aainstitutetext: CERN, Theory Division, 1 Esplanade des Particules,
Geneva 23, CH-1211, Switzerlandbbinstitutetext: Department of Theoretical
Physics, University of Geneva,
24 quai Ernest-Ansermet, 1211 Geneva 4, Switzerlandccinstitutetext:
Dipartimento di Fisica, Università di Milano - Bicocca
I-20126 Milano, Italyddinstitutetext: Perimeter Institute for Theoretical
Physics,
Waterloo, ON N2L 2Y5, Canadaeeinstitutetext: Center for Gravitational Physics
and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japanffinstitutetext:
Department of Physics, University of Illinois, Urbana-Champaign,
Urbana IL 61801, USA

# Complexity Equals Anything 2

Alexandre Belin d Robert C. Myers e Shan-Ming Ruan a Gábor Sárosi f
and Antony J. Speranza<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We expand on our results in Belin:2021bga  to present a broad new class of
gravitational observables in asymptotically Anti-de Sitter space living on
general codimension-zero regions of the bulk spacetime. By taking distinct
limits, these observables can reduce to well-studied holographic complexity
proposals, e.g., the volume of the maximal slice and the action or spacetime
volume of the Wheeler-DeWitt patch. As with the codimension-one family found
in Belin:2021bga , these new observables display two key universal features
for the thermofield double state: they grow linearly in time at late times and
reproduce the switchback effect. Hence we argue that any member of this new
class of observables is an equally viable candidate as a gravitational dual of
complexity. Moreover, using the Peierls construction, we show that variations
of the codimension-zero and codimension-one observables are encoded in the
gravitational symplectic form on the semi-classical phase-space, which can
then be mapped to the CFT.

CERN-TH-2022-159

YITP-22-101

##  1 Introduction

Complexity quantifies how difficult it is to perform a task from a set of
simple operations. For example, in quantum complexity, one is interested in
constructing a unitary operator performing a particular operation by combining
simple gates, which only act on a few qubits, into a quantum circuit
Aaronson:2016vto ; watrous . An important aspect to keep in mind is that in
complexity theory, the interest is always on robust features, such as the
scaling of the complexity with the “size” of the problem, for example, the
dimensionality of the Hilbert space in quantum complexity or the number of
digits if one aims at factoring a number into primes. Extracting an actual
value for complexity is highly sensitive to not only the choice of the gate
set, i.e., allowed simple operations, but also the cost assigned to each gate.
These ambiguities can be seen as a feature of complexity Brown:2021rmz .

Quantum complexity has recently triggered much interest in the context of
black holes and holography as a new twist in the ongoing effort to connect
quantum information theory to quantum gravity. The length of the wormhole for
a two-sided AdS black hole grows linearly in time at late times and continues
growing far beyond times at which entanglement entropies have thermalized
Hartman:2013qma . This suggests that a new quantum information measure is
needed to encode the growth of the wormhole. It is believed that the
holographic dual of circuit complexity, i.e., holographic complexity, could
capture the evolution of the black hole interior at late times
Susskind:2014moa ; Susskind:2014rva . From the viewpoint of the bulk
spacetime, there are three proposals for holographic complexity which have
been studied extensively: complexity=volume (CV) Susskind:2014rva ;
Stanford:2014jda , complexity=action (CA) Brown:2015bva ; Brown:2015lvg  and
complexity=spacetime volume (CV2.0) Couch:2016exn .

The CV conjecture Susskind:2014rva ; Stanford:2014jda  proposes that the
complexity is dual to the maximal volume of a hypersurface anchored on the
boundary time slice $\Sigma_{\textrm{\tiny CFT}}$ on which the CFT state is
defined, i.e.,

${\cal C}_{\textrm{\tiny V}}(\Sigma_{\textrm{\tiny
CFT}})=\,\max_{\Sigma_{\textrm{\tiny
CFT}}=\partial\mathcal{B}}\left[\frac{\mathcal{V(B)}}{G_{\textrm{\tiny
N}}\,\ell_{\rm bulk}}\right]\,,$ (1)

where $G_{\textrm{\tiny N}}$ denotes Newton’s constant in the bulk
gravitational theory and $\mathcal{B}$ corresponds to the bulk hypersurface of
interest. In the CA proposal Brown:2015bva ; Brown:2015lvg , the complexity is
given by evaluating the gravitational action on a region of spacetime, known
as the Wheeler-DeWitt (WDW) patch. The latter can be defined as the causal
development of a spacelike bulk surface anchored on the boundary time slice
$\Sigma_{\textrm{\tiny CFT}}$. The CA proposal is then given by

${\cal C}_{\textrm{\tiny A}}(\Sigma_{\textrm{\tiny
CFT}})=\frac{I_{\textrm{\tiny WDW}}}{\pi\,\hbar}\,.$ (2)

The CV2.0 proposal generalizes and at the same time, simplifies the previous
approach Couch:2016exn . In this case, the holographic complexity is simply
given by the spacetime volume of the WDW patch, namely

${\cal C}_{\textrm{\tiny SV}}(\Sigma_{\textrm{\tiny
CFT}})=\frac{V_{\textrm{\tiny WDW}}}{G_{\textrm{\tiny N}}\,\ell^{2}_{\rm
bulk}}\,.$ (3)

It is worthwhile noting that all of these proposals for holographic complexity
come with ambiguities in their definitions. For example, the definitions of
both ${\cal C}_{\textrm{\tiny V}}$ and ${\cal C}_{\textrm{\tiny SV}}$ require
the introduction of a new length scale $\ell_{\rm bulk}$ in eqs. (1) and (3)
to make the holographic complexity dimensionless.111For simplicity, one
typically chooses $\ell_{\rm bulk}=L$, i.e., the curvature radius of AdS. For
the CA proposal (2), a similar length scale appears in the boundary terms on
the null boundaries of the WDW patch Lehner:2016vdi . There is no natural
prescription to fix these new scales and hence the precise value of the
holographic complexity in any of these proposals is ambiguous. While one may
worry about such ambiguities, it should be seen as a feature that connects to
the fact that in complexity theory, only the scaling of the complexity
matters, and the precise value of prefactors are of little or no interest.

Two universal features which have been argued to hold for any definition of
quantum complexity in a holographic setting are: At late times in the time
evolution of the thermofield-double state, the complexity should grow linearly
in time, and the growth rate will be proportional to the mass of the dual
black hole Susskind:2014moa ; Susskind:2018pmk . The second is a universal
time delay in the response of the complexity to the insertion of shock waves
in the far past, known as the switchback effect Stanford:2014jda .

The various holographic complexity proposals have certainly drawn attention to
new kinds of gravitational observables in the bulk, as well as highlighting
their possible role in the AdS/CFT correspondence. This discussion was
expanded in Belin:2021bga  to a broad new class of codimension-one
observables, as we now briefly review – see appendix A for further details. To
begin, we may observe that the CV proposal (1) actually involves two
independent steps. First, the maximization procedure selects a special
codimension-one hypersurface in the bulk from among all possible spacelike
surfaces whose boundary is fixed at $\Sigma_{\textrm{\tiny CFT}}$. The second
step is to evaluate a particular geometric feature, i.e., the volume, of this
special hypersurface.

Given this perspective, it is natural to construct an infinite class of new
(diffeomorphism-invariant) gravitational observables on codimension-one
surfaces, defined in terms of two scalar functions $F_{1},F_{2}$ Belin:2021bga
. To begin, we consider codimension-one hypersurfaces $\Sigma$ in an
asymptotically AdS bulk spacetime with $d$+1 dimensions and anchored on a
particular boundary time slice $\Sigma_{\textrm{\tiny CFT}}$, i.e.,
$\partial\Sigma=\Sigma_{\textrm{\tiny CFT}}$. Following the previous
discussion, the first step is to select a special bulk surface within this
class using an extremization procedure

$\delta_{\textrm{\tiny
X}}\left[\int_{\Sigma}d^{d}\sigma\,\sqrt{h}\,F_{2}(g_{\mu\nu};X^{\mu})\right]=0\,.$
(4)

Here $F_{2}$ is a scalar functional integrated over the bulk hypersurfaces. In
general, $F_{2}$ may depend on the bulk metric $g_{\mu\nu}$ and also the
embedding functions $X^{\mu}(\sigma^{a})$ of the hypersurfaces. For example,
$F_{2}$ may be a scalar constructed from the background curvature and/or the
extrinsic curvature of the hypersurfaces. The above extremization then allows
for variations of the position of the hypersurfaces while fixing their
boundary $\Sigma_{\textrm{\tiny CFT}}$. This procedure selects a special
codimension-one hypersurface, which we denote $\Sigma_{F_{2}}$. Given this
hypersurface, we evaluate

$O_{F_{1},\Sigma_{F_{2}}}(\Sigma_{\textrm{\tiny
CFT}})=\frac{1}{G_{\textrm{\tiny
N}}L}\int_{\begin{subarray}{c}\Sigma_{F_{2}}\end{subarray}}\\!\\!\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{1}(g_{\mu\nu};X^{\mu})\,,$
(5)

where the new scalar functional $F_{1}$ again depends on the bulk metric and
the embedding functions.

This procedure produces a well-defined diffeomorphism-invariant observable in
the bulk that characterizes some feature of the boundary state on the time
slice $\Sigma_{\textrm{\tiny CFT}}$. We note that in general the scalar
functionals $F_{1}$ and $F_{2}$ need not coincide, i.e., the choice
$F_{1}=F_{2}$ is a subset of the full family of observables defined in
Belin:2021bga . The simplest choice for these functionals would be
$F_{1}=1=F_{2}$, with which we recover the CV proposal (1). However, the most
interesting feature of this family of new observables is that infinitely many
of them can exhibit the universal behaviour (i.e., linear late-time growth for
the thermofield-double state and the switchback effect for perturbations of
this state) that is expected of holographic complexity. Therefore it was
argued in Belin:2021bga  that any of those new observables are equally viable
candidates to be the gravitational dual of complexity.

In the present paper, we build on these ideas to construct yet another
infinite class of gravitational observables now defined in codimension-zero
regions of the bulk spacetime. Further, we will show that the new observables
display the same universal features discussed above. Hence they are also good
candidates for holographic complexity. Moreover, we show that particular
limits of our construction reduce to the CA and CV2.0 proposals, see eqs. (2)
and (3).

We will also show that the variations of these observables can be captured by
the gravitational symplectic form, which can then be pushed to the boundary
where it is given by a symplectic form on the space of Euclidean-path integral
states with sources for single-trace operators Belin:2018fxe .222These states
should be interpreted as coherent states of the quantum gravity theory Botta-
Cantcheff:2015sav ; Marolf:2017kvq ; Belin:2018fxe ; Belin:2020zjb . See also
e.g., Guo:2018kzl ; Bernamonti:2019zyy ; Bernamonti:2020bcf  for more studies
on holographic complexity of coherent states. This is accomplished in terms of
a particular conjugate variation $\delta_{w}$ Belin:2018bpg  which depends on
the codimension-zero or codimension-one observable $W$ and gives

$\delta W=\Omega(\delta,\delta_{w})\,,$ (6)

where $\Omega$ is the symplectic form of general relativity. For codimension-
zero observables, their variations are obtained thanks to a covariant version
of the Poisson bracket, known as the Peierls bracket Peierls:1952cb  (see also
Harlow:2019yfa  for a more recent discussion on the subject).

