# Chaos and fast scrambling delays of dyonic Kerr-Sen-AdS4 black hole and its
ultra-spinning version

Hadyan Luthfan Prihadi<EMAIL_ADDRESS>Theoretical High Energy
Physics Group, Department of Physics, Institut Teknologi Bandung, Jl. Ganesha
10 Bandung, Indonesia.  Indonesia Center for Theoretical and Mathematical
Physics (ICTMP), Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, 40132,
Indonesia.  Freddy Permana Zen<EMAIL_ADDRESS>Theoretical High Energy
Physics Group, Department of Physics, Institut Teknologi Bandung, Jl. Ganesha
10 Bandung, Indonesia.  Indonesia Center for Theoretical and Mathematical
Physics (ICTMP), Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, 40132,
Indonesia.  Donny Dwiputra<EMAIL_ADDRESS>Research Center for
Quantum Physics, National Research and Innovation Agency (BRIN) South
Tangerang 15314, Indonesia.  Theoretical High Energy Physics Group, Department
of Physics, Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, Indonesia.
Indonesia Center for Theoretical and Mathematical Physics (ICTMP), Institut
Teknologi Bandung, Jl. Ganesha 10 Bandung, 40132, Indonesia.  Seramika
Ariwahjoedi<EMAIL_ADDRESS>Asia Pacific Center for Theoretical Physics
Pohang University of Science and Technology Pohang 37673, Gyeongsangbuk-do,
South Korea  Research Center for Quantum Physics, National Research and
Innovation Agency (BRIN) South Tangerang 15314, Indonesia.

###### Abstract

The scrambling time and its delay are calculated using holography in an
asymptotically AdS black hole solution of the gauged Einstein-Maxwell-Dilaton-
Axion (EMDA) theory, the dyonic Kerr-Sen-AdS4 black hole, perturbed by
rotating and charged shock waves along the equator. The leading term of the
scrambling time for a black hole with large entropy is logarithmic in the
entropy and hence supports the fast scrambling conjecture for this black hole
solution, which implies that the system under consideration is chaotic. We
also find that the instantaneous minimal Lyapunov index is bounded by
$\kappa=2\pi T_{H}/(1-\mu\mathcal{L})$, which is analogous to the surface
gravity but for the rotating shock waves, and becomes closer to equality for
the near extremal black hole. For a small value of the AdS scale, we found
that the Lyapunov exponent can exceed the bound for a large value of
$\mathcal{L}$. Due to the presence of the electric and magnetic charge of the
shock waves, we also show that the scrambling process of this holographic
system is delayed by a time scale that depends on the charges of the shock
waves. The calculations also hold for the ultra-spinning version of this black
hole. The result of this paper generalizes the holographic calculations of
chaotic systems which are described by an EMDA theory in the bulk.

##  I Introduction

Due to their extreme properties, such as a strong gravitational field, black
holes provide a good environment for studying the quantum effects/signatures
of gravity. Previous studies of quantum gravity involve the study of string
theory and loop quantum gravity [1, 2]. One of the other ways to investigate
the quantum properties of gravity around a black hole is by understanding
chaotic phenomena of holographic systems that are dual to some black holes,
which has become an interesting topic until very recently [3, 4, 5, 6, 7, 8].
The study about chaotic phenomena of the 1-dimensional Sachdev-Ye-Kitaev (SYK)
model [9] which is conjectured to be dual to the 2-dimensional Jackiw-
Teitelboim (JT) gravity as the effective theory of a near-extremal black hole
near its horizon [10] also leads us to the understanding of a possibility to
study traversable wormholes in the lab [11, 12, 13]. According to the fast
scrambling conjecture [14], black holes are the fastest scrambler in nature,
with the scrambling time proportional to the logarithm of its degrees of
freedom or entropy, $t_{*}\sim\log S$, for systems with a large number of
degrees of freedom (see also [15, 16, 17]). A system can be considered chaotic
when the Out-Of-Time-Ordered Correlators (OTOC) decay exponentially. The time
required for the OTOC to vanish is known as the scrambling time, and for a
fast scrambler, it is logarithmic in entropy.
Another way to diagnose chaos in a quantum system is by using the Mutual
Information $I(A;B)=S_{A}+S_{B}-S_{A\cup B}$, where $S_{A}$ is the von Neumann
entropy for the reduced density matrix $\rho_{A}$ of a subsystem $A$. This can
be done since $I(A;B)$ provides an upper bound for correlators. If $I(A;B)=0$,
then the correlators also vanish [18], and the scrambling time can be obtained
from the time required for $I(A;B)$ to vanish. Using the help from holographic
theories such as the AdS/CFT correspondence [19], the Mutual Information can
be calculated using the holographic entanglement entropy, i.e. the Ryu-
Takayanagi (RT) or Hubeny-Rangamani-Takayanagi (HRT) formula [20, 21, 22].
Entanglement entropy of a subregion in a conformal field theory (CFT) with an
anti-de Sitter (AdS) dual is equal to the area of a minimal surface in the
bulk which is homologous to the region divided by $4G_{N}$. This can be used
to calculate the Mutual Information of a thermofield CFT which is dual to an
eternal black hole in the AdS bulk [23, 24]. Recent calculation of
entanglement entropy using replica trick in the bulk shows that it is also
possible to calculate entanglement entropy of black hole spacetimes with
multiple horizons [25].
The chaotic behavior of a black hole is first studied in [3], by calculating
the Mutual Information in a CFT which is dual to a three-dimensional Banadoz-
Teitelboim-Zanelli (BTZ) black hole (see also [26, 27]). A tiny perturbation
traveling at the speed of light which is sent from the left boundary at a very
early time and near the horizon can get highly blue-shifted and disrupt the
geometry to make the Mutual Information vanish. Such a perturbation is called
the gravitational shock waves which are represented by the Dray-’t Hooft
solution [28]. Adding the perturbation to the bulk geometry also corresponds
to disrupting the thermofield double state in the boundary CFT, causing the
correlator to vanish in late times. From [3], it is found that the scrambling
time for a black hole with large entropy is given by
$t_{*}=\frac{\beta}{2\pi}\log\mathcal{S}$, where $\mathcal{S}$ is the black
hole entropy and $\beta$ is the inverse Hawking temperature. Furthermore, the
Lyapunov exponent is found to saturate the Maldacena, Shenker, and Stanford
(MSS) bound on chaos [29], i.e. $\lambda_{L}=2\pi/\beta$. This observation
supports the fast scrambling conjecture, which states that a black hole is
maximally chaotic. The chaotic feature of black holes can be tested for other
more general black holes such as charged and rotating black holes. The charged
extension is first studied in [4], where the scrambling time for a Reissner-
Nordström-AdS black hole perturbed by neutral shock waves is calculated. It is
also found that the scrambling time is logarithmic in entropy. The calculation
is then generalized to charged and rotating BTZ black holes perturbed by
charged shocks [30]. Aside from the bulk calculations, the chaotic behavior of
black holes has also been studied from the CFT side.
A recent study [8] shows that the charged shell must bounce inside the horizon
and change its null trajectory for the shock wave solution to not violate the
null energy condition from the null-shell formalism [31, 32]. The calculation
is done for a four-dimensional Reissner-Nordström-AdS black hole perturbed by
charged shock waves. This has an important implication for interpreting the
scrambling time obtained. According to standard calculations, the effect of
charged shock waves is to increase the scrambling time by an extra factor that
depends on the charge of the shocks. However, since the shock waves bounce
inside the horizon, there is a maximum time $t_{b}$ for a null particle sent
from the right asymptotic to meet the shock waves inside the horizon. The
difference between $t_{b}$ and the time the shocks are sent is equal to the
difference between scrambling time with neutral shocks and charged ones plus
some terms in order of the thermal time $\beta$. Using quantum circuit
description, [8] concludes that the effect of the charged shock waves is to
delay the start of the scrambling process.
Other than the charged shock waves, one may also consider rotating shock
waves. This is first done in [5] for rotating BTZ black hole and in [6, 7] for
Kerr-AdS black hole in 4 and 5 dimensions. The importance of adding the
angular momentum to the shocks is that the Lyapunov exponent can be larger
than the MSS bound [33] due to the existence of a global conserved charge.
Furthermore, the solution survives the extremal limit even though the black
hole temperature becomes zero at extremality.
Aside from the standard Reissner-Nordström and Kerr black holes, there are
many known black hole solutions that can be described microscopically using
holographic theories [34, 35, 36, 37, 38, 39, 40, 41]. It is very interesting
to study the chaotic behavior of various black holes, particularly in an
asymptotically AdS spacetime, to give us a better understanding of the chaotic
properties of black holes or even holographic theories in general. It is shown
in [6] that the charged version (the Kerr-Newman-AdS) black hole does not
provide much difference compared to the uncharged Kerr-AdS black hole.
However, there is another black hole solution that is quite similar (to some
extent) to the Kerr-Newman-AdS black hole, but with extra dilaton and axion
scalar fields, named the dyonic Kerr-Sen-AdS4 black hole. The chaotic behavior
of such a black hole has not been studied before. Since the dilaton and axion
fields are present, the result will differ from the charged and rotating Kerr-
Newman-AdS black hole. Furthermore, it is widely known that rotating black
holes in AdS have an ultra-spinning limit–a limit where the rotational
parameter reaches its maximum value. It is also interesting to study the
chaotic behavior of rotating black holes in such a limit since it is known
[40] that an ultra-spinning version of a dyonic Kerr-Sen-AdS black hole may
still obey the Reverse Isoperimetric Inequality due to the existence of the
dilaton and axion fields. Charged and rotating perturbations, in addition to
the uncharged rotating perturbations, will also give a significant effect on
the chaotic dynamics of the black holes, and it also has not been studied
before, especially in a four-dimensional rotating and charged black holes with
dilaton and axion fields.
In this work, we extend the calculations for the scrambling time for charged
and rotating black holes in the gauged Einstein-Maxwell-Dilaton-Axion (EMDA)
theory perturbed by (electrically and magnetically) charged rotating shock
waves. The black hole solution to the gauged EMDA theory in 4 dimensions is
called the dyonic Kerr-Sen-AdS4 [42]. The gauged version of the theory is
equipped with a negative cosmological constant which provides us with an
asymptotically AdS spacetime. Therefore, it is suitable for studying chaos in
this black hole background using AdS/CFT. The eternal black hole form of this
dyonic Kerr-Sen-AdS4 black hole can also be represented holographically by two
CFTs in their left and right asymptotic boundaries similar to [23], with extra
chemical potentials which correspond to the rotation and (electric and
magnetic) charges. Studying the chaotic behavior of such a system also lead us
to an even more understanding of the holographic relation in the gauged EMDA
theory. Using the Kerr/CFT correspondence [43] and its extensions [34, 35, 36,
39], the extremal dyonic Kerr-Sen-AdS4 black hole entropy can be described
microscopically by its dual two-dimensional CFT [40, 41]. The dyonic Kerr-Sen-
AdS4 solution is parameterized by its mass, AdS scale, angular momentum,
electric and magnetic charges, and two extra charges which correspond to the
dilaton and axion fields.
Using holographic calculations, we found that the scrambling time of the black
hole is also logarithmic in entropy. The minimal instantaneous Lyapunov
exponent $\lambda_{L}$ is bounded by $\kappa=2\pi T_{H}/(1-\mu\mathcal{L})$,
where $T_{H}$ and $\mu$ are the temperature and the angular momentum of the
black hole respectively, and $\mathcal{L}$ is the angular momentum (per unit
energy) of the shock waves. This indicates that the system in the dyonic Kerr-
Sen-AdS4 black hole is chaotic and thus supports the fast scrambling
conjecture, even for the rotating charged black hole in the EMDA theory. We
show that the ratio $\kappa/\lambda_{L}$ is constant with respect to
$\mathcal{L}$, and approaches 1 as the AdS radius $l$ becomes large. The
constant can vary, depending on the black hole parameters, and thus we suggest
that it should take the form $\kappa/\lambda_{L}=\mathcal{C}$, for some
constant $\mathcal{C}>1$. For example, it has been shown recently that a
three-dimensional rotating BTZ black hole has $\mathcal{C}=2$ [5].
Interestingly, for a small value of $l$, $l=1$, we observe that the bound
$\lambda_{L}<\kappa$ can be violated for large $\mathcal{L}$. This is unusual
and it does not present in the case of an uncharged Kerr-AdS4 black hole [6].
The violations of chaos bound have been observed earlier (see, for example,
[44, 45, 46, 47]) using different methods, although only the standard MSS
bound $2\pi T_{H}$ is used. The violations are present in the charged black
hole cases, for some value of rotation parameter (both black hole’s and
particle’s rotation). In this work, we show that the bound
$\kappa/(1-\mu\mathcal{L})$ can also be violated due to the existence of the
dilaton and axion charges.
The structure of this paper is as follows: In Section II, we review the dyonic
Kerr-Sen-AdS4 black hole solution and its ultra-spinning version in [42, 48]
including their thermodynamics. In Section III, we construct the lightcone
coordinates of the rotating and charged shockwaves and then obtain the
corresponding Dray-’t Hooft shock waves solution for this geometry. In Section
IV, we first briefly explain the holographic CFT model for this dyonic Kerr-
Sen-AdS4 black hole. Using the RT/HRT surfaces, we calculate the corresponding
entanglement entropy and obtain the Mutual Information which is sensitive to
the charged and rotating shock waves in the bulk. From that, we can obtain the
scrambling time $\tau_{*}$ and the scrambling delay $\tau_{d}$ both for the
dyonic Kerr-Sen-AdS4 black hole and its ultra-spinning counterpart. We also
calculate the mutual information using holography, which leads to the
calculations of the Lyapunov exponent. We show how the Lyapunov exponent
behaves as we change the black hole parameters. These observations support the
fast scrambling conjecture for the dyonic Kerr-Sen-AdS4 black hole. We sum up
our work in the Discussions and Conclusions session in Section V.

