††institutetext: aTheory Division, Saha Institute of Nuclear Physics,
Homi Bhaba National Institute (HBNI),
1/AF, Bidhannagar, Kolkata 700064, India.
bInstitute for Theoretical Physics, TU Wien,
Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria.

# Fast Scrambling of mutual information in Kerr-AdS${}_{\textbf{5}}$

Vinay Malvimata, Rohan R. Poojaryb<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the disruption of mutual information in a TFD state dual to a Kerr
black hole with equal angular momenta in $AdS_{5}$ due to an equatorial
shockwave. The shockwave respects the axi-symmetry of the Kerr geometry with
specific angular momenta $\mathcal{L}_{\phi_{1}}$ & $\mathcal{L}_{\phi_{2}}$.
The sub-systems considered are hemispheres in the $left$ and the $right$ dual
CFTs with the equator of the $S^{3}$ as their boundary. We compute the change
in the mutual information by determining the growth of the HRT surface at late
times. We find that at late times leading up to the scrambling time the
minimum value of the instantaneous Lyapunov index $\lambda_{L}^{(min)}$ is
bounded by $\kappa=\frac{2\pi T_{H}}{(1-\mu\,\mathcal{L}_{+})}$ and is found
to be greater than $2\pi T_{H}$ in certain regimes with $T_{H}$ and $\mu$
denoting the black hole’s temperature and the horizon angular velocity
respectively while
$\mathcal{L}_{+}=\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}$. We also find
that for non-extremal geometries the null perturbation obeys
$\mathcal{L}_{+}<\mu^{-1}$ for it to reach the outer horizon from the $AdS$
boundary. The scrambling time at very late times is given by
$\kappa\tau_{*}\approx\log\mathcal{S}$ where $\mathcal{S}$ is the Kerr
entropy. We also find that the onset of scrambling is delayed due to a term
proportional to $\log(1-\mu\,\mathcal{L}_{+})^{-1}$ which is not extensive and
does not scale with the entropy of Kerr black hole.

##  1 Introduction

It has been well known in recent years that quantum chaos exhibited by
large-$N$ thermal field theories is holographically dual to the near horizon
dynamics of black holes Shenker:2013pqa . A hallmark of such systems is that
they exhibit what is known as fast scrambling $i.e.$ the time taken $t_{*}$
for any initial $\mathcal{O}(1)$ perturbation to grow is proportional to the
degrees of freedom $t_{*}\sim\log\mathcal{S}$ Hayden:2007cs ; Sekino:2008he .
Black holes in quantum gravity were then conjectured to be among the fastest
scramblers. This expectation was validated by Shenker and Stanford for the
case of static black holes in AdS by analysing the disruption of the finely
tuned entanglement between the 2 CFTs constituting the TFD state dual to the
black hole geometry Shenker:2013pqa . This was also followed by the analysis
of the _out-of-time-ordered-correlator_ (OTOC) in the field theory computed
holographically for static BTZ which exhibited an exponential decay

$\frac{\langle W(0)V(t)W(0)V(t)\rangle}{\langle WW\rangle\langle
VV\rangle}=1-\epsilon e^{\lambda_{L}t},\hskip 8.5359pt\lambda_{L}=2\pi T_{H}$
(1)

where $\epsilon\sim c^{-1}$ and $T_{H}=\beta^{-1}$ is the temperature of the
geometry Shenker:2014cwa . The exponential index $\lambda_{L}$ is what is
termed as the Lyapunov index and generally determines how quickly the
scrambling of information takes place. Using the analyticity property of the
four point correlators of a thermal QFT in the complex time plane it was
famously shown by Maldacena, Shenker & Stanford Maldacena:2015waa  that the
Lyapunov index for time scales leading up to the scrambling time is bounded to
be

$\lambda_{L}\leq\frac{2\pi}{\beta}$ (2)

thus bounding the scrambling time of such systems. The analysis of the 1d
solvable strongly coupled SYK-like models by Maldacena & Stanford
Maldacena:2016hyu  had revealed similar chaotic behaviour with
$\lambda_{L}=2\pi/\beta$. Here the 1d SYK model possesses a 1d
reparametrization symmetry at zero temperature which is spontaneously broken
at small temperatures, giving rise to an effective theory of 1d
reparametrizations in the IR governed by the Schwarzian action. The
holographic dual of such a 1d theory was found by analysing a 2d dilaton
gravity theory in $AdS_{2}$ first analysed by Jackiw and Teitelboim called the
JT theory Jensen:2016pah ; Maldacena:2016upp . This theory reproduced the
Schwarzian action as its boundary effective action for scalar perturbations
corroborating the behaviour (2) for scalar correlators at its 1d boundary. It
was systematically shown that the JT theory effectively describes dynamics of
small departures from extremality due to excess mass111In all these analysis
the rest of the black hole charges are held constant. for a wide class of
black holes Nayak:2018qej ; Moitra:2018jqs ; Castro:2018ffi ; Moitra:2019bub ;
Ghosh:2019rcj ; Castro:2021fhc ; Castro:2021csm . Here, the JT theory is
obtained as an effective near horizon gravity theory after dimensionally
reducing the higher dimensional Einstein-Hilbert action (with or without
cosmological constant) about a near extremal black hole with excess mass. It
is also worth noting that chaotic behaviour is observed in holographic systems
associated with the dynamics of a string stretched in a black hole back
grounddeBoer:2017xdk , and associated modes have also been identified in this
case and for similar theories involving branes with a Schwarzian effective
action being associated with such modes too Banerjee:2018kwy ;
Banerjee:2018twd . This further suggests that the chaotic behaviour associated
with fast scrambling is a hallmark of holographic theories of gravity.

The case for black holes with rotation is much more subtle. The analysis of
the four point OTOC in BTZ revealed the existence of 2 Lyapunov exponents,
$\lambda_{L}=\frac{2\pi}{\beta_{\pm}}$, each corresponding to the left and
right temperatures of the dual CFT2 of which one survives the extremal limit
Poojary:2018esz ; Stikonas:2018ane ; Jahnke:2019gxr . The generalization of
the arguments leading up to the MSS bound for the case of thermal QFT with
chemical potential $\mu$ revealed that $\lambda_{L}$ obeys a bound of the form

$\lambda_{L}\leq\frac{2\pi T_{H}}{(1-\mu/\mu_{c})}$ (3)

with $\mu_{c}$ being the maximum possible value for the chemical potential
Halder:2019ric . This bound is saturated by the larger of the 2 temperatures
of the CFT2. Similar behaviour was also observed for the case of string
stretched in rotating BTZ and $AdS_{4}$ Kerr geometries, in the former case an
effective theory was also obtained Banerjee:2019vff .

“Pole skipping” is yet another phenomena that is closely associated to chaotic
phenomena in fast scrambling systems where retarded energy 2pt functions in
frequency space have been observed to skip poles at frequency
$\omega_{L}=i\lambda_{L}$ and wave-number $k=\pm iv_{B}$, $v_{B}$ being the
butterfly velocity Grozdanov:2017ajz ; Blake:2017ris . Static branes in a
black hole background in $AdS$ have been shown to exhibit pole skipping at
$\lambda_{L}=2\pi T_{H}$ with an effective theory of hydrodynamics associated
with them Blake:2017ris ; Blake:2018leo . For the case of rotating black holes
in BTZ pole skipping was observed for the 2 temperatures of the dual CFT2
Liu:2020yaf . It is also worth noting that the pole skipping analysis by
investigating ingoing modes of the graviton in rotating Kerr $AdS_{4}$
revealed $\lambda_{L}=2\pi T_{H}$ Blake:2021wqj .

However the scrambling time for the 4pt OTOC in a CFT dual to a rotating BTZ
revealed that the long time averaged exponent is
$|\lambda_{L}|=\frac{2\pi}{\beta}$ Mezei:2019dfv ; Craps:2020ahu , while the
growth in time of the OTOC exhibited a sawtooth like pattern with
$\lambda_{L}^{(max)}$ being the larger of the 2 temperatures of the CFT2.
These were also reproduced from the CFT2 perspective Craps:2021bmz  assuming
vacuum block domination.

The mutual information $I(A:B)$ between the 2 boundary CFTs represented in the
TFD description can serve as a better diagnostic of chaotic phenomena
Wolf:2007tdq . Here $A$ and $B$ are large enough subsystems in the left and
right boundary CFTs respectively. The disruption of $I(A:B)$ due to an
$O(G_{N})$ perturbation at late times can be computed using the RT or the HRT
prescription Ryu:2006bv ; Hubeny:2007xt ; Shenker:2013pqa . The analysis in
Shenker:2013pqa  realises the late time effect of such a perturbation by a
shockwave in the black hole geometry with the rate of disruption of $I(A:B)$,
thus determining $\lambda_{L}$. A crucial physical insight of this analysis
was that the $\lambda_{L}$ turns out to be given by the exponent of the
blueshift suffered at the outer horizon by an in-falling null particle
released from the boundary. For the case of rotating shockwaves in rotating
BTZ it was shown that this blueshift is given by

$E=E_{0}e^{\kappa t_{0}},\hskip 8.5359pt\kappa=\frac{2\pi
T_{H}}{(1-\mu\,\mathcal{L})}$ (4)

where $\mu=r_{\\!{}_{-}}/r_{\\!{}_{+}}$ is the chemical potential and
$\mathcal{L}$ is the angular momentum of the shockwave per unit of its energy
Malvimat:2021itk . It was also shown by an analysis similar to Shenker:2013pqa
that in certain cases $\lambda_{L}>2\pi T_{H}$ and such a $\lambda_{L}$ also
determines the scrambling time222Here we explicitly see that $I(A:B)$ is a
better diagnostic of chaotic phenomena Wolf:2007tdq  than 4pt OTOC where the
later sees an $|\lambda_{L}|=2\pi/\beta$ for large times Mezei:2019dfv . .
Recently the disruption of mutual information due to rotating shockwaves was
analysed for Kerr black holes in $AdS_{4}$ Malvimat:2022oue . Here the $A$ &
$B$ subsystems were chosen to be the hemispheres of the left and right CFTs
respectively such that they respect the axi-symmetry of the geometry. Here too
we find that the blueshift suffered by a rotating shockwave released at time
$t_{0}$ from the boundary is given by (4). In this case we clearly find that
the instantaneous $\lambda_{L}^{(min)}$ leading up to the scrambling time can
be greater than $2\pi T_{H}$ while the late time average $\lambda_{L}$ that
determines the scrambling time is given by