The remainder of the paper is organized as follows: In section 2, we describe
the construction for general codimension-zero observables and the
corresponding linear growth at late times. Section 3 presents the construction
of the conjugate variations $\delta w$ for either codimension-one or
codimension-zero gravitational observables $W$. We will work out some examples
explicitly and construct the appropriate conjugate variation for several
functionals $W$. We conclude in section 4 with a brief discussion of our
results and possible future directions. Appendix A reviews the construction of
codimension-one observables from Belin:2021bga , as well as providing some
additional details. Further, we consider adding extrinsic curvature terms in
codimension-one observables in Appendix B. Concerning the example in section
2, we further discuss the existence and uniqueness of constant mean curvature
slices in asymptotically AdS spacetime in Appendix C. In Appendix D, we
explore the limit of the gravitational action of a codimension-zero subregion
by taking its spacelike boundaries to be null hypersurfaces. In Appendix E, we
derive the conjugate variations $\delta_{w}h^{ab}$ and $\delta_{w}K_{ab}$ for
a general codimension-one observable $W(h^{ab},K_{ab})$. Finally, we discuss
one example of codimension-one observables using three-dimensional rotating
black holes in Appendix F.

##  2 Codimension-Zero Observables

This section will build on the ideas of Belin:2021bga  to construct a broad
class of gravitational observables associated with codimension-zero regions of
the bulk spacetime – see figure 1. Further, we will demonstrate that the new
observables exhibit the universal behaviour, i.e., linear late-time growth and
the switchback effect, desired for holographic complexity. Our construction of
observables associated with spacetime regions is, of course, inspired by the
CA and CV2.0 proposals, which evaluate various geometric functionals on the
WDW patch. In contrast to these proposals, we begin by identifying a
particular bulk spacetime region using an extremization procedure, which
mimics the first step of the generalized approach in eq. (4). In parallel with
the second step (5), the observable is then given by evaluating geometric
features of this particular codimension-zero region. While this approach may
seem more elaborate than that in CA and CV2.0 proposals, we show that eqs. (2)
and (3) can be recovered as a special case of our new approach.

Figure 1: The codimension-zero region $\mathcal{M}$ (orange shaded patch)
illustrated in an eternal AdS black hole background. The future and past
boundaries of $\mathcal{M}$ are denoted by $\Sigma_{\pm}$, respectively, and
are both anchored on the same boundary time slice $\Sigma_{\textrm{\tiny
CFT}}$.

Let us describe our construction in more detail now. As illustrated in figure
1, we begin by choosing a boundary time slice $\Sigma_{\textrm{\tiny CFT}}$.
We then consider bulk regions $\mathcal{M}$ which are bounded by two
codimension-one surfaces $\Sigma_{\pm}$ (i.e., the future and past boundaries
of $\mathcal{M}$, which we assume not to touch or cross in the bulk) anchored
on $\Sigma_{\textrm{\tiny CFT}}$, i.e.,
$\partial\Sigma_{\pm}=\Sigma_{\textrm{\tiny CFT}}$. We now define the
following functional in such regions

$\begin{split}W_{G_{2},F_{2,\pm}}(\mathcal{M})&=\int_{\Sigma_{+}}\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{2,+}(g_{\mu\nu};X^{\mu}_{+})+\int_{\Sigma_{-}}\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{2,-}(g_{\mu\nu};X^{\mu}_{-})\\\
&\qquad+\frac{1}{L}\int_{\mathcal{M}}\\!\\!d^{d+1}x\,\sqrt{g}\
G_{2}(g_{\mu\nu})\,.\end{split}$ (7)

As in eq. (4), $F_{2,\pm}$ are scalar functionals of the bulk metric
$g_{\mu\nu}$ and the embedding functions $X^{\mu}_{\pm}(\sigma^{a})$ of the
corresponding boundaries $\Sigma_{\pm}$. For example, $F_{2,\pm}$ may be
scalars constructed from the Riemann curvature of the bulk geometry and/or the
extrinsic curvature of $\Sigma_{\pm}$. Similarly, $G_{2}$, which is integrated
over the region $\mathcal{M}$, may be a scalar constructed from the background
curvature. As emphasized by our notation, $F_{2,\pm}$ and $G_{2}$ are three
independent scalars, i.e., we may choose completely different functionals on
the future and past boundaries.333Let us note that we assume that these
functionals are dimensionless and so a prefactor of $1/L$ appears in the final
integral in eq. (7) for consistency of the overall dimension of
$W_{G_{2},F_{2,\pm}}$. This factor was chosen to be the inverse of the AdS
curvature scale for simplicity, but any other choice would simply correspond
to a change in the overall normalization of $G_{2}$. The above functional (7)
is constructed to select a special bulk region by extremizing

$\delta_{\textrm{\tiny
X}_{\pm}}\\!\left[\,W_{G_{2},F_{2,\pm}}(\mathcal{M})\,\right]=0\,,$ (8)

where as indicated, we are varying the shape of the two boundaries
$\Sigma_{\pm}$.444If there are more than one extremal solutions, we choose the
one yielding the maximum value of $W_{G_{2},F_{2,\pm}}$, following the CV
proposal (1). We return to this point in section 2.2.3. For codimension-one
observables, the maximization is also discussed in Appendix A.2.3.

Before we move on, let us illustrate that the above extremization procedure
(8) can always be recast as two independent extremizations, which are each
analogous to the codimension-one case (4). This is actually a simple result of
Stokes’ theorem, which allows us to rewrite the bulk integral related to an
arbitrary scalar function $G(g_{\mu\nu})$ as an integral on the boundaries,
i.e., $\Sigma_{\pm}$. More explicitly, it means that one can always find

$\int_{\mathcal{M}}\\!\\!d^{d+1}x\,\sqrt{g}\
G_{2}(g_{\mu\nu})=L\,\int_{\Sigma_{+}\cup\Sigma_{-}=\partial\mathcal{M}}\\!\\!\\!\\!d^{d}\sigma\,\sqrt{h}\
\widetilde{G}_{2}(g_{\mu\nu};X^{\mu})\,.$ (9)

Roughly speaking, this means that we can always find a primitive function
$\widetilde{G}_{2}$ for any continuous function $G_{2}$ and perform Stokes’
theorem.555We note that the primitive function $\widetilde{G}_{2}$ will
generally not be a local functional of background and extrinsic curvatures on
the boundaries $\Sigma_{\pm}$. Of course, this is in contrast with
$F_{2,\pm}(g_{\mu\nu};X^{\mu})$, which are constructed as local functionals.
Another interpretation of this equality is that it is the statement that every
top form (which is always closed by definition) on non-compact orientable
manifold (such as the subregions $\mathcal{M}$ considered here) is always
exact. Substituting the above expression in eq. (7), our extremization
procedure (8) reduces to two independent variations for the future and past
boundaries,

$\begin{split}&\delta_{\textrm{\tiny
X}_{+}}\\!\left[\int_{\Sigma_{+}}d^{d}\sigma\,\sqrt{h}\,\left(F_{2,+}(g_{\mu\nu};X^{\mu}_{+})+\widetilde{G}_{2}(g_{\mu\nu};X_{+}^{\mu})\right)\right]=0\,,\\\
&\delta_{\textrm{\tiny
X}_{-}}\\!\left[\int_{\Sigma_{-}}d^{d}\sigma\,\sqrt{h}\,\left(F_{2,-}(g_{\mu\nu};X^{\mu}_{-})-\widetilde{G}_{2}(g_{\mu\nu};X_{-}^{\mu})\right)\right]=0\,.\\\
\end{split}$ (10)

This procedure then selects out special codimension-one surfaces as the future
and past boundaries of our codimension-zero bulk region. We denote these
boundaries as $\Sigma_{\pm}[G_{2},F_{2,\pm}]$ and the corresponding bulk
region as $\mathcal{M}_{G_{2},F_{2,\pm}}$. Given this special codimension-zero
region, we evaluate

$\begin{split}&O\left[G_{1},F_{1,\pm},\mathcal{M}_{G_{2},F_{2,\pm}}\right](\Sigma_{\textrm{\tiny
CFT}})=\frac{1}{G_{\textrm{\tiny
N}}L}\int_{\Sigma_{+}[G_{2},F_{2,+}]}\\!\\!\\!\\!\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{1,+}(g_{\mu\nu};X^{\mu}_{+})\\\
&\qquad+\frac{1}{G_{\textrm{\tiny
N}}L}\int_{\Sigma_{-}[G_{2},F_{2,-}]}\\!\\!\\!\\!\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{1,-}(g_{\mu\nu};X^{\mu}_{-})+\frac{1}{G_{N}L^{2}}\int_{\mathcal{M}_{G_{2},F_{2,\pm}}}\\!\\!\\!\\!\\!\\!d^{d+1}x\,\sqrt{g}\
G_{1}(g_{\mu\nu})\,.\end{split}$ (11)

As in eq. (4), $F_{1,\pm}$ are scalar functionals which may be constructed
from the bulk curvature and/or the extrinsic curvature of $\Sigma_{\pm}$.
Similarly, $G_{1}$, which is integrated over the region
$\mathcal{M}_{G_{2},F_{2,\pm}}$, is a scalar constructed from the background
curvature. Again as emphasized by our notation, $F_{1,\pm}$ and $G_{1}$ are
three independent scalars, which can, in general, be chosen to be completely
different from the functionals appearing in eq. (7).

Hence our codimension-zero version of the complexity=anything proposal
Belin:2021bga  again follows a two-step procedure. First, we pick out the
codimension-zero region, which is tied to a boundary time slice
$\Sigma_{\textrm{\tiny CFT}}$, by extremizing a geometric functional (7). Then
we evaluate a separate geometric functional (11) on this region. In all, this
construction involves two independent ‘bulk’ scalars, $G_{1}$ and $G_{2}$, and
four independent ‘boundary’ scalars, $F_{1,\pm}$ and $F_{2,\pm}$. Of course,
this new proposal includes the observables originally discussed in
Belin:2021bga , e.g., by setting $G_{1}=G_{2}=F_{1,-}=F_{2,-}=0$ which focuses
our attention on codimension-one observables constructed on the single
boundary $\Sigma_{\pm}[G_{2}=0,F_{2,+}]$. Hence it is perhaps not surprising
that the analysis of Belin:2021bga  is readily extended to show that the new
codimension-zero observables exhibit the universal behaviour expected for
holographic complexity. We now explicitly show these features in the following
subsection.

