Volume complexity of dS bubbles

Roberto Auzzia,b, Giuseppe Nardellia,c,

Gabriel Pedde Ungureanud and Nicolò Zenonie

a Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore,
Via della Garzetta 48, 25133 Brescia, Italy

b INFN Sezione di Perugia, Via A. Pascoli, 06123 Perugia, Italy

c TIFPA - INFN, c/o Dipartimento di Fisica, Università di Trento,
38123 Povo (TN), Italy

d SISSA, Via Bonomea 265, Trieste, Italy

e Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
In de Sitter spacetime, the growth of holographic complexity has a hyperfast
behaviour, which leads to a divergence in a finite time. This is very
different from the AdS spacetime, where instead the complexity rate
asymptotically reaches a constant value. We study holographic volume
complexity in a class of asymptotically AdS geometries which include de Sitter
bubbles in their interior. With the exception of the static bubble case, the
complexity obtained from the volume of the smooth extremal surfaces which are
anchored just to the AdS boundary has a similar behaviour to the AdS case,
because it asymptotically grows linearly with time. The static bubble
configuration has a zero complexity rate and corresponds to a discontinuous
behaviour, which resembles a first order phase transition. If instead we
consider extremal surfaces which are anchored at both the AdS boundary and the
de Sitter stretched horizon, we find that complexity growth is hyperfast, as
in the de Sitter case.

##  1 Introduction

The AdS/CFT correspondence [1] provides an interesting theoretical laboratory
to address many important open questions in quantum gravity. However, our
observed universe is rather different from an AdS background. It is then a
crucial problem to find a quantum gravity formulation for a cosmological
setting. It is natural to investigate generalizations of holography for de
Sitter (dS) spacetime. This is a challenging problem, because in dS there is
no natural notion of timelike boundary contrary to asymptotically AdS
spacetime.

In order to provide a holographic description of dS, the dS/CFT correspondence
[2, 3, 4] proposes a duality between quantum gravity in dSD spacetime and a
$(D-1)$-dimensional CFT living on a spacelike boundary at the future spacelike
infinity in dS. Examples of explicit dS/CFT correspondence have been proposed
for higher spin gravity in four dimensions [5] and for 3-dimensional Einstein
gravity [6, 7]. The boundary theory in these cases is not unitary and rather
exotic compared to the textbook examples of CFTs, because it describes dS from
the perspective of a metaobserver who lives at the future infinity.

In another class of approaches, see for example [8, 9, 10, 11, 12, 13],
gravity in dS is conjectured to be dual to a quantum mechanical system with a
finite number of degrees of freedom. This can be physically motivated by the
finite entropy which can be associated with the dS cosmological horizon [14]
surrounding a static patch observer. Recently, the entropy associated to the
dS cosmological horizon has been related to the theory of a von Neumann
algebra of type II1 [15].

Quantum information provides a useful conceptual framework to implement
several entries of the dictionary of holographic dualities, and might give
precious insights on how to formulate holography for the dS spacetime. An
interesting generalization of the Ryu-Takayanagi [16, 17] entanglement entropy
formula has been proposed in the context of static patch horizon holography
[18, 19, 20]. In this proposal, the AdS boundary is replaced in dS by a
stretched horizon which is taken just inside the cosmological horizon that
surrounds a static patch observer. The Bekenstein-Hawking entropy associated
to the cosmological horizon [14] is then interpreted as the entanglement
entropy between the left and right static patches. In the case of two-
dimensional dS, the proposed dual field theory is a double-scaled SYK model
[18]. Further recent works include [21, 22, 23, 24, 25, 26].

Computational complexity is another concept in quantum information theory
which may play an important role in holography, see [27, 28, 29] for reviews.
Indeed, entanglement entropy saturates too fast to describe the growth of the
Einstein-Rosen bridge inside a black hole horizon [30] in terms of the
boundary CFT. On the other hand, quantum complexity saturates in a much larger
timescale compared to the thermalization one, so it has the correct behavior
[31, 27] to overcome the limitations of entanglement entropy. In theoretical
computer science, complexity [32] measures how hard it is to build a generic
target state from a simple reference one, applying a set of elementary gates.
For quantum systems with a finite number of degrees of freedom, a continuous
notion of complexity was introduced by Nielsen [33] in terms of the length of
geodesics in the space of unitary operators. There is a large amount of
arbitrariness in the definition of complexity, due to the choice of reference
state and of the computational cost of elementary gates. Despite these
ambiguities, quantum complexity is expected to exhibit several robust and
universal properties [34], provided that the computational costs of elementary
gates is chosen in such a way that the complexity metric has negative
curvature [35, 36, 37, 38, 39, 40]. It is still an open problem to generalize
Nielsen’s approach to complexity in quantum field theory. Many advances have
been made in defining complexity in free field theory [41, 42]. The definition
of complexity in CFT is still a work in progress, see [43, 44, 45, 46, 47].

Three main proposals have been extensively studied as a holographic dual of
quantum computational complexity:

  * •

Complexity=volume (CV) [48], in which complexity is proportional to the volume
of the maximal codimension-one spatial surface anchored at a boundary time
slice.

  * •

Complexity=action (CA) [49, 50], in which complexity is proportional to the
gravitational action evaluated on a codimension-zero bulk region, called the
Wheeler DeWitt patch.

  * •

Complexity=volume $2.0$, in which complexity is proportional to the spacetime
volume of the Wheeler DeWitt patch [51].

Further generalizations have been investigated in [52, 53]. All these
holographic proposals reproduce the expected behavior of quantum complexity
for an AdS black hole: in the regime of bulk classical gravity, which should
be appropriate for times which are less than exponential in the entropy of the
system [54], complexity asymptotically grows linearly with time.

As studied in [18, 55], the definition of holographic complexity can be
naturally extended to static patch horizon holography in dS. Much differently
compared to AdS, holographic complexity in dS exhibits a hyperfast growth,
_i.e._ the complexity growth is so fast that it diverges at a finite critical
time. In [18], Susskind proposed the hyperfast growth to be a signature that
the Hamiltonian of the dual of dS is not of the usual $k$-local type. Here
$k$-local means that the Hamiltonian is the sum of terms that simultaneously
act at most on $k$ degrees of freedom, where $k$ is parametrically of order
unity in the limit of a large number of degrees of freedom.

It is tempting to relate the hyperfast growth of complexity in dS to the
exponential growth of spacetime. In order to further investigate the reason of
a different time dependence of holographic complexity compared to AdS, it is
interesting to contemplate intermediate situations. At the crossroad between
AdS and dS holography, it is natural to consider gravity backgrounds with an
asymptotically AdSD boundary which include dSD regions in their interior.
Examples in $D=2$ include the centaur geometry, which can be built in dilaton-
gravity theories [56, 57]. For centaurs, the dS part of the spacetime is not
hidden behind an AdS black hole horizon. Holographic volume complexity in
these backgrounds was recently studied in [58]. In this case, the evolution of
complexity is qualitatively different compared to both AdS and dS cases,
because the rate is a decreasing function of time and diverges to minus
infinity in a finite amount of time.

In this paper, we study volume complexity in higher-dimensional examples of
asymptotically AdS spacetimes with dS bubbles in their interior [59, 60]. This
kind of geometries can be realized, for instance, in an Einstein-scalar theory
where the potential has various minima separated by a domain wall. The $D\geq
3$ case significantly differs from the $D=2$ one, since in higher dimensions
the dS portion of the geometry is always screened by an AdS horizon at late
time. In order to simplify the model, it is useful to consider the limit in
which the thickness of the domain wall surrounding the dS interior is small
compared to the other physical scales, as studied in [61, 62, 63] for a flat
external space and in [59, 60] for an AdS external region. For simplicity, we
specialize to $D=3$ bulk spacetime dimensions and we consider spherically
symmetric geometries consisting of a dS region and an asymptotically AdS
spacetime separated by a domain wall with negligible thickness. We set the AdS
scale to one, and we parameterize the dS cosmological constant by $\lambda$
and the domain wall tension by $\kappa$. Outside the bubble, the solution is a
BTZ black hole [64] with mass proportional to $\mu$. While the parameters
$\lambda$ and $\kappa$ specify the theory, $\mu$ is a property of the state.

We focus on backgrounds which are invariant under time-reversal symmetry
$t\to-t$. For the critical value $\mu=\mu_{0}$, where

$\mu_{0}=\frac{\sqrt{(\kappa^{2}+\lambda-1)^{2}+4\lambda}-(\kappa^{2}+\lambda-1)}{2\lambda}\,,$
(1.1)

the only time-reversal invariant solution is the static bubble, see figure 1.
For $\mu>\mu_{0}$, the bubble starts from zero radius and expands to infinite
size, without entering the external true vacuum region. From the viewpoint of
an external observer, the bubble remains behind a black hole horizon. However,
no such bubble solution exists which enjoys time-reversal invariance. So, in
this paper we focus on the regime $\mu\leq\mu_{0}$.

Figure 1: Penrose diagrams for a static bubble. We show in red and in blue the
bubble trajectory in dS (left) and in AdS (center), respectively. Our geometry
(on the right) is obtained by joining the dS portion to the left of the red
line with the AdS region to the right of the blue line.

For a fixed $\mu<\mu_{0}$, there are two time-reversal invariant solutions:

  * •

A bubble collapsing in a finite proper time, for which the interior portion of
dS has finite spacetime volume. We refer to this case as ”small bubble”
solution. We have to further distinguish between two subcases. For
$0<\mu<{\mu}_{s}$, where

${\mu}_{s}=\frac{1}{\kappa^{2}+\lambda}<\mu_{0}\,,$ (1.2)

the dS bubble is initially on the same side of the Penrose diagram as the AdS
boundary. We refer to this configuration as a ”very small bubble”, see figure
2. For ${\mu}_{s}<\mu<\mu_{0}$ the bubble is initially on the opposite side of
the Penrose diagram with respect to the AdS boundary. We call this situation a
”not so small bubble”, see figure 3.

Figure 2: Penrose diagrams for a very small bubble $0<\mu<\mu_{s}$.

Figure 3: Penrose diagrams for a not so small bubble $\mu_{s}<\mu<\mu_{0}$.

  * •

A bubble expanding for an infinite proper time, which contains an infinite
portion of the dS spacetime. We refer to this case as ”large bubble” solution.
We here introduce

${\mu}_{h}=\frac{1-\kappa^{2}}{\lambda}<\mu_{0}\,.$ (1.3)

For $0<\mu<\mu_{h}$, the interior of the bubble contains as a subset a dS
static patch. We call this situation a ”very large bubble”, see figure 4. By
contrast, we refer to the solution with ${\mu}_{h}<\mu<\mu_{0}$ as a ”not so
large bubble”, see figure 5. Note that very large bubbles can only be obtained
for small enough domain wall tension, _i.e._ $\kappa<1$.

Figure 4: Penrose diagrams for a very large bubble $0<\mu<\mu_{h}$.

Figure 5: Penrose diagrams for a not so large bubble $\mu_{h}<\mu<\mu_{0}$.

In the parameter space, the static bubble configuration is at the border
between the small and the large bubble regimes.

The most straightforward way to apply the CV proposal is to consider the
volume of extremal surfaces which are anchored at the AdS boundary and which
are smooth both in the AdS and in the dS portions of the spacetime. This
prescription can be applied both for large and small bubbles. With the
exception of the static bubble configuration, we find that at late time
holographic complexity grows linearly with the same slope as for the BTZ black
hole. In the fine-tuned case of the static bubble, we find instead that volume
complexity is time-independent.

The time dependence of complexity that we obtain reveals that the hyperfast
growth is not necessarily related to the exponential growth of spacetime,
which is a feature of large bubbles. If we consider the prescription in which
smooth extremal surfaces are anchored just at the AdS boundary, we find that
they always remain in the static patches, without entering the regions behind
the dS horizon. This is the reason why the exponential growth of large bubbles
fails to be detected by volume complexity.

As proposed in [59], the boundary dual of a large bubble configuration should
be a a quantum field theory in a mixed state. This observation can be
justified as follows. For time-reversal symmetric bubbles, eq. (1.1) implies
that

$\lambda\mu\leq\lambda\mu_{0}\leq 1\,,$ (1.4)

which is equivalent to

$S_{\rm BH}\leq S_{\rm dS}\,,$ (1.5)

where $S_{\rm BH}$ is the entropy of the external AdS black hole and $S_{\rm
dS}$ is the entropy of the internal dS static patch $S_{\rm dS}$, _i.e._

$S_{\rm BH}=2\pi\sqrt{\mu}\,,\qquad S_{\rm dS}=\frac{2\pi}{\sqrt{\lambda}}\,.$
(1.6)

For large bubbles, the interior region contains a portion of the dS horizon,
so the number of degrees of freedom accessible from the AdS boundary is less
than the number of degrees of freedom of the internal dS region. We are then
forced to interpret large bubbles configurations in AdS/CFT as gravity duals
of a density matrix, obtained by tracing over a part of the degrees of freedom
of the dS region inside the bubble.

In the context of static patch horizon holography in dS [18, 19, 20], it is
natural to interpret the purification of the dual mixed state as a
generalization of thermofield double state [65] in which the CFT living at the
boundary of AdS and the quantum system living on the stretched dS horizon are
entangled. This suggests another natural way to apply the CV proposal in the
case of very large bubbles: we can anchor extremal surfaces both at the AdS
boundary and at the dS static patch stretched horizon. In this case, we find
that extremal surfaces cross the dS horizon and tend to bend towards the
future infinity. The complexity growth is hyperfast and diverges in finite
time, as in the dS case.

The paper is organized as follows. In section 2 we describe the bubble setup
and the thin wall approximation. In section 3 we present the equations of the
extremal surfaces both in AdS and in dS and we discuss the refraction law for
the extremal surface on the bubble. In section 4 we study volume complexity,
using the prescription in which the extremal surfaces are anchored just at the
AdS boundary and are smooth both in the AdS and in the dS regions of the
spacetime. In section 5 we study volume complexity for very large bubbles,
using extremal surfaces which extend between the AdS boundary and the dS
stretched horizon. We conclude in section 6. Several technical details are
presented in appendices.

##  2 Theoretical setup

We consider a spherically symmetric dS3 bubble inside an asymptotically AdS3
spacetime. The spacetime metric is taken as follows

$ds^{2}_{i,o}=(g_{i,o})_{\mu\nu}dx_{i,o}^{\mu}dx_{i,o}^{\nu}=-f_{i,o}(r)\,dt_{i}^{2}+\frac{dr^{2}}{f_{i,o}(r)}+r^{2}d\theta^{2}\,,$
(2.1)

where the subscripts $i$ and $o$ refer to the inside and outside regions,
respectively. The outside geometry is a BTZ black hole [64] with

$f_{o}(r)=r^{2}-\mu\,,$ (2.2)

where the mass of the black hole is proportional to $\mu$. For simplicity we
set the AdS length $L=1$. The inside geometry is a dS3 spacetime with radius
$r_{dS}=1/\sqrt{\lambda}$, namely

$f_{i}(r)=1-\lambda\,r^{2}\,.$ (2.3)

Introducing the tortoise coordinate

$r^{*}_{i,o}=\int\frac{d\tilde{r}}{f_{i,o}(\tilde{r})}\,,$ (2.4)

we can define the light-cone coordinates $v,u$ as follows

$v_{i,o}=t_{i,o}+r^{*}_{i,o}(r)\,,\qquad u_{i,o}=t_{i,o}-r^{*}_{i,o}(r)\,.$
(2.5)

An explicit evaluation of the integral (2.4) gives

$r^{*}_{o}(r)=\frac{1}{4\sqrt{\mu}}\log\left(\frac{r-\sqrt{\mu}}{r+\sqrt{\mu}}\right)^{2}\,,\qquad
r^{*}_{i}(r)=\frac{1}{4\sqrt{\lambda}}\log\left(\frac{1+r\sqrt{\lambda}}{1-r\sqrt{\lambda}}\right)^{2}\,,$
(2.6)

where the integration constants are chosen in such a way that
$r^{*}_{o}(\infty)=0$ and $r^{*}_{i}(0)=0$.