$\lambda_{L}=\kappa=\frac{2\pi T_{H}}{(1-\mu\,\mathcal{L})}$ (5)

This is aslo not in contradiction with the result obtained in Blake:2021wqj
mentioned before as it investigates pole skipping by looking at non-rotating
in-going graviton modes at the horizon $i.e.$ $\mathcal{L}=0$; the work in
Malvimat:2022oue  predicts that similar analysis with rotating modes would
corroborate the above results. In this paper we provide further evidence of
such phenomena in Kerr black holes in $AdS_{5}$ with equal angular momentum.
We analyse the disruption of $I(A:B)$ where $A$ & $B$ are hemispheres in the
left and right boundary CFT4s respectively $s.t.$ they respect the axi-
symmetry of the geometry. The disruption is caused due to a rotating shockwave
with angular momenta per unit energy $\mathcal{L}_{\phi_{1}}$ and
$\mathcal{L}_{\phi_{2}}$ about the 2 axi-symmetric angular directions of the
geometry. The equal angular momentum Kerr $AdS_{5}$ possesses an $S^{2}$ worth
of isometry which makes the analysis easier. The paper is organized as
follows: we first briefly review the Kerr geometry with equal angular momentum
in section 2. In section 3 we construct the Dray-’t Hooft solution for
rotating shockwaves. This we do by constructing the necessary Kruskal
coordinates suitable for expressing the Dray-’t Hooft solution. In the process
we also obtain the blue shift suffered by such a shockwave released from the
boundary in reaching the outer horizon. We also briefly comment about the
turning point analysis required for rotating null geodesics to reach the
horizon from the boundary and show that the exponent of the blueshift does not
diverge. In section 4 we determine the change in the area of the extremal
surface following the computation in Leichenauer:2014nxa , due to such a
shockwave and plot its instantaneous $\lambda_{L}^{(min)}$ for various values
of shockwave angular momenta along with the respective blueshift exponents
$\kappa$ against $r_{\\!{}_{-}}/r_{\\!{}_{+}}$ in Fig-2. We also compute the
scrambling time $i.e.$ time taken for $I(A:B)$ to go to zero and find that the
late time $\lambda_{L}$ is given by (5) where
$\mathcal{L}=\mathcal{L}_{+}=\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}$.

##  2 Kerr in AdS${}_{\textbf{5}}$ with $J_{\phi_{1}}=J_{\phi_{2}}$

In this section we briefly introduce the geometry of an $AdS_{5}$ Kerr black
hole with equal angular momentum. The asymptotic boundary of $AdS_{5}$ is a
$\mathbb{R}\times S^{3}$ however a Kerr geometry in the bulk does not have the
rich isometries of $S^{3}$ ($SU(2)\times SU(2)$). Choosing the 2 angular
momenta of the Kerr geometry to be equal $J_{\phi_{1}}=J_{\phi_{2}}=J$
preserves an $S^{2}$ fibration of $S^{3}$, therefore the geometry possesses an
$SO(3)\subset SU(2)\times SU(2)$ isometry. This allows for a much simpler
expression for the metric in terms of the line element squared. The equal
angular momentum Kerr black hole in $AdS_{5}$ in Boyer-Lindquist coordinates
is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}dr^{2}}{(r^{2}+a^{2})\Delta(r)}\,-\,\frac{1}{\Xi}\Delta(r)e^{U_{2}-U_{1}}dt^{2}\,+\,e^{-U_{1}}d\Omega_{2}^{2}\,+\,e^{-U_{2}}\left(\sigma^{3}+A\right)^{2},$
(6) $\displaystyle{\rm where}\hskip 85.35826pt$
$\displaystyle\Delta(r)=1+\frac{r^{2}}{l^{2}}-\frac{2mr^{2}}{(r^{2}+a^{2})^{2}},\hskip
8.5359pt\Xi=1-\frac{a^{2}}{l^{2}},$ (13) $\displaystyle
e^{-U_{2}}=\frac{r^{2}+a^{2}}{4\,\Xi}+\frac{2ma^{2}}{2\,\Xi^{2}(r^{2}+a^{2})},\hskip
8.5359pte^{-U_{1}}=\frac{r^{2}+a^{2}}{4\,\Xi},$ $\displaystyle
A=\frac{a}{2\,\Xi}\left(\frac{r^{2}+a^{2}}{l^{2}}-\frac{m}{r^{2}+a^{2}}\right)e^{U_{2}}dt$

The angular line elements are

$\sigma^{3}=d\psi+\cos\theta\,d\phi,\hskip
11.38092ptd\Omega^{2}_{2}=d\theta^{2}+\sin^{2}\theta\,d\phi^{2}$ (14)

where the $S^{2}$ symmetry is manifest. The polar coordinates range from
$\theta\in\\{0,\pi\\}$ , $\phi\in\\{0,2\pi\\}$ and $\psi\in\\{0,4\pi\\}$333The
toric coordinates describing the $S^{3}$ are $\eta=\theta/2$ and
$\phi_{1}=(\psi-\phi)/2$, $\phi_{2}=(\psi+\phi)/2$ where
$d\Omega_{3}^{2}=d\eta^{2}+\sin^{2}\theta\,d\phi_{1}+\cos^{2}\theta\,d\phi_{2}$.
. The axi-symmetry directions of the black hole are $\phi_{1}=(\psi-\phi)/2$
and $\phi_{2}=(\psi+\phi)/2$. The black hole is defined by the mass parameter
$m$ and the angular momentum parameter $a$. The ADM mass and 2 equal angular
momenta are Skenderis:2006uy

$M=M_{C}+\frac{2\pi^{2}m(3+\frac{a^{2}}{l^{2}})}{\Xi^{2}},\hskip
8.5359ptJ=\frac{8\pi^{2}ma}{\Xi^{2}},\hskip
8.5359ptM_{C}=\frac{3\pi^{2}l^{2}}{4}$ (15)

Here $M_{C}$ is the Casimir energy associated with $S^{3}$ and is independent
of the black hole parameters. The entropy and temperature is

$\mathcal{S}=\frac{\pi^{2}(r_{\\!{}_{+}}^{2}+a^{2})^{2}}{2G_{N}r_{\\!{}_{+}}\Xi^{2}},\hskip
14.22636ptT_{H}=\frac{r_{\\!{}_{+}}^{2}-a^{2}+\frac{2a^{2}}{l^{2}}}{2\pi
r_{\\!{}_{+}}(r_{\\!{}_{+}}^{2}+a^{2})}$ (16)

where $l$ is the radius of $AdS_{5}$ an $G_{N}$ is the Newton’s constant. The
Boyer-Lindquist coordinates used above are adapted from that of flat space, in
asymptotically $AdS_{5}$ space these have a non-zero angular velocity at the
boundary about $\phi_{1}$ and $\phi_{2}$. It turns out that the correct
horizon angular velocity about both these directions after cancelling the
relative angular velocity of the boundary is

$\Omega=\Omega_{\phi_{1}}=\Omega_{\phi_{2}}=\frac{a\left(1+\frac{r_{\\!{}_{+}}^{2}}{l^{2}}\right)}{r_{\\!{}_{+}}^{2}+a^{2}}=\mu$
(17)

which plays the role of chemical potential $\mu$ for both the angular momenta
and is relevant for the thermodynamics of the black hole. The reader is
referred to Skenderis:2006uy  for a detailed derivation of asymptotic charges
relevant for thermodynamics for generic Kerr black holes in $AdS_{5}$.

##  3 Dray-’t Hooft solution

In this section we construct the Dray-’t Hooft solution corresponding to a
shockwave along the equator $\theta=\pi/2$ such that the shockwave and thus
the resulting geometry obeys the axi-symmetries of the Kerr black hole. By
symmetry of the Kerr solution it can be shown that there always exists a
geodesic (massive and massless) which stays restricted to the equatorial
plane. As the Dray-’t Hooft solution requires well defined Kruskal coordinates
smooth at the outer horizon we first construct such coordinates by working out
the null geodesics with angular momenta per unit energy
$\mathcal{L}_{\phi_{1}}$ & $\mathcal{L}_{\phi_{2}}$ about the two symmetric
angular coordinates of the Kerr black hole. We then show systematically that
the blueshift suffered by such a shockwave is obtained by demanding that the
associated in-out going Kruskal coordinates be affine at the outer horizon.
The Dray-’t Hooft solution is then obtained by utilizing these Kruskal
coordinates and determining the shift in one of them as a function of time
$t_{0}\gg\beta$ when the perturbation was realised from the left AdS boundary.

###  3.1 Rotating Kruskal Coordinates

We begin with null geodesics parametrized by the constants

$\xi^{2}=0,\hskip
8.5359ptg_{\mu\nu}\xi^{\mu}\zeta^{\nu}_{t}=\mathcal{E}=1,\hskip
8.5359ptg_{\mu\nu}\xi^{\mu}\zeta^{\nu}_{\phi_{1}}=\mathcal{L}_{\phi_{1}},\hskip
8.5359ptg_{\mu\nu}\xi^{\mu}\zeta^{\nu}_{\phi_{2}}=\mathcal{L}_{\phi_{2}}$ (18)

where $\zeta_{\phi_{1}},\zeta_{\phi_{2}}$ and $\zeta_{t}$ are the symmetry
generators with respect to the stationary boundary observer. In terms of the
Boyer-Lindquist coordinates these are

$\displaystyle\zeta_{\phi_{1}}=\partial_{\phi_{1}},\hskip
8.5359pt\zeta_{\phi_{2}}=\partial_{\phi_{2}},\hskip
8.5359pt\zeta_{t}=\partial_{t}-\frac{2a}{l^{2}}\partial_{\psi}$ (19) (20)
$\displaystyle{\rm where}\hskip 8.5359pt\phi_{1}=\frac{\psi-\phi}{2},\hskip
8.5359pt\phi_{2}=\frac{\psi+\phi}{2}$ (21)

There is a further constant that needs to be specified which corresponds to
the angular momentum in the $\theta$ direction. This is done by specifying the
Carter’s constant $\mathcal{Q}$ which is defined by using a higher form of
symmetry present in rotating geometries called the Killing-Yano tensor
$f_{\mu\nu}$. The Killing-Yano tensor can be used to write down a higher form
generalization of the Killing vector called the Killing tensor which satisfies

$\nabla_{\rho}K_{\mu\nu}=0,\,\,\,K_{\mu\nu}=K_{\nu\mu},\,\,\,K_{\mu\nu}=-f_{\mu}^{\,\,\,\sigma}f_{\sigma\nu}$
(22)

The Carter’s constant is then defined as

$\xi^{\mu}\xi^{\nu}K_{\mu\nu}=\mathcal{Q}$ (23)

The Killing tensor for generic Kerr $AdS_{5}$ is worked out in Kunduri:2005fq
. We make use of it in our coordinates for the case of equal angular momentum.
Imposing that the geodesic stays in the equatorial plane $i.e.$
$\dot{\theta}=\xi^{\theta}=0$ implies holding $\mathcal{Q}$ constant at

$\mathcal{Q}=\Xi\,(2(\mathcal{L}_{\phi_{1}}^{2}+\mathcal{L}_{\phi_{2}}^{2})-l^{2}).$
(24)

The precise form of the vector field $\xi^{\mu}\partial_{\mu}$ along the null
geodesic parametrized by the above constants is cumbersome and can be worked
out easily. The null pairs corresponding to in going and out going geodesics
can be obtained by reversing the signs of the angular coordinates, we denote
these as $\xi^{\pm}$ and use their dual one-forms to write the metric at
$\theta=\pi/2$.

