# An expedition to the islands of stability in the first-order causal
hydrodynamics

Rajesh Biswas<EMAIL_ADDRESS>Sukanya Mitra
<EMAIL_ADDRESS>Victor Roy<EMAIL_ADDRESS>School of Physical
Sciences, National Institute of Science Education and Research, An OCC of Homi
Bhabha National Institute, Jatni-752050, India.

###### Abstract

The recently proposed connection between the Lorentz invariance of stability
and the speed of signal propagation has been tested for a first-order
relativistic dissipative hydrodynamic theory. The fact that the stability
situation in different reference frames agrees with each other only as long as
the signal propagation respects causality, has been explicitly established for
the theory, which is microscopically derived from the covariant kinetic
equation in general hydrodynamic frames with arbitrary momentum-dependent
interactions.

##  1 Introduction

Since the inception of the special theory of relativity (STR), there has been
a consensus that faster-than-light communication is not possible. To our best
knowledge, it also has not been achieved in any experiment. However, several
theories and phenomena related to superluminal communication have been
proposed or studied, including tachyons [1], neutrinos, quantum nonlocality,
etc. The superluminal signal propagations are not permitted according to STR
because, in a Lorentz-invariant theory, it could be used to transmit
information into the past. This complicates causality leading to logical
paradoxes of the “kill your own grandfather” type. On the other hand, examples
of matter of very high density allowing wave modes with superluminal group
velocities have been known for a long time [2]. The early formulations of the
relativistic dissipative hydrodynamics (a.k.a the first-order hydrodynamic
theories because the dissipative corrections to the energy-momentum tensor
were constructed from the first-order gradient of fluid variables) by Eckart
and Landau are also known to be inherently acausal due to the spacelike nature
of the characteristics [3, 4]. The acausality problem was later cured by
Mueller-Israel-Stewarts (MIS), where the entropy flux includes the higher
order dissipative terms [5]. At the same time, people also realized that the
causality of relativistic hydrodynamics is closely associated with the
stability of the fluid under small perturbations. Usually, any small
perturbation in fluid propagates with a characteristic speed of sound and
transports energy from the source of the perturbation to distant points; the
amplitude of the perturbation usually diminishes due to the dissipation of
kinetic energy into heat which increases the total entropy, and the system
reaches a new thermodynamic stable state. In some cases, the small
perturbation may grow exponentially, eventually leading to instability in the
fluid [6]. Energy conservation, however, enforces that the instability cannot
grow indefinitely unless the temperature of the system tends to infinity at
the same time [7]. The situation for relativistic fluids became further
complicated since it was found that a state of fluid can be stable to
perturbations in one Lorentz frame and unstable in another, which clearly
violates the principle of relativity. Hence one must make sure that any
consistent relativistic hydrodynamic formulation must testify to the two
benchmark criteria to be physically acceptable: (i) signal propagation must be
subluminal to respect the causality, and (ii) the system must be stable in all
Lorentz frames against any perturbations from the equilibrium state.The close
correlation between them has been studied for MIS theory [8, 9, 10], where the
theory is shown to be unstable because of superluminal modes. In the context
of the recently proposed first-order stable-causal (BDNK) theory, [11] gives a
theorem that asserts that stability in the fluid’s local rest frame implies
stability in any Lorentz boosted frame, provided that the system is causal and
strong hyperbolic. Finally, in [12], the authors provide an intuitive
understanding of this puzzle. They show that in presence of dissipation two
observers can disagree on whether a state is stable or unstable iff the
perturbations can exit the light cone. In conclusion, the stability of a state
is a Lorentz-invariant property of dissipative theories iff the principle of
causality holds. Motivated by these studies, we explicitly demonstrate here
the said connection for a stable-causal first-order theory, derived from a
first principle kinetic theory and valid in general hydrodynamic frames and
with arbitrary momentum-dependent interactions. In [13], a first-order stable-
causal relativistic theory (henceforth called BMR approach for brevity) has
been derived, motivated from the works of [14] (BDNK theory) that presents a
pathology -free first-order relativistic theory via hydrodynamic field
redefinition. The BMR approach showed that along with the general hydrodynamic
frame choice the momentum dependence of microscopic interaction rate are also
imperative for producing a causal and stable first-order relativistic theory.
In the current work, we have checked that for BMR approach, all microscopic
parameters considered to define the hydrodynamic frame and momentum dependence
of the interaction rate, the linear stability analysis can differ from one
Lorentz frame to another, if and only if the signal propagation exits the
light cone. Stability is Lorentz invariant for all time-like perturbations;
hence, testing it in one reference frame suffices for causal wave propagation.

Throughout the manuscript, we have used natural unit ($\hbar=c=k_{B}=1$) and
flat space-time with mostly negative metric signature
$g^{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right)$.

##  2 BMR theory

In [13], a first-order, relativistic, stable and causal hydrodynamic theory
has been derived in general frames from the Boltzmann transport equation,
where the system interactions are introduced via the microscopic particle
momenta captured through momentum-dependent relaxation time approximation
(MDRTA) [15]. The basic idea is to estimate the out-of-equilibrium one-
particle distribution function $f(x,p)$ for general hydrodynamic frame and
arbitrary momentum dependent interactions. $f(x,p)$ can be decomposed in two
components : one is the homogeneous solution which sets the hydrodynamic frame
choice and the second is the inhomogeneous solution that is controlled by the
system interaction. To extract the homogeneous part of the solution, we use
the matching conditions (constraints that set the thermodynamic fields to
their equilibrium values even in the presence of dissipation). For estimating
the inhomogeneous part, the relativistic transport equation,
$p^{\mu}\partial_{\mu}f(x,p)=-\cal{L}[\phi]$, is solved where $\cal{L}[\phi]$
is the linearized collision operator with $\phi$ as the deviation in particle
distribution due to dissipation. $p^{\mu}$ and $x$ denote the particle four-
momenta and space-time variable respectively. We employ here the MDRTA method
to explicitly linearize the collision term, with the relaxation time of single
particle distribution function $\tau_{R}$ expressed as a power law of scaled
particle energy ($\tilde{E_{p}}=p^{\mu}u_{\mu}/T$) in comoving frame,
$\tau_{R}=\tau_{R}^{0}(x)(\tilde{E_{p}})^{\Lambda}$. We propose an appropriate
collision operator (that satisfies all the conservation properties, adjoint
properties and summation invariant properties) for a semi-orthogonal, monic
polynomial basis of the distribution function, instead of the Anderson-Witting
type relaxation kernel,

$\displaystyle{\cal{L}}_{\text{MDRTA}}[\phi]$
$\displaystyle=\frac{\left(p\cdot u\right)}{\tau_{R}}f^{(0)}(1\pm
f^{(0)})\bigg{[}\phi-\tilde{p}_{\langle\nu\rangle}\frac{\langle\frac{\tilde{E}_{p}}{\tau_{R}}\phi\tilde{p}^{\langle\nu\rangle}\rangle}{\frac{1}{3}\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{p}^{\langle\mu\rangle}\tilde{p}_{\langle\mu\rangle}\rangle}$
$\displaystyle-\frac{\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{E}_{p}^{2}\rangle\langle\frac{\tilde{E}_{p}}{\tau_{R}}\phi\rangle-\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{E}_{p}\rangle\langle\frac{\tilde{E}_{p}}{\tau_{R}}\phi\tilde{E}_{p}\rangle}{\langle\frac{\tilde{E}_{p}}{\tau_{R}}\rangle\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{E}_{p}^{2}\rangle-\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{E}_{p}\rangle^{2}}$
$\displaystyle-\tilde{E}_{p}\frac{\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{E}_{p}\rangle\langle\frac{\tilde{E}_{p}}{\tau_{R}}\phi\rangle-\langle\frac{\tilde{E}_{p}}{\tau_{R}}\rangle\langle\frac{\tilde{E}_{p}}{\tau_{R}}\phi\tilde{E}_{p}\rangle}{\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{E}_{p}\rangle^{2}-\langle\frac{\tilde{E}_{p}}{\tau_{R}}\rangle\langle\frac{\tilde{E}_{p}}{\tau_{R}}\tilde{E}_{p}^{2}\rangle}\bigg{]}~{}.$
(1)

This collision operator preserves the conservation laws microscopically. By
virtue of that, the right hand side of the transport equation singularly
excludes the zero modes of the linearized collision operator. Because of the
fact, the left hand side of the transport equation is not necessarily needed
to be orthogonal to zero modes and hence the covariant time derivatives
appearing on the left hand side of the transport equation are not necessarily
required to be exchanged by the spatial gradients. As a result, to solve the
equation, we can safely apply a perturbative method keeping the time
derivatives over the fundamental thermodynamic quantities alive (necessarily
for a stable-causal theory) in the first-order thermodynamic field corrections
as the following,