###  2.1 General Analysis

With our discussion above, we have defined a large class of diffeomorphism-
invariant observables (11) which are tied to a boundary time slice of an
asymptotically AdS spacetime. However, to connect these new observables to
holographic complexity, we need to demonstrate that they exhibit the expected
universal behaviour, i.e., linear late-time growth and the switchback effect
in a black hole background.

For simplicity, we will focus our discussion here on the eternal planar black
hole in $d+1$ dimensions,666For the most part, the following analysis does not
depend on the details of the blackening factor $f(r)$. Hence we could easily
extend the discussion to consider e.g., static black holes with curved event
horizons (i.e., with
$f(r)=k+\frac{r^{2}}{L^{2}}\left(1-\frac{r_{h}^{d}}{r^{d}}\,\right)$) and/or
with electromagnetic charges.

$\begin{split}ds^{2}&=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\,d\vec{x}^{2}\,,\qquad\text{with}\quad
f(r)=\frac{r^{2}}{L^{2}}\left(1-\frac{r_{h}^{d}}{r^{d}}\,\right)\,,\end{split}$
(12)

and where $r=r_{h}$ is the location of the event horizon. The corresponding
temperature and mass of the black hole are given by

$T_{\textrm{\tiny BH}}=\frac{d\,r_{h}}{4\pi L^{2}}\,,\qquad\quad
M=\frac{(d-1)V_{x}\,r_{h}^{d}}{16\pi G_{\mathrm{N}}L^{d+1}}\,,$ (13)

where we have introduced $V_{x}$ as the regulated volume of the
$(d-1)$-dimensional spatial boundary geometry, i.e., $V_{x}=\int d^{d-1}x$.

This two-sided black hole geometry (12) is dual to the thermofield double
(TFD) state of two decoupled boundary CFTs on independent planar background
geometries (i.e., $\mathbb{R}^{d}$),

$\left|\psi_{\textrm{\tiny TFD}}(t_{\textrm{\tiny L}},t_{\textrm{\tiny
R}})\right>=\sum_{E_{n}}e^{-\beta E_{n}/2-iE_{n}(t_{\textrm{\tiny
L}}+t_{\textrm{\tiny R}})/2}\left|n\right>_{\textrm{\tiny
L}}\otimes\left|n\right>_{\textrm{\tiny R}}\,.$ (14)

Here, we have associated the state with the time slices $t=-t_{\textrm{\tiny
L}}$ and $t=t_{\textrm{\tiny R}}$ in the left and right boundaries, as
illustrated in figure 1. Of course, this state is invariant under time shifts
$\delta t_{\textrm{\tiny L}}=-\delta t_{\textrm{\tiny R}}$, which is reflected
in the flow of the Killing vector $\partial_{t}$ in the bulk spacetime, e.g.,
see figure 1. Hence, without loss of generality, we will set $t_{\textrm{\tiny
L}}=t_{\textrm{\tiny R}}=\tau/2$ and consider the time evolution of the CFT
state $\left|\psi_{\textrm{\tiny TFD}}(\tau)\right>$.

Because the regions and surfaces of interest, i.e., $\mathcal{M}$ and
$\Sigma_{\pm}$, extend from the asymptotic boundaries to the interior of the
black hole, our calculations are facilitated by rewriting the metric (12) in
terms of infalling Eddington-Finkelstein coordinates,

$ds^{2}=-f(r)\,dv^{2}+2\,dv\,dr+\frac{r^{2}}{L^{2}}\,d\vec{x}^{2}\,.$ (15)

As usual, the infalling coordinate is defined by $v=t+r_{*}(r)$ with
$r_{*}(r)=-\int^{\infty}_{r}\frac{dr^{\prime}}{f(r^{\prime})}$.

####  2.1.1 Observables with $G_{1}=G_{2}$ and $F_{1,\pm}=F_{2,\pm}$

To illustrate our new proposal, we begin by considering observables with
$G_{1}=G_{2}$ and $F_{1,\pm}=F_{2,\pm}$. That is, the same geometric
functional appears in the extremization (7) and in the final observable (11).
To further simplify the analysis, we focus on functionals that only depend on
the background geometry (and not on the extrinsic or intrinsic geometry of the
boundary surfaces, in the case of $F_{1,\pm}$). We label the corresponding
observables $\mathcal{C}_{\textrm{gen}}(\tau)$ since we anchor them on the
boundary time slice $t_{\textrm{\tiny L}}=\tau/2=t_{\textrm{\tiny R}}$, which
we denote $\Sigma_{\textrm{\tiny CFT}}(\tau)$. Thus eq. (11) reduces to

$\begin{split}\mathcal{C}_{\rm{gen}}(\tau)&=\max_{\partial\Sigma_{\pm}(\tau)=\Sigma_{\textrm{\tiny
CFT}}(\tau)}\bigg{[}\frac{1}{G_{N}L}\int_{\Sigma_{+}[G_{1},F_{1,+}]}\\!\\!\\!\\!\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{1,+}(g_{\mu\nu})\\\
&+\frac{1}{G_{N}L}\int_{\Sigma_{-}[G_{1},F_{1,-}]}\\!\\!\\!\\!\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{1,-}(g_{\mu\nu})+\frac{1}{G_{N}L^{2}}\int_{\mathcal{M}_{G_{1},F_{1,\pm}}}\\!\\!\\!\\!\\!\\!d^{d+1}x\,\sqrt{g}\
G_{1}(g_{\mu\nu})\bigg{]}\,.\end{split}$ (16)

Due to the symmetries of the background geometry (15), we can parametrize the
boundaries $\Sigma_{\pm}$ as
$\left(v_{\pm}(\sigma),r_{\pm}(\sigma),\vec{x}\right)$, where we have
introduced the ‘radial’ coordinate $\sigma$ on these two codimension-one
surfaces. The codimension-zero functional then becomes

$\mathcal{C}_{\rm{gen}}(\tau)=\frac{V_{x}}{G_{N}L}\sum_{\varepsilon=+,-}\int_{\Sigma_{\varepsilon}}d\sigma\,\left[\left(\frac{r_{\varepsilon}}{L}\right)^{d-1}\sqrt{-f(r_{\varepsilon}){\dot{v}_{\varepsilon}}^{2}+2\dot{v}_{\varepsilon}\,\dot{r}_{\varepsilon}}\
a_{\varepsilon}(r_{\varepsilon})-\varepsilon\dot{v}_{\varepsilon}\,b(r_{\varepsilon})\right]\,,$
(17)

where $\dot{v}_{\pm}=dv_{\pm}/d\sigma$, $\dot{r}_{\pm}=dr_{\pm}/d\sigma$ and
the functions $a_{\pm}(r)$ are given by evaluating $F_{1,\pm}$ in the black
hole background (15).777To reduce the clutter in our equations here and in the
following, we have dropped the subscripts from the coordinates specifying the
profiles of the surfaces $\Sigma_{\pm}$. Similarly, $b(r)$ is the primitive
function found by evaluating $\sqrt{g}\,G_{1}$ in the background and
integrating with respective to $r$.888Note the extra minus sign appearing in
front of this term in eq. (17) because radial integral runs from the black
hole interior to the conformal boundary, which implicitly extends from
$\Sigma_{+}$ to $\Sigma_{-}$ in the infalling coordinates (15). That
is,999Here we have assumed that $\sqrt{g}\,G_{1}$ is a continuous function of
$r$ in the domain $\mathcal{M}$, which is a sufficient (but not necessary)
condition for the existence of the primitive function $b(r)$.

$\sqrt{g}\,G_{1}(g_{\mu\nu})=G_{1}(r)\left(\frac{r}{L}\right)^{d-1}\equiv
L\,\frac{\partial b(r)}{\partial r}\,.$ (18)

Hence as expressed in eq. (17), our observable reduces to two independent
contributions coming from the future and past boundaries of the region
$\mathcal{M}$. As observed in eq. (10), the next step involves extremizing
each of these contributions independently. Finding the extremal boundaries can
be recast as a classical mechanics problem with identifying the corresponding
integrand as a Lagrangian, i.e.,

$\mathcal{L}_{\pm}:=\left(\frac{r}{L}\right)^{d-1}\sqrt{-f(r){\dot{v}}^{2}+2\dot{v}\,\dot{r}}\
a_{\pm}(r)\mp\dot{v}\,b(r)\,.$ (19)

Moreover, because spacetime is stationary, the effective Lagrangians
$\mathcal{L}_{\pm}$ are $v$-independent. Hence the conjugate momenta are
conserved, i.e., $P^{\pm}_{v}={\partial{\cal L}_{\pm}}/{\partial\dot{v}}$ are
constants along the entire profiles of $\Sigma_{\pm}$.

Now we will be interested in the time evolution of the observable (17) with
respect to the boundary time $\tau$. Because the boundary time evolution can
be interpreted as the variation from one pair of extremal boundaries
$\Sigma_{\pm}(\tau)$ to another $\Sigma_{\pm}(\tau+\delta\tau)$, where we are
infinitesimally shifting the endpoints of the profiles, we can use our
intuition from classical mechanics to write

$\begin{split}\frac{d}{d\tau}\,\mathcal{C}_{\rm{gen}}(\tau)&=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\left(P_{\tau}^{+}(\tau)+P_{\tau}^{-}(\tau)\right)\Big{|}_{r=\infty}\\\
&=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\left(P_{v}^{+}(\tau)+P_{v}^{-}(\tau)\right)\,.\end{split}$ (20)

As usual, in the first line, only the boundary terms evaluated at the
asymptotic boundary contribute since the bulk integral coming from the
variation along the extremal hypersurface $\Sigma_{\pm}$ vanishes. The second
line follows from the fact that the infalling coordinate $v$ and the usual
Schwarzschild time coordinate $t$ coincide at the asymptotic boundary, and
hence their conjugate momenta are also equal there. Further, as noted above,
$P_{v}^{\pm}$ are conserved along the corresponding surfaces (for fixed
$\tau$), so the last expression need not be evaluated at $r=\infty$. Let us
add that the linear growth at late times will follow from the fact that
$P_{v}^{\pm}(\tau)$ will approach constant values at large $\tau$.

Each of the contributions in eq. (17) is invariant under reparametrizations of
the worldvolume coordinate $\sigma\to\sigma^{\prime}(\sigma)$. One finds that
a convenient constraint to fix the choice of this coordinate on each surface
is101010In general, the functions $a_{\pm}(r)$ may become negative in certain
ranges of $r$ and in this case, we should set
$\sqrt{-f(r)\dot{v}^{2}+2\dot{v}\dot{r}}=|a_{\pm}(r)|\left(\frac{r}{L}\right)^{d-1}$
– see further comments after eq. (172) in Appendix A.

$\sqrt{-f(r)\dot{v}^{2}+2\dot{v}\dot{r}}=a_{\pm}(r)\left(\frac{r}{L}\right)^{d-1}\,.$
(21)

With this constraint, the conserved momenta become

$P^{\pm}_{v}=\frac{\partial{\cal
L}_{\pm}}{\partial\dot{v}}=\dot{r}-f(r)\,\dot{v}\mp b(r)\,.$ (22)

We will choose the worldvolume coordinate $\sigma$, so it increases from the
left AdS boundary to the right AdS boundary. Hence moving to larger (positive)
$\sigma$ corresponds to moving to larger $r$ near the right quadrant of figure
1. With this choice, we will generally find the conserved momentum positive on
the late-time surfaces.

