#### Action growth for AdS black holes

HJE
Action growth for AdS black holes
Rong-Gen Cai 0 1 2 3 4
Shan-Ming Ruan 0 1 2 3 4
Shao-Jiang Wang 0 1 2 3 4
Run-Qiu Yang 0 1 2 3 4
Rong-Hui Peng 0 1 2 3 4
0 No. 55 Zhong Guan Cun East Street, Beijing 100190 , China
1 Chinese Academy of Sciences
2 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics
3 No. 19A Yuquan Road, Beijing 100049 , China
4 School of Physical Sciences, University of Chinese Academy of Sciences
Recently a Complexity-Action (CA) duality conjecture has been proposed, which relates the quantum complexity of a holographic boundary state to the action of a Wheeler-DeWitt (WDW) patch in the anti-de Sitter (AdS) bulk. In this paper we further investigate the duality conjecture for stationary AdS black holes and derive some exact results for the growth rate of action within the Wheeler-DeWitt (WDW) patch at late time approximation, which is supposed to be dual to the growth rate of quantum complexity of holographic state. Based on the results from the general D-dimensional Reissner-Nordstrom (RN)-AdS black hole, rotating/charged Ban~ados-Teitelboim-Zanelli (BTZ) black hole, Kerr-AdS black hole and charged Gauss-Bonnet-AdS black hole, we present a universal formula for the action growth expressed in terms of some thermodynamical quantities associated with the outer and inner horizons of the AdS black holes. And we leave the conjecture unchanged that the stationary AdS black hole in Einstein gravity is the fastest computer in nature.
Black Holes; AdS-CFT Correspondence; Conformal Field Theory
1 Introduction 2
D-dimensional RN-AdS black hole
Setup
Action growth rate
Bound violation Bound restoration
3
4
5
2.1
2.2
2.3
2.4
3.1
3.2
5.1
5.2
5.3
Rotating/charged BTZ black hole
Rotating BTZ black hole
Charged BTZ black hole
Kerr-AdS black hole
Charged Gauss-Bonnet-AdS black hole
Gauss-Bonnet black hole and singularities inside horizon
Charged Gauss-Bonnet-AdS black hole
Neutral limit of charged GB-AdS black hole
6
Conclusions and discussions
A The Gauss-Bonnet-AdS black hole
A.1 Singularity located at r = 0
A.2 Singularity located at r
e
theory [1] motivates a lot of studies in eld theory [2] and gravitational physics [3{7].
Especially, Susskind and his collaborators' works [3{6] shed some lights on the connection
between quantum computational complexity and black hole physics. It is expected that
computational complexity will be helpful for our understanding of black hole physics,
holographic property of gravity, and especially Hawking radiation. And on the other hand, the
holographic principle of gravity will provide us with some useful tools to study problems
of complexity [8].
Maldacena and Susskind [9, 10] have related the Einstein-Podolsky-Rosen (EPR)
correlation in quantum mechanics to wormhole, or more precisely the Einstein Rosen (ER)
bridge in gravity, and proposed the so-called ER = EPR relation that the ER bridge
between two black holes can be considered as EPR correlation. This relation allows Alice at
{ 1 {
one side of ER bridge to communicate with Bob locating at the another side through the
ER bridge.
However, a natural question is how di cult it is for Alice to send signal through ER
bridge. It is worth noting that quantum computational complexity can be understood as
a measure of how di cult it is to implement some unitary operations during computation.
In quantum circuits [11], complexity can also be de ned as the minimal number of gates
used for processing the unitary operation [4]. As a result, Susskind proposed a new duality
to relate the distance from the layered stretched horizon to computational complexity
in [3], which at the rst time shows the connection between horizon and complexity. The
dual connection then is promoted to a conjecture that complexity of quantum state of
and proposed a new one called Complexity-Volume (CV) duality [6],
C(tL; tR)
V
GN L
where V is the spatial volume of the ER bridge that connects two boundaries at the times
tL and tR and L is chosen to be the AdS radius for large black hole and the Schwarzschild
radius for small black hole. The CV duality means the complexity of dual boundary state
j (tL; tR)i is proportional to the V =L rather than the length of the ER bridge. Although
the conjecture has been tested in spherical shock wave geometries [6], the appearance
of L seems unnatural. It is worth noting that there is an alternative de nition for the
holographic complexity [13{15] by the extremal bulk volume of a co-dimension one time
slice enclosed by the extremal surface area of co-dimension two time slice appearing in
the holographic entanglement entropy [16]. Refer to [17, 18] for possible applications of
this de nition.
