-25\left(u-\frac{1}{4}\right)^{2}},\qquad\qquad
a^{(4)}=-\frac{1}{2},\qquad\qquad r_{h}=1,\qquad\qquad N=51,\qquad\qquad
h=10^{-2}$

Plot (Fig. 3) shows the time evolution of the boundary pressure anisotropy
$\delta P\equiv T_{zz}-\frac{1}{2}\left(T_{xx}+T_{yy}\right)$ in units of
$\frac{N^{2}_{c}}{2\pi^{2}}$ for $v_{0}\leq v<v_{0}+4$. It’s value can be
computed from (31) to be $\delta P=-3b^{(4)}\frac{N^{2}_{c}}{2\pi^{2}}$. As
seen from the plot, the pressure anisotropy rapidly decays with exponentially
damped oscillations till equilibrium is reached, i.e. till
$T_{xx}=T_{yy}=T_{zz}=\frac{N^{2}_{c}}{8\pi^{2}}$ where the plasma is
homogenous and isotropic. Plot (Fig. 4) shows the computed total anisotropy
function in the AdS5 bulk - $B(u,v)$ for $0\leq u\leq 1$ and $v_{0}\leq
v<v_{0}+4$. It also decays exponentially with time for all $u$ till the
geometry becomes that of an AdS5-Schwarschild black brane with horizon fixed
at $r_{h}=1$ and Hawking temperature $T_{H}=\frac{1}{\pi}$.

Figure 3: Normalized pressure anisotropy
$\frac{2\pi^{2}}{N^{2}_{c}}\cdot\delta P$ shown as a function of (infalling-
coordinate) time $v$. Figure 4: Time evolution of the anisotropy function
$B(u,v)$ in units of $u^{3}$.

Having seen an illustrative example of how to use pseudospectral methods for
solving the Einstein field equations, the next two sections describe how to
use this approach for computing the QNMs of a) Reisnner-Nordstrom
(electrically charged) and b) Magnetically charged AdS5 black branes.

##  3 Reissner-Nordstrom Black Branes

In this section, we consider the gravitational dual of an electrically charged
$\mathcal{N}=4$ SYM plasma with uniform charge density $C$ [7, 6]. The dual
theory has a corresponding non-zero gauge field
$A_{m}\equiv\\{A_{t},0,0,0\\}$, which from (10) is

$A_{t}=\mu-\frac{C}{r^{2}}$ (43)

in units $\\{c,G_{5},k_{e},L=1\\}$ ($k_{e}$ is the Cuolomb constant). The
corresponding (bulk) electromagnetic stress-energy tensor is

$\displaystyle{T^{m}}_{n}=\frac{1}{4\pi}\left(F^{m\alpha}F_{n\alpha}-\frac{1}{4}{\delta^{m}}_{n}F_{\alpha\beta}F^{\alpha\beta}\right)$
$\displaystyle{T^{v}}_{v}={T^{r}}_{r}=-\frac{1}{4\pi}\frac{2C^{2}}{r^{6}},\qquad\qquad{T^{i}}_{j}=\delta_{i,j}\,\frac{1}{4\pi}\frac{2C^{2}}{r^{6}}\qquad(i,j=x,y,z)$
(44)

The Einstein-Maxwell field equations are

$R_{mn}-\frac{1}{2}g_{mn}R-6g_{mn}=8\pi T_{mn}$ (45)

which yield the following asymptotically-AdS5 Reisnner-Nordstrom black brane
metric (in Eddington-Finkelstein coordinates)

$ds^{2}_{5}=r^{2}\left(-f(r)dv^{2}+d\mathrm{\bf{x}}^{2}\right)+2dvdr$ (46)

where the metric function $f(r)$ encodes the bulk geometry and horizon
information (47). Two radially separated horizons exist with locations given
by the positive roots of $f(r)$. The larger root is the radial location of
event horizon and the smaller root is that of an inner Cauchy horizon. The
black brane has an extremal value of charge density
$\rho_{\mathrm{ex}}=\sqrt{2}\left(\frac{m}{3}\right)^{\frac{3}{4}}$ where the
two roots coincide and the Hawking temperature vanishes $T_{H}=0$ (from (48))

$f(r)=1-\frac{m}{r^{4}}+\frac{\rho^{2}}{r^{6}},\qquad\qquad\rho=\frac{2}{\sqrt{3}}C$
(47)

By requiring regularity of the Euclidean manifold at the event horizon
($r=r_{+}$), the Hawking temperature $T_{H}$ and chemical potential $\mu$ can
be computed as [3]

$T=r^{2}_{+}\frac{\left|f^{\prime}(r_{+})\right|}{4\pi},\qquad\qquad\mu=\frac{C}{r^{2}_{+}}$
(48)

###  3.1 Numerics

The horizon location $r_{+}$ demarcates a boundary for the computational
domain as the metric (46) becomes singular at that point [6]. In order to use
the pseudospectral method for computing the quasinormal mode frequencies for
this metric, it is convenient to transform the radial coordinate as
$r\to\frac{r_{+}}{u}$. This brings the computational domain from
$r\in(r_{+},\infty)$ to $u\in(0,1)$ which is the fundamental domain for
Chebyshev expansions of the metric functions. The metric itself transforms as

$ds^{2}_{5}=\frac{r^{2}_{+}}{u^{2}}\left(-f(u)dv^{2}+d\mathrm{\bf{x}}^{2}\right)-2\frac{r_{+}}{u^{2}}dvdu$
(49)

with the temperature

$T_{H}=r_{+}\frac{\left|f^{\prime}(u=1)\right|}{4\pi}=r_{+}\frac{\left(2-\rho^{2}r^{-6}_{+}\right)}{2\pi}$
(50)

####  3.1.1 QNMs of Reisnner-Nordstrom Black Branes

Here we focus exclusively on gravitational fluctuations for vanishing spatial
momenta ($k=0$) as they are radially infalling and correspond to pressure
anisotropy in the boundary stress-energy tensor. For this case, only the
rank-2 tensor metric perturbations have non-trivial QNMs, while scalar and
vector perturbations only yield trivial solutions completely determined by
boundary data [7]. With $k=0$ and fixing the diffeomorphism gauge $h_{rm}=0$,
independent tensor fluctuations under the full SO(3) rotation group are -
$h_{xy},\,h_{yz},\,h_{zx},\,h_{xx}-h_{yy},\,h_{yy}-h_{zz}$. All of them
satisfy the same equation of motion and are decoupled from the rest.

