††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-$AdS_{4}$

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

We compute the disruption of mutual information between the hemispherical
subsystems on the left and right CFT$s$ of a Thermofield Double state
described by a Kerr geometry in $AdS_{4}$ due to shockwaves along the
equatorial plane. The shockwaves and the subsystems considered respect the
axi-symmetry of the geometry. At late times the disruption of the mutual
information is given by the lengthening of the HRT surface connecting the two
subsystems, we compute the minimum value of the Lyapunov
index-$\lambda_{L}^{(min)}$ at late times and find that it is bounded by
$\kappa=\frac{2\pi T_{H}}{(1-\mu\,\mathcal{L})}$ where $\mu$ is the horizon
velocity and $\mathcal{L}$ is the angular momentum per unit energy of the
shockwave. At very late times we find the the scrambling time for such a
system is governed by $\kappa$ with $\kappa t_{*}=\log\mathcal{S}$ for large
black holes with large entropy $\mathcal{S}$. We also find a term that
increases the scrambling time by $\log(1-\mu\,\mathcal{L})^{-1}$ but which
does not scale with the entropy of the Kerr geometry.

##  1 Introduction

Chaos is a hallmark of large-$N$ many body dynamics. In recent years, the
pursuit for understanding the mechanisms of chaotic phenomena in holographic
systems has led to remarkable progress in quantum gravity. The initial works
of Hayden and Preskill Hayden:2007cs  and Sekino & Susskind Sekino:2008he  had
conjectured that black holes are amongst the fastest scramblers of in-fallen
information based on quantum information theoretic arguments for systems with
a large number of local degrees of freedom. Such systems were known to exhibit
a scrambling time proportional to the local degrees of freedom $i.e.$

$t_{*}\sim\log\mathcal{S},\hskip 11.38092pt{\rm where}\,\,\,\mathcal{S}={\rm
entropy.}$ (1)

A useful diagnostic of chaos is the Lyapunov index $\lambda_{L}$ which can be
discerned from the late time decay of the out of time ordered correlators
$OTOC$

$\frac{\langle W(0)V(t)W(0)V(t)\rangle}{\langle WW\rangle\langle
VV\rangle}=1-\epsilon\,e^{\lambda_{L}t}+\dots$ (2)

where $\epsilon$ is a perturbative parameter in the theory. It can also be
understood in terms of the scrambling of Mutual Information
$I[A:B]=S_{A}+S_{B}-S_{A\cup B}$ between large enough subsystems in a
Thermofield-Double (TFD) state describing an eternal black hole in AdS due to
very early in-fallen perturbation. Here $S_{A}$ is the entanglement entropy of
a sub-region $A$ and the two sub-regions $A$ & $B$ are in the left and right
CFTs describing the Hilbert space of the black hole in $AdS$. Mutual
information can be regarded as a better diagnostic of chaos as it provides a
bound for the correlation function of bounded operators Wolf:2007tdq . For
static black holes in 3d $AdS$ both these measures of chaos revealed a
Lyapunov index given by their temperature $i.e.$ $\lambda_{L}=2\pi T_{H}$
Shenker:2013pqa Shenker:2013yza Shenker:2014cwa . It was further famously
shown by Maldacena, Shenker and Stanford (MSS) Maldacena:2015waa  that large
$N$ QFTs characterised by only temperature possessed an upper bound for chaos
where $\lambda_{L}$ is to bounded by

$\lambda_{L}\leq\frac{2\pi}{\beta},\hskip 14.22636pt{\rm
where}\,\,\beta=T_{H}^{-1}.$ (3)

thus validating the earlier conjecture of black holes as the fastest
scramblers. The case for rotating black holes is more intriguing. Rotating
black holes in $AdS_{3}$ were shown to posses 2 Lyapunov indices governing the
behaviour of the OTOC of 4pt functions, each corresponding to the left and
right moving temperatures of the dual CFT2 Poojary:2018eszzJahnke:2019gxr
Stikonas:2018ane . One of these temperatures is greater than the temperature
of the BTZ and survives the extremality limit. The arguments made for arriving
at the MMS bound were also generalized for large$-N$ thermal QFTs with a
chemical potential $\mu$ associated with a global charge Halder:2019ric , this
revealed that the Lyapunov index to be bounded by

$\lambda_{L}\leq\frac{2\pi/\beta}{(1-\mu/\mu_{c})}$ (4)

where $\mu_{c}$ is the maximum or critical value attainable by $\mu$. It is
also worth noting that such chaotic phenomena have been seen in other
holographic systems such as the dynamics a string suspended in an AdS black
hole deBoer:2017xdk Banerjee:2018kwy Banerjee:2018twd . For the case of
rotating black holes the string worldsheet degrees of freedom also see a
chaotic behaviour consistent with the above bound Banerjee:2019vff . However,
for the case of rotating BTZ the long time behaviour of the OTOC seems to
decay with an average value of $|\lambda_{L}|=2\pi T_{H}$ Mezei:2019dfv ;
Craps:2020ahu . Thus the scrambling time for OTOCs in rotating BTZ and its
holographic CFT2 seems to exhibit the same scrambling time as in the static
case with short bursts of scrambling where instantaneous $\lambda_{L}$ is
greater of the 2 CFT temperatures. However, the recent analysis of the
scrambling of mutual information in rotating BTZ perturbed by shockwaves with
non-zero angular momentum shows a different picture. Here rotating BTZ does
exhibit a $\lambda_{L}>2\pi T_{H}$ even for late times and the scrambling time
is also determined by such a $\lambda_{L}$ Malvimat:2021itk .

Yet another diagnostic of chaos in maximally chaotic systems is the phenomena
of “pole skipping”. Here the retarded energy 2pt functions in frequency space
are seen to have poles except at points where the frequency is
$\omega=i\lambda_{L}$ and wave-number is $k=\pm i\lambda_{L}/v_{B}$; here
$v_{B}$ is the butterfly velocity which measures the spread of perturbation in
space Grozdanov:2017ajz Blake:2017ris . Static black branes in $AdS$ have been
shown to posses this phenomena of pole-skipping with $\lambda_{L}=2\pi T_{H}$.
This peculiar phenomena of pole-skipping has also been justified as arising
from an effective theory of hydrodynamics of maximally chaotic systems
Blake:2021hjj ; Blake:2017ris . It was also shown to be a generic feature of
planar holographic black branes in $AdS$ Blake:2018leo . For rotating BTZ the
pole-skipping phenomena was studied in Liu:2020yaf  where the 2 Lyapunov
indices corresponding to the 2 temperatures of the dual CFT2 were found as the
pole skipping points, corroborating the earlier results in rotating BTZ.
Recent analysis of pole-skipping in Kerr-AdS4 have uncovered $\lambda_{L}=2\pi
T_{H}$ Blake:2021wqj .

In this article we analyse the effect of rotating shockwaves on the Mutual
Information $I(A:B)$ between the left and right CFTs describing the
analytically extended rotating bulk geometry as a thermofield double state.
Here $A$ and $B$ are identical hemispheres which end at the equator at the
left and right boundary CFTs respectively. For simplicity the shockwaves
considered are axi-symmetric $i.e.$ they exist for every value of the
symmetric direction $\phi$, and are confined to the equator. The shockwave
geometry to be solved for is given by the Dray-’tHooft solution provided one
utilizes the Kruskal coordinates along the shockwave trajectory. We therefore
construct such coordinates for Kerr-$AdS_{4}$ metric along in-out going null
rotating geodesics with angular momentum per unit energy $\mathcal{L}$, and
write the Dray-’t Hooft solution at very late times as a function of the time
$t_{0}$ in the past- denoting the time when the shockwaves were released.

Following the general construction by Shenker and Stanford Shenker:2013pqa  we
then analyse the late time behaviour of $I(A:B)$ due to such a shockwave
perturbing the Kruskal extension via the left boundary at some time $t_{0}$ in
the past $w.r.t$ the right CFT. Along the way we note that the blueshift
suffered by such a rotating shockwave as it falls towards the horizon is given
by $E\sim E_{0}e^{\kappa t_{0}}$ with

$\displaystyle\kappa=\frac{2\pi}{\beta(1-\mu\mathcal{L})},\hskip 8.5359pt{\rm
where}$ $\displaystyle{\mu=\Omega_{+}}={\rm horizon\,velocity,}$ (6)
$\displaystyle\mathcal{L}={\rm angular\,mom./\,energy.}$

This quantity is obtained by demanding that the Kruskal coordinates be smooth
as one approaches the horizon and are devoid of any coordinate singularities.
As observed in Shenker:2013pqa  the blueshift essentially dictates the late
time behaviour of the perturbation to $I[A:B]$ and thus the scrambling of
holographic mutual information between the 2 $CFT$s in a Thermofield double
state describing the Kerr black hole in the bulk. The computation of $I[A:B]$
for this geometry is not analytically tractable and hence we determine the
perturbation to $S_{A\cup B}$ Leichenauer:2014nxa  at very late times as it is
the only term in $I(A:B)$ which is sensitive to the shift in horizon along
null coordinates, unlike $S_{A}$ and $S_{B}$ which remain unaffected. We
obtain this perturbation at $t=0$ as function of $t_{0}$ and the black hole
parameters, and find that it grows linearly in $t_{0}$ as expected $i.e.$

$I[A:B]\sim\mathcal{A}\,\lambda_{L}\,t_{0}$ (7)

where $\mathcal{A}$ can be taken to be the unperturbed area capturing
$S_{A\cup B}$. We would regard the above $\lambda_{L}$ as the instantaneous
Lyapunov index computed at late times $i.e.$ $\gg\beta=T_{H}^{-1}$. We find
that for $\mathcal{L}=0$, $\lambda_{L}^{\rm(min)}\leq\kappa_{0}=2\pi T_{H}$
with the equality holding at extremality. For general values of $\mathcal{L}$
we find

$\lambda_{L}^{\rm(min)}\leq\kappa=\frac{2\pi T_{H}}{(1-\mu\mathcal{L})}$ (8)

We plot this as a function of the ratio of the inner to outer horizon
${r_{\\!{}_{-}}}/{r_{\\!{}_{+}}}$; Fig [2]. We also find that as one
approaches extremality, after a finite value of
${r_{\\!{}_{-}}}/{r_{\\!{}_{+}}}$, $\lambda_{L}^{\rm(min)}>2\pi T_{H}$ $i.e.$
the black hole scrambles information at a rate greater than $2\pi$ times its
temperature. The near extremal limit of $\kappa=\kappa_{ext}$ can be non-zero
if $\mathcal{L}\rightarrow\mu^{-1}$ as one approaches extremality. For such a
limit we find

$0<\lambda_{L}^{\rm(min)}<\kappa_{ext}.$ (9)

We find that at very late times the rate of scrambling is given by $\kappa$
and the scrambling time by

$t_{*}=\frac{1}{\kappa}\log[\mathcal{S}]+\dots$ (10)

where $\mathcal{S}$ is the entropy of the black hole and the $\dots$ indicate
terms which do not scale with $\mathcal{S}$ at large values of entropy.

