#### On some universal features of the holographic quantum complexity of bulk singularities

HJE
On some universal features of the holographic quantum complexity of bulk singularities
Stefano Bolognesi 0 1 3 5
Eliezer Rabinovici 0 1 3 4
Shubho R. Roy 0 1 2 3
0 Hyderabad , Kandi, Sangareddy 502285, Medak, Telengana , India
1 91904 , Israel
2 e Witt patch
3 Largo Pontecorvo , 3, Ed. C, 56127 Pisa , Italy
4 Racah Institute of Physics, The Hebrew University of Jerusalem
5 Department of Physics “E. Fermi”, University of Pisa and INFN Sezione di Pisa
We perform a comparative study of the time dependence of the holographic quantum complexity of some space like singular bulk gravitational backgrounds. This is done by considering the two available notions of complexity, one that relates it to the maximal spatial volume and the other that relates it to the classical action of the WheelerWe calculate and compare the leading and the next to leading terms and find some universal features. The complexity decreases towards the singularity for both definitions, for all types of singularities studied. In addition the leading terms have the same quantitative behavior for both definitions in restricted number of cases and the behaviour itself is different for different singular backgrounds. The quantitative details of the next to leading terms, such as their specific form of time dependence, are found not to be universal. They vary between the different cases and between the different bulk definitions of complexity. We also address some technical points inherent to the calculation.
Gauge-gravity correspondence; Spacetime Singularities
1 Introduction 2 3 4
1
Time dependence in the boundary metric: Kasner and Topological
Time dependence in the bulk: De Sitter crunch
Conclusion
Introduction
of the order of exp(S). We remark that the claim for the existence of a maximum value
– 1 –
of the complexity resides in the attempt to reconcile the continuous structure of the space
of all quantum states with the manner by which each such state is reached by a discrete
and finite number of prescribed operations. This is done by assigning an arbitrary required
accuracy for declaring a particular approximation of the target state as successful. Without
such a cutoff the complexity defined in this way has no upper bound. Some authors, [5]
and references therein, have been flirting on and off with considering metrics and geodesics
on the space of states with which the system would have a universal maximum complexity,
however at this stage neither this nor a dynamical cutoff were identified. The need for
an arbitrary cutoff has not been removed and the states of a Quantum Field Theory have
indeed a maximum complexity for given values of the entropy and the cutoff. For other
recent developments on the definition of complexity in QFT see [9–12].
There are presently two suggestions, each with its own motivation, as to what the
complexity of a quantum boundary state should be given by in the bulk. One is that is
should be, up to some proportionality coefficient, the volume of the maximal spatial surface
extending into the bulk and terminating on the boundary at the spacial slice on which the
boundary quantum state is defined [6]. This is referred as the complexity proportional to
volume (C ∝ V) conjecture.
The other is [7, 8] is that the complexity is proportional to the classical value of the
action in a WdW patch. The so called (C ∝ A) conjecture. One takes the spatial slice on
the boundary on which the state is defined. Then one considers the union of all possible
spatial slices which extend into the bulk and terminates on the same spatial slice at the
boundary. The union forms a sub-set of the bulk called the Wheeler-de Witt (WdW) patch.
The Einstein-Hilbert (EH) action evaluated on this patch, with the inclusion of the proper
York-Gibbons-Hawking (YGH) boundary term, is conjectured to be proportional to the
complexity. The spatial slice of maximal volume considered for the C ∝ V conjecture is
contained in the WdW patch but now it does not play any preferential role with respect to
the others. Be the complexity of a state what it may be on the boundary CFT and be the
corresponding bulk quantity what it may be for the purpose of the study of singularities it
is one of its features, its time dependence, that is proposed as a diagnostic tool on its own.
It is on the universal features of this quantity that we focus on in this paper.
Based on examples [7, 8] it is argued that as long as the complexity of a state increases
in a black hole context no singularity is encountered. In addition it is claimed that this
result is robust and remains valid when small perturbations are added to fine tuned states
such as the thermofield double [13]. On the other hand if the complexity does decrease in
time the state may well encounter a singularity that one could associate with the formation
of a firewall like obstacle. However this evolution is claimed, on the basis of examples, not
to be robust, once small perturbations are added to such a system the potential formation
of a firewall type singularity is delayed for at least a time of the order exp(S). There are
other circumstances in which the complexity is expected to decrease [
14
]. The Poincare
recurrences which occur in QFTs, when they obey certain conditions, after the passage very
large times of order exp(exp(S)), bring down the complexity with them. This decrease need
not signal necessarily an approach to a singular configuration. This decay is indeed not
robust itself but once one decides on what the Hamiltonian and the state are these decays
will occur again and again with Poincare time intervals.