Consider a plane wave ansatz for the shear perturbation $h_{xy}\equiv
e^{-i\omega v}g(u)u^{2}$. Using the perturbed metric $g_{mn}=\eta_{mn}+h_{mn}$
in Einstein-Maxwell equations (45), we get the equation of motion

$g^{\prime\prime}(u)+g^{\prime}(u)\frac{\left(uf^{\prime}(u)+5f(u)+2i\tilde{\omega}u\right)}{uf(u)}+g(u)\frac{\left(4f^{\prime}(u)+5i\tilde{\omega}\right)}{uf(u)}=0$
(51)

where $\tilde{\omega}\equiv\omega/r_{+}$ is the rescaled angular frequency,
which removes explicit dependence of (51) on the horizon location $r_{+}$. As
can be readily checked (51) takes the form of an eigenvalue equation

$\Phi\,g(u)=i\tilde{\omega}\,g(u),\qquad\qquad\Phi\equiv\Phi(\partial_{u},f(u),u)$
(52)

One can now use the psuedospectral approach outlined in a previous section and
approximate $g(u)$ by a truncated expansion in the Chebyshev polynomial basis
over $u$ at the collocation grid points (35). This transforms (52) into a
linear algebraic eigenvalue problem with resulting eigenvalues being the QNM
frequencies.

###  3.2 Results and Analysis

We found a grid size of 50 grid points adequate to get stable QNM frequencies
with rapid convergence using pseudospectral approach. Table. 2 lists the
lowest four QNM frequencies ($\tilde{\omega}$), in order of magnitude of their
imaginary part, for charge density varying from 0 to 0.9$\rho_{ex}$.

$\rho/\rho_{\mathrm{ex}}$ | n=1 | n=2 | n=3 | n=4
---|---|---|---|---
0.0 | $\pm 3.119452-2.746676i$ | $\pm 5.169521-4.763570i$ | $\pm 7.187931-6.769565i$ | $\pm 9.179199-8.772481i$
0.1 | $\pm 3.116518-2.752218i$ | $\pm 5.163269-4.774448i$ | $\pm 7.178353-6.786078i$ | $\pm 9.184267-8.794832i$
0.2 | $\pm 3.107594-2.769530i$ | $\pm 5.144302-4.808750i$ | $\pm 7.149378-6.838513i$ | $\pm 9.145265-8.866208i$
0.3 | $\pm 3.092364-2.800902i$ | $\pm 5.112266-4.872204i$ | $\pm 7.101081-6.937009i$ | $\pm 9.081220-9.002002i$
0.4 | $\pm 3.070567-2.851114i$ | $\pm 5.068295-4.977288i$ | $\pm 7.038750-7.104216i$ | $\pm 9.005274-9.236813i$
0.5 | $\pm 3.043124-2.929579i$ | $\pm 5.022870-5.148924i$ | $\pm 6.997738-7.380419i$ | $\pm 8.993865-9.613485i$
0.6 | $\pm 3.017494-3.054959i$ | $\pm 5.029894-5.417824i$ | $\pm 7.089142-7.741854i$ | $\pm 9.145420-10.01256i$
0.7 | $\pm 3.033036-3.254092i$ | $\pm 5.170330-5.703165i$ | $\pm 7.255584-8.055419i$ | $\pm 9.307353-10.40596i$
0.8 | $\pm 3.164806-3.470840i$ | $\pm 5.326875-5.968086i$ | $\pm 7.438105-8.452644i$ | $\pm 9.540560-10.92871i$
0.9 | $\pm 3.322345-3.683569i$ | $\pm 5.557329-6.366283i$ | $\pm 7.757594-9.017011i$ | $\pm 9.946420-11.65867i$
Table 2: Lowest four QNM frequencies (tensorial sector) for $k=0$ and $\rho$
varying from 0 to 0.9$\rho_{\mathrm{ex}}$

From Table. 2, it can be seen that up until $\rho\sim 0.5\rho_{\mathrm{ex}}$,
the real part of each QNM mode’s frequency decreases with increasing $\rho$
and starts increasing somewhere between $\rho$=0.5$\rho_{\mathrm{ex}}$ to
0.6$\rho_{\mathrm{ex}}$. The imaginary parts (damping coefficients), however,
increase monotonically with $\rho$ for each QNM mode’s frequency. Also, the
computation is numerically stable up until $\rho=\rho_{\mathrm{ex}}$.

####  3.2.1 QNMs for Near Extremal Reisnner-Nordstrom Black Branes

Plot (Fig. 5) shows behavior of imaginary part of lowest QNM frequency as the
extremal value of charge density $\rho_{\mathrm{ex}}$ is approached
$\left(\rho/\rho_{\mathrm{ex}}=0.9\right.$ to $\left.(1-10^{-11})\right)$. It
is a log-log plot with
$\log(\left|\mathrm{Im}[\tilde{\omega}-\tilde{\omega}_{\mathrm{ex}}]\right|)$
on the y-axis and $\log(\pi T_{H}/\rho^{1/3})$ on the x-axis represented by
the blue dots. The red line shows a linear fit $y=x+c_{I}$. Similar result is
obtained for the real part as shown in plot (Fig. 6), with the linear fit
$y=x+c_{R}$ ($c_{I}$ and $c_{R}$ are constants) and this pattern is observed
for all modes.

Figure 5: Plot of
$\log(\left|\mathrm{Im}[\tilde{\omega}-\tilde{\omega}_{\mathrm{ex}}]\right|)$
vs. $\log(\pi T_{H}/\rho^{1/3})$ for $\rho/\rho_{\mathrm{ex}}=0.9$ to
$(1-10^{-11})$. The red line shows a linear fit $y=x+c_{I}$ Figure 6: Plot of
$\log(\left|\mathrm{Re}[\tilde{\omega}-\tilde{\omega}_{\mathrm{ex}}]\right|)$
vs. $\log(\pi T_{H}/\rho^{1/3})$ for $\rho/\rho_{\mathrm{ex}}=0.9$ to
$(1-10^{-11})$. The red line shows a linear fit $y=x+c_{R}$

We see that close to extremality, there is a linear relation between
$\tilde{\omega}^{(n)}$ and $\pi T_{H}/\rho^{1/3}$ ($n$ is the mode #,
$\alpha_{n}$ are mode specific constants). From (53), decay times for
tensorial perturbations of near-extremal Reisnner-Nordstrom black branes are
proportional only to their Hawking temperature (in units of charge density)
$T_{H}/\rho^{1/3}$. Table. 3 lists the values of $\alpha_{n}$ for the lowest
ten QNM frequencies.

$\tilde{\omega}^{(n)}\approx\tilde{\omega}^{(n)}_{\mathrm{ex}}+\alpha_{n}\cdot\frac{\pi
T_{H}}{\rho^{1/3}},\qquad\qquad\left(\frac{T_{H}}{\rho^{1/3}}\ll 1\right)$
(53)

n | $\alpha_{n}$
---|---
1 | $\mp 0.720435+0.814211i$
2 | $\mp 1.203368+1.407563i$
3 | $\mp 1.678953+1.993761i$
4 | $\mp 2.152130+2.577889i$
5 | $\mp 2.624237+3.161153i$
6 | $\mp 3.095778+3.743983i$
7 | $\mp 3.566986+4.326567i$
8 | $\mp 4.037981+4.908998i$
9 | $\mp 4.508832+5.491329i$
10 | $\mp 4.979582+6.073592i$
Table 3: $\alpha_{n}$ for lowest ten tensorial QNM frequencies ($k=0$) of
Reissner-Nordstrom black branes