This paper is organized as follows: In section 2 we review some details of the
Kerr black hole in $AdS_{4}$. In section 3 we introduce the rotating null
coordinates appropriate for solving the Dray-’tHooft solution for a rotating
shockwave in this Kerr geometry. In doing so we compute the blueshift along a
null in-falling geodesic with angular momentum $\mathcal{L}$. We also solve
for the Dray-’t Hooft solution for an equatorial axi-symmetric shockwave with
angular momentum $\mathcal{L}$. In section 4 we turn to compute the
perturbation of the Mutual Information between the CFTL and CFTR for identical
hemispheres ending at the equator due to such a shockwave. We end with some
Discussions and Conclusions in section 5.

##  2 Kerr-$AdS_{4}$

The Kerr metric in $AdS_{4}$ in the Boyer-Lindquist coordinates
Plebanski:1976gy ; Carter:1968ks ; Iyer:1994ys  takes the form

$\displaystyle
ds^{2}=\rho^{2}\left(\frac{dr^{2}}{\Delta}+\frac{d\theta^{2}}{\Delta_{\theta}}\right)-\frac{\Delta}{\rho^{2}}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi\right)^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\left(adt-\frac{r^{2}+a^{2}}{\Xi}d\phi\right)^{2}$
(11) $\displaystyle{\rm where}$ $\displaystyle\hskip
56.9055pt\Delta=(r^{2}+a^{2})(1+r^{2}/l^{2})-2mr,\hskip
14.22636pt\Xi=1-\frac{a^{2}}{l^{2}},$ (15) $\displaystyle\hskip
56.9055pt\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,\hskip
14.22636pt\Delta_{\theta}=1-\frac{a^{2}}{l^{2}}\cos^{2}\theta.$

The outer horizon exists as the largest root of $\Delta=0$. Here the boundary
coordinates at $r\rightarrow\infty$ are rotating with an angular velocity of

$\Omega_{\infty}=-\frac{a}{l^{2}}$ (16)

while the horizon rotates at

$\Omega_{\phi}=\frac{a\,\Xi}{{r_{\\!{}_{+}}}^{2}+a^{2}}$ (17)

Therefore the horizon’s velocity $w.r.t.$ a stationary boundary observer is

$\Omega_{+}=\Omega_{\phi}-\Omega_{\infty}=\frac{a(1+{r_{\\!{}_{+}}}^{2}/l^{2})}{{r_{\\!{}_{+}}}^{2}+a^{2}}=\mu$
(18)

which is indeed the chemical potential relevant for the thermodynamics of the
black hole Papadimitriou:2005ii . The temperature and entropy can be given in
terms of $\\{a,{r_{\\!{}_{+}}}\\}$ as

$2\pi
T_{H}=\frac{{r_{\\!{}_{+}}}^{2}-a^{2}+{r_{\\!{}_{+}}}^{2}/\l^{-2}(3{r_{\\!{}_{+}}}^{2}+a^{2})}{2{r_{\\!{}_{+}}}({r_{\\!{}_{+}}}^{2}+a^{2})},\hskip
5.69046pt\mathcal{S}=\frac{\pi({r_{\\!{}_{+}}}^{2}+a^{2})}{G_{N}\Xi}$ (19)

The rotating black hole in $AdS$ would correspond to a boundary large $N$
theory with the above fixed temperature and chemical potential given by
$\beta\mu$ where $T_{H}=\beta^{-1}$. The bulk geometry provides a semi-
classical description of a system with a partition function and density of
states given by

$Z=Tr\left[e^{-\beta H-\beta\mu J}\right],\hskip
11.38092pt\rho=\frac{e^{-\beta H-\beta\mu J}}{Z}$ (20)

Following Maldacena:2001kr , in holography black holes in $AdS$ can be
purified by considering its Kruskal extension, in such a description the two-
sided $AdS$ black hole is described by a pure state in the extended Hilbert
space of the 2 boundary CFTs at the left and right exteriors

$|\Psi_{\rm BH}(\beta,\mu)\rangle=\frac{1}{\sqrt{{\rm Tr}\,e^{-\beta(H+\mu
J)}}}\sum_{i}e^{-\beta(H+\mu J)}\,\,|i_{L}\rangle\otimes|i_{R}\rangle.$ (21)

The black hole entropy in any of the exteriors can therefore be obtained as
the von Neuman entropy of the density of states after having integrated out
one of the CFT Hilbert space. The dual bulk geometry describes a fine grained
atypical correlation between the left and the right CFTs. This non-trivial
entanglement between the 2 CFTs is characterized by the holographic
entanglement entropy which at the semi-classical level is given by the area of
the Ryu-Takayanagi (RT) surface for the case of static geometries and by the
Hubeny-Rangamani-Takayanagi (HRT) surfaces for the case of time dependent
geometries Ryu:2006bv ; Hubeny:2007xt .

A useful diagnostic of this fine correlation is the Mutual Information
$I[A:B]$ between space-like subsystems $A$ in the CFTL and $B$ at the CFTR

$I[A:B]=S_{A}+S_{B}-S_{A\cup B}$ (22)

where $S_{A_{i}}$ in the above equation denotes is the entanglement entropy of
the subsystem-$A_{i}$ obtained by tracing out the rest of the system from the
density matrix. As both RT and HRT surfaces are minimal area co-dimension two
surfaces homologous to the boundary subsystems, the extent they probe the bulk
depends on the sizes of the subsystems. For small enough subsystems $A$ & $B$
this implies that $I[A:B]$ vanishes as $S_{A}=S_{B}$. This is simply because
the other possible minimal area surface homologous to $A\cup B$ traversing the
black hole geometry would be larger than $S_{A}+S_{B}$. However for large
enough sizes of $A$ & $B$ the situation reverses and $S_{A\cup B}$ is given by
an HRT surface traversing the two boundaries. In such cases we get a glimpse
of the fine correlation between the 2 subsystems especially when we observe
the behaviour of $I[A:B]$ for late times after having perturbed it with an
$\mathcal{O}(1)$ operator in the CFT or equivalently an $\mathcal{O}(G_{N})$
perturbation in the bulk Shenker:2013pqa .

The study of perturbations to black hole geometries can be very complicated
and is a fertile area of research in numerical relativity, however certain
well known exact solutions exist when the perturbations are shockwaves $i.e.$
the Vaidya geometry Vaidya:1950  and the Axelburg-Sexl or the Dray-’tHooft
solutions Dray:1984ha ; Sfetsos:1994xa . A small perturbation emanating from
the boundary of $AdS$ in a black hole geometry would get blue-shifted as time
passes and it gravitates towards the horizon. Therefore at very late times
$\gg\beta$, such a perturbation can be regarded as a shockwave and- as
observed by Shenker:2013pqa , can be utilized to compute the change in the
late time behaviour of mutual information for large enough subsystems due to
such a perturbation. As noted in Shenker:2013pqa  the late time change in
$I[A:B]$ is basically governed by the blueshift suffered by the in-falling
shockwave. For static geometries this is simply given by

$E\sim E_{0}e^{\frac{2\pi}{\beta}t_{0}},$ (23)

where $E$ is the energy of the shockwave as it passes the bifurcate horizon
and $E_{0}$ and $t_{0}$ are the energy at the boundary and the time in the
past when the shockwave was released. We release this shockwave from the left
boundary at time $t_{L}=t_{0}\gg\beta$ which can be regarded as perturbing
right system in its far past. The mutual information at $t_{L}=0=t_{R}$ for
large enough subsystems would see the effect of this shockwave due the the
above blueshift and is scrambled at a rate determined by the index of the
blue-shift $i.e.$ the temperature $\frac{2\pi}{\beta}$. The blueshift of the
shockwave is tied in with the Kruskal coordinates used to analyse the problem.
One generally chooses

$\displaystyle U=e^{\frac{2\pi}{\beta}(r_{*}-t)},\hskip
11.38092ptV=e^{\frac{2\pi}{\beta}(r_{*}+t)}\hskip 14.22636pt{\rm
where}\,\,\,r_{*}\sim-\int_{\infty}^{r^{\prime}}\frac{dr}{g_{tt}}$ (24)

is the tortoise coordinate. However, this decides the trajectory of the null
shockwave as the Dray-’tHooft solutions are given for shockwaves at $U=0$ or
$V=0$ $i.e.$ for shockwaves along the past or the future horizons
Sfetsos:1994xa . For a more complete analysis, in the present article we
consider the Kerr geometry perturbed by the rotating shockwaves with angular
momentum per unit energy $\mathcal{L}$. The analysis of such a perturbation
along the lines of Dray-’tHooft in this case would therefore require the
Kruskal coordinates which also follow the shockwave.

##  3 The Dray-’tHooft solution

In this section we construct the Dray-’tHooft solution for a generic null
shockwave as a function of the time $t_{0}$ when it was released from one of
the boundaries. For simplicity we would be considering shockwaves along the
equatorial plane $\theta=\pi/2$ such that they always stay on the equator.
This is guaranteed by the symmetry of the geometry. We first begin by
constructing the Kruskal coordinates along such null rotating geodesics with
angular momentum $\mathcal{L}$ along the equator and then extend it to
arbitrary values of $\theta$. In this process we uncover the index of the
exponential blueshift suffered by the shockwave. We then construct the
Dray-’tHooft solution by determining the shift in one of the Kruskal
coordinates in terms of the initial time $t_{0}\gg\beta$ when it was released
into the bulk.