In a recent letter ref. [8] and a detailed paper ref. [19], these authors further proposed
a Complexity-Action (CA) conjecture that the quantum complexity of a holographic state
is dual to the action of certain Wheeler-DeWitt (WDW) patch in the AdS bulk,
(1.1)
(1.2)
(1.3)
The authors of [8, 19] tested the CA conjecture by computing the growth rate of action
within the WDW patch at late time approximation, which should also obey the quantum
complexity bound (1.3) if the CA conjecture is correct.1 Along with other examples, such
1Therefore we will use \complexity bound" to infer both \growth rate of quantum complexity for
C = A :
~
dC
dt
2E
~
:
neutral BH :
rotating BH :
charged BH :
dA = 2M ;
dt
dA
dt
dA
dt
the ground states subscripted by \gs" are argued to make the combinations (M
and (M
Q)gs to be zero for the last two examples. As already noted in [8, 19] that the
intermediate and large charged black holes apparently violate the action growth bound they
proposed, they argued that only the small charged black holes still obey the action growth
bound due to BPS bound in supersymmetric theory, while in the case of intermediate
and large charged black holes, the RN-AdS black holes are not a proper description of
UV-complete holographic eld theory. As we explicitly show in this paper, even the small
charged black holes also violate the action growth bound. Based on our calculations made
in this paper, we will present a universal formula for the action growth of stationary AdS
black holes.
In this paper, we rst repeat the calculations of action growth rate for general
Ddimensional Reissner-Nordstrom (RN)-AdS black hole (section 2), rotating/charged BTZ
black hole (section 3). It is found that the original action growth bound is inappropriate,
which causes the apparent violation for any size of charged black hole. We then
investigate some other AdS black holes, such as Kerr-AdS black hole (section 4) and charged
Gauss-Bonnet-AdS black hole (section 5). The exact results of growth rate of action are
summarized as
as black hole with static shells and shock waves, the concrete forms of action growth bound
for anti de-Sitter (AdS) black holes (BH) are claimed to be
neutral BH :
rotating BH :
charged BH :
dA
dt
dt
dt
dt
dA = 2M ;
dA = [(M
dA = [(M
J )+
Q)+
(M
(M
J ) ] ;
Q) ] ;
where the subscripts
present evaluations on the outer and inner horizons of the AdS black
holes. We conjecture that the action growth bound for general AdS black holes should be
(M
J
Q)+
(M
J
Q) ;
the equality is saturated for stationary AdS black holes in Einstein gravity and charged
AdS black hole in Gauss-Bonnet gravity as we show in this paper, and for general
nonLet us rst consider the case for a general D-dimensional RN-AdS black hole with its action
A =
1
dDxp
g(R
2
GF 2) +
d
D 1 p
x
hK;
(2.1)
HJEP09(216)
where the cosmological constant
is related to the AdS radius L by
=
(D
1)(D
2)=2L2, h represents the determinant of induced metric on the boundary @M, K is the
trace of the second fundamental form. The trace of the energy-momentum tensor of
electromagnetic eld T = (4
D)F 2=16 is non-vanishing except for the case in four dimensions.
After applying the trace of the equations of motion, the total Einstein-Hilbert-Maxwell bulk action becomes where the eld strength of the Maxwell eld is
R
2
=
2(D
1)
L2
+ G
D
D
4 F 2;
2
AEHM =
1
where the inner and outer horizons are determined by f (r ) = 0 with
f (r) = 1
(D
8
2GM
2) D 2 rD 3 +
(D
8
2) D 2 r2(D 3) +
GQ2
r
2
L2
;
where M and Q are the mass and charge of the black hole, respectively.
2.2
Action growth rate
Following [19], we can calculate the growth rate of Einstein-Hilbert-Maxwell bulk action
within the WDW patch at late-time approximation as
dAEHM =
dt
D 2 Z r+
16 G r
drrD 2
=
8 DGL22 (r+D 1
r
2
{ 4 {
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
With extrinsic curvature associated with metric (2.5),
K =
1
r
D 2pf
=
D
r
2 pf +
f 0(r)
p ;
L2
r
D 1 !
:
The above result can be made more compact, if we rst solve M from f (r+) = 0 as
M =
2
1 2 3 D +
Q r+
(D
and then plug the above expression into f (r ) = 0 to get the expression for Q in terms of
r as
It is easy to see that the growth rate of action can be rewritten as
+
L2 rD 3
+
r
r
D 1 !
dA = Q
2
r
1
When D = 4, the above expression reduces to the one in [19]. The mass can also be
expressed in terms of r , if we plug (2.12) back to (2.11) to obtain
M =
(D
r2(D 2) !
r
;
which will be used below.
