aainstitutetext: Dipartimento di Matematica e Fisica, Università Cattolica del
Sacro Cuore, Via Musei 41, 25121 Brescia, Italy, 2 and INFN Sezione di
Perugia, Via A. Pascoli, 06123 Perugia, Italybbinstitutetext: Department of
Physics E. Fermi, University of Pisa and INFN Sezione di Pisa
Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italyccinstitutetext: Racah Institute
of Physics, The Hebrew University of Jerusalem, 91904, Israelddinstitutetext:
Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-
sur-Yvette, Franceeeinstitutetext: Departamento de Ciencias, Facultad de Artes
Liberales, Universidad Adolfo Ibáñez,
Santiago 7941169, Chile

# On the time dependence of holographic complexity for charged AdS black holes
with scalar hair

Roberto Auzzi b Stefano Bolognesi c,d Eliezer Rabinovici d Fidel I. Schaposnik
Massolo e Gianni Tallarita<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the presence of a scalar hair perturbation, the Cauchy horizon of a
Reissner-Nordström black hole disappears and is replaced by the rapid collapse
of the Einstein-Rosen bridge, which leads to a Kasner singularity
Hartnoll:2020rwq ; Hartnoll:2020fhc . We study the time-dependence of
holographic complexity, both for the volume and for the action proposals, in a
class of models with hairy black holes. Volume complexity can only probe a
portion of the black hole interior that remains far away from the Kasner
singularity. We provide numerical evidence that the Lloyd bound is satisfied
by the volume complexity rate in all the parameter space that we explored.
Action complexity can instead probe a portion of the spacetime closer to the
singularity. In particular, the complexity rate diverges at the critical time
$t_{c}$ for which the Wheeler-DeWitt patch touches the singularity. After the
critical time the action complexity rate approaches a constant. We find that
the Kasner exponent does not directly affect the details of the divergence of
the complexity rate at $t=t_{c}$ and the late-time behaviour of the
complexity. The Lloyd bound is violated by action complexity at finite time,
because the complexity rate diverges at $t=t_{c}$. We find that the Lloyd
bound is satisfied by the asymptotic action complexity rate in all the
parameter space that we investigated.

##  1 Introduction and conclusions

The AdS/CFT correspondence constitutes a tool one can use to relate the
problem of quantum gravity in asymptotically anti de Sitter (AdS) spacetimes
to the study of a Conformal Field Theory (CFT) defined on the boundary of the
spacetime. Quantum information concepts such as entanglement entropy
Ryu:2006bv ; Hubeny:2007xt  play an important role in reconstructing the
geometry close to event horizons. However, despite many advances, we do not
currently have a clear picture of the emergence of various aspects of the
spacetime geometry from the physics of the CFT living on the boundary. For
example, it turns out that the entanglement entropy is not enough
Susskind:2014moa  to probe the growth of the Einstein-Rosen Bridge (ERB)
inside the event horizon of a Black Hole (BH).

In Susskind:2014rva  it was proposed that computational complexity may play an
important role in understanding the black hole interior. Quantum computational
complexity is a quantum information concept. For a quantum system, one can
heuristically define complexity as the minimal number of elementary operations
that are needed to prepare a given state starting from a reference one. A
continuous and bounded version of complexity, following Nielsen Nielsen1 , can
be defined in terms of geodesics in the space of unitary Hamiltonians. There
is a great deal of ambiguity in the definition of complexity, due to the
choice of the reference state and of the computational cost of the elementary
operations. It is not yet clear which one of the many possible choices of
these details of computational complexity might be relevant in its precise
definition for holography. For systems with a finite number of degrees of
freedom, it is expected that the maximal complexity scales exponentially with
the number of qubits. Negative curvature Brown:2016wib ; Brown:2017jil ;
Brown:2019whu ; Auzzi:2020idm ; Brown:2021euk ; Basteiro:2021ene  is an
important ingredient to realise this property. The study of complexity for
quantum systems with an infinite number of degrees of freedom is still in its
preliminary stages, with much progress being made in defining complexity for
free field theories Jefferson:2017sdb ; Chapman:2017rqy ; Khan:2018rzm ;
Hackl:2018ptj . However, a definition of complexity in interacting CFT is
still lacking, see Caputa:2017urj ; Caputa:2018kdj ; Erdmenger:2020sup ;
Flory:2020eot ; Chagnet:2021uvi ; Koch:2021tvp  for some advances in this
direction.

Two main quantities have been proposed as bulk holographic duals to
computational complexity. The Complexity=Volume (CV) proposal Stanford:2014jda
relates the complexity to the maximum volume $V$ of the codimension-one
surface anchored to the given boundary time,

$C_{V}=\frac{V}{GL}\,,$ (1)

where $G$ is the Newton constant and $L$ is the AdS radius. The
Complexity=Action (CA) conjecture Brown:2015bva ; Brown:2015lvg  proposes
instead that complexity is proportional to the classical action $I_{WDW}$
evaluated on the Wheeler-DeWitt patch, which is defined as the bulk domain of
dependence of the maximal slice attached to the boundary time, _i.e._

$C_{A}=\frac{I_{WDW}}{\pi}\,.$ (2)

Due to the large amount of arbitrariness in the definition of computational
complexity, it could be that both of these holographic duals, and also
possible generalisations thereof Couch:2016exn ; Belin:2021bga , may
correspond to different prescriptions to define the computational complexity
on the dual CFT, perhaps sharing some similarities.

In recent years both the CV and CA proposals have been tested in a variety of
holographic models and in different backgrounds. There are a few universal
properties that we would require from any notion of complexity to be
acceptable, and which are reproduced by both prescriptions, such as the
switchback effect Stanford:2014jda ; Susskind:2014jwa , the structure of the
UV divergencies Carmi:2016wjl ; Reynolds:2016rvl ; Akhavan:2019zax ;
Omidi:2020oit  and the linear growth for a parametrically large time after
thermalization. In particular, it is expected that quantum complexity
increases linearly for a time which is exponential in the entropy of the
system Susskind:2015toa . This regime is reproduced by the classical gravity
dual Stanford:2014jda ; Brown:2015bva ; Brown:2015lvg ; Cai:2016xho ;
Carmi:2017jqz ; Yang:2017czx ; Auzzi:2018zdu ; Auzzi:2018pbc ;
Bernamonti:2021jyu , which gives a late-time growth which is linear in the
boundary time $t_{b}$

$\lim_{t_{b}\to\infty}\frac{d\,C_{V}}{dt_{b}}=W_{V}\,,\qquad\lim_{t_{b}\to\infty}\frac{d\,I_{WDW}}{dt_{b}}=W_{A}\,,$
(3)

where the coefficients $W_{V}$ and $W_{A}$ are proportional, up to an order-
one coefficient, to $TS$, where $T$ is the temperature and $S$ the entropy of
the system. After this linear growth epoch, complexity should experience a
plateau Susskind:2015toa  and then, after the recurrence time (which is double
exponential in the entropy of the system) complexity should become small
again. This very late-time behavior is probably related to gravitational
quantum corrections, as studied in Iliesiu:2021ari  for two-dimensional
gravity.

In this work we will test both the volume and the action conjectures in
certain backgrounds of charged black holes with scalar fields, where the inner
Cauchy horizon disappears due to a small perturbation of the BH exterior and
the singularity becomes space-like and of Kasner type. We will focus on the
case of an eternal black hole with two disconnected boundaries, which on the
field theory side is dual to the thermofield double state Maldacena:2001kr ,
which has the following schematic form

$|\Psi_{TFD}\rangle\propto\sum_{n}e^{-E_{n}\beta/2-iE_{n}\,t_{b}}|E_{n}\rangle_{R}|E_{n}\rangle_{L}\,,$
(4)

where $\beta$ is the inverse temperature and $|E_{n}\rangle_{L,R}$ are the
energy eigenstates of left and right boundary theories, $t_{b}$ being again
the boundary time.

In section 2 we will introduce the model and describe various solutions of
charged black holes. This section mostly consists of a review of previous
works. We rederive the numerical solutions in order to use them in the rest of
the paper. We will work with Einstein-Maxwell theory in four dimensions with
negative cosmological constant, and thus an asymptotic AdS4 background. This
is dual to a strongly-coupled $3$-dimensional CFT with a $U(1)$ symmetry. We
will consider black holes in the Poincaré patch with the metric

$ds^{2}=\frac{1}{z^{2}}\left(-fe^{-\chi}dt^{2}+\frac{dz^{2}}{f}+dx^{2}+dy^{2}\right)\,,$
(5)

where $f$ and $\chi$ are functions of the radial AdS coordinate $z$. The
Reissner-Nordström (RN) black hole solution provides a gravity dual to the CFT
states with finite temperature and chemical potential. As in the case of
asymptotically flat spacetime Simpson:1973ua ; Chandrasekhar-Hartle , the
Cauchy horizon inside the event horizon of the black hole is unstable w.r.t.
small perturbations. In order to detect this instability one may consider a
scalar perturbation Hartnoll:2020rwq ; Hartnoll:2020fhc , which on the CFT
side is dual to an operator whose dimension is related to the mass.

The deformation by a neutral scalar field was studied in detail in
Hartnoll:2020rwq . Turning on a relevant deformation in the CFT by adding a
source for the scalar field has a dramatic effect in the inner structure on
the black hole. The Cauchy horizon is no longer present and the causal
structure of the BH becomes closer to that of an eternal Schwarzschild BH. The
singularity is of Kasner type, with a Kasner coefficient that depends on the
various external parameters: the temperature, the chemical potential and the
magnitude of the source for the scalar operator. For small values of the
source, the black hole remains close to the RN solution only up to the “would
be” Cauchy horizon. In this context, another interesting new phenomenon that
has been observed in Hartnoll:2020rwq  is that the scalar field condensation
becomes important and backreacts on the geometry, the most prominent
consequence of this being an exponential decay of $g_{tt}$. This phenomenon
was called the “collapse of the Einstein-Rosen (ER) bridge”. After this period
the metric flows to a Kasner singularity, with

$f=-f_{0}z^{3+\alpha^{2}}\,,\qquad\chi=2\alpha^{2}\log z+\chi_{0}\,,$ (6)

where $\chi_{0}$ and $f_{0}$ are integration constants and $\alpha$ can be
used to parametrize the Kasner exponents, _i.e._

$p_{t}=\frac{\alpha^{2}-1}{3+\alpha^{2}}\,,\qquad
p_{x}=\frac{2}{3+\alpha^{2}}\,.$ (7)

whose definition will be recalled later in (35).

For a charged scalar field, even without an external source, the phenomenon of
spontaneous condensation below a certain temperature is observed,
corresponding to the holographic superconductor Hartnoll:2008vx ;
Hartnoll:2008kx . In this case there is also a resolution of the Cauchy
horizon into a Kasner singularity, and the collapse of the ER bridge
Hartnoll:2020fhc . Other interesting phenomena have been observed inside the
horizon, such as Josephson-type oscillations of the scalar order parameter or
the inversion of the Kasner exponent close to the singularity. As in the
neutral case, the metric nearby the singularity is again of the form (6).

The regime where the scalar hair is small outside the Cauchy horizon is most
challenging for numerical approaches, since it is in these conditions where
the collapse of the bridge is more rapid and abrupt. The case of charged
scalar is particularly interesting due to the possible presence of Kasner
inversions, in which the Einsten-Rosen bridge can experience a period of
expansion followed by a final contraction towards the singularity. In the
limit where the scalar is small outside the Cauchy horizon, the number of
Kasner inversions depends in a very sensitive way on the scalar hair. Close to
the critical temperature of the holographic superconductor this also occurs in
the absence of an external source. For fine-tuned values of the scalar, an
infinite number of transitions can take place, and the solution has a fractal-
like behaviour in the external source and temperature parameters
Hartnoll:2020fhc . Here we also present some new results regarding the
complete phase diagram of the charged scalar field case also in presence of a
source, see Figure 10.

A natural question to ask is how the echos of this bulk gravitational chaotic
behaviour inside the event horizon may be reflected in the physics of the
conformal field theory on the boundary. Complexity is then a natural probe to
consider. For an eternal black hole, the CV proposal implies the computation
of the volume of a maximal spatial slice which is anchored at the two sides of
the black hole. Going from one side to the other inevitably requires crossing
the ER bridge, and thus allows us to glance inside the horizon of the black
hole. The linear growth of volume complexity at late times is indeed related
to the growth of the ER bridge, and it is therefore natural to wonder if it
can detect the collapse of the bridge.

In section 3 we investigate the volume conjecture and find that this proposal
does not probe the collapse of the bridge and the Kasner behavior, because the
extremal codimension one surface gets stuck far away from the would be Cauchy
horizon, well before the new phenomena discovered in Hartnoll:2020rwq ;
Hartnoll:2020fhc  start to take place. Volume compexity for holographic
superconductors have been studyed before in Yang:2019gce . Respect to
Yang:2019gce  we investigate a more general class of solutions which includes
the neutral scalar case and the charged scalar case with external source. We
find that the Lloyd bound Lloyd ; Brown:2015lvg  is always satisfied in the
parameter space that we investigate, both with Dirichlet and Neumann
definition of the mass.

Section 4 contains the study of action complexity in this class of
backgrounds. This is the main new result of the paper. The action prescription
for complexity requires the computation of the action of the Wheeler-DeWitt
(WDW) wedge which is anchored at the two sides of the black hole. It can be
thought of as the union of all possible spatial slices with the given boundary
conditions. In contrast to the maximal spatial slice, the WDW wedge can touch
the singularity and thus, in principle, can also probe the region nearby the
black hole singularity. In the discussion of action complexity we must
distinguish two cases, according to the shape of the Penrose diagram which
describes the causal structure of the black hole solution Fidkowski:2003nf .
Indeed, the Penrose diagram can be schematically drawn as a square, where the
vertical sides correspond to the left and right boundaries where each of the
entangled CFTs of the thermofield double state live. The spacetime singularity
can then bend the top and bottom sides inwards or outwards, compared to the
horizontal side of the square inside the black hole horizon. This will be
reviewed in Appendix C. This distinction is a conformally invariant property
of the black hole solution which does not depend on the arbitrary choice of
functions used in the conformal mapping that was used to construct the
diagram. In this paper we will denote a solution with a lower bending of the
singularity as being of “type $D$”, while one with an upper bending will be
referred to as of “type $U$”. For example, the Schwarzschild solution in AdSd
with $d>3$ is of type $D$. In the case of the black holes with scalar hair
discussed in this paper, both type $D$ and type $U$ solutions can be realised.

Figure 1:  WDW patch for a type $D$ solution. Left: before $t_{c}$. Right:
after $t_{c}$.

The eternal black hole solution is symmetric under time reflection $t_{b}\to-
t_{b}$, and so it is enough to study the behaviour of complexity for
$t_{b}>0$. The time dependence of the WDW patch for type $D$ solutions is
shown in figure 1. At $t_{b}=0$, the WDW patch has a hexagonal shape   . As
the boundary time increases, at some critical time $t_{b}=t_{c}$ the WDW wedge
experiences a discontinuous transition   $\to$   . For $t<t_{c}$, the
complexity rate is identically zero. When the tip of the WDW (a null-like
joint) forms at the singularity inside the white hole horizon, there is a
peculiar logarithmic divergence in the complexity rate

$\frac{dI_{\rm WDW}}{dt_{b}}\approx\frac{V_{0}}{16\pi
G}A(\infty)\,\log(t_{b}-t_{c})\,,\qquad t_{b}>t_{c}\,,$ (8)

where $V_{0}$ is the volume of the boundary theory and $A(\infty)>0$ is given
by eq. (141). When the limit $t_{b}\to t_{c}$ is approached from
$t_{b}>t_{c}$, the complexity rate tends to $-\infty$. This is the same kind
of behaviour as in the Schwarzschild case Carmi:2017jqz ; Yang:2017czx .

Figure 2:  WDW patch for type $U$. Left: before $t_{c}$. Right: after $t_{c}$.

A sketch of the time dependence of the WDW patch for the type $U$ solution is
shown in figure 2. In this case, at $t_{b}=0$ the WDW has the shape   . At a
critical time $t_{b}=t_{c}$ the shape of the WDW changes abruptly as   $\to$
  . The limit in which the scalar perturbation vanishes outside the Cauchy
horizon corresponds to an extreme type $U$ case, in which the singularity
approaches the location of the would be Cauchy horizon in the Penrose diagram.
In this limit we have that the critical time tends to infinity. Denoting by
$\phi_{h}$ the value of the scalar on the event horizon, we find (see appendix
H) that the critical time scales as

$t_{c}=O\left(\frac{1}{\phi_{h}^{2}}\right)\,.$ (9)

See figure 19 for a plot of the critical time as a function of $\phi_{h}$.

When the tip of the WDW touches the singularity there is a logarithmic
divergence in the complexity rate,

$\frac{dI_{\rm WDW}}{dt_{b}}\approx-\frac{V_{0}}{16\pi
G}A(\infty)\,\log(t_{c}-t_{b})\,,\qquad t_{b}<t_{c}\,.$ (10)

If we approach the limit $t_{b}\to t_{c}$ from $t_{b}<t_{c}$, the complexity
rate diverges to $+\infty$. For $t_{b}>t_{c}$ the complexity rate is instead
finite. In the $\phi_{h}\to 0$ limit, the value of $A(\infty)$ in eq. (10) can
be determined analytically:

  * •

For a neutral scalar $q=0$, the quantity $A(\infty)$ tends to a finite value
for $\phi_{h}\to 0$,

$A(\infty)\to-\frac{16\pi G}{V_{0}}T_{c}S_{c}\,,$ (11)

where $S_{c}$ and $T_{c}$ are the formal entropy and negative temperature
computed on the Cauchy horizon, see eq. (134). In this limit the Kasner
parameter $\alpha$ diverges.

  * •

For $q\neq 0$, we find that $A(\infty)$ vanishes for $\phi_{h}\to 0$, so that
the divergence of the complexity rate at the critical time tends to disappear.
In this case the final Kasner parameter $\alpha$ oscillates an infinite number
of times as the limit $\phi_{h}\to 0$ is approached.

Thus, there is no direct relation between the divergence in the complexity
rate in eq. (10) and the Kasner parameter $\alpha$. Also, the divergence of
the complexity rate at critical time in eq. (10) shows that the Lloyd bound
can not hold at finite time in this class of models.

We determined analytically the asymptotic action complexity rate in the
$\phi_{h}\to 0$ limit. We find that the behavior of the asymptotic action
complexity rate is discontinuous for $q\to 0$:

  * •

For the neutral scalar case $q=0$, the asymptotic complexity tends to that of
the RN black hole, i.e.

$\lim_{t_{b}\to\infty}\frac{dI_{\rm WDW}}{dt_{b}}=T\,S\,-T_{c}\,S_{c}\,,$ (12)

where $T$ and $S$ are the temperature and entropy.

  * •

In the charged scalar case $q\neq 0$, we find

$\lim_{t_{b}\to\infty}\frac{dI_{\rm WDW}}{dt_{b}}=T\,S\,.$ (13)

We numerically checked that the Lloyd bound holds for the asymptotic action
complexity rate $W_{A}$ in all the parameter space that we investigated.

In geometries with a naked Kasner singularity, the structure of the time
dependence of complexity is such that complexity decreases as the singularity
is approached Barbon:2015ria ; Bolognesi:2018ion . In that case, however, the
singularity is not hidden behind a protective horizon as in the present setup.
Eventually, one would expect that there is an intermediate setup for which the
complexity ceases to increase for a long time without decreasing, allowing for
a parametrically long period of large and constant complexity. This was not
achieved in this work. However, we do sustain the believe that such a setup
will be eventually found, either semiclassically or perhaps including higher
genus effects Iliesiu:2021ari .