$\displaystyle\left[\begin{array}[]{l}\delta n^{(1)}\\\
\delta\epsilon^{(1)}\\\ \delta P^{(1)}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{l}\nu_{1}\\\ \varepsilon_{1}\\\
\pi_{1}\end{array}\right]\frac{DT}{T}+\left[\begin{array}[]{l}\nu_{2}\\\
\varepsilon_{2}\\\ \pi_{2}\end{array}\right]\left(\partial\cdot
u\right)+\left[\begin{array}[]{l}\nu_{3}\\\ \varepsilon_{3}\\\
\pi_{3}\end{array}\right]D\tilde{\mu}~{},$ (14)
$\displaystyle\left[\begin{array}[]{l}W^{(1)\mu}\\\
V^{(1)\mu}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{l}\theta_{1}\\\
\gamma_{1}\end{array}\right]\left(\frac{\nabla^{\mu}T}{T}-Du^{\mu}\right)+\left[\begin{array}[]{l}\theta_{3}\\\
\gamma_{3}\end{array}\right]\nabla^{\mu}\tilde{\mu}~{}.$ (21)

Here, $\delta n,\delta\epsilon,\delta P,W^{\alpha}$ and $V^{\alpha}$
respectively denote dissipative corrections over particle number density,
energy density, pressure, energy flow and particle current. The field
corrections include first order derivatives over all the fundamental
thermodynamic quantities such as temperature $T$, chemical potential $\mu$ and
hydrodynamic four-velocity $u^{\mu}$. The field correction coefficients turn
out to be elaborate functions of the frame indices $i,j,k$ and the parameter
$\Lambda$ of MDRTA (for details see [13]). These field corrections along with
$\pi^{(1)\mu\nu}=2\eta\sigma^{\mu\nu}$ ($\eta$ is shear viscosity), constitute
the first order out of equilibrium particle four-flow, $N^{\mu}=(n+\delta
n^{(1)})u^{\mu}+V^{\mu(1)}$, and energy-momentum tensor,
$T^{\mu\nu}=\left(\epsilon+\delta\epsilon^{(1)}\right)u^{\mu}u^{\nu}-\left(P+\delta
P^{(1)}\right)\Delta^{\mu\nu}+\left(W^{\mu(1)}u^{\nu}+W^{\nu}u^{\mu}\right)+\pi^{\mu\nu(1)}$,
respectively. Here, $n,\epsilon$ and $P$ indicate particle number density,
energy density and pressure at their local equilibrium values. We observed
that both the conventional frame choices (Landau ($i,j,k=1,2,1$) and Eckart
($i,j,k=1,2,0$)) and the momentum independent $\tau_{R}$ ($\Lambda=0$) lead to
an acausal and unstable theory.

##  3 Stability and causality analysis with boosted background

In [13], the stability and causality of BMR theory have been investigated
(with an affirmation that the theory is indeed stable and causal with a
critical dependence on the system interaction) in the Lorentz rest frame of
the fluid. But it has been observed a number of times [16, 17] that such an
analysis in local rest frame is often inadequate (even sometimes misleading)
because with a boosted background the stability situation can alter
drastically, even completely new modes can appear due to the degree of the
dispersion polynomial changes. We linearize the conservation equations for
small perturbations of fluid variables around the hydrostatic equilibrium,
$\psi(t,x)=\psi_{0}+\delta\psi(t,x)$, with the fluctuations expressed in their
plane wave solutions via a Fourier transformation $\delta\psi(t,x)\rightarrow
e^{i(\omega t-kx)}\delta\psi(\omega,k)$, (subscript $0$ indicates global
equilibrium). The background fluid is now considered to be boosted along
x-axis with a constant velocity v, $u^{\mu}_{0}=\gamma(1,\textbf{v},0,0)$ with
$\gamma=1/\sqrt{1-\textbf{v}^{2}}$. The corresponding velocity fluctuation is
$\delta u^{\mu}=(\gamma\textbf{v}\delta u^{x},\gamma\delta u^{x},\delta
u^{y},\delta u^{z})$ which again gives $u^{\mu}_{0}\delta u_{\mu}=0$ to
maintain velocity normalization. The dispersion relation for linear
perturbations can be obtained in the boosted frame by giving the
transformations, $\omega\rightarrow\gamma(\omega-k\textbf{v})$ and
$k^{2}\rightarrow\gamma^{2}(\omega-k\textbf{v})^{2}-\omega^{2}+k^{2}$ to the
local rest frame. The resulting dispersion relation polynomial turns out to be
too mathematically cumbersome (as expected) and it is only possible to extract
the modes in limiting situations. The transverse or shear channel at low wave
number limit $(k\rightarrow 0)$ gives only one non-hydrodynamic (gapped)
frequency mode,

$\omega^{\perp}=i\frac{(\epsilon_{0}+P_{0})}{\gamma(\theta-\eta\textbf{v}^{2})}+{\cal{O}}(k)~{},$
(22)

where we define $\theta=-\theta_{1}$. The other one is a hydrodynamic
(gapless) mode that vanishes at spatially homogeneous limit ($k=0$). Hence,
from the criteria that the imaginary part of frequency must be positive
definite to give rise to exponentially decaying perturbations, the stability
condition for the shear channel becomes $\theta>\eta\textbf{v}^{2}$. Here, we
recall the asymptotic causality condition for the shear channel in Lorentz
rest frame (LRF) was $\theta>\eta$ [13] which readily reproduces the stability
condition for all $0<\textbf{v}<1$. With boosted background, the group
velocity of the propagating shear mode becomes
$v_{g}^{\perp}=\lim_{k\rightarrow\infty}\bigg{|}\frac{\partial\textrm{~{}Re}({\omega^{\perp}})}{\partial
k}\bigg{|}=(\textbf{v}+\sqrt{\eta/\theta})/(1+\textbf{v}\sqrt{\eta/\theta})$,
which is subluminal as long as the asymptotic causality condition in LRF is
satisfied.

Figure 1: Estimating the value of $v_{g}^{\parallel}$ at LRF with different
$\Lambda$ and in different frame situations. $\textbf{x}=1$ corresponds to the
speed of light in vacuum.

In longitudinal or sound channel the situation becomes quite involved since
the number of modes significantly increases. At the asymptotic limit
$k\rightarrow\infty$, the dispersion relation becomes,

$\displaystyle
H_{6}(\omega^{\parallel})^{6}+H_{5}(\omega^{\parallel})^{5}k+H_{4}(\omega^{\parallel})^{4}k^{2}+H_{3}(\omega^{\parallel})^{3}k^{3}$
$\displaystyle+H_{2}(\omega^{\parallel})^{2}k^{4}+H_{1}(\omega^{\parallel})k^{5}+H_{0}k^{6}=0~{}.$
(23)

The coefficients $H_{0,\cdots,6}$ are elaborate functions of frame indices
$i,j,k$, the MDRTA parameter $\Lambda$ and the boost velocity v. We will give
here the explicit expressions of only those coefficients which are exclusively
required for establishing causality and stability. In the limit of large $k$,
an expansion of the form,
$\omega^{\parallel}=v_{g}^{\parallel}k+\sum_{n=0}^{\infty}c_{n}k^{-n}$ can be
used as a solution [18]. The values of the asymptotic group velocity
$v_{g}^{\parallel}$ are then given by the sixth order polynomial,
$H_{6}(v_{g}^{\parallel})^{6}+H_{5}(v_{g}^{\parallel})^{5}+H_{4}(v^{\parallel}_{g})^{4}+H_{3}(v^{\parallel}_{g})^{3}+H_{2}(v_{g}^{\parallel})^{2}+H_{1}v_{g}^{\parallel}+H_{0}=0.$
Here, for simplicity, in Fig.(1), we extract the value of $v_{g}^{\parallel}$
in LRF for two different general frames, ($i,j,k$)=($4,3,2$) and
($i,j,k$)=($6,4,4$), for which the sixth order polynomial reduces to a cubic
equation,
$(\textbf{x}^{2})^{3}+X_{2}(\textbf{x}^{2})^{2}+X_{1}(\textbf{x}^{2})+X_{0}=0$
with $X_{2}=-A_{4}^{2}/A_{6},X_{1}=A_{2}^{4}/A_{6},X_{0}=-A_{0}^{6}/A_{6}$,
whose roots will give the values of $v_{g}^{\parallel}$. Now, if the
discriminant of this equation,
$\Delta(=18X_{2}X_{1}X-0-4X_{2}^{3}X_{0}+X_{2}^{2}X_{1}^{2}-4X_{1}^{3}-27X_{0}^{2})>0$,
we will have three distinct real roots, while for $\Delta<0$, we have only one
real root and two complex conjugate roots. For small values of $\Lambda$,
($\Lambda<1.2$ for $(i,j,k)=(4,3,2)$ and $\Lambda<0.6$ for $(i,j,k)=(6,4,4)$),
we get three real roots of $\textbf{x}^{2}$, among which two are always
negative and one is always greater than one. So, that range fully excludes any
subluminal root for $v_{g}^{\parallel}$. For the range of $\Lambda$ shown in
Fig.(1), we have only one real, positive root for both the cases. We observe
that increasing $\Lambda$ corresponds to smaller $v_{g}^{\parallel}$ that
eventually becomes subluminal. It can be seen, that the critical value of
$\Lambda(=\Lambda_{c})$ for reaching the asymptotic causality condition is,
$\Lambda_{c}=6.65$ for frame $(i,j,k)=(4,3,2)$ and $\Lambda_{c}=2.9$ for frame
$(i,j,k)=(6,4,4)$. Hence, the theory is causal for $\Lambda>\Lambda_{c}$. So
we conclude, that more general the frame situation becomes (farther away from
Landau or Eckart frame where we can not find any subluminal $v_{g}$ for any
$\Lambda$ value), the system reaches causality with smaller values of
$\Lambda$. But that never reaches $\Lambda=0$ (momentum independent RTA)
whatever large frame indices are. It can be shown, that for the causal
parameter sets shown in Fig.(1), the theory is asymptotically causal for any
boosted frame as well, i.e, the original sixth order polynomial always has a
subluminal root, as suggested by the principle of relativity.