When considering spacetime regions nearby the black hole and the cosmological
horizons, it is convenient to write the metric in Eddington-Finkelstein (EF)
coordinates $\left(v_{i,o},r\right)$ or $\left(u_{i,o},r\right)$:

$\displaystyle ds^{2}_{i,o}$ $\displaystyle=$ $\displaystyle-
f_{i,o}(r)\,dv_{i,o}^{2}+2dr\,dv_{i,o}+r^{2}d\theta^{2}$ (2.7)
$\displaystyle=$ $\displaystyle-
f_{i,o}(r)\,du_{i,o}^{2}-2dr\,du_{i,o}+r^{2}d\theta^{2}\,.$

In order to describe the maximally extended versions of the dS3 and the BTZ
spacetimes, it is necessary to introduce two copies of the EF coordinates $u$
and $v$, which we denote by $u_{\rm L},v_{\rm L}$ and $u_{\rm R},v_{\rm R}$,
where ${\rm L}$ and ${\rm R}$ stand for left and right, respectively. Penrose
diagrams for our geometries with constant $u,v$ lines are shown in figure 6.
Our conventions for Penrose diagrams are discussed in appendix A.

Figure 6:  Penrose diagrams for dS3 spacetime (left) and for the BTZ black
hole (right). The arrows show the directions of increasing coordinate $t$ on
both the sides of the diagrams.

###  2.1 The domain wall

The inside and the outside geometries are patched together along a domain wall
with negligible thickness, whose trajectory on each side of the spacetime is
parametrized by

$r=R(\tau)\,,$ (2.8)

with $\tau$ the proper time measured on the domain wall itself. We will denote
by a dot $\dot{}$ the derivative with respect to the proper time $\tau$.

The equation of motion for $R(\tau)$ follows from Israel’s junction conditions
[66] (see [61, 59, 67, 60] for reviews), _i.e._ :

  * •

The metric must be continuous across the wall. The coordinate $r$ multiplies
the metric of the transverse sphere $S^{1}$, so it is continuous. Instead, the
coordinate $t_{i,o}$ is in general discontinuous in passing from inside to
outside.

  * •

The discontinuity in the extrinsic curvature $K_{ab}$ across the wall is fixed
by the energy-momentum tensor. For spherically symmetric geometries, it is
customary to introduce the quantities

$\beta_{i,o}=(K^{\theta}_{\theta})_{i,o}\,R\,.$ (2.9)

The jump between $\beta_{i}$ and $\beta_{o}$ is

$\beta_{i}-\beta_{o}=\kappa\,R\,,$ (2.10)

where in our case

$\beta_{i}=\pm\sqrt{\dot{R}^{2}+f_{i}(R)}\,,\qquad\beta_{o}=\pm\sqrt{\dot{R}^{2}+f_{o}(R)}\,,\qquad\kappa=8\pi
G\,\sigma\,.$ (2.11)

In eq. (2.11), $\sigma$ is the domain wall tension and $G$ is the Newton’s
constant. The sign of $\beta_{i,o}$ is positive if the coordinate $r$
increases as the wall is approached from the interior or as one moves away
from the wall in the exterior, and negative in the opposite situations. If
both $\beta_{i,o}$ have the same sign, $r$ is monotonic near the wall. If
$\beta_{i,o}$ have different signs, $r$ is locally extremised at the location
of the wall.

Squaring twice eq. (2.10), independently of the choices of signs in eq.
(2.11), the equation of motion can be expressed as

$\dot{R}^{2}+V(R)=0\,,$ (2.12)

where the second term, playing the role of an effective potential, is

$V(R)=f_{o}(R)-\frac{(f_{i}(R)-f_{o}(R)-\kappa^{2}\,R^{2})^{2}}{4\,\kappa^{2}\,R^{2}}\,.$
(2.13)

The equation of motion (2.12) can also be written in one of the following
equivalent ways:

$\beta_{i}=\frac{f_{i}(R)-f_{o}(R)+\kappa^{2}R^{2}}{2\kappa
R}\,,\qquad\beta_{o}=\frac{f_{i}(R)-f_{o}(R)-\kappa^{2}R^{2}}{2\kappa R}\,.$
(2.14)

These expressions fix the sign in eq. (2.11).

Specializing the general expression (2.13) to eqs. (2.3) and (2.2), we find

$V(R)=-A\,R^{2}+B-\frac{C}{R^{2}}\,,$ (2.15)

where

$A=\frac{(\lambda+\kappa^{2}-1)^{2}+4\lambda}{4\kappa^{2}}\,,\qquad
B=\frac{1+\kappa^{2}+\lambda+\mu-\kappa^{2}\mu+\lambda\mu}{2\kappa^{2}}\,,\qquad
C=\frac{(1+\mu)^{2}}{4\kappa^{2}}\,.$ (2.16)

Note that $A>0$ and $C>0$. It is also convenient to introduce

$\beta=\frac{B}{A}\,,\qquad\gamma=\frac{C}{A}\,.$ (2.17)

Depending on the values of parameters, we can have three physically different
situations:

  1. a)

The maximum of $V(R)$ is positive. In this case, the radius of the bubble as a
function of the bubble proper time $\tau$ has either a maximum or a minimum .

  2. b)

The maximum of $V(R)$ is exactly zero. In this case, we have an unstable
static bubble solution, besides other solutions which approach the maximum
from both sides with asymptotically zero velocity $\dot{R}$.

  3. c)

The maximum of $V(R)$ is negative. In this case, the radius of the bubble
monotonically contracts or expands without any maximum or minimum.

The qualitative behavior of $V(R)$ in these three cases is shown in figure 7.

$V$$V$$V$$R$$R$$R$a)b)c) Figure 7: Qualitative plots of the effective
potential $V(R)$ in eq. (2.15) with positive (plot a), vanishing (plot b) and
negative (plot c) maximum. Blue arrows represent the possible evolution of the
bubble radius $R(\tau)$.

Time-reversal invariance selects a potential $V(R)$ with non-negative maximum,
as in cases a) and b). This requires

$\beta^{2}\geq 4\gamma\,.$ (2.18)

For time-reversal symmetric bubbles, an explicit integration of eq. (2.12)
gives

$R(\tau)=\sqrt{\frac{\beta}{2}\mp\frac{\sqrt{\beta^{2}-4\gamma}}{2}\cosh(2\sqrt{A}\tau)}\,.$
(2.19)

The $-$ sign solution corresponds to a small bubble (the bubble has maximal
radius at $\tau=0$ and collapses in a finite proper time), whereas the $+$
sign solution describes a large bubble (the bubble has minimal radius at
$\tau=0$ and expands forever).

The condition in eq. (2.18) is equivalent to

$1+\mu(1-\kappa^{2}-\lambda)-\lambda\mu^{2}\geq 0\,,$ (2.20)

which is satisfied for $0<\mu\leq\mu_{0}$. The limiting value $\mu=\mu_{0}$
corresponds to a static bubble and is defined in eq. (1.1). Eq. (2.20) implies
that $\beta>0$. So, since we also have $\beta^{2}-4\gamma>0$ with $\gamma>0$,
$R(\tau=0)$ in eq. (2.19) is always a real number.

For a given choice of the parameters $\lambda,\mu,\kappa$ with $\mu<\mu_{0}$
it is possible to have both a contracting and an expanding solution. The
maximal radius for the contracting bubble is

$R_{\max}(\mu)=\sqrt{\frac{\beta}{2}-\frac{\sqrt{\beta^{2}-4\gamma}}{2}}\,,$
(2.21)

while the minimal radius for the expanding bubble is

$R_{\min}(\mu)=\sqrt{\frac{\beta}{2}+\frac{\sqrt{\beta^{2}-4\gamma}}{2}}\,.$
(2.22)

Both $R_{\rm max}$ and $R_{\rm min}$ are real positive numbers for
$0<\mu\leq\mu_{0}$. The values of $R_{\rm max}$ and $R_{\rm min}$ satisfy the
following constraints:

  * •

$R_{\rm max}\leq R_{\rm min}$, with the equality saturated for $\mu=\mu_{0}$.

  * •

$R_{\min}\leq 1/\sqrt{\lambda}$, with the equality saturated for the special
value $\mu=\mu_{h}$, see eq. (1.3).

  * •

$R_{\max}\geq\sqrt{\mu}$ with the equality saturated for the special value
$\mu=\mu_{s}$, see eq. (1.2).

In figure 8 we show a plot of $R_{\rm max}$ and $R_{\rm min}$ as functions of
$\mu$, for a fixed value of $\lambda$ and $\kappa$.

Figure 8:  Illustrative plot of $R_{\max}$ and $R_{\min}$ as functions of
$\mu$, for $\lambda=0.1$ and $\kappa=0.8$.

In order to specify the solution, we should also determine the time
coordinates $t_{i,o}$ on the surface of the bubble, both in the inside and in
the outside regions. Such time coordinates, which we denote by
$T_{i,o}(\tau)$, are specified by the equation

$\dot{T}^{2}_{i,o}=\frac{1}{f_{i,o}(R)}\left(1+\frac{\dot{R}^{2}}{f_{i,o}(R)}\right)\,,$
(2.23)

following from the normalization of the bubble velocity vector $w^{\alpha}$

$w^{\alpha}w_{\alpha}=-1\,,\qquad w^{\alpha}=(\dot{T},\dot{R},0)\,.$ (2.24)

Combining the equation of motion (2.12) with eq. (2.23) we find

${dT_{i,o}\over dR}=\pm{\sqrt{f_{i,o}(R)-V(R)}\over
f_{i,o}(R)\sqrt{-V(R)}}\,.$ (2.25)

Plugging the explicit expressions for $f_{i}$ and $f_{o}$ in eq. (2.25), we
get

$\displaystyle\frac{dT_{i}}{dR}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda-\kappa^{2})}{(1-\lambda
R^{2})\sqrt{AR^{4}-BR^{2}+C}}\,,$ $\displaystyle\frac{dT_{o}}{dR}$
$\displaystyle=$
$\displaystyle\pm\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda+\kappa^{2})}{(R^{2}-\mu)\sqrt{AR^{4}-BR^{2}+C}}\,.$
(2.26)

Time-reversal solutions to these equations are obtained by imposing the
boundary condition

$T_{i,o}(R_{\max})=0\,\qquad{\rm or}\qquad T_{i,o}(R_{\min})=0$ (2.27)

in the collapsing and the expanding case, respectively.

###  2.2 Static bubbles

A static bubble solution is realized for $\beta^{2}=4\gamma$, or equivalently
$\mu=\mu_{0}$, see eq. (1.1). The radius of the bubble is thus

$R_{\rm static}=R_{\min}(\mu_{0})=R_{\max}(\mu_{0})\,.$ (2.28)

The matching condition between the $T_{i,o}$ coordinates is

$\frac{dT_{i}}{dT_{o}}=\pm\sqrt{\frac{f_{o}(R_{\rm static})}{f_{i}(R_{\rm
static})}}=\pm\mu_{0}\,.$ (2.29)

The quantities in eq. (2.14) have the following signs

$\beta_{i}>0\,,\qquad\beta_{o}=-\mu_{0}\,\beta_{i}<0\,.$ (2.30)

Consequently, the radius $R$ is locally minimized on the surface of the
bubble. This is consistent with the Penrose diagram shown in figure 1.

###  2.3 Small bubbles

In this case, eq. (2.26) admits a smooth solution for $T_{i}$. On the other
hand, the trajectory is not smooth in the coordinate $T_{o}$, which should
then be replaced by an EF coordinate, see eq. (2.5). Referring to the Penrose
diagrams for the collapsing bubble sketched in figures 2 and 3, we distinguish
between two cases:

  * •

Very small bubble, $0<\mu<\mu_{s}$. The initial position of the bubble
$R=R_{\rm max}$ is on the same side of the AdS boundary. The smooth
coordinates that should be used are $v_{\rm R}$ for the black hole (or $u_{\rm
R}$ for the white hole). Denoting by $U_{{\rm R},o}$ and $V_{{\rm R},o}$ the
right EF coordinates on the bubble surface in BTZ spacetime, the equations of
motion are

$\displaystyle\frac{dV_{{\rm R},o}}{dR}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda+\kappa^{2})}{(R^{2}-\mu)\sqrt{AR^{4}-BR^{2}+C}}+\frac{1}{R^{2}-\mu}\,,$
$\displaystyle\frac{dU_{{\rm R},o}}{dR}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda+\kappa^{2})}{(R^{2}-\mu)\sqrt{AR^{4}-BR^{2}+C}}-\frac{1}{R^{2}-\mu}\,.$
(2.31)

  * •

Not so small bubble, $\mu_{s}<\mu<\mu_{0}$. In this case we should use $u_{\rm
L}$ for the black hole (or $v_{\rm L}$ for the white hole). Denoting by
$U_{{\rm L},o}$ and $V_{{\rm L},o}$ the left EF coordinates on the bubble
surface in BTZ spacetime, the equations of motion are

$\displaystyle\frac{dU_{{\rm L},o}}{dR}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda+\kappa^{2})}{(R^{2}-\mu)\sqrt{AR^{4}-BR^{2}+C}}-\frac{1}{R^{2}-\mu}\,,$
$\displaystyle\frac{dV_{{\rm L},o}}{dR}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda+\kappa^{2})}{(R^{2}-\mu)\sqrt{AR^{4}-BR^{2}+C}}+\frac{1}{R^{2}-\mu}\,.$
(2.32)

###  2.4 Large bubbles

Large bubbles are examples of bags of gold [68, 60], which are defined as
spacetimes in which an eternal black hole exterior is attached by an Einstein-
Rosen bridge to an interior which is a portion of Friedman-Lemaitre-Robertson-
Walker (FLRW) cosmology with an infinite spacetime volume. In these
geometries, the entropy of the interior can exceed the exterior Bekenstein-
Hawking entropy, see eq. (1.5), so the bulk states cannot be put in
correspondence with the CFT states of the dual field theory. It has been
suggested that the Bekenstein-Hawking entropy does not in general count all
the states inside the black hole, but only those which are distinguishable
from the outside [69, 68]. In particular, large bubble solutions has been
proposed to provide the holographic dual of a density matrix [59].