– 2 –
In addition to the black hole case it was found, using the C ∝ V conjecture, that the
complexity does decrease as the system approaches some pre-engineered classes of time
dependent singularities [
17
]. These are intriguing singularities of the type one may be
able to coexist with. This result could be expected and is robust for BKL [
15, 16
] type
singularities but the generality of this feature is less clear and was obtained using the
extremal volume prescription to calculate the complexity.
Given the scarcity of the present knowledge on the nature of these singularities we
explore here if this feature depends on the well motivated, but particular, bulk quantity
which was conjectured to be associated with the QFT complexity. In this work we obtain
the complexity by using the C ∝ A prescription. This way of evaluating the complexity
presents its own challenges. The work will be largely of a technical nature. The result that
we find is qualitatively similar the one obtained with the C ∝ V, in particular complexity
always decreases toward the crunch. The quantitative details and in particular the specific
form of time-dependence of the complexity are instead different.
Complexity is a UV divergent quantity so its regularization is necessary, as we discussed
it is after the time dependence of the complexity that we are after. Both the C ∝ V and
C ∝ A conjectures are sensitive to this divergence and a way to deal with it is to introduce
a UV cutoff close to the boundary. For the C ∝ A there is computational difficulty.
The boundary of the WdW patch is a light-like sub-manifold with joints. Both issues,
being light-like and having joints, means that the YGH boundary term must be properly
defined. This problem has been recently discussed in [
18–20
]. For the present paper we
take a different approach to this problem (see also [
8, 21
]). Since we have anyhow to UV
regularize the WdW patch, we use this opportunity to introduce a particular regularization
that makes the WdW boundary time-like and also smooths-out the joints. In this way we
can compute the YGH term with no ambiguities.
There are two distinct types of time-dependent backgrounds. For the first type time
dependence is explicit in the UV part of the metric. This case corresponds to some time
dependent marginal operator in the dual field theory. The Kasner metric and the
topological crunch are two examples of this type and we will study them in detail. In those cases
also in the UV divergent part of the complexity is time-dependent. For the second class
the UV metric has no explicit time dependence but the whole bulk metric does. The de
Sitter crunch model is the specific example of this type that we will consider here. In those
last cases the divergent piece of the complexity is free from any time dependence and the
time derivative of the complexity is a finite quantity.
The paper is organized as follows. In section 2 we consider the cases of time dependence
on the boundary. In section 3 we consider the bulk time dependence. We conclude in
section 4.
2
Time dependence in the boundary metric: Kasner and Topological
crunch
In this section we perform a comparative calculation and study of the complexity and its
time dependence as obtained according to two of its different bulk definitions. This for
– 3 –
two cases engineered in such a way that the world volume of the boundary theory itself is
singular. For each definition and each class of metrics we obtain the leading and next to
leading terms. The result will coincide with those of [
17
] for the leading term when the
volume prescription is adopted, this for both types of metrics.
We do this by studying two cases in which the metric on the world-volume of the dual
field theory has the form
with i, j = 1, . . . , d. The UV boundary is set at z = 0 and the world-volume frame where the
CFT is realized is ds2CFT = −dt2 + hij (t, x)dxidxj . All cases we consider satisfy everywhere
z2 −dt2 + dz2 + hij (t, x)dxidxj
R − 2Λ = −
2(d + 1)
L2
The first definition of complexity is C ∝ V where V is the spatial slice with maximal
volume whose boundary is at a fixed time t = t∗. We consider any possible spatial slice
defined by
ΛUV =
1
zUV
.
t = f (z, xi)
– 4 –
d(2dL+21) is the cosmological constant. This is true whenever the world-volume
The first specific example is that of the Kasner metric. The Kasner metric is of the
hij (t, x) = diag
t 2p1
l
, . . . ,
t 2pd !
l
X pi =
X pi2 = 1 .
i
i
and l is a dimensional scale. The second specific example is that of the so called topological
crunch. The topological crunch, close to the boundary, is of the form (2.1) with
where Λ = −
frame is Ricci flat.
form (2.1) with
with the relation
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
hij (t, x)dxidxj = l2 dΩ2d−1 + cos2
dφ2
when the CFT world-volume goes to zero, and this is t = 0 for Kasner and t = ±l π2 for the
topological crunch case. The latter is called topological because locally the metric (2.1)
with (2.5) is always the same as AdSd+2. We will perform the computations using the
generic form (2.1) and then adapt to the specific cases at the end.
Complexity, in any of its forms, has always a divergent term that is due to the
contribution coming from close to the UV boundary. This can be regularized by introducing a
UV cutoff ΛUV related to a minimal value of the z coordinate
with the property
z→0
lim f (z, xi) = t∗,
∀xi .