Figure 2: Island of stability for different boost velocities and different
$\Lambda$ values.

Next we turn to the stability analysis of the longitudinal modes. The
dispersion relation at $k\rightarrow 0$ limit turns out to be,

$\displaystyle
G_{6}(i\omega^{\parallel})^{6}+G_{5}(i\omega^{\parallel})^{5}+G_{4}(i\omega^{\parallel})^{4}+G_{3}(i\omega^{\parallel})^{3}+$
$\displaystyle
G_{2}(i\omega^{\parallel})^{2}(ik)+G_{1}(i\omega^{\parallel})k^{2}+G_{0}ik^{3}=0~{}.$
(24)

Eq.(24) gives rise to three hydrodynamic and three nonhydrodynamic modes. The
hydro modes are given by, $\omega^{\parallel}=(a_{1,2,3})k$, where the group
velocities can be estimated from the equation,
$G_{3}a^{3}+G_{2}a^{2}-G_{1}a-G_{0}=0$. The non-hydro modes (growing ones give
rise to instability) are given by the following polynomial equation,

$G_{6}(\omega^{\parallel})^{3}-iG_{5}(\omega^{\parallel})^{2}-G_{4}\omega^{\parallel}+iG_{3}=0~{}.$
(25)

Using Routh-Hurwitz stability criteria, we find the following conditions for
stability of the non-hydro modes,

$G_{6},G_{5},G_{3}>0~{},G_{B2}=(G_{4}G_{5}-G_{3}G_{6})/G_{5}>0~{}.$ (26)

The coefficients are given by,
$G_{6}=\gamma^{6}(A_{6}-A_{4}^{2}\textbf{v}^{2}+A_{2}^{4}\textbf{v}^{4}-A^{6}_{0}\textbf{v}^{6}),~{}G_{5}=\gamma^{5}(A_{5}-A_{3}^{2}\textbf{v}^{2}+A_{1}^{4}\textbf{v}^{4}),~{}G_{4}=\gamma^{4}(A_{4}^{0}-A_{2}^{2}\textbf{v}^{2}+A_{0}^{4}\textbf{v}^{4})$
and $G_{3}=\gamma^{3}(A_{3}^{0}-A_{1}^{2}\textbf{v}^{2})$. The explicit
mathematical expressions of the $A$ functions in terms of ($i,j,k$) and
$\Lambda$, through which the frame situation and the momentum dependence of
the interaction cross section duly control the stability and causality of the
theory, are given in the Appendix.

Among the coefficients of (26),
$G_{3}=\gamma^{3}n_{0}(\epsilon_{0}+P_{0})(1-\textbf{v}^{2}c_{s}^{2})$ is
always positive where $c_{s}$ is the velocity of sound. In the parameter space
spanned by ($i,j,k$), $\Lambda$ and v, the positivity of $G_{6},G_{5}$ and
$G_{B2}$ ensures the stability of the system. In Fig.(2), system stability has
been displayed for two frame conditions, $(i,j,k)=(4,3,2)$ and
$(i,j,k)=(6,4,4)$, as a function of two independent parameters $\Lambda$ and
v. Here, the shaded areas indicate that the system stability holds for those
parameters sets. Here, we notice something very interesting. The $\Lambda$
values for each frame, beyond which the asymptotic causality condition holds,
$v_{g}<1$, ($\Lambda_{c}=6.65$ for $(i,j,k)=(4,3,2)$ and $\Lambda_{c}=2.9$ for
$(i,j,k)=(6,4,4)$, see Fig.(1)), the system is also stable irrespective of the
values of boost velocity as long as $0<\textbf{v}<1$. This is the main result
of this work. In the region where $v_{g}>1$, we observe that there is no
$\Lambda$ value for which the system is either stable or unstable for all
values of v, whereas beyond the $v_{g}=1$ line, the stability situation agrees
in all reference frames.

We now analyse the stability situation at large wave-number limit. With a
general boost velocity v, at large $k$, the shear channel dispersion relation
gives the following modes,

$\omega^{\perp}_{1,2}=k\frac{\bigg{(}\textbf{v}\pm\sqrt{\frac{\eta}{\theta}}\bigg{)}}{\bigg{(}1\pm\textbf{v}\sqrt{\frac{\eta}{\theta}}\bigg{)}}+i\frac{(\epsilon_{0}+P_{0})\bigg{[}1\pm\sqrt{\frac{\eta\textbf{v}^{2}}{\theta}}\bigg{]}}{2\gamma(\theta-\eta\textbf{v}^{2})}+{\cal{O}}(\frac{1}{k})~{}.$
(27)

Clearly, with $0<\textbf{v}<1$ and asymptotic causality condition
$\theta>\eta$, the imaginary part is always positive which ensures stability
of the shear modes. The situation at sound channel becomes somewhat non-
trivial. With general $k$ and v, it is a sixth-order polynomial of $\omega$,
with each coefficient itself as a polynomial over $k$, where the coefficient
of each power of $k$ is again a polynomial of v. This is only possible to
solve at limiting situations. Hence, we present here the solution upto the
order of $k$, i.e, $\omega^{\parallel}=a_{0}+a_{1}k$. $a_{0}$ is purely
imaginary with three roots whose positivity has been checked by Routh-Hurwitz
stability criteria in Eq.(25). We have checked that $a_{1}$ is a real quantity
which does not contribute to the stability of the theory. In view of the non-
trivial structure of the sound channel polynomial, we believe that this result
might suffice for a long-wavelength effective theory such as relativistic
hydrodynamics. It is in principle important to analyse also the larger $k$
limit, but in practice it is beyond the possibilities of the current work.

So, from this analysis we can safely conclude here that stability is a Lorentz
invariant property if and only if the signal propagation within the medium is
causal. Our results corroborates with the finding of [12], where the authors
addressed the physical origin of the relation between “causality” and
“instability” using space-like events connected via perturbation mediating
through a dissipative medium. Fig.(2) gives a very clear pictorial
representation of similar argument for the BMR theory, where we have
established that for a first order relativistic dissipative system in general
frame and with momentum dependent interaction cross section, the principle of
causality is requisite for the stability to be a Lorentz invariant property.

##  4 Conclusion

The idea is that, causality violation can chronologically reorder a
perturbation in different reference frames by the relativity of simultaneity
so that in a dissipative medium, two observers can disagree on whether the
perturbation is growing or decaying. This gives rise to instability in the
medium, so causality is the key signature in this paradox. This phenomenon has
been explicitly demonstrated in this work, where all the stability controlling
factors, such as frame indices and interaction parameters, give rise to a
stable system at all inertial frames iff they respect the asymptotic causality
for the system. This undeniably reduces the task of checking stability in all
the reference frames, and the local rest frame suffices for the analysis. This
is a much desirable situation since, with a boosted background, the dispersion
relation becomes way more complicated than in the local rest frame. Most
importantly, this work singularly establishes the physical argument and the
associated theorems for a coarse-grained system with momentum-dependent
microscopic interactions from the first principle calculation. The analysis of
causality in this work is based on the asymptotic causality condition, which
states that the asymptotic group velocity
$v_{g}=\lim_{k\rightarrow+\infty}\frac{d\operatorname{Re}\omega(k)}{dk}$ must
be subluminal (i.e. $v_{g}\leq 1$). However, there are examples of acausal
theories with subluminal group velocity, such as the diffusion equation:
$\partial_{t}T=\partial_{x}^{2}T$, with dispersion relation $\omega(k)=ik^{2}$
and $v_{g}=0\leq 1$. And yet, we know that the diffusion equation is acausal.
We note that the “asymptotic causality condition” is a necessary condition for
causality, but not sufficient [19]. A rigorous study of causality requires the
study of characteristics and ensuring the strong hyperbolicity condition,
which is beyond the scope of the present work.