##  II Dyonic Kerr-Sen-AdS4 metric

In this section, we briefly review the black hole solution to the gauged
Einstein-Maxwell-Dilaton-Axion theory known as the dyonic Kerr-Sen-AdS4 black
hole [42]. We also write down the thermodynamic quantities and their first-
law-like relation. The ultra-spinning version of the metric with
$\varphi\rightarrow\varphi/\Xi$ and $a\rightarrow l$ is also given along with
its thermodynamics.
The dyonic Kerr-Sen-AdS4 black hole is the solution to the gauged Einstein-
Maxwell-Dilaton-Axion (EMDA) theory with action

$I=\frac{1}{16\pi
G_{N}}\int\sqrt{|g|}d^{4}x\bigg{(}R-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}e^{2\phi}(\partial\chi)^{2}-e^{-\phi}F^{2}+\frac{\chi}{2}F\tilde{F}\bigg{)}+I_{\Lambda},$
(1)

where $\phi$ and $\chi$ are the dilaton scalar and axion pseudoscalar field
respectively, $F=dA$ is the electromagnetic tensor of an abelian gauge
potential one-form $A_{\mu}$ with its Hodge dual $\tilde{F}$. We use the
notation $(\partial\phi)^{2}=(\partial_{\mu}\phi)(\partial^{\mu}\phi)$,
$(\partial\chi)^{2}=(\partial_{\mu}\chi)(\partial^{\mu}\chi)$,
$F^{2}=F_{\mu\nu}F^{\mu\nu}$, $F\tilde{F}=F_{\mu\nu}\tilde{F}^{\mu\nu}$ with
$\tilde{F}^{\mu\nu}=\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ where
$\varepsilon^{\mu\nu\alpha\beta}$ denotes a four-dimensional totally-
antisymmetric tensor. The axion pseudoscalar $\chi$ is defined from
$d\mathcal{B}=-e^{2\phi}\star d\chi$, where $\mathcal{B}_{\mu\nu}$ is an
antisymmetric two-form tensor and $\star$ denotes the Hodge duality operator.
The term $I_{\Lambda}$ represents the cosmological-constant term which comes
from the gauged version of the standard EMDA theory and It is given by

$I_{\Lambda}=\frac{1}{16\pi
G_{N}}\int\sqrt{|g|}d^{4}x(4+e^{-\phi}+e^{\phi}(1+\chi^{2}))/l^{2}.$ (2)

The solution for the dyonic Kerr-Sen-AdS4 black hole is given by [42]

$ds^{2}=-\frac{\Delta}{\Sigma}X^{2}+\frac{\Sigma}{\Delta}dr^{2}+\frac{\Sigma}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\Sigma}Y^{2},$
(3)

where

$\displaystyle X=$ $\displaystyle dt-\frac{a\sin^{2}\theta}{\Xi}d\varphi,$ (4)
$\displaystyle Y=$ $\displaystyle
adt-\frac{(r^{2}-d^{2}-k^{2}+a^{2})}{\Xi}d\varphi,$ $\displaystyle\Delta(r)=$
$\displaystyle\bigg{(}1+\frac{r^{2}-d^{2}-k^{2}}{l^{2}}\bigg{)}(r^{2}-d^{2}-k^{2}+a^{2})$
(5) $\displaystyle-2Mr+p^{2}+q^{2},$ $\displaystyle\Delta_{\theta}=$
$\displaystyle
1-\frac{a^{2}}{l^{2}}\cos^{2}\theta,\;\;\;\Xi=1-\frac{a^{2}}{l^{2}},$ (6)
$\displaystyle\Sigma=$ $\displaystyle r^{2}-d^{2}-k^{2}+a^{2}\cos^{2}\theta.$

Here, $m,a,l,d,k,p,q$ are the mass, rotation, cosmological constant (AdS
length), dilaton charge, axion charge, magnetic charge, and electric charge
parameters respectively with the relations $d=(p^{2}-q^{2})/2m$ and $k=pq/m$.
Note that we use the shifted coordinate $r\rightarrow r+d$ for mathematical
simplicity. The potential $A_{\mu}$, its dual $B_{\mu}$ defined from
$e^{-\phi}\star F+\chi F=-dB$, dilaton scalar and axion pseudoscalar fields
are given by

$\displaystyle A$
$\displaystyle=\frac{q(r+d-p^{2}/m)}{\Sigma}X-\frac{p\cos\theta}{\Sigma}Y,$
$\displaystyle B$
$\displaystyle=\frac{p(r+d-p^{2}/m)}{\Sigma}X+\frac{q\cos\theta}{\Sigma}Y,$
$\displaystyle e^{\phi}$
$\displaystyle=\frac{(r+d)^{2}+(a\cos\theta+k)^{2}}{\Sigma},$
$\displaystyle\chi$ $\displaystyle=2\frac{kr-
da\cos\theta}{(r+d)^{2}+(a\cos\theta+k)^{2}}.$ (7)

The thermodynamic quantities of the dyonic Kerr-Sen-AdS4 black hole are given
as follows:

$\displaystyle M$
$\displaystyle=\frac{m}{\Xi},\;\;\;J=\frac{ma}{\Xi},\;\;\;Q=\frac{q}{\Xi},\;\;\;P=\frac{p}{\Xi},$
$\displaystyle S$
$\displaystyle=\frac{\pi}{\Xi}(r_{+}^{2}-d^{2}-k^{2}+a^{2}),$
$\displaystyle\Omega_{\varphi}$
$\displaystyle=\frac{a\Xi}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},$ $\displaystyle\Phi$
$\displaystyle=\frac{q(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},$
$\displaystyle\Psi$
$\displaystyle=\frac{p(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},$
$\displaystyle 2\pi T_{H}$
$\displaystyle=\frac{r_{+}(2r_{+}^{2}-2d^{2}-2k^{2}+a^{2}+l^{2})-Ml^{2}}{(r_{+}^{2}-d^{2}-k^{2}+a^{2})l^{2}},$
(8)

where $r_{+}$ is the outermost horizon as the largest solution to
$\Delta(r_{+})=0$. Here, $M,J,Q,P$ are the mass, angular momentum, electric
charge, and magnetic charge of the black holes as the corresponding conserved
charges. Furthermore, $S$ is the entropy of the black hole while
$\Omega_{\varphi},\Phi,\Psi$ denotes the angular momentum, electric potential,
and magnetic potential of the horizon respectively. For now, we assume that
the black hole is non-extremal, i.e. $\Delta^{\prime}(r_{+})\neq 0$ where the
prime denotes the derivative with respect to $r$. In this work, we treat the
cosmological constant parameter $l$ as non-dynamical such that the first law
equation becomes

$dM=T_{H}dS+\Omega_{\varphi}dJ+\Phi dQ+\Psi dP+Jd\Xi/(2a).$ (9)

The dyonic Kerr-Sen-AdS4 metric is rotating at $r\rightarrow\infty$ with
angular velocity

$\Omega_{\infty}=-\frac{a}{l^{2}}.$ (10)

Therefore, a stationary observer at infinity can be obtained by a coordinate
transformation $\varphi\rightarrow\varphi+\Omega_{\infty}t$ and some of the
thermodynamic quantities are shifted by

$M\rightarrow\bar{M}=M+\frac{a}{l^{2}},J=\frac{m}{\Xi^{2}},\;\;\;\Omega_{\varphi}\rightarrow\Omega_{\varphi}-\Omega_{\infty},$
(11)

where now the newly shifted black hole angular velocity is defined as

$\mu\equiv\Omega_{\varphi}-\Omega_{\infty}=\frac{a(1+(r_{+}^{2}-d^{2}-k^{2})/l^{2})}{r_{+}^{2}-d^{2}-k^{2}+a^{2}}.$
(12)

Now the angular velocity $\mu$ becomes the new chemical potential for the
first-law relation

$d\bar{M}=T_{H}dS+\mu dJ+\Phi dQ+\Psi dP.$ (13)

###  II.1 Ultra-spinning dyonic Kerr-Sen-AdS4

The redefinition of the coordinate $\varphi\rightarrow\varphi/\Xi$ allows us
to obtain the ultra-spinning limit $a\rightarrow l$ of the dyonic Kerr-Sen-
AdS4 black hole, which now has the metric and the corresponding fields in the
form of

$\displaystyle d\hat{s}^{2}$
$\displaystyle=-\frac{\hat{\Delta}}{\hat{\Sigma}}\hat{X}^{2}+\frac{\hat{\Sigma}}{\hat{\Delta}}dr^{2}+\frac{\hat{\Sigma}}{\sin^{2}\theta}d\theta^{2}+\frac{\sin^{4}\theta}{\hat{\Sigma}}\hat{Y}^{2},$
$\displaystyle A$
$\displaystyle=\frac{q(r+d-p^{2}/m)}{\hat{\Sigma}}\hat{X}=\frac{p\cos\theta}{\hat{\Sigma}}\hat{Y},$
$\displaystyle B$
$\displaystyle=\frac{p(r+d-p^{2}/m)}{\hat{\Sigma}}\hat{X}+\frac{q\cos\theta}{\hat{\Sigma}}\hat{Y},$
$\displaystyle e^{\phi}$
$\displaystyle=\frac{(r+d)^{2}+(l\cos\theta+k)^{2}}{\hat{\Sigma}},$ (14)
$\displaystyle\chi$ $\displaystyle=2\frac{kr-
dl\cos\theta}{r^{2}+(l\cos\theta+k)^{2}},$

with

$\displaystyle\hat{X}$ $\displaystyle=dt-l\sin^{2}\theta d\varphi,$ (15)
$\displaystyle\hat{Y}$ $\displaystyle=ldt-(r^{2}-d^{2}-k^{2}+l^{2})d\varphi,$
$\displaystyle\hat{\Delta}(r)$
$\displaystyle=(r^{2}-d^{2}-k^{2}+l^{2})^{2}/l^{2}-2Mr+p^{2}+q^{2},$
$\displaystyle\hat{\Sigma}$
$\displaystyle=r^{2}-d^{2}-k^{2}+l^{2}\cos^{2}\theta.$ (16)

The periodicity of $\varphi$ is now assumed to be given by $\lambda$ instead
of $2\pi$. Notice that the standard dyonic Kerr-Sen-AdS4 metric $ds^{2}$
cannot be obtained directly from the ultra-spinning metric $d\hat{s}^{2}$.
The thermodynamic quantities of the ultra-spinning dyonic Kerr-Sen-AdS4 black
hole are given by

$\displaystyle\hat{M}$
$\displaystyle=\frac{\mu}{2\pi}m,\;\;\hat{J}=\frac{\mu}{2\pi}ml=Ml,\;\;\hat{Q}=\frac{\mu}{2\pi}q,\;\;\hat{P}=\frac{\mu}{2\pi}p,$
$\displaystyle\hat{S}$
$\displaystyle=\frac{\mu}{2}(r_{+}^{2}-d^{2}-k^{2}+l^{2}),\;\;\hat{\Omega}_{\varphi}=\frac{l}{r_{+}^{2}-d^{2}-k^{2}+l^{2}},$
$\displaystyle\hat{\Phi}$
$\displaystyle=\frac{q(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+l^{2}},\;\;\;\hat{\Psi}=\frac{p(r_{+}+d-p^{2}/2m)}{r_{+}^{2}-d^{2}-k^{2}+l^{2}},$
$\displaystyle 2\pi\hat{T}_{H}$
$\displaystyle=2\frac{r_{+}}{l^{2}}-\frac{m}{r_{+}^{2}-d^{2}-k^{2}+l^{2}}.$
(17)