Since the spaces we are considering are homogeneous in xi, we can always restrict to the
case ∂if = 0 everywhere, so from now on we take f (z) to be just a function of z. The
induced metric on this space slice is
L2
ds2 =
z2 (1 − (∂zf )2)dz2 + hij(f, x)dxidxj
and the space volume is then
where we indicate
V =
Z
dzdVx zd+1
Ld+1
p(1 − (∂zf )2) h(f (z), x)
dVx = dx1 . . . dxd
and
h(t, x) = det hij(t, x) .
The expansion of the metric determinant is
h(t, x) ≃ h(t∗, x) + (t − t∗)h(t∗, x)hij(t∗, x)∂thij(t∗, x) + . . .
Note that for spaces homogeneous in x the quantity hij(t, x)∂thij(t, x) does not depend on
x so we will denote it simply as hij∂thij(t).
The Euler-Lagrange equation for f (z), after some simplification, becomes
∂z2f = 1 − (∂zf )2
(1 + d)∂zf
z
− 2
1 hij∂thij(f (z))
By expanding near z = 0 and solving the equation in a power series of z we find that f (z)
goes to t∗ with zero derivative
The space volume is dominated by the following UV divergent term:
f (z) = t∗ + α(t∗)z2 + . . . .
V ≃
≃
Z
1/ΛUV
ΛdUVLd+1 Z
d
dz
Z
Ld+1
dVx zd+1
Note that the leading divergence is proportional to ΛdUV and does not depend on α(t∗)
or any of the sub-leading terms in the expansion of (2.14). Note also that this term is
proportional to the boundary space volume R dVx ph(t∗, x) and this is what provides the
dependence with respect to anchored time t∗.
We next turn to the sub-leading divergencies which are affected by the coefficient α(t∗)
in (2.14). Continuing the power expansion of the equation (2.13) it can be found, for d > 1
α(t∗) =
hij∂thij(t∗)
4d
.
– 5 –
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
HJEP06(218)
d
Z
dVx
ph(t∗, x) = 2πldVSd cos
t
l
and hij ∂thij = − 2l tan tl . The volume is then
d
V = ΛUV
Ld+12πldVSd cos tl
+ ΛdU−V2 Ld+1π(d − 1)ldVSd sin2 t
l
d2(d − 2)l2 cos tl
+ . . .
(2.21)
(2.18)
(2.19)
(2.20)
Z
dVx
ph(t∗, x) =
V t
l
where V is an IR cutoff for the spatial volume in xi. Moreover for the Kasner metric we
have hij ∂thij = 2t . The volume is then
d
V = ΛUV
Ld+1V t
dl
+ ΛdU−V2 L2dd+21((dd−−21))lVt
+ . . .
For the leading divergence we recover the result of [
17
]. Note that the leading order
divergent term is linear in t and it is decreasing in time till it would have vanished at
the singularity t → 0. The validity of the approximation breaks down as one reaches
to Planckian and string scale distances tUV ≃ ΛU1V . This is also the time by which the
sub-leading divergent term becomes of the same order of the leading one. The complexity
decreases at a rate linear in time as long as the approximation is valid. A linear dependence
of the complexity on time was obtained in the eternal black hole case among other ones [6,
7], although in the latter case it was for a non-divergent piece of the complexity. In the
Kasner case the linear dependence follows directly from the relation (2.4) and is not a
universal feature as we are going to see next.
For the specific case of the topological crunch we have
Note that α(t∗) is uniquely determined once the first constant of integration t∗ is decided.
The second constant of integration appears at orders higher than z2 in the power expansion
for f (z). The sub-leading divergent term of the volume depends only on α(t∗) and is
V = O(ΛdUV) + ΛdU−V2 Ld+1(d − 1)(hij ∂thij (t∗))2 Z
8d2(d − 2)
dVx
ph(t∗, x) + . . .
(2.17)
Sub-leading divergent terms are present whenever hij ∂thij 6= 0.
where O(ΛdUV) is the leading term (2.15) while the sub-leading term is of order O(ΛdU−V2).
For the case of the Kasner metric we have
For the leading divergence we again recover the result of [
17
]. As in the Kasner metric case
the complexity vanishes towards and close to the crunch, but the time dependence is no
longer linear in this case. The considerations to be applied in constraining the proximity
of the time to the singularity while retaining the validity of the approximation are similar
to those presented for the Kasner metrics.
We will now spend a moment on an issue of the C ∝ V not discussed previously.
When we have only one boundary for the CFT frame, a quantum state is defined once
– 6 –
we specify one number, the time t∗ when we take the spatial slice. The maximal volume
equation (2.13) on the other hand is a second order differential equation and needs two
conditions to select a particular volume that extends in the bulk. This second boundary
condition, for d > 1, appears in the expansion (2.14) to higher order than O(z2), so it does
not affect either the leading (2.15) or the sub-leading (2.17) divergences of the volume.