##  5 Appendix

The $A$ functions mentioned in the main text are given by :

$\displaystyle A_{6}=F_{13}(F_{3}F_{6}-F_{1}F_{8})~{},$ (28) $\displaystyle
A_{5}=F_{1}F_{8}-F_{3}F_{6}-(\epsilon_{0}+P_{0})F_{1}F_{13}-n_{0}F_{8}F_{13}~{},$
(29) $\displaystyle
A_{4}^{0}=(\epsilon_{0}+P_{0})F_{1}+n_{0}F_{8}-n_{0}(\epsilon_{0}+P_{0})F_{13}~{},$
(30) $\displaystyle
A_{4}^{2}=F_{13}(F_{4}F_{6}-F_{2}F_{8})+F_{5}(F_{8}F_{11}-F_{6}F_{12})$
$\displaystyle~{}~{}~{}~{}-F_{1}(F_{9}F_{13}-F_{8}F_{14}-F_{10}F_{12})$
$\displaystyle~{}~{}~{}~{}-F_{3}(F_{10}F_{11}+F_{6}F_{14}-F_{7}F_{13})$ (31)
$\displaystyle A_{3}^{0}=n_{0}(\epsilon_{0}+P_{0})~{},$ (32) $\displaystyle
A_{3}^{2}=n_{0}(F_{8}F_{14}+F_{10}F_{12}-F_{9}F_{13}+F_{8}F_{11}-F_{6}F_{12})$
$\displaystyle+(\epsilon_{0}+P_{0})(F_{1}F_{14}+F_{1}F_{12}-F_{2}F_{13}-F_{3}F_{11}+F_{5}F_{11})$
$\displaystyle+F_{1}F_{9}+F_{2}F_{8}-F_{3}F_{7}-F_{4}F_{6}$
$\displaystyle+b(F_{1}F_{10}-F_{5}F_{6})+a(F_{5}F_{8}-F_{3}F_{10})~{},$ (33)
$\displaystyle A_{2}^{2}=(\epsilon_{0}+P_{0})(F_{1}b+F_{2}-aF_{3}+aF_{5})$
$\displaystyle~{}~{}~{}~{}+n_{0}(F_{9}+bF_{10}-bF_{6}+aF_{8})$
$\displaystyle~{}~{}~{}~{}+n_{0}(\epsilon_{0}+P_{0})(F_{11}+F_{12}+F_{14})~{},$
(34) $\displaystyle A_{2}^{4}=F_{2}(F_{8}F_{14}+F_{10}F_{12}-F_{9}F_{13})$
$\displaystyle~{}~{}~{}~{}+F_{4}(F_{7}F_{13}-F_{10}F_{11}-F_{6}F_{14})$
$\displaystyle~{}~{}~{}~{}+F_{5}(F_{9}F_{11}-F_{7}F_{12})+F_{14}(F_{1}F_{9}-F_{3}F_{7})~{},$
(35) $\displaystyle A_{1}^{2}=n_{0}(\epsilon_{0}+P_{0})(a+b)~{},$ (36)
$\displaystyle
A_{1}^{4}=(F_{2}F_{9}-F_{4}F_{7})+n_{0}(F_{9}F_{14}+F_{9}F_{11}-F_{7}F_{12})$
$\displaystyle~{}~{}~{}~{}+(\epsilon_{0}+p_{0})(F_{2}F_{12}+F_{2}F_{14}-F_{4}F_{11})$
$\displaystyle~{}~{}~{}~{}+a(F_{5}F_{9}-F_{4}F_{10})+b(F_{2}F_{10}-F_{5}F_{7})~{},$
(37) $\displaystyle
A_{0}^{4}=(\epsilon_{0}+P_{0})(bF_{2}-aF_{4})+n_{0}(aF_{9}-bF_{7})~{},$ (38)
$\displaystyle A_{0}^{6}=F_{14}(F_{2}F_{9}-F_{4}F_{7})~{}.$ (39)

These $A$’s are the functions of the first order field correction coefficients
(given in Eq.(2) and (3) of the main text) as the following,

$\displaystyle
F_{1}=\nu_{1}f+\nu_{3}c~{},~{}~{}~{}~{}~{}F_{2}=\gamma_{1}f+\gamma_{3}c~{},$
(40) $\displaystyle
F_{3}=\nu_{1}g+\nu_{3}d~{},~{}~{}~{}~{}~{}F_{4}=\gamma_{1}g+\gamma_{3}d~{},$
(41) $\displaystyle
F_{5}=\nu_{2}-\gamma_{1}~{},~{}~{}~{}~{}~{}~{}~{}~{}F_{6}=\varepsilon_{1}f+\varepsilon_{3}c~{},$
(42) $\displaystyle
F_{7}=\theta_{1}f+\theta_{3}c~{},~{}~{}~{}~{}~{}F_{8}=\varepsilon_{1}g+\varepsilon_{3}d~{},$
(43) $\displaystyle
F_{9}=\theta_{1}g+\theta_{3}d~{},~{}~{}~{}~{}~{}F_{10}=\varepsilon_{2}-\theta_{1}~{},$
(44) $\displaystyle
F_{11}=[f(\pi_{1}-\theta_{1})+c(\pi_{3}-\theta_{3})]/(\epsilon_{0}+P_{0})~{},$
(45) $\displaystyle
F_{12}=[g(\pi_{1}-\theta_{1})+d(\pi_{3}-\theta_{3})]/(\epsilon_{0}+P_{0})~{},$
(46) $\displaystyle F_{13}=\theta_{1}/(\epsilon_{0}+P_{0})~{},$ (47)
$\displaystyle
F_{14}=(4\eta/3)/(\epsilon_{0}+P_{0})-\pi_{2}/(\epsilon_{0}+P_{0})~{}.$ (48)

Here, we have used, $a=\frac{n_{0}}{(\epsilon_{0}+P_{0})}(\frac{\partial
P}{\partial n})_{\epsilon}~{},~{}b=(\frac{\partial
P}{\partial\epsilon})_{n}~{},~{}c=J_{0}I_{3}/(I_{2}^{2}-I_{1}I_{3})~{},~{}d=-J_{1}I_{2}/(I_{2}^{2}-I_{1}I_{3})~{},~{}f=-J_{0}I_{2}/(I_{2}^{2}-I_{1}I_{3})$
and $g=J_{1}I_{1}/(I_{2}^{2}-I_{1}I_{3})$. We readily identify that,
$c_{s}^{2}=a+b$. The moment integrals used here are defined as, $I_{n}=\int
dF_{p}\tilde{E}_{p}^{n},~{}\Delta^{\mu\nu}J_{n}=\int
dF_{p}\tilde{p}^{\langle\mu\rangle}\tilde{p}^{\langle\nu\rangle}\tilde{E}_{p}^{n}$,
and $\Delta^{\alpha\beta\mu\nu}K_{n}=\int
dF_{p}\tilde{p}^{\langle\mu}\tilde{p}^{\nu\rangle}\tilde{p}^{\langle\alpha}\tilde{p}^{\beta\rangle}\tilde{E}_{p}^{n}$.
$z=m/T$ is the scaled rest mass of the particles.