Again, $r_{+}$ is defined as the outermost horizon satisfying
$\hat{\Delta}(\hat{r}_{+})=0$ and we assume that the ultra-spinning black hole
is also non-extremal with $\hat{\Delta}^{\prime}(r_{+})\neq 0$. The first-law-
like relation is now given by

$d\hat{M}=\hat{T}_{H}d\hat{S}+\hat{\Omega}_{\varphi}d\hat{J}+\hat{\Phi}d\hat{Q}+\hat{\Psi}d\hat{P}+\hat{K}d\lambda,$
(18)

where $\lambda$ also plays a role as the dynamical variable with a new
chemical potential

$K=m\frac{l^{2}-(r_{+}+d+q^{2}/m)(r_{+}+d-p^{2}/m)}{4\pi(r_{+}^{2}-d^{2}-k^{2}+l^{2})}.$
(19)

##  III Equatorial Rotating Shock wave Solutions

###  III.1 Kruskal coordinates

Consider an equatorial ($\theta=\pi/2$) rotating null particle in the
background of the dyonic Kerr-Sen-AdS4 black hole with unit energy
$\mathcal{E}=1$ and angular momentum per unit energy $\mathcal{L}$. We will
then call this the rotating shock wave. Such a particle will have a geodesic
$\xi^{\mu}$ observed by a stationary observer at infinity which satisfies

$\xi^{2}=0,\;\;\;\xi\cdot\zeta_{t}=-\mathcal{E},\;\;\;\xi\cdot\zeta_{\varphi}=\mathcal{L},\;\;\;\xi^{\theta}=0.$
(20)

The last equality is required for the equatorial orbit, which means that the
particle always stays at the equator $\theta=\pi/2$. The corresponding Killing
vector associated with axisymmetry is given by
$\zeta_{t}=\partial_{t}-a/l^{2}\partial_{\varphi}$ and
$\zeta_{\varphi}=\partial_{\varphi}$. There are two solutions to eq. (20),
$\xi_{\pm}^{\mu}$, where the negative solution is obtained from reversing the
axisymmetry $-\mathcal{E}\rightarrow\mathcal{E}$ and
$\mathcal{L}\rightarrow-\mathcal{L}$. The geodesic solution in the background
of a dyonic Kerr-Sen-AdS4 black hole for unit energy ($\mathcal{E}=1$) is
given by

$\xi_{+}\cdot dx=dr_{*}-d\tau\equiv du,\;\;\;\xi_{-}\cdot
dx=dr_{*}+d\tau\equiv dv,$ (21)

where $\tau=(1-a\mathcal{L}/l^{2})t-\mathcal{L}\varphi$ and $r_{*}$ is the
tortoise-like coordinate given by

$r_{*}(r)=\int\frac{\tilde{f}}{\Delta}dr,$ (22)

with $\tilde{f}$ is defined from

$\tilde{f}^{2}=-\Delta(\mathcal{L}-a)^{2}+[\mathcal{L}a(1+(r^{2}-d^{2}-k^{2})/l^{2})-(r^{2}-d^{2}-k^{2}+a^{2})]^{2}.$
(23)

With vanishing dilaton and axion charge, i.e. $d,k\rightarrow 0$, we recover
the tortoise-like coordinate found in [6] and with vanishing angular momentum
$\mathcal{L}\rightarrow 0$, we recover the standard Kruskal coordinates for
rotating dyonic Kerr-Sen-AdS4 black hole.
The Kruskal-like metric which follows the geodesic of the rotating shock wave
is given by

$ds^{2}=F(r)dudv+\tilde{h}(r)(d\varphi+\tilde{h}_{\tau}(r)d\tau)^{2},$ (24)

with

$\displaystyle F(r)$
$\displaystyle=\frac{\Delta(r^{2}-d^{2}-k^{2})}{\tilde{f}^{2}},\;\;\;\tilde{h}(r)=\frac{(1-a\mathcal{L}/l^{2})^{-2}}{r^{2}-d^{2}-k^{2}}\frac{\tilde{f}^{2}}{\Xi^{2}},$
$\displaystyle\tilde{h}_{\tau}(r)$
$\displaystyle=\frac{\Xi}{\tilde{f}^{2}}\big{(}(\Delta(a-\mathcal{L})+a\big{(}a\mathcal{L}(1+(r^{2}-d^{2}-k^{2})/l^{2})-(r^{2}-d^{2}-k^{2}+a^{2})\big{)}\big{)}.$
(25)

We then shift the axial coordinate $\varphi\rightarrow\eta z+\gamma\tau$ for
two reasons: one is to make sure that $\tilde{h}_{\tau}(r)+\gamma$ behaves
like $\mathcal{O}(r-r_{+})$ near the horizon and to recover the black hole’s
horizon area when integrating $\theta$ from $0$ to $\pi$ and $z$ from $0$ to
$2\pi$. Thus, we have

$\gamma=\frac{\Omega_{\varphi}}{1-\mu\mathcal{L}},\;\;\;\eta=\frac{1}{1-\mu\mathcal{L}}.$
(26)

From the coordinate transformations, we have

$ds^{2}=F(r)dudv+h(r)(dz+h_{\tau}(r)d\tau)^{2},$ (27)

where $h(r)=\eta^{2}\tilde{h}(r)$ and
$h_{\tau}(r)\equiv\eta^{-1}(\tilde{h}_{\tau}(r)+\gamma)$.
Furthermore, we would like to work with coordinates that are affine at the
horizon to generate the Dray-’t Hooft solution later on. In this case,
$\\{u,v\\}$ coordinate is not affine, i.e. the tangent vector
$\chi_{u}=\partial_{u}$ satisfy
$\chi_{u}\cdot\nabla\chi_{u}^{\mu}=\mathcal{K}\chi_{u}^{\mu}$ (similar with
$\chi_{v}^{\mu}$), where

$\mathcal{K}=\frac{1}{2}\xi_{\pm}\cdot\partial F.$ (28)

The new coordinates $\\{U,V\\}$ are

$U=-e^{\kappa u},\;\;\;V=e^{\kappa
v},\;\;\;\kappa\equiv\mathcal{K}|_{r=r_{+}},$ (29)

and it can easily be shown that both $\partial_{U}$ and $\partial_{V}$ are
affine coordinates at $r_{+}$. The metric in the affine coordinates is now
written as

$ds^{2}=\frac{F(r)}{\kappa^{2}UV}dUdV+h(r)(dz+h_{\tau}(r)d\tau)^{2}.$ (30)

The value of $\kappa$ is related to the black hole’s surface gravity
$\kappa_{0}$ by

$\kappa=\frac{\kappa_{0}}{1-\mu\mathcal{L}}=\frac{2\pi
T_{H}}{1-\mu\mathcal{L}}.$ (31)

This $\kappa$ has a form similar to the standard Kerr-AdS4 black hole with
rotating shock waves. However, it differs in the value of the Hawking
temperature $T_{H}$ and the angular momentum $\mu$ which now depends on the
value of the dilaton and axion charge $d,k$.
Next, we would like to generate the Kruskal-like coordinates for the ultra-
spinning counterpart where the metric is given by eq. (II.1). Since the ultra-
spinning version of the dyonic Kerr-Sen-AdS4 has a null $\varphi$ coordinate
at the boundary $r\rightarrow\infty$ and hence the metric is non-diagonal, we
need to adopt the so-called conformal completion technique to compute the
conserved quantities [49, 42]. The mass and the angular momentum of the ultra-
spinning dyonic Kerr-Sen-AdS4 black hole given by eq. (17) is obtained by
calculating the conserved charge $\mathcal{Q}[\zeta]$ defined in eq. (9) of
[42] for a Killing vector $\zeta$. The timelike Killing vector
$\zeta_{t}=\partial_{t}$ gives us the mass $\hat{M}$, while the axial Killing
vector $\zeta_{\varphi}=\partial_{\varphi}$ gives us the angular momentum
$\hat{J}$. Therefore, we also have to use the Killing vectors at infinity
$\zeta_{t}=\partial_{t}$ and $\zeta_{\varphi}=\partial_{\varphi}$ which
corresponds to the conserved energy $\mathcal{E}$ and angular momentum
$\mathcal{L}$ of a rotating shock wave around the ultra-spinning black hole.
The geodesic of a rotating shock wave $\hat{\xi}^{\mu}$ in the background of
an ultra-spinning dyonic Kerr-Sen-Black hole is obtained by solving eq. (20)
with the ultra-spinning black hole metric in eqs (II.1) and (16) and Killing
vectors $\zeta_{t}=\partial_{t},\zeta_{\varphi}=\partial_{\varphi}$. The
solutions for unit energy $\mathcal{E}=1$ and an angular momentum per unit
energy $\mathcal{L}$ are given in the form of

$\hat{\xi}_{+}\cdot dx=d\hat{r}_{*}-d\hat{\tau}\equiv
d\hat{u},\;\;\;\hat{\xi}_{-}\cdot dx=d\hat{r}_{*}+d\hat{\tau}\equiv d\hat{v},$
(32)

where $\hat{\tau}=t-\mathcal{L}\varphi$ and

$r_{*}\equiv\int\frac{\hat{\tilde{f}}dr}{\hat{\Delta}},$ (33)

with

$\hat{\tilde{f}}^{2}=-\hat{\Delta}(\mathcal{L}-l)^{2}+\big{(}\mathcal{L}l-(r^{2}-d^{2}-k^{2}+l^{2})\big{)}^{2}.$
(34)

Notice that both $\hat{\tau}$ and $\hat{\tilde{f}}$ cannot be obtained by
taking $a\rightarrow l$ limit of $\tilde{f}$. The Kruskal-like metric for the
ultra-spinning black hole with rotating shock waves is given by

$d\hat{s}^{2}=\hat{F}(r)d\hat{u}d\hat{v}+\hat{\tilde{h}}(r)(d\varphi+\hat{\tilde{h}}_{\hat{\tau}}(r)d\hat{\tau})^{2},$
(35)

where now we have

$\displaystyle\hat{F}(r)$
$\displaystyle=\frac{\hat{\Delta}(r^{2}-d^{2}-k^{2})}{\hat{\tilde{f}}^{2}},\;\;\;\hat{\tilde{h}}(r)=\frac{\tilde{f}^{2}}{r^{2}-d^{2}-k^{2}},$
$\displaystyle\hat{\tilde{h}}_{\hat{\tau}}(r)$
$\displaystyle=\frac{-\hat{\Delta}(\mathcal{L}-l)^{2}+l(\mathcal{L}l-(r^{2}-d^{2}-k^{2}+l^{2}))}{\tilde{f}^{2}}.$
(36)

Following similar reasoning to the normal dyonic Kerr-Sen-AdS4, we redefine
the coordinate $\varphi\rightarrow\hat{\eta}\hat{z}+\hat{\gamma}\hat{\tau}$,
where now

$\hat{\eta}=\frac{1}{1-\hat{\Omega}_{\varphi}\mathcal{L}},\;\;\;\hat{\gamma}=\frac{\hat{\Omega}_{\varphi}}{1-\hat{\Omega}_{\varphi}\mathcal{L}}.$
(37)

Again, all of the functions appear in the ultra-spinning case
$\\{\hat{F},\hat{\tilde{h}},\hat{\tilde{h}}_{\hat{\tau}},\hat{\eta},\hat{\gamma}\\}$
cannot be directly obtained from
$\\{F,\tilde{h},\tilde{h}_{\tau},\eta,\gamma\\}$ by taking the limit of
$a\rightarrow l$ and thus they are all completely different functions. After
the coordinate transformation
$\varphi\rightarrow\hat{\eta}\hat{z}+\hat{\gamma}\hat{\tau}$, the Kruskal-like
metric for the ultra-spinning black hole now takes the form

$d\hat{s}^{2}=\hat{F}(r)d\hat{u}d\hat{v}+\hat{h}(r)(d\hat{z}+\hat{h}_{\hat{\tau}}(r)d\hat{\tau})^{2},$
(38)

where $\hat{h}(r)=\hat{\eta}^{2}\hat{\tilde{h}}(r)$ and
$\hat{h}_{\hat{\tau}}(r)=\hat{\eta}^{-1}(\hat{\tilde{h}}_{\hat{\tau}}+\hat{\gamma})$.
We also would like to work with affine coordinates $\\{\hat{U},\hat{V}\\}$
instead of $\\{\hat{u},\hat{v}\\}$ for the ultra-spinning black hole.
Following similar reasoning as before, we obtain

$\hat{U}=-e^{\hat{\kappa}\hat{u}},\;\;\;\hat{V}=e^{\hat{\kappa}\hat{v}},$ (39)

where now $\hat{\kappa}$ is given by

$\hat{\kappa}=\frac{2\pi\hat{T}_{H}}{(1-\hat{\Omega}_{\varphi}\mathcal{L})}.$
(40)