We now resume the main new results contained in the paper:

  * •

We numerically compute the Kasner exponent $p_{t}$ as a function of $\phi_{0}$
and $T$ for the charged scalar and we determine if solution is type $U$ or
type $D$. See figure 10.

  * •

We investigate the time dependence of volume complexity in a large class of
hairy black hole solutions and we provide numerical evidence that the Lloyd
bound Lloyd ; Brown:2015lvg  is always satisfied in this model.

  * •

We study the time dependence of action complexity in the same class of hairy
black hole solutions. The time dependence of action complexity can distinguish
between type $D$ and type $U$ solutions. For type $D$ solutions, we find a
similar behavior to the Schwarzschild case studied in Carmi:2017jqz . For type
$U$ solutions, we find that the complexity rate diverges to $+\infty$ at the
critical time $t_{c}$. This is how complexity feels the Kasner singularity.

  * •

The divergence of complexity rate at critical time shows that the Lloyd bound
does not hold for action complexity at finite time in this class of solutions.
We provide numerical evidence that the Lloyd bounds holds for the asymptotic
action complexity rate.

  * •

We find that the Kasner exponents nearby the singularity are not directly
correlated with the coefficient of the linear growth at late time $W_{A}$ and
with the coefficient of the leading singular behavior of action complexity at
the critical time.

Note added: While we were finishing to write this work, Ref. An:2022lvo
appeared on the arXiv. There is some overlap with the results presented in
this work.

##  2 The model and the black hole solutions

###  2.1 Theoretical setting

We consider the Einstein-Maxwell model with a scalar field, with action:

$S=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left\\{R-2\Lambda-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-g^{\mu\nu}D_{\mu}\phi(D_{\nu}\phi)^{*}-m^{2}\phi\phi^{*}\right\\}\,,$
(14)

where the cosmological constant is

$\Lambda=-\frac{3}{L^{2}}\,,$ (15)

and $L$ is the AdS radius. The covariant derivative is

$D_{\mu}\phi=\partial_{\mu}\phi+iqA_{\mu}\phi\,.$ (16)

We will consider both the case of neutral scalar field $q=0$ and charged
scalar field $q\neq 0$. We will take the scalar mass to be above the
Breitenlohner-Freedman (BF) bound Breitenlohner:1982jf

$m^{2}=-\frac{2}{L^{2}}\,,$ (17)

and from now on we will set $L=1$ for simplicity.

A generic metric for a planar black hole is

$ds^{2}=\frac{1}{z^{2}}\left(-fe^{-\chi}dt^{2}+\frac{dz^{2}}{f}+dx^{2}+dy^{2}\right)\,,$
(18)

where $f$ and $\chi$ are functions of $z$ and the ansatz for the scalar and
gauge field is

$\phi=\phi(z)\,,\qquad A=a(z)dt\,.$ (19)

The equations of motion are

$\displaystyle
f^{\prime}-\left(\frac{\chi^{\prime}}{2}+\frac{3}{z}\right)f+\frac{3}{z}-\frac{e^{\chi}z^{3}\left(a^{\prime}\right)^{2}}{4}-\frac{m^{2}\phi^{2}}{2z}=0\,,$
$\displaystyle\chi^{\prime}-\frac{za^{2}q^{2}\phi^{2}}{f^{2}}e^{\chi}-z(\phi^{\prime})^{2}=0\,,$
$\displaystyle
a^{\prime\prime}+\frac{\chi^{\prime}}{2}a^{\prime}-\frac{2q^{2}\,\phi^{2}}{fz^{2}}a=0\,,$
$\displaystyle\phi^{\prime\prime}-\left(\frac{\chi^{\prime}}{2}+\frac{2}{z}-\frac{f^{\prime}}{f}\right)\phi^{\prime}+\left(\frac{a^{2}q^{2}e^{\chi}}{f^{2}}-\frac{m^{2}}{fz^{2}}\right)\phi=0\,,$
(20)

with asymptotic conditions at the boundary

$f|_{z\rightarrow 0}=1\,,\qquad\chi|_{z\rightarrow 0}=0\,,\qquad
a|_{z\rightarrow 0}=\mu\,,$ (21)

where $\mu$ is chemical potential. With our choice of scalar mass, the
expansion of the scalar field $\phi$ near the boundary reads

$\phi=\phi_{0}z+\phi_{1}z^{2}+\dots\,.$ (22)

With Dirichlet boundary conditions, $\phi_{0}$ can be identified with the
source and $\phi_{1}$ with the expectation value of the operator with
dimension $\Delta_{+}=2$. On the other hand, with Neumann boundary conditions
$\phi_{1}$ is proportional to the source and $\phi_{0}$ to the expectation
value of an operator with dimension $\Delta_{-}=1$. We will mainly work with
the choice $\Delta_{-}$ that corresponds to Neumann boundary conditions for
the scalar field. The empty AdS solution of (20) is

$f=1\,,\qquad\chi=a=\phi=0\ .$ (23)

###  2.2 The Reissner-Nordström black hole

For the Reissner-Nordström (RN) solution we have $\phi,\chi=0$ and

$\displaystyle f_{\rm
RN}(z)=1-\left(\frac{z}{z_{h}}\right)^{3}\left(1+\frac{\rho^{2}z_{h}^{4}}{4}\right)+\frac{\rho^{2}}{4}z^{4}\,,\qquad\rho=\frac{\mu}{z_{h}}\,,$
$\displaystyle a(z)=\mu\left(1-\frac{z}{z_{h}}\right)\,,$ (24)

where $z_{h}$ is the external horizon, $\mu$ is the chemical potential as
before, and $\rho$ is the charge density. The RN black hole has a Cauchy
horizon at $z=z_{c}$. In terms of $(z_{h},z_{c})$ the solution is:

$f_{\rm
RN}(z)=1-z^{3}\frac{(z_{h}+z_{c})(z_{c}^{2}+z_{h}^{2})}{z_{h}^{3}z_{c}^{3}}+z^{4}\frac{z_{h}^{2}+z_{c}^{2}+z_{h}z_{c}}{z_{h}^{3}z_{c}^{3}}\,,\qquad\rho=\frac{2\sqrt{z_{h}^{2}+z_{c}^{2}+z_{h}z_{c}}}{z_{h}^{\frac{3}{2}}z_{c}^{\frac{3}{2}}}\,,$
(25)

and has energy density

$\mathcal{E}=\frac{1}{16\pi
G}\frac{2(z_{h}+z_{c})(z_{c}^{2}+z_{h}^{2})}{z_{h}^{3}z_{c}^{3}}\,.$ (26)

The total black hole charge, mass and entropy are

$Q=\rho\,V_{0}\,,\qquad M=\mathcal{E}\,V_{0}\,,\qquad
S=\frac{V_{0}}{4G}\frac{1}{z_{h}^{2}}\,,$ (27)

the black hole temperature being in turn

$T=-\frac{1}{4\pi}f^{\prime}(z_{h})=\frac{(z_{c}-z_{h})(3z_{c}^{2}+2z_{c}z_{h}+z_{h}^{2})}{4\pi
z_{c}^{3}z_{h}}\,.$ (28)

If we send $\rho\rightarrow 0$, we have $z_{c}\to\infty$ and in this limit we
recover the Schwarzschild black hole. The extremal limit corresponds to

$\mu_{e}=\frac{2\sqrt{3}}{z_{h}}\,,$ (29)

for which $z_{c}=z_{h}$ and the temperature vanishes. It is useful to express
$T/\mu$ as a function of $y=z_{c}/z_{h}$, giving

$\frac{T}{\mu}=\frac{1}{8\pi}\frac{(y-1)(3y^{2}+2y+1)}{y^{3/2}\sqrt{1+y+y^{2}}}\,,$
(30)

see figure 3 for a plot.

Figure 3:  $T/\mu$ as a function of $z_{c}/z_{h}$ for the Reissner-Nordström
AdS spacetime.

When we include the scalar backreaction, the inner horizon of the RN black
hole, which is unstable, disappears and there is singularity at
$z\rightarrow\infty$, just as for the Schwarzchild black hole. This was proved
in various theoretical setting in Hartnoll:2020rwq ; Hartnoll:2020fhc ;
Cai:2020wrp ; An:2021plu .

###  2.3 Approximate solution at large $z$: the Kasner limit

The presence of the scalar $\phi$ gives rise to interesting dynamical
phenomena, which have been investigated in detail in several theoretical
setting in Hartnoll:2020rwq ; Hartnoll:2020fhc ; Frenkel:2020ysx ; Cai:2020wrp
; Henneaux:2022ijt ; Mansoori:2021wxf ; Sword:2021pfm ; Caceres:2022smh . The
full system in eq. (20) can in general only be solved numerically. To make the
analysis more robust, it is therefore useful to introduce analytical
approximations.

In particular, it is convenient to consider a regime in which the scalar
$\phi$ scales logarithmically. Numerical analysis confirms that this is a good
approximation when $z$ is large enough. Let us assume that we can neglect the
electric field $a^{\prime}$, the cosmological constant term, the scalar mass
term $m^{2}$ and its charge $q$. The system in eq (20) is then approximately

$\displaystyle
f^{\prime}-\left(\frac{\chi^{\prime}}{2}+\frac{3}{z}\right)f=0\,,\qquad\chi^{\prime}-z(\phi^{\prime})^{2}$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\phi^{\prime\prime}-\frac{\chi^{\prime}\phi^{\prime}}{2}+\left(\frac{f^{\prime}}{f}-\frac{2}{z}\right)\phi^{\prime}$
$\displaystyle=$ $\displaystyle 0\,.$ (31)

The equations above admit the following exact solution:

$f=-f_{0}z^{3+\alpha^{2}}\,,\qquad\phi=\alpha\,\sqrt{2}\log
z\,,\qquad\chi=2\alpha^{2}\log z+\chi_{0}\,,$ (32)

where $\alpha$, $f_{0}$, $\chi_{0}$ are integration constants. The behaviour
for the metric component $g_{tt}$ is

$g_{tt}=-f_{0}e^{-\chi_{0}}z^{1-\alpha^{2}}\,.$ (33)

With the change of variables

$\tau=\frac{2}{(3+\alpha^{2})\sqrt{f_{0}}}z^{-\frac{3+\alpha^{2}}{2}}\,,$ (34)

where $\tau\to 0$ corresponds to $z\to\infty$, the metric with $f$ and $\chi$
given by eq. (32) can be put in the Kasner form

$ds^{2}=-d\tau^{2}+c_{t}\tau^{2p_{t}}dt^{2}+c_{x}\tau^{2p_{x}}(dx^{2}+dy^{2})\,,$
(35)

where $c_{t},c_{x}$ are constants and the Kasner exponents are

$p_{t}=\frac{\alpha^{2}-1}{3+\alpha^{2}}\,,\qquad
p_{x}=\frac{2}{3+\alpha^{2}}\,.$ (36)

where $p_{t}$ and $p_{x}$ obey the modified Kasner constraints

$p_{t}+2p_{x}=1\,,\qquad
p_{t}^{2}+2p_{x}^{2}=1-\frac{4\alpha^{2}}{(3+\alpha^{2})^{2}}\,.$ (37)

The Schwarzchild solution close to the singularity is of Kasner-type, with
$\alpha=0$ and $p_{t}=-\frac{1}{3}$. The exponent $p_{x}$ is always positive,
and so the transverse direction $x,y$ always experiences a crunch inside the
BH in the regime described by eq. (32). The exponent $p_{t}$ is in turn only
positive for $\alpha>1$, which gives a contracting ER bridge. It is negative
for $\alpha<1$, which corresponds instead to an expanding bridge.

The Kasner approximation in eq. (32) is useful in the regime of large $z$. In
particular, for $\alpha>1$ the approximation is stable: if it is satisfied for
some $z$, the fields go in a direction such that the approximation becomes
even better at larger $z$. In contrast, for $\alpha<1$ the approximation is
not stable for a charged black hole. In this case, at large enough $z$ we are
forced to consider in eq. (20) the backreaction of the electric field
$F_{zt}=a^{\prime}(z)$ on the metric. From the numerical study in
Hartnoll:2020fhc  we know that, also for $\alpha<1$, there often exists a
parametrically large regime in $z$ for which eq. (32) provides a good
approximation. However, for $\alpha<1$ we cannot extrapolate the approximation
all the way to $z\to\infty$, because at some point the backreaction due to the
electric field will be important.

###  2.4 The Kasner inversion

For $\alpha=\alpha_{0}<1$, if we take into account the backreaction of the
electric field the solution in eq. (32) experiences a Kasner inversion, _i.e._
at some point the exponent $\alpha$ jumps from $\alpha_{0}$ to its inverse
$\alpha_{f}=1/\alpha_{0}$,

$\alpha_{0}\qquad\to\qquad\alpha_{f}=\frac{1}{\alpha_{0}}\,.$ (38)

For this reason, in the limit $z\to\infty$ we always expect $\alpha>1$. In
terms of the Kasner exponent $p_{t}$, the transition is

$p_{t}\to-\frac{p_{t}}{2p_{t}+1}\,,$ (39)

and transforms a growing ER bridge ($p_{t}<0$) to a contracting one
$(p_{t}>0)$.

This transition could in principle happen both in the case of neutral or
charged scalars, because it does not require a direct coupling between the
scalar $\phi$ and the gauge field $A_{\mu}$. However, for $q=0$ it seems that
this transition is not realised in the parameter space of the model, which was
studied in Hartnoll:2020rwq . On the other hand, for a charged scalar this
transition is indeed realised in many numerical examples Hartnoll:2020fhc .
Due to non-linear effects in the full system of equations of motion (20),
multiple Kasner transitions are also possible for $q\neq 0$ when the values of
the parametersare fine-tuned .

The Kasner transition in eq. (38) can be derived introducing a more refined
approximation to the system in eq. (20), where we keep also the term which is
responsible for the backreaction of the Maxwell field strength on the metric,
_i.e._

$\displaystyle
f^{\prime}-\left(\frac{\chi^{\prime}}{2}+\frac{3}{z}\right)f-\frac{e^{\chi}z^{3}\left(a^{\prime}\right)^{2}}{4\,L^{2}}$
$\displaystyle=$ $\displaystyle
0\,,\qquad\chi^{\prime}-z(\phi^{\prime})^{2}=0\,,$
$\displaystyle\phi^{\prime\prime}-\left(\frac{\chi^{\prime}}{2}+\frac{2}{z}-\frac{f^{\prime}}{f}\right)\phi^{\prime}$
$\displaystyle=$ $\displaystyle 0\,.\qquad
a^{\prime\prime}+\frac{\chi^{\prime}}{2}a^{\prime}=0\,.$ (40)

Here we still neglect the cosmological constant term and the effects due to
the mass $m$ and the charge $q$ of the scalar field. We can integrate the
equation for $a$ as follows

$a^{\prime}=E_{0}e^{-\frac{\chi}{2}}\,,\qquad a=A_{0}+E_{0}\int
e^{-\frac{\chi}{2}}\,dz\,,$ (41)

where $E_{0}$ and $A_{0}$ are integration constants. With this relation, the
equation for $f$ can be put in the form

$f^{\prime}-f\left(\frac{\chi^{\prime}}{2}+\frac{3}{z}\right)-\frac{E_{0}^{2}z^{3}}{4}=0\,,$
(42)

and assuming the Kasner behaviour for $f$ and $\chi$ given by eq. (32), then
the new term ${E_{0}^{2}z^{3}}/{4}$ dominates the others at large $z$ only for
$\alpha<1$.

Equation (42) can be solved as follows

$\left(\frac{fe^{-\chi/2}}{z^{3}}\right)^{\prime}=\frac{E_{0}^{2}}{4}e^{-\chi/2}\,,\qquad\frac{fe^{-\frac{\chi}{2}}}{z^{3}}=C_{0}+\frac{E_{0}^{2}}{4}\int
e^{-\frac{\chi}{2}}\,dz\,,$ (43)

where $C_{0}$ is an integration constant. We can rewrite the equation
involving the second derivative of $\phi$ as

$\left(\frac{e^{-\frac{\chi}{2}}f\phi^{\prime}}{z^{2}}\right)^{\prime}=0\,.$
(44)

Combining eqs. (43) and (44) we find

$\left(\frac{E_{0}^{2}}{4}e^{-\frac{\chi}{2}}\right)z\phi^{\prime}+\left(C_{0}+\frac{E_{0}^{2}}{4}\int
e^{-\frac{\chi}{2}}\,dz\,\right)(\phi^{\prime}+z\phi^{\prime\prime})=0\,.$
(45)

Following Hartnoll:2020fhc , it is convenient to write $\phi$ in terms of an
auxiliary function $\psi$ defined by

$\phi=\sqrt{2}\int\frac{\psi(z)}{z}dz\,.$ (46)

Note that a constant $\psi(z)=\alpha$ mimics the solution without inversion in
eq. (32). The Kasner inversion correponds to a solution which interpolates
from an initial $\psi=\alpha_{0}$ to a final $\psi=\alpha_{f}$. Here the
physically interesting regime is $0<\alpha_{0}<1$, since otherwise there is no
inversion because the backreaction due to the electric field remains small.

In terms of $\psi$, it is possible to show that eq. (45) implies the following
differential equation Hartnoll:2020fhc

$\psi^{\prime\prime}-\frac{2(\psi^{\prime})^{2}}{\psi}+\frac{\psi^{2}\psi^{\prime}}{z}=0\,,$
(47)

whose solution is

$\psi\left(\psi-\alpha_{0}\right)^{-\frac{1}{1-\alpha_{0}^{2}}}\left(\frac{1}{\alpha_{0}}-\psi\right)^{-\frac{1}{1-\frac{1}{a_{0}^{2}}}}=\frac{z_{0}}{z}\,,$
(48)

where $z_{0}$ is an integration constant. In this case $\psi\to\alpha_{0}$ for
$z\ll z_{0}$ and $\psi\to\alpha_{f}=1/\alpha_{0}$ for $z\gg z_{0}$. This is
indeed the behaviour described in eq. (38).

###  2.5 Neutral scalar field

Let us now discuss the $q=0$ model, which was studied in detail in
Hartnoll:2020rwq . In this case, we can solve explicitly (20) for the gauge
field

$a^{\prime}=-\rho\,e^{-\frac{\chi}{2}}\,,$ (49)

where $\rho$ is the charge density. Indeed, eq. (49) can be physically derived
from the conservation of the electric flux, see appendix A. We end up with a
system composed of the remaining differential equations

$\displaystyle-f^{\prime}+\left(\frac{z\left(\phi^{\prime}\right)^{2}}{2}+\frac{3}{z}\right)f+\frac{\rho^{2}z^{3}}{4}-\frac{\phi^{2}}{z}-\frac{3}{z}=0\,,$
$\displaystyle\phi^{\prime\prime}-\frac{z}{2}\left(\phi^{\prime}\right)^{3}+\left(\frac{f^{\prime}}{f}-\frac{2}{z}\right)\phi^{\prime}+\frac{2}{fz^{2}}\phi=0\,,\qquad\chi^{\prime}=z(\phi^{\prime})^{2}\,.$
(50)

Evaluating the equations above at the horizon $z=z_{h}$, which is defined by
the condition $f(z_{h})=0$, we obtain using the expansion $f(z)\approx
f^{\prime}(z_{h})(z-z_{h})$,

$\displaystyle-f^{\prime}(z_{h})+\frac{\rho^{2}z_{h}^{3}}{4}-\frac{\phi^{2}(z_{h})}{z_{h}}-\frac{3}{z_{h}}=0\,,$
$\displaystyle\frac{1}{z-z_{h}}\left(\phi^{\prime}(z_{h})+\frac{2}{f^{\prime}(z_{h})z_{h}^{2}}\phi(z_{h})\right)+{\rm
finite}=0\,.$ (51)

In order to get a smooth solution, we must then impose the following condition
at the horizon

$\phi(z_{h})=\phi_{h}\,,\qquad\phi^{\prime}(z_{h})=-\frac{8\phi_{h}}{z_{h}\left(\rho^{2}z_{h}^{4}-4\phi_{h}^{2}-12\right)}\,.$
(52)

These boundary conditions, together with $f(z_{h})=0$ and the value of $\rho$,
are enough to solve the problem. A solution is then determined by the values
of $z_{h}$, $\rho$ and $\phi_{h}$.