Next, we give the explicit expressions of the first order field correction
coefficients (scaled by $\tau_{R}^{0}$) in terms of the frame indices
($i,j,k$) and MDRTA parameter $\Lambda$, using the notation
$\mathcal{D}_{i,j}^{m,n}=I_{i+m}I_{j+n}-I_{i+n}I_{j+m}$ :

$\displaystyle\nu_{1}=-TI_{\Lambda+2}+\frac{\partial
n}{\partial\tilde{\mu}}\frac{{\cal{D}}_{i,j}^{\Lambda+1,1}}{{\cal{D}}_{i,j}^{0,1}}+T\frac{\partial
n}{\partial T}\frac{{\cal{D}}_{i,j}^{\Lambda+1,0}}{{\cal{D}}_{i,j}^{1,0}}~{},$
(49)
$\displaystyle\nu_{2}=-T\left[\frac{1}{3}I_{\Lambda+2}-\frac{z^{2}}{3}I_{\Lambda}\right]+\frac{\partial
n}{\partial\tilde{\mu}}\left[\frac{1}{3}\frac{{\cal{D}}_{i,j}^{\Lambda+1,1}}{{\cal{D}}_{i,j}^{0,1}}-\frac{z^{2}}{3}\frac{{\cal{D}}_{i,j}^{\Lambda-1,1}}{{\cal{D}}_{i,j}^{0,1}}\right]$
$\displaystyle~{}~{}~{}~{}+T\frac{\partial n}{\partial
T}\left[\frac{1}{3}\frac{{\cal{D}}_{i,j}^{\Lambda+1,0}}{{\cal{D}}_{i,j}^{1,0}}-\frac{z^{2}}{3}\frac{{\cal{D}}_{i,j}^{\Lambda-1,0}}{{\cal{D}}_{i,j}^{1,0}}\right]~{},$
(50) $\displaystyle\nu_{3}=-TI_{\Lambda+1}+\frac{\partial
n}{\partial\tilde{\mu}}\frac{{\cal{D}}_{i,j}^{\Lambda,1}}{{\cal{D}}_{i,j}^{0,1}}+T\frac{\partial
n}{\partial T}\frac{{\cal{D}}_{i,j}^{\Lambda,0}}{{\cal{D}}_{i,j}^{1,0}}~{},$
(51)
$\displaystyle\varepsilon_{1}=-T^{2}I_{\Lambda+3}+\frac{\partial\epsilon}{\partial\tilde{\mu}}\frac{{\cal{D}}_{i,j}^{\Lambda+1,1}}{{\cal{D}}_{i,j}^{0,1}}+T\frac{\partial\epsilon}{\partial
T}\frac{{\cal{D}}_{i,j}^{\Lambda+1,0}}{{\cal{D}}_{i,j}^{1,0}}~{},$ (52)
$\displaystyle\varepsilon_{2}=-T^{2}\left[\frac{1}{3}I_{\Lambda+3}-\frac{z^{2}}{3}I_{\Lambda+1}\right]+\frac{\partial\epsilon}{\partial\tilde{\mu}}\left[\frac{1}{3}\frac{{\cal{D}}_{i,j}^{\Lambda+1,1}}{{\cal{D}}_{i,j}^{0,1}}-\frac{z^{2}}{3}\frac{{\cal{D}}_{i,j}^{\Lambda-1,1}}{{\cal{D}}_{i,j}^{0,1}}\right]$
$\displaystyle~{}~{}~{}~{}+T\frac{\partial\epsilon}{\partial
T}\left[\frac{1}{3}\frac{{\cal{D}}_{i,j}^{\Lambda+1,0}}{{\cal{D}}_{i,j}^{1,0}}-\frac{z^{2}}{3}\frac{{\cal{D}}_{i,j}^{\Lambda-1,0}}{{\cal{D}}_{i,j}^{1,0}}\right]~{},$
(53)
$\displaystyle\varepsilon_{3}=-T^{2}I_{\Lambda+2}+\frac{\partial\epsilon}{\partial\tilde{\mu}}\frac{{\cal{D}}_{i,j}^{\Lambda,1}}{{\cal{D}}_{i,j}^{0,1}}+T\frac{\partial\epsilon}{\partial
T}\frac{{\cal{D}}_{i,j}^{\Lambda,0}}{{\cal{D}}_{i,j}^{1,0}}~{},$ (54)
$\displaystyle\pi_{1}=-T^{2}\left[\frac{1}{3}I_{\Lambda+3}-\frac{z^{2}}{3}I_{\Lambda+1}\right]+\frac{\partial
P}{\partial\tilde{\mu}}\frac{{\cal{D}}_{i,j}^{\Lambda+1,1}}{{\cal{D}}_{i,j}^{0,1}}+T\frac{\partial
P}{\partial T}\frac{{\cal{D}}_{i,j}^{\Lambda+1,0}}{{\cal{D}}_{i,j}^{1,0}}~{},$
(55)
$\displaystyle\pi_{2}=-T^{2}\left[\frac{1}{9}I_{\Lambda+3}-\frac{2}{9}z^{2}I_{\Lambda+1}+\frac{1}{9}z^{4}I_{\Lambda-1}\right]$
$\displaystyle~{}~{}~{}~{}+\frac{\partial
P}{\partial\tilde{\mu}}\left[\frac{1}{3}\frac{{\cal{D}}_{i,j}^{\Lambda+1,1}}{{\cal{D}}_{i,j}^{0,1}}-\frac{z^{2}}{3}\frac{{\cal{D}}_{i,j}^{\Lambda-1,1}}{{\cal{D}}_{i,j}^{0,1}}\right]$
$\displaystyle~{}~{}~{}~{}+T\frac{\partial P}{\partial
T}\left[\frac{1}{3}\frac{{\cal{D}}_{i,j}^{\Lambda+1,0}}{{\cal{D}}_{i,j}^{1,0}}-\frac{z^{2}}{3}\frac{{\cal{D}}_{i,j}^{\Lambda-1,0}}{{\cal{D}}_{i,j}^{1,0}}\right]~{},$
(56)
$\displaystyle\pi_{3}=-T^{2}\left[\frac{1}{3}I_{\Lambda+2}-\frac{z^{2}}{3}I_{\Lambda}\right]+\frac{\partial
P}{\partial\tilde{\mu}}\frac{{\cal{D}}_{i,j}^{\Lambda,1}}{{\cal{D}}_{i,j}^{0,1}}+T\frac{\partial
P}{\partial T}\frac{{\cal{D}}_{i,j}^{\Lambda,0}}{{\cal{D}}_{i,j}^{1,0}}~{},$
(57)
$\displaystyle\theta_{1}=-T^{2}\left[J_{\Lambda+1}+\frac{\epsilon_{0}+P_{0}}{T^{2}}\frac{J_{k+\Lambda}}{J_{k}}\right]~{},$
(58)
$\displaystyle\theta_{3}=-T^{2}\left[J_{\Lambda}+\frac{\epsilon_{0}+P_{0}}{T^{2}}\frac{J_{k+\Lambda-1}}{J_{k}}\right]~{},$
(59)
$\displaystyle\gamma_{1}=-T\left[J_{\Lambda}+\frac{n_{0}}{T}\frac{J_{k+\Lambda}}{J_{k}}\right]~{},$
(60)
$\displaystyle\gamma_{3}=-T\left[J_{\Lambda-1}+\frac{n_{0}}{T}\frac{J_{k+\Lambda-1}}{J_{k}}\right]~{},$
(61) $\displaystyle\eta=\frac{1}{2}T^{2}K_{\Lambda-1}~{}.$ (62)

These $14$ first order transport coefficients, through their dependence on
frame parameters and interaction parameter, critically control the stability
and causality analysis presented in the current work. For the analysis we have
used $T=300$ MeV and $m=300$ MeV.

Acknowledgements.$\textendash$ R.B. acknowledges the financial support from
SPS, NISER planned project RIN4001. S.M and V.R. acknowledges the financial
support from the Department of Atomic Energy,India.

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# Absorption by stringy black holes

Sérgio V. M. C. B. Xavier<EMAIL_ADDRESS>Luís C. B. Crispino
<EMAIL_ADDRESS>Programa de Pós-Graduação em Física, Universidade Federal do
Pará, 66075-110, Belém, Pará, Brazil.  Carolina L. Benone<EMAIL_ADDRESS>Campus Universitário Salinópolis, Universidade Federal do Pará, 68721-000,
Salinópolis, Pará, Brasil

###### Abstract

We investigate the absorption of planar massless scalar waves by a charged
rotating stringy black hole, namely a Kerr–Sen black hole. We compute
numerically the absorption cross section and compare our results with those of
the Kerr-Newman black hole, a classical general relativity solution. In order
to better compare both charged black holes, we define the ratio of the black
hole charge to the extreme charge as $Q$. We conclude that Kerr–Sen and Kerr-
Newman black holes have a similar absorption cross section, with the
difference increasing for higher values of $Q$.

##  I Introduction

The validity of general relativity (GR) in the regime of weak gravitational
fields is unquestionable, since it was extensively tested and verified [1, 2,
3]. In the last decade, following suitable technological advancements, tests
of gravity in the strong field regime became more recurrent. Observations such
as the first detection of gravitational waves (GW) by the LIGO and VIRGO
collaborations [4] and the recent results of the Event Horizon Telescope (EHT)
collaboration [5] made an object stand out as the protagonist of these
experiments: the black hole. Emerged in literature as solutions of Einstein’s
GR equations, black holes are astrophysical objects that have an event
horizon.