##  4 Magnetic Black Branes

In this section, we consider the gravitational dual of a magnetically charged
$\mathcal{N}=4$ SYM plasma with uniform magnetic field $b$ along the $\vec{z}$
direction [6, 7]. The dual theory has a corresponding bulk electromagnetic
tensor $F_{mn}$ with non-zero components

$F_{xy}=-F_{yx}=b$ (54)

Unlike the Reisnner-Nordstrom case, the metric solution to Einstein-Maxwell
equations (45) for this case is not analytically known. We consider an
asymptotically AdS5 metric ansatz (in Eddington-Finkelstein coordinates) with
SO(2) symmetry in the $x\leftrightarrow y$ plane and translational/boost
symmetries

$ds^{2}_{5}=-U(r)dv^{2}+e^{2V(r)}(dx^{2}+dy^{2})+e^{2W(r)}dz^{2}+2dvdr$ (55)

The bulk electromagnetic stress-energy tensor ${T^{\mu}}_{\nu}$ as given by
its non-zero components is

${T^{v}}_{v}={T^{r}}_{r}=-b^{2}e^{-4V(r)},\qquad\qquad{T^{i}}_{i}=b^{2}e^{-4V(r)},\qquad(i,j=x,y,z)$
(56)

The Hawking temperature $T_{H}$ is given by (48). With a transformation of the
radial coordinate $r\to 1/u$, the Einstein-Maxwell field equations (45) form
the system

$\displaystyle
3U^{\prime\prime}(u)u^{4}+3U^{\prime}(u)u^{3}\left(2+2uV^{\prime}(u)+uW^{\prime}(u)\right)-4b^{2}e^{-4V(u)}-24=0$
(57)
$\displaystyle\left(2V^{\prime\prime}(u)+W^{\prime\prime}(u)\right)u+2\left(V^{\prime}(u)^{2}u+2V^{\prime}(u)\right)+\left(W^{\prime}(u)^{2}u+2W^{\prime}(u)\right)=0$
(58) $\displaystyle
U^{\prime}(u)\left(W^{\prime}(u)-V^{\prime}(u)\right)u^{4}+U(u)\left(W^{\prime\prime}(u)u-V^{\prime\prime}(u)u+2W^{\prime}(u)-2V^{\prime}(u)+W^{\prime}(u)^{2}u\right.$
$\displaystyle\left.-2V^{\prime}(u)^{2}u-2V^{\prime}(u)W^{\prime}(u)u\right)u^{3}-2b^{2}e^{-4V(u)}=0$
(59)

We choose the horizon location $r_{+}=1$ by fixing the gauge choice for radial
shift $r\to r+\lambda$ such that $U(u=1)=0$. Following the prescription in
[6], for a chosen renormalization point $\mu\sim O(\left|b\right|^{1/2})$ and
a (renormalized) boundary energy density of $\varepsilon_{b}=3/4$, the near
boundary expansions of the metric functions allow the following redefinitions.
The redefined fields $\\{\alpha(u),\beta(u),\sigma(u)\\}$ are analytic over
the computational interval $u\in(0,1)$.

$\displaystyle
U(u)=\alpha(u)u^{3}-\left(1-\frac{2}{3}b^{2}\log(u)-\frac{1}{3}b^{2}\log(b)\right)u^{2}+\left(\frac{1}{u}+\sigma(u)\right)^{2}$
(60) $\displaystyle V(u)=\beta(u)+\log\left(\frac{1}{u}+\sigma(u)\right)$ (61)
$\displaystyle W(u)=-2\beta(u)+\log\left(\frac{1}{u}+\sigma(u)\right)$ (62)

###  4.1 Numerics

We first need to numerically compute the metric functions using the
pseudospectral approach. The system of equations
$\ref{eq:Eq57}\to\ref{eq:Eq59}$ are to be solved for the redefined fields
$\\{\alpha(u),\beta(u),\sigma(u)\\}$ with AdS5 boundary conditions
$V(u=0)=W(u=0)=0$ and horizon fixing condition $U(u=1)=0$.

To do this, the redefined fields are approximated by a truncated expansion in
Chebyshev polynomials over $u$ at the collocation grid points. As this system
of equations is coupled and highly non-linear, one needs to employ a
multivariate root finding algorithm (e.g. Newton’s method) over the values of
the redefined field at each collocation point. The newton step can be
expressed in block matrix form as

$\begin{pmatrix}\bf{A}_{n+1}\\\ \bf{B}_{n+1}\\\
\bf{C}_{n+1}\end{pmatrix}=\begin{pmatrix}\bf{A}_{n}\\\ \bf{B}_{n}\\\
\bf{C}_{n}\end{pmatrix}-\begin{pmatrix}\bf{J}_{1A}&\bf{J}_{1B}&\bf{J}_{1C}\\\
\bf{J}_{2A}&\bf{J}_{2B}&\bf{J}_{2C}\\\
\bf{J}_{3A}&\bf{J}_{3B}&\bf{J}_{3C}\end{pmatrix}^{-1}\cdot\begin{pmatrix}\bf{f_{1}}\\\
\bf{f_{2}}\\\ \bf{f_{3}}\end{pmatrix}$ (63)

where $\mathrm{A}\equiv\left\\{\alpha(u_{1}),...\alpha(u_{k})\right\\}$,
$\mathrm{B}\equiv\left\\{\beta(u_{1}),...\beta(u_{k})\right\\}$,
$\mathrm{C}\equiv\left\\{\sigma(u_{1}),...\sigma(u_{k})\right\\}$ and
$\mathrm{J}_{ij}=\mathrm{\nabla_{j}}\mathrm{f_{i}}\big{|}_{\vec{u}}$ ($k$ is
the grid size). $\mathrm{f_{i}}$ represents the computed values for
($\ref{eq:Eq57}\to\ref{eq:Eq59}$) over $u$ and boundary conditions at time
step $n$. One starts with $b=0$ where the solution is AdS5-Schwarschild black
brane metric (17) and gradually increases $b$ so that Newton’s method can
converge faster [6]. This process is repeated till the desired magnetic field
strength is reached.

####  4.1.1 QNMs of Magnetic Black Branes

As with the Reisnner-Nordstrom case, we focus exclusively on gravitational
fluctuations for vanishing spatial momenta ($k=0$). The $z$ direction of the
magnetic field breaks SO(3) symmetry down to SO(2) rotational symmetry in the
$x\leftrightarrow y$ plane. All the three type of perturbations viz., the
scalar, vector and tensor channels have non-trivial QNMs in this case.
However, we focus on the scalar and tensor perturbations in this article.
Because of the SO(3) to SO(2) symmetry breaking, the perturbations
corresponding to boundary pressure anisotropy belong to the scalar sector for
magnetic black branes [7]. All perturbations in a sector are decoupled from
other sectors and satisfy a closed set of equations of motion.