2.3
Bound violation
Although the authors of [19] have realized that in 4-dimensions the situation for
intermediate-sized (r+
L) and large charged black holes (r+
L) leads to an
apparent violation of the action growth bound (1.6), they mis-claimed that the small charged
{ 5 {
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
L) precisely saturate the action growth bound (1.6). We will explicitly
show below that the action growth bound (1.6) is always broken for any nonzero size of the
RN-AdS black holes in any dimensions D
4. Fixing the chemical potential
so that the ground state for (M
Q)gs is zero for 2 < 1, one can explicitly show that
the di erence between the growth rate of action (2.13) and the action growth bound (1.6),
= Q=r+D 3
dA
dt
2(M
Q) =
(D
2) D 2 r+
+
r
0;
(2.15)
which is always positive for any nonzero size of the RN-AdS black holes (r+
r
> 0)
HJEP09(216)
in any dimensions D
4, and becomes zero only for the asymptotic at limit L ! 1 or
chargeless limit Q ! 0, namely r
! 0. In this sense it looks then very strange for the
case of RN-AdS black holes to be an exception for the action growth bound made in [8, 19].
2.4
Bound restoration
We can eliminate the unappealing exception mentioned above by simply rewriting the
growth rate of action (2.13) of RN-AdS black hole as
dA
dt !
(D
where the chemical potentials on inner and outer horizons are de ned as
and
expression (
+ = Q=r+D 3, respectively. Although (2.16) can be easily inferred from (2.13) as
+)Q, we prefer the former formulation in order to keep the similar
manner as (1.6). In addition, we would like to stress here that at rst glance, the chemical
potential
at the inner horizon has no corresponding quantity at the boundary, but it
indeed has some relation to the quantities de ned in the boundary eld theory, because
is given by Q=rD 3, and the latter can be expressed by the mass and charge of the black
hole. But we prefer to keep the form (2.16) since it looks more simple.
In the limit of zero charge, Q ! 0, namely r ! 0, we have +Q ! 0 and
Q ! 2M ,
which leads to a very special case that (M
Q ! 0. It recovers the case of Schwarzschild-AdS black hole,
Q) !
(M
+Q) in the neutral limit
dA
This explains why the authors of [8, 19] could nd the saturated bound (1.4) along with
an overall factor of 2.
In the asymptotic at limit, the action growth rate for the RN black hole is
dA
dt !
(D
above bound can only be saturated for stationary black hole for gravity theory without
higher derivative terms of curvature. As a means of illustrating this conjecture,
dt
we suggest to test the above bound for the AdS-Vaidya spacetimes, which is under
investigation.
3
3.1
Rotating/charged BTZ black hole
Rotating BTZ black hole
Let us pause and have a few comments on the asymptotic
at limit. The
conformal boundary of an asymptotic AdS space-time is timelike and dual to a conformal eld
theory, but the conformal boundary of an asymptotic
at space-time is null and dual to
Galilean conformal eld theory [22]. Although the casual structure and Penrose's diagram
are totally di erent for the asymptotic AdS space-time and its asymptotic
at limit, the
contributions to the growth rate from the regions outside the horizon is vanished at late
time approximation, therefore the growth rate for the asymptotic at spacetime can be
obtained by a naive extrapolation limit from the case of AdS spacetime.
We will show in the subsequent sections that a more general result for the action
growth,
HJEP09(216)
(2.20)
(2.21)
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
under usual convention 8G
1. Similarly one can express both the mass and angular
momentum in terms of r as
and de ne the angular velocities on inner and outer horizons as
= J=2r2 and
+ = J=2r+2, then we arrive at
We next consider the case for the rotating/charged BTZ black hole. The action growth
rate of the WDW patch for rotating BTZ black hole in D = 2 + 1 dimensions has been
carried out in [8, 19] as
where the inner and outer horizons are determined by f (r ) = 0 with
dA =
dt
2
which can be expressed in terms of left- and right-moving sectors of dual 2D CFT,
Here the left- and right-moving sectors of dual 2D CFT are of the forms of
TR;L =
; SR;L =
cR;LTR;L =
(r+
r );
4G
2L
3
where the Brown-Henneaux central charges cL = cR = 23GL . Although the action growth
rate (3.5) contains quantity de ned on inner horizon without dual eld theory descriptions,
one can re-express it in terms of the left- and right-moving sectors of dual 2D CFT,
The situation for rotating BTZ black hole is very special because in this case one can
explicitly nd that (M
J ) =
(M
+J ), and this explains why the authors of [8, 19]
could
nd the saturated bound (1.5) along with an overall factor of 2. In the non-rotating
limit, it recovers the neutral result,
dt
dA = 2(M
+J ) ! 2M;
J ! 0:
The action growth rate (3.5) involves simple cancelations of various thermodynamical
quantities on inner and outer horizons, of which the rst law of thermodynamics [23, 24]
can be written as dM =
T dS
+
dJ . Here the temperatures and entropies de ned
on both horizons are of the forms of
which now makes sense from the view point of eld theory side. The same tricks are
expected to be applied to other kinds of black holes [25] with CFT descriptions.