Nevertheless, if we want to make sense of the C ∝ V conjecture we need to specify this
condition and select one particular volume. If we take hij constant, the generic solution
of (2.13) with f (0) = t∗ is
1
d + 2
f (z) = t∗ + zψ βzd+1
= t∗ +
βzd+2 + . . .
(2.22)
where ψ is some known function expressed in terms of an hypergeometric function and β is
the second integration constant which appears only at order zd+2 in the expansion. All of
those solutions are locally maximal volume surfaces and the union of all of them gives the
WdW patch. If we want to select a particular one, in order to associate it to the complexity
of the boundary state, a natural choice would be the constant solution, β = 0, equal to
t∗ everywhere. When hij depends of time this choice becomes less obvious. Let’s take for
example the case of Kasner and we plot in figure 1 various solutions of the maximal volume
equation (2.13) with f (0) = t∗. As before the union of all the solutions gives the WdW
patch. All solutions, save one, become asymptotically null at large z like f (z) = z + const
or f (z) = −z + const. The special one that divides the two types of solutions is a natural
candidate to be associated at the boundary state. Its slope at z → ∞ in neither 1 nor
−1 and in general depends on the anchored time t∗. For the purpose of the present paper
this is not an urgent issue since all those solutions share the same quadratic expansion and
thus have the same leading and sub-leading divergences. So we will not discuss this issue
any further here.
We now turn to obtain the complexity as given by the definition C ∝ A where A is the
action of the WdW patch anchored at t = t∗ on the UV boundary. We need to regularize
the action and to this end we smoothen-out the boundary of the patch. Our choice is to
– 7 –
WdW patch
1 t∗
ΛUV
take, as in figure 2, the region of space-time delimited by the following hyperbole
.
As ΛUV → ∞ we recover the full WdW patch. This regularization achieves two goals
simultaneously. First it regularize the UV divergence. Second the boundary of the regularized
patch is now a smooth time-like boundary with no joints: this allows the computation of
the boundary gravitation action without any ambiguities.
The total action is given by the Einstein-Hilbert (EH) term with the cosmological
constant plus the boundary York-Gibbons-Hawking (YGH) term
A = AEH + AYGH
=
1
16πG
Z
WdW
√−g (R − 2Λ) +
ǫ Z
where ǫ = ±1 according if the boundary is time-like or space-like. The boundary of the
WdW is a null hypersurface, but we will consider it as a limit of a time-like
submanifold (2.23). So from now on we will assume ǫ = +1 and |γ| = −γ.
The EH term evaluated on the regularized patch is
AEH =
=
16πG
1
1
16πG
Z
Z
Z ∞
dt
Z
dVx
dt √1+Λ2UV(t−t∗)2 dz
We are interested in capturing the dominant divergent piece of the action. So we take g to
be equal to its boundary value at t = t∗:
AEH ≃ − 8πG
Γ d+1
2
Note that this term is proportional to the boundary space volume and the divergence is
proportional to ΛdUV. The sign of AEH is negative because of the sign of R − 2Λ , but it
will be overcome by the boundary term which turns out to be positive. To see to the final
answer the reader can go directly to (2.35) and (2.36). In the following we present the
technical details needed for obtaining AYGH.
We need to compute some useful geometrical quantities before the evaluation of the
boundary YGH action. The components of the connection we will use later are:
Γzzz = − z ,
1
Γtzt = Γttz = Γtzt = − z ,
1
Γizj =
1
z hij ,
Γitj =
1
2 ∂thij .
(2.27)
The boundary (2.23) is a time-like sub-manifold and we can choose to parameterize it with
the coordinates xa : t, xi. It is defined by the following embedding:
1 q
The Jacobian of the embedding eaμ = ∂∂xxμa is
ett = 1
etj = 0
etz = q
ejz = 0
ΛUV(t − t∗)
1 + Λ2UV(t − t∗)2
eit = 0
eij = δji .
K = γabeaμebν∇μnν
=
1
L
d − 2
1 hij∂thij(t − t∗)
q
1 + (t − t∗)2Λ2UV .
– 9 –
The induced metric γab on the boundary manifold xa is
The outward unit normal is
ds2 =
L2
z2
1
− (t − t∗)2Λ2UV + 1
dt2 + hij(t, x)dxidxj .
nt = q
LΛ2UV(t − t∗)
1 + Λ2UV(t − t∗)2
nz = −LΛUV
with eaμnμ = 0 and nμnμ = 1. The final result will not depend on the specific way nμ is
extended outside the boundary. The covariant derivative is ∇μnν = ∂μnν − Γρμνnρ, and the
components are
∇tnt = ∂tnt +
∇znz =
nz
z
nz
z
n
t
z
∇znt = ∇tnz =
1
1
∇inj = − 2 ∂thijnt − z hijnz .