###  3.1 Rotating Kruskal coordinates at the equator

In falling null geodesics $\xi\cdot\partial$ would be parametrised by energy
$\mathcal{E}=1$, angular momenta about the $\phi$ direction and the Carter’s
constant $\mathcal{Q}$ given by the Killing-Yano tensor $K_{\mu\nu}$

$\xi^{2}=0,\hskip
28.45274ptg_{\mu\nu}\xi^{\mu}\zeta^{\nu}_{t}=\mathcal{E},\hskip
28.45274ptg_{\mu\nu}\xi^{\mu}\zeta^{\nu}_{\phi}=\mathcal{L},\hskip
28.45274pt\xi^{\mu}K_{\mu\nu}\xi^{\nu}=\mathcal{Q}$ (25)

where $\zeta_{t}=\partial_{t}-\tfrac{a}{l^{2}}\partial_{\phi}$ and
$\zeta_{\phi}=\partial_{\phi}$. The geodesics along the equator stay along the
equator. We define in-falling(out-going) Kruskal coordinates along such
geodesics with arbitrary $\mathcal{L}$. It turns out that $\xi^{\theta}$ can
consistently be put to zero at the equator for

$\mathcal{Q}=-\frac{(a-\mathcal{L})^{2}}{a^{2}}$ (26)

in (25). The metric (15) can then be recast using the one-forms dual to the
in-out double null geodesics to look like

$\displaystyle
ds^{2}_{\theta=\tfrac{\pi}{2}}=F\,\xi_{\mu}^{+}\xi_{\nu}^{-}dx^{\mu}dx^{\nu}+h\,(dz+h_{\tau}d\tau)^{2}$
(27) (28)
$\displaystyle\tau=t\left(1-\tfrac{a\mathcal{L}}{l^{2}}\right)-\mathcal{L}\,\phi,\hskip
28.45274pt\eta
z=(1+\mathcal{L}\gamma)\,\phi-(1-\tfrac{a\mathcal{L}}{l^{2}})\gamma\,t$ (29)

where the first term is obtained by comparing the coefficient of the $dr^{2}$
term in (15). Here the shift of the $z$ coordinate in $t$ is forced by
demanding that $h_{\tau}\sim(r-{r_{\\!{}_{+}}})$ as
$r\rightarrow{r_{\\!{}_{+}}}$ which fixes $\gamma$ to be

$\gamma=\frac{a(1-a^{2}/l^{2})}{(a^{2}+{r_{\\!{}_{+}}}^{2})l^{2}-({r_{\\!{}_{+}}}^{2}+l^{2})a\mathcal{L}}$
(30)

$\eta$ can be chosen to one, however choosing

$\displaystyle\eta=\frac{l^{2}(1+\mathcal{L}\gamma)}{l^{2}-a\mathcal{L}},\hskip
8.5359pt{\rm implies}$ (31) (32) $\displaystyle
t=\frac{\tau+\mathcal{L}\,z}{1-\mu\,\mathcal{L}},\hskip
8.5359pt\phi=\frac{(1-a\mathcal{L}/\l^{2})\,z+\Omega_{\phi}\tau}{1-\mu\,\mathcal{L}}.$
(33)

The choice of $\eta$ or $z$ is such that we recover the horizon area when
$\theta$ is integrated from $0$ to $\pi$ and $z$ from $0$ to $2\pi$ for the
above metric at the outer horizon111The reason we choose the horizon and not
any other point is because in calculating the Dray-’tHooft solution we would
be requiring the transverse components be smooth across the horizon, also the
shockwave would produce the strongest backreaction at $r={r_{\\!{}_{+}}}$ than
at any other value of $r$ and it is this near horizon dynamics we wish to
capture. Note, $z$ is precisely the near horizon co-moving coordinate one
needs to utilize in order to work in the near horizon region of (near)extremal
Kerr geometries $c.f.$ Lu:2008jk Hartman:2008pb .. The above relation can be
inverted

$\tau=\bigg{(}1-\frac{a\mathcal{L}}{l^{2}}\bigg{)}t-\mathcal{L}\,\phi,\hskip
11.38092ptz=\phi-\mu\,t.$ (34)

This choice of $\eta$ also implies that $z$ is a co-moving co-ordinate along
the horizon’s angular velocity as observed by a stationary observer at the
$AdS$ boundary. The co-moving coordinate $z$ also occurs when one takes the
near horizon limit of extremal Kerr solutions Guica:2008mu ; Lu:2008jk . The
null vector fields $\xi_{\pm}$ have their signs of $\mathcal{E}$ &
$\mathcal{L}$ reversed $w.r,t.$ each other. The light-cone coordinates
$\\{u,v\\}$ are given in terms of the one-forms

$\displaystyle du=\xi^{+}_{\mu}dx^{\mu}\implies u=r_{*}-\tau$ (35) (36)
$\displaystyle dv=\xi^{-}_{\mu}dx^{\mu}\implies v=r_{*}+\tau$ (37)

We now note that the tangent vector $\chi=\partial_{u}$ (or $\partial_{v}$) is
not affine at ${r_{\\!{}_{+}}}$ $i.e.$
$\chi\cdot\nabla\chi^{\mu}=\mathcal{K}\chi^{\mu}$, where

$\mathcal{K}=\tfrac{1}{2}\xi^{+}\cdot\partial F$ (38)

We can therefore define new coordinates $\\{U,V\\}$ on the equator such that
they are affine at ${r_{\\!{}_{+}}}$

$U=-e^{\kappa(r_{*}-\tau)},\,\,V=e^{\kappa(r_{*}+\tau)},\hskip 19.91684pt{\rm
where}\,\,\kappa=\mathcal{K}|_{{r_{\\!{}_{+}}}}$ (39)

These are the required Kruskal coordinates with angular momenta $\mathcal{L}$.
The above co-ordinates capture the right exterior while the left exterior is
obtained by changing the signs in front of $\\{U,V\\}$. The metric now takes
the form

$ds^{2}_{\theta=\tfrac{\pi}{2}}=\frac{F}{\kappa^{2}UV}dUdV+h\,(dz+h_{\tau}d\tau)^{2}$
(40)

The blueshift along this in falling null geodesic is given by the exponential
factor $e^{\kappa t_{0}}$ for some particle released from the boundary at time
$t=0$. We obtain the following result for generic values of ${\cal L}$

$\displaystyle\kappa=\frac{\kappa_{0}}{(1-\mu\mathcal{L})},\,\,\,\,{\rm
where}\,\,\mu=\Omega_{+}$ (41)

For the specific case of a shockwave with zero angular momentum $i.e$ ${\cal
L}=0$ we get

$\displaystyle\kappa|_{\mathcal{L}\rightarrow
0}=\kappa_{0}=\frac{2\pi}{\beta}=2\pi T_{H},\,\,\,\,$ (42)

####  3.1.1 Turning point

It must be noted that not any value of $\mathcal{L}$ is allowed as we will
later setup the shockwave to emanate from the $AdS$ boundary and reach at
least the outer horizon. Assuming that we send null perturbations from the
very start222This is not required as one may choose to send in a massive
particle who’s trajectory tends to that of a null particle as it approaches
the horizon. the allowed values of $\mathcal{L}$ for such null geodesices are
obtainable by a turning point analysis. We find that for $\mathcal{L}>0$ the
following general conditions hold

$\displaystyle\mathcal{L}$ $\displaystyle\leq$
$\displaystyle\mathcal{L}_{max}<\mu^{-1},\hskip 11.38092pt{\rm
for}\,\,T_{H}>0$ (43) $\displaystyle\mathcal{L}$ $\displaystyle\leq$
$\displaystyle\mathcal{L}_{max}=\mu^{-1},\hskip 11.38092pt{\rm
for}\,\,T_{H}=0$ (45)

where $\mathcal{L}_{max}$ is the maximum allowed limit of angular momenta (per
unit energy) beyond which the turning point is outside the horizon. The above
condition at extremality can be checked explicitly by plugging in
$\mathcal{L}=\mu^{-1}$ into the radial component $\xi^{r}=\dot{r}$ of the null
geodesic to find $\xi^{r}({r_{\\!{}_{+}}})=0$. Therefore $\kappa$ doesn’t blow
up and is finite and non-zero even at extremality. We obtain the extremal
value of $\kappa$ by scaling
${r_{\\!{}_{-}}}\rightarrow\epsilon{r_{\\!{}_{+}}}$,
$\mathcal{L}\rightarrow\epsilon\mu^{-1}$ and taking the limit
$\epsilon\rightarrow 1$

$\kappa_{\rm
ext}=\frac{1+6r_{0}^{2}/l^{2}-3r_{0}^{4}/l^{4}}{4r_{0}(1+r_{0}^{2}/l^{2})}$
(46)

where $r_{0}$ is the extremal horizon radius. This is akin to taking the
extremal limit of
$\mathcal{L}=\frac{{r_{\\!{}_{-}}}}{{r_{\\!{}_{+}}}}\mu^{-1}$. We find that

$\mathcal{L}=\frac{{r_{\\!{}_{-}}}}{{r_{\\!{}_{+}}}}\mu^{-1}\leq\mathcal{L}_{max}$
(47)

with the equality holding at extremality. We also observe that the extremal
limit of $\kappa$ for the above form of $\mathcal{L}$ is related to the left
moving Frolov-Thorne temperature $T_{L}$ observed in the Kerr-CFT literature
for Kerr-$AdS_{4}$ at extremality Lu:2008jk

$\kappa_{\rm ext}=\frac{T_{L}}{2\,\Omega_{\phi\,{\rm ext}}}.$ (48)

Here $\Omega_{\phi\,{\rm ext}}$ is the extremal value of the horizon velocity
(17) measured in the rotating Boyer-Lindquist frame at the boundary. Note that
$\kappa_{\rm ext}$ depends on how we scale $\mathcal{L}$ $w.r.t.$ the black
hole parameters as we approach near extremal configuration, in general it can
easily be seen that for ${r_{\\!{}_{-}}}\rightarrow\epsilon{r_{\\!{}_{+}}}$
and $\mathcal{L}\rightarrow\epsilon^{\sigma}\mu^{-1}$ we find the above
$\kappa_{\rm ext}$ changes to

$\kappa_{\rm ext}\rightarrow\frac{\kappa_{\rm ext}}{\sigma}$ (49)

It would be interesting to understand how different possible ways of scaling
$\mathcal{L}$ relate with regards to the near horizon perspective.

###  3.2 Full Coordinates

We would like to extend the above set of coordinates along the entire sphere.
Although this would not be required for the later computations we demonstrate
here that this is indeed possible. The above set of coordinates were written
out for the equatorial plane at $\theta=\pi/2$, however in order to truly
realize $\\{U,V\\}$ as Kruskal coordinates we need to show that they are exact
$i.e.$

$\xi^{-}_{\mu}dx^{\mu}=dV,\,\,\,\xi^{+}_{\mu}dx^{\mu}=dU,\,\,\,\forall\,\theta\in[0,\pi]$
(50)

For this it is enough to show

$\partial_{\theta}\xi^{\pm}_{r}=\partial_{r}\xi^{\pm}_{\theta}$ (51)

This implies $\mathcal{Q}=const$. As we would be interested in the scenario
wherein the shock-wave exists only on the equatorial plane we choose this to
constant to be

$\mathcal{Q}=-\frac{(a-\mathcal{L})^{2}}{a^{2}}$ (52)

as this condition is equivalent to choosing the $\xi^{\theta}|_{\pi/2}=0$.
Note that this implies that the non-equatorial geodesics have
$\xi^{\theta}\neq 0$ $i.e.$ they do not move along constant values of
$\theta$. It turns out that for constant value of $\mathcal{Q}$ above we have
$\partial_{\theta}\xi^{\pm}_{r}=\partial_{r}\xi^{\pm}_{\theta}=0$. We further
have to have to reverse the sign of the $\theta$ component between in and out
going null geodesics $i.e.$ $\xi^{\theta}_{+}=-\xi^{\theta}_{-}$ as simply
reversing the signs of $\mathcal{E}$ & $\mathcal{L}$ doest seem to do this.
This is important as we expect in going geodesics to have the opposite rate of
change in $\theta$ as compared to the out going one.