$ds^{2}_{\theta=\frac{\pi}{2}}=F(\xi^{+}_{\mu}dx^{\mu})(\xi^{-}_{\nu}dx^{\nu})+h_{\phi}(d\phi+h_{\tau}d\tau)^{2}+h_{z}(dz+g_{\phi}d\phi+A_{\tau}d\tau)^{2}$
(25)

where

$\tau=\left(1-\frac{a}{l^{2}}(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}})\right)t+\frac{\mathcal{L}_{\phi_{1}}-\mathcal{L}_{\phi_{2}}}{2}\phi-\frac{\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}}{2}\psi$
(26)

Here the $\tau$ naturally occurs as the component of the one-form in
$\xi^{\pm}_{\mu}dx^{\mu}$ in the non-radial direction. The rest of the
functions in the line elements written above can be obtained uniquely by
completing the squares. $z$ is a linear function of $\tau$ and $\psi$ which we
elaborate below. Although the forms of these functions are important, their
precise expressions are cumbersome and hence it serves little purpose to
explicitly give them here. Their behaviour close to the horizon would be
significant for our analysis as we shall see further. The $\tau$ coordinate in
terms of the Kruskal coordinates reads

$d\tau\sim\frac{UdV-VdU}{UV}$ (27)

therefore the coefficient of the $d\tau$ term in the line elements better
vanish at the horizons. The function
$h_{\tau}\sim\mathcal{O}(r-r_{\\!{}_{+}})$, therefore the $\phi$ coordinate
need not be redefined. However the $\psi$ coordinate is shifted to $z$ to
demand that $A_{\tau}\sim\mathcal{O}(r-r_{\\!{}_{+}})$.

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\eta\,z-\frac{2a(a^{2}-l^{2})}{l^{2}(a^{2}+r_{\\!{}_{+}}^{2})-a(r_{\\!{}_{+}}^{2}+l^{2})(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}})}\tau$
(28)

The coefficient of $z$ denoted by $\eta$ in the above expression is fixed by
demanding that the horizon area remain the same in the coordinates
$\\{\theta,\phi,z\\}$ and that $z$ and $\psi$ have the same domain
$\\{0,2\pi\\}$.

$\eta=\frac{1-\tfrac{a}{l^{2}}(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}})}{1-\mu(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}})},\hskip
22.76228pt{\rm where\,\,\,\,}\mu=\Omega$ (29)

The $z$ coordinate is basically a co-rotating coordinate close to the horizon.
Light cone coordinates can be constructed by defining

$\displaystyle du=\xi^{+}_{\mu}dx^{\mu}\implies u=r_{*}-\tau$ (30) (31)
$\displaystyle dv=\xi^{-}_{\mu}dx^{\mu}\implies v=r_{*}+\tau$ (32)

where the radial tortoise coordinate is defined by integrating
$dr_{*}=\xi_{r}dr$ and choosing the boundary at $r_{*}=0$. We can see that
these $\\{u,v\\}$ coordinates are not smooth at the horizons; for instance for
the vector field $\chi$ that generates translations along $v$ $i.e.$
$\chi=\partial_{u}$ we find $\chi\cdot\nabla\chi=\mathcal{K}\chi$ where

$\mathcal{K}=\frac{1}{2}\xi^{+}\cdot\partial F$ (33)

Here the non-vanishing of $\mathcal{K}$ at the outer horizon signals the non-
smoothness of the $\\{u,v\\}$ coordinates. One can therefore define the smooth
Kruskal coordinates as

$U=-e^{\kappa(r_{*}-\tau)},\,\,V=e^{\kappa(r_{*}+\tau)},\hskip 19.91684pt{\rm
where}\,\,\kappa=\mathcal{K}|_{r_{\\!{}_{+}}}$ (34)

Rewriting the line element along the equator in terms of $\\{U,V\\}$ we find

$ds^{2}_{\theta=\frac{\pi}{2}}=\frac{F}{\kappa^{2}UV}dUdV+h_{z}(dz+g_{\phi}d\phi+A_{\tau}d\tau)^{2}+h_{\phi}(d\phi+h_{\tau}d\tau)^{2}$
(35)

where the value of $\kappa$ is

$\displaystyle\kappa=\frac{\kappa_{0}}{(1-\mu(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))},\,\,\,\,{\rm
where}\,\,\mu=\Omega\,\,\,\,\,\&\,\,\,\,\,\kappa|_{(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}})\rightarrow
0}=\kappa_{0}=\frac{2\pi}{\beta}=2\pi T_{H}$ (36)

Here we denote $\kappa_{0}$ to be $2\pi$ times the black hole temperature.

###  3.2 Full Coordinates

We would like to extend these coordinates along the angular coordinate
$\theta$ so as to describe the full black hole geometry. Therefore we first
write down the null geodesics $\xi^{\mu}\partial_{\mu}$ for unit energy with
charges $\mathcal{L}_{\phi_{1}},\mathcal{L}_{\phi_{2}}$ and arbitrary
$\mathcal{Q}$ with its $\theta$ component. The in-out going vector fields have
opposite signs for their $\theta$ component. Next we demand that their dual
one-forms be exact

$\xi^{+}_{\mu}dx^{\mu}=du,\hskip 8.5359pt\xi^{+}_{\mu}dx^{\mu}=dv$ (37)

This implies

$\partial_{\theta}\xi^{\pm}_{r}=\partial_{r}\xi^{\pm}_{\theta}=0,\hskip
11.38092pt\implies\mathcal{Q}=const$ (38)

We then choose the value of $\mathcal{Q}$ to be (24) such that the family of
geodesics that start at the equator stay at the equator. Note that this
implies $\dot{\theta}=\xi^{\theta}=0$ only at $\theta=\pi/2$. Thus the whole
metric can be written as

$ds^{2}=\frac{F}{\kappa^{2}UV}dUdV+h_{z}(dz+g_{\phi}d\phi+A_{\tau}d\tau)^{2}+h_{\phi}(d\phi+h_{\tau}d\tau)^{2}+h_{\theta}(d\theta+g_{\tau}d\tau)^{2}$
(39)

where we use the same functions as in the previous sub-section to denote the
non-trivial $\theta$ dependence. The line elements in the above metric are
obtained uniquely by completing squares as before. We also note that
$g_{\tau}\sim\mathcal{O}(r-r_{\\!{}_{+}})$ implying that $\theta$ need not be
redefined as expected. Note that the redefinition of $\psi$ to $z$ is
independent of $\theta$. The Kruskal coordinates can now be defined at any
value of $\theta$ to be

$U=-e^{\kappa(r_{*}-\tau-\tilde{g}(\theta))},\,\,V=e^{\kappa(r_{*}+\tau+\tilde{g}(\theta))},\hskip
19.91684pt{\rm where}\,\,\kappa=\mathcal{K}|_{r_{\\!{}_{+}}}$ (40)

with an extra $\theta$ dependence $s.t.$ $\tilde{g}(\pi/2)=0$. $\mathcal{K}$
is similarly defined as

$\mathcal{K}=\tfrac{1}{2}\xi^{+}\cdot\partial F$ (41)

The extra $\theta$ dependence is absorbed into $F$ however $\kappa$ is
independent of $\theta$ and is still given by

$\kappa=\frac{\kappa_{0}}{(1-\mu(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))},\hskip
19.91684pt{\rm where}\,\,\,\,\kappa_{0}=\frac{2\pi}{\beta},\,\,\mu=\Omega$
(42)

The above value of $\kappa$ as in Malvimat:2021itk ; Malvimat:2022oue
suggests an upper bound for the angular momentum of the shock-wave to be
$\mathcal{L}<\mu^{-1}=\Omega_{+}^{-1}$. Here $\mu$ is the same along
$\phi_{1}$ and $\phi_{2}$.

The functions $A_{\tau}$, $h_{\tau}$ and $g_{\tau}$ vanish at the future or
the past outer horizon. The redefinition of $\psi\rightarrow z$ is precisely
dictated by this condition. Note that $\theta$ and $\phi$ do not need to be
redefined. This requirement would be crucial for writing down the Dray-’t
Hooft solutions in response to a shockwave.

The exact functional forms of the functions
$\\{F,h_{z},g_{\phi},A_{\tau},h_{\phi},h_{\tau},h_{\theta},g_{\tau}\\}$ in the
metric (39) are complicated and can be arrived at by following the above
procedure after obtaining the in-out going null geodesics (18). We note the
expansion of the function $F$ at the outer horizon

$F(r,\theta)=(r-r_{\\!{}_{+}})F^{\prime}(r_{\\!{}_{+}})+\tfrac{1}{2}(r-r_{\\!{}_{+}})^{2}F^{\prime\prime}(r_{\\!{}_{+}},\theta)+\dots$
(43)

where we write

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

as the exteriors have either $U$ or $V$ negative. Note that $F^{\prime}$ is
independent of $\theta$. Here $\mathbb{A}$ is some proportionality constant
which we assume does not scale with the size or the entropy of the black hole.
It would be useful to write $F/UV$ as an expansion in $UV$ using the above
expansion

$\frac{F}{\mathbb{A}UV}=-1-\frac{\mathbb{A}UV}{2}\frac{F^{\prime\prime}}{F^{\prime
2}}.$ (45)

We will find that such an expansion would be useful for analysing the
smoothness condition across the shockwave.