3.2
Charged BTZ black hole
from [28]. The total action reads
Now we turn to the case of charged BTZ black hole [26, 27]. We follow the conventions
A =
1
d x
2 p
hK;
and the metric is given by
ds2 =
{ 8 {
where the inner and outer horizons are de ned by f (r ) = 0 with
under usual convention 8G
1. After applying the on-shell condition, f (r) =
where the eld strength should be
Then one can easily compute that
F 2 =
Q2
thus the total growth rate of action reads
the total Einstein-Hilbert-Maxwell bulk action becomes
HJEP09(216)
dA =
dt
2
r2 ln rL+ ;
dA
{ 9 {
Analogy with the case of RN-AdS black hole, the mass and charge can be expressed in
terms of r as
If we further de ne the chemical potential on the inner and outer horizon as
=
2 Q ln(r =L) and
+ =
2 Q ln(r+=L), respectively, we can rewrite (3.21) as
dt
which shares exactly the same form with (2.16). As usual, the neutral charge limit Q ! 0,
! 0, the mass M ! r+2=L2, and
+Q ! 0, while
Q ! 2r+2=L2. As a result,
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
The Kerr-AdS black hole shares similar Penrose diagrams as the RN-AdS black hole,
therefore the same region from inner horizon to outer horizon contributes to the growth rate of
action within the WDW patch at late time approximation.2 We use the conventions and
results in [29] for the thermodynamics of Kerr-AdS black holes. Here we only focus on the
case in four dimensions for simplicity and clarity and the results can be easily generalized
to the higher dimensional case. We start with the total action,
A =
1
(4.1)
HJEP09(216)
The four dimensional Kerr-(anti)-de Sitter metric is obtained by Carter in [30] and can
be written as
ds2 =
where
sin
2
:
dM = T dS +
dJ;
It is easy to obtain the determinant of Kerr-AdS metric as
The outer and inner horizons are determined by the equation
(r ) = 0, respectively. The
rst law of thermodynamics holds at both horizons,
where the physical mass M , angular momentum J , the angular velocity
and the area
A
of outer and inner horizons can be expressed as
M =
m
G 2
;
=
a(1 + r2 L 2
)
r2 + a2
J =
A
=
ma
G 2
;
4 (r2 + a2)
:
AEH =
1
By directly integrating the on-shell Einstein-Hilbert bulk action,
2Although we don't analyze the spacetime structure for the WDW patch because it will be very similar
to the case in the paper [19], it is actually very important to get the reasonable contribution to the growth
region of WDW patch by careful and complicated cancellation of corners and surface regions.
we have
It is worth noting that the induced metric on the null hypersurface r should be de ned
by the induced metric on a timelike hypersurface with constant r approaching r ,
dAEH =
dt
(r3 + a2r) r+
2GL2
r
:
p
h =
r
g
grr
K = r n = p
=
1
sin
p
gn );
Here we have used
(r ) = 0 to get the second line. Combining the bulk action and
boundary term, we have the growth rate of total action,
dAYGH =
dt
1
4G
=
4G
Z
0
r
d sin
=
r
dA =
dt
r3 + rL2 r+
2G L2
r
mr+2
(r+2 + a2)G
=
= (M
mr2
(r2 + a2)G
+J )
(M
J ):
From the de nition of extrinsic curvature, its trace K can be written as
where the normal vector n = 0; q
2 ; 0; 0 . Then we can obtain the YGH boundary term,
(4.12)
(r ) = 0 to get the second line and the thermodynamical quantities
to rewrite the nal result, which shares exactly the same form (3.5) as the rotating BTZ
black hole case.