The metric then takes form

$\displaystyle
ds^{2}=\frac{F}{\kappa^{2}UV}dUdV+h\,(dz+h_{\tau}d\tau)^{2}+g\,(d\theta+g_{\tau}d\tau)^{2},\hskip
11.38092ptg_{\tau}(\pi/2)=0$ (53)

where

$\displaystyle F(r,\theta)=$ (54) (55) $\displaystyle\hskip
5.69046pt=\frac{\ell^{2}\left(a^{2}\left(r^{2}+\ell^{2}\right)+r\ell^{2}(r-2M)+r^{4}\right)\left(a^{2}\cos^{2}(\theta)+r^{2}\right)}{r\left(\ell^{4}\left(a^{2}(2M+r)+r^{3}\right)+\mathcal{L}^{2}\left(a^{2}r\left(r^{2}+\ell^{2}\right)-\ell^{2}\left(\ell^{2}(r-2M)+r^{3}\right)\right)-a^{2}r\ell^{2}\left(a^{2}+r^{2}\right)-4aM\ell^{4}\mathcal{L}\right)}$
(56)

Here we have used the functions $\\{F,h,h_{\tau},g,g_{\tau}\\}$ which now
depend on $r$ and $\theta$. We use the same letters to denote these functions
as in the previous section with the understanding the analysis in the previous
section was carried out at $\theta=\pi/2$. The rest of the functions can be
easily deduced from the transformation (33). The transverse directions can be
worked out in detail and we do not explicitly write them here. We concern
ourselves with the smoothness of the $\\{U,V\\}$ directions. It is
straightforward to see that the Kruskal coordinates are defined as before to
be

$U=-e^{\kappa(r_{*}-\tau-\tilde{g}(\theta))},\,\,V=e^{\kappa(r_{*}+\tau+\tilde{g}(\theta))},\hskip
19.91684pt{\rm where}\,\,\kappa=\mathcal{K}|_{{r_{\\!{}_{+}}}}$ (58)

with an extra $\theta$ dependence

$\widetilde{g}(\theta)=\frac{\sqrt{-\Xi\cot^{2}\theta\left(\ell^{2}(\Xi-\Delta_{\theta})+\Delta_{\theta}\mathcal{L}^{2}\right)}}{\Delta_{\theta}}\implies\widetilde{g}(\pi/2)=0$
(59)

$\mathcal{K}$ is similarly defined as

$\mathcal{K}=\tfrac{1}{2}\xi^{+}\cdot\partial F$ (60)

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

$\kappa=\frac{\kappa_{0}}{(1-\mu\mathcal{L})},\hskip 19.91684pt{\rm
where}\,\,\,\,\kappa_{0}=\frac{2\pi}{\beta},\,\,\mu=\Omega_{+}$ (61)

The above value of $\kappa$ suggests an upper bound for the angular momentum
of the shock-wave to be $\mathcal{L}<\mu^{-1}=\Omega_{+}^{-1}$.

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

The specific forms of the functions $\\{F,h,h_{\tau},g,g_{\tau}\\}$ are quite
cumbersome, it is useful to remember their behaviours close to the outer
horizon

$F\sim F^{\prime}({r_{\\!{}_{+}}})(r-{r_{\\!{}_{+}}}),\hskip
11.38092pth_{\tau}\sim\mathcal{O}(r-{r_{\\!{}_{+}}}),\hskip
11.38092ptg_{\tau}\sim\mathcal{O}(r-{r_{\\!{}_{+}}})$ (62)

while the rest approach constant values. We note the values of
$F^{\prime}({r_{\\!{}_{+}}})$ below

$\displaystyle F^{\prime}({r_{\\!{}_{+}}})=\frac{\pi
T_{H}\,({r_{\\!{}_{+}}}^{2}+a^{2}\cos^{2}\,\theta)}{(a^{2}+{r_{\\!{}_{+}}}^{2})(1-\mu\,\mathcal{L})^{2}}\hskip
11.38092pt$ (63)

It would be necessary for later purposes to expand $F(r,\theta)$ around
${r_{\\!{}_{+}}}$ and re-write it in terms of $UV$. Expanding $F$ as

$F(r,\theta)=(r-{r_{\\!{}_{+}}})F^{\prime}({r_{\\!{}_{+}}})+\tfrac{1}{2}(r-{r_{\\!{}_{+}}})^{2}F^{\prime\prime}({r_{\\!{}_{+}}})+\dots$
(64)

we choose

$(r-{r_{\\!{}_{+}}})F^{\prime}({r_{\\!{}_{+}}})=-\mathbb{A}\,\,UV.$ (65)

The $-$ve sign is chosen as $lhs$ above is positive and $UV$ is negative in
the exterior regions. The proportionality constant $\mathbb{A}$ is an unknown
function of the black hole parameters who’s precise form is not required for
the analysis, however we do require it not scale $w.r.t.$ the entropy of the
black hole. This is expected as the above equation is simply a relation
between coordinates. Therefore we find

$\frac{F}{\mathbb{A}\,UV}=-1-\frac{\mathbb{A}\,UV}{2}\frac{F^{\prime\prime}({r_{\\!{}_{+}}})}{F^{\prime}({r_{\\!{}_{+}}})^{2}}+\dots$
(66)

where $\dots$ denote $\mathcal{O}(U^{2}V^{2})$ terms.

####  3.2.1 Kerr-Newman $AdS_{4}$

The generalization to the charged case is straightforward. The metric is the
same as in (15) with the exception of

$\Delta=(r^{2}+a^{2})(1+r^{2}/l^{2})-2mr+q^{2}$ (67)

and a gauge field given by

$A=-\frac{2qr}{\rho^{2}}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi\right).$
(68)

The electric charge is given by

$Q=\frac{2q}{\Xi}$ (69)

The expressions for the angular velocities is the same as in the Kerr case in
terms of $a$ and ${r_{\\!{}_{+}}}$. The blueshift along a null in-falling
geodesic is then given by

$\kappa=\frac{2\pi T_{H}}{(1-\mu\mathcal{L})},\hskip 5.69046pt{\rm
where}\,\,\,\,2\pi
T_{H}=\frac{{r_{\\!{}_{+}}}^{2}-(a^{2}+q^{2})+{r_{\\!{}_{+}}}^{2}/\l^{-2}(3{r_{\\!{}_{+}}}^{2}+a^{2})}{2{r_{\\!{}_{+}}}({r_{\\!{}_{+}}}^{2}+a^{2})}.$
(70)

It is interesting to note that the form of the blueshift is unchanged and is
only dependent on the coupling $\mu\,\mathcal{L}$ and $T_{H}$. Note, the
trajectory of charged and un-charged in-falling massive particles at
sufficiently late times would still be that of a null geodesic. However, the
amount of time required for a charged massive particle to approach a light-
like trajectory would be different from that of an uncharged one. Massless
particles on the other hand would follow light-like trajectories irrespective
of their charge. We also repeat similar analysis for the case of RN $AdS_{4}$
in the appendix A and find that the angular momentum of the shockwave has no
effect on the blueshift seen at the horizon. In the rest of the paper we only
consider the Kerr $AdS_{4}$ case for simplicity as the analysis can be readily
generalised.

###  3.3 Shockwaves along the equator

We next consider in-falling shockwaves along the equatorial plane with angular
momentum $\mathcal{L}$. The coordinate system developed in the previous
subsection is precisely suited for studying the back reaction generated by
these shockwaves. Like in the case studied in Shenker:2013pqa  we choose some
time $t_{0}$ in the far future of the left exterior at which we send in an
axisymmetric shockwave. We begin with the Dray-’t Hooft solution for the
metric (53) for a axi-symmetric shockwave at $U=0$ $i.e.$ along the $V$
coordinate which is in-falling333It is regarded in-falling $w.r.t.$ the time
in the right exterior. in the left exterior. The metric for $U<0$ is defined
by (53) while the metric for $U>0$ is given by shifting

$V\rightarrow\widetilde{V}=V+\alpha\,\Theta(U)f(\theta)$ (71)

where the constant $\alpha$ is proportional to the strength of the shockwave
at $\theta=\pi/2$; its precise form will be obtained by satisfying a
smoothness condition. The response function $f(\theta)$ captures the response
of the shockwave in the transverse direction and is independent of $\phi$ due
to axi-symmetry. The metric (53) in the coordinates $\\{U,V,z\\}$ can be
obtained by noting

$d\tau=\frac{1}{2\kappa UV}(UdV-VdU)-\widetilde{g}^{\prime}d\theta$ (72)

yielding

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{F}{\kappa^{2}UV}dUdV+h\,\left[dz+\frac{h_{\tau}}{2\kappa
UV}(UdV-VdU)-h_{\tau}\widetilde{g}^{\prime}d\theta\right]^{2}+$ (75)
$\displaystyle\hskip 79.6678pt+g\,\left[d\theta+\frac{g_{\tau}}{2\kappa
UV}(UdV-VdU)-g_{\tau}\widetilde{g}^{\prime}d\theta\right]^{2}$

The Dray-’tHooft solution is then simply obtained by implementing the shift
(71)

$ds^{2}\rightarrow\widetilde{ds}^{2}-\frac{F}{\kappa^{2}UV}\delta(U)fdU^{2}$
(76)

where $\widetilde{ds}$ denotes the line element (53) with
$V\rightarrow\widetilde{V}$ Sfetsos:1994xa . The above metric solves the
Einstein’s equation in presence of a stress-tensor sourced by a shockwave

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}-\Lambda
g_{\mu\nu}=R_{\mu\nu}+\frac{3}{l^{2}}g_{\mu\nu}=-8\pi G_{N}T_{\mu\nu}$ (77)
$\alpha\mathcal{D}_{\theta}\,f=-8\pi
G_{N}T_{UU},\,\,T_{UU}=p^{V}\left(\frac{F^{2}}{\kappa^{2}UV}\right)_{U_{0}}\delta(\theta-\pi/2)$
(78)

Here $\mathcal{D}_{\theta}$ is a first order differential operator of rank 2.
We normalize $f$ by choosing $f(\pi/2)=1$, thus $\alpha$ captures the strength
of the back reaction while $f(\theta)$ captures the profile of the
perturbation around the sphere. Here $p^{V}$ denotes the strength of such a
shockwave. We will determine the value of $\alpha$ by demanding smoothness
along the transverse volume element as the perturbation tends to the past
horizon (past from the right exterior $pov$).

The problem we want to analyse is the one studied first in Shenker:2013pqa
wherein one sends an infinitesimally small ($\mathcal{O}(G_{N})$) perturbation
into the black hole from the left exterior at a time $t_{0}$ which is in the
past of $t=0$ slice of the right exterior. This perturbation slowly grows in
strength as it falls into the black hole and at late time develops enough
energy (as measured locally) that the back reaction of the metric is of
$\mathcal{O}(1)$. It is useful to work with $\\{\tau,z\\}$ coordinates, as
$\tau=t-\mathcal{L}\phi$ for very large times $t_{0}\gg\beta$ implies
$\tau_{0}\gg\beta$. The Kerr geometry is also periodic in terms of the co-
moving coordinate $z=\phi-\mu\,t$. We therefore parametrize the axi-symmetric
shockwave by $U=U_{0}$

$U_{0}=e^{-\kappa\tau_{0}}$ (79)

where $\tau_{0}\gg\beta$ is the time on the left boundary. As
$\tau_{0}\rightarrow\infty$ $U_{0}\rightarrow 0$ therefore the above Dray-’t
Hooft solution is recovered in this limit. It is important to note that the
Dray-’tHooft solution can only describe the late time behaviour of the back-
reacted metric. Imposing smoothness of the volume element $H=\sqrt{{\rm
det}[g_{\\{\theta,z\\}}]}$ transverse to the shockwave at $U=U_{0}$ implies

$H\big{|}_{U_{0}^{+}}=H\big{|}_{U_{0}^{-}}$ (80)

Where we use $\widetilde{V}$ coordinate for $U>U_{0}$ and $V$ for $U<U_{0}$.
At late times $i.e.$ as $\tau_{0}\rightarrow\infty$ we expect $\widetilde{V}$
to be shifted as compared to $V$ by a step function. We determine the
magnitude of this shift by demanding the above smoothness condition holds as
we approach the Dray-’t Hooft solution.