Combining this pair of equations, the gauge constraint (21) and the conserved
momentum eq. (22), we can solve for the profile of the future and past
boundaries (i.e., $v_{\pm}(\sigma)$ and $r_{\pm}(\sigma)$) with the first
order equations

$\begin{split}\dot{r}&=\lambda\left[(P^{\pm}_{v}\pm\,b(r))^{2}+f(r)\,a^{2}(r)\,\left(\frac{r}{L}\right)^{2(d-1)}\right]^{1/2}\,,\\\
\dot{v}&=\frac{\dot{r}-P^{\pm}_{v}\mp b(r)}{f(r)}\\\
&=\frac{1}{f(r)}\left(\lambda\left[(P^{\pm}_{v}\pm\,b(r))^{2}+f(r)\,a^{2}(r)\,\left(\frac{r}{L}\right)^{2(d-1)}\right]^{1/2}-(P^{\pm}_{v}\pm
b(r))\right)\,,\end{split}$ (23)

with $\lambda=\pm$. Let us note that the surfaces of interest at the right
asymptotic AdS boundary (i.e., at large $r$) fall inward to some minimum
radius $r=r_{\textrm{\tiny min}}$ (see figure 1, as well as eq. (26)). With
our choice of $\sigma$ (see comments below eq. (22)), this portion of
$\Sigma_{\pm}$ corresponds to the branch $\lambda=+$ with $\dot{r}\geq 0$
above. The branch $\lambda=-$ with $\dot{r}\leq 0$ corresponds to the portion
where the surfaces continue from $r=r_{\textrm{\tiny min}}$ and climb out to
the left asymptotic boundary. In addition, we note that the coordinate time
$t$ along these “extremal” surfaces satisfies

$\dot{t}:=\dot{v}-\frac{\dot{r}}{f(r)}=-\frac{P^{\pm}_{v}\pm b(r)}{f(r)}\,,$
(24)

which applies both inside and outside of the black hole horizon.

Following the discussion in appendix A, the determination of $r_{\pm}(\sigma)$
can be cast as a simple classical mechanics problem, viz,

$\begin{split}&\dot{r}^{2}+\mathcal{U}_{\pm}(P^{\pm}_{v},r)=0\\\ {\rm
where}\qquad\mathcal{U}_{\pm}(P^{\pm}_{v}&,r):=U_{0}(r)-(P^{\pm}_{v}\pm
b(r))^{2}\,.\end{split}$ (25)

and as in eq. (173), $U_{0}(r)=-f(r)\,a^{2}(r)\,({r}/{L})^{2(d-1)}$. Hence we
are looking for a zero-energy trajectory in a potential that is adjusted as we
vary the conserved momentum (22).111111In contrast to eq. (173), we have not
separated out $P^{\pm}_{v}{}^{2}$ as the effective energy here. Even with the
latter choice, the cross term $2P^{\pm}_{v}\,b(r)$ would still result in the
effective potential being dependent on $P^{\pm}_{v}$, hence varying this
parameter would change both the effective energy and the effective potential
at the same time. It is clear from eq. (25) that for large $r$, the effective
potential is negative and that the trajectory reverses at

$\mathcal{U}_{\pm}(P^{\pm}_{v},r_{\pm,\textrm{\tiny min}})=0\,.$ (26)

That is, we switch from the $\lambda=+$ branch to that with $\lambda=-$ in eq.
(23). It is straightforward to show that this reversal occurs inside the
horizon, i.e., $r_{\pm,\textrm{\tiny min}}\leq r_{h}$, since this is the only
region where $U_{0}(r)$ is positive. Further, since the trajectories begin at
large $r$, i.e., at the asymptotic AdS boundary, we always choose the largest
value of $r_{\pm,\textrm{\tiny min}}$ when there is more than one possible
solution for the turning point. For a given value of $P^{\pm}_{v}$, the
minimal radius may be determined by

$P^{\pm}_{v}=\mp b(r_{\pm,\textrm{\tiny
min}})+{\tilde{\lambda}}\sqrt{U_{0}(r_{\pm,\textrm{\tiny min}})}\,,$ (27)

where ${\tilde{\lambda}}=\pm$. Alternatively, given $r_{\pm,\textrm{\tiny
min}}$, we can determine the corresponding conserved momentum with this
equation. With our conventions, late times (i.e., $\tau\to+\infty$) will arise
with the ${\tilde{\lambda}}=+$ branch.121212The ${\tilde{\lambda}}=-$ branch
yields observables anchored at negative $\tau$ and in particular,
$\tau\to-\infty$.

We are interested in the time evolution of the surfaces $\Sigma_{\pm}$ with
respect to the boundary time $\tau$. Recall that both surfaces are anchored on
the boundary time slice $t_{\textrm{\tiny L}}=\tau/2=t_{\textrm{\tiny R}}$.
For a surface with the turning point $r=r_{\pm,\textrm{\tiny min}}$ (and from
eq. (27), the corresponding conserved momentum), we can determine the boundary
time by integrating eq. (24), i.e.,

$\tau\equiv 2\,t_{\textrm{\tiny R}}=-2\int^{\infty}_{r_{\pm,\textrm{\tiny
min}}}\\!\\!\\!dr\,\frac{P^{\pm}_{v}\pm b(r)}{f(r)\,\sqrt{(P^{\pm}_{v}\pm
b(r))^{2}-U_{0}(r)}}\,.$ (28)

Note that the $1/f(r)$ factor introduces a pole in the integrand at the
horizon $r=r_{h}$, and the integral is defined with the Cauchy principal value
at this singularity. Further, the radial integral contains another singularity
at $r=r_{\pm,\textrm{\tiny min}}$ where $\mathcal{U}_{\pm}(P^{\pm}_{v},r)$
vanishes. However, this singularity is integrable as long as
$\partial_{r}\,\mathcal{U}_{\pm}(P^{\pm}_{v},r_{\pm,\textrm{\tiny min}})\neq
0$, and hence the corresponding boundary time remains finite. However, one
finds that $\tau\to\infty$ when the turning point corresponds to an extremum
of the effective potential, i.e.,
$\partial_{r}\,\mathcal{U}_{\pm}(P^{\pm}_{v},r_{\pm,\textrm{\tiny min}})=0$.

We saw in eq. (17) that codimension-zero observable
$\mathcal{C}_{\rm{gen}}(\tau)$ can be expressed as a sum of two integrals over
the codimension-one boundaries $\Sigma_{\pm}$. Now using the gauge constraint
(21), the $\dot{v}$ equation (23) and the symmetry of the surfaces about
$r=r_{\pm,\textrm{\tiny min}}$, this expression can be rewritten as

$\mathcal{C}_{\rm{gen}}(\tau)=-\frac{2V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\int^{\infty}_{r_{\varepsilon,\textrm{\tiny
min}}}\frac{U_{0}(r)-\varepsilon\,b(r)\left(P^{\varepsilon}_{v}+\varepsilon\,b(r)\right)}{f(r)\,\sqrt{-\mathcal{U}_{\varepsilon}(P^{\varepsilon}_{v},r)}}\,dr\,.$
(29)

Of course, using eq. (28), we have tuned the minimal radius for each surface
so that they reach the asymptotic AdS boundary on the same time slice. Further
applying eqs. (27) and (28), we can write the above expression as

$\mathcal{C}_{\rm{gen}}(\tau)=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\,\sum_{\varepsilon=+,-}\left(P^{\varepsilon}_{v}\,\tau+2\int^{\infty}_{r_{\varepsilon,\textrm{\tiny
min}}}\frac{\sqrt{-\mathcal{U}_{\varepsilon}(P^{\varepsilon}_{v},r)}}{f(r)}\,dr\right)\,.$
(30)

Now it is straightforward to calculate the time derivative of
$\mathcal{C}_{\rm{gen}}$ as

$\begin{split}\frac{d}{d\tau}\mathcal{C}_{\rm{gen}}(\tau)&=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\,\sum_{\varepsilon=+,-}\left(P^{\varepsilon}_{v}+\frac{dP^{\varepsilon}_{v}}{d\tau}\left[\tau+2\int^{\infty}_{r_{\varepsilon,\textrm{\tiny
min}}}\frac{(P^{\varepsilon}_{v}+\varepsilon\,b(r))}{f(r)\sqrt{-\mathcal{U}_{\varepsilon}(P^{\varepsilon}_{v},r)}}dr\,\right]\right.\\\
&\qquad\qquad\qquad\qquad\left.-2\,\frac{dr_{\varepsilon,\textrm{\tiny
min}}}{d\tau}\,\frac{\sqrt{-\mathcal{U}_{\varepsilon}(P^{\varepsilon}_{v},r)}}{f(r)}\bigg{|}_{r=r_{\varepsilon,\textrm{\tiny
min}}}\right)\,,\\\ &=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\,\left(P_{v}^{+}(\tau)+P_{v}^{-}(\tau)\right)\,.\end{split}$ (31)

where in the first line, the two terms multiplying
$\frac{dP^{\varepsilon}_{v}}{d\tau}$ precisely cancel using eq. (28) and the
last term in the second line vanishes because
$\mathcal{U}_{\varepsilon}(P^{\varepsilon}_{v},r)$ vanishes at
$r=r_{\varepsilon,\textrm{\tiny min}}$ by eq. (26). Hence we have recovered
our result in eq. (20).

####  2.1.2 Linear Growth

We would like to show that the growth rate (31) becomes constant at late times
and that the constant is proportional to the mass of the black hole. Hence, we
examine the late-time behaviour here, i.e., as $\tau\to\infty$. As we noted
below eq. (28), we reach this regime when the conserved momenta are chosen
such that $\partial_{r}\,\mathcal{U}_{\pm}(P^{\pm}_{v},r_{\pm,\textrm{\tiny
min}})=0$. Following the notation of Appendix A, we denote the radius at this
critical point as $r_{\pm,f}$ – see eq. (176). Now recall that solving eq.
(25) requires the effective potential to be negative, and so the critical
point must correspond to a local maximum of the effective potential, i.e.,

$\mathcal{U}_{\pm}(P^{\pm}_{\infty},r_{\pm,f})=0\,,\quad\partial_{r}\,\mathcal{U}_{\pm}(P^{\pm}_{\infty},r_{\pm,f})=0\,,\quad\partial^{2}_{r}\,\mathcal{U}_{\pm}(P^{\pm}_{\infty},r_{\pm,f})\leq
0\,.$ (32)

where we denote the conserved momentum in this late time limit as
$P^{\pm}_{\infty}=\lim_{\tau\to\infty}P^{\pm}_{v}(\tau)$.