Simple extension to the case of Kerr-Newman-AdS black holes [31] should be
straightforward, and the action growth rate in the form of
dt
dA = (M
+J
+Q)
(M
J
Q)
is expected. However the non-rotating limit of Kerr-AdS black hole might be a little tricky.
It seems that the naive limit a ! 0, namely r
! 0 of growth rate (4.15) of total action,
dA =
dt
r3 + rL2 r+
2G L2
=
2mL2
2GL2
r !0
M;
would not recover the result 2M of Schwarzschild-AdS black hole. The rst term in
parenthesis of (4.13) is zero for a 6= 0 due to
would give an extra M to the total growth rate,
(r ) = 0, however, in the case of a = 0, this term
1
4G
Z
0
d sin
r
r+
hence we recover the result 2M for the non-rotating limit.
In this section, we investigate the growth rate of the action in Gauss-Bonnet gravity in
ve dimensions. Gauss-Bonnet term naturally appears in the low energy e ective action
of heterotic string theory [32, 33] and can be derived from eleven-dimensional supergravity
limit of M theory [34, 35].
We will con rm a reduced contribution of complexi cation rate in the presence of stringy corrections and propose a method to deal with singularities behind the horizon, both of which are mentioned as open questions in section 8.2.4 and section 8.2.6 of ref. [19].
Gauss-Bonnet black hole and singularities inside horizon
The whole action of the Gauss-Bonnet gravity is
HJEP09(216)
A =
1
is the Gauss-Bonnet term. The appropriate
The exact vacuum solution follows from [38] as
ds2 =
where hij dxidxj represents the line element of (D
2)-dimensional hypersurface with con
stant curvature (D
2)(D
3)k and volume
D 2, and the metric function f (r) can be
expressed as
where
e =
AdS solution with
f (r) = k +
r
2
2
e
s
1
1 +
(D
4) and M represents the gravitational mass. Under the limit of
! 0, one can nd that the minus branch solution will become the standard
Schwarzschild
J
GabKab
b
;
1
3
where Gbab is the Einstein tensor related to the induced metric hab and J is the trace of
tensor Jab de ned as
Jab =
(2KKacKcb + KcdKcdKab
2KacKcdKdb
K2Kab):
Using the Gauss-Codazzi equations [37], we can get
where RG2B = R2
boundary term was derived in [36, 37] as
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
1
3
J
GabKab =
b
KKabKab
K3 +
KacKcdKda + K2habKab
KcpKpcKbdhbd
2Raqcphcahqbh d
p Kbd + RaqcphcahqphbdKbd:
3
4
1
f (r)
(D
16 GM
2) D 2rD 3 +
r
2
L2
:
rh =
L2 + pL4 + 4L2(m
2
Unlike the case of the Schwarzschild-AdS solution, there are not only the singularity located
is the solution of equation
at r = 0 but also a singularity located at re if the Gauss-Bonnet coupling
> L2=8,3 which
By solving the equation f (rh) = 0, one can
nd the event horizon of the black hole is
located at
Hence we only consider the case with k = 1 in ve dimensions and the minus branch with
expected asymptotical behavior. In order to simplify the calculation, we choose
Due to the presence of singularities r = 0 or r = r~ behind the event horizon r = rh,
the Penrose diagram is generally di erent from the case of Schwarzschild-AdS black hole.
As we show in appendix A, a reasonable result can not be obtained by directly integrating
the action (5.1) in the region with its boundary approaching any of these two singularities.
Therefore we will handle the case of singularities by hiding them behind the inner horizon
introduced by adding charge into the Gauss-Bonnet-AdS black hole, namely the charged
Gauss-Bonnet-AdS black hole, which will be calculated below. The case of the
GaussBonnet-AdS black hole should be deduced from the zero charge limit, where the above
singularities will always be behind the inner horizon.
5.2
Charged Gauss-Bonnet-AdS black hole
The charged Gauss-Bonnet-AdS black hole solution reads [39, 40],
f (r) = k
s
1
1 + 4
e
m
rD 1
1
L2
q2
r2D 4
!
;
r
2
2
e
At =
s
s
1
q
c rD 3
c =
r 2(D
D
3)
2
:
;
;
and the potential form is de ned by
The parameter m and q can be respectively related to the physical mass M and charge Q by
m =
3As noted in [38], in order to have a well-de ned vacuum solution with m = 0, the Gauss-Bonnet coupling
should satisfy
L2=8. In that case the singularity at r~
does no longer appear. However, for our aim
here, we relax this condition and consider the case with an additional singularity at r~ .