The metric in the affine coordinates $\\{\hat{U},\hat{V}\\}$ is now given by

$d\hat{s}^{2}=\frac{\hat{F}(r)}{\hat{\kappa}^{2}\hat{U}\hat{V}}d\hat{U}d\hat{V}+\hat{h}(r)(d\hat{z}+\hat{h}_{\hat{\tau}}(r)d\hat{\tau})^{2}.$
(41)

In this coordinate system, we then generate the Dray-’t Hooft solution for the
ultra-spinning dyonic Kerr-Sen-AdS4 black hole.
After we obtain the metric in Kruskal coordinates for the dyonic Kerr-Sen-AdS4
and its ultra-spinning counterpart, we now see how the functions $F(r)$ and
$\hat{F}(r)$ behave near the horizon. Indeed, by expanding $r$ near $r_{+}$ up
to the second order, we have

$\displaystyle F(r)=$ $\displaystyle F^{\prime}(r_{+})(r-r_{+})+$ (42)
$\displaystyle\frac{1}{2}F^{\prime\prime}(r_{+})(r-r_{+})^{2}+...\;,$
$\displaystyle\hat{F}(r)=$ $\displaystyle\hat{F}^{\prime}(r_{+})(r-r_{+})+$
(43)
$\displaystyle\frac{1}{2}\hat{F}^{\prime\prime}(r_{+})(r-r_{+})^{2}+...\;,$

with all
$F^{\prime}(r_{+}),F^{\prime\prime}(r_{+}),\hat{F}^{\prime}(r_{+}),\hat{F}^{\prime\prime}(r_{+})$
are non zero. Using the definitions of the Kruskal coordinates in eq. (29) and
the tortoise-like coordinates in (22) for the dyonic Kerr-Sen-AdS4 black hole,
we may write that, near the horizon,

$\displaystyle(r-r_{+})F^{\prime}(r_{+})$ $\displaystyle=-\mathbb{A}UV,$ (44)

where $\mathbb{A}$ is some dimensionless proportionality constant which
depends on the dyonic Kerr-Sen-AdS4 black hole parameters.
One need to be more cautious when considering $\hat{U}\hat{V}$ near the
horizon for the ultra-spinning black hole because $\hat{\kappa}$ is now
equipped with $\hat{\Omega}_{\varphi}$ instead of $\mu$. However, it turns out
that the value of $\hat{U}\hat{V}$ near the horizon is still linear in
$r-r_{+}$ since the tortoise coordinate near the horizon behaves as

$\displaystyle r_{*}$
$\displaystyle\approx\frac{(1-\hat{\Omega}_{\varphi}\mathcal{L})(r_{+}^{2}-d^{2}-k^{2})}{\hat{\Delta}^{\prime}(r_{+})}\ln|r-r_{+}|+C$
(45) $\displaystyle=\frac{1}{2\hat{\kappa}}\ln|r-r_{+}|+C,$

where $C$ is an integration constant in which the explicit value depends on
the geometry of the black hole. Therefore,
$\hat{U}\hat{V}\sim\mathcal{O}(r-r_{+})$ near the horizon and we may also
write

$(r-r_{+})\hat{F}^{\prime}(r_{+})=-\hat{\mathbb{A}}\hat{U}\hat{V},$ (46)

where $\hat{\mathbb{A}}$ is a dimensionless proportionality constant that
depends on the ultra-spinning black hole parameters.
The expansions of the functions $F(r)$ and $\hat{F}(r)$ near the horizon can
then be extended to the second order of $r-r_{+}$. For the standard dyonic
Kerr-Sen-AdS4, we have

$\frac{F}{\mathbb{A}UV}=-1-\frac{\mathbb{A}UV}{2}\frac{F^{\prime\prime}(r_{+})}{F^{\prime}(r_{+})}+\mathcal{O}(U^{2}V^{2}),$
(47)

while for the ultra-spinning black hole, we have

$\frac{\hat{F}}{\hat{\mathbb{A}}\hat{U}\hat{V}}=-1-\frac{\hat{\mathbb{A}}\hat{U}\hat{V}}{2}\frac{\hat{F}^{\prime\prime}(r_{+})}{\hat{F}^{\prime}(r_{+})}+\mathcal{O}(\hat{U}^{2}\hat{V}^{2}).$
(48)

###  III.2 Extremal limits

From the turning-point analysis, the maximum value that $\mathcal{L}$ can
achieve if we need the perturbation to reach the horizon from infinity is
bounded by $\mu^{-1}$ ($\mathcal{L}_{\text{max}}\leq\mu^{-1}$) and the
equality holds at extremality, i.e. $\mathcal{L}_{\text{max}}=\mu^{-1}$ for
$T_{H}\rightarrow 0$ [6]. For the dyonic Kerr-Sen-AdS4 black hole, the value
of $\kappa$ also survives the extremal limit. By expanding the value of
$r_{+}$ near the extremal horizon radius $r_{0}$, where
$\Delta(r_{0})=\Delta^{\prime}(r_{0})=0$, to the first-order approximation, we
have

$T_{H}\approx\frac{\partial T_{H}}{\partial
r_{+}}\bigg{|}_{r_{0}}(r_{+}-r_{0}).$ (49)

On the other hand, the value of $(1-\mu\mathcal{L})$ can also be expanded near
the extremal value $r_{0}$,

$1-\mu\mathcal{L}\approx\bigg{(}\frac{1}{\mu}\frac{\partial\mu}{\partial
r_{+}}\bigg{)}_{r_{0}}(r_{+}-r_{0}).$ (50)

Therefore, by taking the extremal limit $r_{+}\rightarrow r_{0}$, we obtain
the extremal value of $\kappa$,

$\kappa_{\text{ext}}=\frac{2\pi(\partial T_{H}/\partial
r_{+})_{r_{0}}}{(\mu^{-1}\partial\mu/\partial r_{+})_{r_{0}}}=-2\pi
T_{L}\mu_{\text{ext}},$ (51)

where $\mu_{\text{ext}}$ is given by eq. (12) with $r_{+}$ is replaced by
$r_{0}$ and $\kappa_{\text{ext}}$ is related by the definition of the left-
moving Frolov-Thorne temperature found by the Kerr/CFT correspondence for the
extremal dyonic Kerr-Sen-AdS4 black hole $T_{L}\equiv-\frac{(\partial
T_{H}/\partial r_{+})_{r_{0}}}{(\partial\mu/\partial r_{+})_{r_{0}}}$ derived
in [40, 41].
For the dyonic Kerr-Sen-AdS4 black hole, we have

$\kappa_{\text{ext}}=\frac{(1+(r_{0}^{2}-d^{2}-k^{2})/l^{2})(1+a^{2}/l^{2}+(6r_{0}^{2}-2d^{2}-2k^{2})/l^{2})}{2r_{0}\Xi},$
(52)

while for the ultra-spinning counterpart (following the same procedure), we
have

$\displaystyle\hat{\kappa}_{\text{ext}}$
$\displaystyle=-2\pi\hat{T}_{L}\hat{\Omega}^{\text{ext}}_{\varphi}$ (53)
$\displaystyle=\frac{(6r_{+}^{2}-2d^{2}-2k^{2}+2l^{2})}{2r_{+}l^{2}}.$ (54)

Both $\kappa_{\text{ext}}$ and $\hat{\kappa}_{\text{ext}}$ scales with the
left-moving Frolov-Thorne temperature calculated by the Kerr/CFT. The
difference lies in the type of angular momentum which appear in both
$\kappa_{\text{ext}}$ and $\hat{\kappa}_{\text{ext}}$ where it is given by
$\mu_{\text{ext}}$ for the former and $\hat{\Omega}^{\text{ext}}_{\varphi}$
for the latter (instead of $\hat{\mu}_{\text{ext}}$). This difference lies in
the definition of angular velocities. For the standard black hole, $\mu$ is
the difference between the horizon’s angular velocity and the angular velocity
of a stationary observer at infinity. For the ultra-spinning black hole,
however, we do not have a notion of the angular velocity of a stationary
observer at infinity because the $\varphi$ coordinate becomes null. Instead,
what appears in $1-\hat{\Omega}_{\varphi}\mathcal{L}$ is the horizon’s angular
velocity $\hat{\Omega}_{\varphi}$ which is the chemical potential for
$\hat{J}$. Up until this point, all of the hatted functions for the ultra-
spinning black hole do not depend on the new dynamical variable $\lambda$.

###  III.3 Dray-’t Hooft solution

After we obtain the Kruskal-like coordinates for rotating null geodesic for
both standard dyonic Kerr-Sen-AdS4 black hole and its ultra-spinning
counterpart, we are now able to generate the Dray-’t Hooft solution. For a
Kruskal-like metric perturbed by an in-falling rotating shock wave at the
equator located at $U_{0}$ for the standard black hole and at $\hat{U}_{0}$
for the ultra-spinning black hole, the metric is given by the shift for
$U>U_{0}$,

$V\rightarrow\tilde{V}=V+\alpha\Theta(U-U_{0}),$ (55)

for the dyonic Kerr-Sen-AdS4, and

$\hat{V}\rightarrow\hat{\tilde{V}}=\hat{V}+\hat{\alpha}\Theta(\hat{U}-\hat{U}_{0}),$
(56)

for its ultra-spinning version. It is known that the shift should contain a
function $f(\theta)$ which captures the perturbation away from the equator.
However, when we only consider $\theta=\pi/2$, we may normalize the function
such that $f(\pi/2)=1$. In this section, we will calculate the Dray-’t Hooft
solution and the strength of the shock waves $\alpha$ for the standard dyonic
Kerr-Sen-AdS4 first, then followed by its ultra-spinning counterpart later on.

#### Standard Kerr-Sen-AdS4

We first present the coordinate $\tau$ in terms of $U$ and $V$. The metric in
eq. (30) can then be written as

$\displaystyle ds^{2}=$ $\displaystyle\frac{F(r)}{\kappa^{2}UV}dUdV+$ (57)
$\displaystyle h(r)\bigg{(}dz+\frac{h_{\tau}(r)}{2\kappa UV}(UdV-
VdU)\bigg{)}^{2}.$

After applying the Dray-’t Hooft solution which gives the shift in eq. (56)
due to the shock, the metric becomes

$ds^{2}\rightarrow\tilde{ds}^{2}-\frac{F}{\kappa^{2}UV}\delta(U)dU^{2},$ (58)

after the shift, where $\tilde{ds}^{2}$ is given by eq. (30) with $V$ is
replaced by $\tilde{V}$.
The strength of the shift, $\alpha$ and $\hat{\alpha}$ can be obtained by
requiring that the transverse volume element $H=\sqrt{g_{\theta\theta}g_{zz}}$
(in the full coordinates involving $\theta$) to be smooth at the location of
the shock. Suppose that after the shock wave enters the black hole horizon, it
increases the black hole mass by a small amount $E_{0}$. To make the case more
general, we may also consider a perturbation which increases the value of the
electric charge of the black hole by a small amount $\delta q$ and the
magnetic charge by a small amount $\delta p$ as well although we assume that
the trajectory of the perturbation still follows the rotating shock waves
geodesic given by eq. (21). In this case, the area of the horizon changes
according to the first law of thermodynamics, which also change $H$ at the
horizon, since $H$ is nothing but the area of the horizon, divided $4\pi$. By
imposing $H$ to be smooth at the horizon, we have

$H\big{|}_{U_{0}^{+}}=H\big{|}_{U_{0}-}.$ (59)

Using eq. (44) and taking the limit $E_{0}\rightarrow 0$ and $U_{0}\rightarrow
0$ simultaneously, we have

$\alpha=\frac{(\tilde{H}_{r_{+}}-H_{r_{+}})F^{\prime}(r_{+})}{U_{0}\mathbb{A}\tilde{H}^{\prime}(r_{+})},$
(60)

where $\tilde{H}_{r_{+}}$ is the shifted horizon area divided by $4\pi$. Upon
taking the limit of $E_{0}\rightarrow 0$, we have
$(\tilde{H}_{r_{+}}-H_{r_{+}})\rightarrow 0$ as well. However, $\alpha$ can be
fixed by taking the limit $U_{0}\rightarrow 0$, which means that the shocks
hover very close to the horizon, and were sent from the AdS boundary in the
far past $\tau_{0}\rightarrow\infty$.
To be more precise, since $H_{r_{+}}$ is proportional to the area of the
horizon, we have

$\tilde{H}_{r_{+}}-H_{r_{+}}\propto\delta S=\beta_{H}(\delta\bar{M}-\mu\delta
J-\Phi\delta Q-\Psi\delta P),$ (61)

by the first law of thermodynamics given by eq. (13), where
$\beta_{H}=T_{H}^{-1}$ is the inverse of the Hawking temperature. The change
in mass $\delta\bar{M}$ is proportional to the energy of the shock wave
$E_{0}$ and the change in the angular momentum $\delta
J=\delta\bar{M}\mathcal{L}$ is proportional to $E_{0}\mathcal{L}$. From the
thermodynamic relation between $Q,P$ and $q,p$, we have $\delta Q\sim\delta q$
and $\delta P\sim\delta p$ which is small, i.e. $\delta q,\delta p\rightarrow
0$, but can be fixed by setting $\mathcal{Q}=\delta q/E_{0}$ and
$\mathcal{P}=\delta p/E_{0}$ with $E_{0}\rightarrow 0$ as well. Therefore, we
may write

$\tilde{H}_{r_{+}}-H_{r_{+}}\sim\beta_{H}\delta\bar{M}(1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P}),$
(62)

where now, aside from the angular momentum per unit energy $\mathcal{L}$, we
also have the electric and magnetic charge per unit energy,
$\mathcal{Q},\mathcal{P}$. Next, we need to calculate
$F^{\prime}(r_{+})/\tilde{H}^{\prime}(r_{+})$. It is straightforward to show
that, for large black hole entropy $S\rightarrow\infty$, it is given by

$\frac{F^{\prime}(r_{+})}{\tilde{H}^{\prime}(r_{+})}=\frac{\mathbb{B}}{S},$
(63)

where $\mathbb{B}$ is some function of the black hole parameters and it is
dimensionless.
From the thermodynamic analysis, we obtain (by absorbing all of the
proportionality constants $\mathbb{A},\mathbb{B},2\pi$ into the definition of
$\delta\bar{M}$)

$\alpha=\frac{\beta_{H}E_{0}(1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P})}{U_{0}S},$
(64)

which is set to be fixed upon taking $E_{0}\rightarrow 0$ and
$U_{0}\rightarrow 0$ for large $S$.