Let us first consider the tensorial sector. We take a plane wave ansatz for
shear perturbation $h_{xy}\equiv e^{-i\omega v}g(u)u^{2}$. Using the perturbed
metric $g_{mn}=\eta_{mn}+h_{mn}$ in Einstein-Maxwell equations (45), we get
the equation of motion

$\displaystyle g^{\prime\prime}(u)+\Phi_{1}(u)g^{\prime}(u)+\Phi_{2}(u)g(u)=0$
(64)
$\displaystyle\Phi_{1}(u)=\frac{6U(u)u+2i\omega+U^{\prime}(u)u^{2}}{U(u)u^{2}}-2V^{\prime}(u)+W^{\prime}(u)$
$\displaystyle\Phi_{2}(u)=\frac{2U^{\prime}(u)\left(1-V^{\prime}(u)u\right)u^{2}+i\omega\left(4-2V^{\prime}(u)u+W^{\prime}(u)u\right)+2U(u)\left(3+W^{\prime}(u)u-V^{\prime}(u)(4+W^{\prime}(u)u)u+V^{\prime\prime}(u)u^{2}\right)u}{U(u)u^{3}}$

which is an eigenvalue equation to be solved once the metric functions are
computed for a given value of $b$. The resulting eigenvalues are the QNM
frequencies in the tensorial sector.

Similarly consider the case of scalar perturbations. Three of the scalar
perturbations $h_{vv},h_{xx}+h_{yy},h_{zz}$ are coupled in the equations of
motion. We use a plane wave ansatz for each of the three scalar perturbations

$h_{vv}=e^{-i\omega v}g_{1}(u)u^{2},\,\,\,h_{xx}=h_{yy}=e^{-i\omega
v+2V(u)}\left(g_{2}(u)-g_{3}(u)\right)u^{2},\,\,\,h_{zz}=-2e^{-i\omega
v+2W(u)}\left(g_{2}(u)+g_{3}(u)\right)$ (65)

The metric ansatz perturbed with these scalar perturbations is substituted in
the Einstein-Maxwell equations (45) which produce three coupled equations of
motion (omitted for brevity). These are eigenvalue equations and the computed
eigenvalues are the QNM frequencies for the scalar sector.

###  4.2 Results and Analysis

We used a (radial) grid of 60 points for stable convergence. Table. 4 lists
the lowest two QNM frequencies ($\omega/\pi T_{H}$) for the tensorial
perturbations, in order of decreasing imaginary part. The (dimensionless)
field strength $b/T^{2}_{H}$ varies from 0 to $\sim 34$ and boundary energy
density (renormalized) is set to $\varepsilon_{b}=3/4$. Table. 5 lists the
lowest two QNM frequencies ($\omega/\pi T_{H}$) for scalar perturbations of
magnetic black brane ($\varepsilon_{b}=3/4$).

$b/T^{2}_{H}$ | n=1 | n=2
---|---|---
0 | $\pm 3.119452-2.746676i$ | $\pm 5.169521-4.763570i$
2.48 | $\pm 3.124071-2.795568i$ | $\pm 5.173049-4.843428i$
5.17 | $\pm 3.132475-2.937596i$ | $\pm 5.173807-5.072583i$
8.44 | $\pm 3.129093-3.178006i$ | $\pm 5.141546-5.447566i$
12.95 | $\pm 3.075832-3.548793i$ | $\pm 5.006620-5.971654i$
20.17 | $\pm 2.841178-4.114296i$ | $\pm 4.670453-6.541204i$
34.84 | $\pm 2.021158-4.684788i$ | $\pm 4.547208-7.121793i$
Table 4: Lowest two QNM frequencies (tensorial sector) ($k=0$) for magnetic black brane $b/T^{2}_{H}$ | n=1 | n=2
---|---|---
0 | $\pm 3.119452-2.746676i$ | $\pm 5.169521-4.763570i$
2.48 | $\pm 3.143638-2.761258i$ | $\pm 5.192327-4.808130i$
5.17 | $\pm 3.211341-2.803189i$ | $\pm 5.246431-4.930215i$
8.44 | $\pm 3.319144-2.873182i$ | $\pm 5.309472-5.114889i$
12.95 | $\pm 3.474557-2.980537i$ | $\pm 5.376798-5.362916i$
20.17 | $\pm 3.706464-3.154053i$ | $\pm 5.468700-5.739936i$
34.84 | $\pm 4.105908-3.490995i$ | $\pm 5.556184-6.661389i$
Table 5: Lowest two QNM frequencies (scalar sector) ($k=0$) for magnetic black
brane

####  4.2.1 QNMs of Near-Extremal Magnetic Black Branes

Plot (Fig. 7) shows behavior of imaginary part of lowest QNM frequency for
tensorial sector as the value of magnetic field is varied $b/T^{2}_{H}\approx
1.5\times 10^{3}$ to $3.8\times 10^{3}$. It is a log-log plot with
$\log(\left|\mathrm{Im}[\omega-\omega_{0}]\right|)$ on the y-axis and
$\log(\pi T_{H}/b^{1/2})$ on the x-axis represented by the blue dots.
$\omega_{0}$ is the lowest QNM fequency computed for the highest value of
$b/T^{2}_{H}\approx 3.8\times 10^{3}$ before computational errors get too
large to yield meaningful results. The red line shows a linear fit with slope
1 and this is observed for all modes. Unlike the Reisnner-Nordstrom case, the
real part $\mathrm{Re}[\omega-\omega_{0}]$ does not show such scaling with
$\pi T_{H}/b^{1/2}$.

Figure 7: Plot of $\log(\left|\mathrm{Im}[\omega-\omega_{0}]\right|)$ vs.
$\log(\pi T_{H}/b^{1/2})$ for $b/T^{2}_{H}\approx 1.5\times 10^{3}$ to
$3.8\times 10^{3}$. The red line shows a linear fit $y=x+c_{I}$

Thus, a linear scaling between $\mathrm{Im}[\omega^{(n)}]$ and $\pi
T_{H}/b^{1/2}$ is seen close to extremality.

$\mathrm{Im}\left[\omega^{(n)}\right]\approx\mathrm{Im}\left[\omega^{(n)}_{0}\right]+\gamma_{n}\cdot\left(\frac{\pi
T_{H}}{b^{1/2}}\right),\qquad\qquad\left(\frac{T_{H}}{b^{1/2}}\ll 1\right)$
(66)

where $n$ is the mode # and $\gamma_{n}$ are mode specific constants. From
(66), decay times for tensorial perturbations (proportional to
$\mathrm{Im}[\omega^{(n)}]$) of near-extremal magnetic black branes only
depend on their Hawking temperature (in units of magnetic field)
$T_{H}/b^{1/2}$. Table. 6 lists the values of $\gamma_{n}$ for the lowest ten
QNM frequencies.

n | $\gamma_{n}$
---|---
1 | $-5.627282$
2 | $-11.338257$
3 | $-15.788416$
4 | $-19.799393$
5 | $-24.860070$
6 | $-30.175361$
7 | $-34.071120$
8 | $-37.788665$
9 | $-43.785514$
10 | $-48.479225$
Table 6: $\gamma_{n}$ for lowest ten tensorial QNM frequencies ($k=0$) of
magnetic black branes

##  5 Conclusion

The study of quasinormal modes in AdS5 gravity duals of $\mathcal{N}=4$ SYM
plasmas provides important information on transport coefficients and other
dynamic properties otherwise not easily computable in lattice-gauge models.
Literature, especially on QNMs of magnetically charged black branes, is scarce
and does not have a high amount of detail. In this article, we employ a
pseudospectral approach using Chebyshev polynomial basis to solve Einstein
field equations for perturbations of asymptotically AdS5 spacetimes. In
particular, we compute the QNMs for tensor metric perturbations of Reisnner-
Nordstrom black branes as well as tensor and scalar metric perturbations of
magnetic black branes in the limit of vanishing spatial momenta. For both
cases, we find that the imaginary part of tensorial sector (non-hydrodynamic)
QNM frequencies, and therefore perturbation decay times, display a linear
scaling with temperature as extremality is approached which is only known to
be true in general for hydrodynamic QNMs.