(5.8)
(5.9)
are presented with wiggly lines. The growth rate of WDW patch at late time
approximation comes from the spacetime region that lies outside the inner horizon and inside the
In the following calculations we only consider the case with k = 1 and D = 5. Then the
horizons of the solution are determined by the equation f (r ) = 0,4 namely,
r
4
L2 + r2 +
q
r
2
2 = m
Here we only consider the case of grand canonical ensemble and x the potential
= Q=rD 3. Therefore no boundary term is needed for the Maxwell eld.
The Penrose diagram of the charged Gauss-Bonnet-AdS black hole is presented in
gure 1. As in the case of RN-AdS black hole in [19], the contribution to the growth rate
of total action at late time approximation comes from the WDW patch that lies outside
the inner horizon and inside the outer horizon. The contribution from the extra matter
eld reads
AMaxwell =
action is
1
gF
F
=
d
D 1
S p
hA F
;
(5.15)
where we have used Maxwell equation and Stokes's theorem. The growth rate of matter
dAMaxwell =
dt
=
Q2 r+
2r2 r
The contribution from the Einstein-Hilbert-Gauss-Bonnet (EHGB) action is
dAEHGB =
dt
3
4We only consider the case which allows the equation to have two positive roots and share the similar
Penrose diagram to the one in RN-AdS spacetime. In [40] there are general discussions about the parameters,
solutions and corresponding spacetime structures for the Gauss-Bonnet gravity with electric charge.
r+
r
:
;
(5.16)
(5.17)
dt
3
Combining the above results (5.16)(5.17)(5.18), one can nd that the total action growth
rate of the charged Gauss-Bonnet-AdS black hole reads,
and the contribution from the boundary term is
Recall that the boundary is located at r satisfying f (r ) = 0, we have a remarkably
simple result,
2
q
r2
2q2 r+
r2
r
r+
r
r+
r
:
r+
r
:
dA =
dt
3
16 G
Neutral limit of charged GB-AdS black hole
Now we come back to the case of the Gauss-Bonnet-AdS black hole, which we argued should
be deduced from zero charge limit of the charged Gauss-Bonnet-AdS black hole to avoid
the encounter with singularities. One can consider the inner horizon in the limit of q ! 0
as the cut-o screen for the spacetime near the singularities. Maybe one can also use other
cut-o screen but need a reasonable method to take limit in order to avoid of approaching
e
the singularity located at r if exists. We choose the inner horizon just because it will be
easy to deal with from the charged case. From (5.14) one can express q2 in terms of r as
where we have used (5.14) to get the second line and (5.13) to get the fourth line. The nal
result shares exactly the same form as the general D-dimensional RN-AdS black hole (2.16)
as well as charged BTZ black hole (3.24). When
! 0, it naturally goes to the result of
the RN-AdS black holes. In the following subsection, we will discuss the limit when q ! 0.
q2 = r+2r2
1 + r+2 + r2
L2
:
Substituting (5.22) into (5.20), we arrive at
lim
dA = lim
3
;
where we have used (5.14) in the zero charge limit q ! 0 to get the third line. When
the above results goes to the case of a Schwarzschild-AdS black hole, as expected. Therefore
we claim that the growth rate of action for uncharged and non-rotating AdS black hole in
Gauss-Bonnet gravity is smaller than its Einstein gravity counterpart, namely,
dt
dA = 2M
3
4 G
3 < 2M;
where we only consider
> 0 inspired by string theory. This con rms the speculation that
stringy corrections should reduce the computation rate of black hole solutions mentioned as
an open question in section 8.2.4 of ref. [19]. It seems that the neutral bound (1.7) can only
be saturated for Einstein gravity and it would be interesting to investigate whether higher
order stringy corrections or some correction terms from other kinds of gravity theories like
Lovelock gravity theory [41] will arrive at the same conclusion.
6
Conclusions and discussions
We have investgated in this paper the original action growth rate (1.4)(1.5)(1.6) proposed in
the recent papers [8, 19], which passed for various examples of stationary AdS black holes.
In the example of general D-dimensional RN-AdS black hole, it is found that the original
action growth rate (1.6) is apparently violated even for the case of small charged black hole
in addition to the cases of intermediate and large charged black holes. It is also found that
the precise saturations of original action growth rate (1.4)(1.5) for Schwarzschild-AdS black
hole and rotating BTZ black hole along with an overall factor of 2 are purely coincidence
in the view point of the results presented in this paper (1.8)(1.9).