Alternatively, a given solution can be determined in terms of the temperature
$T$ and the sources $\mu$ and $\phi_{0}$ in the dual CFT. The temperature is:

$T=-\frac{1}{4\pi}f^{\prime}(z_{h})e^{-\frac{\chi(z_{h})}{2}}\,,$ (53)

and the chemical potential is

$\mu=a(0)=\rho\int_{0}^{z_{h}}e^{-\frac{\chi}{2}}\,dz\,.$ (54)

The limit $\phi_{h}\to 0$ can be studied analytically. In this case, the
backreaction of the scalar is negligible up to the location of the Cauchy
horizon $z=z_{c}$ in the unperturbed RN solution, for which the scalar field
$\phi$ identically vanishes. More precisely, the solution is with good
approximation the same as the $RN$ for $z<z_{i}$, where

$z_{i}=z_{c}-\epsilon\,,\qquad\epsilon\to 0\qquad{\rm for}\qquad\phi_{h}\to
0\,.$ (55)

For $z\approx z_{i}$, the dynamics enters in a highly non-linear regime in
which $g_{tt}$ drops exponentially towards zero. This regime was called in
Hartnoll:2020rwq  the collapse of the Einstein-Rosen (ER) bridge.111In some
sense the collapse is also there in the RN solution, because $g_{tt}\to 0$ at
the location of the Cauchy horizon. The main difference here is that the inner
horizon disappears. As can be checked _a posteriori_ , in this limit we can
neglect the mass of the scalar field $\phi$. With this approximation, the
equations of motion (50) take the following form

$\left(\frac{e^{-\chi/2}f}{z^{3}}\right)^{\prime}=e^{-\chi/2}\left(\frac{\rho^{2}}{4}-\frac{3}{z^{4}}\right)\,,\qquad\chi^{\prime}=z(\phi^{\prime})^{2}\,,\qquad\left(\frac{e^{-\chi/2}f\phi^{\prime}}{z^{2}}\right)^{\prime}=0\,.$
(56)

The collapse of the ER bridge happens in a small range of the coordinate $z$,
and so it is consistent to set $z\approx z_{i}$ in the equations of motion
above, keeping $f$, $\chi$ and $\phi$ as functions of $z-z_{i}$, _i.e._

$\left({e^{-\chi/2}f}\right)^{\prime}=e^{-\chi/2}\left(\frac{\rho^{2}}{4}z_{i}^{3}-\frac{3}{z_{i}}\right)\,,\qquad\chi^{\prime}=z_{i}(\phi^{\prime})^{2}\,,\qquad\left({e^{-\chi/2}f\phi^{\prime}}\right)^{\prime}=0\,.$
(57)

The equations for the metric functions can be written as

$\chi^{\prime}f^{2}e^{-\chi}=\tilde{A}\,,\qquad
f^{\prime}-\frac{\chi^{\prime}}{2}f=\tilde{B}\,,$ (58)

where $\tilde{A}$ is an integration constant and

$\tilde{B}=\frac{\rho^{2}}{4}z_{i}^{3}-\frac{3}{z_{i}}\approx\frac{z_{h}^{2}z_{c}+z_{c}^{3}+z_{h}z_{c}^{2}-3z_{h}^{3}}{z_{h}^{3}z_{c}}>0\,.$
(59)

Introducing

$H=z_{i}^{2}g_{tt}=-fe^{-\chi}\,,\qquad G=e^{-\chi}\,,$ (60)

the solution to (58) takes the form

$\frac{\tilde{A}}{2\tilde{B}}\log H+H=-\tilde{C}(z-z_{0})\,,\qquad
G=\frac{(H^{\prime})^{2}}{\tilde{B}\tilde{C}}\,,$ (61)

where $z_{0}=z_{c}$ and $\tilde{C}$ are integration constants. In the limit
$\phi_{h}\to 0$, we expect that $\tilde{A}\to 0$ while $\tilde{C}\approx
f^{\prime}_{RN}(z_{c})=O(\phi_{h}^{0})$. The scalar $\phi$ is given by

$\phi=-\frac{\tilde{A}^{1/2}}{(\tilde{B}\tilde{C})^{1/2}}\frac{1}{z_{i}^{1/2}}\log(H\tilde{D})\,,$
(62)

where $\tilde{D}$ is another integration constant. In the limit $\phi_{h}\to
0$, the equation for the scalar is approximately linear for $z<z_{i}$, because
the scalar is small. Then the amplitude of the scalar in eq. (62) must scale
linearly in $\phi_{h}$, _i.e._

$\tilde{A}^{1/2}=O(\phi_{h})\,,\qquad{\rm for}\qquad\phi_{h}\to 0\,.$ (63)

For $z\gg z_{0}$ we find that $g_{tt}$ approaches to zero exponentially,

$g_{tt}=\frac{H}{z_{i}^{2}}\approx\frac{1}{z_{i}^{2}}\exp\left(-\frac{2\tilde{B}\tilde{C}}{\tilde{A}}(z-z_{0})\right)\,,$
(64)

This approximation can then be matched with the Kasner approximation, which is
valid at large $z$. In particular, matching eqs. (32) and (61) we find

$e^{-\chi_{0}/2}f_{0}=-z^{3}\,e^{-\chi/2}f=z^{3}\frac{H}{G^{1/2}}\approx
z_{c}^{3}\frac{\tilde{A}}{2(\tilde{B}\tilde{C})^{1/2}}\,.$ (65)

From eq. (63) we find that $e^{-\chi_{0}/2}f_{0}=O(\phi_{h}^{2})$ for
$\phi_{h}\to 0$.

A numerical approach (see appendix B for our numerical techniques) can be used
to solve (50) also away from the limit $\phi_{h}\to 0$. In general, there is a
crossover at $z\approx z_{c}$ from a solution qualitatively similar to the RN
for $z\ll z_{c}$ and a Kasner regime for $z\gg z_{c}$. In the intermediate
region, the metric $g_{tt}$ drops exponentially to a value very close to zero.
This drop is very sudden in the $\phi_{h}\to 0$ limit. In all the parameter
space investigated in Hartnoll:2020rwq , the solution stabilizes to a Kasner
regime with $\alpha>1$ just after the collapse of the ER bridge. For this
reason, the Kasner inversion in eq. (38) never takes place. In figure 4 we
reproduce the numerical analysis of Hartnoll:2020rwq , showing how the Kasner
coefficient $p_{t}$ depends on the parameter space of solutions.

Figure 4:  Kasner exponent $p_{t}$ for the neutral scalar case ($q=0$). Left:
$p_{t}$ as a function of ${T}/{\mu}$ and ${\phi_{h}}/{\mu}$. Right: $p_{t}$ as
a function of ${T}/{\mu}$ and ${\phi_{0}}/{\mu}$. Here we have $p_{t}>0$ and
$\alpha>1$ in all the parameter space, and so there is no Kasner inversion.
The points marked with letters A and B correspond to the specific examples
shown in fig. 5.

For the $q=0$ case it is necessary to introduce a source to get solutions with
non-zero scalar profile, _i.e._ the $\phi_{h}=0$ line corresponds to the
$\phi_{0}=0$ line in the two diagrams in figure 4. There is a region at very
small temperatures $T/\mu\approx 10^{-4}$ which is not explored in fig. 4. In
the extremal RN limit, the near horizon geometry is described by AdS2
$\times\mathbb{R}^{2}$ with a different AdS radius, and the scalar with mass
$m^{2}=-{2}/{L^{2}}$ is below its BF bound, such that condensation can occur
Hartnoll:2008kx .

Two examples of numerical solutions are presented in figure 5. In the example
(A) we take $\phi_{h}$ rather large and far away from the limit $\phi_{h}\to
0$. Example (B) is closer to the $\phi_{h}\to 0$ limit, and indeed we see that
the collapse of the Einstein-Rosen bridge is very fast in the coordinate $z$.
Various interesting features can be seen in these plots. It is clear that for
all of the solutions the late Kasner behaviour predicted in (32) is present,
with $\alpha$ controlling the linear behaviour of $\log f$ and $\phi$ vs.
$\log z$. The example (B) is a typical case of small $\phi_{h}$ which is
almost identical to RN up to $z=z_{c}$. In the $\phi_{h}\to 0$ limit, the
backreaction of the scalar is negligible only up to the “would be” Cauchy
horizon $z_{c}$ of the RN solution. At $z_{c}$ there is a transition to the
Kasner regime, which is very sharp in the $z$ coordinate. Approaching
$\phi_{h}=0$ is not a continuous limit for $z>z_{c}$, since in this limit we
have $\alpha\to\infty$ and $p_{t}\to 1$. Then there is a crossover between the
RN solution and the Kasner regime discussed in eq. (32). This transition
becomes very sharp around $z_{c}$ as $\phi_{h}\to 0$.

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(A) | (B)
Figure 5: Sample solutions for the neutral $q=0$ case. For illustration
purposes, we have chosen the values $(z_{h},\rho,\phi_{h})=(1,0.9,1.45)$ for
(A) and $(1,0.35,0.035)$ for (B), where the letters match the ones figure 4.
The vertical dotted line is the “would be” Cauchy horizon $z_{g_{tt}}$.

In order to describe the causal structure of the solution, it is convenient to
use Penrose diagrams. In these diagrams there is always an ambiguity in the
choice of the function used in the conformal mapping, so for completeness we
explain our conventions in appendix C. There is an important conformally
invariant property of the Penrose diagrams which is universal and does not
depend on this arbitrary choice Fidkowski:2003nf . Schematically, the Penrose
diagram of an eternal asymptotically AdS black hole with a spacelike
singularity can be drawn as a square, where the vertical sides are the
boundaries of AdS. The spacelike singularity can then bend inwards or outwards
the horizontal sides of the square inside the black hole horizon. The
direction of this bending is a conformally invariant property, which can be
established as follows: let us prepare a couple of ingoing radial lightlike
geodesics from the left and the right boundaries at time $t_{b}=0$, and ask
whether they meet before reaching the singularity or not. If the singularity
coincides with the horizontal line of the square, the two light rays meet
exactly at the singularity. If instead the singularity is bending upwards the
top side of the square, the two rays meet before intersecting the singularity;
finally, if the singularity is bending the top side of the square downwards,
the two light rays never meet. This shows that the bending of the singularity
is a physical property and does not depend on the choice of conventions used
to construct the Penrose diagram.

For the non-rotating BTZ black hole in AdS3 Banados:1992wn , the singularity
lies exactly on the horizontal line. For Schwarzschild AdSd black holes with
$d>3$ the singularity bends the top side downwards. In the case of the black
holes discussed in this section, we can in general achieve in the parameter
space both upwards and downwards bending of the top side of the Penrose
diagram. The kind of bending of the singularity in the Penrose diagram, as we
will find in section 4, will influence the time dependence of the action
complexity. For convenience, we will denote a solution with the singularity
with upper bending as type $U$ and one with the singularity with lower bending
as type $D$.

The black dashed line in figure 4 separates two shapes of Penrose diagram: the
solutions on the left side of the line are of type $U$, while the ones on the
right side are of type $D$. In particular, in the $\phi_{h}\to 0$ limit, the
singularity approaches the Cauchy horizon of the RN black hole, and so the
bending is up. In figure 6 we show two examples of these Penrose diagram; we
plot on the diagram the value of $e^{-\chi}$, which gives a measure of how
much the solution is different from the RN limit, which has $\chi=0$.

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(A) | (B)
Figure 6: Penrose diagrams for the sample solutions A and B, marked in fig. 4
and whose field profiles are shown in fig. 5. On the right, we see that for
case B, which is close to the limit $\phi_{h}\to 0$, the collapse of the ER
bridge is very fast, as mentioned in the main text. We refer to appendix C for
the definition of the coordinates of the Penrose diagram.

###  2.6 Charged scalar field

The explicit examples studied in Hartnoll:2020fhc  correspond to the zero
source limit, which is known as for holographic superconductors
Hartnoll:2008vx ; Hartnoll:2008kx . In the charged case, the main term which
can induce condensation of the scalar close to the horizon is the term in the
Lagrangian coming from scalar covariant derivative,

$-q^{2}g^{tt}(A_{t})^{2}\phi\phi^{*}\,.$ (66)

This term induces a tachyonic mass term for the scalar in the region laying
just outside the horizon, where $g^{tt}\to-\infty$ and $A_{t}\to 0$. This is
the reason for the condensation of the scalar in the bulk to be rather common
in the charged case, even with a vanishing boundary source. Two well studied
boundary conditions for holographic superconductors are

  * •

Dirichlet, which is realised for $\phi_{0}=0$ in eq. (22), with expectation
value $\phi_{1}$.

  * •

Neumann, which is realised by $\phi_{1}=0$ in eq. (22), with expectation value
$\phi_{0}$.

For both boundary conditions above, at fixed $\rho$ there is a critical
temperature $T_{c}$ such that the expectation values are non-zero for
$T<T_{c}$. By dimensional analysis, the critical temperature must be
proportional to $\sqrt{\rho}$, _i.e._

$T_{c}=C(q)\,{\rho}^{1/2}\,,$ (67)

where $C(q)$ is a function of the charge. For a plot of $C(q)$ for Neumann and
Dirichlet boundary conditions, see fig. 2 of Hartnoll:2008kx . In the regime
of large $q$, we have

$\displaystyle C(q)$ $\displaystyle\approx$ $\displaystyle
0.12\,\sqrt{q}\qquad{\rm for\,Dirichlet}\,,$ $\displaystyle C(q)$
$\displaystyle\approx$ $\displaystyle 0.23\,\sqrt{q}\qquad{\rm
for\,Neumann}\,.$ (68)

Just below the critical temperature $T_{c}$ the condensate is negligible on
the solution, at least outside the Cauchy horizon. Then, very close to $T_{c}$
the solution is given in good approximation by the RN solution. Using
expressions (28) and (25) to express $(T,\rho)$ in terms of $(z_{c},z_{h})$,
we find that the critical temperature $T_{c}$ corresponds to the value of
$y=z_{c}/z_{h}$ obtained by solving the following equation

$\frac{3y^{3}-y^{2}-y-1}{4\sqrt{2}\pi\sqrt[4]{y^{9}\left(y^{2}+y+1\right)}}=C(q)\,.$
(69)

This expression, in combination with the plot of $C(q)$ in Hartnoll:2008kx ,
can be use to find the holographic superconductor regime in the parameter
space.

As for the uncharged case, the $\phi_{h}\to 0$ limit can also be studied
analytically when $q\neq 0$. Once more, in this limit the backreaction of the
scalar is negligible up to $z=z_{i}$, see eq. (55). For $z\approx z_{i}$, we
have again the collapse of the Einstein-Rosen bridge. As can be checked _a
posteriori_ , to study the collapse of the bridge we can neglect the mass of
the scalar field $\phi$ and the electric source term in Maxwell’s equation. In
particular, the gauge field $a$ still satisfies with good approximation eq.
(49). The equations of motion (20) then take the form

$\displaystyle\left(\frac{e^{-\chi/2}f}{z^{3}}\right)^{\prime}={e^{-\chi/2}}\left(\frac{\rho^{2}}{4}-\frac{3}{z^{4}}\right)\,,\qquad\chi^{\prime}=\frac{za^{2}q^{2}\phi^{2}}{f^{2}}e^{\chi}+z(\phi^{\prime})^{2}\,,$
$\displaystyle\left(\frac{e^{-\chi/2}f\phi^{\prime}}{z^{2}}\right)^{\prime}=-\frac{1}{z^{2}}\frac{a^{2}q^{2}e^{\chi/2}}{f}\phi\,.$
(70)

The collapse takes place in a small range of the coordinate $z$, so that as in
the uncharged case it is consistent to set $z\approx z_{i}$ in the equations
of motion above. In correspondence with the collapse, the field $\chi$ rapidly
becomes large so that from eq. (49) the gauge field $a$ can be taken to be
practically constant, _i.e._ $a(z)=a_{0}$. With these simplifications, we must
solve the system

$\displaystyle\left({e^{-\chi/2}f}\right)^{\prime}={e^{-\chi/2}}\left(\frac{\rho^{2}}{4}z_{i}^{3}-\frac{3}{z_{i}}\right)\,,\qquad\chi^{\prime}=z_{i}\left(\frac{a_{0}^{2}q^{2}\phi^{2}}{f^{2}}e^{\chi}+(\phi^{\prime})^{2}\right)\,,$
$\displaystyle\left({e^{-\chi/2}f\phi^{\prime}}\right)^{\prime}=-\frac{a_{0}^{2}q^{2}e^{\chi/2}}{f}\phi\,.$
(71)

It turns out that the solution for $f$ and $\chi$ has the same functional form
as for $q=0$. The functional form of the scalar field $\phi$ is now different,
however. We can solve for $\phi$ in terms of $\chi$ and $f$,

$\phi(z)=\Phi_{0}\,\cos\left(q\,a_{0}\,\int_{z_{i}}^{z}\frac{e^{\chi/2}}{f}dz+\varphi_{0}\right)\,,$
(72)

where $\Phi_{0}$ and $\varphi_{0}$ are integration constants. Using the
relation

$q^{2}\phi^{2}a_{0}^{2}+e^{-\chi}f^{2}(\phi^{\prime})^{2}=q^{2}\Phi_{0}^{2}a_{0}^{2}\,,$
(73)

we get the same system as for $q=0$, see eq. (58), with $\tilde{B}$ given by
eq. (59) and

$\tilde{A}=z_{i}q^{2}\Phi_{0}^{2}a_{0}^{2}\,.$ (74)

The solution is given by eq. (61), with eq. (65) still holding. Note that in
the $\phi_{h}\to 0$ limit we still have
$e^{-\chi_{0}/2}f_{0}=O(\phi_{h}^{2})$, because $\Phi_{0}=O(\phi_{h})$.

The system can be solved numerically also away from the limit $\phi_{h}\to 0$,
as done in Hartnoll:2020fhc . See appendix B for a description of our
numerical methods.