We can now compute the extrinsic scalar curvature of the boundary sub-manifold
(2.29)
(2.30)
(2.31)
(2.32)
(2.33)
The YGH boundary term is then
AYGH =
As before, to extract the dominant divergent term we take g to be equal to its boundary
value at t = t∗:
AYGH ≃ 8πG
≃ 8πG
The final result for the dominant divergent term of the action is then the sum of (2.26)
and (2.35):
The coefficient is always positive for d > 1 despite the EH being negative, the positive
YGH term is always the dominant contribution. The result for the leading UV divergence
of the complexity is thus the same obtained for the C ∝ V in (2.15). In both cases the
complexity decreases towards the singularity in the region of validity of the approximation.
We next study the sub-leading divergences. For this we need the expansion of the
metric determinant up to the second order around t∗
where H1(t) and H2(t) are given by
h(t, x) ≃ h(t∗, x) 1 + (t − t∗)H1(t) + (t − t∗)2H2(t) + . . .
1
2
H1(t) = hij ∂thij ,
H2(t) = hij ∂t2hij − hjl∂thlkhki∂thij + H12 .
(2.35)
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
(2.41)
The expansion for the EH terms (2.25) gives
AEH ≃ O
d
ΛUV
AYGH ≃ O
d
ΛUV
+
LdΛd−2 √πΓ d2 − 1
UV
− 128πG
where O(ΛdUV) is the leading term (2.35). The final result is
A ≃ O
d
ΛUV
+
LdΛd−2 √πΓ d2 − 1
UV
128πG
where O(ΛdUV) is the leading term (2.36).
2H2 − H12 Z
dVx
2(d − 1)H2 − (d + 1)H12 Z
dVx
det hij (t∗, x)
where O(ΛdUV) is the leading term (2.26). The expansion for the YGH terms (2.34) gives
q
q
For the Kasner metric H1 = 2t , H2 = t22 and the action is then
.
In this case, due to the fact that √
h is linear in t, we essentially recover the same result
see some difference in the subleading divergence because √
we had for the volume prescription (2.19). For the case of topological crunch instead we
h is not linear in time. In the
latter case we have H1 = − 2l tan tl , H2 = l22 tan2 t
l − 1 and the action is
HJEP06(218)
We see that the subleading divergence is different from the one obtained with the volume
prescription (2.21).
In the next part of the paper we will test if the result we have found in this section,
that the main features of the complexity for space like singularities, are independent on
the choice among the two bulk prescriptions, is valid also for the de Sitter case.
3
Time dependence in the bulk: De Sitter crunch
We next turn to cases in which a Big Crunch occurs semi-classically in the bulk. This
can be represented by either a boundary theory, with particular time dependent relevant
deformations, living on a time independent world volume which exists for only a finite time
(denoted EF) or by a conformally related time independent boundary system living on an
expanding de Sitter world volume (denoted by dSF). This duality was studied in [
1, 3
]. In
this case the time derivative of the complexity is a finite quantity, both for the volume and
the action prescriptions. It does not depend on the UV regularization.
We first make a more generic analysis in the absence of divergent terms in the time
derivative of the complexity. Consider a metric is of the form
(3.1)
(3.2)
L2
z2
ds2 =
dz2 + hμν (t, x)dxμdxν
with µ, ν
= 0, 1, . . . , d and the UV boundary is at z = 0. The expansion of the metric near
the boundary is in general of the following form
hμν = h(μ0ν) + z2h(μ2ν) + · · · + zdh(μdν) + zd log z2h(μdν) + . . .
For the EF case one has h(μ0ν) equal to the EF metric and the entire time dependence is
due to a domain wall moving in the bulk. We thus consider the case in which h(μ0ν) is static
while sub-leading terms in the expansion (3.2) may contain information about the time
dependence. Let’s assume for generality that h(μ2νk)(t, x) is the first term to contain a time
dependence, then the contribution to the complexity from the term (2.41) is
A ≃ O
d
ΛUV
+ O
ΛdU−V4k
So, for small enough dimension d < 5 we do not have any divergent terms.1 In this case
to compute the time dependence of the complexity, which is a finite term, we have to
consider the full metric and also the precise shape of the WdW patch; considering only the
asymptotic form of the metric close to the UV is not enough. This leads us to perform the
computations on a case-by-case basis.