$H=H_{0}+H_{1}(r-{r_{\\!{}_{+}}})=H_{0}-H_{1}\left(\mathbb{A}\frac{U_{0}V}{F^{\prime}({r_{\\!{}_{+}}})}\right)$
(81)

Denoting parameters after the shockwave with a tilde we have

$\displaystyle
H_{0}-H_{1}\left(\mathbb{A}\frac{U_{0}V}{F^{\prime}({r_{\\!{}_{+}}})}\right)=\widetilde{H}_{0}-\widetilde{H}_{1}\left(\widetilde{\mathbb{A}}\frac{U_{0}\widetilde{V}}{\widetilde{F}^{\prime}({r_{\\!{}_{+}}})}\right)$
(82) $\displaystyle\implies$
$\displaystyle\widetilde{V}=\frac{\widetilde{F}^{\prime}_{+}\mathbb{A}H_{1}}{F^{\prime}_{+}\widetilde{\mathbb{A}}\widetilde{H}_{1}}V+\frac{(\widetilde{H}_{0}-H_{0})\widetilde{F}^{\prime}_{+}}{\mathbb{A}\widetilde{H}_{1}U_{0}},\hskip
8.5359pt{\rm where}\,\,F^{\prime}_{+}\equiv F^{\prime}({r_{\\!{}_{+}}})$ (84)

Note that $H_{0}$ is simply the horizon area divided $4\pi^{2}$. Therefore we
expect $\delta H_{0}$ to vary according to the first law as

$\delta H_{0}\sim\beta(\delta\mathcal{M}-\mu\,\delta J)=\delta\mathcal{S}$
(85)

where $\mathcal{M}$ and $J$ are the ADM mass and angular momentum of the black
hole computed appropriately Papadimitriou:2005ii .

$\mathcal{M}=\frac{M}{\Xi^{2}},\hskip 8.5359ptJ=\frac{Ma}{\Xi^{2}}$ (86)

We work in the limit where the black hole has large entropy, thus evaluating
$\frac{F^{\prime}_{+}}{H_{1}}$ in the limit $\mathcal{S}\rightarrow\infty$ we
find

$\frac{F^{\prime}_{+}}{H_{1}}=\frac{\mathbb{B}}{\mathcal{S}}$ (87)

where $\mathbb{B}$ depends only on ${r_{\\!{}_{-}}}/{r_{\\!{}_{+}}}$ and
$\theta$. We further have the limit $U_{0}\rightarrow 0$ which is simply the
limit that the perturbation was sent in the far past as
$\tau_{0}\rightarrow\infty$ on the boundary of the left exterior implies far
past $w.r.t.$ the CFT on the right boundary. We consider the following scaling
in this limit with $\mathcal{S}\rightarrow\infty$

$U_{0}\rightarrow 0,\hskip 8.5359pt\frac{(\delta\mathcal{M}-\mu\,\delta
J)}{T_{H}\mathcal{S}}\rightarrow 0,\hskip
8.5359pt\frac{(\delta\mathcal{M}-\mu\,\delta
J)}{U_{0}\,T_{H}\,\mathcal{S}}\rightarrow{finite}.$ (88)

In this limit the above shift in $\widetilde{V}$ (92) has a finite
contribution. The perturbation we consider further has a specified angular
momentum per unit energy $\mathcal{L}$, this implies that if the change in
black hole’s ADM mass is $\delta\mathcal{M}\sim E_{0}$ \- $E_{0}$ being the
energy of the shockwave measures at the $AdS$ boundary; then $\delta
J=\delta\mathcal{M}\mu\mathcal{L}\sim E_{0}\mu\mathcal{L}$. Therefore we have

$\delta\mathcal{M}-\mu\delta J=\delta\mathcal{M}(1-\mu\,\mathcal{L})$ (89)

Absorbing the non-extensive parameters $(\mathbb{B},\mathbb{A},2\pi)$ in the
variation of ADM mass and angular momentum we can write

$\widetilde{V}=V-\frac{\Theta(U-U_{0})}{U_{0}}\frac{\delta\mathcal{M}(1-\mu\,\mathcal{L})}{T_{H}\mathcal{S}}$
(90)

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

Working with the above limits imply our analysis holds for large black holes
with large entropy as compared to the perturbation. It is worth mentioning
that the shift $\delta\mathcal{M}$ has complicated $\theta$ dependence and is
further supposed to solve a differential equation implied by (78). This should
not be surprising as 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 dependence444The 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 (71) we see that

$\alpha
f(\theta)=\frac{\beta\,\delta\mathcal{M}(1-\mu\,\mathcal{L})}{U_{0}\,\mathcal{S}}=\frac{\beta(1-\mu\,\mathcal{L})E_{0}}{U_{0}\,\mathcal{S}}f(\theta)$
(91)

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 measured at the boundary at left
exterior time $\tau_{0}$. Therefore we have

$\alpha=\frac{1}{U_{0}}\frac{\beta(1-\mu\,\mathcal{L})E_{0}}{\mathcal{S}},\hskip
8.5359pt{\rm and}\hskip
8.5359ptV\rightarrow\widetilde{V}=V-\frac{1}{U_{0}}\frac{\beta(1-\mu\,\mathcal{L})E_{0}}{\mathcal{S}}f(\theta)$
(92)

The inverse $U_{0}^{-1}=e^{\kappa\tau_{0}}$ dependence is indicative of the
perturbation growing at an exponential rate dictated by $\kappa\geq T_{H}$ for
an arbitrary value of angular momentum $\mathcal{L}\geq 0$. We can in
principle solve for $f(\theta)$ to find the how the backreaction spreads away
from the equator; a similar analysis can also be done for a point null
particle perturbation at the equator. This must indeed reveal interesting
dynamics and butterfly velocity associated with such perturbations to the
rotating geometry; see Blake:2021hjj  for a analysis of the butterfly velocity
in a slowly rotating Kerr $AdS_{4}$. However for the case at hand we need not
be concerning ourselves with dynamics away from the equator and choose
$f(\theta=\pi/2)=1$.

In the above analysis we have assumed that the shockwave emanates at the
boundary of $AdS_{4}$ at a near horizon time $\tau_{0}$, the $\\{\tau,z\\}$
coordinates are related to the Boyer-Lindquist coordinates $\\{t,\phi\\}$ by
(34). The choice of the $\\{\tau,z\\}$ coordinates is forced upon us from
arguments presented subsection 3.1. It is important to note that demanding the
smoothness of the Dary-’tHooft solution at the outer horizon we are forced to
work with the periodic coordinate $z$. Had we been analysing the backreaction
due to a single in-falling particle localized in $z$ the resultant response
function $f(\theta,z)$ would have to periodic in $z$ and not in $\phi$. One
way to understand this is that $z$ is the co-moving coordinate outside the
horizon and in the ergo-region where no stationary observer can exist who is
time like. Since the response function has to periodic in $z$ it cannot grow
exponentially in $z$ (with a real coefficient). Therefore $\alpha f(\theta,z)$
occurring instead in (91) would still have an exponential behaviour only in
$\tau_{0}$.

For the case at hand of a shockwave existing at every point in the $z$
coordinate and starting out from the $AdS_{4}$ boundary at $\tau_{0}$ we would
have a spread in the time
$t^{(s)}=\left(1-\frac{a\mathcal{L}}{\ell^{2}}\right)t$ of the static boundary
coordinate given by (34)

$\tau_{0}=t_{0}^{(s)}-\mathcal{L}\,\phi_{0},\hskip
8.5359pt\forall\,\,\phi_{0}\in[0,2\pi]$ (93)

and the spread in $[0,2\pi]$ in $\phi$ ensures an equivalent spread in $z$ by
the time the shockwave reaches the outer horizon555This is because we demanded
$\int_{0}^{2\pi}\sqrt{-g}d\phi=\int_{0}^{2\pi}\sqrt{-g}dz$ in the near horizon
region in defining $z$. As the backreaction is maximum closer to
${r_{\\!{}_{+}}}$ it is this periodicity which is the most relevant for the
Dray-’tHooft solution. . Thus fixing a fixed time $\tau_{0}$ at the boundary
for the start of perturbation implies sending the first pulse at $\phi=0$ at
$t_{0}^{(s)}=\tau_{0}$. We will henceforth mention only the behaviour with
regards to $\tau_{0}$ as this signals the start of the perturbation and linear
shifts of the form indicated above would not affect the qualitative statements
made towards the end.

##  4 Extremal Surfaces

We next compute the disruption of Mutual Information between the TFD states
due to such a shockwave along the equator. We consider a hemispherical
subsystem with the equator at $\theta=\pi/2$ as it’s boundary, and consider
identical subsystems $A$ and $B$ at both the $left$ and $right$ Kruskal
boundaries. The Mutual Information is then given by

$I[A:B]=S_{A}+S_{B}-S_{A\cup B}$ (94)

with the appropriate HRT surface used to compute the $rhs$ above Hubeny:2007xt
. The shockwave geometry is locally unchanged from the Kerr metric in each of
the exteriors in the Kruskal diagram except for the shift in the ingoing $V$
coordinate (92) along the past horizon denoting the change due to the
perturbation at very late times. Observe that only the HRT surface
corresponding to $S_{A\cup B}$ traversing the geometry from $right$ to $left$
is sensitive to this shift and hence detects the effect of this shockwave. It
can be explicitly shown that the HRT surface dual to $S_{A\cup B}$ lies in the
equatorial plane as it is a co-dimension two extremal surface in both the
exteriors.

The extremal surface inherits the axi-symmetry of the Kerr geometry along the
$z$ coordinate. We extremize the surface $w.r.t.$ the co-ordinates
$\\{r,\tau,z\\}$ and the metric given in (29). As the extremal surface is
symmetric in the $z$ coordinate we only need to extremize it along the
$\\{r,\tau\\}$ direction. We would only be interested in the late time
behaviour of the dependence of this extremal surface on the shift in the $V$
coordinate. The procedure for obtaining this change in the extremal surface
would be identical to the one followed in Leichenauer:2014nxa .