#### Ultra-spinning dyonic Kerr-Sen-AdS4

We may follow similar reasoning to calculate $\alpha$ for the ultra-spinning
case. The value of $\hat{\alpha}$ is still given by the form of eq. (60), with
all functions hatted, after taking the limit of $E_{0}\rightarrow 0$ and
$\hat{U}_{0}\rightarrow 0$. The difference lies in the first-law relation,
where for the ultra-spinning case, it is now given by

$\displaystyle\hat{\tilde{H}}_{r_{+}}-\hat{H}_{r_{+}}$
$\displaystyle\propto\delta\hat{S}$ (65)
$\displaystyle=\hat{\beta}_{H}(\delta\hat{M}-\hat{\Omega}_{\varphi}\delta\hat{J}-\hat{\Phi}\delta\hat{Q}-\hat{\Psi}\delta\hat{P}-\hat{K}\delta\lambda),$

from eq. (18). The last term $\hat{K}\delta\lambda$ appears from the new
dynamical variable $\lambda$. However, if we treat $\lambda$ as a constant,
and remain unchanged by the shock waves, we have $\hat{K}\delta\lambda=0$. The
perturbation changes the black hole mass such that $\delta\hat{M}$ is also
proportional to $E_{0}$, and for the other dynamical variables, we have
$\delta\hat{J}\sim E_{0}\mathcal{L},\mathcal{Q}\sim\delta
q/E_{0},\mathcal{P}\sim\delta p/E_{0}$, and the first-law equation becomes

$\hat{\tilde{H}}_{r_{+}}-\hat{H}_{r_{+}}\sim\hat{\beta}_{H}\delta\hat{M}(1-\hat{\Omega}_{\varphi}\mathcal{L}-\hat{\Phi}\mathcal{Q}-\hat{\Psi}\mathcal{P}).$
(66)

In this case, from eq. (17), $\delta\hat{M}$ is also proportional to
$\lambda$, while all $\lambda$ inside the parenthesis of eq. (66) cancel out.
Therefore, eq. (66) is also proportional to $\lambda$ and can be expressed as
$\delta\hat{M}\sim\lambda E_{0}$.
Since the value of $\hat{F}^{\prime}(r_{+})/\hat{H}^{\prime}(r_{+})$ also
scales as the inverse of the black hole entropy $\hat{S}$ for large $\hat{S}$,
i.e.

$\frac{\hat{F}^{\prime}(r_{+})}{H^{\prime}(r_{+})}=\frac{\hat{\mathbb{B}}}{\hat{S}},$
(67)

for some $\hat{\mathbb{B}}$, the value of the shift

$\alpha=\frac{\hat{\beta}_{H}\lambda
E_{0}(1-\hat{\Omega}_{\varphi}\mathcal{L}-\hat{\Phi}\mathcal{Q}-\hat{\Psi}\mathcal{P})}{\hat{U}_{0}\hat{S}},$
(68)

does not depends on $\lambda$. This is desired because the physical scrambling
time $t_{*}$ which is computed later on should not depend on some arbitrary
parameter $\lambda$.

##  IV Lyapunov Index, Scrambling Time and its Delay

###  IV.1 Dyonic Kerr-Sen-AdS4 black hole as the holographic model

The dyonic Kerr-Sen-AdS4 black hole is an asymptotically AdS rotating black
hole with a complete commuting set of conserved charges given by (aside from
the conserved energy/mass $M$) $J,Q,P$ given in eq. (II) with their
corresponding chemical potentials are respectively given by $\mu,\Phi,\Psi$.
This eternal black hole in AdS is similar to [23] and, using the AdS/CFT
correspondence, assumed to be dual to two copies of large-$N$ CFT in its
asymptotic boundaries which are described by the thermo-field double (TFD)
state

$|\psi\rangle=\frac{1}{\sqrt{Z[\beta_{H},\mu,\Phi,\Psi]}}\sum_{n}e^{-\beta_{H}(H-\mu
J-\Phi Q-\Psi P)}|\psi_{n}\rangle_{L}\otimes|\psi_{n}\rangle_{R},$ (69)

where $L$ and $R$ denote the CFT who lives in the left and right asymptotic
boundaries respectively. The partition function is given by

$Z[\beta_{H},\mu_{i}]=\text{Tr}(e^{-\beta_{H}(H-\mu J-\Phi Q-\Psi P)}),$ (70)

where $\mu_{i}$ are the chemical potentials, and the density matrix is given
by

$\rho=\frac{e^{-\beta_{H}(H-\mu J-\Phi Q-\Psi P)}}{Z[\beta_{H},\mu_{i}]}.$
(71)

The entanglement (von Neumann) entropy corresponds to the reduced density
matrix from tracing out one of the asymptotic boundaries that recovers the
Bekenstein-Hawking entropy.
Chaos in the CFT can be scrutinized by the Lyapunov exponent $\lambda_{L}$
which appears in the decay of the out-of-time ordered correlators (OTOC)
corresponding to the TFD state in eq. (69),

$\frac{\langle W(0)V(t)W(0)V(t)\rangle}{\langle WW\rangle\langle
VV\rangle}=1-\varepsilon e^{\lambda_{L}t}+...,$ (72)

where $\varepsilon\sim 1/N$ is the perturbation parameter which is assumed to
be inversely proportional to the number of degrees of freedom of the CFT $N$.
Thus, the OTOC vanishes at

$t_{*}\sim\log N,$ (73)

which is defined to be the scrambling time. The system admits fast scrambling
for $t_{*}$ which depends logarithmically on $N$. The decay of OTOC can also
be associated with the decay of mutual information defined as

$I(A;B)=S_{A}+S_{B}-S_{A\cup B},$ (74)

where $S_{A}$ is the entanglement entropy associated with a reduced density
matrix $\rho_{A}$. This can be understood from the fact that the mutual
information provides an upper bound for correlations between two subsystems
[18]

$I(A;B)\geq\frac{(\langle\mathcal{O}_{A}\mathcal{O}_{B}\rangle-\langle\mathcal{O}_{A}\rangle\langle\mathcal{O}_{B}\rangle)^{2}}{2||\mathcal{O}_{A}||^{2}||\mathcal{O}_{B}||^{2}}.$
(75)

The dual CFT associated with this black hole has non-trivial entanglement
between the degrees of freedom in the left and right boundaries. The
entanglement entropy of a sub-region in the CFT can be calculated
holographically using the Ryu-Takayanagi (RT) or Hubeny-Rangamani-Takayanagi
(HRT) surfaces from the AdS/CFT correspondence. The RT/HRT surface is minimal
in the bulk which is homologous to the sub-region of the CFT. The entanglement
entropy is then given by the area of the RT/HRT surface divided by $4G_{N}$.
This can also be used to calculate the mutual information in eq. (74), and
from eq. (75), $I(A;B)\rightarrow 0$ indicates scrambling and thus the
scrambling time $t_{*}$ can be determined from there. In this work, we
calculate $I(A;B)$ for the CFT given by the TFD state in eq. (69)
holographically, and extract its scrambling time $t_{*}$ and the Lyapunov
exponent $\lambda_{L}$.
The scrambling phenomena happens after we mildly perturb our TFD state at a
very early time on the left asymptotic boundary. In the bulk description, the
perturbation corresponds to a shock wave that propagates from the boundary to
the black hole. The geometry can be understood as the Dray-’t Hooft solution.
The following section shows that even an infinitesimally small perturbation
can still cause scrambling, disrupting the entanglement.

###  IV.2 Lyapunov Exponent and Scrambling Time

Figure 1: Plot for the ratio between $\kappa$ and $\lambda_{L}$ with respect
to $\mathcal{L}$ up to $\mathcal{L}\rightarrow 1/\mu$. We use an example with
$m=1,a=0.5,p=0.2,q=0.1$ and varying $l$ from $l=1$ (top-left), $l=5$ (top-
right), $l=10$ (bottom-left), and $l=100$ (bottom-right). Those examples are
quite far from extremality with the outer and inner horizons ratio given by
$r_{-}/r_{+}=\\{0.199265,0.100242,0.092043,0.0887473\\}$ for
$l=\\{1,5,10,100\\}$ respectively. For the first case with $l=1$,
$\lambda_{L}$ and $\kappa$ intersect each other at $\mathcal{L}=0.759852$ or
in another perspective $\mathcal{L}\approx 0.677925/\mu$. For the other cases
($l=5,10,100$) the ratio is almost constant w.r.t $\mathcal{L}$. As $l$ gets
higher, the ratio approaches 1.

We consider two sub-regions $A$ and $B$ to be identical at their left and
right asymptotic boundaries, respectively, with the equator $\theta=\pi/2$
serving as the boundary, following [6]. The RT/HRT surface for both $S_{A}$
and $S_{B}$ does not depend on the Dray-’t Hooft shift $\alpha$ since they lie
outside the outer horizon $r_{+}$. However, the RT/HRT surface corresponding
to $S_{A\cup B}$ penetrates the horizon; we will call this surface
$\mathcal{A}_{A\cup B}$. It has a turning point inside it, then connects to
the other side of the asymptotic boundary. Thus, $\alpha$ plays an important
role here.
Due to its symmetry, the surface $\mathcal{A}_{A\cup B}$ can be obtained by
extremizing the following integral

$\mathcal{A}=2\pi\int
d\tau\sqrt{h}\sqrt{-F+F\frac{\tilde{f}^{2}\dot{r}^{2}}{\Delta^{2}}}.$ (76)

This surface consists of three segments, I, II, and III, where I stretches
from the left asymptotic boundary $(U,V)=(1,-1)$ to a point which intersects
$V=0$, i.e. $(U,V)=(U_{1},0)$, II stretches from $(U,V)=(U_{1},0)$ to the
turning point at $r_{\star}$, and II stretches from the turning point to the
intersection point at $U=0$ and $V=\alpha/2$. The area $\mathcal{A}_{A\cup B}$
is then equal to four times $\mathcal{A}$. Since the area functional does not
depend on $\tau$, there exists a conserved quantity defined as

$\frac{-F\sqrt{h}}{\sqrt{-F+F\tilde{f}^{2}\dot{r}^{2}/\Delta^{2}}}=\sqrt{F_{\star}h_{\star}},$
(77)

where $F_{\star}\equiv F(r_{\star})$ and $h_{\star}\equiv h(r_{\star})$ with
$r_{\star}$ is defined as the turning point where $\dot{r}=0$ (not to be
confused with the tortoise coordinate $r_{*}$ defined in eq. (22)).
By following similar calculations done in [6] (and earlier by [4]), we have

$\alpha=2\exp\bigg{(}Q_{1}+Q_{2}+Q_{3}\bigg{)},$ (78)

where the functions $Q_{1},Q_{2},Q_{3}$ are defined as

$\displaystyle Q_{1}$
$\displaystyle=-2\kappa\int_{\bar{r}}^{r_{0}}\frac{\tilde{f}dr}{-\Delta},$
(79) $\displaystyle Q_{2}$
$\displaystyle=2\kappa\int_{r_{\star}}^{\infty}\frac{\tilde{f}dr}{\Delta}\bigg{(}1-\frac{1}{\sqrt{1+(Fh/F_{\star}h_{\star})}}\bigg{)},$
(80) $\displaystyle Q_{3}$
$\displaystyle=2\kappa\int_{r_{\star}}^{r_{+}}\frac{\tilde{f}dr}{\Delta}\bigg{(}1+\frac{1}{\sqrt{1+(Fh/F_{\star}h_{\star})}}\bigg{)},$
(81)

where $\bar{r}$ is defined as the location where $r_{*}=0$. Both $Q_{1}$ and
$Q_{2}$ diverge as $r_{\star}\rightarrow r_{+}$, which correspond to
$\alpha\rightarrow 0$. On the other hand, $Q_{3}$ diverges as
$r_{\star}\rightarrow r_{c}$, where $r_{c}$ satisfies

$h(r_{c})F^{\prime}(r_{c})+h^{\prime}(r_{c})F(r_{c})=0.$ (82)

Figure 2: Similar plot with Figure 1 for near-extremal case. We choose
$m=1.03865,a=1,p=0.2,q=0.2$ which gives two almost degenerate horizons
$r_{+}=1.04562$ and $r_{-}=1.03103$ with ratio $r_{-}/r_{+}\approx 0.986$. In
this case, the ratio also constant w.r.t. $\mathcal{L}$ and it is close to 1.