###  3.3 Turning point

We would be imagining the perturbation to begin at some point on the (left)
AdS boundary and follow a null geodesic towards the outer horizon along
$\theta=\pi/2$ Fig-1. This is consistent as $\xi^{\theta}(r,\theta=\pi/2)=0$.
Not all null perturbations would be able to reach the outer horizon as the
geometry is rotating and further the geodesics have a angular momenta about
both the symmetric directions. In order to determine whether a null geodesic
reaches at least the outer horizon we need to look at the zeros of the radial
component of the geodesic vector field (18) as it represents the change
$\dot{r}$ along the null trajectory. An existence of a zero (of $\xi^{r}$)
outside the outer horizon would indicate that the null perturbation would
reverse direction and start moving away from the black hole444Complicated
motions like orbiting the black hole would also be possible, however we would
only be interested if the zero exists outside of the horizon and not the fate
of the geodesic after it reaches the turning point. and therefore is called
the turning point. The turning point analysis in general is quite complicated
however for our analysis we only require a few qualitative properties
describing the behaviour of turning point as a function of the shockwave’s
angular momenta $\mathcal{L}_{\phi_{1}}$ and $\mathcal{L}_{\phi_{2}}$, the
black hole’s ratio of inner and outer horizon
$\mu_{T}=r_{\\!{}_{-}}/r_{\\!{}_{+}}$ and its outer radius $r_{\\!{}_{+}}$. We
choose to work with

$\mathcal{L}_{\pm}=\mathcal{L}_{\phi_{1}}\pm\mathcal{L}_{\phi_{2}}$ (46)

as only $\mathcal{L}_{+}$ determines the blueshift suffered by the null
geodesic. We define $\mathcal{L}_{\pm}$ in terms of the inverse of the horizon
velocity $\mu$ (17)

$\mathcal{L}_{\pm}=\epsilon_{\pm}\mu^{-1}$ (47)

We plot $\xi^{r}(r)$ for $\mu_{T}\in\\{0,1\\}$, and
$\epsilon_{\pm}\in\\{0,1\\}$ for different values of $r_{\\!{}_{+}}$ ranging
from very small (compared to AdS radius $l$) to $l/\sqrt{2}$ as this is the
largest value of the outer horizon for which one can have an extremal Kerr
black hole in $AdS_{5}$. We find the following behaviour for the largest zero
$r_{tp}$:

  * 1

For $\epsilon_{+}=1$ $i.e.$ $\mathcal{L}_{+}=\mu^{-1}$ we find that
$r_{tp}>r_{\\!{}_{+}}$ for non extremal black holes and $r_{tp}\rightarrow
r_{\\!{}_{+}}$ as $\mu_{T}\rightarrow 1$. This holds true for any value of
$\mathcal{L}_{-}$ and $r_{\\!{}_{+}}$. Thus we find that the blueshift
exponent $\kappa$ (42) does not blow up for non-extremal geometries.

  * 2

As $\epsilon_{+}$ is reduced from $1$ at some point we do see that $r_{tp}\leq
r_{\\!{}_{+}}$ thus indicating that the geodesic can cross the horizon. This
limiting value of $\epsilon_{+}$ is difficult to determine analytically and
does depend on $\mu_{T}$, $r_{\\!{}_{+}}$ and $\epsilon_{-}$. In general we
find that if $r_{tp}>r_{\\!{}_{+}}$ for a certain value of $\epsilon_{+}$,
then increasing $\epsilon_{-}$ from $0$ to $1$ would further increase the
value of $r_{tp}$ $i.e.$ making it difficult for the geodesic to reach the
outer horizon.

  * 3

For $\epsilon_{+}=0$ there are no zeros outside the horizon for
$\epsilon_{-}=0$. For a certain window of $\mu_{T}\in\\{0,\mu_{T}^{\prime}\\}$
(where $\mu_{T}^{\prime}<1$) we find an $r_{tp}>r_{\\!{}_{+}}$ after a certain
value of $\epsilon_{-}$ as it is increased from $0$ to $1$. Here
$\mu_{T}^{\prime}$ is sufficiently far fro extremality and depends in general
on all the other parameters. For $\mu>\mu_{T}^{\prime}$ changing
$\epsilon_{-}$ between $\\{0,1\\}$ has no effect on the turning point and
there exist no $r_{tp}>r_{\\!{}_{+}}$.

Generally one analyses the turning points by studying the solution to
$\xi^{r}=0$. However in this case the behaviour of the zeros with regards to
$\epsilon_{\pm},\mu_{T}$ and $r_{\\!{}_{+}}$ are too cumbersome and we leave a
more complete analysis of the properties of the geodesics in rotating $AdS$
black hole geometries for future investigations. Furthermore, the first point
mentioned above would be the only characteristic of the null geodesics of
direct relevance for us as it clearly rules out the possibility of the
exponent of the blueshift $\kappa$ (42) blowing up for a non extremal
configuration.

Given the results of Halder:2019ric  we expect Lyapunov index we wish to
compute to be bounded by (3). As we would later see that since the Lyapunov
index turns out to be dictated by the exponent of the blueshift suffered along
the null in-falling geodesic we expect $r_{tp}>r_{\\!{}_{+}}$ for ${\cal
L}_{+}>\mu_{c}^{-1}$ where $\mu_{c}$ is the horizon velocity for an extremal
black hole with the same ADM mass. This would require a further numerical
analysis of the turning point and we leave the this thorough analysis for the
near future.

####  3.3.1 Superradiance

It is worthwhile to comment about Superradiance due to an in-falling rotating
perturbation into the Kerr black hole under consideration. Superradiance can
be understood as a classical phenomena555For discussion regarding the quantum
effects of superradiance refer to Brito:2015oca . wherein the particle falling
into a Kerr space-time with an angular momentum enters the ergo-region
surrounding the black hole horizon and exists with an energy greater than
which it began at an early time. This process is also called the Penrose
process whereby one can extract the rotational energy of a Kerr black hole.
For a detailed review describing the current understanding of numerous such
cases, we refer the reader to the extensive lecture notes of Brito:2015oca
which is continuously updated666We thank Indranil Halder for this excellent
reference.. This process can be seen as a simple consequence of the first law
and the area increase theorem (second law) for black hole dynamics. Let us
assume that the particle has an angular momentum per unit of its energy
$\mathcal{L}$. The change in the mass $M$, entropy $S$ and angular momentum
$J$ of the black hole due to this interaction would be

$\displaystyle\delta M=T_{H}\,\delta\mathcal{S}+\mu\,\delta J$ (48)
$\displaystyle{\rm for}\hskip 11.38092pt\delta J=\mathcal{L}\delta M$
$\displaystyle\implies$ $\displaystyle\delta
M=\frac{T_{H}}{(1-\mu\,\mathcal{L})}\delta\mathcal{S}$ (50)

The area increase theorem for black holes implies that $\delta\mathcal{S}>0$
in any such process. Therefore for the particle to exit the ergo-region with
greater energy the change in the black hole’s mass must be $\delta M<0$, which
is possible only if $\mathcal{L}>\mu^{-1}$. For our case of Kerr black hole in
$AdS_{5}$ with $\mu=\mu_{1}=\mu_{2}$ we simply have $\delta J=\delta
J_{1}+\delta J_{2}$ and $\mathcal{L}=\mathcal{L}_{+}$ defined above. In other
words the perturbation ought to have a specific angular momentum $\mathcal{L}$
such that its trajectory enters the ergo-region and leaves it $i.e.$ doesn’t
fall into the black hole. This is consistent with the above turning point
analysis where we found that the necessary condition for the geodesic to fall
into the non-extremal geometry is $\mathcal{L}_{+}<\mu^{-1}$. For specific
angular momentum greater than this the turning point lies outside the horizon,
thus the null particle reverses its direction in terms of the radial
coordinate and depending on the geodesic may either escape to a region far
from the black hole or simple be trapped in an orbit around it. In the former
case we get to see the Penrose process while in the later the null trajectory
gets trapped. Either of these scenarios would not be of concern to us as we
would be interested in only those values of $\mathcal{L}_{+}$ wherein the
geodesic makes it past the outer horizon.

###  3.4 Shockwaves along the equator

The Dray-’t Hooft solution is given by

$V\rightarrow\widetilde{V}=V+\alpha\Theta(U)f$ (51)

where $f$ in general is function of the spatial coordinates
$\\{\theta,z,\phi\\}$ transverse to the shockwaves. For the sake of simplicity
we would send shockwaves with axi-symmetry of the geometry $i.e.$ the
shockwave would be symmetric in the $z$ and $\phi$ directions and be localized
on the equatorial plane $\theta=\pi/2$. Thus $f$ would only depend on
$\theta$.

The metric in $\\{U,V,\theta,\phi,z\\}$ coordinates can be obtained by first
noting $d\tau$ in terms of Kruskal coordinates

$d\tau=\frac{1}{2\kappa\,UV}(UdV-VdU)-\widetilde{g}^{\prime}d\theta$ (52)

yielding the Kerr metric in Kruskal coordinates as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{F}{\kappa^{2}UV}dUdV+h_{z}\left(dz+g_{\phi}d\phi+\frac{A_{\tau}}{2\kappa\,UV}(UdV-
VdU)+A_{\tau}\widetilde{g}^{\prime}d\theta\right)^{2}+$ (55)
$\displaystyle+h_{\phi}\left(d\phi+\frac{h_{\tau}}{2\kappa\,UV}(UdV-
VdU)+h_{\tau}\widetilde{g}^{\prime}d\theta\right)^{2}+h_{\theta}\left(d\theta+\frac{g_{\tau}}{2\kappa\,UV}(UdV-
VdU)+g_{\tau}\widetilde{g}^{\prime}d\theta\right)^{2}$

Note that along and across the horizon the above metric is smooth, this is
guaranteed by the behaviour of the functions $h_{\tau},g_{\tau}\,\&\,A_{\tau}$
all vanishing as $\mathcal{O}(r-r_{\\!{}_{+}})$ at the outer horizon. The
Dray-’t Hooft solution for a shockwave parametrized by $U=0$ can now be
written following the general arguments presented in Sfetsos:1994xa  as

$ds^{2}\rightarrow
d\widetilde{s}^{2}-\frac{F}{\kappa^{2}UV}\delta(U)f(\theta)dU^{2}$ (57)

The above form of the metric implies that the Einstein’s equation

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}-\Lambda
g_{\mu\nu}=R_{\mu\nu}+\frac{4}{l^{2}}g_{\mu\nu}=-8\pi G_{N}T_{\mu\nu}$ (58)

with the $rhs$ corresponding to that of a stress-tensor due to shockwave with
momentum $p^{V}$ along the $V$ coordinate. The above equation then reduces to
that of source equation for the response function $f(\theta)$ in the
transverse direction

$\alpha\mathcal{D}_{\theta}\,f=-8\pi
G_{N}T_{UU},\,\,T_{UU}=p^{V}\left(\frac{F^{2}}{\kappa^{2}UV}\right)_{U_{0}}\delta(\theta-\pi/2)$
(59)

where $\mathcal{D}_{\theta}$ is a differential operator obtained by evaluating
the Einstein’s equation at $U=0$. Note, that its is important here that the
metric (LABEL:metric_kruskal_L_0) be smooth at $U=0$. Failure to ensure this
would imply coordinate singularities in the analysis of the back reaction.