In particular, if we consider the limit $\phi_{h}\to 0$ keeping $T$ and $\rho$
fixed, we find that after the collapse of the ER bridge the solution flows to
a Kasner regime, with parameter $\alpha(\phi_{h})$. As a function of
$\phi_{h}$, the Kasner parameter $\alpha(\phi_{h})$ is not continuous for
$\phi_{h}\to 0$, and it oscillates between values of $\alpha$ which can be
bigger or lower than one. In particular, if the value of $\alpha$ after the
collapse is less than one we see that there is a Kasner inversion, which in
the approximation discussed in 2.4 brings $\alpha\to 1/\alpha$. Something
special happens in the limit $\alpha\to 0$: in this case the approximations
used in section 2.4 fail, because we should also take into account terms
involving the charge of the scalar field. In this case multiple Kasner
inversions are possible, see Henneaux:2022ijt  for a recent study with the
billiard approach.

In figure 7 we show an example of a solution without inversions, while in
figure 8 we show an example with one inversion. Penrose diagrams for these
solutions are shown in figure 9.

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Figure 7: Sample solution for the charged case, without Kasner inversion. For illustration purposes, we have chosen the values $\rho=2,\phi_{h}=0.16,z_{h}=1,q=1$ and $\chi_{h}=0.1$. The vertical dotted line is the “would be” Cauchy horizon $z_{g_{tt}}$. In the little boxes we display a magnification of the plots near $z_{h}=1$.  |
---|---
|
Figure 8: Sample solution for the charged case, with a single Kasner inversion. For illustration purposes, we have chosen the values $\rho=2,\phi_{h}=0.3,z_{h}=1,q=1$ and $\chi_{h}=0.1$. The vertical dotted line is the “would be” Cauchy horizon $z_{g_{tt}}$. |
---|---
Figure 9: Penrose diagrams for the charged solutions shown in Figures 7 left,
and 8 right.

Figure 10 displays a sketch of the Kasner exponent as a function of
$\phi_{0}/\mu$ and $T/\mu$. Something similar could be obtained using
$\sqrt{q\phi_{1}}/\mu$ instead, however the value of this condensate is very
large for $q=1$ as previously found in Hartnoll:2008kx , meaning the plot
becomes less clear to visualize. For this reason we choose not to display it
here.

Figure 10:  The Kasner exponent $p_{t}$ as a function of the parameter space
for $q=1$. The black solid line is the line of zero source $\phi_{1}=0$ with
spontaneous condensation. The white region in this plot is where our numeric
procedure didn’t converge. The plot shows the last value of $p_{t}$ displayed
by our solutions at the maximum possible value of $z$. The dashed line divides
Type U (below) and type D (above) solutions.

###  2.7 A conserved quantity

Inserting the ansatz (18,19) into the action (14) we obtain

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{V_{0}}{16\pi G}\int
dz\left\\{-\frac{e^{-\frac{\chi}{2}}f\chi^{\prime}}{z^{3}}-\frac{12fe^{-\frac{\chi}{2}}}{z^{4}}+\frac{4f^{\prime}e^{-\frac{\chi}{2}}}{z^{3}}+\left(\frac{e^{-\frac{\chi}{2}}f\chi^{\prime}}{z^{2}}-\frac{e^{-\frac{\chi}{2}}f^{\prime}}{z^{2}}\right)^{\prime}\right.$
$\displaystyle\qquad\qquad\quad\left.+\frac{6e^{-\frac{\chi}{2}}}{z^{4}}+\frac{e^{\frac{\chi}{2}}(a^{\prime})^{2}}{2}-\frac{e^{-\frac{\chi}{2}}f|\phi^{\prime}|^{2}}{z^{2}}+\frac{e^{-\frac{\chi}{2}}}{z^{4}}\left(\frac{z^{2}e^{\chi}q^{2}a^{2}}{f}-m^{2}\right)|\phi|^{2}\right\\}.$

This action is invariant under the symmetry Gubser:2009cg

$z\to\lambda z\,,\qquad\chi\to\chi-6\log\lambda\,,\qquad a\to\lambda^{2}a\,,$
(76)

with fixed $f$ and $\chi$. From this invariance, it follows that the quantity

$\tilde{Q}=e^{\frac{\chi}{2}}\left(\frac{(fe^{-\chi})^{\prime}}{z^{2}}-aa^{\prime}\right)\,,$
(77)

is conserved, i.e. $\tilde{Q}^{\prime}=0$, as can be checked from the
equations of motion (20). For the RN solution, the conserved quantity in eq.
(77) is

$\tilde{Q}=\frac{z_{h}\rho^{2}}{4}-\frac{3}{z_{h}^{3}}\,.$ (78)

In this case $\tilde{Q}$ is generically negative and vanishes in the extremal
limit.

In order to compute $\tilde{Q}$ in the Kasner regime eq. (32), let us first
discuss the screening of electric charge. We can write in general $a^{\prime}$
as

$a^{\prime}=-\tilde{\rho}(z)\,e^{-\frac{\chi}{2}}\,,$ (79)

in such a way that $\tilde{\rho}(0)=\rho$. We can introduce
$\rho_{f}=\tilde{\rho}(\infty)$. Using the equations of motion, we find:

$\tilde{\rho}^{\prime}(z)=-2L^{2}q^{2}\,a\,\frac{\phi^{2}e^{\frac{\chi}{2}}}{fz^{2}}\,,\qquad\tilde{\rho}(z)=\rho-2L^{2}q^{2}\,\int_{0}^{z}a\,\frac{\phi^{2}e^{\frac{\chi}{2}}}{f\tilde{z}^{2}}\,d\tilde{z}\,.$
(80)

If we insert the approximation in eq. (32) in the conserved quantity in eq.
(77), we get

$\tilde{Q}=f_{0}e^{-\frac{\chi_{0}}{2}}(\alpha^{2}-3)+\tilde{\rho}(z)a(z)\,.$
(81)

We will see that this relation plays an important role in determining the
asymptotic complexity rate for the action conjecture.

In particular, it is interesting to consider the $\phi_{h}\to 0$ limit. In
this case, for $z<z_{i}$ defined in eq. (55), the profiles $f$, $\chi$ and $a$
are given by the RN solution. We can then compare the conserved charge
$\tilde{Q}$ of the RN solution in eq. (78) with the one of the final Kasner in
eq. (81), i.e.

$\tilde{Q}=\frac{z_{h}\rho^{2}}{4}-\frac{3}{z_{h}^{3}}=f_{0}e^{-\frac{\chi_{0}}{2}}(\alpha^{2}-3)+\rho_{f}\,a(\infty)\,.$
(82)

In the limit $\phi_{h}\to 0$, from the analytic solutions in eq. (61), we have
that $\chi$ becomes suddenly very large at the would be Cauchy horizon
$z_{c}$. Moreover, from the equations of motion eq. (20) we know that $\chi$
is a monotonic function of $z$. Then, from eq. (79), we have that $a$
approaches a constant for $z>z_{c}$. Then it follows that

$a(\infty)\approx a_{\rm RN}(z_{c})=-(z_{c}-z_{h})\rho\,.$ (83)

Let us now distinguish the two cases:

  * •

For $q=0$, there is no screening of electric charge, i.e.
$\tilde{\rho}(z)=\rho=\rho_{f}$. Using this result in eq. (82), we find a
useful relation between $\alpha$, $f_{0}$ and $\chi_{0}$ valid in the
$\phi_{h}\to 0$ limit:

$f_{0}e^{-\frac{\chi_{0}}{2}}(\alpha^{2}-3)=\frac{(z_{c}-z_{h})(z_{c}^{2}+2z_{h}z_{c}+3z_{h}^{2})}{z_{h}^{3}z_{c}^{3}}\,.$
(84)

From eq. (65) we find

$f_{0}e^{-\frac{\chi_{0}}{2}}=O(\phi_{h}^{2})\,.$ (85)

As a consequence, $\alpha=O(1/\phi_{h})$, as can be checked from numerical
calculations. We can approximate the previous relation for $\phi_{h}\to 0$ as
follows

$\alpha^{2}=\frac{1}{f_{0}e^{-\frac{\chi_{0}}{2}}}\frac{(z_{c}-z_{h})(z_{c}^{2}+2z_{h}z_{c}+3z_{h}^{2})}{z_{h}^{3}z_{c}^{3}}\,.$
(86)

  * •

For the first Kasner region we have that eq. (85) still holds because the
solution in eq. (61) and the estimate eq. (65) still hold also for $q\neq 0$.
During a Kasner inversion, we expect that the value of
$f_{0}e^{-\frac{\chi_{0}}{2}}$ stays almost constant, because from equations
of motion we have

$\left(\frac{e^{-\chi/2}f}{z^{3}}\right)^{\prime}={e^{-\chi/2}}\left(\frac{\tilde{\rho}^{2}}{4}-\frac{3}{z^{4}}\right)\,,$
(87)

and, also, we have that $\chi$ is large in the $\phi_{h}\to 0$ limit. The
exponent $\alpha$ does not tend to infinity; instead, the value of $\alpha$
after the final inversion oscillates wildly for $\phi_{h}\to 0$ among all
possible values from $1$ to $\infty$. As a consequence of eq. (82) we have

$\frac{\rho_{f}}{\rho}=\frac{1+2y+3y^{2}}{4(1+y+y^{2})}\,,\qquad
y=\frac{z_{c}}{z_{h}}\,.$ (88)

##  3 Volume complexity

In this section we will first treat with a unified approach different extremal
surfaces with different dimensions $k$, and then we will focus on the $k=3$
case, which is the one relevant for the volume conjecture. Our results are
consistent with Yang:2019gce , which focuses on the charged scalar case with
vanishing external sources.

###  3.1 Extremal bulk surfaces

It is interesting to probe the asymptotically AdS4 black hole geometry with
extremal bulk surfaces with different dimensions. In particular, the length of
dimension one spacelike curves is related to correlators of heavy scalar
operators in the WKB approximation Balasubramanian:1999zv ; Fidkowski:2003nf ;
Kraus:2002iv ; Festuccia:2005pi ; Festuccia:2008zx . The area of dimension two
extremal surfaces is the holographic dual of the entanglement entropy
Ryu:2006bv ; Hubeny:2007xt ; Hartman:2013qma . In the CV conjecture, the
complexity is dual to the volume of dimension three extremal bulk surfaces
Susskind:2014rva ; Stanford:2014jda ; Carmi:2017jqz . In this section we will
briefly discuss how to evaluate these three functionals in the black hole
solution that we discussed in section 2.

It is useful to change variables, introducing the lightcone coordinate $v$
defined by

$dv=dt-\frac{e^{\frac{\chi}{2}}}{f}dz\,,$ (89)

where $v={\rm constant}$ correspond to the ingoing radial null geodesics. The
metric (18) in the coordinates $(v,z,x,y)$ reads

$ds^{2}=\frac{1}{z^{2}}\left(-fe^{-\chi}dv^{2}-2e^{-\frac{\chi}{2}}dvdz+dx^{2}+dy^{2}\right)\,.$
(90)

Let us then consider the following particular cases of extremal surfaces with
different dimensions:

  * •

A spacelike radial geodesic (with constant $x$ and $y$), parametrized by
$z=z(\lambda)$ and $v=v(\lambda)$. The length is

$\ell=\int
d\lambda\,\frac{1}{z}\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}\,.$
(91)

  * •

A dimension two surface with constant $y$, parametrized by $(\lambda,x)$. The
area functional is

$A=L_{x}\int
d\lambda\,\frac{1}{z^{2}}\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}\,.$
(92)

where $L_{x}=\int dx$ is a cutoff length in the $x$ direction.

  * •

A dimension three surface, parametrized by $(\lambda,x,y)$. The volume
functional is

$V=V_{0}\int
d\lambda\,\frac{1}{z^{3}}\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}\,.$
(93)

where $V_{0}=\int dxdy$ is a cutoff area in the $x,y$ plane.

###  3.2 Equations of motion

We can treat the three functionals as particular cases of the same functional

$S_{k}=\int
d\lambda\,\frac{1}{z^{k}}\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}=\int
d\lambda\,\mathcal{L}_{k}\,.$ (94)

Since $S_{k}$ is translationally invariant in $v$, we can obtain the conserved
quantity

$E_{k}=-\frac{\partial\mathcal{L}_{k}}{\partial\dot{v}}=\frac{1}{z^{k}}\frac{\dot{z}\,e^{-\frac{\chi}{2}}+\dot{v}fe^{-\chi}}{\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}}\,.$
(95)

Following Belin:2021bga , it is convenient to choose the parameter $\lambda$
in this way

${z^{k}}\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}=e^{-\frac{\chi}{2}}\,.$
(96)

With this choice, from eq. (95) we obtain

$\dot{v}=(E_{k}-\dot{z})\frac{e^{\frac{\chi}{2}}}{f}\,,$ (97)

which, inserted in eq. (96) gives

$\dot{z}^{2}+V_{k}(z)=E_{k}^{2}\,,\qquad
V_{k}(z)=-\frac{e^{-\chi}\,f}{z^{2k}}\,.$ (98)

The problem is recasted as the motion of a classical non-relativistic particle
in a potential $V_{k}(z)$.

###  3.3 The time dependence

Using eqs. (96, 98), the functional $S_{k}$ in eq. (94) can be written as
follows

$S_{k}=\int_{z_{\epsilon}}^{z_{t}}\frac{dz}{z^{k}\sqrt{f+E_{k}^{2}z^{2k}e^{\chi}}}\,.$
(99)

$z_{\epsilon}$ is the UV cutoff and $z_{t}$ is the turning point. The value of
$z_{t}$ can be obtained setting $\dot{z}=0$ in eq. (98):

$V_{k}(z_{t})=E_{k}^{2}\,.$ (100)

The turning point $z_{t}$ requires $f(z_{t})\leq 0$, and so it is inside the
horizon $z_{h}$. The quantity $E_{k}$ is function of the boundary time $t_{b}$
at which the probe is anchored. Note also that for $z_{t}=z_{h}$, we have that
$E_{k}=0$. As a convention, we set $t_{b}=0$ for $E_{k}=0$.

The difference in $v$ coordinates

$v\left(z_{t}\right)-v\left(z_{\epsilon}\right)=\int_{z_{\epsilon}}^{z_{t}}\frac{dz}{f}\left(\frac{e^{\chi}z^{k}E_{k}}{\sqrt{f+e^{\chi}E_{k}^{2}z^{2k}}}-e^{\frac{\chi}{2}}\right)\,.$
(101)

The boundary time can be obtained by integrating eq. (89)

$\int_{z_{\epsilon}}^{z_{t}}dt=\int_{z_{\epsilon}}^{z_{t}}dv+\int_{z_{\epsilon}}^{z_{t}}\frac{dz}{f}e^{\frac{\chi}{2}}\,.$
(102)

which gives

$t(z_{t})-t(z_{\epsilon})=v(z_{t})-v(z_{\epsilon})+\int_{z_{\epsilon}}^{z_{t}}\frac{dz}{f}e^{\frac{\chi}{2}}\,,$
(103)

By symmetry argument at the turning point $t(z_{t})=0$. We can then solve for
the boundary time:

$t_{b}=t(z_{\epsilon})=-\int_{z_{\epsilon}}^{z_{t}}\frac{dz}{f}\frac{e^{\chi}z^{k}E_{k}}{\sqrt{f+e^{\chi}E_{k}^{2}z^{2k}}}\,.$
(104)

Using eq. (104), we can rewrite $S_{k}$ in eq. (99) as follows

$S_{k}=\int_{z_{\epsilon}}^{z_{t}}\frac{\sqrt{f+E_{k}^{2}z^{2k}e^{\chi}}}{z^{k}f}dz+E_{k}t_{b}$
(105)

Taking time derivative of (105) and using eq. (100), we find

$\frac{dS_{k}}{dt_{b}}=E_{k}\,.$ (106)

This proves that the derivative of $S_{k}$ with respect $t_{b}$ is exactly
$E_{k}$. Using the potential $V_{k}$ is defined in eq. (98), we can express
the time derivative of $S_{k}$ as a function of $z_{t}$ as follows:

$\frac{dS_{k}}{dt_{b}}=\sqrt{V_{k}(z_{t})}\,.$ (107)

The relation between boundary time $t_{b}$ and $z_{t}$ must be determined
numerically, by integrating eq. (104).

We can write eq. (104) in the following form

$t_{b}=\int_{z_{\epsilon}}^{z_{t}}\,\frac{1}{(-fe^{-\chi/2})}\frac{1}{\sqrt{E_{k}^{2}-V_{k}(z)}}\,dz\,.$
(108)

Figure 11 shows a plot of $\sqrt{V_{3}}$ as a function of $t_{b}$.

Figure 11: Volume complexity rate as a function of time for a black hole with
$\rho=2$, $z_{h}=1$ and $q=1$.

Note that the integrand in eq. (108) is diverging for $z=z_{t}$, since
$E_{k}^{2}=V_{k}(z_{t})$. In order for the result of the integral to diverge
logarithmically, we have to impose that $E_{k}^{2}-V_{k}(z)$ is quadratic in
$z$ for $z=z_{t}$. For this reason, the late time limit $t_{b}\to\infty$
corresponds to a maximum of $V_{k}(z_{t})$.

For the Schwarzschild case, $V_{2,3}$ have always a unique extremal point,
which is a maximum. On the other hand $V_{1}$ is a monotonic function which
has no extremal points. For this reason, spacelike geodesic can pass
arbitrarily near the singularity Fidkowski:2003nf . Instead dimension two and
three extremal surfaces are respectively stuck at the value of $z$
corresponding to the maximum of $V_{2}$ and $V_{3}$.

In the RN case, we have that $V_{k}$ has always a unique extremal point, which
is a maximum, at a finite value of $z<z_{c}$. So, no extremal surface can
approach the singularity.

If we consider the model in eq. (14), we have that, in the $\phi_{h}\to 0$
limit, the field profiles tend to the RN solution for $z<z_{c}$. So, in this
limit, the potential is the same as in the RN case and it has a maximum for
$z_{h}<z<z_{c}$. The potential may have additional extremal points for
$z>z_{c}$. Using the Kasner approximation in eq. (32), we get that at large
$z$

$V_{k}(z)=f_{0}e^{-\chi_{0}}z^{3-2k-\alpha^{2}}\,.$ (109)

For $k>1$, we get that $V_{k}(z)$ has no maximum in the Kasner region. As
pointed out in Hartnoll:2020fhc , for geodesics ($k=1$) we can get another
maximum of $V_{k}(z)$ in correspondence of the Kasner inversion. In this case
there exist values of the boundary time for which multiple extremal surfaces
are possible. It would be interesting to understand the physical meaning of
these multiple solutions.

###  3.4 Asymptotic complexity rate

The asymptotic volume complexity rate is given by

$W_{V}=\lim_{t_{b}\to\infty}\frac{d\,C_{V}}{d\,t_{b}}=\frac{V_{0}}{G}\sqrt{V_{3}(z_{\rm
max})}\,,$ (110)

where $z_{\rm max}$ is the position of the maximum of $V_{3}(z)$, see eq.
(98). For the Schwarzchild solution we have

$W_{V}=\frac{V_{0}}{2G}\frac{1}{z_{h}^{3}}=\frac{8\pi}{3}\,TS\,.$ (111)

In the extremal limit, for the RN solution the rate vanishes and at the first
order in $z_{c}-z_{h}$ is

$W_{V}\approx\sqrt{\frac{3}{2}}\,\frac{V_{0}}{G}\frac{z_{c}-z_{h}}{z_{h}^{4}}=\sqrt{\frac{3}{2}}\,\frac{8\pi}{3}\,TS\,.$
(112)

We expect that the asymptotic complexity rate, in units of $TS$, is a slowly
varying function of the parameters of the model. This is consistent with the
numerical results shown in figure 12. In figure 13 we show the behavior at
constant temperature of the asymptotic volume complexity, as a function of
$\phi_{0}/\mu$.