This limit corresponds to $\alpha\rightarrow\infty$, and hence it is the limit
where the shift $\alpha$ becomes important. The area $\mathcal{A}_{A\cup B}$
can then be obtained to be proportional to $Q_{3}$, i.e.

$\displaystyle\mathcal{A}_{A\cup B}$
$\displaystyle\approx\frac{4\pi}{\kappa}\sqrt{-F(r_{c})h(r_{c})}Q_{3}$ (83)
$\displaystyle=\frac{4\pi}{\kappa}\sqrt{-F(r_{c})h(r_{c})}\log\alpha.$

By plugging in the value of $\alpha$ in eq. (64) into $\mathcal{A}_{A\cup B}$
and use $U_{0}=e^{-\kappa\tau_{0}}$, we obtain

$\displaystyle\mathcal{A}_{A\cup B}\approx 4\pi\tau_{0}$
$\displaystyle\sqrt{-F(r_{c})h(r_{c})}$ (84)
$\displaystyle+\frac{4\pi}{\kappa}\sqrt{-F(r_{c})h(r_{c})}$
$\displaystyle\times\log\bigg{(}\frac{\beta_{H}E_{0}(1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P})}{S}\bigg{)}.$

The area $\mathcal{A}_{A\cup B}$ grows linearly in time and picks up a
contribution from the perturbations $\mathcal{L},\mathcal{Q},\mathcal{P}$ in
the second term. The rate of growth of $\mathcal{A}_{A\cup B}$ depends on
$\sqrt{-F(r_{c})h(r_{c})}$, which also gives us the information about the
quantum Lyapunov exponent. Following [6], the instantaneous Lyapunov exponent
is given by

$\lambda^{\text{(min)}}_{L}=\frac{4\pi\sqrt{-F(r_{c})h(r_{c})}}{\mathcal{A}_{H}},$
(85)

where $\mathcal{A}_{H}$ is the horizon’s area. This Lyapunov exponent depends
on the angular momentum $\mathcal{L}$, as shown in Figure 1 for some
particular example with two horizons, away from extremality. This gives us
enough insight that the Lyapunov exponent $\lambda_{L}$ is approximately equal
to $\kappa$ and the ratio $\kappa/\lambda_{L}$ hardly depends on the angular
momentum $\mathcal{L}$ of the shocks. However, for smaller $l$,
$\kappa/\lambda_{L}$ becomes more likely to depends on $\mathcal{L}$ and for
the case $l=1$, we have $\kappa/\lambda_{L}<1$ for $\mathcal{L}>0.759852$. In
this particular case, the Lyapunov exponent seems to violate its upper bound
$\kappa$ for large $\mathcal{L}$. At near-extreme conditions, the difference
between $\lambda_{L}$ and $\kappa$ becomes even tighter, as shown in Figure 2.
This also indicates that the Lyapunov exponent of the dyonic Kerr-Sen-AdS4 is
also bounded by $\kappa$, and approaches its maximal value when the black hole
approaches extremality.
More detailed behaviors of the Lyapunov exponent can be seen in Figure 3. Now,
we scale up the angular momentum of the particle as
$\mathcal{L}=s\frac{r_{-}}{r_{+}}\mu^{-1}$, following [6], so that it
approaches $\mu\mathcal{L}\rightarrow s$ when the black hole becomes extremal
at $\frac{r_{-}}{r_{+}}\rightarrow 1$. Note that, in this paper, we work with
the solution in which there are only two real and positive horizons, $r_{+}$
and $r_{-}$ with $r_{+}>r_{-}$. For $l=1$, we observe the violation of
$\lambda_{L}\leq\kappa$ bound for large $\mathcal{L}$. Interestingly, such a
violation has been observed earlier in various black holes involving charges
and rotations (See, for example, [44, 45, 46, 47]), for some large values of
the rotation parameter. In this work, although we are already using
$\kappa=\kappa_{0}/(1-\mu\mathcal{L})>\kappa_{0}$, the Lyapunov exponent can
still surpass the value of $\kappa$ at large $\mathcal{L}$. This violation was
not observed in the standard (uncharged) Kerr-Sen-AdS black hole [6]. Such a
violation might occur due to the existence of extra charges such as the
dilaton and axion charges. From the plot, we also see that the Lyapunov
exponent approach $\kappa$ as $l$ becomes large, i.e. the ratio
$\kappa/\lambda_{L}$ approaches 1 in such cases. All of the values of
$\lambda_{L}$ and $\kappa$ approach zero as we reach extremality
$\frac{r_{-}}{r_{+}}$ since $T_{H}\rightarrow 0$ at this limit. There is an
exception for $s=1$, since $\mathcal{L}\rightarrow\mu^{-1}$ at extremality and
$\kappa$ becomes $\kappa_{\text{ext.}}$. This feature is interesting in
studying the behavior of chaos in extremal black holes [50].
The violation of the standard MSS chaos bound in the EMDA theory has been
investigated earlier in [47] using particle’s geodesic. They find that the
violation is more likely when the particle rotates in the opposite direction
from the black hole. Using our analysis, we also see how the Lyapunov exponent
behaves as we change the sign of $\mathcal{L}$ to its negative counterpart.
The result can be seen in Figure 4. However, in this case, the violation does
not occur. For the negative value of $\mathcal{L}$, all $\lambda_{L}$ and
$\kappa$ approach zero as the black hole approaches extremality. This is
because the numerator $(1-\mu\mathcal{L})$ will never reach zero, while the
temperature of the black hole approaches zero at extremality. We suggest that
the chaotic behavior with a negative value of $\mathcal{L}$ is also crucial in
understanding the chaotic behavior of an extremal black hole. The behavior of
$\lambda_{L}$ and $\kappa$ at extremality, whether they are zero or not,
depends on the sign of $\mathcal{L}$. This needs to be investigated further.
The violation of the bound in our case can be understood from the behavior of
the black hole’s entropy for large and small $l$. For small $l$ with $l\approx
a$, the temperature becomes

$2\pi T_{H}\approx\frac{2r_{+}}{l^{2}}-\frac{M}{r_{+}^{2}-d^{2}-k^{2}+l^{2}}.$
(86)

Here, the role of the dilaton and axion charges $d,k$ is to lower the
temperature, and hence, also lower the value of $\kappa$. On the other hand,
for large $l$ with $l\gg a$, we have

$2\pi T_{H}=\frac{r_{+}-M}{r_{+}^{2}-d^{2}-k^{2}+a^{2}}.$ (87)

In contrast with the previous case, here, the dilaton and axion charges
enlarge the black hole’s temperature. For the black hole with non vanishing
$d,k$ we expect lower value of $\kappa$ for smaller $l$, and hence the
violation of the chaos bound may occur.
From the formula of the Lyapunov exponent given by eq. (85) alone, it is hard
to see whether the violation of the $\kappa$ bound for a small value of $l$ is
caused by the dilaton or axion charges. However, this might be physically
interesting since such a violation does not occur in other analogous black
holes such as the standard Kerr-AdS4 calculations [6]. Conducting further
investigations on different types of black holes could be crucial in gaining
insights that allow us to pinpoint the source of the violation.

Figure 3: Plot of both $\lambda_{L}$ and $\kappa$ with $\mathcal{L}$ is scaled
as $\mathcal{L}=s\frac{r_{-}}{r_{+}}\mu^{-1}$ from
$\frac{r_{-}}{r_{+}}\rightarrow 0$ to $\frac{r_{-}}{r_{+}}\rightarrow 1$, for
$s=\\{0.1,0.5,0.9,1\\}$. We vary $l$ from $l=1$ (top-left), $l=5$ (top-right),
$l=10$ (bottom-left), and $l=100$ (bottom-right). We gradually decrease the
black hole’s mass while keeping other parameters fixed: $a=0.5,p=0.2,q=0.1$.
The corresponding mass are $m\in[1.5,0.65474]$ for $l=1$, $m\in[2,0.55163]$
for $l=5$, $m\in[1,0.54729]$ for $l=10$, and $m\in[1,0.54582]$ for $l=100$.
Figure 4: Plot of both $\lambda_{L}$ and $\kappa$ for $l=1$, with remaining
parameters are identical to the ones in Figure 3 (top-left). In this case, we
reverse the sign of $\mathcal{L}$ by performing $s\rightarrow-s$.

#### Ultra-spinning Case

Figure 5: Plot similar to Figure 1 for the ultra-spinning cases. We use
$m=100,p=0.2,q=0.1,l=1$ for the first case (top) which violates the RII,
$\mathcal{R}=0.995016$. For the second case (bottom), we use
$m=5,p=2,q=1,l=0.5$ which does not violate the RII, $\mathcal{R}=1$. Both
cases are non-extremal with $r_{-}/r_{+}=\\{0.000915922,0.478328\\}$ for the
first and the second cases respectively.

For the ultra-spinning case, we should be careful when integrating the
transverse volume since now $\phi$ takes arbitrary periodicity, which is
denoted by $\lambda$. In our derivation around eq. (37), we define
$\hat{\eta}$ such that we recover the horizon area when we integrate $\hat{z}$
from 0 to $\lambda$. Therefore, the area functional in the ultra-spinning case
should be written as

$\mathcal{A}=\lambda\int
d\tau\sqrt{\hat{h}}\sqrt{-\hat{F}+\hat{F}\frac{\hat{\tilde{f}}^{2}\dot{r}^{2}}{\hat{\Delta^{2}}}}.$
(88)

Following similar derivations as before, we obtain the area
$\mathcal{A}_{A\cup B}$ in the limit of $r_{\star}\rightarrow r_{c}$ which
corresponds to large $Q_{3}$. The value of $r_{c}$ can be obtained by solving

$\hat{h}(r_{c})\hat{F}^{\prime}(r_{c})+\hat{h}^{\prime}(r_{c})\hat{F}(r_{c})=0.$
(89)

Thus, the area $\mathcal{A}_{A\cup B}$ is again proportional to $Q_{3}$:

$\displaystyle\mathcal{A}_{A\cup B}$
$\displaystyle\approx\frac{2\lambda}{\kappa}\sqrt{-\hat{F}(r_{c})\hat{h}(r_{c})}Q_{3}$
(90)
$\displaystyle=\frac{2\lambda}{\kappa}\sqrt{-\hat{F}(r_{c})\hat{h}(r_{c})}\log\hat{\alpha}.$

Although the area $\mathcal{A}_{A\cup B}$ depends on $\lambda$, interestingly,
the minimal instantaneous Lyapunov index $\lambda_{L}^{\text{(min)}}$ does not
depend on $\lambda$. This is so because the black hole horizon area
$\mathcal{A}_{H}$ also depends linearly on $\lambda$ and the two cancel each
other. For the ultra-spinning case, we can write the Lyapunov index as

$\lambda_{L}^{(\text{min})}=\frac{\sqrt{-\hat{F}(r_{c})\hat{h}(r_{c})}}{(r_{+}^{2}-d^{2}-k^{2}+l^{2})}.$
(91)

We can then study the behavior of $\lambda_{L}^{(\text{min})}$ using graphs
for the ultra-spinning black hole and compare it with the value of
$\hat{\kappa}$.
It is known that for ultra-spinning black holes, there exist super-entropic
cases which violate the Reverse Isoperimetric Inequality (RII) [51], which is
given by

$\mathcal{R}=\bigg{(}\frac{r_{+}^{2}}{r_{+}^{2}-d^{2}-k^{2}+l^{2}}\bigg{)}^{1/6}\geq
1,$ (92)

for the dyonic Kerr-Sen-AdS4 black hole [42, 40, 41]. An ultra-spinning dyonic
Kerr-Sen-AdS4 black hole always violates the RII for $0\leq q^{2}+p^{2}<2ml$
or $0\leq d^{2}+k^{2}<l^{2}$. However, for other values of $q,p$ (or
equivalently $d,k$), we can have cases in which the RII is not violated. In
this work, we see the behavior of $\lambda_{L}^{(\text{min})}$ and
$\hat{\kappa}$ for two cases: obeys RII and violates RII. The result is shown
in Figure 5. It is interesting that for both cases,
$\lambda_{L}^{(\text{min})}$ is bounded by $\kappa$ and the ratio
$\kappa/\lambda_{L}^{(\text{min})}$ does not depends on $\mathcal{L}$ up to
the fourth decimal places. For the case which violates the RII, the gap
between $\kappa$ and $\lambda_{L}^{(\text{min})}$ is wider than the one which
does not violate the RII. It is interesting to learn more about the
relationship between the Lyapunov index and the violation of the RII for
ultra-spinning black holes.