As noted below eq. (20), linear growth of the new observable at late times
requires that $P_{v}^{\pm}(\tau)$ is nearly constant at large $\tau$. That is,

$\lim_{\tau\to\infty}\frac{dP^{\pm}_{v}}{d\tau}\to
0\qquad\text{or}\qquad\lim_{\tau\to\infty}\frac{d\tau}{dP^{\pm}_{v}}\to\infty\,.$
(33)

We now show that this asymptotic behaviour arises with some more detailed
analysis.

First, we expand the effective potential around the critical point using eq.
(32),

$\lim_{r\to
r_{\pm,f}}\mathcal{U}_{\pm}(P^{\pm}_{\infty},r)\simeq\frac{1}{2}\,\mathcal{U}_{\pm}^{\prime\prime}(P^{\pm}_{\infty},r_{\pm,f})\,(r-r_{\pm,f})^{2}+\mathcal{O}((r-r_{\pm,f})^{3})\,,$
(34)

where
$\mathcal{U}_{\pm}^{\prime\prime}=\partial_{r}^{2}\,\mathcal{U}_{\pm}(P^{\pm}_{v},r)$.
Next, one can expand the potential in the late-time limit where
$r_{\pm,\textrm{\tiny min}}$ approaches $r_{\pm,f}$ to find

$\lim_{r_{\pm,\textrm{\tiny min}}\to
r_{\pm,f}}\mathcal{U}_{\pm}(P^{\pm}_{\infty},r_{\pm,\textrm{\tiny min}})\simeq
2(P^{\pm}_{v}-P^{\pm}_{\infty})(P^{\pm}_{\infty}\pm
b(r_{\pm,f}))+\mathcal{O}((P^{\pm}_{\infty}-P^{\pm}_{v})(r_{\pm,\textrm{\tiny
min}}-r_{\pm,f}))\,.$ (35)

Combining the above two limits, we arrive at the asymptotic behaviour for the
conserved momenta, i.e.,

$\lim_{\tau\to\infty}(P^{\pm}_{\infty}-P^{\pm}_{v})\simeq-\frac{\mathcal{U}_{\pm}^{\prime\prime}(P^{\pm}_{\infty},r_{\pm,f})}{4(P^{\pm}_{\infty}\pm
b(r_{\pm,f}))}(r_{\pm,\textrm{\tiny
min}}-r_{\pm,f})^{2}+\mathcal{O}((r_{\pm,\textrm{\tiny
min}}-r_{\pm,f})^{3})\,.$ (36)

Now it is straightforward to differentiate eq. (28) to find

$\begin{split}\frac{d\tau}{dP^{\pm}_{v}}&=\frac{dr_{\pm,\textrm{\tiny
min}}}{dP^{\pm}_{v}}\,\frac{2(P^{\pm}_{v}\pm
b(r))}{f(r)\sqrt{\mathcal{U}_{\pm}(P^{\pm}_{v},r)}}\Bigg{|}_{r\to
r_{\pm,\textrm{\tiny min}}}+\int^{\infty}_{r_{\pm,\textrm{\tiny
min}}}\\!\\!\\!dr\,\frac{2U_{0}(r)}{f(r)\left(\mathcal{U}_{\pm}(P^{\pm}_{v},r)\right)^{3/2}}\,.\end{split}$
(37)

Let us note that the two terms on the right-hand side above are individually
divergent since from eq. (26),
$\mathcal{U}_{\pm}(P^{\pm}_{v},r_{\pm,\textrm{\tiny min}})=0$. However, one
can show that these divergences precisely cancel at any finite time $\tau$ for
which $\partial_{r}\mathcal{U}_{\pm}(P^{\pm}_{v},r_{\pm,\textrm{\tiny
min}})\neq 0$. On the other hand, this cancellation fails in the late-time
limit, i.e., with $r_{\pm,\textrm{\tiny min}}\to r_{\pm,f}$ which yields
$\partial_{r}\mathcal{U}_{\pm}(P^{\pm}_{\infty},r_{\pm,\textrm{\tiny
min}})=0$. After a careful analysis of the leading divergences, we obtain the
following asymptotic behaviour

$\lim_{\tau\to\infty}\frac{dP^{\pm}_{v}}{d\tau}\simeq\frac{-f(r_{\pm,f})\,(-\mathcal{U}_{\pm}^{\prime\prime}(P^{\pm}_{\infty},r_{\pm,f})^{3/2}}{2\sqrt{2}\,(P^{\pm}_{\infty}\pm
b(r_{\pm,f}))^{2}}(r_{\pm,\textrm{\tiny min}}-r_{\pm,f})^{2}\,,$ (38)

Combining eqs. (35) and (38), we find

$\lim_{\tau\to\infty}P^{\pm}_{v}\simeq
P^{\pm}_{\infty}-N_{\pm}\,e^{-\kappa_{\pm}\tau}\,,$ (39)

with $N_{\pm}$ are positive coefficients (independent of $\tau$) and
$\kappa_{\pm}$ given by

$\kappa_{\pm}=-f(r_{\pm,f})\frac{\sqrt{-2\,\mathcal{U}_{\pm}^{\prime\prime}(P^{\pm}_{\infty},r_{\pm,f})}}{P^{\pm}_{\infty}\pm
b(r_{\pm,f})}\,.$ (40)

Recalling that
$\frac{d}{d\tau}\,\mathcal{C}_{\rm{gen}}\propto(P_{v}^{+}(\tau)+P_{v}^{-}(\tau))$
from eq. (20), we see that at late times, the growth rate is constant up to
exponentially suppressed corrections.

Of course, the analysis and results presented above are analogous to that
given for codimension-one observables in Belin:2021bga . Further, following
the discussion there, we can also show that $P^{\pm}_{\infty}$ are both
proportional to the mass, where $M\propto r_{h}^{d}/L^{d}$ from eq. (13).
Here, we recall that we have focused on the case where the functionals
appearing in eqs. (7) and (11) involve only background curvature invariants
(and not extrinsic or intrinsic curvatures of the boundary surfaces for
$F_{1,\pm}$ and $F_{2,\pm}$).

Let us then consider the various elements comprising the effective potentials
$\mathcal{U}_{\pm}(P^{\pm}_{v},r)$ in eq. (25). With the restriction to
background curvatures, one finds that

$U_{0}=\left(\frac{r_{h}}{L}\right)^{2d}(w-w^{2})\,a^{2}(w)=\left(\frac{r_{h}}{L}\right)^{2d}\widehat{U}_{0}(w)\,,$
(41)

where we introduced the dimensionless radial coordinate $w=(r/r_{h})^{d}$ –
compare to eq. (168). The fact that $a(w)$ has no explicit dependence on
$r_{h}$ when written in terms of $w$ is inherited from curvature invariants
which have the same property for the planar black hole background (15). We are
assuming that the dimensional coefficients appearing in the functions
$F_{1,\pm}$ and $F_{2,\pm}$ are independent of $r_{h}$ because the observable
should be defined in a state independent way.131313As in Appendix A, it is
natural to absorb the dimensions in these couplings with the AdS scale. For
the primitive function $b(r)$, we have from eq. (18)

$b(r)=\int\frac{dr}{L}\,\left(\frac{r}{L}\right)^{d-1}G_{1}(r)=\frac{1}{d}\left(\frac{r_{h}}{L}\right)^{d}\int
dw\,G_{1}(w)=\left(\frac{r_{h}}{L}\right)^{d}\,\hat{b}(w)\,,$ (42)

where $G_{1}(w)$ has no explicit dependence on $r_{h}$ when written in terms
of $w$, as above for $a(w)$.141414Let us note here that $b(r)$ is only defined
up to a constant shift, i.e., the integration constant in eq. (42). Eq. (27)
shows that this shift produces a shift in the corresponding momenta, i.e.,
with $b(r)\to b(r)+c$, the same boundary surfaces $\Sigma_{\pm}$ are described
by $P^{\pm}_{v}\to P^{\pm}_{v}\mp c$. However, the latter shifts cancel in the
sum $P^{+}_{v}(\tau)+P^{-}_{v}(\tau)$ which determines
$\partial_{t}\mathcal{C}_{\rm{gen}}(\tau)$, as shown in eq. (20).

Combining these expressions, we may rewrite the effective potential (25) as

$\mathcal{U}_{\pm}(P^{\pm}_{v},w)=\left(\frac{r_{h}}{L}\right)^{2d}\widehat{U}_{0}(w)-\left(P^{\pm}_{v}\pm\left(\frac{r_{h}}{L}\right)^{d}\,\hat{b}(w)\right)^{2}\,.$
(43)

Now the critical value $w_{f}=(r_{f}/r_{h})^{d}$ can be found by combining
$\mathcal{U}_{\pm}(P^{\pm}_{\infty},w_{f})=0=\partial_{w}\mathcal{U}_{\pm}(P^{\pm}_{\infty},w_{f})$
and takes a numerical value which depends only on the couplings appearing in
the various functions (i.e., $w_{f}$ is independent of $r_{h}$). Then the
corresponding conserved momenta can be determined using eq. (27),

$P^{\pm}_{\infty}=\left(\frac{r_{h}}{L}\right)^{d}\,\left(\sqrt{\widehat{U}_{0}(w_{f})}\mp\hat{b}(w_{f})\right)\,.$
(44)

As desired, we have found that $P^{\pm}_{\infty}$ are both proportional to the
mass, $M\propto r_{h}^{d}/L^{d}$ as in eq. (13). Hence from eq. (20), the late
time growth rate of this class of observables will also be proportional to the
mass defined in eq. (13).

####  2.1.3 Switchback effect

The analysis from the previous subsection reveals that the gravitational
observable $\mathcal{C}_{\rm gen}$ in the black hole background exhibits a
universal linear growth at late times, namely,

$\mathcal{C}_{\rm{gen}}{(t_{\textrm{\tiny L}},t_{\textrm{\tiny
R}}})\simeq\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\left(P_{\infty}^{+}+P_{\infty}^{-}\right)|t_{\textrm{\tiny
R}}+t_{\textrm{\tiny L}}|+\mathcal{O}(1)\,,$ (45)

where the first “subleading” term is actually a UV divergent constant
(independent of the boundary times, $t_{\textrm{\tiny L}}$ and
$t_{\textrm{\tiny R}}$) Carmi:2016wjl . Of course, this universal linear
growth is the first evidence that our new gravitational observables could be
related to holographic complexity. In addition, holographic complexity is also
expected to exhibit the so-called switchback effect Stanford:2014jda , which
describes the time dependence of complexity under perturbations. The
derivation of the switchback effect for the codimension-zero observables is
similar to that in Belin:2021bga  for the codimension-one case. In the
following, we only sketch the two salient properties of the codimension-zero
observables needed to support the switchback effect.