Figure 12:  Asymptotic volume complexity rate $W_{V}$, as defined in eq. (110), in units of $TS$. Left: Neutral scalar case ($q=0$). Blue ploints denote type U solutions, red points denote type D solutions. Right: Charged scalar case with $q=1$. Blue points denote type U solutions, yellow points denote type D solutions. In both cases, there is no direct correlation between the complexity rate and the coefficient $p_{t}$ plotted in figures 4 and 10. The white region of the plots is not covered by our numerics.  |
---|---
Figure 13:  Left: Asymptotic volume complexity for charged solutions with
$q=1$ at constant $T/\mu=0.15$. Right: Asymptotic volume complexity for
uncharged solution at $T/\mu=0.25$.

The Lloyd bound Lloyd  conjectures that the rate of computation of a quantum
computer is bounded by a quantity which is proportional to the total energy of
the system. In the context of holography, in Brown:2015lvg  it was conjectured
that the Lloyd bound for the complexity rate is saturated by the uncharged
planar BH in AdSd+1, i.e.

$\frac{d\,C_{V}}{dt_{b}}\leq\frac{8\pi}{d-1}M\,,$ (113)

where $M$ is the mass of the black hole. In the complexity=volume conjecture,
the complexity rate is a monotonically increasing function of the time and so
it is enough to check it at late time.

In our setting there is an extra subtlety, because the value of the black hole
mass depends on the choice of the boundary conditions, see appendix D for
details. We checked the Lloyd bound both for the Dirichlet and Neumann choices
of boundary conditions, that for $d=3$ is

$W_{V}\leq 4\pi M_{D,N}\,,$ (114)

where $M_{D,N}$ are in eqs. (199,201). We find that the bound is satisfied in
all the parameter space that we explored, see fig. 14 for some sample plots
for fixed value of $\phi_{0}$. This is consistent with Yang:2019gce  , who
numerically checked the Lloyd bound in the case of zero sources (for which the
Neumann and Dirichlet masses coincide).

Figure 14:  In these pictures, we show $W_{V}/(TS)$ in solid line, $(4\pi
M_{D})/(TS)$ in blue dashed line and $(4\pi M_{N})/(TS)$ in red dashed line
(see eqs. (110),(114)) as a function of $T/\mu$ for a fixed $\phi_{0}$. Left:
neutral scalar. Right: charged $q=1$ scalar. The Lloyd bound is always
satisfied for both the Dirichlet and the Neumann definition of mass.

###  3.5 Generalised volume functionals

A broader class of complexity duals were proposed in Belin:2021bga . In
particular, a larger class of functionals was considered, i.e.

$\hat{S}=\int d^{3}\sigma\,\sqrt{h}\,F(g_{\mu\nu},X^{\mu})\,,$ (115)

where $F$ is a scalar function of the background metric and of the embedding
$X^{\mu}(\sigma^{a})$ of the codimension-one surface. These functionals may
provide an infinite class of alternative holographic definitions of
complexity; the CV conjecture is recovered by setting $F$ equal to a constant.

If we consider a surface parameterized by $(\lambda,x,y)$ in our metric
ansatz, we have that a functional of the form eq. (115) can be written as
follows

$\hat{S}=\int
d\lambda\,\frac{a(z)}{z^{3}}\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}=\int
d\lambda\,\hat{\mathcal{L}}\,,$ (116)

where the function $a(z)$ can be found by evaluating $F(g_{\mu\nu},X^{\mu})$
on the surface.

In a similar way to eq. (95), due to translation invariance in $v$, we can
again define the conserved quantity

$E=-\frac{\partial\hat{\mathcal{L}}}{\partial\dot{v}}=\frac{a(z)}{z^{k}}\frac{\dot{z}\,e^{-\frac{\chi}{2}}+\dot{v}fe^{-\chi}}{\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}}\,.$
(117)

As in eq (96), it is convenient to fix the $\lambda$ parameterization as
follows

${z^{3}}\sqrt{-2\dot{v}\dot{z}e^{-\frac{\chi}{2}}-fe^{-\chi}\dot{v}^{2}}=a(z)\,e^{-\frac{\chi}{2}}\,.$
(118)

In this gauge, the conserved quantity $E$ takes the form

$E=\dot{z}+\dot{v}fe^{-\chi/2}\,,\qquad\dot{v}=(E-\dot{z})\frac{e^{\frac{\chi}{2}}}{f}\,,$
(119)

which, inserted back in (118) gives an effective potential as in eq. (98)

$\dot{z}^{2}+V(z)=E^{2}\,,\qquad V(z)=-\frac{e^{-\chi}\,f\,a^{2}}{z^{6}}\,.$
(120)

The details of complexity evolution depend crucially on the choice of
$F(g_{\mu\nu},X^{\mu})$. Let us briefly discuss an example. Let us choose, as
in the example explicitly discussed in Belin:2021bga , the following function

$F(g_{\mu\nu},X^{\mu})=1+\kappa\,W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}\,,$
(121)

where $\kappa$ is a constant and $W_{\mu\nu\rho\sigma}$ is the spacetime Weyl
tensor. In this case, we find

$a=1+\kappa\,\frac{z^{4}}{12}\left(2f^{\prime\prime}-3f^{\prime}\chi^{\prime}+f(\chi^{\prime})^{2}-2f\chi^{\prime\prime}\right)\,.$
(122)

If we specialize to the RN back hole case as in eq. (24), we find:

$a=1+\kappa\,\frac{12z^{6}\left(\left(z_{c}+z_{h}\right)\left(z_{c}^{2}+z_{h}^{2}\right)-2z\left(z_{c}z_{h}+z_{c}^{2}+z_{h}^{2}\right)\right){}^{2}}{z_{c}^{6}z_{h}^{6}}\,.$
(123)

Figure 15:  Examples of potentials $V(z)$ the RN black solution for with
$z_{h}=1$, $z_{c}=5$. In the black curve, we choose the volume functional,
i.e. $F=1$. In the blue and red curves, we choose the functional in eq. (121)
with $\kappa=0.0003$ and $\kappa=0.003$, respectively.

We plot some examples of potentials in figure 15. In these cases the potential
in eq. (120) has two maxima. Each of these maxima correspond to an extremal
surface of the given functional which is attached to the boundary at a time
$t_{b}\to\infty$.

This shows that, changing $F(g_{\mu\nu},X^{\mu})$, for a given value of the
time we can have multiple extremal surfaces. The asymptotic complexity rate
for each of the extremal surfaces is proportional to the value of
$\sqrt{V(z)}$ evaluated on each of the maxima. With a generic choice of the
parameter $\kappa$, we expect that, in the hairy black hole case, the extremal
surface remains generically far from the would be Cauchy horizon. It would be
interesting to study other choices of $F(g_{\mu\nu},X^{\mu})$ in a more
systematic way. We leave this as a topic for further investigation.

##  4 Action complexity

To compute the action complexity we must first identify the Wheeler-DeWitt
(WDW) patch Brown:2015bva , which is given by the union of all the spatial
slices that can be attached to the left and right boundaries at some given
pair of times $t_{L}$ and $t_{R}$. The Killing vector corresponding to a
translation in $t$ in the metric (18) shifts

$t_{L}\to t_{L}+\delta t\,,\qquad t_{R}\to t_{R}+\delta t\,.$ (124)

For this reason, if we take the left and the right time translation in
opposite direction, the thermofield double state is time-independent. In order
to study the time dependence of the complexity of the thermofield double, we
take as boundary conditions for the WDW patch

$t_{L}=t_{R}=\frac{t_{b}}{2}\,.$ (125)

The boundary of the WDW patch can be obtained by sending null rays from the
left and right boundary.

Various shapes of the WDW region are possible. If the future light rays meet
at a point we have a future joint, otherwise the future light rays end in the
singularity and thus we have a space-like side of the WDW. The same is true
for the past light rays so in total we have four possible shapes: the diamond
  , a five-sides polygon with one side on the singularity of the future
or the past   , or a six- sides polygon with sides on both singularities
.

Once the WDW patch is obtained for a given boundary time $t_{b}$, we need to
compute the action Lehner:2016vdi  which is given by the sum of the following
contributions

$I_{\rm WDW}=I_{\rm V}+I_{\rm GHY}+I_{\rm J}+I_{\rm b}+I_{\rm ct}\,,$ (126)

where $I_{\rm V}$ is the bulk actions in eq. (14), $I_{\rm GHY}$ is the
Gibbons-Hawking-York boundary term eq. (207), $I_{\rm J}$ is the joint term
(212), $I_{\rm b}$ is the null boundary term (216) and $I_{\rm ct}$ its
counter term (218). Details on how to compute these four terms are given in
appendix E. We will always compute the derivative of the action with respect
the boundary time $\frac{dI_{\rm WDW}}{dt_{b}}$. This removes various UV
divergences222In our setting divergencies are time-independent because they
come from the region of spacetime outside the horizon, and so they are
invariant under the Killing vector $\partial_{t}$ (which is timelike for
$z<z_{h}$ and spacelike for $z>z_{h}$). This is different from the case where
the metric have no timelike Killing vector Bolognesi:2018ion . present in
$I_{\rm WDW}$.

In addition to the gravitational constant $G$, the action (126) contains a
scale $L_{\rm ct}$ which is needed in a counterterm that restores the
reparameterization invariance of the action. This term was introduced in
Lehner:2016vdi  with the motivation to remove an ambiguity in the
parameterization of the null hypersurfaces which delimit the WDW patch. In
many static situations, such as in Carmi:2017jqz , the asymptotic complexity
rate at $t_{b}\to\infty$ is independent on $L_{ct}$, which affects only the
action rate during a finite transient period. In out-of-equilibrium
situations, such as Vaidya spacetime, the counterterm is important in order to
reproduce many expected properties of complexity, such as the late-time growth
and the switchback effect Chapman:2018dem ; Chapman:2018lsv . The counterterm
scale introduces an ambiguity in defining holographic complexity that could be
related to the details of how complexity is defined in the dual field theory,
such as the reference state or the choice of gates.

In order to evaluate the action of the WDW patch, it is convenient to
introduce the tortoise coordinate

$z^{*}(z)=\int_{0}^{z}\frac{d\tilde{z}}{f(\tilde{z})}e^{\frac{\chi(\tilde{z})}{2}}\,,$
(127)

in which the radial-time part of the metric is conformally flat

$ds^{2}=\frac{1}{z^{2}}\left(fe^{-\chi}(-dt^{2}+(dz^{*})^{2})+dx^{2}+dy^{2}\right)\,,$
(128)

and the coordinates $u$ and $v$

$v=t-z^{*}\,,\qquad u=t+z^{*}\,.$ (129)

The time dependence of the action complexity depends on the type of Penrose
diagram that we are considering, which can have a singularity with upper
bending (type $U$) or lower bending (type $D$). It is useful to introduce a
quantity to discriminate the two kinds of behaviour. An ingoing null geodesic
which leaves the boundary at $t_{b}=0$ reaches the singularity at a time
$t_{\infty}$ given by

$t_{\infty}=z^{*}(\infty)=\int_{0}^{\infty}\frac{e^{\frac{\chi}{2}}}{f}d\tilde{z}\,,$
(130)

which is a convergent integral at $z\to\infty$, as can be shown using the
asymptotic Kasner behaviour in eq. (32). This shows that the integral is
converging at large $z$. If the quantity $t_{\infty}$ is positive, the
solution is of type $D$. Instead, if $t_{\infty}$ is negative, the solution is
of type $U$. In both cases, there is a critical boundary time $t_{c}$ at which
the structure of the WDW patch changes in a discontinuous way.

###  4.1 RN case

The time dependence of action complexity for the RN black hole was studied in
Carmi:2017jqz . We briefly review their results in appendix F, where we give a
closed-form expression for the time dependence of the complexity. In general,
the details of the evolution during the transient time range area function the
counterterm scale $L_{\rm ct}$. The complexity rate at late time is instead
independent of $L_{\rm ct}$.

Nearby the extremal limit, at small $T/\mu$, we have that the time dependence
of the rate of the action complexity is a smooth monotonic function of the
time, and it is at first approximation independent of the counterterm scale
$L_{\rm ct}$. By expanding the results in Carmi:2017jqz  (see appendix F.1) we
find the compact expression

$\frac{dI}{dt_{b}}\approx\frac{3\,V_{0}}{4\pi
G}\frac{z_{c}-z_{h}}{z_{h}^{4}}\tanh\frac{3t_{b}(z_{c}-z_{h})}{2z_{h}^{2}}\,.$
(131)

For a comparison with numerical results, see the top panel of figure 16.

As we increase $T/\mu$, the qualitative features of the plot change. For
$T/\mu\to\infty$, the Schwarzschild case must be reproduced. In this case, the
complexity rate is zero up to the critical time $\hat{t}_{c}$ in eq (2.9) of
Carmi:2017jqz

$\hat{t}_{c}=\frac{1}{2T}\frac{1}{\sqrt{3}}=\frac{2\pi}{3\sqrt{3}}z_{h}\approx
1.21z_{h}\,.$ (132)

where $T$ is the temperature $T=3/(4\pi z_{h})$. In appendix F.2 we check that
this is indeed the case, by studying the $z_{c}/z_{h}\to\infty$ of the RN
case. Just after $\hat{t}_{c}$, the rate drops to $-\infty$ in a discontinuous
way, and after that raises to approach a positive constant at late time.

At large but finite $T/\mu$, we have that the action rate is still to a good
approximation constant up to a timescale $t_{b}\approx\hat{t}_{c}$. Then,
there is a sudden drop of the action rate to a large negative value (which in
the limit $T/\mu\to\infty$ tends to $-\infty$). After that the action rate
raises and approaches a positive constant at $t_{b}\to\infty$. Figure 16 shows
and example of this behavior from the numerical solution.

Figure 16:  Action complexity rate as a function of time for some examples of
RN case, with $V_{0}/(16\pi G)=1$. Top: almost extremal case with
$T/\mu=1.4\cdot 10^{-3}$, $\rho=3.43$ and $z_{h}=1$. The approximate
analytical expression in eq. (131) in red is compared with numerical one, in
black. In the scale of the plot, the curves with ${L}_{\rm ct}^{2}=0.1,1,10$
all coincide with the black curve. Bottom: case with $T/\mu=0.28$, $\rho=0.8$
and $z_{h}=1$ (for various ${L}_{\rm ct}^{2}$).

At late time, from a direct calculation it follows that the bulk contribution
in eq. (228) vanishes, i.e. ${dI_{V}}/{dt_{b}}\approx 0$. Also, we can
approximate eq. (229) as

$\frac{dI_{J}}{dt_{b}}\approx\frac{V_{0}}{8\pi G}\left(\left.\frac{f_{\rm
RN}^{\prime}}{2z^{2}}\right|_{z=z_{c}}-\left.\frac{f_{\rm
RN}^{\prime}}{2z^{2}}\right|_{z=z_{h}}\right)\,.$ (133)

Defining a formal temperature $T_{c}$ and an entropy $S_{c}$ computed on the
Cauchy horizon

$T_{c}=-\frac{1}{4\pi}f_{\rm
RN}^{\prime}(z_{c})=-\frac{(z_{c}-z_{h})(3z_{h}^{2}+2z_{c}z_{h}+z_{c}^{2})}{4\pi
z_{c}z_{h}^{3}}\,,\qquad S_{c}=\frac{1}{4G}\frac{V_{0}}{z_{c}^{2}}\,,$ (134)

the action rate at late time is

$W_{A}=\lim_{t_{b}\to\infty}\frac{dI_{\rm WDW}}{dt_{b}}=\frac{V_{0}}{4\pi
G}\frac{z_{c}^{3}-z_{h}^{3}}{z_{c}^{3}z_{h}^{3}}=T\,S\,-T_{c}\,S_{c}\,,$ (135)

where $T$ and $S$ are the RN temperature and entropy in eqs (27) and (28). A
plot of the asymptotic action rate is shown on the left hand side of figure
17. On the right hand side of the same figure, we compare the action and the
volume rates for different $T/\mu$.

Figure 17:  Left: Plot of ${W_{A}}/{TS}$ as a function of $T/\mu$ for the RN
solution. Right: Ratio between the asymptotic CV and the asymptotic CA rate as
a function of $T/\mu$ for the RN solution. Combining eqs. (111), (112) and
(135), we find that for $T/\mu\to\infty$ we have $W_{V}/W_{A}=2\pi$, while for
$T/\mu\to 0$ we have $W_{V}/W_{A}=2\pi\sqrt{2/3}$. This is in agreement with
numerical calculations for generic $T/\mu$.

###  4.2 Type $D$

This case corresponds to a singularity with a lower bending and to
$t_{\infty}>0$. Here the WDW starts at $t_{b}=0$ in the shape   , with two
sides on the past and future singularity. There is a critical time:

$t_{c}=2z^{*}(\infty)=2t_{\infty}\,.$ (136)

For $t_{b}<t_{c}$, the WDW remains hexagonal, see figure 1 on the left. In
this case the complexity rate is zero, because the time dependence
contribution from regions $1$ and $3$ in figure turns out to cancel each
other, and the contribution from region $2$ is time independent. See appendix
G.1 for the details of the calculation. The same cancellation is present also
for the Schwarzschild case Carmi:2017jqz .

At $t_{b}=t_{c}$ the shape becomes   with the lower tip touching the past
singularity. For $t_{b}>t_{c}$ the shape remains   with only one side at the
future singularity. See figure 1 on the right. Here the complexity rate is non
zero and has a non trivial time dependence on $t_{b}$. First of all, we should
find the $z_{m1}$ of the lower tip of the WDW as a function of $t_{b}$. For
$t_{b}=t_{c}$, we have $z_{m1}\rightarrow\infty$ since it is touching the past
singularity. For $t_{b}\rightarrow\infty$, we have $z_{m1}=z_{h}$. For generic
time $t>t_{c}$, we can find $z_{m1}(t_{b})$ by inverting the equation:

$z^{*}(z_{m1})=\frac{t_{b}}{2}\,;$ (137)

it is the same as the equation (226) for the RN case, but now only the lower
tip is relevant. For the complexity rate we should evaluate three
contributions

$\frac{dI}{dt_{b}}=\frac{d\,I_{V}}{dt_{b}}+\frac{d\,I_{\rm
GHY}}{dt_{b}}+\frac{d\,(I_{J}+I_{\rm ct})}{dt_{b}}\,.$ (138)

A detailed derivation is provided in appendix G.3. The bulk contribution is
given by

$\frac{d\,I_{V}}{dt_{b}}=\frac{V_{0}}{16\pi
G_{N}}\int^{\infty}_{z_{m1}}dz\,s(z)\,,$ (139)

where $s(z)$ is given by eq. (206). The GHY contribution is a constant and
gives:

$\frac{dI_{\rm GHY}}{dt_{b}}=\frac{V_{0}}{16\pi G}\lim_{z\to\infty}A(z)\,,$
(140)

where $A(z)$ is given by eq. (211). Using the asymptotic Kasner solution in
eq. (32) in the expression (211) we find

$A(\infty)=f_{0}e^{-\frac{\chi_{0}}{2}}(3+\alpha^{2})\,.$ (141)

The joints and counterterm contributions are

$\frac{d(I_{J}+I_{\rm ct})}{dt_{b}}=\frac{V_{0}}{16\pi
G}\left\\{fe^{-\frac{\chi}{2}}\left[\frac{2}{z^{3}}\log\left(-4f\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-fe^{-\chi}}{z^{2}}\right)\right]\right\\}_{z=z_{m1}}$
(142)

Examples of total complexity rate as function of time are shown in figure 18.