Equation (2.26) for the outside time $T_{o}$ gives a smooth solution, while
$T_{i}$ is singular because the dS horizon is crossed at some time by the
bubble. We then pass to EF coordinates, see eq. (2.5). With reference to the
Penrose diagrams for the expanding bubble sketched in figures 4 and 5, we
discriminate between two cases:

  * •

Very large bubble, $0<\mu<\mu_{h}$. The initial position of the bubble
$R=R_{\rm min}$ is in the right static patch. Denoting by $U_{{\rm R},i}$ and
$V_{{\rm R},i}$ the right EF coordinates on the bubble surface in dS, the
equations of motion are

$\displaystyle\frac{dV_{{\rm R},i}}{dR}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda-\kappa^{2})}{(1-\lambda
R^{2})\sqrt{AR^{4}-BR^{2}+C}}+\frac{1}{1-\lambda R^{2}}\,,$
$\displaystyle\frac{dU_{{\rm R},i}}{dR}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda-\kappa^{2})}{(1-\lambda
R^{2})\sqrt{AR^{4}-BR^{2}+C}}-\frac{1}{1-\lambda R^{2}}\,.$ (2.33)

  * •

Not so large bubble, $\mu_{h}<\mu<\mu_{0}$. The initial position of the bubble
$R=R_{\rm min}$ is in the left static patch. Denoting by $U_{{\rm L},i}$ and
$V_{{\rm L},i}$ the left EF coordinates on the bubble surface in dS, the
equations of motion are

$\displaystyle\frac{dU_{{\rm L},i}}{dR}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda-\kappa^{2})}{(1-\lambda
R^{2})\sqrt{AR^{4}-BR^{2}+C}}-\frac{1}{1-\lambda R^{2}}\,,$
$\displaystyle\frac{dV_{{\rm L},i}}{dR}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\kappa}\frac{1+\mu-R^{2}(1+\lambda-\kappa^{2})}{(1-\lambda
R^{2})\sqrt{AR^{4}-BR^{2}+C}}+\frac{1}{1-\lambda R^{2}}\,.$ (2.34)

##  3 Volume functional

According to the CV conjecture [48], complexity of the boundary state is
proportional to the volume of a maximal codimension-one surface anchored at
the given boundary time. The volume complexity $C_{V}$ is usually normalized
as

$C_{V}=\frac{\mathcal{V}}{GL}\,,$ (3.1)

where $\mathcal{V}$ is the volume of the maximal slice, $G$ the Newton’s
constant and $L$ the AdS radius. In this section we discuss the volume of
extremal surfaces in both the AdS and the dS parts of the geometry. Then, we
address the matching condition on the domain wall.

Due to spherical symmetry, the extremal surface can be parameterized as

$r=r(l)\,,\qquad t=t(l)\,,$ (3.2)

where $l$ is a single-valued coordinate along the surface. Since the
coordinate $t$ is singular nearby the horizons, it is useful to write the
volume functional in both the versions of the EF coordinates $u$ and $v$:

$\displaystyle\mathcal{V}_{i,o}$ $\displaystyle=$ $\displaystyle
2\pi\int\mathcal{L}\,dl\,,$ $\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle
r\,\sqrt{-f_{i,o}(v^{\prime}_{i,o})^{2}+2r^{\prime}v^{\prime}_{i,o}}=r\,\sqrt{-f_{i,o}(u^{\prime}_{i,o})^{2}-2r^{\prime}u^{\prime}_{i,o}}\,,$
(3.3)

where ′ denotes a derivative with respect to $l$. The extremal surface is
characterized by a conserved quantity $P_{i,o}$, which in the $(v,r)$ and in
the $(u,r)$ coordinates reads

$\displaystyle P_{i,o}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
v^{\prime}_{i,o}}=\frac{r\,(-f_{i,o}v^{\prime}_{i,o}+r^{\prime})}{\sqrt{-f_{i,o}(v^{\prime}_{i,o})^{2}+2r^{\prime}v^{\prime}_{i,o}}}\,,$
$\displaystyle P_{i,o}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
u^{\prime}_{i,o}}=\frac{r\,(-f_{i,o}u^{\prime}_{i,o}-r^{\prime})}{\sqrt{-f_{i,o}(u^{\prime}_{i,o})^{2}-2r^{\prime}u^{\prime}_{i,o}}}\,,$
(3.4)

respectively.

The volume functional in eq. (3.3) is invariant under reparameterization in
$l$. To fix this gauge freedom, it is convenient to impose the conditions

$\sqrt{-f_{i,o}(v^{\prime}_{i,o})^{2}+2r^{\prime}v^{\prime}_{i,o}}=r\qquad{\rm
or}\qquad\sqrt{-f_{i,o}(u^{\prime}_{i,o})^{2}-2r^{\prime}u^{\prime}_{i,o}}=r\,,$
(3.5)

in such a way that the volume functional becomes

$\mathcal{V}_{i,o}=2\pi\int r^{2}\,dl\,.$ (3.6)

With this gauge choice, the conserved quantity $P_{i,o}$ can be expressed as
follows

$\displaystyle P_{i,o}$ $\displaystyle=$ $\displaystyle-
f_{i,o}v^{\prime}_{i,o}+r^{\prime}\,,\qquad
v^{\prime}_{i,o}=\frac{r^{\prime}-P_{i,o}}{f_{i,o}}\,,$ $\displaystyle
P_{i,o}$ $\displaystyle=$ $\displaystyle-
f_{i,o}u^{\prime}_{i,o}-r^{\prime}\,,\qquad
u^{\prime}_{i,o}=\frac{-r^{\prime}-P_{i,o}}{f_{i,o}}\,.$ (3.7)

By inserting this back into the gauge constraint, we get

$\displaystyle(r^{\prime})^{2}+U_{i,o}(r)=P_{i,o}^{2}\,,\qquad
U_{i,o}(r)=-f_{i,o}(r)\,r^{2}\,,$ $\displaystyle
r^{\prime}=\pm\sqrt{P_{i,o}^{2}+f_{i,o}(r)\,r^{2}}\,,$ (3.8)

which is valid in both the EF coordinate systems $(v,r)$ and $(u,r)$. The
conserved quantity $P_{i,o}$ can also be written as

$P_{i,o}=-f_{i,o}\,t^{\prime}_{i,o}\,.$ (3.9)

Note that $U_{i,o}(r)$ in eq. (3.8) can be interpreted as effective
potentials, see figure 9 for qualitative plots.

Figure 9:  Effective potential for the extremal surface in the dS region
(left) and in the AdS one (right).

The extremum of $r(l)$ represents a turning point $r_{t}$ for the extremal
surface. We can have a turning point both in the external BTZ region and in
the internal dS one:

  * •

In the external asymptotically AdS region, $r_{t,{\rm AdS}}$ is a minimum of
$r(l)$, given by

$r_{t,{\rm AdS}}=\sqrt{\frac{\mu+\sqrt{\mu^{2}-4P_{o}^{2}}}{2}}\,.$ (3.10)

For a turning point to exist, it is necessary to require

$P_{o}^{2}\leq{P}_{\rm max}^{2}=\frac{\mu^{2}}{4}\,.$ (3.11)

  * •

In the internal dS region, $r_{t,{\rm dS}}$ is a maximum of $r(l)$, given by

$r_{t,{\rm dS}}=\sqrt{\frac{1+\sqrt{1+4\lambda\,P_{i}^{2}}}{2\lambda}}\,.$
(3.12)

The difference in the time coordinate $t$ between two points on the extremal
surface can be expressed as

$t_{i,o}(r_{2})-t_{i,o}(r_{1})=\mp\int^{r_{2}}_{r_{1}}\frac{P_{i,o}}{f_{i,o}\,\sqrt{P_{i,o}^{2}+f_{i,o}(r)\,r^{2}}}dr\,,$
(3.13)

where the $-$ sign should be chosen for a parameterization with
$r^{\prime}(l)>0$, while the $+$ sign for a parameterization with
$r^{\prime}(l)<0$. In the integral in eq. (3.13), the Cauchy principal value
prescription should be used when crossing the horizon.

With the convention $r_{2}>r_{1}$, the volume of the extremal surface reads

$\mathcal{V}_{i,o}=2\pi\int_{r_{1}}^{r_{2}}\,\frac{r^{2}}{\sqrt{P_{i,o}^{2}+f_{i,o}(r)\,r^{2}}}\,dr\,.$
(3.14)

###  3.1 A refraction law for the extremal surface

To determine the codimension-one extremal surfaces, we solve eqs. (3.8) and
(3.7) both in the interior and in the exterior of the bubble. Then, we match
the two solutions on top of the domain wall, imposing that the total volume is
extremal. Physically, the extremal surface is somehow ”refracted” by the
domain wall. In appendix B, by introducing a coordinate system which describes
both the interior and the exterior of the bubble in terms of the same time
coordinate $t_{i}$, we derive the refraction condition in the thin wall
approximation.

By spherical symmetry, it is not restrictive to focus just on the time and
radial coordinates. We denote by $x^{\mu}_{i,o}(l)$ the coordinates of the
extremal codimension-one surface and by $X^{\mu}_{i,o}(\tau)$ the trajectory
of the domain wall. We then introduce the tangent vector to the extremal
surface $\frac{dx^{\mu}_{i,o}}{dl}$ and the velocity vector of the domain wall
$\frac{dX^{\mu}_{i,o}}{d\tau}$, namely

$\frac{dx^{\mu}_{i,o}}{dl}=(t^{\prime}_{i,o}(l),r^{\prime}_{i,o}(l))\,,\qquad\frac{dX^{\mu}_{i,o}}{d\tau}=(\dot{T}_{i,o}(\tau),\dot{R}(\tau))\,.$
(3.15)

The matching condition on top of the domain wall is

$(g_{i})_{\mu\nu}\frac{dx^{\mu}_{i}}{dl}\frac{dX^{\nu}_{i}}{d\tau}=(g_{o})_{\mu\nu}\frac{dx^{\mu}_{o}}{dl}\frac{dX^{\nu}_{o}}{d\tau}\,.$
(3.16)

The details of the derivation are in appendix B. A similar result was derived
in [70] for geodesics.

Given an extremal surface intersecting the domain wall at some value of the
radial coordinate $R$, we denote by $\rho_{i,o}(R)$ the value of
$r^{\prime}_{i,o}(l)$ computed at the intersection, _i.e._

$\rho_{i,o}(R)=r^{\prime}_{i,o}(l_{0})\,\qquad{\rm where}\qquad
r_{i,o}(l_{0})=R\,.$ (3.17)

By means of eq. (3.9), we can write the matching condition (3.16) as

$P_{i}\frac{d{T}_{i}}{dR}+\frac{\rho_{i}(R)}{f_{i}(R)}=P_{o}\frac{d{T}_{o}}{dR}+\frac{\rho_{o}(R)}{f_{o}(R)}\,.$
(3.18)

The setup with an extremal surface crossing a null shell of matter with
negligible thickness was studied in [71] for geodesics and in [72, 73] for
codimension-one surfaces. This case formally corresponds to a domain wall
moving at the speed of light and the result is consistent with eq. (3.18).

##  4 Complexity from smooth extremal surfaces

The most natural way to apply the CV conjecture in asymptotically AdS
geometries with an internal dS bubble is to consider extremal surfaces which
are anchored at some given time $t_{b}$ at the AdS boundary and which are
smooth in the interior. By convention, we choose $l$ in eq. (3.2) to be
positive and to vanish at the center of the dS static patch.

In order to avoid a singularity at $r=0$ in the dS interior, we must impose
the condition $P_{i}=0$. This can be checked as follows. Using eqs. (3.9) and
(3.8), the induced metric on the extremal surface in the gauge (3.5) is

$d\tilde{s}_{i}^{2}=r^{2}(l)\left(dl^{2}+d\theta^{2}\right)\,.$ (4.1)

From eq. (3.8), $r(l)$ can have two qualitatively different behaviors nearby
$l\to 0$:

  * •

For $P_{i}=0$, $r(l)\approx r^{\prime}(l)$, so the induced metric

$d\tilde{s}_{i}^{2}\approx dr^{2}+r^{2}d\theta^{2}$

is smooth at $l\to 0$, with $0\leq\theta\leq 2\pi$.

  * •

For $P_{i}\neq 0$, $r^{2}(l)\approx P_{i}^{2}l^{2}$, so the induced metric

$d\tilde{s}_{i}^{2}\approx P_{i}^{2}(l^{2}dl^{2}+l^{2}d\theta^{2})$

has a singular scalar curvature at $l\to 0$.

A similar property holds in the case of AdS Vaidya spacetime, see [72].

Equation (3.9) implies that for $P_{i}=0$ the extremal surface in the interior
lies at constant $t_{i}$ coordinate.

###  4.1 Small bubbles

For small bubbles, the inside geometry is just a region of the left static
patch. Then, there is no turning point in the dS portion of the geometry, or
equivalently $r_{i}^{\prime}(l)>0$. As a direct consequence, we have
$\rho_{i}(R)>0$ everywhere on the domain wall. With the condition $P_{i}=0$,
the refraction law in eq. (3.18), which must be implemented at the bubble
surface $r=R(\tau)$, is

$\frac{R}{\sqrt{f_{i}(R)}}=\frac{\rho_{o}(R)}{f_{o}(R)}+\frac{dT_{o}}{dR}P_{o}\,,\qquad
P_{o}=\pm\sqrt{\rho_{o}^{2}(R)-f_{o}(R)\,R^{2}}\,.$ (4.2)

Depending on the sign of the product $\frac{dT_{o}}{dR}P_{o}$, two physically
different solutions to the constraint (4.2) exist. Let us fix for convenience
the sign in eq. (2.26) as follows

$\frac{dT_{o}}{dR}=-\frac{1}{2\kappa}\frac{w_{o}(R)}{(R^{2}-\mu)\sqrt{AR^{4}-BR^{2}+C}}\,,\qquad
w_{o}(R)=1+\mu-R^{2}(1+\lambda+\kappa^{2})\,.$ (4.3)

Note that111This can be checked by the following properties: a) the unique
positive solution of the equation $w_{o}(R_{\rm max})=0$ in the variable $\mu$
is $\mu=\mu_{s}$; b) for $\mu=0$, $w_{o}(R_{\rm max})>0$. the quantity
$w_{o}(R_{\rm max})$ is positive for $0<\mu<\mu_{s}$ and negative for
$\mu_{s}<\mu<\mu_{0}$. In both cases, the sign choice in eq. (4.3) is such
that, for $T_{o}\geq 0$, the bubble always moves towards the upper direction
of the Penrose diagram. The configuration in which the bubble moves in the
opposite direction can be recovered by a time reflection $t\to-t$.

With the assumption of positive $P_{o}$, eq. (4.2) has the following solution

$\displaystyle\rho_{o}(R)=-\frac{R\left(\mu-1+R^{2}\left(\kappa^{2}+\lambda-1\right)\right)}{2\sqrt{1-\lambda
R^{2}}}\,.$ (4.4)

From a direct calculation, we can check that $\rho_{o}(R_{\rm max})$ vanishes
just for $\mu=\mu_{s}$. Also, we have that $\rho_{o}(R_{\rm max})>0$ for
$\mu\to 0$. This shows that $\rho_{o}(R_{\rm max})$ is positive for
$0<\mu<\mu_{s}$ and negative for $\mu_{s}<\mu<\mu_{0}$. In both cases, this
sign is consistent with a refraction of the extremal surface through the
domain wall. For negative $P_{o}$, the solution to eq. (4.2) corresponds to a
”reflection” of the extremal surface, see appendix C.1. Thus, we discard this
solution.