## References

  * [1] I.. Aref’eva  “Holographic approach to quark-gluon plamsa in heavy ion collisions”  In _Physics-Uspekhi_ 57.6, 2014
  * [2] S. Nakamura and S.. Sin  “A holographic dual of hydrodynamics”  In _Journal of High Energy Physics_ 2006.9, 2006
  * [3] O. Aharony et al.  “Large N field theories, string theory and gravity”  In _Physics Reports_ 323.3, 2000, pp. 183–386
  * [4] J. Casalderrey-Solana et al.  “Gauge/string duality, hot QCD and heavy ion collisions”  Cambridge University Press, 2014
  * [5] E. Berti, V. Cardoso and A.. Starinets  “Quasinormal modes of black holes and black branes”  In _Classical and Quantum Gravity_ 26.16, 2009
  * [6] J.. Fuini and L.. Yaffe  “Far-from-equilibrium dynamics of a strongly coupled non-Abelian plasma with non-zero charge density or external magnetic field”  In _Journal of High Energy Physics_ 2015.7, 2015, pp. 1–48
  * [7] S. Janiszewski and M. Kaminski  “Quasinormal modes of magnetic and electric black branes versus far from equilibrium anisotropic fluids”  In _Physical Review D_ 93.2, 2016
  * [8] P.. Chesler and L.. Yaffe  “Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes”  In _Journal of High Energy Physics_ 2014.7, 2014, pp. 1–68
  * [9] J.. Boyd  “Chebyshev and Fourier spectral methods”  Courier Corporation, 2001


# Robotic and Generative Adversarial Attacks in Offline Writer-independent
Signature Verification

Jordan J. Bird
Computational Intelligence and Applications Research Group (CIA)
Department of Computer Science
Nottingham Trent University
Nottingham, United Kingdom
<EMAIL_ADDRESS>
https://jordanjamesbird.com/

###### Abstract

This study explores how robots and generative approaches can be used to mount
successful false-acceptance adversarial attacks on signature verification
systems. Initially, a convolutional neural network topology and data
augmentation strategy are explored and tuned, producing an 87.12% accurate
model for the verification of 2,640 human signatures. Two robots are then
tasked with forging 50 signatures, where 25 are used for the verification
attack, and the remaining 25 are used for tuning of the model to defend
against them. Adversarial attacks on the system show that there exists an
information security risk; the Line-us robotic arm can fool the system 24% of
the time and the iDraw 2.0 robot 32% of the time. A conditional GAN finds
similar success, with around 30% forged signatures misclassified as genuine.
Following fine-tune transfer learning of robotic and generative data,
adversarial attacks are reduced below the model threshold by both robots and
the GAN. It is observed that tuning the model reduces the risk of attack by
robots to 8% and 12%, and that conditional generative adversarial attacks can
be reduced to 4% when 25 images are presented and 5% when 1000 images are
presented.

_Keywords_ Offline Signature Verification $\cdot$ Writer-independent Signature
Verification $\cdot$ Robotic Adversarial Attacks $\cdot$ Generative
Adversarial Networks.

##  1 Introduction

To forge a signature with the aim of deceiving is a serious crime throughout
the world[1, 2] with financial implications and risks to personal identity.
There are countless examples of signature forgery and the issues it causes
such as fake historical documents fraudulently signed with Abraham Lincoln’s
signature[3], the forgery and subsequent sale of celebrity signatures[4], as
well as the forging and cashing of cheques[5]. To have a signature verified by
a human expert is an expensive endeavour, and is thus oftentimes another step
in the process which bares even further financial implications for an
individual or a business. In the UK alone, cheque fraud resulted in losses of
£12.3 million in 2020 following £53.6 million losses in 2019[6]. The COVID-19
lockdown introduced a level of fraud that had not been seen for several years.
In this study, losses of £558.8 million were also shown to be prevented from
cheque fraud in 2019. According to the American Bankers Association, cheque
fraud amounted to $15.1 billion in 2018 and affected around half a million
individuals[7].

Modern computing proposes several solutions to the detection of forged
signatures through autonomous verification as an added layer of protection
against such attempts. Much state-of-the-art work in signature verification is
to detect when a human being has forged another’s signature. Given the rapid
growth of consumer robotics due to their ease of use and low cost, the level
of detail now possible via robotic signature forgery is a growing concern.
Rapid analysis and near-perfect replication of a signature is now possible by
machines that cost a fraction of the price of the average smartphone. In terms
of generative approaches, it is possible to train models such as GANs on home
computers; that is, with hardware already found in the home, signature
verification systems can be succesfully attacked and fooled by data generated
by a neural network. In this study, we explore how robots and generative
adversarial methods can be used to fool offline writer-independent signature
verification systems. Following this, efforts are made to tune and improve
such systems to provide a preliminary line of defence against such adversarial
attacks.

The multiple scientific contributions presented by this work are as follows:
(i) tuning of a vision-based system inspired by the current State-of-the-Art
for accurate signature verification. (ii) a pipeline to analyse signatures
(raster images), produce vectors, and then G-code for execution of two pen-
holding robotic arms (Line-us and iDraw 2.0). (iii) a Conditional Generative
Adversarial Network to discern and generate real and forged signatures. (iv)
Successful adversarial attacks on the verification system by both robots and
generative approaches. (v) Successful defence of the verification system by
fine-tune transfer learning from examples produced during adversarial attack.
To the author’s knowledge, this article proposes the first case of an attack
on a signature verification system by using robots to physically copy and
write signatures.

The remainder of this article is as follows: Section 2 reviews the background
of the field, including the state of the art in signature verification and
adversarial attacks. Following this, Section 3 outlines the method followed by
the experiments in this work, and the results are presented and discussed in
Section 3. Finally, concluding remarks and future work are discussed in
Section 5.

##  2 Background

This section explores the current state of the art related to this work. This
includes biometrics, signature verification with visual and deep learning
approaches, and methods of attacking verification methods.

Biometrics are systems that recognise an individual based on a given input.
For example, recognition of an individual based on their fingerprint[8],
speech patterns[9, 10], EEG[11] and ECG[12] signals, or the iris of their
eye[13] to name a few. Signature verification is a biometric for the
recognition and verification of an individual based on the way that they sign
their name[14]. Given the ability that a genuine signature has, successful
forgeries can therefore have major implications related to personal identity
and finances. Online signature verification deals with smart devices, such as
tablets and pens, that record the resultant signature along with features such
as pressure, azimuth, velocity, and inclination[15]. Offline signature
verification is based on the resultant signature alone, and is more common
given that signed paper documents are often signed.