The subsystem we consider is the union of upper hemispheres at the 2 boundary
$CFT_{3}$s and the corresponding extremal surface homologous to it is the
$\theta=\frac{\pi}{2}$ surface for which it can be shown that the trace of
extrinsic curvature vanishes666Here we consider the trace of the extrinsic
curvature constructed out of the unit normal in the $\theta$ direction. The
HRT surface further needs to be extremised in the time direction $\tau$.. The
induced metric on $\theta=\frac{\pi}{2}$ surface given by (29) can be
expressed in $\\{r,\tau\\}$ coordinates as follows

$ds^{2}_{\theta=\tfrac{\pi}{2}}=F(-d\tau^{2}+\frac{dr^{2}}{f^{2}})+h\,(dz+h_{\tau}d\tau)^{2}$
(95)

This implies that the area of the extremal surface ${\cal A}_{\theta=\pi/2}$
is given by extremizing the following functional

$\displaystyle{\cal A}_{\theta=\tfrac{\pi}{2}}=S_{z}\int
d\tau\sqrt{h}\sqrt{-F+Ff^{-2}\dot{r}^{2}}$ (96)

where $S_{z}=2\pi$ corresponds to the transverse volume obtained by
integrating along $z$ direction. Note that the area functional is independent
of $\tau$ coordinate which leads to a conserved quantity given as

$\displaystyle\gamma=\frac{-F\sqrt{h}}{\sqrt{-F+Ff^{-2}\dot{r}^{2}}}=\sqrt{-F_{0}h_{0}}.$
(97)

Figure 1: Figure depicting the extremal surface (Red) at $t=0$ in extended
Kerr spacetime which is analysed by dividing into three segments denoted as I,
II and III. The shift $V$ coordinate at $U=0$ is $\alpha$. Note, the lengths
II and III are the same and so are the lengths along the red curve on the
either sides of $U=0\,\,\&\,\,V=\alpha/2$.

The above expression can also be found by solving for $\dot{r}$ from its
$eom$. $\gamma$ is thus a constant that describes the extremal surface. Here
$H_{0}$ and $F_{0}$ are simply the values of function $H,F$ at a point where
$\dot{r}=0$. Following Leichenauer:2014nxa  we would try to learn about the
nature of the extremal surface from the above expression from it’s dependence
on $\dot{r}$.

Observe that $F_{0}$ is considered to be negative as $r_{0}$ lies behind the
horizon. When $r_{0}$ approaches horizon $i.e.$ $r_{0}\to r_{+}$, $\gamma\to
0$ (as $F\sim\mathcal{O}(r-{r_{\\!{}_{+}}})$) which is also the limit
$\alpha\to 0$ $i.e.$ when the shock wave is absent. Note that the (97) can be
inverted to find $\tau$ as a function of $r$ as follows

$\displaystyle\tau(r)=\int dr\frac{1}{f\sqrt{1+\gamma^{-2}Fh}}$ (98)

The tortoise coordinate $r_{*}$ relevant for our analysis can also be obtained
as a function of $r$ through the following integral

$\quad r_{*}(r)=-\int_{\infty}^{r}\frac{dr^{\prime}}{f(r^{\prime})},$ (99)

however the exact form of this integral would not be needed. It suffices to
state that the $AdS$ boundary occurs at $r_{*}=0$ and the outer horizon at
$r_{*}=-\infty$ for non-extremal black hole parameters.

Following Leichenauer:2014nxa  we compute the area of the extremal surface by
dividing its left half three parts as depicted in Figure-1 and finally we
obtain the full area by symmetry which is twice that of the left half. Note,
when $\alpha=0$ the above figure would not have a shift along the $V$
coordinate $i.e.$ the $V=0$ surface would be a continuous straight line.
Further the extremal surface in red would be a horizontal straight line
beginning at $t_{R}=0$ at the right boundary, passing through the bifurcate
horizon $(U=0=V)$ and ending at $t_{L}=0$ on the left boundary. Let us first
consider the segment I- starting from the boundary $(U,V)=(1,-1)$ to
$(U,V)=(U_{1},0)$ the point where extremal surface intersects $V=0$. For this
surface we have

$\displaystyle U_{1}^{2}$
$\displaystyle=\exp{\bigg{[}2\kappa(dr_{*}-d\tau)\bigg{]}}$
$\displaystyle=\exp\bigg{[}-2\kappa\int_{r_{+}}^{\infty}\frac{dr}{f}\bigg{(}1-\frac{1}{\sqrt{1+\gamma^{-2}Fh}}\bigg{)}\bigg{]}$
(100)

The second segment II begins at $(U,V)=(U_{1},0)$ and ends at $(U_{2},V_{2})$
which lies on the surface $r=r_{0}$ which is the turning point for the
extremal surface. Here we have

$\displaystyle\frac{U_{2}^{2}}{U_{1}^{2}}=\exp\bigg{[}-2\kappa\int_{r_{0}}^{r_{+}}\frac{dr}{f}\bigg{(}1-\frac{1}{\sqrt{1+\gamma^{-2}Fh}}\bigg{)}\bigg{]}$
(101)

This in turn leads to

$\displaystyle
U_{2}^{2}=\exp\bigg{[}-2\kappa\int_{r_{0}}^{\infty}\frac{dr}{f}\bigg{(}1-\frac{1}{\sqrt{1+\gamma^{-2}Fh}}\bigg{)}\bigg{]}$
(102)

In order to find $V_{2}$ as in Leichenauer:2014nxa  we consider a reference
surface described by $r=\bar{r}$ at which $r_{*}=0$ and hence we get

$\displaystyle
V_{2}=\frac{1}{U_{2}}\exp\bigg{[}-2\kappa\int^{r_{0}}_{\bar{r}}\frac{dr}{-f}\bigg{]}$
(103)

Similarly considering the segment III we find $V_{2}$ in terms of the
intersect at $U=0$ and $V=\alpha/2$ as

$\displaystyle\frac{\alpha^{2}}{4V_{2}^{2}}=\exp\bigg{[}2\kappa\int_{r_{0}}^{r_{+}}\frac{dr}{f}\bigg{(}1+\frac{1}{\sqrt{1+\gamma^{-2}Fh}}\bigg{)}\bigg{]}$
(104)

Hence $\alpha$ is described by a combination of three integrals as follows

$\displaystyle\alpha=2\exp\bigg{(}Q_{1}+Q_{2}+Q_{3}\bigg{)}$ (105)

where the integrals $Q_{1},Q_{2}$ and $Q_{3}$ take the form

$\displaystyle Q_{1}$
$\displaystyle=-2\kappa\int^{r_{0}}_{\bar{r}}\frac{dr}{-f}$ (106)
$\displaystyle Q_{2}$
$\displaystyle=2\kappa\int_{r_{0}}^{\infty}\frac{dr}{f}\bigg{(}1-\frac{1}{\sqrt{1+\gamma^{-2}Fh}}\bigg{)}$
(107) $\displaystyle Q_{3}$
$\displaystyle=2\kappa\int_{r_{0}}^{r_{+}}\frac{dr}{f}\bigg{(}1+\frac{1}{\sqrt{1+\gamma^{-2}Fh}}\bigg{)}$
(108)

Observe that integrals $Q_{1}$ and $Q_{2}$ diverge as $r_{0}\to r_{+}$ which
is also the limit in which $\alpha\to 0$. Notice that $Q_{3}$ diverges as
$r_{0}\to r_{c}$ where $r_{c}$ corresponds to the point at which the following
expression holds

$\displaystyle h(r_{c})F^{\prime}(r_{c})+h^{\prime}(r_{c})F(r_{c})=0$ (109)

The above equation is cubic in $r$ and $r_{c}$ can be obtained analytically.
However, notice that $Q_{1}$ and $Q_{2}$ are finite as $r_{0}\to r_{c}$. Hence
as $r_{0}\to r_{c}$, $Q_{3}$ alone diverges and corresponds to
$\alpha\to\infty$. Let us now try to compute the divergent part of the area in
(96) which can be re-expressed using (97) as follows

$\displaystyle{\cal
A}_{\theta=\tfrac{\pi}{2}}=2\pi\int\frac{dr}{f}\frac{Fh/\gamma}{\sqrt{1+\gamma^{-2}Fh}}$
(110)

Note that at late times when $\alpha$ becomes large the dominant contribution
to the above extremal surface area comes from its segment near $r=r_{c}$ and
in this regime the area above can be approximated to be proportional to the
$Q_{3}$ integral given in (108). Note that the total area contributing to the
extremal surface corresponding to $S_{A\cup B}$ is given by four times the
segment evaluated above which gives the dominant contribution for large
$\alpha$. This is expressed as

$\displaystyle{\cal A}_{A\cup B}$
$\displaystyle\approx\frac{4\pi}{\kappa}\sqrt{-F_{c}h_{c}}\,Q_{3}=\frac{4\pi}{\kappa}\sqrt{-F_{c}h_{c}}\,\log\alpha$
(111)

Upon substituting the expression for $\alpha$ we derived in (92), the above
result for the extremal area reduces as follows

$\displaystyle{\cal A}_{A\cup B}$ $\displaystyle\approx
4\pi\tau_{0}\sqrt{-F_{c}h_{c}}\,+\frac{4\pi}{\kappa}\sqrt{-F_{c}h_{c}}\,\log\bigg{[}\frac{\beta
E_{0}\,(1-\mu\mathcal{L})}{\mathcal{S}}\bigg{]}$ (112)

The above equation clearly indicates that the rate of growth of area of the
extremal surface corresponding to $S_{A\cup B}$ $w.r.t.$ the time-stamp
$\tau_{0}$ of the perturbation is controlled by the parameter
$\sqrt{-F_{c}h_{c}}$. Also note that the above expression is only valid for
large $\alpha$ and hence the unperturbed value of the area
$\mathcal{A}_{\theta=\frac{\pi}{2}}$ cannot be inferred from this expression.