Plugging eq. (4.4) into eq. (4.2), we get

$P_{o}^{2}(R)=R^{2}(\mu-R^{2})+\frac{1}{4}R^{2}\frac{\left(\mu-1+R^{2}\left(\kappa^{2}+\lambda-1\right)\right)^{2}}{1-\lambda
R^{2}}\,,$ (4.5)

see figure 10 for a plot. Note that $P_{o}^{2}(0)=P_{o}^{2}(R_{\rm max})=0$.

Figure 10:  Behavior of $P_{o}^{2}(R)$ in eq. (4.5) for small bubbles.

Using the fact that $R<1/\sqrt{\lambda}$ for small bubbles, we obtain the
following inequality

$P_{o}^{2}(R)\geq U_{o}(R)=R^{2}(\mu-R^{2})\,,$ (4.6)

where $U_{o}(R)$ is the effective potential in eq. (3.8). The maximum of
$U_{o}(R)$ is at $R=\hat{R}=\sqrt{\mu/2}$ and its value is
$U_{o}(\hat{R})=\mu^{2}/4=P_{\rm max}^{2}$, defined in eq. (3.11). Then, the
maximum of $P_{o}^{2}(R)$ is bigger than $P_{\rm max}^{2}$, as it is clear
from figure 10. Let us define the radii $R_{\pm}$ as the two solutions to the
equation

$P_{o}^{2}(R_{\pm})=P_{\rm max}^{2}=\frac{\mu^{2}}{4}\,.$ (4.7)

From the property $P_{o}^{2}(\sqrt{\mu/2})>\mu^{2}/4$, it follows that

$R_{-}\leq\sqrt{\frac{\mu}{2}}\leq R_{+}\,.$ (4.8)

As we will see, the radii $R_{\pm}$ determine the behavior of the complexity
rate at large time $t_{b}$.

####  4.1.1 Complexity rate

Depending on the parameter values and on $R$, the quantity $\rho_{o}(R)$ in
eq. (4.4) can be either positive or negative. A detailed analysis is deferred
to appendix C.2. Based on the sign of $\rho_{o}(R)$, the details of the
extremal surface are slightly different. Let us distinguish between the two
situations:

  * •

$\rho_{o}(R)>0$. The radial coordinate of the extremal surface is monotonic
between $r=R$ and the AdS UV cutoff $r=\Lambda$. From eq. (3.14), the volume
is

$\frac{\mathcal{V}}{2\pi}=\int_{0}^{R}\,\frac{r}{\sqrt{f_{i}(r)}}\,dr+\int_{R}^{\Lambda}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr\,.$
(4.9)

According to eq. (3.13), the boundary time is

$t_{b}=T_{o}(R)-\int^{\Lambda}_{R}\frac{1}{f_{o}}\left(\frac{P_{o}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\right)dr\,.$
(4.10)

In order to find the complexity rate, we use the same strategy as in [77].
Namely, we sum and subtract the quantity $P_{o}t_{b}$ to the volume expressed
in eq. (4.9):

$\displaystyle\frac{\mathcal{V}}{2\pi}$ $\displaystyle=$
$\displaystyle\int_{0}^{R}\,\frac{r}{\sqrt{f_{i}(r)}}\,dr+\int_{R}^{\Lambda}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr+P_{o}t_{b}-P_{o}t_{b}$
(4.11) $\displaystyle=$ $\displaystyle
P_{o}t_{b}+\int_{0}^{R}\,\frac{r}{\sqrt{f_{i}(r)}}\,dr+\int_{R}^{\Lambda}\,\frac{\sqrt{P_{o}^{2}+f_{o}r^{2}}}{f_{o}}\,dr-
P_{o}T_{o}(R)\,.$

The derivative of the volume with respect to the boundary time $t_{b}$ is then

$\displaystyle W=\frac{1}{2\pi}\frac{d\,\mathcal{V}}{d\,t_{b}}$
$\displaystyle=$ $\displaystyle
P_{o}+\frac{d\,P_{o}}{d\,t_{b}}\left(t_{b}+\int_{R}^{\Lambda}\,\frac{P_{o}}{f_{o}\sqrt{P_{o}^{2}+f_{o}r^{2}}}\,dr-
T_{o}(R)\right)$ (4.12)
$\displaystyle+\frac{dR}{dt_{b}}\left(\frac{R}{\sqrt{f_{i}(R)}}-\,\frac{\sqrt{P_{o}^{2}+f_{o}(R)R^{2}}}{f_{o}(R)}-P_{o}\frac{d\,T_{o}(R)}{d\,R}\right)$
$\displaystyle=$ $\displaystyle
P_{o}+\frac{dR}{dt_{b}}\left(\frac{R}{\sqrt{f_{i}(R)}}-\,\frac{\rho_{o}(R)}{f_{o}(R)}-P_{o}\frac{d\,T_{o}(R)}{d\,R}\right)=P_{o}\,,$

where in the last step use has been made of eq. (4.2).

The asymptotic linear growth corresponds to the $\mathcal{V}\to\infty$ and
$t_{b}\to\infty$ limit, which formally comes from the divergence of the
integrands in eqs. (4.9) and (4.10) in correspondence of the turning point
$r_{t,\rm{AdS}}$, see eq. (3.10). In particular, the turning point
$r_{t,\rm{AdS}}$ satisfies

$P_{o}^{2}+f_{o}\,r^{2}=0\,.$

This singularity is integrable except for $P_{o}^{2}\to P_{\rm
max}^{2}=\mu^{2}/4$. In order to find a divergent $t_{b}$ in eq. (4.10), we
need the turning point $r_{t,\rm{AdS}}=\sqrt{\mu/2}$ to lie inside the
integration domain $[R,\infty]$. Thus, from eq. (4.8), we find that $t_{b}$ is
regular for $R\to R_{+}$ and diverges for $R\to R_{-}$. In other words, the
late time limit $t_{b}\to\infty$ corresponds to $R\to R_{-}$.

  * •

$\rho_{o}(R)<0$. For the extremal surfaces to be attached to the AdS boundary
there must be a turning point $r_{t,{\rm AdS}}<R$ inside the black hole. From
eq. (3.14), the volume is

$\frac{\mathcal{V}}{2\pi}=\int_{0}^{R}\,\frac{r}{\sqrt{f_{i}(r)}}\,dr-\int_{R}^{r_{t,{\rm
AdS}}}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr+\int_{r_{t,{\rm
AdS}}}^{\Lambda}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr\,.$
(4.13)

According to eq. (3.13), the boundary time reads

$t_{b}=T_{o}(R)+\int_{R}^{r_{t,{\rm
AdS}}}\frac{1}{f_{o}}\left(\frac{P_{o}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\right)dr-\int^{\Lambda}_{r_{t,{\rm
AdS}}}\frac{1}{f_{o}}\left(\frac{P_{o}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\right)dr\,.$
(4.14)

The rate $W$ can be evaluated by the same strategy as in eq. (4.12). The
result is again

$W=\frac{1}{2\pi}\frac{d\,\mathcal{V}}{d\,t_{b}}=P_{o}\,.$ (4.15)

As in the previous case, the asymptotic linear growth of complexity is in
correspondence of the divergence of the integrands in eq. (4.14) at
$r=r_{t,\rm{AdS}}$. The singularity is generally integrable, except for
$P_{o}=P_{\rm max}$, for which $r_{t,\rm{AdS}}=\sqrt{\mu/2}$. In this case,
the late $t_{b}$ limit corresponds to $R\to R_{+}$. For $0<R<R_{+}$, instead,
the extremal surface never reaches the AdS boundary.

Extremal surfaces can be found numerically by integrating the equations of
motion (3.7) and (3.8). Two prototypical solutions are plotted in the Penrose
diagrams in figures 11 and 12.

Figure 11:  Left panel: AdS portion of the extremal surfaces for a very small
bubble. Right panel: plot of the complexity rate. For every value of the
boundary time $t_{b}$ there exists a single extremal surface.

Figure 12:  Left panel: AdS portion of the extremal surfaces for a not so
small bubble. Right panel: plot of the complexity rate. Note that multiple
extremal surfaces exist for boundary times $t_{b}$ inside the window in eq.
(4.16).

In the example in figure 11, for a given value of the boundary time $t_{b}$
there exists a single extremal surface anchored at the AdS boundary. Instead,
in the example in figure 12 in a time window centered at $t_{b}=0$, _i.e._

$-T_{b}<t_{b}<T_{b}\,,$ (4.16)

three extremal surfaces anchored at the same value of the boundary time
$t_{b}$ exist. The rate $W$ is then a multivalued function of the time
$t_{b}$.

To discriminate between the two possible behaviors, it is useful to introduce
the quantity

$K=\left.\frac{dt_{b}}{dP_{o}}\right|_{t_{b}=0}\,,$ (4.17)

representing the reciprocal of the slope $\frac{dW}{dt_{b}}$ computed at the
origin of the plots in the right panel of figures 11 and 12. The complexity
rate in figure 11 is characterized by $K>0$, contrary to the one in figure 12,
where $K<0$. An explicit expression for $K$ is given in appendix C.3. From
eqs. (C.8) and (C.11), we find that $K$ is a decreasing function of $\mu$ at
fixed $\lambda$ and $\kappa$ and that $K\to-\infty$ for $\mu\to\mu_{0}$. In
the left panel of figure 13, we show $K$ as a function of $\mu$ for a fixed
value of $\lambda$ and $\kappa$. In the parameter region

$\lambda\geq 2+\kappa-\kappa^{2}\,,$ (4.18)

the quantity $K$ is always negative, so multiple extremal surfaces exist for
every $\mu$, see the right panel of figure 13.

Figure 13:  Left panel: plot of the quantity $K$ defined in eq. (4.17) as a
function of $\mu$ for $\lambda=1$, $\kappa=0.5$. In this numerical example $K$
is negative for $\mu>0.53$ and approaches $-\infty$ for $\mu\to\mu_{0}\approx
0.88$. In order to have multiple extremal surface for the same value of
$t_{b}$ as in figure 12, we need $K<0$. The blue portion of the plot
corresponds to very small bubbles, while the red part to not so small bubbles.
Right panel: the white region represents the portion of the $(\lambda,\kappa)$
plane defined in eq. (4.18), for which $K$ is negative for every value of
$\mu$.

According to the CV conjecture, when multiple extremal surfaces exist, the one
with maximal volume should be picked. To deal with this, let us consider the
prototypical result in the left panel of figure 14. The three points $C$, $D$,
and $E$ are obtained by extremal surfaces anchored at the same boundary time,
but whose volumes $\mathcal{V}$ and values of the rate differ. To properly
choose the maximal solution, let us first consider the two points $C$ and $D$.
The volumes $\mathcal{V}_{C}$ and $\mathcal{V}_{D}$ of the corresponding
surfaces can be inferred by starting from the volume $\mathcal{V}_{A}$ of the
surface located at the point $A$, and following the two arrows which bring to
$C$ and $D$, respectively. Since the rate is positive, the volume increases as
we move along these arrows. Hence, both $\mathcal{V}_{C}$ and
$\mathcal{V}_{D}$ are higher than $\mathcal{V}_{A}$. Precisely, since the
growth rate as we move from $A$ to $C$ is larger than the one going from $A$
to $D$, we conclude that $\mathcal{V}_{C}>\mathcal{V}_{D}$. Moreover, by time-
reversal symmetry, $\mathcal{V}_{C}=\mathcal{V}_{F}$. Moving along the curve
from $E$ to $F$ the rate is negative, which means that the volume is
decreasing: $\mathcal{V}_{F}<\mathcal{V}_{E}$. From this argument, we should
choose the lowest value of the rate for $t_{b}<0$ and the highest value of the
rate for $t_{b}>0$, see the right panel of figure 14. Consequently, the
complexity rate $W$ experiences a discontinuous jump at $t_{b}=0$. A similar
analysis was performed in [55] for complexity in dS spacetime.

Figure 14: Left panel: schematic plot of the complexity rate in the parameter
region where there are multiple extremal surfaces anchored at the same
boundary time. Right panel: the requirement of maximal volume selects the
step-like function rate represented by the solid line.

####  4.1.2 Late time complexity rate

We now discuss the late time behavior of the rate $W$. In terms of the bubble
radius, we will show that $t_{b}\to\infty$ corresponds to $R\to R_{+}$ or to
$R\to R_{-}$, depending on the point in the parameter space. First note that
$\rho_{o}(R_{+})$ has the same sign as $\rho_{o}(R_{-})$. In fact, if the
signs of $\rho_{o}(R_{\pm})$ were different, we would have
$R_{-}<R_{0}<R_{+}$, where $R_{0}$ is defined by the condition
$\rho_{o}(R_{0})=0$, see eq. (C.3). Then we should have
$P^{2}_{o}(R_{0})>\frac{\mu^{2}}{4}$, see figure 10, but this is impossible
because

$P^{2}_{o}(R_{0})-\frac{\mu^{2}}{4}=-\frac{\left(\kappa^{2}\mu+\lambda\mu+\mu-2\right)^{2}}{4\left(\kappa^{2}+\lambda-1\right)^{2}}\leq
0\,.$ (4.19)

We can then check that the $t_{b}\to\infty$ limit corresponds to $R\to R_{+}$
or to $R\to R_{-}$. In particular, we can have

  * •

Case 1: If $\rho_{o}(R_{+})<0$, the $t_{b}\to\infty$ limit corresponds to
$R\to R_{+}$. In this case both the integrals in eq. (4.14) diverge, due to a
singularity in correspondence of $r_{t,{\rm AdS}}$. For $t_{b}>0$ the
complexity rate is an increasing function of time.

  * •

Case 2: If $\rho_{o}(R_{+})>0$, then also $\rho_{o}(R_{-})>0$. From eq.
(3.10), both for $R=R_{+}$ and $R=R_{-}$, we have that $r_{t,{\rm
AdS}}=\sqrt{\mu/2}$, because $P^{2}_{o}=\mu^{2}/4$. The integral which gives
$t_{b}$, see eq. (4.10), is then divergent for $R=R_{-}$ and not for
$R=R_{+}$, because $R_{-}\leq r_{t,{\rm AdS}}\leq R_{+}$, see eq. (4.8). In
this case, at large enough $t_{b}>0$ the complexity rate is a decreasing
function of time.

Either way, the complexity rate at late time is insensitive to the presence of
the bubble

$\frac{1}{2\pi}\lim_{t_{b}\to\infty}\frac{d\,\mathcal{V}}{d\,t_{b}}=\frac{\mu}{2}\,.$
(4.20)

However, the sign of $\rho_{o}(R_{+})$ discriminates between a growing or a
decreasing complexity rate at late time. It is a complicated problem to
discuss the sign of $\rho_{o}(R_{+})$ in a generic point of the parameter
space. Still, in some particular regions of parameters we can determine the
sign of $\rho_{o}(R_{+})$ with some simple arguments. For example, for $\mu\to
0$ we are always in case 2, because in this limit

$R_{+}\to R_{\rm
max}\,,\qquad\rho(R_{+})\to\frac{1}{(\kappa+1)^{2}+\lambda}>0\,.$ (4.21)

Instead, depending on the value of $\lambda$ and $\kappa$, for $\mu\to\mu_{0}$
we can have both case 1 (for example $\lambda=0.5$, $\kappa=0.5$) and case 2
(for example $\lambda=1$, $\kappa=1$). See appendix C.2 for further details.