We now consider in detail the de Sitter crunch in the case that the thin wall
approximation is valid. This occurs when the boundary field theory in its dSF coordinates
contains a negative relevant operator whose coefficient is much larger than its expansion
rate [
1, 3
]. The metric in the UV is AdS+ with curvature length l+ = 1 which we fix to
one for convenience
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
ds2 = dρ2 + sinh2 (ρ) −dτ 2 + cosh2 (τ )dΩ2d
This is valid for ρ > ρW where ρW is the position of the domain wall. The metric on the
other side of the wall ρ < ρW is AdS− with a smaller curvature length l− < 1. In order
to match the two metrics at the same value of ρ we need to perform some rescaling of the
standard form. The expression for the interior metric is
with
ds2 = l−2α2dρ2 + l−2 sinh2 (αρ) −dτ 2 + cosh2 (τ )dΩ2d
1
ρW
α =
arcsinh
sinh (ρW )
l−
> 1
This is valid for ρ < ρW . In this coordinate choice the two metrics are patched so that
they are continuous at ρ = ρW . The position of the wall ρW is constant in the time τ .
In the dual field theory this correspond to an RG flow between two different CFT’s. The
limit we are discussing is that of the thin-wall approximation.
Now we focus on the exterior region, the one that extends to the UV. By changing
coordinates to
the metric (3.4) becomes
r = sinh (ρ) cosh (τ )
cos (t) =
cosh (ρ)
q
dr2
ds2 =
(1 + r2) − 1 + r2 dt2 + r2dΩ2d
1Rember that d is the number of space dimensions on the boundary, for example d = 3 for AdS5. For
d ≥ 5, so from AdS7 onward, a bulk time dependence of the metric can in general provide a time-dependent
divergent piece in the action trough its tail at z → 0. We will not consider those cases in the present paper.
and the wall is not static but follows the trajectory
r(t)W =
cosh2 (ρW )
cos2 (t)
At the time t → π2 where r(t)W → ∞ we have a bulk crunch. We will work in this EF
frame where the metric does not depend on time, although it exists only for a finite time,
and the bulk crunch is visible on the boundary. Note that the metric time dependence is
only through the wall trajectory. In the UV the metric is constant.
We then have consider the interior part AdS− with L = l−. We first rewrite (3.5) the
The domain wall is then at ρW = αρW . Then we change coordinates to
r = sinh (ρe) cosh (τ )
e
cos et =
q
cosh (ρe)
The wall trajectory is these coordinates is
dr2
e
ds2 = l−2
1 + r2 − 1 + r2 det2 + r2dΩ2d
e e e
re(et)W =
cosh2 (αρW )
cos2 et
1
! 2
− 1
The metric is AdS− in the region re ≤ re(et)W . The crunch is at et → π2 where re(et)W → ∞.
In what follows we will use two different coordinate systems: for the exterior (3.8)
and for the interior (3.13). The two patches are connected at the wall trajectory which is
different in the two coordinates systems, respectively (3.9) and (3.14). This is the price we
pay in order to have the two metrics writen in a simple form. When passing from one patch
to the other we have to change also coordinates. A point on the domain wall in AdS+ at
time t and radius r(t)W in the AdS− patch will be seen at time and radius
et = arccos q
cos (t) cosh (αρW )
cos2 (t) + ssininhh((αρρWW)) cosh2 (ρW ) − cos2 (t)
,
r =
e
r
l−
.
(3.15)
We work in the thin wall approximation. In general we can construct a wall with
a scalar field and a suitable potential with two minima, the values at the two minima
correspond to the cosmological constant at the two sides of the wall. We can then scale
the microscopic parameters and send the wall thickness to zero while keeping the tension
fixed. Assuming that the curvature in the middle of the wall remains of the same order
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
V+ =
The total volume is then
V = V+ + V− = VSd ld+1 Z reW (t∗)
−
dr √
r
d
1 + r2
+
The derivative with respect to t∗ is free from UV divergence and, using (3.15), it becomes
ddtV∗ = VSd dt
drW rW (t∗)d q
1
1 + rW (t∗)2/l−2 − p1 + rW (t∗)2
1
This is always negative for l− < 1 and t > 0 which means that complexity(volume) is
decreasing going toward the crunch. For t∗ → π2 we have
ddtV∗ = VSd
coshd (ρw)(−1 + l−)
We now calculate the Complexity using the WdW prescription. For this task it is
convenient to change again coordinates
!
of the cosmological constants, the thin wall limit allows us to neglect the terms from the
action coming from the wall itself.
The space volume anchored at t = t∗ in the UV part is
The space volume anchored at t∗(t∗) (the relation is given by (3.15)) in the IR part is
r = cot(z)
π
with 0 ≤ z ≤ 2
. The metric (3.8) becomes
ds2 = 1 + cot2(z)
−dt2 + dz2 +
The domain wall trajectory is
Close to the crunch
In these coordinates the WdW patch is delimited by
zW (t) =
π
2 − arctan r(t)W
zW (t) ≃ cosh (ρW )
z = |t − t∗|
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
t∗
zW (t)
z
A
AdS+
HJEP06(218)
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
show the WdW patch anchored at t∗ enclosed in this region and the intersections zA(t∗) and zB(t∗).