We would next like to imagine that the shockwave emanated from the left
boundary at a time $\tau_{0}>0$, moves along

$U_{0}=e^{-\kappa\tau_{0}}.$ (60)

As the above Dray-’t Hooft solution is only valid for the shockwave
travaelling along $U=0$ we have to ensure that we work in the limit
$U_{0}\rightarrow 0$ which translates to the $\tau_{0}\rightarrow\infty$. In
this limit we would like to determine the value of $\alpha$ $i.e.$ the
magnitude of the back reaction induced on the metric. We do this following
Shenker:2013pqa  by demanding the smoothness of the transverse volume density
$H=\sqrt{{\rm det}[g_{\\{\theta,z,\phi\\}}]}$ along the shockwave

$H\big{|}_{U_{0}^{+}}=H\big{|}_{U_{0}^{-}}.$ (61)

Expanding this about the outer horizon we have

$H=H_{0}+H_{1}(r-r_{\\!{}_{+}})=H_{0}-H_{1}\left(\mathbb{A}\frac{U_{0}V}{F^{\prime}(r_{\\!{}_{+}})}\right)$
(62)

where we express radial distances using the eq(44). The smoothness of
transverse volume density across the shockwave thus implies

$H_{0}-H_{1}\left(\mathbb{A}\frac{U_{0}V}{F^{\prime}(r_{\\!{}_{+}})}\right)=\widetilde{H}_{0}-\widetilde{H}_{1}\left(\widetilde{\mathbb{A}}\frac{U_{0}\widetilde{V}}{\widetilde{F}^{\prime}(r_{\\!{}_{+}})}\right)$
(63)

where we use tilde to denote quantities after the shockwave $U>U_{0}$. To the
leading order we can therefore find the shift in the $V$ coordinate to be

$\displaystyle\widetilde{V}=\frac{\widetilde{F}^{\prime}_{+}\mathbb{A}H_{1}}{F^{\prime}_{+}\widetilde{\mathbb{A}}\widetilde{H}_{1}}V+\frac{(\widetilde{H}_{0}-H_{0})\widetilde{F}^{\prime}_{+}}{\mathbb{A}\widetilde{H}_{1}U_{0}},\hskip
8.5359pt{\rm where}\,\,F^{\prime}_{+}\equiv F^{\prime}(r_{\\!{}_{+}},\pi/2)$
(64)

Note as $H_{0}$ is simply the transverse volume density at the outer horizon,
integrating this along the periodic coordinates gives us the black hole
entropy times $4G_{N}$. Therefore we can write

$\widetilde{H}_{0}-H_{0}=\delta H_{0}\sim\beta(\delta\mathcal{M}-\mu\,(\delta
J_{\phi_{1}}+\delta J_{\phi_{2}}))=\delta\mathcal{S}$ (65)

where we use the ADM mass defined in AdS spaces and
$\mu_{\phi_{1}}=\mu_{\phi_{2}}=\mu$

$\mathcal{M}=\frac{M}{\Xi^{2}},\hskip 8.5359ptJ=\frac{Ma}{\Xi^{2}}$ (66)

We would like to view the in-falling perturbation into the black hole as
increasing the mass and angular momentum of the geometry and thus its entropy
according to the first law. Moreover we would be interested in black holes
with large entropy, in the limit of large entropy we observe that

$\frac{F^{\prime}_{+}}{H_{1}}=\frac{\mathbb{B}}{\mathcal{S}}$ (67)

where $\mathbb{B}$ can be obtained by expanding $H$ and $F$ and is found to
depend only on $r_{\\!{}_{-}}/r_{\\!{}_{+}}$ and $\theta$. We imagine the
shockwave to have very small energy as compared to the black hole mass,
therefore the shockwave adds infinitesimally small entropy $\delta\mathcal{S}$
to the black hole where

$\frac{\delta\mathcal{S}}{\mathcal{S}}\rightarrow 0\hskip 5.69046pt\&\hskip
5.69046pt\delta J_{\phi_{1}}+\delta J_{\phi_{2}}=\mathcal{L}\,\delta M$ (68)

Note we still can only work in the limit $U_{0}\rightarrow 0$, therefore we
work in the scaling limit

$U_{0}\rightarrow 0,\hskip 8.5359pt\frac{(\delta\mathcal{M}-\mu\,(\delta
J_{\phi_{1}}+\delta J_{\phi_{2}}))}{T_{H}\mathcal{S}}\rightarrow 0,\hskip
8.5359pt\frac{\delta\mathcal{M}(1-\mu\,(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))}{U_{0}\,T_{H}\,\mathcal{S}}\rightarrow{finite}.$
(69)

where we made use of the first law and the previous relation. In such a limit
the shift in $V$ would still be finite. The parameters
$(\mathbb{B},\mathbb{A},2\pi)$ are non-extensive and we can adsorb them in the
variation of ADM mass and angular momentum we can write

$\widetilde{V}=V-\frac{\Theta(U-U_{0})}{U_{0}}\frac{\delta\mathcal{M}(1-\mu\,(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))}{T_{H}\mathcal{S}}$
(70)

where we have introduced the step function to indicate the shift in $V$ across
the shock-wave.

It is crucial to note that the shift $\delta\mathcal{M}$ has a $\theta$
dependence as it is supposed to solve a differential equation implied by (59).
This can be easily seen at $\delta H_{0}$ is simply the change in horizon area
density which needs to be integrated along the periodic coordinates to give
$\delta\mathcal{S}$. This reflects the fact that the geometry after the
shockwave is no longer a stationary solution to Einstein’s equations hence the
$\delta\mathcal{M}$ can and must have spatial dependence777The time dependence
comes $via$ the dependence on $V$ and would play an important role if one
relaxes the limit $U_{0}\rightarrow 0$. . Comparing this with (51) we see that

$\alpha
f(\theta)=\frac{\beta\,\delta\mathcal{M}(1-\mu\,(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))}{U_{0}\,\mathcal{S}}=\frac{\beta(1-\mu\,(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))E_{0}}{U_{0}\,\mathcal{S}}f(\theta)$
(71)

where the transverse $\theta$ dependence on the right is captured by
$\delta\mathcal{M}$. We have taken $\delta\mathcal{M}$ to be proportional to
the total energy $E_{0}$ of the perturbation measured888The absorption of non-
extensive constants such as $(\mathbb{B},\mathbb{A},2\pi)$ would not effect
this identification in the final result. at the boundary at left exterior time
$\tau_{0}$. Therefore we have

$V\rightarrow\widetilde{V}=V-\alpha f(\theta)\hskip 8.5359pt{\rm
where}\,\,\,\alpha=\frac{1}{U_{0}}\frac{\beta(1-\mu\,(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))E_{0}}{\mathcal{S}},\hskip
8.5359pt$ (72)

We can see that the shockwave grows exponentially as
$U_{0}^{-1}=e^{\kappa\tau_{0}}$ $i.e.$ as $\tau_{0}$ tends towards infinity.
The rate of growth of this exponential is governed by $\kappa$ for
$(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}})<1/\mu$. In what usually
follows in finding a Dray-’t Hooft solution one finds the form of the response
function $f(\theta)$ constrained by (59) which can be solved with recourse to
spherical harmonics. For the case of single null particle falling in with a
specific angular momentum $\mathcal{L}$ the response function would would be a
function of $\\{\theta,\phi,z\\}$ and would likewise be constrained by a
sourced differential equation of the type (59) with the $rhs$ localizing the
particle on $S^{3}$. In either cases the response function $f$ would determine
how fast the backreaction spreads along the sphere and thus determine the
‘butterfly velocity’ $v_{B}$. For the case at hand of an equatorial shockwave
at every point in $\phi$ and $z$ it is apparent that the backreaction would be
peaked at the equator and falloff gradually towards the poles in a manner
dictated by (59). Though interesting physics of the butterfly velocity $v_{B}$
can be discerned from such an analysis we would not concerning ourselves with
it in this paper. We would therefore choose a normalization where
$f(\theta=\pi/2)=1$ as we would only be concerned with dynamics of the metric
at the equator.

Throughout this paper we assume that the shockwave emanates at the left
$AdS_{5}$ boundary at a time $\tau_{0}$ at $\theta=\pi/2$ from every point in
the $\\{\phi,\psi\\}$ plane. As is apparent we choose to work with the non
radial coordinates $\\{\tau,\theta,\phi,z\\}$ given by (26)(28) and (29).
These coordinates are forced upon us by demanding that the components of the
metric as seen along the infalling null rotating perturbation be smooth in the
near horizon region, which is precisely the region in the bulk where the
backreaction is the strongest. Therefore in the Dary-’t Hooft analysis we have
to demand that the response function is periodic in the $z$ coordinate999There
is also a periodicity in the $\phi$ coordinate which remains the same. and not
the Boyer-Lindquist coordinate $\psi$. Assuming otherwise would lead to
singularities in the metric components which end up in determining the
differential equation (59) constraining the response function $f$. This can be
understood as working with the co-moving coordinates once we are in the ergo-
region as there are no possible stationary time like observers possible in
this region. Therefore for the case of the backreaction due to a single in-
falling null particle the equivalent shift in one of the Kruskal coordinates
is given by $\alpha f(\theta,\phi,z)$ where $\alpha$ takes the form given
above. Crucially, the exponential growth is only due to the presence of
$U_{0}$.