According to the Lloyd’s bound [74], the maximum allowed growth rate of
quantum computational complexity should be proportional to the total energy.
In quantum systems with holographic duals, it was proposed [50] that the
Lloyd’s bound is saturated at late time by the uncharged planar black hole
solutions in AdS. In the parameter space portion of case 2 this version of the
proposal is violated, because the asymptotic value is approached from above.
Violations of the Lloyd’s bound have been previously found for the CA
conjecture, see [75, 76, 77, 78, 79]. In holographic models including just AdS
boundaries, we do not know about any other violation of the bound for the CV
proposal. See [81, 82] for examples of AdS hairy black holes in which the
bound is instead satisfied.

###  4.2 Large bubbles

Contrary to small bubbles, for large bubbles the dS part of the geometry
contains the region beyond the cosmological horizon $r=1/\sqrt{\lambda}$.
However, recalling that $P_{i}=0$, we point out that extremal codimension-one
surfaces in the dS part cannot enter the region with $r>1/\sqrt{\lambda}$,
where the potential $U_{i}$ is positive, see figure 9. Therefore, the extremal
surfaces are confined into the static patches. In particular, for
$0<\mu<\mu_{h}$ the extremal surfaces extend into the right static patch.
Since in this case the domain wall never enter the left static patch, we
necessarily have $\rho_{i}(R)<0$. On the other hand, for $\mu>\mu_{h}$ no
portion of the right static patch is present in the geometry, so the extremal
surfaces remain into the left static patch. From this argument we conclude
that $\rho_{i}(R)>0$.

Some technical details in the calculations slightly differ in the two cases:

  * •

Very large bubble, $0<\mu<\mu_{h}$. Equation (4.2), which determines $P_{o}$,
is replaced by

$-\frac{R}{\sqrt{f_{i}}}=\frac{1}{f_{o}}\rho_{o}+\frac{dT_{o}}{dR}P_{o}\,,\qquad
P_{o}=\pm\sqrt{\rho_{o}^{2}(R)-f_{o}(R)\,R^{2}}\,.$ (4.22)

We fix the sign of eq. (2.26) as in eq. (4.3). The solution for positive
$P_{o}$ is

$\displaystyle\rho_{o}=\frac{R\left(\mu-1+R^{2}\left(\kappa^{2}+\lambda-1\right)\right)}{2\sqrt{1-\lambda
R^{2}}}\,.$ (4.23)

From here, we obtain the same value of $P_{o}$ as in eq. (4.5). It can be
checked that $\rho_{o}(R_{\rm min})$ in eq. (4.23) is always negative. Also,
for fixed $\lambda$, $\kappa$, and $\mu$, $\rho_{o}(R)$ is a decreasing
function of $R$. Therefore, $\rho_{o}(R)$ is always negative, which means that
there is a turning point $r_{t,{\rm AdS}}<R$ in AdS. Note that
$\rho_{o}(R)\to-\infty$ for $R\to 1/\sqrt{\lambda}$.

The extremal surface always crosses the dS horizon on the bifurcation sphere,
because $P_{i}=0$. Consequently, the volume is given by

$\displaystyle{\mathcal{V}\over 2\pi}$ $\displaystyle=$
$\displaystyle\int_{0}^{\frac{1}{\sqrt{\lambda}}}{r\over\sqrt{f_{i}(r)}}dr-\int^{R}_{\frac{1}{\sqrt{\lambda}}}{r\over\sqrt{f_{i}(r)}}dr$
(4.24) $\displaystyle-\int^{r_{t,{\rm
AdS}}}_{R}{r^{2}\over\sqrt{P_{o}^{2}+f_{o}(r)r^{2}}}dr+\int_{r_{t,{\rm
AdS}}}^{\Lambda}{r^{2}\over\sqrt{P_{o}^{2}+f_{o}(r)r^{2}}}dr\,.$

The boundary time $t_{b}$ can be obtained by eq. (4.14). Using the same
methods as in eqs. (4.11) and (4.12), we again find that

$W={1\over 2\pi}{d\mathcal{V}\over dt_{b}}=P_{o}\,,$ (4.25)

where eq. (4.22) has been used instead of eq. (4.2).

  * •

Not so large bubble, $\mu_{h}<\mu<\mu_{0}$. Equation (4.2) still holds. We fix
the sign of eq. (2.26) as in eq. (4.3), in which $w_{o}(R)$ is always
negative.222 As $w_{o}(R)$ is a decreasing function of $R$, it is enough to
check this property for $R=R_{\rm min}$, see eq. (C.14). This choice
corresponds to a bubble which, for $T_{o}\geq 0$, moves in the lower direction
of the Penrose diagram. Assuming $P_{o}$ to be positive,333If $P_{o}$ is
negative, $\rho_{o}$ is given by eq. (C.1). This solution corresponds to a
reflection, so we discard it. the $\rho_{o}$ solution is given by eq. (4.4).

The quantity $\rho_{o}(R)$ changes sign at $R=R_{0}$, given in eq. (C.3).
However, for $\mu>\mu_{h}$ we have $R_{0}<R_{\rm min}$. In addition, in the
present case the equation $\rho_{o}(R_{\rm min})=0$ has no solution, so
$\rho_{o}(R_{\rm min})$ is always negative, in accordance with the refraction
interpretation. Putting all together, we conclude that $\rho_{o}(R)$ remains
negative in the range $R_{\rm min}<R<1/\sqrt{\lambda}$. In particular, for
$R\to 1/\sqrt{\lambda}$ we get $\rho_{o}\to-\infty$.

Due to the negativity of $\rho_{o}(R)$, there is a turning point at $r_{t,{\rm
AdS}}<R$ in the AdS portion. We can then use eq. (4.13) for the volume, eq.
(4.14) for the boundary time, and eq. (4.15) for the complexity rate. The
value of $P_{o}^{2}$ is still given by eq. (4.5). From a direct calculation

$P_{o}(R_{\rm min})=0\,,\qquad P_{o}(1/\sqrt{\lambda})=\infty\,.$ (4.26)

Moreover, for $R_{\rm min}<R<1/\sqrt{\lambda}$, $P_{o}$ is a monotonic
function, a plot of which is displayed in figure 15. The unique solution
$R^{*}$ to the equation

$P_{o}^{2}(R^{*})=P_{\rm max}^{2}=\mu^{2}/4\,,$ (4.27)

with $R_{\rm min}<R^{*}<1/\sqrt{\lambda}$, corresponds to the extremal surface
at $t_{b}\to\infty$.

Figure 15:  Behavior of $P_{o}^{2}(R)$ in eq. (4.5) for large bubbles.

In figure 16 and 17 we show the extremal surfaces and the complexity rate for
a very large and a not so large bubble, respectively. In these examples, a
single extremal surface exists for any given boundary time $t_{b}$ and the
complexity rate is an increasing function of time.

Figure 16:  Left panel: AdS portion of the extremal surfaces for an example of
very large bubble. Right panel: plot of the complexity rate.

Figure 17:  Left panel: AdS portion of the extremal surfaces for an example of
not so large bubble. Right panel: plot of the complexity rate.

In order to check if multiple extremal surfaces can emerge for a given
$t_{b}$, we look at the behavior of $K$, expressed in eq. (C.12). A plot of
$K$ as a function of $\mu$ for fixed $\lambda$ and $\kappa$ is shown in figure
18. From the discussion below eq. (C.13), we find that for large bubbles $K$
is always positive, thus there are no multiple extremal surfaces attached at
the same boundary time.

The complexity rate at large $t_{b}$ is still given by eq. (4.20). The rate is
always an increasing function of $t_{b}$ at late time, because $P_{o}^{2}(R)$
is always an increasing function of $R$ in the range $R_{\rm min}\leq R\leq
R^{*}$, see figure 15. Referring to subsection 4.1.2, the late time rate of
complexity behaves as in case 1 of small bubbles.

Figure 18:  Plot of $K$, defined in eq. (4.17), as a function of $\mu$ for
$\lambda=1$ and $\kappa=0.5$ in the case of large bubbles.

###  4.3 Static bubble limit

For the strictly static bubble configuration $\mu=\mu_{0}$, eq. (B.13) tells
us that $P_{o}=0$, so the complexity rate identically vanishes. In this setup,
the time-translation symmetry is not broken by the presence of the bubble and
$\partial/\partial t$ is an everywhere well-defined Killing vector. Also, the
extremal surfaces never enter the black and white hole regions of the
geometry.

Figure 19:  Left panel: AdS portion of the extremal surfaces for a not so
small bubble. The value of $\mu$ is rather close to the static bubble
configuration, which, with the chosen values of $\lambda$ and $\kappa$, is
realized for $\mu_{0}\approx 0.88278$. Right panel: plot of the the complexity
rate.

Figure 20:  Left panel: AdS portion of the extremal surfaces for an example of
not so large bubbles with a $\mu$ rather close to the static bubble value
$\mu_{0}\approx 0.88278$. Right panel: plot of the the complexity rate.

Let us now comment on how the static bubble is realized as a limit of the
small and large bubbles:

  * •

As $\mu$ approaches the critical value $\mu_{0}$ from the small bubble
direction, the time $T_{b}$ in eq. (4.16) tends to $+\infty$, and the slope of
the complexity rate in the central unstable branch tends to zero, see figure
19. This is consistent with $K\to-\infty$ for $\mu\to\mu_{0}$, as shown in the
left panel of figure 13. In the static bubble limit $\mu\to\mu_{0}$, just the
central branch solution survives and the complexity rate is zero.
Interestingly, this limit is discontinuous, since the central branch is
discarded by the maximum volume prescription, as indicated by the dashed line
in the right panel of figure 14.

  * •

As $\mu$ approaches the static bubble value $\mu_{0}$ from the large bubble
direction, at small $t_{b}$ the complexity rate grows very slowly, see figure
20. When $\mu\to\mu_{0}$, the complexity rate remains frozen at zero for a
large time. This is consistent with $K$ diverging to $+\infty$ in this limit,
as shown in figure 18.

###  4.4 Complexity of formation

While it is possible to look at the collapse of a very small bubble from the
outside of the black hole, for all the other kinds of bubble an external
observer just see a black hole horizon. It is thus natural to ask whether
complexity can help us to discriminate between a large and small bubble state
with the same $\mu$. To look for an answer, we consider the complexity of
formation [80] at $t_{b}=0$, at which time $P_{i}=P_{o}=0$. To get a finite
quantity, we subtract from the volume of the bubble $\mathcal{V}_{\rm large}$
(or $\mathcal{V}_{\rm small}$) at $t_{b}=0$ the outside volume of the BTZ
black hole at the same boundary time:

$\mathcal{V}_{\rm
BTZ}=2\pi\int_{\sqrt{\mu}}^{\Lambda}\frac{r}{\sqrt{f_{o}}}dr=2\pi\,\sqrt{\Lambda^{2}-\mu}=2\pi\,\Lambda+O\left(\frac{1}{\Lambda}\right)\,,$
(4.28)

where $\Lambda$ is the AdS UV cutoff. The complexity of formation is thus
proportional to

$\Delta\mathcal{V}_{\rm large}=\mathcal{V}_{\rm large}-\mathcal{V}_{\rm
BTZ}\,,\qquad\Delta\mathcal{V}_{\rm small}=\mathcal{V}_{\rm
small}-\mathcal{V}_{\rm BTZ}\,.$ (4.29)

A direct evaluation from eqs. (4.9), (4.13), and (4.24) gives

$\displaystyle\frac{\Delta\mathcal{V}_{\rm small}}{2\pi}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}\frac{1-\sqrt{1-R_{\rm
max}^{2}\lambda}}{\lambda}-\sqrt{R_{\rm max}^{2}-\mu}\qquad 0<\mu<\mu_{s}\\\
\frac{1-\sqrt{1-R_{\rm max}^{2}\lambda}}{\lambda}+\sqrt{R_{\rm
max}^{2}-\mu}\qquad\mu_{s}<\mu<\mu_{0}\end{array}\right.\,,$ (4.32)
$\displaystyle\frac{\Delta\mathcal{V}_{\rm large}}{2\pi}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}\sqrt{R_{\rm
min}^{2}-\mu}+\frac{1+\sqrt{1-R_{\rm min}^{2}\lambda}}{\lambda}\qquad
0<\mu<\mu_{h}\\\ \sqrt{R_{\rm min}^{2}-\mu}+\frac{1-\sqrt{1-R_{\rm
min}^{2}\lambda}}{\lambda}\qquad\mu_{h}<\mu<\mu_{0}\end{array}\right.\,.$
(4.35)

Figure 21:  Plot of $\Delta\mathcal{V}_{\rm small}$ and
$\Delta\mathcal{V}_{\rm large}$ in eq. (4.35) as functions of $\mu$ for
$\lambda=1$ and $\kappa=0.5$.

A plot of $\Delta\mathcal{V}_{\rm large}$ and $\Delta\mathcal{V}_{\rm small}$
as function of $\mu$ is shown in figure 21. The large bubble has always a
larger complexity than the small one. This confirms the intuition that a large
bubble state is more complex compared to a small bubble one with the same
mass. Also, $\Delta\mathcal{V}_{\rm small}$ is negative for $\mu\to 0$. Very
small bubbles in the limit $\mu\to 0$ then correspond to less complex states
compared to the BTZ eternal black hole.

##  5 Complexity with a dS stretched horizon

For a very large bubble with $0<\mu<\mu_{h}$, we can consider a generalization
of the thermofield double state with both an AdS and a dS boundary. This is a
hybrid case between the standard AdS thermofield double state [65] and the dS
generalization [18, 58, 55]. As in [18], the dS boundary is defined as a
stretched dS horizon located at constant $r={r_{\rm sh}}$, where

$r_{\rm sh}=\frac{1}{\sqrt{\lambda}}\,\left(1-\epsilon\right)\,,$ (5.1)

and $\epsilon\to 0$ is the horizon cutoff. An $\epsilon>0$ is necessary to
define a notion of time on the dS boundary, as the horizon is a null
hypersurface for $\epsilon=0$.

The presence of two boundaries suggests an alternative way to apply the CV
conjecture, by computing the volume of an extremal surface anchored both at
the dS and at the AdS boundaries. We choose the parameter $l$ in eq. (3.2) in
such a way that it is positive and that $r(l=0)=r_{\rm sh}$. The dS boundary
removes the conical singularity presented by extremal surfaces at the center
of the static patch for $P_{i}\neq 0$. For this reason, we are allowed to
consider an arbitrary value of $P_{i}$.