Earlier works such as Kalera et al.[16] proposed methods such as distance
metrics between examples of real and forged signatures. The identification of
two data sets was found to reach an accuracy of around 93%, while verification
was possible for around 78% of the data objects in one dataset and around 68%
in the second set. In [17], the authors proposed a machine learning approach
for least-squares support vector machines with respect to the grey-level
features observed. Consideration of grey levels within pixel values provides
pseudo-online insight into pen velocity and pressure. Such features are noted
to be important given that a human will sign their own signature quickly,
while a forgery is oftentimes more thought-out and therefore takes longer to
replicate.

State-of-the-art research on signature verification relies largely on deep
learning-based approaches. In 2019, Sam et al.[18] proposed an offline
verification technique with the Inception-v3 Convolutional Neural Network
(CNN) that achieved 88% validation accuracy on 170 images after training on
370 signatures. A similar approach was explored in [19], with a CNN model
achieving 62.5% writer-independent and 75% writer-dependent signature
verification. In this study, the writer-independent dataset contained 300
images for training and 240 for validation, and the writer-dependent dataset
contained 30 images for training and 24 for validation. In 2018, Souza et
al.[20] proposed a feature extraction technique with a CNN prior to
classification by Support Vector Machines; achieving an equal error rate of
1.48% on the Brazilian PUC-PR dataset.

Given that there is a growing reliance on automated systems for verification,
therein lies the risk of attack. An adversarial attack is a method of fooling
a machine learning model by engineering input data to force a chosen output
prediction. In signature verification, a successful attack would allow for an
individual to pass another’s signature without their participation or
knowledge. Work by the Autonomous University of Madrid’s Biometric Recognition
Group showed that a Bayesian hill-climbing attack could overcome a signature
verification system 95% of the time[21]. Similarly, Li et al.[22] proposed
generating invisible perturbations on signatures which led to a verification
system being succesfully attacked in 92.1% of cases. In [23] the authors
proposed to generate synthetic signatures by spectral analysis of trajectory
functions, presenting results of up to 0.04% brute-force attack successes.
Scheidat et al. [24] proposed a strategy of distance-level fusion to overcome
brute-force attacks on verification systems, noting a lower error rate when
combining online verification experts into a unified system. In [25], an
attack is described which can implement digital signatures into documents, by
embedding PDF and TIFF files within a file which goes largely undetected. In
another related study, Alonso-Fernandez et al.[26] described how forgeries do
not only simply attack a system while remaining static, rather, forger skills
are improved over time. This is similar to a generative adversarial learning
framework, which inspired the added approach to this study alongside the two
robots.

##  3 Method

Figure 1: The general approach for robotic signature forgery. The raster image
is converted to a centerline vector, which is then in turn converted to
G-Code. The robots then execute the code and physically forge the given
signature with a pen. The outputs of this diagram are the paper signatures
scanned and pre-processed.

In this section, the method followed by the proposed approach is described.
Initially, data pre-processing and augmentation are considered, followed by
topology engineering and learning for signature verification models. Following
this, adversarial attacks and defences are then outlined.

The data used in this study is the CEDAR Signature Verification
Dataset111https://cedar.buffalo.edu/signature/ from [27]. In the study, 55
individuals contributed 24 of their own signatures, and some of the
participants forged the signatures of others. The dataset comprises 1,320
genuine and 1,320 forged signatures.

Figure 2: An example of a signature before (left) and after (right)
preprocessing.

The dataset was found to contain images that varied in size and aspect ratio.
Given the nature of the input topologies of convolutional neural networks,
data pre-processing was performed. All images were resized to 256px and
binarized; An example of an image before and after processing can be found in
Figure 2. Some signatures were also observed to be signed at a rotation of up
to approximately 20%. Random rotation is therefore explored as a potential
method of data augmentation along with a vanilla convolutional neural network.
Convolutional Neural Networks were the main learning method within the
experiments. During an 80/20 split, topology tuning was performed via CNN
layers of {1, 2, 3} in number, each containing {16, 32, 64} filters. Dense
interpretation multilayer perceptrons were then tuned, involving 1 or 2 layers
with {64, 128, 256} rectified linear units. Since some methods would likely
converge sooner than others, the CNNs are therefore given an infinite number
of epochs to train, instead stopping when there were no observed improvements
on the validation metrics within a 25 epoch window. The Conditional GAN
approach to generating signature forgeries was conditioned on the class label.
The GAN generator began with an $8\times 8$ random noise matrix before 5
transpose layers of 128 filters each upscaled the matrix to a $256\times 256$
image. Each generator layer had a Leaky Rectified Linear Unit activation
function. The discriminator network consisted of two convolutional layers with
128 filters and a 2,2 stride. A dropout value of 0.4 was implemented prior to
output to prevent mode collapse. Adversarial attacks are then tested with 25
and 1000 images output by the GAN, conditioned with a class label
corresponding to the real signature class.

Figure 3: The two robots used in this study; Line-us (left) and iDraw 2.0
(right).

In addition to the generative adversarial attack, this study also explores
whether robotic arms can forge signatures. Two robots are used, the Line-us
robotic arm and the iDraw 2.0 handwriting plotter. These two robots can be
seen in Figure 3. Each robot was tasked with replicating the same 25
signatures from the dataset, and these images provide robotic attacks on the
verification model. Figure 1 shows the general approach to enabling physical
replication of the signature; The raster scan is vectorised through a
centerline trace, which is then converted to robot-compatible G-Code. The
G-Code is executed on the robots, which then write the signature on paper.

For defence, a further 25 real signatures are signed by the two robots (and
generated by the GAN). The model is fine-tuned for one epoch, with the
resultant signatures included within the dataset labelled as forgeries.
Therefore, this strategy exposes the verification model to robot-forged
signatures with the aim of providing further enhanced awareness. The aim thus
is to classify behaviours that the GAN and robots exhibit which humans
necessarily do not.

##  4 Results

In this section, the results of the proposed approach are presented. The
verification model is first tuned and forgery methods are discussed. Following
this, the robotic and generative adversarial attacks are performed and
defended against with fine-tune transfer learning.