Let us pause here to compare the above result for generic values of
$\mathcal{L}$. Note that for $\mathcal{L}<\mu^{-1}$ we have
$F\sim\mathcal{O}(r-{r_{\\!{}_{+}}})$. However only for
$0\leq\mathcal{L}<\mu^{-1}$ we have $F^{\prime}({r_{\\!{}_{+}}})\rightarrow 0$
as ${r_{\\!{}_{-}}}\rightarrow{r_{\\!{}_{+}}}$ $i.e.$ as the configuration
approaches extremality. This implies that for such shockwaves
$r_{c}\rightarrow{r_{\\!{}_{+}}}$ as the condition (109) is satisfied as we
take the configuration arbitrarily close to extremality. Let us compare this
with the case where
$\mathcal{L}=\frac{{r_{\\!{}_{-}}}}{{r_{\\!{}_{+}}}}\mu^{-1}$ where
$\mathcal{L}\rightarrow\mu^{-1}$ as we approach an extremal configuration. For
such a shockwave the condition (109) is not similarly satisfied at
$r=r_{c}={r_{\\!{}_{+}}}$ and consequently $F_{c}$ does not approach zero as
${r_{\\!{}_{-}}}\rightarrow{r_{\\!{}_{+}}}$. In order to compare
quantitatively, it is useful to divide the above expression for
$\mathcal{A}_{A\cup B}$ by a quantity which behaves similarly in terms of
black hole parameters for $0\leq\mathcal{L}<\mu_{-1}$, we choose this to be
the unperturbed value for a similar area $\mathcal{A}^{(0)}_{A\cup B}$.
However since the exact analytic expression for the unperturbed
$\mathcal{A}^{(0)}_{A\cup B}$ is not easily tractable, we utilize the relation
that for large enough subsystems, entanglement entropy scales like the entropy
of the system. Hence, we take it to be proportional to the black hole’s
entropy

$\mathcal{A}^{(0)}_{A\cup B}=\rho\,\mathcal{A}_{H},\hskip 11.38092pt{\rm
where}\,\,\rho\in[0,1].$ (113)

Therefore we can write the term proportional to $\tau_{0}$ in (112) as

$\mathcal{A}_{A\cup B}=\mathcal{A}^{(0)}_{A\cup
B}\,\lambda_{L}\,\tau_{0}\implies\lambda_{L}=\frac{4\pi\sqrt{-h_{c}F_{c}}}{\rho\,\mathcal{A}_{H}}$
(114)

Figure 2: Plots for various values of
$\mathcal{L}=s\left(\frac{{r_{\\!{}_{-}}}}{{r_{\\!{}_{+}}}}\right)\mu^{-1}$
with ${r_{\\!{}_{+}}}=0.57$ and $\ell=1$. The blue, orange and green curves
correspond to $s=\\{0,\frac{2}{3},\frac{9}{10}\\}$ respectively. These curves
approach zero as ${r_{\\!{}_{-}}}\rightarrow{r_{\\!{}_{+}}}$. The red curve
corresponds to $s=1$ and approaches a constant since
$\mathcal{L}\rightarrow\mu^{-1}$ as
${r_{\\!{}_{-}}}\rightarrow{r_{\\!{}_{+}}}$.

where $\lambda_{L}$ can be regarded as the instantaneous Lyapunov exponent at
late times $i.e.$ $\gg\beta^{-1}$. For $\rho=1$ we get the minimum possible
instantaneous $\lambda_{L}$

$\lambda^{\rm(min)}_{L}=\frac{4\pi\sqrt{-h_{c}F_{c}}}{\,\mathcal{A}_{H}}.$
(115)

We plot this for different values of $\mathcal{L}$ in Fig.(2). Here we do not
plot the values exactly at the extremal point. It is clear from the above plot
that if one were to choose an angular momenta $\mathcal{L}$ for the shockwave
such that it approaches the value $\mathcal{L}\rightarrow\mu^{-1}$ as the
configuration approaches extremality, then the rate of scrambling does not
approach zero. Note, away from extremality one can not choose
$\mathcal{L}=\mu^{-1}$ as this value is not allowed by turning point analysis
$i.e.$ the shockwave does not reach the horizon. We can also compare the value
of $\lambda^{(\rm min)}_{L}$ to the temperature $\kappa_{0}=2\pi T_{H}$ and
$\kappa$ which dictates the blueshift suffered by the in-falling null rotating
particle generating the shockwave. We plot these in Fig (3) against the ratio
${r_{\\!{}_{-}}}/{r_{\\!{}_{+}}}$. It is apparent that the instantaneous
$\lambda_{L}^{(\rm min)}$ seems to be bounded effectively by $\kappa$. One can
observe that temperature $\kappa_{0}=2\pi T_{H}$ indicated by the red curve
can be less than $\lambda^{\rm(min)}_{L}$ when the geometry is sufficiently
non-extremal. Thus clearly suggesting that the instantaneous rate of
scrambling can be greater than the temperature of the Kerr geometry but
bounded by the exponent $\kappa$ determining the blueshift for a rotating null
in-falling shockwave.

Having obtained the rate of growth of the HRT surface let us now try to
estimate the scrambling time through mutual information given by

$\displaystyle I(A:B)=S_{A}+S_{B}-S_{A\cup B}.$ (116)

Figure 3: Plots for various values of
$\mathcal{L}=s\left(\frac{{r_{\\!{}_{-}}}}{{r_{\\!{}_{+}}}}\right)\mu^{-1}$
with ${r_{\\!{}_{+}}}=0.57$ and $\ell=1$. The blue, orange and green curves
are for$\lambda^{(\rm min)}_{L}$ corresponding to $s=\\{0,\frac{9}{10},1\\}$
respectively while the red, violet and brown curves correspond to the value of
$\kappa$ for similar respective values of $s$. The red curve $\kappa_{0}$
$i.e.$ the temperature of the geometry is greater than only the $\lambda^{(\rm
min)}_{L}$ for $\mathcal{L}=0\,(s=0)$; the values of $\lambda^{\rm(min)}_{L}$
for $\mathcal{L}>0$ can be greater than this curve.

Note that for large enough $\tau_{0}$ or $\alpha$, the extremal surfaces
corresponding to the individual subsystems $A$ and $B$ remain unaffected by
the change in the geometry.

$\displaystyle I(A:B)=S_{A}+S_{B}-\frac{{\cal A}_{A\cup B}}{4G_{N}}$ (117)

where $S_{A},S_{B}$ are the unperturbed entanglement entropies of the
subsystems $A$ and $B$. Hence, using (112) we have

$\displaystyle I(A:B)\approx
S_{A}+S_{B}-\frac{\pi\tau_{0}}{G_{N}}\sqrt{-F_{c}h_{c}}\,-\frac{\pi}{\kappa
G_{N}}\sqrt{-F_{c}h_{c}}\,\log\bigg{[}\frac{\beta(1-\mu\,\mathcal{L})E_{0}}{\mathcal{S}}\bigg{]}.$
(118)

We will take the energy $E_{0}$ of the perturbation to be that of the order of
the energy of a few Hawking quanta, thus $\beta E_{0}\sim 1$. Further, as the
turning point analysis had revealed (45) for non-extremal configurations
$\mathcal{L}_{max}<\mu^{-1}$, therefore $0<(1-\mu\,\mathcal{L})<1$ and does
not scale extensively as compared to the the black hole’s entropy777This
quantity can be large but would only depend on how one chooses the approach of
$\mathcal{L}$ to $\mu^{-1}$ or a fraction of $\mu^{-1}$, as the geometry tends
to extremality. It would not therefore depend on the size of the black hole..
This in turn leads to the following expression for the scrambling time

$\displaystyle\kappa\tau_{*}\approx\log\mathcal{S}+\frac{\kappa
G_{N}(S_{A}+S_{B})}{\pi\sqrt{-F_{c}h_{c}}}+\log\frac{1}{1-\mu\,\mathcal{L}}$
(119)

Figure 4: Plot of $\kappa/\lambda^{(min)_{L}}$ against
${r_{\\!{}_{-}}}/{r_{\\!{}_{+}}}$ with $\ell=1$. This plot is independent of
the shockwaves specific angular momentum $\mathcal{L}$.

which can be re-expressed as

$\displaystyle\kappa\tau_{*}\approx\log\mathcal{S}\,+\,\frac{\kappa({\cal
A}_{A}+{\cal
A}_{B})}{4\pi\sqrt{-F_{c}h_{c}}}\,+\,\log\frac{1}{1-\mu\,\mathcal{L}}$ (120)

where ${\cal A}_{A}+{\cal A}_{B}$ are the unperturbed areas of the extremal
surfaces homologous to the subsystems $A$ and $B$ respectively computed at
late times. Notice that as all the terms in the above expression are
dimensionless, the second term is non-extensive (${\cal A}_{A}$ and
$\sqrt{-F_{c}h_{c}}/\kappa$ both have dimensions of area.) $i.e.$ it does not
scale with the entropy of the system

$\frac{\kappa(\mathcal{A}_{A}+\mathcal{A}_{B})}{4\pi\sqrt{-F_{c}h_{c}}}=\frac{\kappa}{\lambda_{L}^{(min)}}\frac{(\mathcal{A}_{A}+\mathcal{A}_{B})}{\mathcal{A}_{H}}\approx\kappa\frac{S_{A}+S_{B}}{S_{A\cup
B}}$ (121)

where we used (115) and $\kappa/\lambda^{(min)}_{L}$ although complicated is
found to be independent of $\mathcal{L}$ $c.f.$ Fig-4. It is also worth noting
that this term survives the limit ${r_{\\!{}_{-}}}\rightarrow 0$ in Fig-4,
therefore it exists even for the case of static black holes in $AdS_{4}$.

The third term in (119) does depend on $\mathcal{L}$ and is positive. This
term only depends on the chemical potential and the angular momentum (per unit
energy of the shockwave) $\mathcal{L}$ and has a tendency to increase the
scrambling time. However as this term -like the second, does not scale with an
increasing function of the black hole entropy $\mathcal{S}$ the effect of such
a decrease is parametrically small for large black holes. This term has
recently made an appearance in the charged shockwave analysis in RN-$AdS_{4}$
Horowitz:2022ptw  where the effect of delay in the scrambling time due to a
charged massless shell of in-falling matter was obtained to be

$t_{*}^{(q)}-t_{*}=\frac{\beta}{2\pi}\log\left(1-\mu\frac{q}{E_{0}}\right)^{-1}$
(122)

where the $lhs$ is the difference in the scrambling times of a charged
shockwave to an uncharged one. Here $\mu$ and $\beta^{-1}$ is the chemical
potential and temperature of the RN-$AdS_{4}$ respectively and $q/E_{0}$ -like
$\mathcal{L}$, is the charge per unit energy of the shockwave. As was shown in
section 2 (appendix A) the Lyapunov index for RN-$AdS_{4}$ is not expected to
change if the shockwave has an angular momentum and is given by the
temperature of the static geometry.

For large black holes where $\log\mathcal{S}$ is much larger than the rest of
the terms in (119), the scrambling time is given by

$\displaystyle\tau_{*}\approx\frac{1}{\kappa}\log\mathcal{S}$ (123)

Although the rate of growth of the extremal surface for the $A\cup B$
subsystem is controlled by the parameter $\sqrt{-F_{c}h_{c}}$ as described
earlier, we observe from the above equation that the scrambling time is
controlled by $\kappa$. Thus $\kappa$ can be regarded as the rate of growth of
the disruption of mutual information at very large time scales $i.e.$ when
$I(A:B)\rightarrow 0$. This time scale is larger than the late times at which
$\lambda^{(min)}_{L}$ was computed for.

We must also note that the above analysis is applicable for generic non-
extremal black holes and the behaviour of scrambling close to zero temperature
for large black holes is determined by their near extremal limit. Here the
wormhole connecting the two asymptotic boundaries becomes infinitely long, in
contrast with the strict extremal case wherein we have a disconnected
boundary.