This also implies that there is spread in time stamp associated with the start
of the perturbation at the left asymptotic boundary as seen by a stationary
boundary observer $i.e.$ the proper time $t^{(s)}$ of the dual CFT given by
(26)

$\tau_{0}=t_{0}^{(s)}+\frac{\mathcal{L}_{\phi_{1}}-\mathcal{L}_{\phi_{2}}}{2}\phi_{0}-\frac{\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}}{2}\psi_{0}.$
(73)

Given that $z$ is defined such that $\psi\in[0,2\pi]$ implies $z\in[0,2\pi]$
in the near horizon region101010This is fixed by by demanding $\eta$ take the
specific value in (29). the above relation gives the spread in $t_{0}^{(s)}$
for a fixed $\tau_{0}$ for shockwave originating at evry point in the
$\\{\phi,\psi\\}$ plane. In what follows we will only deal with $\tau_{0}$ as
working with $t_{0}^{(s)}$ only produces a shift which would not effect the
results qualitatively.

Before we proceed to compute the change in the extremal surface due the above
shockwave geometry it would be useful to understand how the shockwave travels
within the (left) exterior of the geometry given the limits (69). The Kruskal
coordinates used helped us to write down the solution to the shockwave present
along the past horizon, however this is used as a limiting case to the
scenario where the shockwave emanates at $U_{0}\rightarrow 0$ from the left
$AdS$ boundary. The limit (69) can be seen to define how close the shockwave
is to the past horizon ($i.e.$ as close as $\delta{\cal S}/{\cal S}$ is to $0$
in terms of $U_{0}$.). The shockwave intersects the future horizon at
$U=U_{0}$ and goes past it depending on the turning point behaviour discussed
in subsection-3.3.1. Upon reaching the turning point the shockwave reverses
direction (in terms of its radial coordinate) and begins to travel along
$V=const$. The location of the turning point requires further numerical
analysis ($c.f.$ subsection-3.3.1) and can be behind even the inner horizon.
However this would not be of concern to us as we would dealing with the change
in $I(A:B)$ up until the time the shockwave just reaches the outer horizon at
$t=0$. This above scenario is similar to the one studied recently in
Horowitz:2022ptw  where a charged non-rotating shockwave was analysed in RN
$AdS_{4}$ extending the analysis of Leichenauer:2014nxa . We will therefore
see a similar effect as seen in Horowitz:2022ptw  exhibiting a delayed onset
of chaos.

##  4 Extremal surface along the equator

In this section, we determine the effect of shockwave on the holographic
mutual information between subsystems in the left and right CFT. The
holographic mutual information involves three terms each of which corresponds
to the holographic entanglement entropy of a subsystem and may be computed
through the HRT proposal as follows

$\displaystyle I(A:B)$ $\displaystyle=S_{A}+S_{B}-S_{AB}$ $\displaystyle
S_{x}$ $\displaystyle=\frac{{\cal A}_{\gamma_{x}}}{4G_{N}^{(5)}}$ (74)

where ${\cal A}_{\gamma_{x}}$ corresponds to the area of a co-dimension two
extremal surface $\gamma_{x}$ homologous to $x$ and $G_{N}^{(5)}$ is the five
dimensional Newtons’s gravitational constant. The areas of the extremal
surfaces and the corresponding holographic entanglement entropies for
subsystems $A$ and $B$ are unaffected by the shockwave at late times as they
are sufficiently away from the black hole horizon. Here, as mentioned in the
introduction the subsystems $A$ -in the $left$ $CFT$, & $B$ in the $right$
$CFT$, are hemispheres ending on the equator described by $\theta=\pi/2$.
Therefore the HRT surface homologous to $A\cup B$ always lies on the
equatorial plane $\theta=\pi/2$, this can be easily seen by the vanishing of
the extrinsic curvature of any co-dim 2 equatorial surface along the $\theta$
direction. We further need to extremize the co-dim 2 surface by moving it in
the time direction. Given that the remaining non-radial spatial coordinates
are symmetric directions of the black hole and also of the back-reacted
geometry111111This is because a shockwave is not localized at any values of
$\phi$ and $z$., this effectively involves extrimizing a space-like length in
a particular 2d metric along the $r$ and $\tau$ directions.

In order to perform this computation we utilize the method described in
Leichenauer:2014nxa  to estimate the growth of the area of the extremal
surface which the present authors utilized in Malvimat:2022oue  to obtain the
change in holographic mutual information of the subsystems in dual CFTs due to
shockwaves in Kerr-$AdS_{4}$ spacetime. The induced metric in equatorial plane
is given by

$ds_{\theta=\frac{\pi}{2}}^{2}=F\left(-d\tau^{2}+\frac{dr^{2}}{f^{2}}\right)+h_{z}(dz+g_{\phi}d\phi+A_{\tau}d\tau)^{2}+h_{\phi}(d\phi+h_{\tau}d\tau)^{2}$
(75)

The area of the required extremal surface corresponding to the holographic
entanglement entropy $S_{AB}$ may be expressed as follows

$\displaystyle{\cal A}_{\theta=\frac{\pi}{2}}$
$\displaystyle=S_{z}S_{\phi}\int
d\tau\sqrt{h_{z}h_{\phi}}\sqrt{-F+Ff^{-2}\dot{r}^{2}}$ (76)

Notice that the integrand is independent of the $\tau$ coordinate which leads
to a conserved quantity $\gamma$ expressed as follows

$\displaystyle\gamma=\frac{-F\sqrt{h_{z}h_{\phi}}}{\sqrt{-F+Ff^{-2}\dot{r}^{2}}}=\sqrt{-F_{0}h_{z_{0}}h_{\phi_{0}}}.$
(77)

where $F_{0},h_{z_{0}},h_{\phi_{0}}$ are the respective values of the
functions $F,h_{z},h_{\phi}$ where $\dot{r}=0$. We may invert the above
equation to obtain an expression for $\tau(r)$ as follows

$\displaystyle\tau(r)=\int dr\frac{1}{f\sqrt{1+\gamma^{-2}Fh_{z}h_{\phi}}}$
(78)

The area may now be computed by dividing it into two symmetric halves and
separately evaluating three different segments of each half. The first part is
described by a segment which starts from boundary and ends at $(U_{a},0)$ the
intersection of the extremal surface with $V=0$. The second segment begins at
($U_{a},0)$ and ends at $(U_{b},V_{b})$ which is a constant $r=r_{0}$ surface
where $r_{0}$ corresponds to the turning point of the extremal surface whereas
the third segment begins from $(U_{b},V_{b})$ and terminates at the radial
point $(0,\alpha/2)$ which is the other horizon. We will not describe the full
computational details here as it is comprehensively explained in
Leichenauer:2014nxa ; Malvimat:2022oue . Basically we obtain the following
expression for $alpha$ which involves three different integrals
$G_{1},G_{2},G_{3}$ as follows

Figure 1: The extremal surface (Red) at $\tau=0$ in the Kruskal extension of
Kerr spacetime divided into three segments denoted as I, II and III. The shift
in $V$ coordinate at $U=0$ is $\alpha$. Note, the lengths II and III are the
same (the diagram is skewed to enlarge the sections) and so are the lengths
along the red curve on the either sides of $U=0\,\,\&\,\,V=\alpha/2$.

$\displaystyle\alpha$ $\displaystyle=2\exp\left(G_{1}+G_{2}+G_{3}\right)$ (79)
where, $\displaystyle\,\,\quad\quad G_{1}$
$\displaystyle=-2\kappa\int_{\bar{r}}^{r_{0}}\frac{dr}{-f}$ (80)
$\displaystyle G_{2}$
$\displaystyle=2\kappa\int_{r_{0}}^{\infty}\frac{dr}{f}\left(1-\frac{1}{\sqrt{1+\gamma^{-2}Fh_{z}h_{\phi}}}\right)$
(81) $\displaystyle G_{3}$
$\displaystyle=2\kappa\int_{r_{0}}^{r_{+}}\frac{dr}{f}\left(1+\frac{1}{\sqrt{1+\gamma^{-2}Fh_{z}h_{\phi}}}\right)$
(82)

where $\bar{r}$ corresponds to the point where $r_{*}$ vanishes inside the
horizon i.e $r_{*}(\bar{r})=0$. Note that the first two integrals given by
$G_{1},G_{2}$ diverge only as $r_{0}\to r_{+}$ which also corresponds to the
limit in which the shift parameter $\alpha\to 0$ whereas the third integral
diverges only as $r_{0}\to r_{c}$, $r_{c}$ being the solution to the following
equation

$\displaystyle
F^{\prime}(r_{c})h_{z}(r_{c})h_{\phi}(r_{c})+F(r_{c})h^{\prime}_{z}(r_{c})h_{\phi}(r_{c})+F(r_{c})h_{z}(r_{c})h^{\prime}_{\phi}(r_{c})=0$
(83)

Therefore as $r_{0}\to r_{c}$ the $G_{3}$ integral diverges which in turn
implies that $\alpha\to\infty$ from eq.(79). By utilizing this result we may
compute the leading contribution to the area integral in eq.(76). To this end
we may first utilize eq.(78) in eq.(76) to obtain the following expression for
the area integral

$\displaystyle\mathcal{A}_{\theta=\frac{\pi}{2}}=8\pi^{2}\int\frac{dr}{f}\frac{Fh_{z}h_{\phi}/\gamma}{\sqrt{1+\gamma^{-2}Fh_{z}h_{\phi}}}$
(84)

Here we used $S_{z}=4\pi$ and $S_{\phi}=2\pi$. We may now re-express the above
area integral by expanding around $r=r_{c}$ in terms of the divergent integral
$G_{3}$ in eq.(82). Note that the required extremal surface gets dominant
contribution from four times the area of the segment as expressed below

$\displaystyle\mathcal{A}_{\theta=\frac{\pi}{2}}$ $\displaystyle\approx
8\pi^{2}\sqrt{-F_{c}\,h^{c}_{z}\,h^{c}_{\phi}}\frac{G_{3}}{2\kappa}$ (85)
$\displaystyle\mathcal{A}_{A\cup B}=4\mathcal{A}_{\theta=\frac{\pi}{2}}$
$\displaystyle\approx\frac{16\pi^{2}}{\kappa}\sqrt{-F_{c}\,h^{c}_{z}\,h^{c}_{\phi}}\log[\alpha]$
(86)

where we have also utilized the relation between $\alpha$ and $G_{3}$ in
eq.(79). Having obtained the area of the required HRT surface we may now
substitute the expression for $\alpha$ given by eq.(72) in the above equation
to obtain