Figure 18:  Time dependence of the action complexity rate in various type D
examples. Top: $q=0$ and $\rho=2$. Down: $q=1$ and $\rho=0.6$. In these
examples we set $z_{h}=1$ and ${L}_{\rm ct}=1$.

It is interesting to investigate more in detail the behaviour of the
complexity rate nearby the critical time $t_{c}$ in eq. (136). Before $t_{c}$,
the action rate vanishes. Just after the critical time $t_{c}$, the joint
inside the past horizon sits nearby the singularity, in the Kasner region at
$z_{m}\rightarrow\infty$. Using the approximations in eq. (32), we find the
negative log divergence in the joint part of the action rate

$\frac{dI_{J}}{dt_{b}}\approx-\frac{V_{0}}{8\pi G}A(\infty)\,\log z_{m}\,.$
(143)

where $A(\infty)$ is given in eq. (141). Combining eq. (247) with the
$z\to\infty$ behaviour in (32), we obtain

$\frac{dz_{m1}}{dt_{b}}\approx\frac{-f_{0}e^{-\frac{\chi_{0}}{2}}\,z_{m1}^{3}}{2}\,.$
(144)

Integrating this differential equation nearby critical time (where $\Delta
t_{b}=t_{b}-t_{c}>0$), we find:

$z_{m1}=\frac{1}{\sqrt{f_{0}e^{-\frac{\chi_{0}}{2}}\Delta t_{b}}}\,,$ (145)

which gives the following behaviour as a function of time

$\frac{dI_{J}}{dt_{b}}\approx\frac{V_{0}}{16\pi G}A(\infty)\,\log\Delta
t_{b}\,.$ (146)

So, for small $\Delta t_{b}>0$ the rate diverges $\to-\infty$. This divergence
is visible in the type $D$ example of figure 18 on as $t\to t_{c}^{+}$. The
divergence at $t_{b}=t_{c}$ is similar to the one that we have for the
Schwarzschild case for $t_{b}=\hat{t}_{c}$ in eq. (132).

###  4.3 Type $U$

This case corresponds to $t_{\infty}<0$ and to a singularity with an upper
bending. This situation is realised in the limit of $\phi_{h}\to 0$, which is
closer to the unperturbed RN solution. Here the WDW starts at $t_{b}=0$ in the
shape   . There is a critical time, for which the WDW changes shape, given
by

$t_{c}=-2z^{*}(\infty)=-2t_{\infty}\,.$ (147)

As we approach the limit $\phi_{h}\to 0$, the critical time $t_{c}$ goes to
infinity. In appendix H we perform an analytic estimate of the divergence of
the critical time, which give the result

$t_{c}=O\left(\frac{1}{\phi_{h}^{2}}\right)\,.$ (148)

This is consistent with the numerical calculations, see figure 19.

Figure 19: Critical time for the $q=0$ case as function of $\phi_{h}$, for
several $\rho$ and $z_{h}=1$. For each curve, the minimum in $t_{c}$ separates
the solutons of types U and D. Dashed lines are extrapolations following the
law $t_{c}={\rm const.}\times\left(\phi_{h}/\mu\right)^{-2.003_{3}}$.

For $t<t_{c}$, , the WDW is a diamond, see figure 2 on the left for an
example. It is important to know first the coordinate of the two joints inside
the horizons as a function of the boundary time $t_{b}$. Let us denote
respectively by $z_{m1}$ and $z_{m2}$ the coordinates of the joints inside the
white and the black hole horizons. They can be found by solving the equations

$z^{*}(z_{m1})=\frac{t_{b}}{2}\,,\qquad z^{*}(z_{m2})=-\frac{t_{b}}{2}\,.$
(149)

For the complexity rate we should evaluate two contributions:

$\frac{dI}{dt_{b}}=\frac{d\,I_{V}}{dt_{b}}+\frac{d(I_{J}+I_{\rm
ct})}{dt_{b}}\,.$ (150)

See appendix G.2 for details. The bulk contribution is

$\frac{d\,I_{V}}{dt_{b}}=\frac{V_{0}}{16\pi
G_{N}}\int^{z_{m2}}_{z_{m1}}s(z)dz\,,$ (151)

$s(z)$ is given by eq. (206). The joints and the counterterm give

$\displaystyle\frac{d(I_{J}+I_{\rm ct})}{dt_{b}}$ $\displaystyle=$
$\displaystyle\frac{V_{0}}{16\pi
G}\left(\left\\{fe^{-\frac{\chi}{2}}\left[\frac{2}{z^{3}}\log\left(-4f\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-fe^{-\chi}}{z^{2}}\right)\right]\right\\}_{z=z_{m1}}\right.$
(152)
$\displaystyle-\left.\left\\{fe^{-\frac{\chi}{2}}\left[\frac{2}{z^{3}}\log\left(-4f\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-fe^{-\chi}}{z^{2}}\right)\right]\right\\}_{z=z_{m2}}\right)$

At $t_{b}=t_{c}$ the shape is still   with the higher tip touching the
future singularity. For $t_{b}>t_{c}$ the shape becomes   with only one side
at the future singularity. See figure 2 on the right for an example. For
$t>t_{c}$, only $z_{m1}$ exists and can be found solving the first of (149).
During this stage, the structure of WDW patch is the same of case $D$ and also
the expression for the complexity rate eqs. (138)-(142). Examples of
complexity rate in case $U$ for different $\phi_{h}$ as function of time are
given in figure 20. In figure 21 we show a plot of total complexity rate for
the uncharged case as a function of choice of ${L}_{\rm ct}$.

Figure 20:  Time dependence of the action complexity rate in various type U
examples for various $\phi_{h}$, with $q=0$, $z_{h}=1$ and ${L}_{\rm ct}=1$.
Figure 21:  Time dependence of the action complexity rate as a function of
$t_{b}$ for various ${L}_{\rm ct}$ with $q=0$, $\phi_{h}=0.7$, $z_{h}=1$ and
$\rho=2.2$.

It is interesting to investigate the behaviour of the complexity rate nearby
the critical time in eq. 147. Just before the critical time $t_{c}$ , as the
joint inside the future horizon approaches the singularity, the action rate
get a positive divergent contribution

$\frac{dI_{J}}{dt_{b}}\approx\frac{V_{0}}{8\pi G}\,A(\infty)\,\log z_{m2}\,,$
(153)

where $A(\infty)$ is given in eq. (141). Just after $t_{c}$ there is no
divergence in the rate. Combining eq. (240) with the $z\to\infty$ behaviour in
(32), we obtain

$\frac{dz_{m2}}{dt_{b}}\approx\frac{f_{0}e^{-\frac{\chi_{0}}{2}}\,z_{m2}^{3}}{2}\,.$
(154)

Following the same steps as for case $D$, we obtain

$z_{m1}=\frac{1}{\sqrt{-f_{0}e^{-\frac{\chi_{0}}{2}}\Delta t_{b}}}\,,$ (155)

which gives, as a function of time

$\frac{dI_{J}}{dt_{b}}\approx-\frac{V_{0}}{16\pi G}A(\infty)\,\log(-\Delta
t_{b})\,.$ (156)

So, for small $\Delta t_{b}<0$ the rate diverges $\to\infty$. This divergence
is visible in the type $U$ example of figure 20 on as $t\to t_{c}^{-}$.

The divergence of the action complexity rate at the critical time $t_{c}$ then
is determined by the quantity $A(\infty)$ in eq. (141), which contains the
parameters of the metric in the $z\to\infty$ Kasner regime, defined in eq.
(32). So one may think that the coefficient of the logarithmic divergence of
the complexity rate at $t_{b}=t_{c}$ is directly related to the Kasner
exponent in the black hole interior. This does not seem to be the case, at
least in the regime of small $\phi_{h}$. In particular, let us distinguish to
cases:

  * •

for $q=0$ and $\phi_{h}\to 0$, we have that $A(\infty)$ is not directly
sensitive to the Kasner parameter $\alpha$ (which in this limit tends to
infinity), but to a quantity that can be defined in term of the unperturbed RN
geometry. This can be shown combining eqs. (141) and (86), which give

$A(\infty)=\frac{(z_{c}-z_{h})(z_{c}^{2}+2z_{c}z_{h}+3z_{h}^{2})}{z_{c}^{3}z_{h}^{3}}=-\frac{16\pi
G}{V_{0}}T_{c}S_{c}\,,$ (157)

where $T_{c}$ and $S_{c}$ are the formal temperature and entropy computed on
the Cauchy horizon of the RN solution, see eq. (134).

  * •

for $q\neq 0$ and $\phi_{h}\to 0$, we have that $A(\infty)$ vanishes. This can
be checked from eq. (85) and eq. (87) we find that for the final Kasner region
$f_{0}e^{-\chi_{0}/2}\to 0$. Also, as the limit $\phi_{h}\to 0$ is approached,
$\alpha$ oscillates many times, remaining finite. From from eq. (141) we find

$\lim_{\phi_{h}\to 0}A(\infty)=\lim_{\phi_{h}\to
0}f_{0}e^{-\frac{\chi_{0}}{2}}(3+\alpha^{2})=0\,.$ (158)

So in this limit the divergence of the complexity rate at $t_{b}=t_{c}$ tends
to disappear.

In figure 22 we plot for the complexity rate in some type U examples, with
charged scalar field $q=1$.

Figure 22: Complexity rate for $q=1$ and $\rho=2$ as a function of $t_{b}$ for
various values of $\phi_{h}$ and ${L}_{\rm ct}=1$. The divergence in the
complexity rate occurs at $t_{b}=t_{c}$ and is shown here for a particular
value of $\phi_{h}=0.4$. Type U cases.

###  4.4 Asymptotic complexity rate

In the late time limit $z_{m1}\to z_{h}$ and the contribution in eq. (142) is

$\left.\frac{d(I_{J}+I_{ct})}{dt_{b}}\right|_{t_{b}\to\infty}=-\frac{V_{0}}{8\pi
G}\left\\{\frac{e^{-\frac{\chi}{2}}}{2z^{2}}f^{\prime}\right\\}_{z=z_{h}}=TS\,,$
(159)

where $T$ ans $S$ are respectively the temperature and the entropy

$T=-\frac{1}{4\pi}f^{\prime}(z_{h})e^{-\frac{\chi(z_{h})}{2}}\,,\qquad
S=\frac{1}{4G}\frac{V_{0}}{z_{h}^{2}}\ .$ (160)

We can write the asymptotic action rate including also the volume term as

$W_{A}=\lim_{t_{b}\to\infty}\frac{dI_{\rm WDW}}{dt_{b}}=TS+\frac{V_{0}}{16\pi
G}\left(A(\infty)+\int^{\infty}_{z_{h}}\,s(z)\,dz\right)\,.$ (161)

For small $\phi_{h}$, the bulk contribution in eq. (161) tends to zero. To
show this, let us separate the bulk integral in to pieces: first, the integral
of $s(z)$ up to the Cauchy horizon of the undeformed RN. This integral is
exactly zero if evaluated on the RN solution, and so it is approximately zero
for $\phi_{h}\to 0$:

$\int_{z_{h}}^{z_{c}}s(z)\,dz=0\,.$ (162)

Then, there is the integral for $z>z_{c}$, which, using eq. (206), is given by
the approximation:

$\int_{z_{c}}^{\infty}s(z)\,dz=\int_{z_{c}}^{\infty}\frac{e^{-\frac{\chi}{2}}}{z^{4}}\left(-6-2\phi^{2}+\frac{1}{2}\tilde{\rho}^{2}z^{4}\right)dz\,.$
(163)

This integral is converging because at large $z$

$s(z)\approx\frac{1}{2}e^{-\frac{\chi_{0}}{2}}\frac{-12+z^{4}\rho^{2}-8\alpha^{2}(\log
z)^{2}}{z^{4+\alpha^{2}}}$ (164)

and $\alpha^{2}>1$ for the final Kasner region. Moreover, for $\phi_{h}\to 0$
we have that $\chi$ tends to $\infty$ very fast for $z>z_{c}$, and so the
integral (163) tends to zero.

We give now an analytic expression for the asymptotic complexity rate in the
$\phi_{h}\to 0$ limit. We distinguish two case:

  * •

For $q=0$ and in the $\phi_{h}\to 0$ limit, we have that the contribution due
to $A(\infty)$ reproduces the contribution of the second joint $z_{m2}$ for
the RN, see eq. (157). So, for $q=0$ and in the $\phi_{h}\to 0$ limit, we
recover the asymptotic complexity rate of the RN solution eq. (135), i.e.

$W_{A}=W_{A}^{(RN)}=T\,S\,-T_{c}\,S_{c}\,,$ (165)

  * •

For $q\neq 0$ and $\phi_{h}\to 0$, from eq. (158) that the contribution due to
$A(\infty)$ vanishes. In this case, we find

$W_{A}=TS\,.$ (166)

For generic $\phi_{h}$, the asymptotic complexity rate can be computed
numerically using eq. (161). We numerically checked that in the $\phi_{h}\to
0$ limit the analytic expressions (165) and (166) are reproduced, see figure
23 for some illustrative plots.

Figure 23:  Asymptotic complexity rate $W_{A}$ as function of $\rho$ for a few
values of $\phi_{h}$, with $16\pi G=1$, $V_{0}=1$ and $z_{h}=1$. For
comparison, in black we show $W_{A}$ for the RN black hole. Left: neutral
scalar case. In this case $W_{A}$ approaches the RN limit for $\phi_{h}\to 0$.
Right: charged scalar case with $q=1$. In this case $W_{A}$ approaches $TS$,
which is shown in the dashed line, in the $\phi_{h}\to 0$ limit.

We show the result of a scan in the parameter space of the model in figure 24.
This confirms the expectation that the order of magnitude of $W_{A}$ is $TS$.
In figure 25 we show the dependence of $W_{A}$ on $\phi_{0}/\mu$ at constant
$T/\mu$. For $T/\mu=0.15$, we see that $W_{A}$ has an interesting oscillatory
behavior as a function of $\phi_{0}/\mu$. The amplitude of these oscillations
vanishes for $\phi_{0}\to 0$. It is tempting to associate these oscillations
to the Josephson oscillations of the scalar field, since these do not appear
in the neutral scalar case, where the solution profile of the scalar field has
no oscillations.

Figure 24:  Left: neutral scalar case $q=0$. Right: charged scalar case with $q=1$. Red points denote type D solutions, black points denote type U solutions. The white region is not covered by our numerics.  |
---|---
Figure 25:  Left: Asymptotic action complexity for charged solutions with
$q=1$ at constant $T/\mu=0.15$. The dashed line corresponds to the Temperature
at which the solution changes type. Right: Asymptotic action complexity for
uncharged solution at $T/\mu=0.25$.

It has been conjectured Brown:2015lvg  that the Lloyd bound is saturated by
the uncharged planar BH in AdSd+1, i.e.

$\frac{d\,I_{WDW}}{dt_{b}}\leq{2M}\,,$ (167)

where $M$ is the mass of the BH. This bound in eq. (167) is violated by the
action conjecture Carmi:2017jqz ; Yang:2017czx , because the asymptotic value
(which is supposed to saturate the bound) is approached from above. In the
examples studied in this paper, we have that for type $U$ black holes the
complexity rate approaches infinity at critical time, and so this also
provides another example of violation of the bound.

One may wonder if a weaker version of the Lloyd bound holds for the asymptotic
complexity rate $W_{A}$, i.e.

$W_{A}\leq 2\,M\,.$ (168)

Violations to the bound of eq. (168) have been previously found in
Couch:2017yil ; Swingle:2017zcd ; An:2018xhv ; Alishahiha:2018tep ;
Mahapatra:2018gig ; Babaei-Aghbolagh:2021ast ; Avila:2021zhb . In our model,
we checked that the bound given by eq. (168) holds in all the parameter space
that we explored, with both choices $M=M_{D},M_{N}$ corresponding to Dirichlet
and Neumann boundary conditions. See figure 26 for some sample plot.

Figure 26:  In these pictures, we show $W_{A}/(TS)$ in solid line,
$(2M_{D})/(TC)$ in dashed blue line and $(2M_{N})/(TS)$ in dashed red line
(see eqs. (161) and (168)) as a function of $\rho$ for a fixed $\phi_{0}$ and
$z_{h}=1$. Left: neutral scalar. Right: charged $q=1$ scalar.  Figure 27:
Asymptotic value of complexity at late time as function of the scalar source,
for $z_{h}$. The dashed lines are extrapolations following the law
$\left.\frac{dI}{dt_{b}}\right|_{t_{b}\gg 1}={\rm
const.}+0.88_{1}\log\tfrac{\phi_{0}}{\mu}$.

We see from figure 27 that, for large $\phi_{0}/\mu$ the asymptotic action
complexity rate has a logarithmic dependence on $\phi_{0}/\mu$ for fixed
values of $\rho$ and $z_{h}$.

###### Acknowledgements.

This work of R.A. and S.B. is supported by the INFN special research project
grant “GAST" (Gauge and String Theories). G.T is funded by the Fondecyt
Regular grant number 1200025. The work of E.R. is partially supported by the
Israeli Science Foundation Center of Excellence.

##  Appendix A Conservation of electric flux in the $q=0$ case

From Stokes theorem, we have

$\int_{\Sigma}D_{\beta}F^{\alpha\beta}n_{\alpha}\,\sqrt{|h|}d^{3}y=\int_{\partial\Sigma}F^{\alpha\beta}\,n_{[\alpha}r_{\beta]}\sqrt{\sigma}d^{2}\theta$
(169)

where $n^{\alpha},r^{\beta}$ are unit normals, ${\Sigma}$ denotes a
codimension 1 submanifold, $\partial{\Sigma}$ denotes the boundary of
$\Sigma$, which is codimension 2, and $\sqrt{|h|}d^{3}y$ and
$\sqrt{\sigma}d^{2}\theta$ are the induced metric measures on codimension $1$
and $2$ submanifolds. For $q=0$ we have from Maxwell eqs. that
$D_{\beta}F^{\alpha\beta}=0$. Then, the right hand side of eq. (169) is zero
if integrated on a closed boundary $\partial\Sigma$. The only non-vanishing
component of the gauge field strength is

$F^{tz}=a^{\prime}(z)e^{\chi}z^{4}\,,$

Using the unit normals and codimension 2 measure

$n_{\alpha}=(1,0,0,0)\frac{\sqrt{|f|}e^{-\frac{\chi}{2}}}{z}\,,\qquad
r_{\alpha}=(0,1,0,0)\frac{1}{z\sqrt{|f|}}\,,\qquad\sqrt{\sigma}d^{2}\theta=\frac{dx\,dy}{z^{2}}\,,$

we get

$a^{\prime}(z)e^{\frac{\chi}{2}}={\rm constant}=-\rho\,.$ (170)

##  Appendix B Numerical techniques

For the neutral scalar case, we solve the system in eq. (50). Introducing both
an IR and a UV cutoff, $z_{\rm max}$ and $z_{\epsilon}$ respectively, we then
choose a value $z_{0}>z_{h}$ and split the full domain in two parts,
$(z_{\epsilon},z_{0})$ and $(z_{0},z_{\rm max})$. We solve the problem in the
small-$z$ regime imposing at the horizon

$\displaystyle f(z_{h})$ $\displaystyle=0\,,$
$\displaystyle\chi(z_{h})=\chi_{h}\,,$ $\displaystyle\phi(z_{h})$
$\displaystyle=\phi_{h}\,,$
$\displaystyle\phi^{\prime}(z_{h})=-\frac{8\phi_{h}}{z_{h}\left(\rho^{2}z_{h}^{4}-4\phi_{h}^{2}-12\right)}\,.$
(171)

The parameter $\chi_{h}$ is not physical. Once a solution is found, it can be
removed by a reparameterization of the time coordinate $t$ at infinity. We can
define

$\hat{\chi}_{h}=\chi_{h}-\chi(0)\,.$ (172)

Then, solving again with an initial condition $\chi(z_{h})=\hat{\chi}_{h}$, we
can find the solution with the usual normalization of the time coordinate,
where $\chi(0)=0$.