Extremal surfaces are stretched between the dS horizon and the AdS boundary,
so we can define the left and the right boundary times $t_{L}$, $t_{R}$ as the
boundary conditions

$\displaystyle t_{L}=t_{i}(l=0)\,,\qquad{\rm{where}}\qquad r(l=0)=r_{\rm sh}$
$\displaystyle t_{R}=t_{o}(l=l_{\Lambda})\,,\qquad{\rm{where}}\qquad
r(l=l_{\Lambda})=\Lambda$ (5.2)

and $\Lambda$ is the AdS UV cutoff. In the definition of thermofield double
state, we consider an arbitrary linear relation between $t_{L}$ and $t_{R}$

$t_{L}=\alpha_{t}\,t_{R}\,,$ (5.3)

where $\alpha_{t}$ is some opportune numerical constant. By convention, we
define the boundary time $t_{b}$ as

$t_{b}=-t_{L}=-\alpha_{t}\,t_{R}\,.$ (5.4)

As a technical difference compared to the usual thermofield double state,
there is no time-translation Killing vector $\partial_{t}$ which is globally
defined in all the geometry, because time invariance is broken by the
trajectory of the bubble. If such a symmetry existed, it would provide an
appropriate value of $\alpha_{t}$ which would give a zero complexity rate. We
then expect a non-trivial time dependence of complexity for every value of
$\alpha_{t}$.

###  5.1 Extremal surfaces

In the AdS part of the geometry, the domain wall is located into the left
exterior of the black hole. Thus, for the extremal surface to cross the black
or white hole interior, we must require $\rho_{o}(R)<0$. Instead, the sign of
$\rho_{i}(R)$ might be positive or negative. Explicitly,

$\rho_{o}(R)=-\sqrt{P_{o}^{2}+f_{o}(R)\,R^{2}}\,,\qquad\rho_{i}(R)=\pm\sqrt{P_{i}^{2}+f_{i}(R)\,R^{2}}\,.$
(5.5)

The condition for the extremal surface to reach the AdS boundary without
falling into the black hole singularity is $-\mu/2\leq P_{o}\leq\mu/2$ (see
the plot of the potential in the right panel of figure 9). By time-reflection
invariance, we take $P_{o}=P_{i}=0$ at $t_{b}=0$, when the extremal surface
meets the domain wall at $R=R_{\rm min}$.

Under these assumptions, the matching condition in eq. (3.18) reads

$P_{i}\frac{d{T}_{i}}{dR}\pm\frac{\sqrt{P_{i}^{2}+f_{i}(R)\,R^{2}}}{f_{i}(R)}=P_{o}\frac{d{T}_{o}}{dR}-\frac{\sqrt{P_{o}^{2}+f_{o}(R)\,R^{2}}}{f_{o}(R)}\,,$
(5.6)

where the $\pm$ sign corresponds to $\rho_{i}(R)>0$ or $\rho_{i}(R)<0$,
respectively. We choose the sign of $dT_{o}/dR$ as in eq. (4.3), and the sign
of $dT_{i}/dR$ in eq. (2.26) as follows

$\frac{dT_{i}}{dR}=-\frac{1}{2\kappa}\frac{w_{i}(R)}{(1-\lambda
R^{2})\sqrt{AR^{4}-BR^{2}+C}}\,,\qquad
w_{i}(R)=1+\mu-R^{2}(1+\lambda-\kappa^{2})\,.$ (5.7)

As a function of $\mu$, the quantity $w_{i}(R_{\rm min})$ is positive for
$\mu>\mu_{h}$ and negative for $\mu<\mu_{h}$. For $\mu=\mu_{h}$, $w_{i}(R_{\rm
min})$ vanishes and the bubble initially sits exactly at the dS bifurcation
point.

Equation (5.6) is solved by

$P_{i}=\frac{P_{o}(w_{i}\,w_{o}\pm\xi)+\sqrt{\xi}(w_{i}\pm
w_{o})\sqrt{P_{o}^{2}+f_{o}R^{2}}}{4\kappa^{2}R^{2}f_{o}}\,,\qquad\xi=-4\kappa^{2}\,V(R)\,R^{2}\,,$
(5.8)

where $V(R)$ is given in eq. (2.15). We point out that the solution with
$P_{i}=0$ reproduces the smooth extremal surfaces studied in section 4. The
$-$ sign describes an extremal surface experiencing a refraction, see eq.
(4.5), while the $+$ sign denotes an extremal surfaces undergoing a
reflection, see eq. (C.2). Specializing to the physical refracted solution,
eq. (5.8) can be further simplified to

$P_{i}=\frac{1}{2f_{o}}\left(P_{o}\,(1-\mu+R^{2}(1-\lambda-\kappa^{2}))+{\sqrt{\xi}\sqrt{P_{o}^{2}+f_{o}R^{2}}}\right)\,.$
(5.9)

An extremal surface is specified by the conserved quantities $P_{i}$ and
$P_{o}$, which are related by eq. (5.9). Their values for given boundary times
are fixed by the thermofield double boundary condition in eq. (5.3), as we
will explain below.

For the extremal surface to reach the AdS boundary, the turning point defined
in eq. (3.12) must be at $r_{t,{\rm dS}}\geq R$, which holds true for

$P_{i}^{2}\geq\tilde{P}_{i}^{2}\,,\qquad\tilde{P}_{i}^{2}=R^{2}(\lambda\,R^{2}-1)\,.$
(5.10)

For $R_{\rm min}<R<1/\sqrt{\lambda}$, the domain wall is located on the right
side of the dS static patch, so $\rho_{i}(R)<0$. We may expect that at some
point $R=\tilde{R}>1/\sqrt{\lambda}$, whose location depends on the bubble
parameters and on the choice of $\alpha_{t}$, the function $\rho_{i}(R)$
vanishes. Then, $\rho_{i}(R)>0$ for $R>\tilde{R}$.

The condition $\rho_{i}(\tilde{R})=0$ is equivalent to
$P_{i}^{2}=\tilde{P}_{i}^{2}$, see eq. (5.10). In terms of $P_{o}$, this gives

$P_{o}=\tilde{P}_{o}=\frac{R(\mu-1+R^{2}(\lambda+\kappa^{2}-1))}{2\sqrt{\lambda
R^{2}-1}}\,,$ (5.11)

which is well defined just for $R>1/\sqrt{\lambda}$. Note that
$\tilde{P}_{o}\to-\infty$ for $R\to 1/\sqrt{\lambda}$. For
$R>1/\sqrt{\lambda}$ and $P_{o}>\tilde{P}_{o}$ we have $\rho_{i}(R)<0$, while
for $P_{o}<\tilde{P}_{o}$ we have $\rho_{i}(R)>0$.

###  5.2 Complexity

For the calculation of volume and boundary time, let us distinguish between
two cases:

  * •

$\rho_{i}(R)<0$.  The volume is given by

$\displaystyle\frac{\mathcal{V}}{2\pi}$ $\displaystyle=$
$\displaystyle\int_{r_{sh}}^{r_{t,\rm
dS}}\,\frac{r^{2}}{\sqrt{P_{i}^{2}+f_{i}(r)\,r^{2}}}\,dr-\int^{R}_{r_{t,\rm
dS}}\,\frac{r^{2}}{\sqrt{P_{i}^{2}+f_{i}(r)\,r^{2}}}\,dr$ (5.12)
$\displaystyle-\int_{R}^{r_{t,\rm{AdS}}}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr+\int_{r_{t,\rm{AdS}}}^{\Lambda}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr\,.$

From eq. (3.13), we can write the following expression for the AdS boundary
time

$t_{R}=T_{o}(R)+\int_{R}^{r_{t,\rm{AdS}}}\frac{P_{o}}{f_{o}\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}dr-\int^{\Lambda}_{r_{t,\rm{AdS}}}\frac{P_{o}}{f_{o}\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}dr\,,$
(5.13)

and the following expression for the dS boundary time

$t_{L}=T_{i}(R)+\int_{{r_{\rm sh}}}^{r_{t,{\rm
dS}}}\frac{P_{i}}{f_{i}\sqrt{P_{i}^{2}+f_{i}(r)\,r^{2}}}dr-\int^{R}_{r_{t,{\rm
dS}}}\frac{P_{i}}{f_{i}\sqrt{P_{i}^{2}+f_{i}(r)\,r^{2}}}dr\,.$ (5.14)

  * •

$\rho_{i}(R)>0$.  The volume can be written as

$\frac{\mathcal{V}}{2\pi}=\int_{r_{\rm
sh}}^{R}\,\frac{r^{2}}{\sqrt{P_{i}^{2}+f_{i}(r)\,r^{2}}}\,dr-\int_{R}^{r_{t,\rm{AdS}}}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr+\int_{r_{t,\rm{AdS}}}^{\Lambda}\,\frac{r^{2}}{\sqrt{P_{o}^{2}+f_{o}(r)\,r^{2}}}\,dr\,.$
(5.15)

Equation (5.13) is still valid. Instead, from eq. (3.13) we find that the time
on the dS boundary is

$t_{L}=T_{i}(R)+\int_{{r_{sh}}}^{R}\frac{P_{i}}{f_{i}\sqrt{P_{i}^{2}+f_{i}(r)\,r^{2}}}dr\,.$
(5.16)

In order to find values of $P_{i}$ and $P_{o}$ consistent with the thermofield
double boundary condition, we need to solve eq. (5.3) where $t_{R}$ and
$t_{L}$ are specified by (5.13) and (5.14) or (5.16), respectively. This
integral equation can be solved numerically by the shooting method.

We can then compute the extremal surfaces and their volume numerically. In
figure 22 we show the result for $\alpha_{t}=1$, while in figure 23 we display
the result for the choice $\alpha_{t}=-1$. In both cases we find a hyperfast
complexity rate, because the volume of the extremal surface diverges for a
finite value of the boundary time $t_{b}$.

Figure 22:  Upper left and right panels: extremal surfaces in the dS and in
the AdS portion of the Penrose diagram, respectively. Lower panel: plot of the
volume as a function of boundary time $t_{b}$. We set the thermofield double
parameter to $\alpha_{t}=1$, see eq. (5.3), and we choose $\kappa=0.2$,
$\lambda=1.5$, $\mu=0.4$, and $\epsilon=0.06$.

Figure 23:  Plots of the same quantities as in figure 22, with the choice
$\alpha_{t}=-1$.

###  5.3 Critical time behaviour

Let us now discuss the limit of hyperfast complexity growth. Expanding eq.
(5.9) for large $R$, we find that

$P_{i}(R)\approx\frac{\sqrt{(\lambda+(\kappa-1)^{2})(\lambda+(\kappa+1)^{2})}}{2}R^{2}+O(R^{0})\,.$
(5.17)

The dependence of $P_{i}$ on $P_{o}$ shows up just at order $O(R^{0})$. At
large $R$, we can approximate $T_{i}(R)$ in eq. (2.26) as

$\frac{dT_{i}}{dR}\approx\frac{1}{f_{i}}\frac{1+\lambda-\kappa^{2}}{\sqrt{(\lambda+\kappa^{2}-1)^{2}+4\lambda}}\,,\qquad
T_{i}(R)\approx\frac{1+\lambda-\kappa^{2}}{\sqrt{(\lambda+\kappa^{2}-1)^{2}+4\lambda}}r^{*}_{i}(R)+Q\,,$
(5.18)

where $Q$ is an integration constant. Note that $1+\lambda-\kappa^{2}>0$,
because $\kappa<1$ in the very large bubble regime.

Let us discriminate between two cases:

  * •

$\lambda+\kappa^{2}>1$. At large $R$, from eq. (5.11) we have
$\tilde{P}_{o}\to+\infty$, then $\rho_{i}(R)>0$. At large $P_{i}$, we can
approximate eq. (5.16) as

$\displaystyle t_{L}\approx T_{i}(R)+\int_{{r_{\rm
sh}}}^{R}\frac{dr}{f_{i}}=T_{i}(R)+r_{i}^{*}(R)-r_{i}^{*}(r_{sh})\,,$ (5.19)

which gives

$t_{L}\approx\omega\frac{1}{R\lambda}+Q-\frac{1}{4\sqrt{\lambda}}\log\frac{4}{\epsilon^{2}}\,,\qquad\omega=1+\frac{1+\lambda-\kappa^{2}}{\sqrt{(\lambda+\kappa^{2}-1)^{2}+4\lambda}}\,.$
(5.20)

In the above expression, we have $\omega>0$ because for very large bubbles
$\kappa<1$. For large $R$, the dS part of the volume in eq. (5.15) diverges
linearly:

$\frac{\mathcal{V}_{dS}}{2\pi}=\int_{r_{\rm
sh}}^{R}\,\frac{r^{2}}{\sqrt{P_{i}^{2}+f_{i}(r)\,r^{2}}}\,dr\,\approx
R\int_{\frac{r_{\rm
sh}}{R}}^{1}\,\frac{y^{2}}{\sqrt{\frac{(\lambda+(\kappa-1)^{2})(\lambda+(\kappa+1)^{2})}{4}-\lambda
y^{4}}}\,dy\,,$ (5.21)

in which the argument of the square root in the integrand denominator is
always positive, since

$\frac{(\lambda+(\kappa-1)^{2})(\lambda+(\kappa+1)^{2})}{4}-\lambda=\frac{(\lambda+\kappa^{2}-1)^{2}}{4}\,.$

The divergence of the volume for large $R$ is at the finite value of $t_{L}$
given in eq. (5.20).

  * •

$\lambda+\kappa^{2}<1$. At large $R$ we have $\tilde{P}_{o}\to-\infty$, which
gives $\rho_{i}(R)<0$. In this limit, combining eq. (3.12) with eq. (5.17), we
find

$r_{t,{\rm
dS}}\approx\frac{R}{\sqrt{2}}\left(\frac{(\lambda+(\kappa-1)^{2})(\lambda+(\kappa+1)^{2})}{\lambda}\right)^{1/4}\,.$
(5.22)

At large $R$, corresponding to large $P_{i}$, we can approximate $t_{L}$ in
eq. (5.14) as

$t_{L}\approx T_{i}(R)+\int_{{r_{\rm sh}}}^{r_{t,{\rm
dS}}}\frac{dr}{f_{i}}+\int_{R}^{r_{t,{\rm dS}}}\frac{dr}{f_{i}}\,,$ (5.23)

which gives

$t_{L}\approx\tilde{\omega}\frac{1}{\lambda
R}+Q-\frac{1}{4\sqrt{\lambda}}\log\frac{4}{\epsilon^{2}}\,,$ (5.24)

where

$\tilde{\omega}=\frac{1+\lambda-\kappa^{2}}{\sqrt{(\lambda+\kappa^{2}-1)^{2}+4\lambda}}-1+\frac{2\sqrt{2}\lambda^{1/4}}{[(\lambda+(\kappa-1)^{2})(\lambda+(\kappa+1)^{2})]^{1/4}}\,.$
(5.25)

In the case at hand, the quantity $\tilde{\omega}$ is always positive. At
large $R$ and at the finite time in eq. (5.24), the volume again diverges
linearly as in eq. (5.21).

Summarizing, in both cases the volume complexity diverges for $t_{b}=t_{\rm
cr}$, where the critical time is

$t_{\rm cr}=\frac{1}{4\sqrt{\lambda}}\log\frac{4}{\epsilon^{2}}-Q\,,$ (5.26)

and $Q$ is the integration constant defined in eq. (5.18). The divergence at
$t_{b}=t_{\rm cr}$ takes place independently of the thermofield double
parameter $\alpha_{t}$ in eq. (5.3). For $t_{b}>t_{\rm cr}$, no extremal
surface exists which connects the left dS and right AdS boundaries. As in
[55], we may regularize the divergence in the complexity rate by introducing a
cutoff surface at large $r=r_{\rm cut}$ nearby the future dS infinity. With
this regularization, the complexity rate would remain finite and at $t>t_{\rm
cr}$ it would saturate at a value divergent in the UV cutoff $r_{\rm cut}$.