Studies about black holes started with the first exact solution of Einstein’s
field equations, namely the Schwarzschild solution [6], which describes a
spherically symmetric black hole parametrized only by its mass. Over the
years, other solutions emerged, as a charged [7, 8], a rotating [9] and a
charged and rotating generalization of the Schwarzschild solution [10].
Despite appearing for the first time in the context of GR, black holes are not
exclusive predictions of this theory. Other theories of gravity also present
such astrophysical objects as possible solutions to their field equations,
making black holes optimal candidates for investigating which gravitational
theory is realized in nature.

On the other hand, the search for a quantum gravity theory has also increased
along the years. In this context, one of the candidates for such theory is the
heterotic string theory. In the low-energy limit of this model of gravity,
there exists a charged rotating black hole solution, namely the Kerr-Sen (KS)
black hole [11], which has been attracting a lot of attention. Several studies
on the features of KS black holes were made, including shadow [12, 13],
Hawking radiation [14], absorption of charged particles and clouds [15, 16],
hidden symmetries [17], cosmic censorship [18, 19] and merger estimates [20].

Although it is expected that black holes in astrophysical scenarios do not
have significant electric charge, in certain contexts it is reasonable to
consider charged black holes [21]. Besides that, recent studies, as the one
fulfilled by Zhang [22], point out that the assumption of a charged rotating
black hole could explain some astrophysical phenomena.

Nowadays, the major branches of experimental search for black holes are GW
astronomy, shadow images and x-ray spectroscopy. All of these three branches
depend, in a sense, on the effect of the black hole in its environment. This
motivates the study of absorption and scattering of waves and particles by
black holes. There is a vast literature about this topic in many different
black hole scenarios (cf., e. g., [23, 24, 25, 26] and references therein).

Scalar fields stand out as the simplest proposed candidates to explain some
issues in physics, as, for instance, the abundance of dark matter [27] and the
strong-CP problem [28, 29]. Related to the latter, some attempts suggested the
axion hypothesis to elucidate the recent reported results of North American
Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration [30,
31]. Moreover, scalar waves have been extensively studied in the investigation
of absorption and scattering by black holes. [32, 33, 34]

We address the problem of the absorption of fields by a charged rotating black
hole in the context of heterotic string theory. We focus on analyzing the
absorption cross section of planar massless scalar waves by a KS black hole.
Due to the similarity between KS black holes and Kerr-Newman (KN) black holes,
the charged rotating black hole solution of GR, we shall compare our results
with those of Ref. [33].

The remainder of this paper is organized as follows. In Sec. II we review the
spacetimes of the charged rotating black holes treated in this paper, namely,
KS and KN black holes. In Sec. III we study the dynamics of a massless scalar
field impinging upon a KS black hole. In Sec. IV we use the partial wave
analysis to express the absorption cross section and investigate the low and
high-frequency regime. In Sec. V we exhibit a selection of our numerical
results of the absorption cross section for different values of the black hole
charge and spin. We compare our results with those obtained in the KN case.
Our final remarks are presented in Sec. VI. Throughout this paper we use
natural units ($c=G=\hbar=1$) and the metric signature ($+,-,-,-$).

##  II Charged rotating black holes

The KS black hole is a solution of the low-energy limit of the heterotic
string theory. The action that describes this theory in four dimensions, in
the Einstein frame, is the following [35]:

$\begin{split}S=\int
d^{4}x\sqrt{-g}\left(R-e^{-2\tilde{\Phi}}F_{\mu\nu}F^{\mu\nu}-2\partial_{\mu}\tilde{\Phi}\partial^{\mu}\tilde{\Phi}\right.\\\
\left.-\frac{1}{12}e^{-4\tilde{\Phi}}H_{k\mu\nu}H^{k\mu\nu}\right),\end{split}$
(1)

where $\tilde{\Phi}$ is the dilaton field and $H_{k\mu\nu}$ is a third-rank
tensor field defined as

$\displaystyle H_{k\mu\nu}\equiv$
$\displaystyle\partial_{k}B_{\mu\nu}+\partial_{\nu}B_{k\mu}+\partial_{\mu}B_{\nu
k}$ $\displaystyle-2\left(A_{k}F_{\mu\nu}+A_{\nu}F_{k\mu}+A_{\mu}F_{\nu
k}\right),$ (2)

with $B_{\mu\nu}$ being a second-rank antisymmetric tensor gauge field, known
as Kalb-Ramond field.

Written in Boyer-Lindquist coordinates $(t,r,\theta,\varphi)$, the line
element of the KS solution is:

$\displaystyle ds^{2}=$
$\displaystyle\left(1-\frac{2Mr}{\rho^{2}}\right)dt^{2}-\rho^{2}\left(\frac{dr^{2}}{\Delta}+d\theta^{2}\right)$
$\displaystyle+\frac{4Mra\sin^{2}\theta}{\rho^{2}}dtd\varphi$
$\displaystyle-\left[r\left(r+2d\right)+a^{2}+\frac{2Mra^{2}\sin^{2}\theta}{\rho^{2}}\right]\sin^{2}\theta\,d\varphi^{2},$
(3)

where

$\displaystyle d\equiv\frac{q^{2}_{\scriptscriptstyle KS}}{2M},$ (4)
$\displaystyle\Delta\equiv r\left(r+2d\right)-2Mr+a^{2},$ (5)
$\displaystyle\rho^{2}\equiv r\left(r+2d\right)+a^{2}\cos^{2}\theta.$ (6)

As can be seen from the action (1), there is a coupling between gravity and
other fields, such as the dilaton field and gauge fields. The solutions for
the non-gravitational fields of this spacetime are:

$\displaystyle\tilde{\Phi}=-\frac{1}{2}\ln\frac{\rho^{2}}{r^{2}+a^{2}\cos^{2}\theta},$
(7) $\displaystyle A_{\mu}dx^{\mu}=\frac{q_{\scriptscriptstyle
KS}}{\rho^{2}}r(dt-a\sin^{2}\theta d\varphi),$ (8) $\displaystyle
B_{t\varphi}=\frac{dra\sin^{2}\theta}{\rho^{2}}.$ (9)

The parameters $(M,q_{\scriptscriptstyle KS},a)$ are the mass, the electric
charge and the angular momentum per unity mass of the KS black hole,
respectively. The event horizon is localized at

$r_{h_{\scriptscriptstyle KS}}\equiv M-d+\sqrt{\left(M-d\right)^{2}-a^{2}}.$
(10)

In the context of GR theory, the black hole that we shall use to compare our
results is the charged rotating KN black hole. In Boyer–Lindquist coordinates,
its line element reads:

$\displaystyle ds^{2}=$ $\displaystyle\left(1-\frac{2Mr-q_{\scriptscriptstyle
KN}^{2}}{\rho^{2}_{KN}}\right)dt^{2}-\rho^{2}_{KN}\left(\frac{dr^{2}}{\Delta_{KN}}+d\theta^{2}\right)$
$\displaystyle+\frac{4Mar\sin^{2}\theta-2aq_{\scriptscriptstyle
KN}^{2}\sin^{2}\theta}{\rho^{2}_{KN}}dtd\varphi$
$\displaystyle-\frac{\left[(r^{2}+a^{2})^{2}-\Delta_{KN}a^{2}\sin^{2}\theta\right]\sin^{2}\theta}{\rho^{2}_{KN}}d\varphi^{2}.$
(11)

The only non-gravitational field is associated with the electromagnetic vector
potential, given by

$A_{\mu}dx^{\mu}=\frac{q_{\scriptscriptstyle
KN}}{\rho^{2}_{KN}}r(dt-a\sin^{2}\theta d\varphi),$ (12)

where

$\displaystyle\Delta_{KN}\equiv r^{2}-2Mr+a^{2}+q_{\scriptscriptstyle
KN}^{2},$ (13) $\displaystyle\rho^{2}_{KN}\equiv r^{2}+a^{2}\cos^{2}\theta.$
(14)

The KN black hole presents an event horizon at:

$r_{h_{\scriptscriptstyle KN}}\equiv M+\sqrt{M^{2}-a^{2}-q_{\scriptscriptstyle
KN}^{2}}.$ (15)

Both spacetimes have very similar geometrical aspects, but also present some
distinctive features. For instance, the KS solutions allow a larger range of
values of electric charge. Therefore, in order to compare our results, we
shall use a normalized charge

$\text{{Q}}_{(i)}=q_{(i)}/q^{ext}_{(i)},$ (16)

where $q^{ext}_{(i)}$ is the value of charge of an extreme black hole, with
the subscript $(i)=KS,KN$ indicating the corresponding spacetime.