To determine the solution for the charged scalar case we take the full system
of equations (20). We again split the full domain in two parts,
$(z_{\epsilon},z_{0})$ and $(z_{0},z_{\rm max})$. We solve the system in the
small-$z$ regime imposing at the horizon

$\displaystyle f(z_{h})$ $\displaystyle=0\,,$
$\displaystyle\chi(z_{h})=\chi_{h}\,,$ $\displaystyle a(z_{h})$
$\displaystyle=0\,,$ $\displaystyle a^{\prime}(z_{h})=a_{1}\,,$
$\displaystyle\phi(z_{h})$ $\displaystyle=\phi_{h}\,,$
$\displaystyle\phi^{\prime}(z_{h})=\frac{8\phi_{h}}{z_{h}\left(-a_{1}^{2}e^{\chi_{h}}z_{h}^{4}+4\phi_{h}^{2}+12\right)}\,.$
(173)

We can then perform a reparameterization of the time coordinate $t$ at
infinity,

$\hat{\chi}_{h}=\chi_{h}-\chi(0)\,,\qquad\hat{a}_{1}=e^{\chi(0)/2}a_{1}\,.$
(174)

and solve again with the initial condition

$\chi(z_{h})=\hat{\chi}_{h}\,,\qquad a^{\prime}(z_{h})=\hat{a}_{1}\,.$ (175)

This procedure fixes the normalization of the time coordinate with
$\chi(0)=0$.

The value of $z_{0}$ is dynamically chosen such that $f(z_{0})<0$ with
$|f(z_{0})|\approx\mathcal{O}(1)$. Then, for the large-$z$ regime we rewrite
our equations in terms of $\log\left|f\right|$ and compactify the $z$
coordinate letting $z\mapsto\tfrac{2}{\pi}\arctan\log z$. This puts the
boundary at the finite value of $z=1$. The boundary conditions are set at
$z=z_{0}$ and used to impose continuity and smoothness of the solutions.

The procedure described above was implemented in Wolfram _Mathematica 13_
using the NDSolve framework333For best results, it is convenient to use the
stiffness-switching method when solving for the small-$z$ region.. In
practice, we set $z_{h}=\chi_{h}=1$, $z_{\epsilon}=10^{-10}$ and
$\tfrac{2}{\pi}\arctan z_{\rm max}\in[0.95,0.99]$ depending on the physical
parameters, leading to $z_{\rm max}/z_{h}\approx 10^{5}$ to $10^{27}$. Once
numerical solutions have been found all relevant physical quantities can be
computed straightforwardly. Those defined at $z=0$ are in practice evaluated
at $z_{\delta}=10^{-5}$ to avoid boundary effects near $z=z_{\epsilon}$. All
numerical integrations are perfomed using a standard locally adaptive method
in the same variables as the solutions are found, so that $z$ is compactified
(not compactified) for $z>z_{0}$ ($z<z_{0}$). Furthermore, for the computation
of action complexities, which involves integrals that are divergent near
$z=z_{h}$, we introduce another cutoff at $|z-z_{h}|=10^{-6}$.

The value of $\alpha$ coming into the definition of the Kasner exponent (32)
is found by fitting the functional forms (32) to the solutions at the regions
of interest, _i.e._ $z\approx z_{\rm max}$ and possibly an intermediate regime
$z_{h}\ll z_{\rm int}\ll z_{\rm max}$ before a Kasner inversion, when present.
Error bars are computed from these fits and also the variation of results
obtained using the functions $f$, $\phi$ and $\chi$.

##  Appendix C Penrose diagrams

Let us start from the metric (18) and let us pass to lightlike coordinates
$u$, $v$, as defined in eq. (129). We now introduce the lightcone coordinates
$(U,V)$. Let us define

$\kappa=-f^{\prime}(z_{h})e^{-\frac{\chi(z_{h})}{2}}>0\,,$ (176)

(remember that $f^{\prime}(z_{h})<0$). The quantity $\kappa$ is proportional
to the temperature of the BH. We pass to the coordinates

$U=\exp\left(-\frac{u}{2}\kappa\right)\,,\qquad
V=\exp\left(\frac{v}{2}\kappa\right)\,.$ (177)

These coordinates are related to the original ones $(z,t)$ as follows

$UV=e^{-\kappa\,z^{*}(z)}\,,\qquad\frac{V}{U}=e^{\kappa\,t}\,.$ (178)

The metric in the $U$, $V$ coordinates is

$ds^{2}=\frac{1}{z^{2}}\left(fe^{-\chi}\frac{4}{\kappa^{2}}\exp(\kappa\,z^{*})dU\,dV+dx^{2}+dy^{2}\right)\,.$
(179)

Nearby $z\to z_{h}$, we have $f\approx f^{\prime}(z_{h})(z-z_{h})$ and
$z^{*}(z)$ is divergent. The divergence can be estimated as follows

$z^{*}(z)\approx\frac{e^{\frac{\chi(z_{h})}{2}}}{f^{\prime}(z_{h})}\int_{0}^{z}\frac{d\tilde{z}}{\tilde{z}-z_{h}}\approx\frac{1}{\kappa}\log\frac{1}{|z-z_{h}|}\,,$
(180)

in such a way that the metric nearby $z_{h}$ is

$ds^{2}\approx\frac{1}{z_{h}^{2}}\left(\frac{4}{f^{\prime}(z_{h})}{\rm
sign}\,{(z-z_{h})}dU\,dV+dx^{2}+dy^{2}\right)\,$ (181)

So the coordinate $(U,V)$ are also non-continuous at the horizon. We can now
introduce the smooth Kruskal coordinates $(T,R)$ in this way:

$\displaystyle{\rm for}\,\,\,\,$ $\displaystyle z<z_{h}\,,\qquad
T=\frac{U-V}{2}\,,\qquad R=\frac{U+V}{2}\,,$ $\displaystyle{\rm for}\,\,\,\,$
$\displaystyle z>z_{h}\,,\qquad T=\frac{U+V}{2}\,,\qquad R=\frac{U-V}{2}\,.$
(182)

Indeed, nearby the horizon, both from outside and from the inside, we find

$ds^{2}\approx\frac{1}{z_{h}^{2}}\left(-\frac{2}{f^{\prime}(z_{h})}\big{(}-dT^{2}+dR^{2}\big{)}+dx^{2}+dy^{2}\right)\
$ (183)

At this point we introduce

$\hat{U}={R+T}\,,\qquad\hat{V}={R-T}$ (184)

in such a way that the metric nearby the horizon is

$ds^{2}\approx\frac{1}{z_{h}^{2}}\left(-\frac{8}{f^{\prime}(z_{h})}\,d\hat{U}d\hat{V}+dx^{2}+dy^{2}\right)\,.$
(185)

Then we pass to compact variables:

$\tilde{U}=\tan^{-1}\hat{U}\,,\qquad\tilde{V}=\tan^{-1}\hat{V}$ (186)

with

$-\frac{\pi}{2}<\tilde{U},\tilde{V}<\frac{\pi}{2}$ (187)

and then we define the coordinates of the Penrose diagram

$\tilde{T}=\frac{\tilde{U}-\tilde{V}}{2}\,,\qquad\tilde{R}=\frac{\tilde{U}+\tilde{V}}{2}\,.$
(188)

The choice of the function $\tan^{-1}$ in eq. (186) is a convention, and can
be replaced by any other $C^{\infty}$ functions which maps the real line to
the segment $(-\pi/2,\pi/2)$.

We finally get the expression which relate the coordinates $\tilde{T}$ and
$\tilde{R}$ of the Penrose diagram to the original coordinates $t$, $z$

  * •

for $z<z_{h}$, then we get

$\tan\left(\frac{\tilde{R}+\tilde{T}}{2}\right)\tan\left(\frac{\tilde{R}-\tilde{T}}{2}\right)=e^{-\kappa\,z^{*}}\,,\qquad\frac{\tan\left(\frac{\tilde{R}-\tilde{T}}{2}\right)}{\tan\left(\frac{\tilde{R}+\tilde{T}}{2}\right)}=e^{\kappa\,t}\,,$
(189)

which, for $\tilde{R},\tilde{T}>0$ should be in the quadrant
$\tilde{R}>\tilde{T}$

  * •

for $z>z_{h}$, then we get

$\tan\left(\frac{\tilde{R}+\tilde{T}}{2}\right)\tan\left(\frac{\tilde{T}-\tilde{R}}{2}\right)=e^{-\kappa\,z^{*}}\,,\qquad\frac{\tan\left(\frac{\tilde{T}-\tilde{R}}{2}\right)}{\tan\left(\frac{\tilde{R}+\tilde{T}}{2}\right)}=e^{\kappa\,t}\,.$
(190)

which, for $\tilde{R},\tilde{T}>0$ should be in the quadrant
$\tilde{T}>\tilde{R}$

The boundary of AdS is realised for $z^{*}=0$, which is at

$\tilde{R}=\pm\frac{\pi}{2}\,.$ (191)

The horizons are at $z^{*}\to\infty$,

$\tilde{T}=\pm\tilde{R}\,.$ (192)

The value $z^{*}(\infty)$ determines the "concavity" of the singularity. If
$z^{*}(\infty)=0$, the singularity is at $\tilde{T}=\frac{\pi}{2}$. If
$z^{*}(\infty)>0$, the concavity is below the line $\tilde{T}=\frac{\pi}{2}$
(type $D$). If $z^{*}(\infty)<0$, the concavity is above the line
$\tilde{T}=\frac{\pi}{2}$ (type $U$).

##  Appendix D Holographic renormalisation

We take the $\phi$ expansion around the boundary $z=0$ as in eq. (22). Let us
expand the metric profile functions as follows

$f=1+f_{2}z^{2}+f_{3}z^{3}+f_{4}z^{4}+\dots\qquad\chi=\chi_{2}z^{2}+\chi_{3}z^{3}+\dots$
(193)

The equations of motion fix:

$\chi_{2}=\frac{\phi_{0}^{2}}{2}\,,\qquad\chi_{3}=\frac{4\phi_{0}\phi_{1}}{3}\,,\qquad
f_{2}=\frac{\phi_{0}^{2}}{2}\,.$ (194)

In Fefferman-Graham (FG) coordinates, the metric has the following form

$ds^{2}=\frac{d\hat{z}^{2}}{\hat{z}^{2}}+\frac{1}{\hat{z}^{2}}g_{ab}(\hat{z},x^{a})dx^{a}dx^{b}\,,$
(195)

where the boundary coordinates are $x^{a}=(t,x,y)$. A direct calculation gives

$z=\hat{z}+\frac{f_{2}}{4}\hat{z}^{3}+\frac{f_{3}}{6}\hat{z}^{4}+\frac{1}{16}\left(f_{2}^{2}+2f_{4}\right)\hat{z}^{5}+\dots\,,$
(196)

The energy-momentum tensor (with Dirichlet boundary conditions) can be
obtained from the results in Balasubramanian:1999re ; deHaro:2000vlm ;
Caldarelli:2016nni

$T_{mn}^{(D)}=\frac{1}{8\pi G}\,\lim_{{\hat{z}}\rightarrow
0}\frac{1}{{\hat{z}}}\left(K_{mn}-g_{mn}K-2\,g_{mn}-\frac{1}{2}\,g_{mn}\,\phi^{2}\right)\,.$
(197)

This gives energy density and pressure

$\mathcal{E}=T_{00}^{(D)}=\frac{\phi_{0}\phi_{1}-f_{3}}{8\pi G}\,\qquad
p=T_{ii}^{(D)}=\frac{\phi_{0}\phi_{1}-f_{3}/2}{8\pi G}\,.$ (198)

which gives the Dirichlet mass

$M_{D}=\frac{V_{0}}{8\pi G}\left(-f_{3}+\phi_{0}\phi_{1}\right)\,.$ (199)

The energy momentum tensor with Neumann boundary condition instead is

$T_{mn}^{(N)}=T_{mn}^{(D)}-\eta_{ij}\frac{\phi_{0}\phi_{1}}{8\pi G}\,,$ (200)

which gives the Neumann mass

$M_{N}=\frac{V_{0}}{8\pi G}\left(-f_{3}+2\phi_{0}\phi_{1}\right)\,.$ (201)

If we consider the case of a system with spontaneous symmetry breaking (zero
sources) with Dirichlet or Neumann boundary conditions, as in Yang:2019gce ,
indeed both the definitions of mass give the same result.

##  Appendix E Terms in the WDW action

We discuss here in detail the evaluation of the terms in the action of the WDW
patch (126).

###  E.1 Bulk action density $I_{V}$

The bulk term is given by the Lagrangian of the model in eq. (14). Using the
Einstein equations, we can express the scalar curvature as follows

$R=-12-4\phi^{2}+D_{\mu}\phi(D^{\mu}\phi)^{*}\,,$ (202)

Using the ansatz in eqs. (18) and (19), we find

$-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{z^{4}e^{\chi}(a^{\prime})^{2}}{2}\,,\qquad
D_{\mu}\phi(D^{\mu}\phi)^{*}=fz^{2}(\phi^{\prime})^{2}-q^{2}\frac{z^{2}e^{\chi}}{f}a^{2}\phi^{2}\,.$
(203)

The on-shell bulk action density then is

$\mathcal{L}=\frac{1}{16\pi
G}\left(-6+\frac{(a^{\prime})^{2}z^{4}e^{\chi}}{2}-2\phi^{2}\right)\,.$ (204)

and the bulk action is

$I_{V}=\int d^{4}x\sqrt{-g}\,\mathcal{L}=\frac{1}{16\pi G}\int
d^{4}x\,s(z)\,,\qquad
s(z)=-\frac{e^{-\frac{\chi}{2}}}{z^{4}}\left(6+2\phi^{2}\right)+\frac{e^{\frac{\chi}{2}}(a^{\prime})^{2}}{2}\,,$
(205)

Using eq. (79), we find

$s(z)=\frac{e^{-\frac{\chi}{2}}}{z^{4}}\left(-6-2\phi^{2}+\frac{1}{2}\tilde{\rho}^{2}z^{4}\right)\,.$
(206)

###  E.2 Gibbons-Hawking-York term $I_{\rm GHY}$

The Gibbons-Hawking-York term is

$I_{\rm GHY}=\mp\frac{1}{8\pi G}\int_{\partial}d^{3}x\sqrt{|h|}K\,,$ (207)

where $-$ sign is chosen for space-like and $+$ for time-like boundaries.
$h_{\alpha\beta}$ is the induced metric on the manifold $\partial$ as embedded
in the space-time and $K$ is the extrinsic curvature. For diagonal metric, one
can use a useful formula for the extrinsic curvature

$K=\frac{n^{\mu}\partial_{\mu}\sqrt{|h|}}{\sqrt{|h|}}\,,$ (208)

where $n^{\alpha}$ is the unit outward-pointing normal. The contributions that
we need for the WDW patch are on portions space-like surfaces at the
singularity eventually present at constant $z\to\infty$. Thus we always use
the minus sign in (207) and $h$ is positive. We find:

$\displaystyle h_{\alpha\beta}={\rm
diag}\left(\frac{-fe^{-\chi}}{z^{2}},\frac{1}{z^{2}},\frac{1}{z^{2}}\right)\
,\qquad\sqrt{h}=\frac{e^{-\frac{\chi}{2}}\sqrt{-f}}{z^{3}}\,,$ $\displaystyle\
n^{\mu}=\left(0,-z\sqrt{-f},0,0\right)\,.$ (209)

Using (208) we obtain

$\sqrt{h}K=-z\sqrt{-f}\partial_{z}\left(\frac{e^{-\frac{\chi}{2}}\sqrt{-f}}{z^{3}}\right)=\frac{e^{-\frac{\chi}{2}}}{2z^{3}}\left(zf^{\prime}-(6+z\chi^{\prime})f\right)\,.$
(210)

It is useful to introduce

$A(z)=2\sqrt{h}K=\frac{e^{-\frac{\chi}{2}}}{z^{2}}\left(f^{\prime}-\chi^{\prime}f-\frac{6f}{z}\right)\,.$
(211)

To compute the contribution of GHY we need to determine
$A(\infty)=\lim_{z\to\infty}A(z)$.

###  E.3 Joint terms $I_{\rm J}$

For joints between space-like and time-like boundary surfaces, the action
contribution was studied in Hayward:1993my ; the case of joint involving null
surfaces was discussed in Lehner:2016vdi . The joint action is

$I_{\rm J}=\frac{1}{8\pi G}\int_{\Sigma}d\theta\sqrt{\sigma}\,\mathfrak{a}\,,$
(212)

where $\sigma$ is the determinant of induced metric on the joint and
$\mathfrak{a}$ depends on normals in the following way. Let us denote
$k^{\mu}$ the future directed null normal to a null surface, $n^{\mu}$ the
normal to a space-like surface and $s^{\mu}$ the normal to a time-like
surface, both directed outwards the volume of interest. In the case of
intersection of two null surfaces with normals $k^{\mu}_{1}$ and
$k^{\mu}_{2}$:

$\mathfrak{a}=\eta\log\left|\frac{k_{1}\cdot k_{2}}{2}\right|\,,$ (213)

while in the case of intersection of a null surface and a space-like surface
or a time-like surface with normal:

$\mathfrak{a}=\eta\log\left|k\cdot
n\right|\,,\qquad\mathfrak{a}=\eta\log\left|k\cdot s\right|\,.$ (214)

The $\eta$ is a sign defined as follows. In eqs. (213)-(214), if the outward
direction to the region of interest is pointing along the future, we should
set $\eta=1$ if the joint lies in the future of the spacetime volume of
interest, and $\eta=-1$ if the joint lies in the past. If instead the outward
direction is pointing along the past, we should set $\eta=1$ if the joint lies
in past of the spacetime volume of interest, and $\eta=-1$ if the joint lies
in the future. There is a subtlety for null surfaces: the normalisation of
$k^{\mu}$ can not be fixed by its length and we thus need a regulator. For the
cases we need in this paper

$(k_{1})_{\mu}=\Big{(}-1,\frac{e^{\frac{\chi}{2}}}{f},0,0\Big{)}a_{\rm
reg}\,,\qquad(k_{2})_{\mu}=\Big{(}1,\frac{e^{\frac{\chi}{2}}}{f},0,0\Big{)}a_{\rm
reg}\,.$ (215)

For this reason, the terms in eqs. (213) and (214) contain an ambiguous
coefficient $a_{\rm reg}$ inside the logarithm. This dependence on $a_{\rm
reg}$ is cancelled by the null boundary counterterm Carmi:2017jqz  discussed
next.