Figure 6: Plot similar to Figure 3 for ultra-spinning black hole. The fixed
parameters are $a=0.5,p=0.2,q=0.1$ while decreasing the mass to approach
extremality.

After we obtain the area connecting two boundaries $\mathcal{A}_{A\cup B}$,
now we are ready to calculate the Mutual Information:

$\displaystyle I(A;B)\approx$ $\displaystyle
S_{A}+S_{B}-\frac{\pi\tau_{0}}{G_{N}}\sqrt{-F(r_{c})h(r_{c})}$ (93)
$\displaystyle-\frac{\pi}{\kappa G_{N}}\sqrt{-F(r_{c})h(r_{c})}$
$\displaystyle\times\log\bigg{(}\frac{\beta_{H}E_{0}(1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P})}{S}\bigg{)}.$

The scrambling time $\tau_{*}$ obtained from $I(A;B)\rightarrow 0$ is then
given by

$\displaystyle\kappa\tau_{*}^{(\mathcal{L},\mathcal{Q},\mathcal{P})}\approx$
$\displaystyle\log
S+\frac{\kappa(\mathcal{A}_{A}+\mathcal{A}_{B})}{4\pi\sqrt{-F(r_{c})h(r_{c})}}$
(94)
$\displaystyle+\log\frac{1}{1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P}},$

while the ultra-spinning version, following a similar derivation, is given by

$\displaystyle\hat{\kappa}\hat{\tau}_{*}^{(\mathcal{L},\mathcal{Q},\mathcal{P})}\approx$
$\displaystyle\log\frac{\hat{S}}{\lambda}+\frac{\hat{\kappa}(\mathcal{A}_{A}+\mathcal{A}_{B})}{4\pi\sqrt{-\hat{F}(r_{c})\hat{h}(r_{c})}}$
(95)
$\displaystyle+\log\frac{1}{1-\hat{\Omega}_{\varphi}\mathcal{L}-\hat{\Phi}\mathcal{Q}-\hat{\Psi}\mathcal{P}}.$

For a black hole with large entropy, the first term is substantially larger
than the rest and thus the scrambling time is approximately given by

$\tau_{*}^{(\mathcal{L},\mathcal{Q},\mathcal{P})}\sim\frac{1}{\kappa}\log
S,\;\;\;\hat{\tau}_{*}^{(\hat{\mathcal{L}},\hat{\mathcal{Q}},\hat{\mathcal{P}})}\sim\frac{1}{\hat{\kappa}}\log\frac{\hat{S}}{\lambda}.$
(96)

This indicates that the dyonic Kerr-Sen-AdS4 and its ultra-spinning
counterpart also follow the fast scrambling conjecture [14]. For the ultra-
spinning case, $\hat{\tau}_{*}$ actually does not depends on $\lambda$ because
the entropy $\hat{S}$ is proportional to $\lambda$. For an ultra-spinning
black hole with large entropy $\hat{S}\rightarrow\infty$ while keeping the
parameter $\lambda$ fixed, the term proportional to $\log\lambda^{-1}$ can be
ignored, and hence the parameter $\lambda$ does not contribute to the
scrambling time.
However, despite being much smaller than the leading $\log S$ term, the last
term of eq. (94) and eq. (95) provide insight into the delay of the scrambling
time compared to the standard scrambling time when
$\mathcal{Q},\mathcal{P}=0$. Such a delay was first investigated by [8] for a
non-rotating charged black hole. This term does depend on the thermodynamic
potentials (or chemical potentials) of the black hole: $\mu,\Phi,\Psi$, and
the external perturbations $\mathcal{Q},\mathcal{P}$. If the trajectory of the
shock waves bounces near the horizon, this term can be large. This will be
further explored in the following subsection.
The second term of eq. (96) does not scale with entropy and hence it is non-
extensive since both $\mathcal{A}_{A}+\mathcal{A}_{B}$ and
$\sqrt{-F(r_{c})h(r_{c})}/\kappa$ have dimensions of the area. Furthermore, it
also not depends on the perturbations $\mathcal{Q},\mathcal{P}$ and also the
plot in Figures 1 and 2 imply that it is also approximately constant in
$\mathcal{L}$, at least for large value of $l$. For the ultra-spinning case,
those arguments also hold (see Figure 5). Therefore, the second term, along
with the first term, will not contribute to the scrambling delays.

###  IV.3 Scrambling time delays of the dyonic Kerr-Sen-AdS4

In this section, we investigate the phenomena of scrambling delays in the
dyonic Kerr-Sen-AdS4 and its ultra-spinning counterpart. Scrambling time
delays come from the difference between the scrambling time from (electrically
and magnetically) charged and rotating shock waves
$\tau_{*}^{(\mathcal{L},\mathcal{Q},\mathcal{P})}$ and the standard scrambling
time from rotating neutral shock waves $\tau_{*}^{(\mathcal{L},0,0)}$, or in
other words,

$\displaystyle\Delta\tau_{*}$
$\displaystyle\equiv\tau_{*}^{(\mathcal{L},\mathcal{Q},\mathcal{P})}-\tau_{*}^{(\mathcal{L},0,0)}$
(97)
$\displaystyle=\frac{1}{\kappa}\log\frac{1-\mu\mathcal{L}}{1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P}},$
$\displaystyle\Delta\hat{\tau}_{*}$
$\displaystyle\equiv\hat{\tau}_{*}^{(\mathcal{L},\mathcal{Q},\mathcal{P})}-\hat{\tau}_{*}^{(\mathcal{L},0,0)}$
(98)
$\displaystyle=\frac{1}{\hat{\kappa}}\log\frac{1-\hat{\Omega}_{\varphi}\mathcal{L}}{1-\hat{\Omega}_{\varphi}\mathcal{L}-\hat{\Phi}\mathcal{Q}-\hat{\Psi}\mathcal{P}},$

for the dyonic Kerr-Sen-AdS4 and its ultra-spinning counterpart respectively.
From the first law of black hole thermodynamics, a positive $\delta S$ implies
that the value within the logarithm is greater than one. Therefore,
$\Delta\tau$ is positive, which means that it prolongs the scrambling time. We
do not compare $\tau_{*}^{(\mathcal{L},\mathcal{Q},\mathcal{P})}$ with
$\tau_{*}^{(0,0,0)}$ since $\kappa$ depends on $\mathcal{L}$ and therefore
$\Delta\tau_{*}$ (and $\Delta\hat{\tau}_{*}$) becomes zero already as
$\mathcal{Q},\mathcal{P}\rightarrow 0$ even though $\mathcal{L}$ is non zero.
In [8], the time difference corresponds to the delay of the start of the
scrambling process, or in other words, the scrambling delay. They found that
the time difference (similar to $\Delta\tau$) for a Reissner-Nordström black
hole is equal to the difference between the bounce time $t_{b}$ and the time
when the shell is sent from the left boundary $t_{wL}$, for time difference
much larger than the thermal time $\beta$. Using the quantum circuit
description, the time difference $t_{d}=-t_{wL}-t_{b}$ is then interpreted as
the scrambling delay in [8].
In this work, we show that $\Delta\tau_{*}$ and $\Delta\hat{\tau}_{*}$ also
corresponds to the scrambling delays for the dyonic Kerr-Sen-AdS4 black hole
and its ultra-spinning counterpart. We begin by calculating the stress tensor
for the rotating charged shockwaves using the null junction formalism [31,
32], which is given by (Appendix A for derivation):

$S^{\mu\nu}\propto[\sigma(r)]k^{\mu}k^{\nu},$ (99)

up to some proportionality constant, where

$[\sigma(r)]\equiv\sigma_{R}(r)-\sigma_{L}(r),$ (100)

with the $L$ and $R$ indexes correspond to the function in the perturbed left
region with parameters $M+\delta M,q+\delta q,p+\delta p,J+\delta J$ and
unperturbed right region with original parameters $M,q,p,J$. For the dyonic
Kerr-Sen-AdS4 black hole, $\sigma_{i}(r)$ is given by

$\sigma_{i}(r)=\frac{\Delta_{i}(r)h^{\prime}_{i}(r)}{\tilde{f}_{i}(r)},$ (101)

with $i=L,R$. If $[\sigma(r)]<0$, the null energy condition is violated.
Therefore, the location of the bounce, $r_{b}$ can be obtained from
$[\sigma(r_{b})]=0$.
Although it is difficult to exactly solve $[\sigma(r_{b})]=0$, we show that we
do not need the explicit form of $r_{b}$ to find $\tau_{b}$ (the ”bounce
time”). The increase of the black hole parameters $\delta M,\delta Q,\delta
P,\delta J$ is directly related to the increase of the black hole entropy
$\delta S$ by the thermodynamic first-law relation. To make sure that the
perturbations increase the black hole entropy according to the second law,
$\delta S>0$, thus the perturbations also increase the horizon’s radius as
$r_{+}\delta r_{+}\sim\delta S$. For small perturbation parameters $\delta
M,\delta q,\delta p,\delta J$, to the first order approximation,
$[\sigma(r_{b})]=0$ can be written as

$\sigma(S+\delta
S)-\sigma(S)=0\;\;\;\Longrightarrow\;\;\;\frac{\partial\sigma}{\partial
S}\delta S=0.$ (102)

We are interested in the regime where $\Delta\tau_{*}$ is particularly large,
much larger than the thermal time $\beta\sim\frac{1}{\kappa}$. This can be
achieved when the bounce happens very close to the horizon, or $r_{b}\approx
r_{+}$. Therefore, eq. (102) can be expanded near $r_{+}$ such as

$\bigg{(}\frac{\partial\sigma}{\partial S}\delta
S\bigg{)}\bigg{|}_{r_{+}}+(r_{b}-r_{+})\frac{d}{dr_{b}}\bigg{(}\frac{\partial\sigma}{\partial
S}\delta S\bigg{)}\bigg{|}_{r_{+}}=0.$ (103)

Using eq. (101) and the horizon’s temperature formula, the first term of eq.
(103) can be written as

$\displaystyle\bigg{(}\frac{\partial\sigma}{\partial S}\delta
S\bigg{)}\bigg{|}_{r_{+}}$ $\displaystyle=\frac{\partial\sigma}{\partial
r_{b}}\frac{\partial r_{b}}{\partial S}\delta S\bigg{|}_{r_{+}}$ (104)
$\displaystyle=\frac{2\Xi}{r_{+}}\frac{h^{\prime}(r_{+})}{1-\mu\mathcal{L}}T_{H}\delta
S,$

which has a dimension of $r_{+}$. Therefore,
$\frac{d}{dr_{b}}\bigg{(}\frac{\partial\sigma}{\partial S}\delta
S\bigg{)}\bigg{|}_{r_{+}}$ in the second term of eq. (103) is dimensionless.
We may extract the location of the bounce $r_{b}$ to be (approximately)

$r_{b}-r_{+}=\tilde{\mathbb{R}}T_{H}\delta
S=\mathbb{R}(1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P}),$ (105)

where $\mathbb{R}$ has a dimension of $r_{+}$ and in the order of
$\mathcal{O}(\delta S)$. We see that $r_{b}\rightarrow r_{+}$ corresponds to
setting the condition such that
$(1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P})\rightarrow 0$. From eq.
(IV.3), setting the bounce near the horizon corresponds to enlarging the
scrambling time difference $\Delta\tau_{*}$.
Using the Kruskal coordinate relation evaluated at $r_{b}$, we have

$U_{b}V_{b}=e^{2\kappa r_{*}(r_{b})},$ (106)