##  6 Conclusions

In this paper we investigated the time dependence of volume complexity in a
class of asymptotically AdS3 spacetimes which include a dS3 bubble in their
interior. We first focused on extremal surfaces attached just at the AdS
boundary and smooth everywhere into the interior spacetime. With the exception
of the static bubble configuration, we found that complexity asymptotically
grows linearly as a function of time, with the same rate as for the BTZ black
hole. For large bubbles, the asymptotic value of the complexity rate is always
reached from below. For small bubbles, the asymptotic limit can be instead
reached either from below (case 1) or from above (case 2), depending on the
parameter space (see section 4.1.2). In [50] the Lloyd’s bound [74], which is
the conjectured upper limit to the complexity rate, has been proposed to be
saturated at late time by the uncharged planar black hole solution in AdS. The
region of parameter space in case 2 then violates this version of the bound.
As far as we know, this is the first violation of the bound by the CV proposal
in holographic models including just AdS boundaries. For the CA conjecture,
similar violations have already been discussed in [77, 78].

The static bubble configuration gives rise to a time-independent complexity,
so it does not match the expectation, generically satisfied by AdS black
holes, that complexity rate at late time is of the same order of magnitude as
$TS$, with $T$ the temperature and $S$ the entropy of the system [27]. This
fine-tuned solution, which is realized for $\mu=\mu_{0}$, interpolates between
the small and the large bubble regimes. As soon as a small perturbation is
introduced, the static bubble limit can be achieved in two different ways:

  * •

If the limit $\mu\to\mu_{0}$ in the parameter space is approached from the
large bubble configuration, the complexity rate remains frozen to zero for an
initial amount of time which tends to infinity for $\mu\to\mu_{0}$, see the
right panel of figure 20. From this side, the asymptotic complexity rate of
the BTZ black hole is recovered in a continuous way after an arbitrarily large
time.

  * •

If the limit $\mu\to\mu_{0}$ is approached from the small bubble region of the
parameter space, the static behavior of complexity emerges from a class of
extremal surfaces with non-maximal volume, see figure 19. According to the CV
prescription, the non-maximal extremal solutions should be discarded for
$\mu\neq\mu_{0}$ in favor of the global maximum, as shown by the dashed line
in the right panel of figure 14. For $\mu\to\mu_{0}$ the complexity rate
resembles a step function, thus the limit is discontinuous, as in a first
order phase transition.

Interestingly, the discarded solutions would give rise to a negative
complexity rate, as for the two-dimensional centaur geometries studied in
[58]. We may contemplate the possibility that other physical configurations
could exist, which resemble our geometry nearby the discarded extremal
surfaces and in which the global maximum is cut away. If such a surgery were
performed, it would give rise to a negative complexity rate, resembling the
two-dimensional case studied in [58], in which no black hole horizon is
present in the AdS region of the geometry.

With both procedures, for $\mu\to\mu_{0}$ the BTZ asymptotic complexity rate
is recovered at late time. The limit is continuous from the large bubble
direction, while it is discontinuous from the small bubble side. In this
sense, the static bubble can be seen as a fine-tuned critical configuration
separating qualitatively different behaviors in the parameter space.

From the point of view of local observables on the AdS boundary, the static
bubble configuration is not distinguishable from the eternal BTZ black hole.
The same holds for entanglement entropy of subregions located on the AdS
boundary. To detect the discontinuous nature of the static bubble limit, other
quantum information probes, such as holographic complexity, are required.

Contrary to the result found in [18, 55] for dS, if we consider smooth
extremal surfaces attached just to the AdS boundary there is no hyperfast
growth of complexity. In section 5, we checked that hyperfast growth is
recovered in the very large bubble case if we consider extremal surfaces
anchored both at the AdS boundary and at the dS stretched horizon. This choice
should correspond to a thermofield double state which involves both an AdS and
a dS boundary. Defining the boundary time $t_{b}$ as in eq. (5.4), we find
that the volume complexity diverges at $t_{b}=t_{\rm cr}$ given in eq. (5.26),
independently of the thermofield double state parameter $\alpha_{t}$ in eq.
(5.3).

It is tempting to suggest the following interpretations for the two different
ways to apply the CV conjecture in AdS geometries with very large dS bubbles:

  * •

the volume of smooth extremal surfaces anchored just at the boundary of AdS is
proportional to the complexity of a mixed CFT state, obtained by tracing over
the dS degrees of freedom in the thermofield double state. This situation is
reminiscent of the subregion complexity proposal [83, 84, 85].

  * •

the volume of the extremal surface anchored both at the AdS boundary and at
the dS static patch horizon is proportional to the complexity of the pure
product thermofield double state.

Why the latter choice of boundary conditions is possible just for very large
bubbles is an interesting question. Small bubble solutions do not contain a
stretched horizon in the dS portion of the geometry, so the only implementable
prescription is to attach the extremal surfaces just to the AdS boundary. In
this case, we can conjecture that the dual CFT state does not arise from a
partial trace over a thermofield double state involving a dS boundary.

## Acknowledgements

We thank S. Baiguera for useful discussions and comments on the draft.

## Appendix

##  Appendix A Details about Penrose diagrams

In this appendix we describe our conventions for Penrose diagrams. Using the
lightcone coordinates $(u,v)$ in eq. (2.5), instead of the static coordinates
$(t,r)$, the metric (2.1) reads

$ds^{2}=-f(r)\,du\,dv+r^{2}d\theta^{2}$ (A.1)

We then pass to the coordinates

$p=\exp\left(-\frac{u}{2}\eta\right)\,,\qquad
q=\exp\left(\frac{v}{2}\eta\right)\,,$ (A.2)

where $\eta$ is a constant to be fixed later. In the $(p,q)$ coordinates the
metric reads

$ds^{2}=f\frac{4}{\eta^{2}}\exp(-\eta\,r^{*})dp\,dq+r^{2}d\theta^{2}\,.$ (A.3)

Let us first specialise to the external region, which is a BTZ black hole. In
order to make the metric in eq. (A.3) smooth nearby the horizon, we need to
choose

$\eta=2\sqrt{\mu}\,,$ (A.4)

and to introduce the smooth lightcone coordinates $(\hat{p},\hat{q})$ as
follows

$\displaystyle{\rm for}\,\,\,\,$ $\displaystyle
r>\sqrt{\mu}\,,\qquad\hat{p}=p\,,\qquad\hat{q}=q\,,$ $\displaystyle{\rm
for}\,\,\,\,$ $\displaystyle
r<\sqrt{\mu}\,,\qquad\hat{p}=p\,,\qquad\hat{q}=-q\,.$ (A.5)

Next, we introduce the compact coordinates $(\tilde{p},\tilde{q})$ defined as

$\tilde{p}=\tan^{-1}\hat{p}\,,\qquad\tilde{q}=\tan^{-1}\hat{q}\,,$ (A.6)

and we define the coordinates of the Penrose diagram

$\tilde{t}=\frac{\tilde{q}-\tilde{p}}{2}\,,\qquad\tilde{r}=\frac{\tilde{p}+\tilde{q}}{2}\,.$
(A.7)

The function $\tan^{-1}$ in eq. (A.6) is chosen conventionally, and can be
replaced by any other $C^{\infty}$ function which maps the real line to a
segment. We finally get the expressions relating the coordinates
$(\tilde{t},\tilde{r})$ of the Penrose diagram to the original coordinates
$(t,r)$

$\displaystyle{\rm
for}\,\,\,r>\sqrt{\mu}\,:\qquad\tan\left({\tilde{r}+\tilde{t}}\right)\tan\left(\tilde{r}-\tilde{t}\right)=e^{\eta\,r^{*}}\,,\qquad\frac{\tan\left({\tilde{r}+\tilde{t}}\right)}{\tan\left({\tilde{r}-\tilde{t}}\right)}=e^{\eta\,t}\,,$
$\displaystyle{\rm
for}\,\,\,r<\sqrt{\mu}\,:\qquad\tan\left({\tilde{t}-\tilde{r}}\right)\tan\left(\tilde{t}+\tilde{r}\right)=e^{\eta\,r^{*}}\,,\qquad\frac{\tan\left({\tilde{t}+\tilde{r}}\right)}{\tan\left({\tilde{t}-\tilde{r}}\right)}=e^{\eta\,t}\,.$
(A.8)

Finally, the relation between the coordinates $(\tilde{t},\tilde{r})$ of the
Penrose diagram and the EF coordinates $(u,v)$ can be expressed as

$\tilde{t}=\frac{\pm\tan^{-1}(e^{v\sqrt{\mu}})-\tan^{-1}(e^{-u\sqrt{\mu}})}{2}\,,\qquad\tilde{r}=\frac{\pm\tan^{-1}(e^{v\sqrt{\mu}})+\tan^{-1}(e^{-u\sqrt{\mu}})}{2}\,,$
(A.9)

where the first sign is for $r>\sqrt{\mu}$ and the second sign for
$r<\sqrt{\mu}$.

For the dS spacetime interior, we can follow the same procedure as for the BTZ
external region. The only difference is that, in order for the metric in eq.
(A.3) to be smooth nearby the horizon, we need to choose

$\eta=-2\sqrt{\lambda}\,.$ (A.10)

With similar steps as for the BTZ spacetime, we can define a choice of Penrose
diagram coordinates as follows

$\tilde{t}=\frac{\tan^{-1}(e^{u\sqrt{\lambda}})\mp\tan^{-1}(e^{-v\sqrt{\lambda}})}{2}\,,\qquad\tilde{r}=\frac{\tan^{-1}(e^{u\sqrt{\lambda}})\pm\tan^{-1}(e^{-v\sqrt{\lambda}})}{2}\,,$
(A.11)

where the first sign is for $r<1/\sqrt{\lambda}$ and the second sign for
$r>1/\sqrt{\lambda}$.

In this paper, Penrose diagrams are obtained by a parametric plot of the
coordinates $(\pm\tilde{r},\pm\tilde{t})$, using eq. (A.9) for the external
BTZ region and eq. (A.11) for the dS interior. The direction of increasing $t$
on each side of the diagrams is shown in figure 6.

##  Appendix B Derivation of the refraction law for extremal surfaces

Let us assume that the change of variable between $t_{i,o}$ is of the form

$t_{o}=G(r)\,t_{i}\,.$ (B.1)

The function $G$ will be specified later by requiring the proper time on the
bubble to be continuous. In terms of the interior time coordinate $t_{i}$, the
outside metric in eq. (2.1) reads

$\displaystyle ds^{2}_{o}$ $\displaystyle=$ $\displaystyle-
f_{o}\,G^{2}\,dt_{i}^{2}-f_{o}\frac{d(G^{2})}{dr}t_{i}\,dr\,dt_{i}+\frac{1-f_{o}^{2}\left(\frac{dG}{dr}\right)^{2}t_{i}^{2}}{f_{o}}\,dr^{2}+r^{2}d\theta^{2}\,,$
(B.2)

where $f_{i}$ and $f_{o}$ are functions of $r$ given in eqs. (2.2) and (2.3).
Hence, both the interior and the exterior metrics have the following form:

$ds^{2}=-g(r,t_{i})\,dt_{i}^{2}+\frac{dr^{2}}{f(r,t_{i})}+2h(r,t_{i})\,dr\,dt_{i}+r^{2}d\theta^{2}\,\,,$
(B.3)

where for the interior metric

$g(r,t_{i})=f_{i}\,,\qquad f(r,t_{i})=f_{i}\,,\qquad h(r,t_{i})=0\,,$ (B.4)

while for the exterior metric

$g(r,t_{i})=G^{2}\,f_{o}\,,\qquad
f(r,t_{i})=\frac{f_{o}}{1-f_{o}^{2}\left(\frac{dG}{dr}\right)^{2}t_{i}^{2}}\,,\qquad
h(r,t_{i})=-f_{o}G\,\frac{dG}{dr}\,t_{i}\,.$ (B.5)

For later purposes, it is useful to evaluate the following quantities both
inside and outside the bubble:

$\left.\frac{g+fh^{2}}{f}\right|_{i}=1\,,\qquad\left.\frac{g+fh^{2}}{f}\right|_{o}=G^{2}\,.$
(B.6)

For the metric in eq. (B.3), the volume functional is

$\mathcal{V}_{i,o}=2\pi\int\mathcal{L}\,dl\,,\qquad\mathcal{L}=r\,\sqrt{-g(r,t_{i})\,t_{i}^{\prime
2}+\frac{r^{\prime 2}}{f(r,t_{i})}+2h(r,t_{i})\,r^{\prime}t_{i}^{\prime}}\,.$
(B.7)

It is convenient to fix the gauge as in eq. (3.6), which is equivalent to

$\sqrt{-g(r,t_{i})\,t_{i}^{\prime 2}+\frac{r^{\prime
2}}{f(r,t_{i})}+2h(r,t_{i})\,r^{\prime}t_{i}^{\prime}}=r\,.$ (B.8)

###  B.1 Static bubble

At constant $r$, continuity of the proper time on the domain wall fixes

$G(r)=\pm\sqrt{\frac{f_{i}(r)}{f_{o}(r)}}\,.$ (B.9)

Let us discuss the extremal codimension-one surfaces. In this case, the
functions $f,g,h$ do not depend on $t_{i}$, so there is a conserved quantity

$\hat{P}=\frac{\partial\mathcal{L}}{\partial
t_{i}^{\prime}}=\frac{-gt_{i}^{\prime}+hr^{\prime}}{\sqrt{-g(r,t_{i})\,t_{i}^{\prime
2}+\frac{r^{\prime
2}}{f(r,t_{i})}+2h(r,t_{i})\,r^{\prime}t_{i}^{\prime}}}\,r\,.$ (B.10)

With our gauge choice, the conserved quantity can be written as

$\hat{P}^{2}=\left(\frac{g+fh^{2}}{f}\right)(r^{\prime})^{2}-gr^{2}\,.$ (B.11)

Using eqs. (B.6) and (B.9), together with the fact that $g$ is continuous on
the bubble, we find

$(r^{\prime}_{i})^{2}=\frac{f_{i}}{f_{o}}(r^{\prime}_{o})^{2}\,.$ (B.12)

From eq. (2.29), we finally get

$P_{o}^{2}=\frac{f_{o}(R)}{f_{i}(R)}P^{2}_{i}=\mu_{0}^{2}\,P_{i}^{2}\,.$
(B.13)

###  B.2 Dynamical bubble

For dynamical bubbles, the functions $f,g,h$ depend on $t_{i}$. All the
derivatives

$\partial_{t_{i}}f\,,\qquad\partial_{t_{i}}h\,,\qquad\partial_{r}f\,,\qquad\partial_{r}h\,,$

will have a Dirac delta contribution localized on the surface of the bubble

$t_{i}=T_{i}(\tau)\,,\qquad r=R(\tau)\,.$

We impose the condition that this delta function contribution is constant on
the surface of the bubble:

$\left.\frac{d\,f(r,t_{i})}{d\tau}\right|_{\rm
bubble}=\frac{d\,f(R(\tau),T_{i}(\tau))}{d\tau}=\dot{T}_{i}\,\partial_{t_{i}}f+\dot{R}\,\partial_{r}f=0\,,\qquad\partial_{t_{i}}f=-\frac{dR}{dT_{i}}\,\partial_{r}f\,.$
(B.14)

So we expect

$\partial_{r}f=\frac{1}{\sqrt{1+\left(\frac{dR}{dT_{i}}\right)^{2}}}\,\delta(r-R(\tau))\,\Delta
f\,,\qquad\partial_{t_{i}}f=-\frac{dR}{dT_{i}}\frac{1}{\sqrt{1+\left(\frac{dR}{dT_{i}}\right)^{2}}}\,\delta(r-R(\tau))\,\Delta
f\,,$ (B.15)

where $\Delta f$ is the discontinuity of $f$ on the surface of the bubble. An
analogous equation holds for $h$.