###  E.4 Null boundary term $I_{\rm b}$ and counterterm $I_{\rm ct}$

The null boundary terms is

$I_{\rm b}=\int dS\,d\lambda\,\sqrt{\sigma}\,\kappa\,.$ (216)

In our case, the null normals $k^{\mu}$ satisfy the geodesic equation

$k^{\mu}D_{\mu}k^{\nu}=\kappa\,k^{\nu}\,,$ (217)

with $\kappa=0$. For this reason, the term in eq. (216) is zero in our
parametrization.

We need also to add a counterterm Lehner:2016vdi , which is needed to restore
parameterisation invariance

$I_{\rm ct}=\frac{1}{8\pi G}\int d\lambda\,dS\,\Theta\log|{L}_{\rm
ct}\Theta|\,,$ (218)

where

$\Theta=\partial_{\lambda}\log\sqrt{\gamma}=D_{\mu}k^{\mu}$ (219)

for each of the null boundaries with normals $k=k_{1},k_{2}$. For both the
normals

$\Theta=-2a_{\rm reg}\,z\,e^{\frac{\chi}{2}}\,.$ (220)

Using geodesics equations, the integration measure in the null affine
parameter $\lambda$ can be written as follows

$d\lambda=\pm\frac{1}{a_{\rm reg}}\frac{dz}{z^{2}e^{\frac{\chi}{2}}}\,,$ (221)

so we get

$I_{\rm ct}=\mp\frac{V_{0}}{4\pi G}\int dz\,\frac{1}{z^{3}}\,\log\left|2a_{\rm
reg}\,{L}_{\rm ct}\,z\,e^{\frac{\chi}{2}}\right|,\,$ (222)

where $\pm$ depend on which of the four null boundaries we are considering.
For consistency, the dependence on $a_{\rm reg}$ should cancel with the joint
term. The counterterm ${L}_{\rm ct}$ does not affect the late-time limit of
the complexity, but just the finite-time behaviour.

##  Appendix F Details of the RN action complexity

For the RN black hole the tortoise coordinate $z^{*}$ is given by eq. (127),
with $\chi=0$:

$z_{\rm RN}^{*}(z)=\int_{0}^{z}\frac{d\tilde{z}}{f_{\rm RN}(\tilde{z})}\,.$
(223)

where $f_{\rm RN}$ is in eq. (24) or alternatively (25). The integral in eq.
(223) is divergent at the horizon $z=z_{h}$ and must be regularised using the
Cauchy principal value method.

We are interested in the function $z^{*}_{\rm RN}(z)$ is defined in the
interval $z_{h}\leq z\leq z_{c}$. By direct evaluation of eq. (223), we can
write the function $z^{*}(z)$ as follows

$z^{*}_{\rm
RN}(z)=\frac{z_{h}z_{c}}{z_{c}-z_{h}}\left[\frac{z_{h}^{2}}{(z_{c}+z_{h})^{2}+2z_{h}^{2}}\log\left(1-\frac{z}{z_{c}}\right)-\frac{z_{c}^{2}}{(z_{c}+z_{h})^{2}+2z_{c}^{2}}\log\left(\frac{z}{z_{h}}-1\right)\right]+\tilde{z}^{*}_{\rm
RN}(z)\,,$ (224)

where $\tilde{z}_{\rm RN}^{*}(z)$ is finite in the interval $z_{h}\leq z\leq
z_{c}$ and is

$\displaystyle\tilde{z}^{*}_{\rm RN}(z)$ $\displaystyle=$
$\displaystyle\frac{z_{c}z_{h}}{2\left(2z_{c}z_{h}+3z_{c}^{2}+z_{h}^{2}\right)\left(2z_{c}z_{h}+z_{c}^{2}+3z_{h}^{2}\right)}\left\\{\left(z_{c}+z_{h}\right)^{3}\log\left(z\left(\frac{z+z_{h}}{z_{h}z_{c}}+\frac{z}{z_{c}^{2}}+\frac{z_{h}+z}{z_{h}^{2}}\right)+1\right)\right.$
(225)
$\displaystyle\left.-2\,\frac{6z_{c}^{3}z_{h}+10z_{c}^{2}z_{h}^{2}+6z_{c}z_{h}^{3}+3z_{c}^{4}+3z_{h}^{4}}{\sqrt{2z_{c}z_{h}+3z_{c}^{2}+3z_{h}^{2}}}\left[\tan^{-1}\left(\frac{z_{c}+z_{h}}{\sqrt{2z_{c}z_{h}+3z_{c}^{2}+3z_{h}^{2}}}\right)\right.\right.$
$\displaystyle\left.\left.-\tan^{-1}\left(\frac{z_{c}z_{h}\left(z_{c}+z_{h}\right)+2z\left(z_{c}z_{h}+z_{c}^{2}+z_{h}^{2}\right)}{z_{c}z_{h}\sqrt{2z_{c}z_{h}+3z_{c}^{2}+3z_{h}^{2}}}\right)\right]\right\\}$

For $z\to z_{h}$, the function $z^{*}_{\rm RN}$ diverges to $+\infty$, while
for $z\to z_{c}$ it diverges to $-\infty$.

Here it is important to know first the coordinate of the two joints inside the
horizons as a function of the boundary time $t_{b}$. Let us denote
respectively by $z_{m1}$ and $z_{m2}$ the coordinates of the joints inside the
white and the black hole horizons. They can be found by solving the equations

$z^{*}_{\rm RN}(z_{m1})=\frac{t_{b}}{2}\,,\qquad z^{*}_{\rm
RN}(z_{m2})=-\frac{t_{b}}{2}\,.$ (226)

At late time, $z_{m1}=z_{h}$ and $z_{m2}=z_{c}$. The WDW patch is always of a
diamond form   , and never reaches the singularity. Its boundaries are only
null-like so there is no GHY term in the action. The rate of increase of the
WDW action is then given by

$\frac{dI_{\rm WDW}}{dt_{b}}=\frac{dI_{V}}{dt_{b}}+\frac{d(I_{J}+I_{\rm
ct})}{dt_{b}}\,,$ (227)

where

$\frac{d\,I_{V}}{dt_{b}}=\frac{V_{0}}{16\pi G_{N}}\int^{z_{m2}}_{z_{m1}}s_{\rm
RN}(z)dz\,,\qquad s_{\rm
RN}=\frac{1}{z^{4}}\left(-6+\frac{1}{2}\rho^{2}z^{4}\right)\,,$ (228)

and

$\displaystyle\frac{d(I_{J}+I_{\rm ct})}{dt_{b}}$ $\displaystyle=$
$\displaystyle\frac{V_{0}}{16\pi G}\left(\left\\{f_{\rm
RN}\left[\frac{2}{z^{3}}\log\left(-4f_{\rm RN}\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-f_{\rm
RN}}{z^{2}}\right)\right]\right\\}_{z=z_{m1}}\right.$ (229)
$\displaystyle-\left.\left\\{f_{\rm RN}\left[\frac{2}{z^{3}}\log\left(-4f_{\rm
RN}\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-f_{\rm
RN}}{z^{2}}\right)\right]\right\\}_{z=z_{m2}}\right)\,.$

###  F.1 Extremal limit

In the extremal limit $z_{c}\to z_{h}$, we can find some compact approximate
expressions. In this limit, the contribution of the function
$\tilde{z}^{*}(z)$ is negligible. We can approximate $z^{*}$ as follows:

$z^{*}_{\rm
RN}=\frac{z_{h}^{2}}{6(z_{c}-z_{h})}\log\frac{z_{c}-z}{z-z_{h}}\,.$ (230)

So we can find the analytical expressions

$z_{m1}=\frac{z_{h}e^{\frac{3t_{b}\left(z_{c}-z_{h}\right)}{z_{h}^{2}}}+z_{c}}{e^{\frac{3t_{b}\left(z_{c}-z_{h}\right)}{z_{h}^{2}}}+1}\,,\qquad
z_{m2}=\frac{z_{c}e^{\frac{3t_{b}\left(z_{c}-z_{h}\right)}{z_{h}^{2}}}+z_{h}}{e^{\frac{3t_{b}\left(z_{c}-z_{h}\right)}{z_{h}^{2}}}+1}\,.$
(231)

Plugging in the expressions for the action rate, we find eq. (131). Note that
in this limit the dependence on ${L}_{\rm ct}$ tends to disappear. See left of
figure 16.

###  F.2 Schwarzschild limit

In the large $z_{c}$ limit with fixed $z_{h}$, eq. (227) must reproduce the
case of a black brane with zero charge. Here we will reproduce the result that
the rate of CA is almost zero for $t_{b}<\hat{t}_{c}$ in eq. (132) from the
$z_{c}\to\infty$ limit of the RN expression. In this approximation, we find:

$\displaystyle z^{*}_{RN}(z)$ $\displaystyle=$ $\displaystyle
z_{h}\left[\frac{z_{h}^{2}}{z_{c}^{2}}\log\left(1-\frac{z}{z_{c}}\right)-\frac{1}{3}\log\left(\frac{z}{z_{h}}-1\right)\right]+\tilde{z}^{*}(z)\,,$
$\displaystyle\tilde{z}^{*}(z)$ $\displaystyle=$
$\displaystyle\frac{z_{h}}{6}\left\\{\log\left(1+\frac{z}{z_{h}^{2}}(z_{h}+z)\right)-2\,\sqrt{3}\left[\frac{\pi}{6}-\tan^{-1}\left(\frac{z_{h}+2z}{z_{h}\sqrt{3}}\right)\right]\right\\}$
(232)

In the limit of small time, one can write explicit expressions for $z_{m1}$
and $z_{m2}$. In this limit, both $z_{m1}$ and $z_{m2}$ are nearby $z_{c}$.
Nearby $z_{c}$, we can approximate $z^{*}$ as follows:

$\displaystyle z^{*}_{RN}(z)$ $\displaystyle=$ $\displaystyle
z_{h}\left\\{\frac{z_{h}^{2}}{z_{c}^{2}}\log\left(1-\frac{z}{z_{c}}\right)-\frac{1}{3}\log\left(\frac{z_{c}}{z_{h}}-1\right)+\frac{1}{6}\log\left(1+\frac{z_{c}}{z_{h}^{2}}(z_{h}+z_{c})\right)\right.\,,$
(233)
$\displaystyle\left.-\frac{1}{\sqrt{3}}\left[\frac{\pi}{6}-\tan^{-1}\left(\frac{z_{h}+2z_{c}}{z_{h}\sqrt{3}}\right)\right]\right\\}\approx$
$\displaystyle\approx$
$\displaystyle\frac{z_{h}^{3}}{z_{c}^{2}}\log\left(1-\frac{z}{z_{c}}\right)+\frac{\pi}{3\sqrt{3}}z_{h}+\frac{z_{h}^{3}}{2z_{c}^{2}}\,.$

This solution then is valid up to the time given by the plateau of the
function in eq. (233), i.e.

$\hat{t}_{c}=\frac{2\pi}{3\sqrt{3}}z_{h}+\frac{z_{h}^{3}}{z_{c}^{2}}\,,$ (234)

which indeed reproduces the result in eq. (132).

We can then solve for $z_{m1}$ and $z_{m2}$:

$z_{m1}=z_{c}\left(1-e^{\frac{t_{b}z_{c}^{2}}{2z_{h}^{3}}-\frac{\pi
z_{c}^{2}}{3\sqrt{3}z_{h}^{2}}-\frac{1}{2}}\right)\,,\qquad
z_{m2}=\left(1-e^{-\frac{t_{b}z_{c}^{2}}{2z_{h}^{3}}-\frac{\pi
z_{c}^{2}}{3\sqrt{3}z_{h}^{2}}-\frac{1}{2}}\right)\approx z_{c}\,.$ (235)

At very early time, both $z_{m1}\approx z_{m2}\approx z_{c}$. At time of order
$\hat{t}_{c}$, $z_{m1}$ drops rather suddenly, while $z_{m2}$ remains
approximately constant. The complexity rate then is

$\frac{d\,I}{dt_{b}}\approx\frac{V_{0}}{16\pi
G_{N}}\frac{1}{z_{h}^{3}}\,w\,e^{-w/2}\,,\qquad
w=\frac{2\sqrt{3}\pi}{9}\frac{z_{c}^{2}}{z_{h}^{2}}+1-\frac{z_{c}^{2}}{z_{h}^{3}}t_{b}\,.$
(236)

The complexity rate is then exponentially suppressed for $t<\hat{t}_{c}$.

##  Appendix G Details of action calculation

###  G.1 Type $D$ before $t_{c}$

Bulk term: Let us split the integral in the three regions $1,2,3$, see fig. 1
on the left:

$\displaystyle I_{V}^{1}$ $\displaystyle=$ $\displaystyle
V_{0}\int^{\infty}_{z_{h}}dz\int_{0}^{\frac{t_{b}}{2}+z^{*}}dt\,s(z)=V_{0}\int^{\infty}_{z_{h}}dz\left[\frac{t_{b}}{2}+z^{*}(z)\right]s(z)\,,$
$\displaystyle I_{V}^{2}$ $\displaystyle=$ $\displaystyle
V_{0}\int^{z_{h}}_{\epsilon}dz\int_{\frac{t_{b}}{2}-z^{*}}^{\frac{t_{b}}{2}+z^{*}}dt\,s(z)=V_{0}\int^{z_{h}}_{\epsilon}dz[2z^{*}(z)]s(z)\,,$
$\displaystyle I_{V}^{3}$ $\displaystyle=$ $\displaystyle
V_{0}\int^{\infty}_{z_{h}}dz\int_{\frac{t_{b}}{2}-z^{*}}^{0}dt\,s(z)=V_{0}\int^{\infty}_{z_{h}}dz\left[-\frac{t_{b}}{2}+z^{*}(z)\right]s(z)\,.$
(237)

The sum of the three terms is time-independent.

GHY term: From eq. (210), there are two contributions, respectively nearby the
future and past singularity, that we cutoff at $z=z_{\max}$

$\displaystyle I_{\rm GHY}^{1}$ $\displaystyle=$ $\displaystyle
2V_{0}\sqrt{|h|}K(z_{\max})\left(z^{*}(z_{\max})+\frac{t_{b}}{2}\right)\,,$
$\displaystyle I_{\rm GHY}^{2}$ $\displaystyle=$ $\displaystyle
2V_{0}\sqrt{|h|}K(z_{\max})\left(z^{*}(z_{\max})-\frac{t_{b}}{2}\right)\,.$
(238)

The total is time-independent.

Joint and counterterm contributions: These contributions are time-independent.

###  G.2 Type $U$ before $t_{c}$

Here it is important to know first the coordinate of the two joints inside the
horizons as a function of the boundary time $t_{b}$. Let us denote
respectively by $z_{m1}$ and $z_{m2}$ the coordinates of the joints inside the
white and the black hole horizons. They can be found by solving the equations

$z^{*}(z_{m1})=\frac{t_{b}}{2}\,,\qquad z^{*}(z_{m2})=-\frac{t_{b}}{2}\,.$
(239)

The time derivative gives

$\frac{dz_{m1}}{dt_{b}}=\frac{1}{2}\frac{1}{\frac{dz^{*}}{dz_{m1}}}=\frac{f(z_{m1})e^{-\chi(z_{m1})/2}}{2}\,,\qquad\frac{dz_{m2}}{dt_{b}}=-\frac{f(z_{m2})e^{-\frac{\chi(z_{m2})}{2}}}{2}\,.$
(240)

Bulk term: Let us split the integral in the three regions $1,2,3$, see figure
2 on the left:

$\displaystyle I_{V}^{1}$ $\displaystyle=$ $\displaystyle
V_{0}\int^{z_{m2}}_{z_{h}}dz\int_{0}^{\frac{t_{b}}{2}+z^{*}}dt\,s(z)=V_{0}\int^{z_{m2}}_{z_{h}}dz\left[\frac{t_{b}}{2}+z^{*}(z)\right]s(z)\,,$
$\displaystyle I_{V}^{2}$ $\displaystyle=$ $\displaystyle
V_{0}\int^{z_{h}}_{\epsilon}dz\int_{\frac{t_{b}}{2}-z^{*}}^{\frac{t_{b}}{2}+z^{*}}dt\,s(z)=V_{0}\int^{z_{h}}_{\epsilon}dz[2z^{*}(z)]s(z)\,,$
$\displaystyle I_{V}^{3}$ $\displaystyle=$ $\displaystyle
V_{0}\int^{z_{m1}}_{z_{h}}dz\int_{\frac{t_{b}}{2}-z^{*}}^{0}dt\,s(z)=V_{0}\int^{z_{m1}}_{z_{h}}dz\left[-\frac{t_{b}}{2}+z^{*}(z)\right]s(z)\,.$
(241)

The time derivative of the sum of these three terms (including a factor of
$2$) gives

$\frac{1}{V_{0}}\frac{d\,I_{V}}{dt_{b}}=\int^{z_{m2}}_{z_{m1}}s(z)dz+\frac{dz_{m1}}{dt_{b}}[-t_{b}+2z^{*}(z_{m1})]\,s(z_{m1})+\frac{dz_{m2}}{dt_{b}}[t_{b}+2z^{*}(z_{m2})]\,s(z_{m2})\,.$
(242)

From eq. (239), we finally find

$\frac{1}{V_{0}}\frac{d\,\Delta
I_{V}}{dt_{b}}=\int^{z_{m2}}_{z_{m1}}s(z)dz\,.$ (243)

Joint contributions: The joint contributions come from eq. (212). Both the
future and the past joint have $\eta=1$ and they give the following
contribution

$I_{J}=-\frac{V_{0}}{8\pi
G}\frac{1}{z_{m1}^{2}}\log\left|\frac{fe^{-\chi}}{z^{2}a_{\rm
reg}^{2}}\right|_{z=z_{m1}}-\frac{V_{0}}{8\pi
G}\frac{1}{z_{m2}^{2}}\log\left|\frac{fe^{-\chi}}{z^{2}a_{\rm
reg}^{2}}\right|_{z=z_{m2}}\,,$ (244)

The time derivative gives

$\frac{dI_{J}}{dt_{b}}=\frac{V_{0}}{8\pi
G}\left\\{\left[\frac{fe^{-\frac{\chi}{2}}}{2}\frac{d}{dz}\left(\frac{1}{z^{2}}\log\frac{-fe^{-\chi}}{z^{2}a_{\rm
reg}^{2}}\right)\right]_{z=z_{m2}}-\left[\frac{fe^{-\frac{\chi}{2}}}{2}\frac{d}{dz}\left(\frac{1}{z^{2}}\log\frac{-fe^{-\chi}}{z^{2}a_{\rm
reg}^{2}}\right)\right]_{z=z_{m1}}\right\\}\,.$

Null boundaries counterterms: From eq. (222), including a factor of $2$

$\frac{dI_{ct}}{dt_{b}}=\frac{V_{0}}{4\pi
G}\left(\left\\{fe^{-\frac{\chi}{2}}\frac{1}{z^{3}}\,\log\left(2a_{\rm
reg}\,{L}_{\rm
ct}\,z\,e^{\frac{\chi}{2}}\right)\right\\}_{z=z_{m1}}-\left\\{fe^{-\frac{\chi}{2}}\frac{1}{z^{3}}\,\log\left(2a_{\rm
reg}\,{L}_{\rm
ct}\,z\,e^{\frac{\chi}{2}}\right)\right\\}_{z=z_{m2}}\right)\,.$

Combining the counterterm with the joint term, we get

$\displaystyle\frac{d(I_{J}+I_{ct})}{dt_{b}}$ $\displaystyle=$
$\displaystyle\frac{V_{0}}{16\pi