##  III Scalar field in the KS background

The dynamics of a massless scalar field $\Psi(x^{\mu})$ in KS spacetime is
described by the Klein Gordon equation, which can be written in a covariant
form as:

$\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi)=0,$
(17)

where $g$ is the determinant of the KS metric, whose contravariant components
are $g^{\mu\nu}$.

In order to solve this equation, one can assume a separation form of $\Psi$ in
terms of wavelike solutions, as given by:

$\Psi=\sum_{l=0}^{+\infty}\sum_{m=-l}^{+l}\frac{U_{\omega
lm}(r)}{\sqrt{r(r+2d)+a^{2}}}S_{\omega lm}(\theta)e^{i(m\varphi-\omega t)}.$
(18)

The functions $S_{\omega lm}(\theta)$ are the oblate spheroidal harmonics,
111The KS solution belongs to a class of axisymmetric and asymptotically flat
spacetimes that allow the separation of variables of the Klein-Gordon equation
solutions as in Eq. (18), with the angular part being given in terms of the
spheroidal harmonics [36]. that satisfy the following equation [37]:

$\displaystyle\left(\frac{d^{2}}{d\theta^{2}}+\cot\theta\frac{d}{d\theta}\right)S_{\omega
lm}$
$\displaystyle+\left(\lambda_{lm}+a^{2}\omega^{2}\cos^{2}\theta-\frac{m^{2}}{\sin^{2}\theta}\right)S_{\omega
lm}=0,$ (19)

where $\lambda_{lm}$ are the eigenvalues of the spheroidal harmonics.

The radial part of the solution (18) satisfies a lengthy equation, which can
be reshaped in terms of the tortoise coordinate $r_{\star}$, defined, in the
KS spacetime, as

$r_{\star}\equiv\int dr\left(\frac{r(r+2d)+a^{2}}{\Delta}\right).$ (20)

With the aid of the tortoise coordinate, it is possible to rewrite the radial
equation in a simpler form:

$\left(\frac{d^{2}}{dr^{2}_{\star}}+V_{\omega lm}\right)U_{\omega
lm}(r_{\star})=0,$ (21)

where the function $V_{\omega lm}$ is given by:

$\displaystyle V_{\omega lm}=$
$\displaystyle-\frac{\Delta\left(a^{2}\left(-d^{2}-2dM+2dr-4Mr+r^{2}\right)+a^{4}+r\left(d^{2}(2M-r)-2d^{3}+2dMr+2Mr^{2}\right)\right)}{\left(a^{2}+2dr+r^{2}\right)^{4}}$
$\displaystyle-(\lambda_{lm}+a^{2}\omega^{2}-2ma\omega)\frac{\Delta}{\left(a^{2}+2dr+r^{2}\right)^{2}}+\frac{\left(a^{2}\omega-
am+2dr\omega+r^{2}\omega\right)^{2}}{\left(a^{2}+2dr+r^{2}\right)^{2}}.$ (22)

In Fig 1, we plot $V_{\omega lm}$ for different values of the normalized
charge $Q_{\scriptscriptstyle KS}$ (top panel) and rotation parameter $a$
(bottom panel), setting $M\omega=0.1$ and $l=m=1$ in both cases. One can
notice that $V_{\omega lm}$ is larger for higher values of the BH charge and
spin. In the asymptotic limit, $V_{\omega lm}$ tends to $\omega^{2}$, as in
the Kerr-Newman case [33].

Figure 1: $V_{\omega lm}$ as a function of $r$, for different values of the KS
black hole normalized charge (top panel) and different values of the KS black
hole angular momentum (bottom panel). For both panels we have chosen $l=m=1$
and $M\omega=0.1.$

When dealing with absorption and scattering phenomena, it is common to use a
set of independent solutions called in modes, which correspond to incoming
waves from the past null infinity, part of which is transmitted to the event
horizon while the remaining part is reflected to the future null infinity. In
the asymptotic limits (infinity and event horizon), these modes have the
following behavior:

$U_{\omega lm}(r_{\star})~{}\begin{cases}\mathcal{I}_{\omega
lm}U_{I}+\mathcal{R}_{\omega lm}U^{*}_{I}\hskip
8.5359pt(r_{\star}/M\rightarrow+\infty),\\\ \mathcal{T}_{\omega lm}U_{T}\hskip
54.06006pt(r_{\star}/M\rightarrow-\infty),\end{cases}$ (23)

with the symbol $*$ denoting complex conjugation. The terms $U_{I}$ and
$U_{T}$ can be expanded as:

$\displaystyle U_{I}=e^{-i\omega r_{\star}}\sum_{j=0}^{N}\frac{h_{j}}{r^{j}},$
(24) $\displaystyle
U_{T}=e^{-i(\omega-m\Omega_{H})r_{\star}}\sum_{j=0}^{N}g_{j}(r-r_{h_{\scriptscriptstyle
KS}})^{j},$ (25)

where

$\Omega_{H}=\frac{a}{2Mr_{h_{\scriptscriptstyle KS}}}$ (26)

is the angular velocity of the event horizon. The coefficients $h_{j}$ and
$g_{j}$ are constants found by solving Eq. (21). The terms $\mathcal{I}$,
$\mathcal{R}$ and $\mathcal{T}$ are related to the incident, reflected and
transmitted parts of the wave, respectively. It can be show that they obey the
relation:

$\left|\frac{\mathcal{R}_{\omega lm}}{\mathcal{I}_{\omega
lm}}\right|^{2}=1-\frac{\omega-m\Omega_{H}}{\omega}\left|\frac{\mathcal{T}_{\omega
lm}}{\mathcal{I}_{\omega lm}}\right|^{2}.$ (27)

For $\omega<m\Omega$ the reflection coefficient is greater than 1,
corresponding to the superradiance phenomena. The equality $\omega=m\Omega$
also corresponds to the minimum value of frequency for which the black hole
can be overspun into a naked singularity [38].

##  IV Absorption cross section

Considering a scalar massless plane wave whose incidence angle is $\gamma$,
one can find the partial absorption cross section through partial wave methods
[39], as:

$\sigma_{lm}=\frac{4\pi^{2}}{\omega^{2}}|S_{\omega
lm}(\gamma)|^{2}\left(1-\left|\frac{\mathcal{R}_{\omega
lm}}{\mathcal{I}_{\omega lm}}\right|^{2}\right).$ (28)

The total absorption cross section is obtained as a sum of the partial
contributions $\sigma_{lm}$ over all allowed values of $l$ and $m$:

$\sigma=\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}\sigma_{lm}.$ (29)

It is possible to decompose the total absorption cross section in a sum of
corotating modes, given by

$\sigma^{co}=\sum_{l=1}^{\infty}\sum_{m=1}^{l}\sigma_{lm},$ (30)

and counterrotating modes, given by

$\sigma^{counter}=\sum_{l=1}^{\infty}\left(\sigma_{l0}+\sum_{m=1}^{l}\sigma_{l(-m)}\right).$
(31)

###  IV.1 Low-frequency regime

There are some analytical approximations in the literature for the absorption
cross sections that allow us to test the accuracy of our numerical results.
For the low-frequency regime, Higuchi found in 2001 [40] that the absorption
cross section of a stationary and circularly symmetric black hole with any
dimensions tends to the area of the event horizon.

The event horizon area of the KS black hole is given by

$A_{h}=\int_{0}^{\pi}\int_{-\pi}^{\pi}\sqrt{g_{\theta\theta}g_{\varphi\varphi}}d\varphi
d\theta=4\pi\left(r_{h_{\scriptscriptstyle
KS}}^{2}+a^{2}+2dr_{h_{\scriptscriptstyle KS}}\right).$ (32)

As we shall see in Sec. V, our numerical results are in excellent agreement
with the corresponding low-frequency approximation.