The action growth rate (1.8)(1.9) are further tested in the context of charged BTZ
black hole and Kerr-AdS black hole, which are shown explicitly sharing the exactly same
manner as in the case of RN-AdS black hole and rotating BTZ black hole. Both the action
growth rates (1.8)(1.9) can reduce to (1.7) for neutral static case, which is also true for the
original action growth rate (1.4). In the end, we test the action growth rate (1.9) in the
case of charged Gauss-Bonnet-AdS black hole and
nd the exactly same equality as well.
Furthermore we also con rm in the neutral limit the action growth rate (1.7) of black hole
is slowed down in the presence of stringy corrections unless it is charged. We thus conclude
that the stationary AdS black hole in Einstein gravity is the fastest computer in nature.
(5.23)
HJEP09(216)
(5.24)
Here some remarks are in order on what we did in this paper. Firstly according
to the holographic principle of gravity, a complexity bound of a boundary state should be
expressed in terms of physics quantities well de ned in the boundary eld theory. However,
some quantities in (1.10) like
and
are de ned on the inner horizon of a black hole,
and those quantities have no corresponding de nitions in the dual eld theory. As we
stressed in the context, at rst glance, it is true. But after a second look, we know that
those quantities are all can be expressed in terms of black hole parameters like mass, charge
and angular momentum, according to no hair theorem of black hole. Therefore it looks
unphysical at rst sight for the presence of those quantities de ned on the inner horizon,
but they can always be expressed as certain combinations of those quantities de ned on
the outer horizon, which is totally acceptable from the view point of eld theory side.
Furthermore, for those black holes with dual CFT descriptions, for example, the rotating
BTZ black hole, the growth rate of action can be simply re-expressed as 2
p
TLSLTRSR,
where TL;R and SL;R are the temperatures and entropies from the left- and right-moving
sectors of dual 2D CFT.
Secondly, the action growth rates (1.8)(1.9), when compared with the original action
growth rate (1.5)(1.6), have no necessity for the notion of ground state, which saves us the
argument made in appendix A of [19]. Nevertheless, if the notion of ground state means a
frozen complexity growth, then the \ground state" of our revised version of action growth
rate is nothing but the zero temperature state, namely the extremal black hole with inner
and outer horizons coincided. Therefore, one can rewrite the action growth rate (1.8),
for example, as (M
J )j+
[(M
J )j+]gs=extremal, which will reduce to the original
result (1.5) for the rotating BTZ black hole by noting that (M
J ) =
(M
+J ).
For an extremal black hole, following our calculations, the action growth rate goes to zero.
This is an expected result since the complexi cation rate must vanish for a ground state.
Thirdly, both the action growth rates (1.5)(1.6) and our results (1.8)(1.9) are nothing
but conjectures if the Complexity-Action duality (the complexity of a boundary state is
dual to the action of the corresponding Wheeler-DeWitt patch in the bulk) is correct.
Without further progress made in how to de ne the complexity of boundary state from
eld theory side alone, one can only test this conjecture by computing the growth rate
of its bulk dual. In this work, we just follow the logic in refs. [8, 19] and calculate some
exact results for the action growth rate of the WDW patch at late-time approximation in
some AdS black holes. It worth noting that it is by no means that we have found any
new non-trivial bound for the complexity growth other than the works done in [8, 19]. In
such calculations some subtleties arise as in [19]. One of them is the contribution from
the singularity which have been stressed in [19] and in this work. The other two concern
with the inner horizon of black hole and the contribution from the part of the WDW patch
behind the past horizon.
Fourthly, the presence of inner horizon is intriguing since the whole point of the growth
of complexity is its duality to the growth of black hole interior, which takes concrete form
of WDW patch served as the spacetime region dual of computational complexity of the
boundary CFT state. As argued in the section 3.2 of [19], before taking the limit of
late-time approximation, the entire WDW patch lies outside of the inner horizon, which
means the action is not sensitive to quantum instabilities of the inner horizon so long as
the horizon remains null. The classical instability of inner horizon is not considered here
just as in [19], since we all rely on the assumption that the black hole interior from static
solution is trustable as long as the complexity is concerned, which certainly calls for further
investigation. Usually the inner horizon will turn to a curvature singularity when black
hole gets perturbations or some matter is added into the theory under consideration. In
that case, one has to re-calculate the action growth rate since the Penrose diagrams for
those black holes are totally di erent from the one like the RN-AdS black hole.