##  5 Conclusions & Discussions

We study the butterfly effect for rotating geometries in $AdS_{4}$ by
computing systematically the rate of disruption of mutual information at very
late times due to in-falling rotating shockwaves with angular momenta per unit
energy $\mathcal{L}$. We find like the analysis in Shenker:2013pqa  that this
rate is controlled by the blueshift suffered by the in-falling shockwave which
is given by $\kappa$ (61) which for angular momentum $\mathcal{L}>0$ can be
greater than the temperature of the black hole $\kappa_{0}=2\pi T_{H}$. We
also find that the rate of growth of the wormhole $i.e.$ the HRT surface
connecting the two boundaries, as a response to the shockwave can grow at rate
$\lambda_{L}^{\rm(min)}$ which can survive the extremal limit for a particular
values of angular momentum $\mathcal{L}$ Fig(2). This is in contrast with
similar analysis in the static case or for the case with $\mathcal{L}=0$ where
this rate drops to zero and is always bounded by the temperature of the black
hole. We also find that $\lambda^{\rm(min)}_{L}$ can easily be greater than
the temperature of the black hole for $\mathcal{L}>0$ for sufficiently non-
extremal geometries. This rate is at best bound by the blueshift $\kappa$ even
when it survives the extremal limit. The scrambling time also behaves as
$t_{*}\sim\kappa^{-1}\log\mathcal{S}$ in such cases. We also find terms that
increase the scrambling time but these terms do not scale like the black hole
entropy for large black holes. We find one such term which increases the
scrambling time to be $\log(1-\mu\,\mathcal{L})^{-1}$ which expressly depends
on the angular momentum $\mathcal{L}$ of the shockwave. Such a delay has
recently been observed for the case of charged shockwaves in RN-$AdS_{4}$
Horowitz:2022ptw . Interestingly for near extremal black holes as $\kappa$
survives this limit for values of $\mathcal{L}$ which tend to
$\mathcal{L}\rightarrow\mu^{-1}$, the scrambling time is still proportional to
the extremal degrees of freedom of the Kerr geometry $i.e.$
$t_{*}\sim\kappa_{ext}^{-1}\log\mathcal{S}_{ext}$ (46). This particular
feature is very interesting given that string theory is able to rightly
predict the microscopic degeneracy of extremal Strominger:1996sh  and near
extremal Maldacena:1996ds  super-symmetric black holes.

There are some interesting features of the above result when contrasted with a
similar study of scrmabling of mutual information due to rotating shockwaves
in BTZ Malvimat:2021itk . In it the turning point analysis allowed for
$\mathcal{L}=1=\mu^{-1}_{ext}$, which is a peculiarity of the BTZ as for
higher dimensional black holes $\mathcal{L}<\mu^{-1}$. The scrambling time and
the Lyapunov index in BTZ can be computed analytically for such a case
Malvimat:2021itk  and was found to be controlled by
$\lambda_{L}=\kappa/2=\frac{\pi T_{H}}{1-\mu}={r_{\\!{}_{+}}}$. In contrast
here we find that the scrambling time is controlled by $\kappa$ (123) for
allowed values of $\mathcal{L}$. In bulk dimensions greater than 3 we can
choose the boundary subsystems in a way which respects the axi-symmetry of the
Kerr black hole, this may explain why the mutual information in the Kerr
geometry sees a different dependence on the exponent of the blueshift
$\kappa$.

The physics of scrambling of mutual information in a TFD state is very similar
to that of the decay of the OTOC in the same state Shenker:2013pqa . In fact
one can regard the mutual information to be a better estimate of the
entanglement between the 2 CFTs of the TFD state than the correlation
functions of local operators Wolf:2007tdq . The study of SYK-like models
suggests that late time scrambling is associated with a Schwarzian action at
very low temperatures, this picture has also since been holographically
realised in the near horizon dynamics of the JT model for near extremal black
holes Jensen:2016pah ; Maldacena:2016hyu . However the rate of scrambling in
the JT model is controlled by the small infinitesimal temperature of the near
extremal geometry. The above result suggests that for near extremal Kerr black
holes there has to exist a similar IR description in terms of an effective 2d
theory explaining scrambling at a rate greater than the temperature of the
black hole. Note, this is has to be true for any allowed value $\mathcal{L}$.
This is to be expected as the JT model studied till now in the context of
higher($d>2$) dimensional black holes describe the $J=const$ 888$Q=const$ for
RN Nayak:2018qej  and $J=const,\,Q=const$ for Kerr-Newman. sector
Moitra:2019bub ; Moitra:2018jqs ; Castro:2018ffi ; Castro:2019crn ;
Castro:2021fhc  . However, the rotating shockwaves with angular momentum per
unit energy $\mathcal{L}$ tend to explore a different sector as the geometry
is perturbed along $\delta J=\mathcal{L}\,\delta\mathcal{M}$. In other words,
if one were to study the IR effective gravity theory for near horizon dynamics
of near extremal Kerr or Kerr-Newman geometries for $\delta
J-\mathcal{L}\,\delta\mathcal{M}=0$, the above results suggest that the
effective theory must account for scrambling at a rate greater than the
temperature of the black hole and controlled by $\mathcal{L}$. This is also in
some sense suggestive from the first law of black hole mechanics which in such
a case takes the form

$\delta\mathcal{M}-\mu\,\delta
J=T_{H}\delta\mathcal{S}\implies\delta\mathcal{M}=\frac{T_{H}}{1-\mu\,\mathcal{L}}\delta\mathcal{S}.$
(124)

Further in case of RN black holes as the blueshift for rotating and non-
rotating shockwaves stay the same, the near horizon dynamics describing the
$Q\neq const$ sector should see the scrambling governed by the temperature of
the black hole.

The phenomena of pole-skipping also serves as a hallmark of maximally chaotic
many body quantum systems. Here the poles in the frequency space of the
retarded energy density 2pt functions are skipped at points which directly
correspond to $\lambda_{L}$ and $v_{B}$\- the butterfly velocity Blake:2018leo
. For the case of rotating BTZ this was first analysed in Liu:2020yaf  where
pole skipping points implied 2 possible Lyapunov indices corresponding to left
and right temperatures of the dual CFT2 as found in Poojary:2018eszz;
Jahnke:2019gxr . The case for Kerr-AdS4 was recently analysed Blake:2021hjj
and the Lyapunov index and the butterfly velocity were inferred by analysing
the ingoing solutions to metric perturbations along null directions at the
horizon. Here the Lyapunov index was found to be $\lambda_{L}=2\pi
T_{H}=\kappa_{0}$ $i.e.$ the temperature of the black hole. This is consistent
with our result as one expects to find the scrambling time governed by the
temperature of the black hole $i.e.$ $\kappa_{0}$, for non-rotating
shockwaves. The in-falling null coordinates used in Blake:2021hjj  are
precisely non-rotating and- according to the general arguments presented in
section 2, must see a blueshift given by $2\pi T_{H}$. Our results further
suggest that adapting the analysis of Blake:2021hjj  to rotating coordinates
provided in section 2 one must be able see pole-skipping at points
$\omega_{*}=i\kappa=2\pi iT_{H}/(1-\mu\,\mathcal{L})$999The work in
Blake:2021hjj  also tries to check the OTOC behaviour by analysing the growth
of perturbations at the horizon due to a non-rotating shockwave.. This amounts
to studying the energy density or the stress-tensor response for in-falling
perturbations with angular momentum per unit energy $\mathcal{L}$ at the
horizon. It would be worth checking this explicitly as one also extracts the
butterfly velocity from such an analysis.

## Acknowledgement

RP would like to thank Daniel Grumiller and Prashanth Kocherlakota for
discussions related to certain aspects of the project. RP is supported by the
Lise Meitner project FWF M-2882 N.

##  Appendix A RN-AdS4

Here we simply note the index of the blueshift suffered by a null rotating
shockwave in Reissner-Nordström black hole in $AdS_{4}$. The metric for
RN-$AdS_{4}$ in Boyer Lindquist coordinates takes the form

$\displaystyle ds^{2}_{RN}$ $\displaystyle=$
$\displaystyle\frac{dr^{2}}{f(r)^{2}}-dt^{2}f(r)^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})^{2}$
(125) $\displaystyle{\rm where}\,\,\,f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{4\pi Q^{2}}{r^{2}}+r^{2},\hskip
11.38092ptF_{rt}=\frac{Q}{r^{2}}$ (127)

RN-AdS4 black holes with magnetic charges too have been considered
Nayak:2018qej , however the charge and $U(1)$ dynamics would play no role in
deciding the blueshift at very late times. The solutions for null geodesic are
likewise obtained by solving the conditions (25) on the vector fields
$\xi\cdot\partial$. As the geometry is spherically symmetric the analysis is
easier and there is no Carter’s constant to be specified, thus implying we can
set $\xi^{\theta}$ component to be zero. The in-out going null pairs are
similarly constructed by reversing the signs of $\mathcal{E}$ & $\mathcal{L}$.
Taking $\mathcal{E}=1$ and writing the metric line element in terms of the
duals to such null pairs of vector fields we have

$\displaystyle ds^{2}_{RN}$ $\displaystyle=$ $\displaystyle
F(r)dudv+h(r)\sin^{2}\theta(d\phi+h_{1}(r)d\tau)^{2}+r^{2}d\theta^{2}$ (128)
$\displaystyle{\rm where}\hskip 5.69046pt\tau$ $\displaystyle=$ $\displaystyle
t-\mathcal{L}\,\phi,$ (130) $\displaystyle F(r)$ $\displaystyle=$
$\displaystyle\frac{r^{2}\left(r\left(-2M+r^{3}+r\right)+4\pi
Q^{2}\right)}{r^{4}-\mathcal{L}^{2}\csc^{2}\theta\left(r\left(-2M+r^{3}+r\right)+4\pi
Q^{2}\right)},$ (132) $\displaystyle h(r)$ $\displaystyle=$ $\displaystyle
r^{2}\sin^{2}\theta-\frac{\mathcal{L}^{2}\left(r\left(-2M+r^{3}+r\right)+4\pi
Q^{2}\right)}{r^{2}},$ (134) $\displaystyle h_{1}(r)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{L}\csc^{2}\theta\left(r\left(-2M+r^{3}+r\right)+4\pi
Q^{2}\right)}{r^{4}-\mathcal{L}^{2}\csc^{2}\theta\left(r\left(-2M+r^{3}+r\right)+4\pi
Q^{2}\right)}$ (136)

Note that here there is no redefinition of the $\phi$ coordinate required as
$h_{1}$ goes to zero at the horizon. $\kappa$ is then similarly defined as in
(60) by demanding smoothness (affines) of the parameter along the null
geodesics at the outer horizon

$\kappa=\left.\mathcal{K}\right|_{{r_{\\!{}_{+}}}}=\left|\tfrac{1}{2}\xi_{\pm}.\partial
F\right|_{{r_{\\!{}_{+}}}}=\frac{2\pi}{\beta}$ (137)

Thus we see that rotating null geodesics see the same blueshift at the horizon
as non-rotating ones $i.e.$ the exponent of the blueshift $\kappa$ is the
temperature of the RN-$AdS_{4}$ irrespective of the geodesics angular
momentum.

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