Table 1: Topology tuning for signature verification without data augmentation. | CNN
---
Layers
# Filters | Max Epoch | Acc. | Prec. | Rec. | F1
1 | 16 | 41 | 75.38 | 0.75 | 0.761 | 0.756
2 | 16 | 52 | 78.41 | 0.791 | 0.773 | 0.782
3 | 16 | 52 | 81.63 | 0.82 | 0.811 | 0.815
1 | 32 | 31 | 73.3 | 0.734 | 0.731 | 0.732
2 | 32 | 35 | 77.84 | 0.78 | 0.777 | 0.778
3 | 32 | 47 | 82.39 | 0.833 | 0.811 | 0.822
1 | 64 | 28 | 74.81 | 0.73 | 0.788 | 0.758
2 | 64 | 29 | 78.79 | 0.82 | 0.739 | 0.777
3 | 64 | 43 | 84.09 | 0.838 | 0.845 | 0.842
Mean |  | 39.78 | 78.52 | 0.79 | 0.78 | 0.78
Table 2: Topology tuning for signature verification with data augmentation (random rotation, max. 20%). | CNN
---
Layers
# Filters | Max Epoch | Acc. | Prec. | Rec. | F1
1 | 16 | 96 | 80.11 | 0.777 | 0.845 | 0.809
2 | 16 | 75 | 85.04 | 0.832 | 0.879 | 0.855
3 | 16 | 83 | 85.04 | 0.829 | 0.883 | 0.855
1 | 32 | 99 | 78.03 | 0.766 | 0.807 | 0.786
2 | 32 | 88 | 87.12 | 0.855 | 0.894 | 0.874
3 | 32 | 99 | 87.12 | 0.926 | 0.807 | 0.862
1 | 64 | 115 | 81.06 | 0.793 | 0.841 | 0.816
2 | 64 | 98 | 86.55 | 0.846 | 0.894 | 0.87
3 | 64 | 60 | 83.9 | 0.833 | 0.849 | 0.841
Mean |  | 90.33 | 83.77 | 0.83 | 0.86 | 0.84
Table 3: Interpretation network topology tuning with data augmentation. | MLP
---
Layers
| MLP
---
Neurons
| Max
---
Epoch
Acc. | Prec. | Rec. | F1
1 | 16 | 53 | 80.3 | 0.808 | 0.796 | 0.802
2 | 16 | 26 | 50 | - | - | -
1 | 32 | 74 | 82.58 | 0.812 | 0.849 | 0.83
2 | 32 | 64 | 81.63 | 0.808 | 0.83 | 0.819
1 | 64 | 80 | 84.28 | 0.842 | 0.845 | 0.843
2 | 64 | 122 | 85.23 | 0.832 | 0.883 | 0.857
1 | 128 | 88 | 87.12 | 0.855 | 0.894 | 0.874
2 | 128 | 86 | 86.55 | 0.861 | 0.871 | 0.866
1 | 256 | 101 | 85.8 | 0.829 | 0.9 | 0.864
2 | 256 | 80 | 87.5 | 0.856 | 0.9 | 0.878

Initially, Tables 1 and 2 present results for topology tuning with and without
data augmentation, respectively. On average, augmentation led to a mean
increase in accuracy by 5.25%, but required a mean of 50.55 additional epochs
to meet the early stopping criteria. The best convolutional models for
signature verification in terms of classification accuracy (87.12%) were found
to be those with 2 and 3 convolutional layers, both with 32 filters. In terms
of metrics, 32 filters within 2 layers were slightly more stable. As can be
observed from Table 3, other dense interpretation networks were also tested;
leading to an overall best score of 87.12%.

Figure 4: Real signatures compared to the forgeries by the iDraw and Line-us
robots. Forgeries have been written physically before being scanned and pre-
processed.

Figure 4 shows a real signature and the two counterparts forged by the iDraw
2.0 and Line-us robots. It can be observed that the iDraw 2.0 arguably creates
the most realistic forgery, but features such as line intersections can
differentiate it from the genuine image. Both robots seem to lose some detail,
seen here via the minute squiggles in the line within the middle name; though
this is not necessarily indicative of forgery given that human beings will
also change the level of detail given due to reasons such as situationally
dependent writing speed.

Figure 5: Generator loss per epoch for the Conditional Generative Adversarial
Network. Figure 6: Discriminator losses per epoch for the Conditional
Generative Adversarial Network. Figure 7: Early per-epoch examples of
Conditional GAN samples. It was observed that more realistic shapes
reminiscent of signatures began to form following epoch 300. Figure 8:
Examples of late generalisations by the GAN conditioned with the “Real" class
label.

Figures 5 and 6 show the generator and discriminator losses for the
Conditional GAN, respectively. Mode collapse did not occur with the chosen
hyperparameters, and it was observed that relatively little change in the
outputs occurred beyond epoch 1000. Realistic images reminiscent of signatures
were also observed to form following the first 300 epochs, which can be seen
in Figure 7. Figure 8 shows examples of how the GAN generalises “Real"
signatures within the latter stages of training. In some examples letters can
be observed (e.g. the capital B in the first output), but it seems that the
generator produces generalised clouds of pixels which can fool the
discriminator.

Figure 9: A possible GAN replication of a human-written letter compared to a
similar sample from the dataset.

Figure 9 shows a comparison of the first GAN example and capital B within a
signature in the data set.

Table 4: Successful attacks on writer-independent signature verification by the robots and GAN prior to tuning.  Method | Successful Attacks | Success (%)
---|---|---
iDraw 2.0 | 8/25 | 32
Line-us | 6/25 | 24
cGAN | 10/25 | 40
cGAN | 297/1000 | 29.7
Table 5: Successful defence of writer-independent signature verification by robots and GAN after tuning.  Method | Successful Attacks | Success (%) | Defense (%)
---|---|---|---
iDraw 2.0 | 2/25 | 8 | +24
Line-us | 3/25 | 12 | +12
cGAN | 1/25 | 4 | +36
cGAN | 50/1000 | 5 | +24.7

Table 4 shows the results of attacks on the system through robotic and GAN-
based forgeries. All successful attacks are outside of the model baseline
(87.12%) and show the dangers of signature forgeries using such methods.
Following tuning, a successful defence is then mounted against these
approaches as shown by Table 3, wherein signature verification ability is
improved by a minimum of 12% and maximum of 24.7%. Fine-tuning with examples
brings the success of attack within the expected margin of error of the model.

##  5 Conclusion and Future Work

This work has shown the dangers surrounding two adversarial attack methods on
vision-based writer-independent signature verification models. Although the
base model experienced over 87% classification accuracy, successful attacks by
robotic and GAN-based approaches were far above this margin of error. The
Line-us and iDraw 2.0 robots were observed to fool the system 24% and 32% of
the time, respectively. The Conditional GAN fooled the system 40% of the time
with a set of 25 signatures, and 29.7% of the time with a set of 1000
signatures. In conclusion of this part of the study, the results have brought
to the forefront information security issues when human versus human
signatures forgeries are focused upon while a robot can achieve much more
similar forgeries to the real human. Given this, biometric security could
therefore be overcome by using these robots, which are low in cost and easily
accessible. In the second part of the study, the verification model was fine-
tuned with some examples generated by the approaches. Given exposure to
robotic and GAN behaviours, the model could then prevent far more of these
attacks; the accuracy of the attack was reduced by 12 and 24 percentage points
for the Line-us (12%) and iDraw 2.0 robots (8%), respectively. The generative
approach was successful only 4% of the time for 25 images and 5% of the time
for a set of 1000 images. Given that these results were below the margin of
error of the model, an acceptable defence has been mounted.

In future, the study could be extended by allowing robots to generate a larger
set of signatures. This is argued for by the two sets of results from the GAN
(25 and 1000 images), which have a noticeable difference. In addition, further
image classification approaches could be explored to form a stronger base
model, such as attention modelling and residual information.