Let us start with a quick review: We begin by perturbing the TFD state (14) as
follows

$|\Psi(t_{\textrm{\tiny L}},t_{\textrm{\tiny R}})\rangle=e^{-iH_{\textrm{\tiny
L}}t_{\textrm{\tiny L}}-iH_{\textrm{\tiny R}}t_{\textrm{\tiny
R}}}\mathcal{O}_{\textrm{\tiny L}}(t_{n})...\mathcal{O}_{\textrm{\tiny
L}}(t_{1})|\psi_{\rm TFD}(0)\rangle\,.$ (46)

Here $\mathcal{O}_{\textrm{\tiny L}}(t_{i})$ denotes a thermal-scale operator
acting on the left boundary at the boundary time $t_{i}$. Further, the times
$t_{1},\cdots,t_{n}$ are in an alternating “zig-zag” order (i.e.,
$t_{2k+1}>t_{2k}$ but $t_{2k}<t_{2k-1}$). Further the perturbing operators are
chosen to be “small”. That is, they only act on a small number of degrees of
freedom, and so they only affect a small perturbation of the overall state
when they are initially applied. Assuming all of the intervals
$|t_{i+1}-t_{i}|$ are much larger than the scrambling time $t_{*}$, the
complexity of this perturbed TFD state is proportional to Stanford:2014jda

$|t_{\textrm{\tiny R}}+t_{1}|+|t_{2}-t_{1}|+\cdots+|t_{\textrm{\tiny
L}}-t_{n}|-2n\,t_{*}\,.$ (47)

In this context, $n$ is referred to as the number of switchbacks or time-
folds. The contribution $-2n\,t_{*}$ reflects that the effect of the
perturbing operators is initially small and only grows to spread through the
entire system over an interval of the scrambling time $t_{*}$. Hence the
forward and backward time evolution of the state near the time-folds cancels,
and these partial cancellations result in a reduction of the complexity
compared to the naively expected linear growth.

In order to demonstrate this behaviour arises for our new gravitational
observables, we recall that the dual bulk geometry of the perturbed TFD state
(46) is given by a shockwave geometry Shenker:2013yza . Taking a single
shockwave produced by massless null matter located at $U=0$ as an explicit
example, the back-reacted black hole geometry becomes Sfetsos:1994xa ;
Shenker:2013yza ; Stanford:2014jda

$\begin{split}ds^{2}&=-2A(U[V+\alpha_{i}\Theta(U)])dUdV+B(U[V+\alpha_{i}\Theta(U)])d{\vec{x}}^{\,2}\,,\\\
&\ \ \ \ A(UV)=-\frac{2}{UV}\frac{f(r)}{f^{\prime}(r_{h})^{2}}\,,\qquad
B(UV)=r^{2}/L^{2}\,,\end{split}$ (48)

where $|U|=e^{-\frac{2\pi}{\beta}u}$, $|V|=e^{\frac{2\pi}{\beta}v}$ are the
usual Kruskal coordinates, $\Theta(U)$ denotes the Heaviside step function,
and $\alpha_{i}=2e^{-\frac{2\pi}{\beta}(t_{*}\pm t_{i})}$ is the null shift
caused by the $i$’th shockwave.

The first property we need is that the gravitational observables (16) are
additive from the left side to the right side of the shockwave, i.e., there
are no contributions resulting from the shockwave located at $U=0$. This fact
can be proven in the limit of strong shockwaves Stanford:2014jda , by
observing that all delta function contributions appearing in any scalar
functional take the form of $U\delta(U),\,U^{2}\delta^{\prime}(U)$ and so on
Belin:2021bga . Hence these terms naturally vanish on the shockwave, and both
the bulk and boundary contributions appearing in eq. (16) are additive, as
desired. However, in the following, we would like to work with the expression
(17) where the codimension-zero observables are given by integrals along the
surfaces $\Sigma_{\pm}$, and hence we want the additivity to extend to this
form as well. Here, one needs to be careful since the terms proportional to
$\dot{v}_{\varepsilon}b(r_{\varepsilon})$ are boundary terms arising from
integrating the bulk contribution by parts. Since the shockwave divides the
codimension-zero subregion $\mathcal{M}$ into the left and right parts, one
might expect to pick up additional boundary terms there from this bulk
integration. Hence we need to make sure that these potential boundary terms
are the same on the two sides of the shockwave, i.e., they cancel one another.
This can be realized by shifting one of the primitive functions $b(r)$ by a
constant to match the value on the opposite side. Recall from eq. (18) that
the primitive functions are only defined up to an additive constant and so we
are free to make such a shift. More generally, we believe this is related to
the regularization of the shockwave, i.e., this shift would arise naturally if
one began with a smooth background where the shockwave was replaced by a
finite matter distribution – see also footnote 9.

Given the above additivity, the gravitational observable $\mathcal{C}_{\rm
gen}$ (which we implicitly write in the form of eq. (17), i.e., as the sum of
surface contributions from $\Sigma_{\pm}$) in the multiple shockwave geometry
is simply given by the sum of each patch, viz,

$\begin{split}\mathcal{C}_{\rm{gen}}{(t_{\textrm{\tiny L}},t_{\textrm{\tiny
R}}})=\mathcal{C}_{\rm gen}(t_{\textrm{\tiny R}},V_{1})&+\mathcal{C}_{\rm
gen}(V_{1}+\alpha_{1},U_{2})+\cdots\\\ &+\mathcal{C}_{\rm
gen}(U_{n-1}-\alpha_{n-1},V_{n})+\mathcal{C}_{\rm
gen}(V_{n}+\alpha_{n},t_{\textrm{\tiny L}})\,.\end{split}$ (49)

Here $\mathcal{C}_{\rm gen}(.,.)$ denotes the contributions from
$\Sigma_{\pm}$ with two fixed endpoints and all endpoints are located either
on the left/right horizon or on the boundary. For the purpose of evaluating
the gravitational observables $\mathcal{C}_{\rm gen}$ in the shockwave
geometry, we finally need to maximize eq. (49) by varying the joint points on
the shockwaves. The maximization is easily performed at late times with strong
shockwaves by noting that the contributions between the left/right horizon
also present the linear growth but in terms of the ingoing/outgoing time as
follows

$\mathcal{C}_{\rm{gen}}{(u_{\textrm{\tiny L}},v_{\textrm{\tiny
R}}})\approx\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\left(P_{\infty}^{+}\,|v_{\textrm{\tiny R}}^{+}+u_{\textrm{\tiny
L}}^{+}|+P_{\infty}^{-}\,|v_{\textrm{\tiny R}}^{-}+u^{-}_{\textrm{\tiny
L}}|\right)+\mathcal{O}(1)\,,$ (50)

where $u^{\pm}_{\textrm{\tiny L}}/v_{\textrm{\tiny R}}^{\pm}$ present the
coordinate value of the intersection between $\Sigma_{\pm}$ and left/right
horizon. In order to derive the linear growth in eq. (50), one can focus on
the right part and first rewrite the value of the intersection , i.e.,
$v_{\textrm{\tiny R}}^{\varepsilon}$ in terms of

$\begin{split}v_{\textrm{\tiny
R}}^{\varepsilon}-v_{\varepsilon,\rm{min}}&=\int^{r_{\rm{h}}}_{r_{\rm{min}}}dr\,\frac{\dot{v}}{\dot{r}}=\int^{r_{\rm{h}}}_{r_{\varepsilon,\rm{min}}}dr\,\frac{1}{f(r)}\left(1-\frac{P^{\varepsilon}_{v}+{\varepsilon}b(r)}{\sqrt{(P^{{\varepsilon}}_{v}+{\varepsilon}b(r))^{2}-U_{0}(r)}}\right)\,,\end{split}$
(51)

by using the definition of infalling coordinate $v$ and the extremization
equations derived in eqs. (23) for $\Sigma_{\pm}$. At late times, the extremal
surface $\Sigma_{\pm}$ inside the horizon approaches the final slice at
$r=r_{\pm,f}$. It is then obvious that the dominant contributions to
$v_{\textrm{\tiny R}}^{\varepsilon}$ (i.e., a logarithmic divergence) at late
times originate from the integral around the regime $r\sim r_{\rm{min}}\sim
r_{\pm,f}$, i.e.,

$\lim\limits_{r_{\varepsilon,\rm{min}}\to r_{\varepsilon,f}}v_{\textrm{\tiny
R}}^{\varepsilon}\approx\left(\frac{P^{\varepsilon}_{\infty}+\varepsilon\,b(r_{\varepsilon,f})}{f(r_{\varepsilon,f})}\right)\frac{\log\left(r_{\varepsilon,\rm{min}}-r_{\varepsilon,f}\right)}{\sqrt{-\frac{1}{2}\mathcal{U}_{\varepsilon}^{\prime\prime}(P_{\infty}^{\varepsilon},r_{\varepsilon,f})}}+\mathcal{O}(1)\,.$
(52)

A similar formula holds for the outgoing coordinate $u_{\textrm{\tiny
L}}^{\varepsilon}$. On the other hand, the volume contribution
$\mathcal{C}_{\rm{gen}}{(u_{\textrm{\tiny L}},v_{\textrm{\tiny R}}})$ from the
portion connecting any two points on the left and right horizon can be
simplified as

$\begin{split}\mathcal{C}_{\rm{gen}}{(u^{\varepsilon}_{\textrm{\tiny
L}},v^{\varepsilon}_{\textrm{\tiny R}}})=-\frac{2V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\int^{r_{\varepsilon,h}}_{r_{\varepsilon,\textrm{\tiny
min}}}\frac{U_{0}(r)-\varepsilon\,b(r)\left(P^{\varepsilon}_{v}+\varepsilon\,b(r)\right)}{f(r)\,\sqrt{-\mathcal{U}_{\varepsilon}(P^{\varepsilon}_{v},r)}}\,dr\,,\end{split}$
(53)

which is similar to eq. (29). The leading contribution of
$\mathcal{C}_{\rm{gen}}{(u^{\varepsilon}_{\textrm{\tiny
L}},v^{\varepsilon}_{\textrm{\tiny R}}})$ at late times is also given by a
logarithmic divergence but with a different coefficient, viz,

$\lim\limits_{r_{\varepsilon,\rm{min}}\to
r_{\varepsilon,f}}\mathcal{C}_{\rm{gen}}{(u^{\varepsilon}_{\textrm{\tiny
L}},v^{\varepsilon}_{\textrm{\tiny
R}}})\approx\sum_{\varepsilon=+,-}\left(\frac{U_{0}(r_{\varepsilon,f})-\varepsilon\,b(r_{\varepsilon,f})\left(P^{\varepsilon}_{\infty}+\varepsilon\,b(r_{\varepsilon,f})\right)}{f(r_{\varepsilon,f})}\right)\frac{\log\left(r_{\varepsilon,\rm{min}}-r_{\varepsilon,f}\right)}{\sqrt{-\frac{1}{2}\mathcal{U}_{\varepsilon}^{\prime\prime}(P_{\infty}^{\varepsilon},r_{\varepsilon,f})}}\,.$
(54)

Noting that the final slice is determined by

$\mathcal{U}_{\pm}(P^{\varepsilon}_{\infty},r_{\varepsilon,f})=U_{0}(r_{\varepsilon,f})-\left(P^{\varepsilon}_{\infty}+\varepsilon\,b(r_{\varepsilon,f})\right)^{2}=0\,,$
(55)

we finally arrive at the desired linear growth shown in eq. (50) by comparing
the two expressions in eq. (52) and eq. (54). Of course, one can also consider
the extremal surface with one endpoint on the horizon and one on the
asymptotic boundary, for which we obtain

$\mathcal{C}_{\rm{gen}}{(t_{\textrm{\tiny L}},v_{\textrm{\tiny
R}}})\approx\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\left(P_{\infty}^{+}\,|v_{\textrm{\tiny R}}^{+}+t_{\textrm{\tiny
L}}|+P_{\infty}^{-}\,|v_{\textrm{\tiny R}}^{-}+t_{\textrm{\tiny
L}}|\right)+\mathcal{O}(1)\,,$ (56)

Besides the linear growth, we would like to highlight that another interesting
and important fact for the switchback effect is that the growth rate (i.e.,
$P^{\pm}_{\infty}$) is also universal for all pieces of the piecewise maximal
surface $\Sigma_{\pm}$. Substituting the linear growth exhibited in eqs. (50)
and (56) into eq. (49), and extremizing with respect to
$V_{1},U_{2},...U_{n-1},V_{n}$, we find that $V_{i}=-\alpha_{i}/2$ and
$U_{i}=\alpha_{i}/2$. Finally, we can conclude that the gravitational
observables $\mathcal{C}_{\rm gen}$ associated with the bulk subregion whose
boundaries are maximal surfaces $\Sigma_{\pm}$ display the desired switchback
effect (47).