$\displaystyle\mathcal{A}_{A\cup B}\approx\
16\pi^{2}\tau_{0}\sqrt{-F_{c}\,h^{c}_{z}\,h^{c}_{\phi}}+\frac{16\pi^{2}}{\kappa}\log[\frac{\beta
E_{0}(1-\mu\,(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))}{\mathcal{S}}]$
(87)

Figure 2: Plots for $\lambda_{L}^{(min)}$(blue,orange) with and without
angular momentum along with that for $\kappa$ (green) & $\kappa_{0}=2\pi
T_{H}$ (red) against $r_{\\!{}_{-}}/r_{\\!{}_{+}}$. Here
$\mathcal{L}_{+}=\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}\mu^{-1}$ and
$\mathcal{L}_{-}=\varepsilon\mathcal{L}_{+}$ for the green curve and the blue
curve corresponding to $\lambda_{L}^{(min)}$ with angular momentum. We have
taken $\ell=1$. The plot is found to independent of $\varepsilon$. The
bottommost (orange) curve is the plot for $\lambda_{L}^{(min)}$ when
$\mathcal{L}_{\pm}=0$ $i.e.$ when the shockwave has no angular momentum. The
above values of $\mathcal{L}_{\pm}$ are allowed by turning point analysis
except for for very small values of $r_{\\!{}_{-}}/r_{\\!{}_{+}}$ for the blue
and orange plots. It is quite apparent that the instantaneous
$\lambda_{L}^{(min)}$ can be greater than $\kappa_{0}$ for non-extremal
geometries. Here we exclude the extremal point
$\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}=1$.

This completes the computation of the growth of extremal surface due to the
perturbation. We now use the above expression to approximate the holographic
mutual information $I(A:B)$ through the application of the $RT/HRT$ formula as
described by eq.(4). We would like to have some idea about the instantaneous
Lyapunov exponent $\lambda_{L}$ as the shockwave affects the geometry at very
late times. As we have seen at such time scales $I(A:B)$ changes only due to
change in entanglement entropy $S_{A\cup B}$ as the HRT surface corresponding
to it traverses the horizons in the Kruskal diagram. Therefore the
instantaneous $\lambda_{L}$ at late times can be discerned by knowing the rate
of growth of the perturbation in $S_{A\cup B}$ due to the shockwave as
compared to it’s unperturbed value. However in geometries such as Kerr the
HRT/RT surfaces corresponding to $S_{A\cup B}$ are difficult to compute. In
such cases we can estimate the magnitude of $S_{A\cup B}$ by the entropy of
the black hole $\mathcal{S}$. The understanding being that for purification of
large enough systems the entanglement entropy roughly peaks when one
integrates out half of the degrees of freedom121212The entropy of the black
hole $\mathcal{S}$ can be thought of as obtained by integrating out one of the
2 CFTs corresponding to the TFD description.. Therefore for the unperturbed
value of $S^{(0)}_{A\cup B}=\mathcal{A}^{(0)}_{A\cup B}/(4G_{N})$ we can write

$\mathcal{A}_{A\cup B}^{(0)}=\rho\mathcal{A}_{H},\hskip
8.5359pt\rho\in\\{0,1\\}.$ (88)

Defining the instantaneous Lyapunov index as

$\mathcal{A}_{A\cup B}=\lambda_{L}\tau_{0}\mathcal{A}^{(0)}_{A\cup B}$ (89)

we find the minimum value of $\lambda_{L}$ for $\rho=1$ as

$\lambda^{(min)}_{L}=\frac{\mathcal{A}_{A\cup
B}}{\mathcal{A}_{H}}=\frac{16\pi^{2}\sqrt{-F_{c}h^{c}_{z}h_{\phi}^{c}}}{\mathcal{A}_{H}}.$
(90)

Figure 3: Plots various $\lambda_{L}^{(min)}$(blue,orange,green) and blueshift
exponents $\kappa$(red,violet,brown) for different values of
$\mathcal{L}_{+}<\mu^{-1}$. Here $r_{\\!{}_{+}}=0.521$ and $\ell=1$. The red
plot labelled $\kappa$ and the blue plot for $\lambda_{L}^{(min)}$ have
$\mathcal{L}_{+}=\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}\mu^{-1}$. The violet plot
labelled $\kappa_{1}$ and the orange plot for $\lambda_{L}^{(min)}$ have
$\mathcal{L}_{+}=\frac{2}{3}\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}\mu^{-1}$. The
brown plot labelled $\kappa_{0}$ and the green plot for $\lambda_{L}^{(min)}$
have $\mathcal{L}_{+}=0$ $i.e.$ $\kappa_{0}=2\pi T_{H}$. $\mathcal{L}_{-}=0$
for all the plots above as it seen not to affect the functional behaviour. It
is apparent that the blue plot for $\lambda_{L}^{(min)}$ is only bounded by
the plot labelled kappa(red) and not by $\kappa_{1}$ or $\kappa_{0}$.
Similarly the orange plot for $\lambda_{L}^{(min)}$ is not bounded by
$\kappa_{0}$ but by those labelled $\kappa_{1}$ and $\kappa$. Here the plots
exclude the extremal point $\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}=1$.

This is the minimum value of the instantaneous Lyapunov index attained at very
late times, the actual value can be greater than this. We plot131313Unlike the
case in Kerr $AdS_{4}$ here the value of $r_{c}$ and consequently the minimum
value of $F_{c}h^{c}_{z}h^{c}_{\phi}$ is determined numerically. this
$\lambda_{L}^{(min)}$ against $r_{\\!{}_{-}}/r_{\\!{}_{+}}$ for $\ell=1$ in
Fig-2 along with the values for $\kappa$ and $\kappa_{0}=2\pi T_{H}$. Here the
sum of the angular momenta of the shockwave scales as
$\mathcal{L}_{+}=\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}=\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}\mu^{-1}$.
The orange plot is for the case when the shockwave has no angular momenta
along any of the directions, we see that this is clearly bounded by
$\kappa_{0}$ and goes to zero as $r_{\\!{}_{-}}/r_{\\!{}_{+}}\rightarrow 1$.
However the blue curve for $\lambda_{L}^{(min)}$ for
$\mathcal{L}_{+}=\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}\mu^{-1}$ is clearly
greater than $\kappa_{0}$ but is always bounded by $\kappa$. It is interesting
to note that $\lambda_{L}^{(min)}$ does not seem to depend on
$\mathcal{L}_{-}$ while the function $Fh_{z}h_{\phi}$ in general depends on
$\mathcal{L}_{\pm}$.

Figure 4: Plot of $\kappa/\lambda_{L}^{(min)}$ with $\ell=1$ against
$r_{\\!{}_{-}}/r_{\\!{}_{+}}$. This plot is independent of both
$\mathcal{L}_{+}$ and $\mathcal{L}_{-}$ and depends only on the black hole
outer horizon radius $r_{\\!{}_{+}}$.

To get a better idea on the dependence of $\lambda_{L}^{(min)}$ on
$\mathcal{L}_{+}$ we plot in Fig-3 the blueshift exponents $\kappa$ and
$\lambda_{L}^{(min)}$ with
$\mathcal{L}_{+}=\varepsilon_{+}\frac{r_{\\!{}_{-}}}{r_{\\!{}_{+}}}\mu^{-1}$
for $\varepsilon_{+}\in\\{1,\frac{2}{3},0\\}$. We find the
$\lambda_{L}^{(min)}$ for a particular value of $\varepsilon_{+}$ is
effectively bounded by the $\kappa$ for the same value of $\varepsilon_{+}$ or
higher. One can plot $\kappa$ and $\lambda_{L}^{(min)}$ against
$r_{\\!{}_{-}}/r_{\\!{}_{+}}$ for larger number of fractions between
$\varepsilon_{+}\in[1,0]$ and reach the same conclusion. We also find
empirically that $\kappa/\lambda_{L}^{(min)}$ does not depend on
$\mathcal{L}_{+}$ and approaches $1$ for small values of the outer horizon. It
is physically understandable that $\lambda_{L}$ be bounded by the blueshift
suffered by the shockwave in each of the cases above as the later determines
the rate at which the metric changes. However since these are the plots for
$\lambda_{L}^{(min)}$ the actual value of instantaneous $\lambda_{L}$ can be
greater than the respective $\kappa$.

There is another way one can discern $\lambda_{L}$, by determining the
scrambling time $\tau_{*}$. In this case it is the time required by the
shockwave to completely disrupt the mutual information between the subsystems
$A$ and $B$. Substituting the area of the extremal surface homologous to the
subsystem-$AB$ given by eq.(87) in (4) for the holographic mutual information
we obtain

$\displaystyle I(A:B)\approx S_{A}+S_{B}-\
\frac{16\pi^{2}\tau_{0}}{G_{N}}\sqrt{-F_{c}\,h^{c}_{z}\,h^{c}_{\phi}}-\frac{16\pi^{2}}{\kappa
G_{N}}\log[\frac{\beta
E_{0}(1-\mu\,(\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}))}{\mathcal{S}}]$
(91)

We may now utilize the above expression to find out the scrambling time
$\tau_{0}=\tau_{*}$ at which the holographic mutual information vanishes. This
leads us to

$\displaystyle\kappa\tau_{*}\approx\log\mathcal{S}+\log\bigg{[}\frac{1}{(1-\mu\,\mathcal{L}_{+})}\bigg{]}+\frac{\kappa(\mathcal{A}_{A}+\mathcal{A}_{B})}{16\pi^{2}\sqrt{-F_{c}\,h^{c}_{z}\,h^{c}_{\phi}}},$
(92) $\displaystyle{\rm where}\hskip 14.22636pt\kappa=\frac{2\pi
T_{H}}{(1-\mu\,\mathcal{L}_{+})},\,\,\,\,\mathcal{L}_{+}=\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}$
(93)

where we have take $\beta E_{0}\approx 1$ implying that the shockwave has
energy as measured at the $AdS$ boundary to be of the order of the temperature
of the black hole. The last term in (92) can be written using (90) as