To finally conclude, this study has shown that it is relatively easy to attack
a signature verification model by generating forgeries physically with robots
and digitally with generative approaches. This study has also shown a method
to defend against such attacks through fine-tune transfer learning. The
results after fine-tuning are effective, showing that such attacks can be
prevented now, rather than after the first consumer robot-based forgery crime
has been committed.

## References

  * [1] D. Cowley, “The forgery and counterfeiting act 1981,” _The Journal of Criminal Law_ , vol. 47, no. 1, pp. 61–70, 1983.
  * [2] M. B. Hemraj, “The crime of forgery,” _Journal of Financial Crime_ , 2002\.
  * [3] J. Nickell, _Detecting forgery: forensic investigation of documents_. University Press of Kentucky, 2005.
  * [4] J. D. McMillen and R. G. Schneider, “Sports celebrity photographs and copyright law in the united states,” _Entertainment Law_ , vol. 1, no. 1, pp. 52–66, 2002.
  * [5] E. A. Farnsworth, “Insurance against check forgery,” _Columbia Law Review_ , vol. 60, no. 3, pp. 284–325, 1960.
  * [6] UK finance, 2021. [Online]. Available: https://www.ukfinance.org.uk/system/files/Fraud%20The%20Facts%202021-%20FINAL.pdf
  * [7] N. Trentmann, “Rise in check fraud could motivate treasurers to switch to other payment tools,” Jan 2020. [Online]. Available: https://www.wsj.com/articles/rise-in-check-fraud-could-motivate-treasurers-to-switch-to-other-payment-tools-11579131505
  * [8] V. Praseetha, S. Bayezeed, and S. Vadivel, “Secure fingerprint authentication using deep learning and minutiae verification,” _Journal of Intelligent Systems_ , vol. 29, no. 1, pp. 1379–1387, 2020.
  * [9] A. Boles and P. Rad, “Voice biometrics: Deep learning-based voiceprint authentication system,” in _2017 12th System of Systems Engineering Conference (SoSE)_. IEEE, 2017, pp. 1–6.
  * [10] J. J. Bird, D. R. Faria, C. Premebida, A. Ekárt, and P. P. Ayrosa, “Overcoming data scarcity in speaker identification: Dataset augmentation with synthetic mfccs via character-level rnn,” in _2020 IEEE International Conference on Autonomous Robot Systems and Competitions (ICARSC)_. IEEE, 2020, pp. 146–151.
  * [11] E. Maiorana, “Learning deep features for task-independent eeg-based biometric verification,” _Pattern Recognition Letters_ , vol. 143, pp. 122–129, 2021\.
  * [12] N. Ibtehaz, M. E. Chowdhury, A. Khandakar, S. Kiranyaz, M. S. Rahman, A. Tahir, Y. Qiblawey, and T. Rahman, “Edith: Ecg biometrics aided by deep learning for reliable individual authentication,” _IEEE Transactions on Emerging Topics in Computational Intelligence_ , 2021.
  * [13] A. S. Al-Waisy, R. Qahwaji, S. Ipson, S. Al-Fahdawi, and T. A. Nagem, “A multi-biometric iris recognition system based on a deep learning approach,” _Pattern Analysis and Applications_ , vol. 21, no. 3, pp. 783–802, 2018.
  * [14] N. Houmani, A. Mayoue, S. Garcia-Salicetti, B. Dorizzi, M. I. Khalil, M. N. Moustafa, H. Abbas, D. Muramatsu, B. Yanikoglu, A. Kholmatov _et al._ , “Biosecure signature evaluation campaign (bsec’2009): Evaluating online signature algorithms depending on the quality of signatures,” _Pattern Recognition_ , vol. 45, no. 3, pp. 993–1003, 2012.
  * [15] E. A. Soelistio, R. E. H. Kusumo, Z. V. Martan, and E. Irwansyah, “A review of signature recognition using machine learning,” in _2021 1st International Conference on Computer Science and Artificial Intelligence (ICCSAI)_ , vol. 1. IEEE, 2021, pp. 219–223.
  * [16] M. K. Kalera, S. Srihari, and A. Xu, “Offline signature verification and identification using distance statistics,” _International Journal of Pattern Recognition and Artificial Intelligence_ , vol. 18, no. 07, pp. 1339–1360, 2004.
  * [17] M. A. Ferrer, J. F. Vargas, A. Morales, and A. Ordonez, “Robustness of offline signature verification based on gray level features,” _IEEE Transactions on Information Forensics and Security_ , vol. 7, no. 3, pp. 966–977, 2012.
  * [18] S. M. Sam, K. Kamardin, N. N. A. Sjarif, N. Mohamed _et al._ , “Offline signature verification using deep learning convolutional neural network (cnn) architectures googlenet inception-v1 and inception-v3,” _Procedia Computer Science_ , vol. 161, pp. 475–483, 2019.
  * [19] M. M. Yapici, A. Tekerek, and N. Topaloglu, “Convolutional neural network based offline signature verification application,” in _2018 International Congress on Big Data, Deep Learning and Fighting Cyber Terrorism (IBIGDELFT)_. IEEE, 2018, pp. 30–34.
  * [20] V. L. Souza, A. L. Oliveira, and R. Sabourin, “A writer-independent approach for offline signature verification using deep convolutional neural networks features,” in _2018 7th Brazilian Conference on Intelligent Systems (BRACIS)_. IEEE, 2018, pp. 212–217.
  * [21] J. Galbally, J. Fierrez, and J. Ortega-Garcia, “Bayesian hill-climbing attack and its application to signature verification,” in _International Conference on Biometrics_. Springer, 2007, pp. 386–395.
  * [22] H. Li, H. Li, H. Zhang, and W. Yuan, “Black-box attack against handwritten signature verification with region-restricted adversarial perturbations,” _Pattern Recognition_ , vol. 111, p. 107689, 2021.
  * [23] J. Galbally, J. Fierrez, M. Martinez-Diaz, and J. Ortega-Garcia, “Evaluation of brute-force attack to dynamic signature verification using synthetic samples,” in _2009 10th International Conference on Document Analysis and Recognition_. IEEE, 2009, pp. 131–135.
  * [24] T. Scheidat, C. Vielhauer, and J. Dittmann, “Distance-level fusion strategies for online signature verification,” in _2005 IEEE International Conference on Multimedia and Expo_. IEEE, 2005, pp. 1294–1297.
  * [25] F. Buccafurri, G. Caminiti, and G. Lax, “Fortifying the dalì attack on digital signature,” in _Proceedings of the 2nd International Conference on Security of Information and Networks_ , 2009, pp. 278–287.
  * [26] F. Alonso-Fernandez, J. Fierrez, A. Gilperez, J. Galbally, and J. Ortega-Garcia, “Robustness of signature verification systems to imitators with increasing skills,” in _2009 10th International Conference on Document Analysis and Recognition_. IEEE, 2009, pp. 728–732.
  * [27] S. N. Srihari, H. Srinivasan, S. Chen, and M. J. Beal, “Machine learning for signature verification,” in _Machine learning in document analysis and recognition_. Springer, 2008, pp. 387–408.