####  2.1.4 General observables with $G_{1}\neq G_{2}$ and $F_{1,\pm}\neq
F_{2,\pm}$

Now we return to consider the general case with the functional (7) which is
extremized to define the bulk region and the functional (11) evaluated on this
region are defined independently. That is, the corresponding integrals involve
different scalar functions, i.e., $(F_{1,\pm},G_{1})$ and $(F_{2,\pm},G_{2})$.
The codimension-zero complexity observable is then reduced to

$O\\!\left[G_{1},F_{1,\pm},\mathcal{M}_{G_{2},F_{2,\pm}}\right]=\frac{V_{x}}{G_{N}L}\sum_{\varepsilon=+,-}\int_{\Sigma_{\varepsilon}(a_{2},b_{2})}\\!\\!\\!\\!\\!\\!\\!d\sigma\,\left[\left(\frac{r}{L}\right)^{d-1}\sqrt{-f(r\,){\dot{v}}^{2}+2\dot{v}\,\dot{r}}\
a_{1}(r)-\varepsilon\dot{v}\,b_{1}(r)\right]\,,$ (57)

when evaluated in the planar black hole spacetime (15). Similar to the
calculations around eq. (29), one can recast the integral along the extremal
surfaces $\Sigma_{\pm}(a_{2},b_{2})$ in terms of a radial integral with

$O\\!\left[G_{1},F_{1,\pm},\mathcal{M}_{G_{2},F_{2,\pm}}\right]=-\frac{2V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\int^{\infty}_{r_{\pm,\textrm{\tiny
min}}}\frac{dr}{f(r)}\left[\frac{\sqrt{U_{1}(r)U_{2}(r)}-b_{1}(r)(b_{2}(r)+\varepsilon
P^{\varepsilon}_{v})}{\sqrt{(P^{\varepsilon}_{v}+\varepsilon\,b_{2}(r))^{2}-U_{2}(r)}}\right]\,,$
(58)

where $U_{1,2}(r)=-f(r)\,a_{1,2}^{2}(r)\,({r}/{L})^{2(d-1)}$. It is worth
noting that the conjugate momenta $P^{\pm}_{v}$ and the corresponding minimal
radii $r_{\pm,\textrm{\tiny min}}$ are associated with $(a_{2}(r),b_{2}(r))$,
i.e.,

$P^{\pm}_{v}=\mp\bar{b}_{2}+{\tilde{\lambda}}\,\sqrt{\bar{U}_{2}}\,.$ (59)

where ${\tilde{\lambda}}=\pm$ as in eq. (27), and we have introduced the
‘overbar’ notation to indicate that the corresponding function is evaluated at
$r=r_{\pm,\textrm{\tiny min}}$, i.e., $\bar{y}=y(r_{\pm,\textrm{\tiny min}})$.
As before, our conventions are such that the ${\tilde{\lambda}}=+$ branch
yields the observable for late times (i.e., $\tau\to+\infty$) and we adopt
this sign choice throughout the following. In the same spirit of the
decomposition in eq. (30), one can re-express the above as

$\begin{split}&O\\!\left[G_{1},F_{1,\pm},\mathcal{M}_{G_{2},F_{2,\pm}}\right]=-\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\left[\left(\varepsilon\bar{b}_{1}-\sqrt{\bar{U}_{1}}\right)\tau\right.\\\
&\qquad\left.+2\int^{\infty}_{r_{\pm,\textrm{\tiny
min}}}\frac{dr}{f(r)}\left(\frac{\sqrt{U_{1}(r)U_{2}(r)}-(b_{1}(r)-\bar{b}_{1}+\varepsilon\sqrt{\bar{U}_{1}}))(b_{2}(r)+\varepsilon
P^{\varepsilon}_{v})}{\sqrt{(P^{\varepsilon}_{v}+\varepsilon\,b_{2}(r))^{2}-U_{2}(r)}}\right)\right]\,.\end{split}$
(60)

It is crucial to note the integrand on the minimal radius exactly vanishes
since

$\left[\sqrt{U_{1}(r)U_{2}(r)}-(b_{1}(r)-\bar{b}_{1}\pm\sqrt{\bar{U}_{1}}))(b_{2}(r)\pm
P^{\pm}_{v})\right]\bigg{|}_{r=r_{\pm,\textrm{\tiny min}}}=0\,.$ (61)

Now, taking the time derivative of the observable
$O\\!\left[G_{1},F_{1,\pm},\mathcal{M}_{G_{2},F_{2,\pm}}\right]$, we
find151515Note that there is a factor of 2 missing in the analogous equations
in Belin:2021bga , i.e., eqs. (23) and (42).

$\begin{split}&\frac{d}{d\tau}\Big{(}O\\!\left[G_{1},F_{1,\pm},\mathcal{M}_{G_{2},F_{2,\pm}}\right]\Big{)}=-\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\Bigg{[}\left(\varepsilon\bar{b}_{1}-\sqrt{\bar{U}_{1}}\right)\\\
&+2\frac{dP^{\varepsilon}_{v}}{d\tau}\int^{\infty}_{r_{\pm,\textrm{\tiny
min}}}\frac{dr}{f(r)}\frac{(P^{\varepsilon}_{v}+\varepsilon\,b_{2}(r))\sqrt{U_{1}(r)U_{2}(r)}+\left(\varepsilon(b_{1}(r)-\bar{b}_{1})+\sqrt{\bar{U}_{1}}\right)\,U_{2}(r)}{\left((P^{\varepsilon}_{v}-\varepsilon\,b_{2}(r))^{2}-U_{2}(r)\right)^{3/2}}\Bigg{]}\,,\end{split}$
(62)

where the bulk integral term does not vanish since the surfaces $\Sigma_{\pm}$
were determined by extremizing with respect to the functional
$W_{G_{2},F_{2,\pm}}$ in eq. (7). However, we can find that the late-time
limit of the corresponding growth rate is dominated by the linear terms, i.e.,

$\lim_{\tau\to\infty}\frac{d}{d\tau}\,\Big{(}O\\!\left[G_{1},F_{1,\pm},\mathcal{M}_{G_{2},F_{2,\pm}}\right]\Big{)}=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\left(P_{\infty}^{+}(F_{1},G_{1})+P_{\infty}^{-}(F_{1},G_{1})\right)\,,$
(63)

where we have defined

$P^{\pm}_{\infty}(F_{1},G_{1})\equiv\mp\bar{b}_{1}+\sqrt{\bar{U}_{1}}\,.$ (64)

To produce this result (63), we have used the asymptotic expansion (38) for
$dP^{\pm}_{v}/d\tau$ and the suppression of the divergence in the bulk
integral by the vanishing of

$\left[(P^{\pm}_{v}\pm
b_{2}(r))\sqrt{U_{1}(r)U_{2}(r)}+\left(\pm(b_{1}(r)-\bar{b}_{1})+\sqrt{\bar{U}_{1}}\right)\,U_{2}(r)\right]\bigg{|}_{r=r_{\pm,\textrm{\tiny
min}}}=0\,.$ (65)

###  2.2 Spacetime volume between CMC slices

In this section, we study an explicit example of the codimension-zero
observable $\mathcal{C}_{\rm gen}$ in eq. (16) with $G_{1}=G_{2}$ and
$F_{1,\pm}=F_{2,\pm}$. In fact, we examine the simplest example where $G_{1}$
and $F_{1,\pm}$ are all taken to be constants. That is, we consider the
following functional:

$\mathcal{C}_{\rm gen}=\frac{1}{G_{\textrm{\tiny
N}}L}\left[\alpha_{+}\int_{\Sigma_{+}}\\!\\!\\!d^{d}\sigma\,\sqrt{h}+\alpha_{-}\int_{\Sigma_{-}}\\!\\!\\!d^{d}\sigma\,\sqrt{h}+\frac{\alpha_{\textrm{\tiny
B}}}{L}\,\int_{\mathcal{M}}\\!\\!d^{d+1}x\,\sqrt{-g}\right]\,,$ (66)

where $\alpha_{\pm}$ and $\alpha_{\textrm{\tiny B}}$ are dimensionless
(positive) constants. In a moment, we will consider evaluating this observable
in a planar black hole background as discussed in the previous section.
However, we begin with a few general remarks. Extremizing over the shape of
the boundary surfaces following eq. (10), one finds that the extremal
boundaries $\Sigma_{\pm}$ are constant mean curvature (CMC) slices, e.g., see
marsden1980maximal ; witten2017 . That is, the extremization equations can be
simply expressed as161616The extrinsic curvature is defined as
$K_{\mu\nu}\equiv h\indices{{}^{\alpha}_{\mu}}\nabla_{\alpha}n_{\nu}$ where
the (timelike) normal vector $n^{\mu}$ is chosen here to be future-directed
for both spacelike hypersurfaces $\Sigma_{\pm}$, and
$h\indices{{}^{\alpha}_{\mu}}$ is the projector onto the tangent space of the
surfaces $\Sigma_{\pm}$.

$K_{\Sigma_{+}}=-\frac{\alpha_{\textrm{\tiny B}}}{\alpha_{+}L}\,,\qquad
K_{\Sigma_{-}}=\frac{\alpha_{\textrm{\tiny B}}}{\alpha_{-}L}\,,$ (67)

Given this geometric framework for the extremal equations, one can argue that
the boundary surfaces $\Sigma_{\pm}$ always exist and are unique, as long as
the cosmological constant is negative and no matter fields are excited. We
leave this discussion to appendix C.