Intersection with the domain wall trajectory is at (tA(t∗), zA(t∗)) and (tB(t∗), zB(t∗)) with
tB(t∗) > tA(t∗) and
intersections are
zA(t∗) > zB(t∗)
for
t∗ > 0
In general we cannot solve analytically for zA(t∗) and zB(t∗). Close to the crunch the two
tA(t∗) ≃
tB(t∗) ≃
π
cosh (ρW )t∗ − 2
cosh (ρW ) − 1
cosh (ρW )t∗ + π2
cosh (ρW ) + 1
zA(t∗) ≃ cosh (ρW ) − 1
zB(t∗) ≃ cosh (ρW ) + 1
The geometry of AdS+ with the WdW patch is pictured in figure 3.
Now we compute the action of the WdW patch anchored at the UV boundary t∗. The
EH term is
AEH+ =
= −
1
16πG
Z
WdW
(d + 1)VSd Z tB
8πG
tA
√
dt
Z zW (t)
|t−t∗|
dz cotd(z) 1 + cot2(z)
where we used R − 2Λ = −2(d + 1). The derivative with respect to t∗ is
The boundary YGH term is obtained next . The metric (3.41) is
with
dAEH+ =
dt∗
VSd
8πG
cotd+1(zB) − cotd+1(zA)
ds2 = g(z) −dt2 + dz2 + f (z)hij (x)dxidxj
g(z) = 1 + cot2(z)
f (z) = cot2(z)
and hij the metric of Sd. The component of the connection of the metric that we need are:
Γzzz =
g′(z)
2g(z)
Γtzt = Γttz = Γtzt =
g′(z)
2g(z)
f ′(z)
Γizj = − 2g(z) hij .
We use the same time-like hyperboloid as before and then send the cutoff to infinity to
obtain the light-like boundary of WdW. The boundary is defined by (2.28). The induced
metric γab on xa is
ds2 = − (t − t∗)2Λ2UV + 1
g(z)
dt2 + f (z)hij(t, x)dxidxj
The Jacobian of the embedding eaμ = ∂∂xxμa is the same as in (2.29). The outward unit
normal is
nt = pg(z)ΛUV(t − t∗)
q
nz = −
g(z) 1 + Λ2UV(t − t∗)2 .
The components of the covariant derivative different from zero are
and the extrinsic curvature is then
∇znt = ∂znt − 2g(z) nt
∇znz = ∂znz − 2g(z) nz
g′(z)
g′(z)
K = γabeaμebν∇μnν = − g(z) 2
ΛUV
dzΛUVf ′(z)
1 − 2f (z)g(z) 12 −
ΛUVzg′(z)
2g(z) 2
3
.
Putting all these together allows to be obtain the YGH boundary term:
AYGH+ =
1 Z
√−γK
=
VSd Z
8πG
dt
d
g(z) 12 f (z) 2
(t − t∗)2Λ2UV + 1 2
1 K
The terms is simplified by removing the cutoff, that is sending ΛUV → ∞
AYGH+ = − 16πG tA
VSd Z tB
dtf (z) 2
d−2 2f (z)
+ df ′(z) +
f (z)g′(z)
g(z)
where z = |t − t∗|. The time derivative takes contributions only from the boundary of the
boundary since the bulk of AdS+ in time independent and is
dAYGH+ = − 16πG
VSd
dt∗
ddztB∗ f (zB) 2
d−2 2f (zB)
+ df ′(zB) +
f (zB)g′(zB)
g(zB)
+
ddztA∗ f (zA) 2
d−2 2f (zA) + df ′(zA) +
zA
f (zA)g′(zA)
g(zA)
.
(3.39)
zB
dinates as
and the metric (3.13) is then
The domain wall trajectory is
AEH− =
= −
1
16πG
Z
WdW
(d + 1)VSd l−d
8πG
e
cotd(z) 1 + cot2(z)
√
−g(R − 2Λ)
Z etA
etmin
e
Next we evaluate the WdW action in AdS− interior. It is convenient to change
coorr = cot (ze)
e
ds2 = l−2 1 + cot2(z)
e
−det2 + dz2 +
e
e
e
dΩ2d
zfW (te)
A
AdS−
π
π
HJEP06(218)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
!
dz
e
(3.45)
together with zW (et). The minimun and maximum values of et are
e
z = −et+ etA + zA
e e
z = et− etB + zB
e e
etmin = etA + zeA − 2
etmax = etB − zeB +
π
2
The geometry of AdS− is pictured in figure 4.