###  IV.2 High-frequency regime

In the high-frequency regime, the absorption cross section is intimately
related with the properties of the photon orbits. This relation is very well
represented by the so-called sinc approximation. Firstly proposed by Sanchez
[41], for the Schwarzschild black hole, this approximation was later derived
using complex angular momentum techniques [42] (c. f. also [32, 43]). One can
show that:

$\sigma\approx\sigma_{geo}\left[1-\frac{8\pi\beta
e^{-\pi\beta}}{\Omega_{N}^{2}b^{2}_{c}}\text{sinc}\left(\frac{2\pi\omega}{\Omega_{N}}\right)\right],$
(33)

where $\beta\equiv\Lambda/\Omega_{N}$, with $\Lambda$ being the Lyapunov
exponent, $\Omega_{N}$ the orbital frequency for the null orbit, $b_{c}$ the
critical impact parameter, and $\sigma_{geo}\equiv\pi b_{c}^{2}$. For the on-
axis case of the KS black hole, we have:

$r_{c}=M-d+2\sqrt{M^{2}-\left(a^{2}+4Md+2d^{2}\right)/3}\cos\left[\frac{1}{3}\arccos\left[\frac{M\left((M-d)^{2}-a^{2}\right)}{\left(M^{2}-\left(a^{2}+4Md+2d^{2}\right)/3\right)^{3/2}}\right]\right],$
(34)
$b_{c}^{2}=\frac{2(r_{c}+d)\left[r_{c}(r_{c}+2d)+a^{2}\right]}{r_{c}+d-M},$
(35)
$\Lambda=\frac{4\sqrt{2a^{2}-b_{c}^{2}+4d^{2}+12dr_{c}+6r^{2}}}{T_{0}\sqrt{\mathcal{K}}}K\left(-\frac{a^{2}}{\mathcal{K}}\right),$
(36)
$T_{0}=\frac{4}{\sqrt{\mathcal{K}}}\left[\mathcal{K}E\left(-\frac{a^{2}}{\mathcal{K}}\right)+\left(\frac{(r_{c}(r_{c}+2d)+a^{2})^{2}}{\Delta}-(a^{2}+\mathcal{K})\right)K\left(-\frac{a^{2}}{\mathcal{K}}\right)\right],$
(37)

where $K(k)$ and $E(k)$ are the complete elliptic integrals of the first and
second kind, respectively,
$\mathcal{K}\equiv\mathcal{C}-(a-\mathcal{L}_{z})^{2}$ is a Carter-like
constant, $\mathcal{L}_{z}\equiv L_{z}/E$ is the azimuthal angular momentum
per unit energy, $\mathcal{C}$ is a constant of motion, $r_{c}$ is the
critical radius and $T_{0}$ is the latitudinal period.

As we shall see in Sec. V, our numerical results are in excellent agreement
with the sinc approximation.

##  V Results

In this section we show a selection of our results for the total absorption
cross section of the KS black hole in the Einstein frame for different values
of the incidence angle $\gamma$ of the scalar wave.

In Fig. 2, we exhibit the total absorption cross section for polar incidence
($\gamma=0$) with fixed values of charge (top panel) and fixed values of the
rotation parameter (bottom panel). One can notice that as the values of charge
and angular momentum increase, the total absorption cross section decreases.
Furthermore, one can see a regular oscillatory behavior around the high-
frequency limit.

Figure 2: Total absorption cross section for fixed charge (top) and angular
momentum per unity mass (bottom) of the KS black hole. Both panels are plotted
for polar incidence ($\gamma=0$).

In Fig. 3 we compare our numerical results for polar incidence with the
corresponding ones obtained via the sinc approximation, showing excellent
agreement.

Figure 3: Sinc approximation for the on-axis ($\gamma=0$) case. The solid
lines are the numerically computed absorption cross sections and the dashed
lines represent the sinc approximation.

Illustrated by Fig. 4, the equatorial incidence case ($\gamma=90$ deg) reveals
a distinctive behavior when compared with in the on-axis case. One can
identify an oscillatory pattern less regular than the one exhibited in Fig. 2.
On the top panel the total absorption cross section is plotted for fixed value
of the KS black hole charge ($q_{\scriptscriptstyle KS}/M=0.3$) and $a/M=0.3$,
$0.6$ and $0.9$, while in the bottom panel we consider KS black holes with
same angular momentum ($a/M=0.9$) and different values of the charge, namely
$q_{\scriptscriptstyle KS}/M=0.1$, $0.3$ and $0.5$. Note that although the
increase of spin and charge leads generally to a smaller absorption cross
section, the plots in the bottom panel preserves its essential shape, what
does not occur in the top panel.

Figure 4: Total absorption cross section for different values of the rotation
parameter (top) and electric charge (bottom), plotted for equatorial
incidence.

In Fig. 5 we show the total absorption cross section for different choices of
the scalar wave incidence angle. One can note that as we move away from the
on-axis case ($\gamma=0$ deg), the behavior of the absorption cross section is
less regular.

Figure 5: Total absorption cross section of a KS black hole with
$q_{\scriptscriptstyle KS}/M=0.3$ and $a/M=0.9$ for different values of the
incidence angle.

Superradiance is associated to the amplification of the reflected waves, what
in this context is characterized by a negative partial absorption cross
section. In Fig. 6 we plot the partial absorption cross section for $l=m=1$,
highlighting the negative values in the insets as a clear manifestation of the
superradiance phenomenon.

Figure 6: Partial absorption cross section with $l=m=1$, for fixed angular
momentum per unity mass (top) and normalised charge (bottom). Both panels are
plotted for equatorial incidence ($\gamma=90$ deg). The insets emphasize the
occurrence of superradiance.

We plot the absorption cross section divided by the horizon area of KN and KS
black holes in Fig. 7, for the on-axis ($\gamma=0$ deg) and equatorial
incidence ($\gamma=90$ deg) cases. We see that, in general, the KS black hole
has a larger total absorption cross section than the corresponding KN case.

Figure 7: Comparison between the ratio of absorption cross section and area of
the black holes as a function of the normalised charge
$\text{{Q}}_{(i)}=q_{(i)}/q^{ext}_{(i)}$. The top panel corresponds to a wave
with polar incidence ($\gamma=0$ deg) whereas the bottom panel corresponds to
a wave with equatorial incidence ($\gamma=90$ deg).

This is in agreement with the fact that the function $V_{\omega lm}$ of the KS
spacetime is generally smaller than the corresponding one of the KN spacetime.
222We notice that for some low-frequency values and black holes close to
extremality, the KN potential can be smaller than the KS one. We also notice
that in the low-frequency limit the absorption cross section agrees with the
result of Ref. [40], i.e., tends to the area of the black hole horizon.

In Fig. 8 we plot the ratio between the absorption cross section and the areas
of the KS and KN black holes, namely
$(\sigma_{KN}/A_{KN})/(\sigma_{KS}/A_{KS})$. We see that this quantity is very
close to unity, implying that the difference between the absorption cross
section of both black holes is very small. Furthermore, we see that, in
general, this quantity is larger for KS black holes than for KN black holes.

Figure 8: Ratio between the absorption cross section and the areas of the
black holes for a wave with incidence angle $\gamma=0$ deg (top), $\gamma=45$
deg (middle) and $\gamma=90$ deg (bottom).

##  VI Final remarks

We computed numerically the absorption cross section for KS black holes. We
have shown that, for the on-axis case ($\gamma=0$), there is a well behaved
oscillatory pattern, resulting from the contributions of waves with different
angular momentum, $l$. Besides that, the increase of the parameters of the
black hole (charge and angular momentum), decreases the absorption cross
section.

For the off-axis case, it can be noted that a less well behaved pattern is
present. This is a feature usually seen in stationary and axisymmetric
spacetimes [32, 33]. It is explained by the distinct contributions coming from
the co- and counterrotating modes. Moreover, the absorption cross section
becomes more irregular for higher values of the black hole charge and angular
momentum.

When the condition $\omega<m\Omega_{H}$ is obeyed, the partial absorption
cross section is negative. This is due to the superradiant scattering, which
implies in a reflection coefficient greater than 1. For the scalar case, this
effect is more pronounced for the $l=m=1$ wave [44]. Although the partial
absorption cross section can be negative, the total absorption cross section
remains positive, due to the fact that to compute the total absorption cross
section we sum over both the positive and negative $m$ contributions.

In order to compare the absorption cross section of the KS black hole with the
one of a KN black hole, we defined the normalized charge Q. We have shown that
both spacetimes have qualitatively similar behaviors of the massless
absorption cross section, although the KS case generally presents a larger
absorption cross section than the KN case, for the same values of the pair
($\omega M,\text{{Q}}$). The difference between the two cases increases as the
black holes approach the extremality, albeit being very small. A small
difference between physical quantities associated to KN and KS black holes
with the same normalized charge is also manifest in the case of their shadows
[17, 20, 13, 45].

The electromagnetic sector of the Kerr-Sen solution is more involved than the
Kerr-Newman case, due to the interaction with additional fields present in the
former case. It would be interesting to analyze the case of the incidence of
an electrically charged scalar field, and the absorption cross section will
depend on whether the electromagnetic interaction between the black hole and
the field is attractive or repulsive.

###### Acknowledgements.

The authors thank Fundação Amazônia de Amparo a Estudos e Pesquisas (FAPESPA),
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) - Finance
Code 001, in Brazil, for partial financial support. We are grateful to Carlos
Herdeiro for useful discussions. This work has further been supported by the
European Union’s Horizon 2020 research and innovation (RISE) programme
H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.

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