Let us introduce the variables $s$ and $w$ such that

$\left(\begin{array}[]{c}t_{i}\\\
r\end{array}\right)=\left(\begin{array}[]{cc}\cos\psi&-\sin\psi\\\
\sin\psi&\cos\psi\\\ \end{array}\right)\left(\begin{array}[]{c}s\\\
w\end{array}\right)\,,$ (B.16)

where

$\sin\psi=\frac{\dot{R}}{\sqrt{\dot{R}^{2}+\dot{T}_{i}^{2}}}\,,\qquad\cos\psi=\frac{\dot{T}_{i}}{\sqrt{\dot{R}^{2}+\dot{T}_{i}^{2}}}\,,\qquad\tan\psi=\frac{dR}{dT_{i}}\,.$
(B.17)

The derivative of $f$ with respect to $s$ is

$\frac{\partial f}{\partial s}=\frac{\partial t_{i}}{\partial
s}\partial_{t_{i}}f+\frac{\partial r}{\partial
s}\partial_{r}f=\frac{\dot{T}_{i}\partial_{t_{i}}f+\dot{R}\,\partial_{r}f}{\sqrt{\dot{R}^{2}+\dot{T_{i}}^{2}}}\,.$
(B.18)

A similar property is valid for $h$. From eq. (B.14), we thus find that the
relations $\partial_{s}f=0$ and $\partial_{s}h=0$ are satisfied by the Dirac
delta term.

Going back to the volume functional in eq. (B.7), we can express it in the
$(s,w)$ coordinates. In the approximation in which we consider just the ”fast”
dependence of the Lagrangian due to discontinuities at the two sides of the
domain wall, the Lagrangian density is independent of $s$. We have then a
conserved quantity of the form

$\hat{P}=\frac{\partial\mathcal{L}}{\partial
s^{\prime}}=(hr^{\prime}-gt_{i}^{\prime})\cos\psi+\left(\frac{r^{\prime}}{f}+ht_{i}^{\prime}\right)\sin\psi\,.$
(B.19)

The quantity $\hat{P}$ is not globally conserved on the codimension-one
extremal surface, but just before and after the collision with the Dirac delta
domain wall. Plugging the gauge fixing condition (B.8) in, we get

$\hat{P}^{2}=\left(\frac{g+fh^{2}}{fg}r^{\prime}\sin\psi\pm\frac{(g\cos\psi-h\sin\psi)}{g}\sqrt{\frac{g+fh^{2}}{f}(r^{\prime})^{2}-g\,r^{2}}\right)^{2}\,.$
(B.20)

The conserved quantities inside and outside the domain wall are

$\displaystyle\hat{P}_{i}^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\sin\psi}{f_{i}}r_{i}^{\prime}\pm\sqrt{(r_{i}^{\prime})^{2}-f_{i}r^{2}}\cos\psi\right)^{2}\,,$
$\displaystyle\hat{P}^{2}_{o}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{f_{o}}r_{o}^{\prime}\sin\psi\pm{\left(G\cos\psi+\frac{dG}{dr}\,t_{i}\,\sin\psi\right)}\sqrt{(r_{o}^{\prime})^{2}-f_{o}\,r^{2}}\right)^{2}\,.$
(B.21)

Combining the condition $\hat{P}_{i}^{2}=\hat{P}^{2}_{o}$ with eqs. (B.17) and
(B.1), we find

$\left(\frac{1}{f_{i}}r_{i}^{\prime}\pm\sqrt{(r_{i}^{\prime})^{2}-f_{i}\,r^{2}}\frac{dT_{i}}{dR}\right)^{2}=\left(\frac{1}{f_{o}}r_{o}^{\prime}\pm\frac{dT_{o}}{dR}\sqrt{(r_{o}^{\prime})^{2}-f_{o}\,r^{2}}\right)^{2}\,.$
(B.22)

We can consider the following crosschecks of eq. (B.22):

  * •

For $\psi=0$, it reproduces the static bubble result in eq. (B.13).

  * •

For a bubble moving at the speed of light, we have

$\frac{dT_{i,o}}{dR}=\pm\frac{1}{f_{i,o}}\,.$ (B.23)

Specializing eq. (B.22) to eq. (B.23) and combining with eq. (3.7), we find
$V^{\prime}_{i}=V^{\prime}_{o}$ or $U^{\prime}_{i}=U^{\prime}_{o}$, depending
on the choice of sign. This is in agreement with the results in [72, 73].

Using the notation in eq. (3.15), the refraction condition (B.22) can be
written in the covariant form

$(g_{i})_{\mu\nu}\frac{dx^{\mu}_{i}}{dl}\frac{dX^{\nu}_{i}}{d\tau}=\pm(g_{o})_{\mu\nu}\frac{dx^{\mu}_{o}}{dl}\frac{dX^{\nu}_{o}}{d\tau}\,.$
(B.24)

By continuity with the case where there is no bubble into the system, the
physical solution should be the one with the $+$ sign, which we consider in
eq. (3.16).

##  Appendix C Technical details for smooth extremal surfaces

###  C.1 Solution with $P_{o}<0$

With the assumption of negative $P_{o}$ and with the choice of sign for
$\frac{dT_{o}}{dR}$ in eq. (4.3), eq. (4.2) is solved by

$\rho_{o}=-\frac{R^{4}\left(\kappa^{2}(\lambda-1)+(\lambda+1)^{2}\right)-R^{2}\left(2(\lambda+1)(\mu+1)-\kappa^{2}(\mu-1)\right)+(\mu+1)^{2}}{2\kappa^{2}R\sqrt{1-\lambda
R^{2}}}\,,$ (C.1)

and

$P_{o}^{2}=-R^{4}+\mu
R^{2}+\frac{\left((\mu+1)^{2}+R^{4}\left(\kappa^{2}(\lambda-1)+(\lambda+1)^{2}\right)+R^{2}\left(\kappa^{2}(\mu-1)-2(\lambda+1)(\mu+1)\right)\right)^{2}}{4\kappa^{4}R^{2}\left(1-\lambda
R^{2}\right)}\,.$ (C.2)

A direct calculation gives that $\rho_{o}(R_{\rm max})$ in eq. (C.1) is the
opposite of $\rho_{o}(R_{\rm max})$ in eq. (4.4). The same holds true for
$\rho_{o}(R_{\rm min})$. So, the solution in eq. (C.1) corresponds to a
”reflection” of the codimension-one extremal surface, and as such should be
discarded. Note that both the solutions in eqs. (4.5) and (C.2) vanish for
$r=R_{\rm max}=R_{\rm min}$.

###  C.2 Sign of $\rho_{o}(R)$ for small bubbles

The function $\rho_{o}(R)$ in eq. (4.4) vanishes for $R=R_{0}$, where

$R_{0}(\mu)=\sqrt{\frac{1-\mu}{\kappa^{2}+\lambda-1}}\,.$ (C.3)

Therefore, if either $R_{0}(\mu)$ is not a real number or $R_{0}>R_{\rm max}$,
the sign of $\rho_{o}(R)$ does not change along the surface of the bubble.
Conversely, if $R_{0}(\mu)$ is real and $R_{0}<R_{\rm max}$, $\rho_{o}(R)$
changes sign along the bubble. With a direct calculation, it can be checked
that

$R_{0}(\mu_{s})=\sqrt{\mu_{s}}=R_{\rm max}(\mu_{s})\,.$ (C.4)

There are then four possible behaviors of $\rho_{o}(R)$ on the domain wall:

  * A)

$\mu<\mu_{s}$ and $\lambda+\kappa^{2}>1$. The function $\rho_{o}(R)$ is always
positive.

  * B)

$\mu<\mu_{s}$ and $\lambda+\kappa^{2}<1$. When $R_{0}$ is real, $\rho_{o}(R)$
is positive for $R_{0}<R\leq R_{\rm max}$ and negative for $0<R<R_{\rm 0}$.
When $R_{0}$ is imaginary, $\rho_{o}(R)$ is always positive.

  * C)

$\mu>\mu_{s}$ and $\lambda+\kappa^{2}>1$. When $R_{0}$ is real, $\rho_{o}(R)$
is negative for $R_{0}<R\leq R_{\rm max}$ and positive for $0<R<R_{\rm 0}$.
When $R_{0}$ is imaginary, $\rho_{o}(R)$ is always negative.

  * D)

$\mu>\mu_{s}$ and $\lambda+\kappa^{2}<1$. The function $\rho_{o}(R)$ is always
negative.

With reference to subsection 4.1.2, from these considerations we find that the
region of parameter space in bullet A) belongs to case 2, while the part of
parameter space in bullet D) belongs to case 1.

###  C.3 Behavior of the complexity rate for $t_{b}=0$

In this appendix, we determine the explicit value of the quantity $K$ defined
in eq. (4.17).
Let us first consider very small bubbles. Since for $t_{b}\to 0$ we have
$\rho_{o}(R)>0$, we should use eq. (4.10) to compute $\frac{dt_{b}}{dP_{o}}$.
By symmetry, $t_{b}=0$ corresponds to $P_{o}=0$, so

$K=\left.\frac{dt_{b}}{dP_{o}}\right|_{P_{o}=0}=\left.\frac{dT_{o}}{dR}\frac{dR}{dP_{o}}\right|_{R=R_{\rm
max}}-\int_{R_{\rm max}}^{\Lambda}\frac{1}{r(f_{o}(r))^{3/2}}dr\,,$ (C.5)

where

$\left.\frac{dT_{o}}{dR}\frac{dR}{dP_{o}}\right|_{R=R_{\rm
max}}=\frac{2\sqrt{1-\lambda R_{\rm max}^{2}}\,(1+\mu-R_{\rm
max}^{2}(1+\lambda+\kappa^{2}))}{R_{\rm max}^{2}(R_{\rm min}^{2}-R_{\rm
max}^{2})(R_{\rm
max}^{2}-\mu)(\lambda+(\kappa-1)^{2})(\lambda+(\kappa+1)^{2})}\equiv H\,.$
(C.6)

We can use the explicit integral

$F(r)=\int\frac{-1}{r(r^{2}-\mu)^{3/2}}dr=\frac{1}{\mu\sqrt{r^{2}-\mu}}-\frac{2}{\mu^{3/2}}\tan^{-1}\sqrt{\frac{r+\sqrt{\mu}}{r-\sqrt{\mu}}}$
(C.7)

to obtain

$K=H-\frac{\pi}{2\mu^{3/2}}-F(R_{\rm max})\,.$ (C.8)

In the case of not so small bubbles, for $t_{b}\to 0$ we have $\rho_{o}(R)<0$,
so $\frac{dt_{b}}{dP_{o}}$ is given by eq. (4.14). To perform the computation,
we first send $r_{t,{\rm AdS}}\to r_{t,{\rm AdS}}+\delta$, then we use the
Leibniz integral rule:

$\displaystyle K$ $\displaystyle=$ $\displaystyle H-\int^{\Lambda}_{r_{t,{\rm
AdS}}+\delta}\frac{r^{2}}{(P_{o}^{2}+f_{o}(r)r^{2})^{3/2}}dr-\int^{R}_{r_{t,{\rm
AdS}}+\delta}\frac{r^{2}}{(P_{o}^{2}+f_{o}(r)r^{2})^{3/2}}dr$ (C.9)
$\displaystyle+2\frac{dr_{t,{\rm AdS}}}{dP_{o}}\frac{P_{o}}{f_{o}(r_{t,{\rm
AdS}})}\frac{1}{\sqrt{P_{o}^{2}+(r_{t,{\rm AdS}}+\delta)^{2}\,f_{o}(r_{t,{\rm
AdS}}+\delta)}}\,.$

At small $P_{o}$, the following approximations are valid:

$\displaystyle P_{o}^{2}+f_{o}(r)\,r^{2}\approx(r-r_{t,{\rm
AdS}})(r+\sqrt{\mu})r^{2}\,,\qquad r_{t,{\rm
AdS}}\approx\sqrt{\mu}-\frac{P_{o}^{2}}{2\mu^{3/2}}+O(P_{o}^{4})\,,$
$\displaystyle\frac{dr_{t,{\rm
AdS}}}{dP_{o}}\approx-\frac{P_{o}}{\mu^{3/2}}\,,\qquad\frac{1}{f_{o}(r_{t,{\rm
AdS}})}\approx-\frac{\mu}{P_{o}^{2}}+O(P_{o}^{0})\,.$ (C.10)

Hence, we get

$\displaystyle K$ $\displaystyle=$ $\displaystyle H-\int^{\Lambda}_{r_{t,{\rm
AdS}}+\delta}\frac{1}{r[(r+\sqrt{\mu})(r-r_{t,{\rm AdS}})]^{3/2}}dr$ (C.11)
$\displaystyle-\int^{R_{\rm max}}_{r_{t,{\rm
AdS}}+\delta}\frac{1}{r[(r+\sqrt{\mu})(r-r_{t,{\rm
AdS}})]^{3/2}}dr+\frac{\sqrt{2}}{{\mu}^{5/4}}\frac{1}{\sqrt{\delta}}$
$\displaystyle=$ $\displaystyle H+F(R_{\rm max})+\frac{3\pi}{2\mu^{3/2}}\,.$

Consistently, eqs. (C.8) and (C.11) matches for $\mu=\mu_{s}$.

In the case of large bubbles, similar calculations give

$K=\tilde{H}+F(R_{\rm min})+\frac{3\pi}{2\mu^{3/2}}\,,$ (C.12)

where

$\tilde{H}=\left.\frac{dT_{o}}{dR}\frac{dR}{dP_{o}}\right|_{R=R_{\rm
min}}=-\frac{2\sqrt{1-\lambda R_{\rm min}^{2}}\,(1+\mu-R_{\rm
min}^{2}(1+\lambda+\kappa^{2}))}{R_{\rm min}^{2}(R_{\rm min}^{2}-R_{\rm
max}^{2})(R_{\rm
min}^{2}-\mu)(\lambda+(\kappa-1)^{2})(\lambda+(\kappa+1)^{2})}\,.$ (C.13)

Note that $K$ in eq. (C.12) is always positive, because both
$F(r)+\frac{3\pi}{2\mu^{3/2}}>0$ and $\tilde{H}>0$. The latter property
follows from

$w_{o}(R_{\rm min})<0\,,$ (C.14)

where $w_{o}$ is defined in eq. (4.3). The inequality (C.14) arises from the
negativity of $w_{o}(R_{\rm min})$ at $\mu=0$. Indeed, no real solutions for
$\mu$ to the equation $w_{o}(R_{\rm min})=0$ exist. Consequently,
$w_{o}(R_{\rm min})$ never changes sign as a function of $\mu$.

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