The action of the WdW patch that anchored at UV boundary t∗. The EH term is
zW (et) =
e
2 − arctan re(et)W
Intersection with the domain wall trajectory is at etA, zeA
and etB, zB to be obtained
e
using (3.15) and (3.40) from the corresponding values in AdS+ which are (tA(t∗), zA(t∗))
and (tB(t∗), zB(t∗)). In these coordinates the WdW patch is again delimited by
Z π/2
dt
e −et+etA+zeA
dz +
e
Z etB
e
tA
Z π/2
dt
e zeW (et) e
dz +
Z etmax
e
tB
Z π/2
dt
e et−etB+zeB
where we used R − 2Λ = − l−
dAEH− = − 8πG
dt∗
VSd l−d
d etB − zeB cotd+1(zeB) −
dt∗
d etA + zA cotd+1(zeA)
e
dt∗
!
(3.46)
The YGH boundary term is
The time derivative is
dAYGH− =
dt∗
AYGH− = − 16πG
VSd l−d
f (z) 2
e
d−2
Z etA
etmin
2f (z)
+
z
e
Z etmax !
e
tB
dt
e
e + df ′ (ze) +
f (ze) g′ (z)
e
h′ (z)
e
VSd l−d
16πG
dzA
+ e f (zeA) 2
dt∗
d−2
dzB
dt∗
e f (zeB) 2
d−2
2f (zeA)
zA
e
2f (zeB)
zB
e
+ df ′ (zeA) +
+ df ′ (zeB) +
f (zeA) g′ (zeA)
h′ (zeA)
f (zeB) g′ (zeB)
h′ (zeB)
Finally we can now sum all contributions to the derivative of the action
which is the sum of the four terms (3.29), (3.39), (3.46), and (3.48).
We present a numerical evaluation of ddtA∗ in figure 5. The plots are done for the specific
values of d = 2, l− = .9 and ρW = .5. We also plot the asymptotic formula close to the
crunch (3.50). The time derivative is negative and becomes larger close to the singularity.
That is qualitatively the same behaviour as obtained for the volume prescription (3.20)
ddAt∗ =
dAEH +
dt∗
dAYGH
dt∗
dA
dt∗ ∝
1
The degree of approach to the singularity is not same as is shown in figure 6 where we plot
the ratio ddtV∗ / ddtA∗ . However the origin of the negative sign is the same in both cases, it follows
because l−, the curvature length in the IR, is smaller than l+ = 1 which is the curvature
length at the UV. This allows one to identify that the source of the decrease in complexity
is the result of the renormalization group flow, there are less degrees of freedom in the IR
than in the UV. Thus both definitions suggested for the Holographic complexity capture
the same physical aspects as far the decrease of complexity in these classes of singular cases.
4
Conclusion
Different motivations lie behind each of the two suggestions for the bulk quantity which
corresponds to the quantum holographic complexity of the boundary state. Given that
one does not have yet a precise and universal definition of complexity in quantum field
theory itself, both can be considered as suggestive holographic probes for the notion of
how complex is a quantum state. There is no a priori reason for them to coincide in a
(3.47)
HJEP06(218)
(3.48)
(3.49)
(3.50)
generic situation and indeed, while we have shown that the degree of divergence is the
same, we have shown also that there are quantitative differences. In some cases only
for the next to leading term and in other cases for both the leading and next to leading
term. The quantitative behaviour itself, even when the same for both suggestions, differed
according the type of singular background. However both deliver the same verdict as far
as the manner in which the complexity of several classes of time-dependent holographic
backgrounds, which have crunch singularities, evolves. The complexity decreases. Another
universal feature that we have found that the time derivative of the complexity contains
a UV divergent part if the boundary metric depends explicitly on time while it is finite
otherwise. Another pattern emerged from the action calculation. For the cases in which the
time dependence character was driven mainly from a divergent term , it was the boundary
YGH term which determined the sign of the time derivative while for the case where the
time dependence was driven by a finite term in the evaluation of the complexity it was
the EH volume term which dictated the final sign. The universality we have found adds
credence to the identification of the physical source of Complexity decrease given in [
17
].
This leaves us with yet one more indication that for some probes space like singularities
may less of a bite than expected.
Acknowledgments
We thank J. Barbon and J. Martin for useful discussions. We thanks Jie Ren for
collaboration in the early stage of this project. S. B. work is supported by the INFN special research
project grant GAST (“Gauge and String Theories”). The work of E. R. was partially
supported by the American-Israeli Bi-National Science Foundation. The work of E. R. and
S. R. was partially supported by the Israel Science Foundation Center of Excellence and
the I Core Program of the Planning and Budgeting Committee and The Israel Science
Foundation The Quantum Universe. The work of S. R. is supported by the IIT Hyderabad
seed grant SG/IITH/F171/2016-17/SG-47.
Open Access.
This article is distributed under the terms of the Creative Commons
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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