Fifthly, the contribution to the action growth rate from the corner term of the WDW
patch inside the past horizon is negligible so long as the late-time approximation is
concerned as argued in [19]. We expect that the GB gravity makes no di erence for this point.
However we cry for a systematic investigation of regulating the action growth rate in that
patch within di erent gravity theories in future.
Finally, in the calculation of the action for the Gauss-Bonnet gravity, we found that
the results are di erent if one takes the contribution from di erent singularities at r = 0
and r = r~ , respectively. To avoid such an ambiguity, we add the Maxwell eld to the
theory and in that case an inner horizon appears and both singularities are hidden behind
the inner horizon. Then the result for the Gauss-Bonnet black hole is obtained by taking
the limit of vanishing charge. Then a natural question arises, what is the guiding principle
for dealing with the singularity when the action growth rate within the WDW path at
late-time approximation is concerned? It is worth noting that, the action growth rate
of Schwarzschild-AdS black hole has saturated the neutral static case (1.4)(1.7), which
is the consensus for both the original proposal and our revised version. We argue that
whether or not the neutral static action growth rate could come back to 2M at the leading
order term of gravity correction is our guiding principle when dealing with singularity.
In Einstein gravity, the neutral limit of action growth rate for RN-AdS black hole and
charged BTZ black hole naturally reduces to the neutral static case (1.7), and the
nonrotating limit of Kerr-AdS black hole and rotating BTZ black hole also reduces to the
neutral static case (1.7). Therefore, when dealing with singularity within Einstein gravity,
one can either directly calculate the neutral static case, or rst shield the singularity with
some cuto screen, which is conveniently chosen as the inner horizon generated by adding
charge or angular momentum into the neutral static black hole, and then take the neutral
non-rotating limit. However, this is not the case for gravity theories other than Einstein
gravity, for example, the Gauss-Bonnet gravity. The direct calculation of action growth
rate for neutral GB-AdS black hole in appendix A can not come back to the neutral static
case (1.7) at the leading order term of GB coupling. Therefor the reasonable approach to
deal with singularity in GB gravity is to rstly screen the singularity with inner horizon in
case of charged GB-AdS black hole, and then take the limit of neutral charge, because this
will give us the neutral static result at the leading order term of GB coupling. Alternative
approach by perturbatively computing the action growth rate might not work out, since the
action growth rate is calculated on-shell which requires the full solution of Gauss-Bonnet
equation of motion.
can be ended either on both singularities r = 0 or r = re .
A
The Gauss-Bonnet-AdS black hole
In this appendix, we present the direct calculation of the growth rate of action for the
Gauss-Bonnet-AdS black hole instead of taking zero charge limit from charged
GaussBonnet-AdS black hole. Unlike the case of charged Gauss-Bonnet-AdS black hole (5.24),
the growth rate of action for the Gauss-Bonnet AdS black hole cannot come back to the
case of Schwarzschild-AdS black hole in the limit of zero Gauss-Bonnet coupling
The Penrose diagram of the neutral Gauss-Bonnet-AdS black hole is presented in gure 2.
! 0.
A.1
Singularity located at r = 0
As mentioned, from the GB-AdS black hole case, there are two singularities. Let us rst
consider the singularity r = 0 as the inner boundary. In this case, using (5.19) with the
range of evaluation replaced by (0; rh), we can easily get
(A.1)
(A.2)
(A.3)
dA0 =
dt
=
=
3
3
3
Lh2 + rh2
where we have used
Finally, we write down the growth rate of action for Gauss-Bonnet-AdS black hole as
We see that the above result cannot return back to the case of Schwarzschild-AdS black
hole in the limit of
! 0. This indicates that the above calculation is not trustable.
Singularity located at re
Taking the singularity r as the inner boundary, one can solve (5.10) and nd that
e
r
e
Similarly we only need to replace the range of evaluation in (5.19) with (r ; r+) and get
dAre
dt
=
=
3
3
3m
16
8
s
8 L2m
(8
L2)
8
r
4
L2
h +rh2
m
L2 (5 2 + 24 ) :
#
e
3
r
4
L2 + r2 +r2
e
e e
1+ e
r
4
2
(A.4)
(A.5)
In this case, the condition 8
from taking the limit
! 0.5 But if naively takes the limit of
! 0, one can see that the
> L2 for the presence of singularity re simply prevents us
above result also cannot return back to the case of Schwarzschild-AdS black hole, which
indicates that the above approach is problematic.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China
under Grants No. 11375247 and No. 11435006.
Open Access.
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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