$\frac{\kappa(\mathcal{A}_{A}+\mathcal{A}_{B})}{16\pi^{2}\sqrt{-F_{c}\,h^{c}_{z}\,h^{c}_{\phi}}}=\frac{\kappa(\mathcal{A}_{A}+\mathcal{A}_{B})}{\lambda^{(min)}_{L}\mathcal{A}_{H}}$
(94)

which we for large enough subsystems $A$ and $B$ can be seen not to scale with
the entropy of the system. Plotting $\kappa/\lambda_{L}^{(min)}$ against
$r_{\\!{}_{-}}/r_{\\!{}_{+}}$ for various values of $r_{\\!{}_{+}}$ we find
that this is independent of
$\mathcal{L}_{+}=\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}$ see Fig-4.
Therefore this term exists as it is for even when the shockwave has no angular
momentum in the axi-symmetry directions corresponding to the exponential
blueshift with index $\kappa_{0}=2\pi T_{H}$. Further (94) can be regarded as

$\kappa\frac{S_{A}+S_{B}}{S_{A\cup B}}$ (95)

and for large enough thermal systems like that of a large black hole each of
$S_{A}$, $S_{B}$ and $S_{A\cup B}$ scale as some fraction of the black hole
entropy $\mathcal{S}$. Therefore the last term in (92) depends on black hole
parameters like $r_{\\!{}_{-}}/r_{\\!{}_{+}}$ only and not extensively on
$\mathcal{S}$ or on details of the shockwave such as its angular momenta. We
thus find that at very late times leading up to the scrambling time the
Lyapunov index is basically

$\lambda_{L}=\kappa=\frac{2\pi T_{H}}{(1-\mu\,\mathcal{L}_{+})}$ (96)

which can be regarded as its large time average value, where as the
$\lambda_{L}^{(min)}$ can be regarded as the minimum value of $\lambda_{L}$ at
large times leading up to the scrambling time given above.

The second term in (92) is the only term which depends on $\mathcal{L}_{+}$
and the fact that the geometry is Kerr via $\mu$. Although this term does not
scale with $\mathcal{S}$ for large black holes like the last term in (92), it
does however depend on $\mathcal{L}_{+}$ and is positive as
$\mathcal{L}_{+}<\mu^{-1}$. The origin of this term can be traced back to the
first law of black hole dynamics used in the previous section $c.f.$ (69).
This term accounts for the delay in the onset of scrambling due to
$\mathcal{L}_{+}$ and is also found to exist in scrambling times recently
computed for Reisnner-Nordstöm black holes Horowitz:2022ptw . Here for the
case of a charged shockwave falling into the RN black hole the authors find a
delay in the onset of scrambling proportional to

$t^{(q)}_{*}-t_{*}=\frac{1}{2\pi
T_{H}}\log\left(1-\mu\frac{q}{E_{0}}\right)^{-1}$ (97)

where $q/E_{0}$ is the charge per unit energy of the shockwave and $\mu$ is
the chemical potential associated with the RN charge and $t_{*}$ is the
scrambling time measured for an uncharged shockwave. As is apparent from the
analysis in section 3 that this term is accounted for by the thermodynamics of
black holes it is immediately clear that if the charge of the shockwave is
reversed $q\rightarrow-q$ this would lead to a corresponding advance in the
scrambling time $i.e.$ $t_{*}^{(q)}<t_{*}$.

##  5 Conclusion & Discussions

In this paper we study the butterfly effect in Kerr $AdS_{5}$ with equal
angular momentum by analysing the disruption caused to the finely tuned mutual
information $I(A:B)$ between large subsystems of the $left$ and $right$ CFTs
due to in-falling rotating shockwaves. Both the shockwave and the subsystems
are such that they respect the axi-symmetry of the geometry along periodic
angular coordinates $\phi_{1}$ and $\phi_{2}$. Further like in the analysis of
the butterfly effect in Kerr $AdS_{4}$ Malvimat:2022oue  we choose the
shockwave to exist for every value of $\phi_{1}$ and $\phi_{2}$ and only
confined to the equator. For late times $>>T_{H}^{-1}$ leading up to the
scrambling time we are able to find the minimum value for the instantaneous
Lyapunov index $\lambda_{L}^{(min)}$. This is basically the instantaneous
minimum rate of growth of the HRT surface homologous to $A\cup B$ and
traversing the Kruskal diagram. We plot this for different values of the
shockwave’s angular momentum $\mathcal{L}_{\phi_{1}}$ &
$\mathcal{L}_{\phi_{2}}$ Fig-2,3. We find $\lambda_{L}^{(min)}$ to be bounded
by $\kappa$ (93) and independent of the
$\mathcal{L}_{-}=\mathcal{L}_{\phi_{1}}-\mathcal{L}_{\phi_{2}}$. It is easily
seen that $\lambda_{L}^{(min)}>2\pi T_{H}$ for sufficiently non-extremal
geometries and can survive the limit to extremality in certain cases. We also
find that the exponent of the blueshift suffered by such a shockwave by the
time it reaches the outer horizon from the $AdS_{5}$ boundary is given by
$\kappa$ with $\kappa=\kappa_{0}=2\pi T_{H}$ for
$\mathcal{L}_{+}=\mathcal{L}_{\phi_{1}}+\mathcal{L}_{\phi_{2}}=0$. Further we
find that the ratio $\kappa/\lambda_{L}^{(min)}$ to be independent of
$\mathcal{L}_{\pm}$. We are also able to estimate the scrambling time $i.e.$
time taken for the mutual information to go to zero (92) and the late time
Lyapunov exponent is given by $\kappa$ i.e.
$\kappa\tau_{*}\sim\log\mathcal{S}$ where $\mathcal{S}$ is the black hole’s
entropy. We also find corrections to this scrambling time which is positive
and proportional to $\log(1-\mu\,\mathcal{L}_{+})^{-1}$ causing a delay in the
onset of scrambling, however this term is not extensive as
$\kappa^{-1}\log\mathcal{S}$ and does not scale with the black hole’s entropy.
A similar term has also recently appeared in the analysis of RN black holes in
$AdS_{4}$ Horowitz:2022ptw  where the phenomena of delay in the scrambling
times due charged shockwaves have been investigated.

The analysis presented here corroborates a similar one presented for Kerr
black holes in $AdS_{4}$ Malvimat:2022oue . It is also interesting to contrast
it with the similar analysis done for rotating shockwaves in BTZ
Malvimat:2021itk  in which $\lambda_{L}$ was found to be greater than $2\pi
T_{H}$ only for certain specific values of chemical potential $\mu$, shockwave
angular momenta $\mathcal{L}$ and subsystems size. Analytic result was only
obtainable for $\mathcal{L}=1$ where for $\mu>1/2$ we found $\lambda_{L}=\pi
T_{H}/(1-\mu\,\mathcal{L})>2\pi T_{H}$. For the case of Kerr $AdS_{4}$ and
$AdS_{5}$ we see a much more cleaner result in terms of the behaviour of the
instantaneous $\lambda_{L}^{(min)}$ and the scrambling time. This may be due
to higher dimensions allowing a choice of subsystems and HRT surfaces in the
bulk which respect the rotational(axial) symmetry of the geometry.

It has been well known that the low energy chaotic behaviour of large$-N$
holographic systems is captured by the near horizon dynamics and effective
theory of black holes. The JT theory of gravity has been shown to reproduce
the near extremal thermodynamics of wide class of black holes Moitra:2018jqs ;
Castro:2018ffi  for fluctuations in mass and entropy over extremality with the
other black hole charges held fixed. Such a JT theory can be used to deduce
the $\lambda_{L}$ due to interactions with 2d probe scalar fields in the near
horizon region and is given by $\lambda_{L}=2\pi T_{H}$ Jensen:2016pah ;
Maldacena:2016upp . However the phenomena explored in this paper (and in
Malvimat:2022oue ; Malvimat:2021itk ) concerns with perturbing a Kerr black
hole with $\delta M$ and
$\delta(J_{\phi_{1}}+J_{\phi_{2}})=\mathcal{L}_{+}\delta M$. The first law in
this case takes the form

$\delta M=\frac{T_{H}}{(1-\mu\,\mathcal{L}_{+})}\delta\mathcal{S}$ (98)

as $\mu_{1}=\mu_{2}=\mu$. It would be interesting to see if a JT like
effective theory of gravity can capture the thermodynamics of such departures
form extremality Kerr black holes. This question has recently been answered
for the case of near extremal BTZ geometries where such departures form
extremality are also seen to be governed by the JT theory Poojary:2022meo .
Here the near horizon geometry and the JT theory is parametrized by specific
angular momentum $\mathcal{L}$ used to obtain the null ingoing geodesics, the
near horizon limit is then taken along such null geodesics. Such a JT
description captures the relevant near extremal thermodynamics and the
temperature $T_{H}^{(2)}$ of the $AdS_{2}$ geometry in the JT theory is
related to the near extremal BTZ temperature $T_{H}$ as

$T^{(2)}_{H}=\frac{T_{H}}{1-\mu\,\mathcal{L}}$ (99)

It would therefore be interesting to obtain similar effective theories in the
near horizon region of near extremal Kerr black holes in dimensions greater
than 3 Poojary:2022vsz . It must also be noted that as far as thermodynamics
is concerned the JT description for Kerr black holes as in Moitra:2018jqs
captures the behaviour in an ensemble where angular momenta
$J_{\phi_{1},\phi_{2}}$ are held fixed. The perturbation studied in this work
would require one to work in a canonical ensemble where
$M-\mathcal{L}_{\phi_{1}}J_{\phi_{1}}$ and
$M-\mathcal{L}_{\phi_{2}}J_{\phi_{2}}$ are held fixed instead.

The work of Castro $et\,al$ Castro:2021fhc  explores the near horizon
effective action for near extremal Myers-Perry type black hole in $AdS_{5}$
and studies the corrections obtained to the (dynamical part of the) JT theory
in the near horizon and near extremal limit. The above Lyapunov exponent (96)
could therefore probably be obtained from the near horizon picture used in
Castro:2018ffi ; Moitra:2018jqs  if such corrections are sufficiently
accounted for in the effective graviton exchanges between probe matter fields.
Given that sufficiently late time dynamics of perturbation is able to see a
$\lambda_{L}$ given by (96), its quite probable that the deep IR behaviour of
probe matter in near extremal Kerr geometries in $AdS_{5}$ is not simply
obtainable by the JT model as studied in Castro:2018ffi ; Moitra:2018jqs ;
Castro:2021fhc .

## Acknowledgements

RP would like to thank Daniel Grumiller and Prashant Kocherlakota for
discussions pertaining to some aspects of this work. RP is supported by the
Lise Meitner grant M-2882 N of the FWF.

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