However, since we set the bounce to be close to the horizon, we have

$r_{*}(r_{b})\approx\frac{1}{2\kappa}\ln\bigg{|}\frac{r_{b}-r_{+}}{\mathbb{R}^{\prime}}\bigg{|}+C,$
(107)

with $\mathbb{R}^{\prime}$ is some constant of dimension $r_{+}$ and $C$ is a
dimensionless constant that depends on the black hole geometry. Also, for
$r_{b}\approx r_{+}$, we can approximate $U_{b}\approx U_{0}$, where $U_{0}$
is the location where the original rotating shock wave is sent. Using
equations (105), (106), (107), as well as the relation between $U_{0}$ and
$\alpha$, we obtain

$V_{b}\approx\frac{\alpha
S}{\beta_{H}E_{0}}\frac{\mathbb{R}}{\mathbb{R}^{\prime}}e^{C},$ (108)

where the factor $\frac{\mathbb{R}}{\mathbb{R}^{\prime}}e^{C}$ is an
$\mathcal{O}(1)$ dimensionless parameter which depends on the black hole’s
geometry and does not scale with $S$ or $\delta S$. We see that $V_{b}\gg 1$
since $\alpha\sim\mathcal{O}(1)$ and $\beta_{H}E_{0}/S\rightarrow 0$.
Again, by plugging in the value of $\alpha$ in eq. (64) into eq. (108), we
obtain the time delay $\tau_{d}=\tau_{0}-\tau_{b}$ for dyonic Kerr-Sen-AdS4
black hole, which is given by

$\displaystyle\tau_{d}=$
$\displaystyle\frac{1}{\kappa}\log\bigg{(}\frac{1}{1-\mu\mathcal{L}-\Phi\mathcal{Q}-\Psi\mathcal{P}}\bigg{)}$
(109)
$\displaystyle+\frac{1}{\kappa}\log\bigg{(}\frac{\mathbb{R}^{\prime}}{\mathbb{R}}e^{-C}\bigg{)},$
$\displaystyle=$
$\displaystyle\Delta\tau_{*}+\frac{1}{\kappa}\log\bigg{(}\frac{1}{1-\mu\mathcal{L}}\bigg{)}+\frac{1}{\kappa}\log\bigg{(}\frac{\mathbb{R}^{\prime}}{\mathbb{R}}e^{-C}\bigg{)}.$

Since $r_{b}\approx r_{+}$, the first term dominates all the remaining terms
and we see that $\tau_{d}$ also approximately equals to $\Delta\tau_{*}$ for
the dyonic Kerr-Sen-AdS4 black hole. The remaining terms are in the order of
thermal time $\beta\sim\frac{1}{\kappa}$ and do not scale like
$\Delta\tau_{*}$ if we take $r_{b}\approx r_{+}$. If we set
$\mathcal{L}\rightarrow 0$, i.e. for the non-rotating shock waves, the result
reduces to the scrambling delay found in [8]. This indicates that the role of
the charges (electric and magnetic) in the dyonic Kerr-Sen-AdS4 black hole is
more to delay the start of the scrambling, rather than to prolong the
scrambling process, at least for $r_{b}\approx r_{+}$. For the ultra-spinning
black hole, the scrambling delay is also similar but with all of the
parameters hatted:

$\hat{\tau}_{d}=\Delta\hat{\tau}_{*}+\frac{1}{\hat{\kappa}}\log\bigg{(}\frac{1}{1-\hat{\Omega}_{\varphi}\mathcal{L}}\bigg{)}+\frac{1}{\hat{\kappa}}\log\bigg{(}\frac{\hat{\mathbb{R}}^{\prime}}{\hat{\mathbb{R}}}e^{-\hat{C}}\bigg{)}.$
(110)

With vanishing $\mathcal{Q},\mathcal{P}$, only the terms with
$\sim\mathcal{O}(\beta)$ remain. Away from extremality, these terms are small.
However, once we approach the extremal limit, the second and the third terms
of $\tau_{d}$ may diverge, and therefore $\Delta\tau_{*}$ can only be
interpreted as the scrambling delay away from extremality. Furthermore, it is
interesting to ask whether the bounce still happens for
$\mathcal{Q},\mathcal{P}\rightarrow 0$ but we keep $\mathcal{L}$. From the
previous analysis, $\tau_{d}$ becomes very small for small $\beta$ which means
that we can send the perturbation from the right asymptotic at the time close
to $\tau_{0}$ and still meet the left perturbations inside the horizon. This
indicates that the scrambling process starts immediately and the delay almost
did not happen. Therefore, the charges $\mathcal{Q},\mathcal{P}$ are the ones
that play the important role in delaying the scrambling process.

##  V Conclusions

We study chaos in the dyonic Kerr-Sen-AdS4 black hole by calculating the
scrambling time $\tau_{*}$ and find that the leading term is logarithmic in
entropy $\tau_{*}\sim\log S$ (see eq. (96)). We use holographic entanglement
entropy calculations to calculate the mutual information $I(A;B)$ and obtain
both scrambling time $\tau_{*}$ and the Lyapunov exponent $\lambda_{L}$ by
perturbing the black hole with rotating and charged shock waves. This work
generalizes the chaotic behavior of black holes to the rotating black holes in
the EMDA theory. Both $\tau_{*}$ and $\lambda_{L}$ depend on the new
parameters, the dilaton and axion charges.
From Figure 3, we show that the instantaneous minimal Lyapunov exponent, for
most of the cases, is bounded by $\kappa=2\pi T_{H}/(1-\mu\mathcal{L})$, and
approaches $\kappa$ when $\frac{r_{-}}{r_{+}}\rightarrow 1$, i.e. when the
black hole approaches extremality. Furthermore, for larger values of AdS
length $l$, the Lyapunov exponent also approaches $\kappa$ and saturates the
bound. Only in the case where $l=1$ that we found the Lyapunov exponent
exceeds $\kappa$, for some large values of $\mathcal{L}$. This violation may
be caused by the existence of charges, especially the dilaton and axion
charges. The fact that the Lyapunov exponent is approximately equal to
$\kappa$ also supports the fast scrambling conjecture. For nonrotating black
holes, the Lyapunov exponent should saturate the MSS bound [29], which is
given by the black hole temperature. We see that from earlier works on
rotating black holes, the upper bound for the Lyapunov index is modified by a
factor of $\frac{1}{1-\mu\mathcal{L}}$, which can be greater than $2\pi T_{H}$
[33, 5, 6].
We suspect that the cause for the violation of the $\kappa$ bound here is due
to the existence of the dilaton or axion charges and in this work, we show
such evidence using the plot given by Figure 3. We plan to do the analytical
calculations of the correction to the Lyapunov bound with the existence of
dilaton and axion charges in future works. The fact that such a violation does
not occur in the standard Kerr-AdS4 black hole [6] might help us to pinpoint
the origin of the violation and this carries significant physical interests.
In future works, we also plan to show that such a violation does not occur in
a Kerr-NUT-AdS black hole [52] at similar conditions (for a small value of
$l$). Furthermore, an investigation of the butterfly velocity and entanglement
velocity due to localized shock waves [53] in the Kerr-Sen-AdS4 background
might also help us to gain insight into the violation of the bound 111We thank
V. Jahnke for pointing this out and bringing the reference about the butterfly
velocity of a static anisotropic black brane.. We plan to investigate it in
future works as well.
We also calculate the scrambling time and the Lyapunov exponent for the ultra-
spinning version of the dyonic Kerr-Sen-AdS4, i.e. the solution where the
rotation parameter has its maximal value, $a\rightarrow l$. The ultra-spinning
solution provides us with new functions that cannot be obtained by simply
taking the limit of $a\rightarrow l$ from the standard dyonic Kerr-Sen-AdS4.
Due to the logarithmic behavior of the scrambling time, the ultra-spinning
version also admits fast scrambling. However, the ratio between $\kappa$ and
the Lyapunov exponent is quite large. For example, we have
$\kappa/\lambda_{L}=\mathcal{C}\sim 12$ for the case which violates the RII
with $\mathcal{R}=0.995016$ and $\mathcal{C}\sim 2.5$ for the case with
$\mathcal{R}=1$ (see Figure 5). From Figure 6, we see that even though $l$ is
small, the violation of the chaos bound does not present in the ultra-spinning
case. We conclude that the ultra-spinning counterpart of the dyonic Kerr-Sen-
AdS4 black hole also admits chaotic behavior but the Lyapunov exponent is
relatively smaller than the standard black hole, with larger $\mathcal{C}$.
Due to the existence of charges, both in the black hole and the shock waves,
we also have the scrambling time delay $\tau_{d}$. The subleading term of
$\tau_{*}$ in eq. (94) and eq. (95) represent the difference between the
scrambling time of neutral black holes with the charged ones, and is denoted
by $\Delta\tau_{*}$. For a perturbation that obeys $\delta S>0$, the
aforementioned term is positive and hence prolongs the scrambling process.
However, since the shock wave bounces inside the horizon at $r_{b}$ (that is
assumed to be very close to the horizon), the role of the subleading term is
to delay the onset of scrambling, following the analysis of [8]. We show that
the scrambling process of a dyonic Kerr-Sen-AdS4 black hole is also delayed by
calculating the time that is needed to send a signal from the right asymptotic
$\tau_{d}$ for the signal to meet the shock waves inside the black hole
interior. The difference between $\tau_{d}$ and $\tau_{0}$, i.e. the time when
the shock wave is sent from the left asymptotic, is equal to $\Delta\tau_{*}$
up to some corrections in the order of $\beta$. For the bounce to happen near
the horizon, $\Delta\tau_{*}$ is much larger than the thermal time $\beta$.
The scrambling delay time depends on the charges of the shock waves,
$\mathcal{Q}$ and $\mathcal{P}$. This is the only function that depends on
$\mathcal{Q},\mathcal{P}$, indicating that the role of the charges is to delay
the scrambling. If we take $\mathcal{L}\rightarrow 0$ (and also
$\mathcal{P}\rightarrow 0$), we recover the scrambling delay found in the non-
rotating Reissner-Nordström black hole [8].
In conclusion, our calculations support the fast scrambling conjecture for the
dyonic Kerr-Sen-AdS4 black hole to some extent. However, since our
calculations only focus on the non-extremal and near-extremal limits of the
black hole, it is also important to understand the chaotic behavior of the
extremal black hole. It is known that an extremal black hole and a non-
extremal black hole are two distinct objects, and taking the extremal limit of
a non-extremal black hole does not obtain the properties of the actual
extremal black hole [55, 25]. Therefore, the chaotic behavior of extremal
dyonic Kerr-Sen-AdS4 black hole deserves more investigation for future works.

### Appendix: Stress tensor from null-shell formalism

In this Appendix, we derive the stress tensor of the rotating charged shock
waves using the null-shell formalism in [32] and hence we adopt their notation
here. Starting with metric in eq. (57), we construct the normal vector to the
shocks following constant $u=r_{*}-\tau$ path,
$k^{\mu}=g^{\mu\nu}\partial_{\nu}u$, which is given by

$k^{\mu}\partial_{\mu}=\frac{1}{F}\bigg{(}\frac{\Delta}{\tilde{f}}\partial_{r}+\partial_{\tau}-h_{\tau}\partial_{z}\bigg{)},$
(111)

while the other tangent generator is given by $e_{z}^{\mu}=\delta_{z}^{\mu}$.
From here, we can obtain the transverse null vector by solving

$N_{\mu}N^{\mu}=0,\;\;\;N_{\mu}k^{\mu}=-1,\;\;\;N_{\mu}e^{\mu}_{z}=0,$ (112)

which results in

$N_{\mu}dx^{\mu}=\frac{F}{2}\bigg{(}d\tau+\frac{\tilde{f}}{\Delta}dr\bigg{)}.$
(113)

The transverse extrinsic curvature can then be calculated from

$C_{AB}=\nabla_{\nu}N_{\mu}e^{\mu}_{A}e^{\nu}_{B}=\frac{\Delta
h^{\prime}}{2\tilde{f}}\delta_{A}^{z}\delta_{B}^{z}\equiv\frac{\sigma}{2}\delta_{A}^{z}\delta_{B}^{z},$
(114)

with $h^{\prime}(r)=dh(r)/dr$. The difference between $C_{AB}$ evaluated in
the perturbed manifold with the unperturbed one is proportional to the stress-
energy tensor, as given by eq. (99). For vanishing all black hole parameters
except those corresponding to the Reissner-Nordström black hole, the result
recovers the stress tensor in [8].

## Acknowledgement

FPZ would like to thank Kemenristek, the Ministry of Research, Technology, and
Higher Education, Republic of Indonesia for financial support. HLP would like
to thank GTA Institut Teknologi Bandung for financial support. HLP would like
to thank the members of the Theoretical Physics Groups of Institut Teknologi
Bandung for their hospitality. S. A. was supported by an appointment to the
Young Scientist Training Program at the Asia Pacific Center for Theoretical
Physics (APCTP) through the Science and Technology Promotion Fund and Lottery
Fund of the Korean Government. This was also supported by the Korean Local
Governments - Gyeongsangbuk-do Province and Pohang City.

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