This functional (66) is also an interesting object since it can be easily
related to the original holographic complexity proposals in eqs. (1 – 3). For
example, one can recover the CV proposal (1) by taking the limit
$\alpha_{+}\to+\infty$ (while keeping $\alpha_{-}$ and $\alpha_{\textrm{\tiny
B}}$ fixed. That is,

${\cal C}_{\textrm{\tiny
V}}=\lim_{\alpha_{+}\to+\infty}\left(\frac{1}{\alpha_{+}}\,\mathcal{C}_{\rm
gen}\right)\,.$ (68)

Of course, this is a simple illustration of how the codimension-one
observables introduced in Belin:2021bga  are related to the new codimension-
zero observables presented here.

Figure 2: Left: Sketch of a codimension-zero region where boundaries
$\Sigma_{\pm}$ are constant mean curvature slices, determined by the
functional in eq. (66). Right: Taking the limit $\alpha_{\pm}\to 0$, the
boundaries $\Sigma_{\pm}$ become null surfaces, and the region becomes the WDW
patch.

Another interesting limit is to take $\alpha_{\pm}\to 0$ with
$\alpha_{\textrm{\tiny B}}$ fixed. From eq. (67), we see the extrinsic
curvatures of the boundaries $\Sigma_{\pm}$ diverge in this limit, i.e.,
$K_{\Sigma_{\pm}}\rightarrow\mp\infty$. This divergence indicates that these
boundary surfaces are being pushed to the future and past light sheets
emanating from the boundary time slice $\Sigma_{\textrm{\tiny CFT}}$, as
illustrated in Figure 2. That is, the bulk codimension-zero region
$\mathcal{M}$ becomes the Wheeler-DeWitt (WDW) patch in this limit!

Hence, one recovers the CV2.0 proposal (3), i.e., the spacetime volume of the
WDW patch, from eq. (66) above by taking $\alpha_{\pm}\to 0$ while keeping
$\alpha_{\textrm{\tiny B}}$ fixed viz,

${\cal C}_{\textrm{\tiny SV}}=\lim_{\alpha_{\pm}\to
0}\left(\frac{1}{\alpha_{\textrm{\tiny B}}}\,\mathcal{C}_{\rm gen}\right)\,.$
(69)

We can also recover the CA proposal (2) with this limit and using our general
proposal (11). That is, we first would extremize the functional
$\mathcal{C}_{\rm gen}$ in eq. (66) and take the limit $\alpha_{\pm}\to 0$.
Having obtained the WDW patch as the region $\mathcal{M}$, we then evaluate
the gravitational action on this region, i.e., take $G_{1}$ as the standard
Einstein-Hilbert term (with cosmological constant contribution) and
$F_{1,\pm}$, as the corresponding null boundary action terms Parattu:2015gga ;
Lehner:2016vdi ; Hopfmuller:2016scf ; Chandrasekaran:2020wwn . An interesting
question is to examine the procedure where we extremize eq. (66) with finite
$\alpha_{\pm}$ first, then evaluate the gravitational action on the resulting
region $\mathcal{M}$ with the Gibbons-Hawking-York boundary terms on the
spacelike surfaces $\Sigma_{\pm}$, and finally take the limit $\alpha_{\pm}\to
0$. Interestingly, this null limit procedure for the action yields a finite
result, although corrections to the standard null boundary terms involving the
surface gravities of the null surfaces arise from this limit. The details of
this limiting procedure and additional discussion of the treatment of Hayward
terms in the action for codimension-two corners are given in appendix D, and
further discussion is given in section 4.

In the following discussion, we will keep $\alpha_{\pm}$,
$\alpha_{\textrm{\tiny B}}$ as independent constants and explicitly explore
the maximization and time evolution of the functional (66) in the planar black
hole background (15).

####  2.2.1 Extremal Surfaces $\Sigma_{\pm}$

Now, let us explicitly evaluate the codimension-zero observable given in eq.
(66) in the black hole background given by the infalling coordinates (15). We
begin with the bulk term in eq. (66), which becomes

$\begin{split}\frac{\alpha_{\textrm{\tiny
B}}}{L}\,\int_{\mathcal{M}}\sqrt{g}&=\frac{\alpha_{\textrm{\tiny B}}}{L}\int
d^{d-1}x\int dv\int^{r_{-}(v)}_{r_{+}(v)}dr\left(\frac{r}{L}\right)^{d-1}\\\
&=\frac{\alpha_{\textrm{\tiny
B}}\,V_{x}}{d\,L^{d}}\left(\int_{\Sigma_{-}}d\sigma\,\dot{v}_{-}(\sigma)\,r_{-}(\sigma)^{d}-\int_{\Sigma_{+}}d\sigma\,\dot{v}_{+}(\sigma)\,r_{+}(\sigma)^{d}\right)\,,\end{split}$
(70)

where we have parametrized the surfaces as $r_{\pm}(\sigma),v_{\pm}(\sigma)$
and integrated by parts to get the second line. As we illustrated before, this
bulk contribution splits into a sum of two independent terms associated to the
boundary surfaces $\Sigma_{\pm}$. The extremality conditions (10) are then
equivalent to solving for two decoupled particle motions governed by the
Lagrangian
$\mathcal{L}=\mathcal{L}_{+}({\Sigma_{+}})+\mathcal{L}_{-}({\Sigma_{-}})$,
where

$\mathcal{L}_{\pm}=\left(\frac{r}{L}\right)^{d-1}\sqrt{-f(r){\dot{v}}^{2}+2\dot{v}\,\dot{r}}\mp\frac{\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\pm}L^{d}}\,\dot{v}\,r^{d}\,,$ (71)

where we have dropped the subscripts $\pm$ on $(v(\sigma),r(\sigma))$ to
reduce the clutter. Further, to simplify the following equations, we have
absorbed the factors $\alpha_{\pm}$ as part of the overall coefficients in the
observable, e.g., along with ${V_{x}}/(G_{\textrm{\tiny N}}L)$. Compared to
the analysis for the codimension-one observables (see appendix A), the new
feature here is a ‘magnetic field’-like term in the Lagrangian.

Comparing the expressions (71) to eq. (19), we see that here we have

$a_{\pm}(r)=1\,,\qquad b(r)=\frac{\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\pm}}\left(\frac{r}{L}\right)^{d}\,.$ (72)

Gauge fixing as before, with

$\sqrt{-f(r)\dot{v}^{2}+2\dot{v}\dot{r}}=(r/L)^{d-1}\,,$ (73)

the conserved momentum becomes

$P_{v}^{\varepsilon}=\dot{r}-\dot{v}f(r)-\varepsilon\,\frac{\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\,\left(\frac{r}{L}\right)^{d}\,,$ (74)

where as before, $\varepsilon=\pm$ for the future/past boundaries
$\Sigma_{\pm}$. Combining eqs. (73) and (74), we can express the equation
solving for the radial profile $r(\sigma)$ as

$\dot{r}^{2}+\mathcal{U}(P_{v}^{\varepsilon},r)=0\,,$ (75)

where

$\begin{split}\mathcal{U}(P_{v}^{\varepsilon},r)&=U_{0}(r)-\left(P_{v}^{\varepsilon}+\varepsilon\,\frac{\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\,\left(\frac{r}{L}\right)^{d}\right)^{2}\,,\\\
{\rm with}\qquad
U_{0}(r)&\equiv-f(r)\left(\frac{r}{L}\right)^{2(d-1)}=\left(\frac{r}{L}\right)^{2d}\left(\frac{r_{h}^{d}}{r^{d}}-1\right)\,.\\\
\end{split}$ (76)

That is, as before, we have expressed this as a simple classical mechanics
problem. However, in the present case, the potential depends on
$P_{v}^{\varepsilon}$. This is not unexpected since canonical momentum differs
from kinetic momentum due to the magnetic field. For convenience, we have
combined all of the $P_{v}^{\varepsilon}$ terms in the potential
$\mathcal{U}(P_{v}^{\varepsilon},r)$ and hence the effective energy (on the
right-hand side of eq. (75)) is zero. Recall that $U_{0}(r)$ is the effective
potential that appears when solving for extremal surfaces for
complexity=volume (and which can be recovered with the limit
$\alpha_{+}\to\infty$ – see eq. (68)).

Figure 3: The effective potentials $\mathcal{U}(P_{v}^{\varepsilon},r)$ for
the future boundary $\Sigma_{+}$ (left) and the past boundary $\Sigma_{-}$
(right), respectively. Hence $\varepsilon=+\,(-)$ in the left (right) panel.
For this figure, we have chosen $d=2$ and
$\alpha_{\pm}=1=\alpha_{\textrm{\tiny B}}$.

We note that

$\begin{split}\mathcal{U}(P_{v}^{\varepsilon},r=0)&=-(P_{v}^{\varepsilon}/\alpha_{\varepsilon})^{2}<0\,,\\\
\mathcal{U}(P_{v}^{\varepsilon},r\to\infty)&\simeq-\left(1+\frac{\alpha_{\textrm{\tiny
B}}^{2}}{d^{2}\alpha_{\varepsilon}^{2}}\right)\left(\frac{r}{L}\right)^{2d}<0\,,\\\
\end{split}$ (77)

In general, we will be interested in the situation where $P_{v}^{\varepsilon}$
is chosen such that the potential is positive over some range of $r$. The
extremal surfaces of interest are then given by the zero-energy trajectories
where the particle comes in from $r=\infty$, reflects off of the potential at
the minimal radius $r_{\pm,\textrm{\tiny min}}$ where
$\mathcal{U}(P_{v}^{\pm},r_{\pm,\textrm{\tiny min}})=0$, and then returns to
$r=\infty$. We illustrate some typical potentials in figure 3.

Before proceeding, let us first show that the surfaces satisfying eqs. (73)
and (75) with any conserved momentum $P_{v}^{\epsilon}$ are precisely the
constant mean curvature slices given in eq. (67). Using the infalling
coordinates (15), the trace of the extrinsic curvature of the hypersurface
$v(r)$ can be expressed as171717As discussed in footnote 16, the normal vector
$n^{\mu}$ is taken to be future-directed for both $\Sigma_{\pm}$, meaning that
its components after lowering an index satisfy
$n_{\mu}\propto(-1,v^{\prime}(r),\vec{0})$.

$K=\frac{4(d-1)v^{\prime}(r)-\left(2\left(d-1\right)f(r)-rf^{\prime}(r)\right)(3-f(r)v^{\prime}(r))(v^{\prime}(r))^{2}-2rv^{\prime\prime}(r)}{2r\left(2v^{\prime}(r)-f(r)v^{\prime}(r)^{2})\right)^{3/2}}\,,$
(78)

where $v^{\prime}\equiv\partial v(r)/\partial r$. Using our intrinsic
parametrization of the boundary surfaces, i.e., $(v(\sigma),r(\sigma))$, we
can rewrite the above result as

$\begin{split}K&=\frac{4(d-1)\dot{v}\dot{r}^{2}-\left(2(d-1)f(r)+rf^{\prime}(r)\right)\left(3\dot{r}-f(r)\dot{v}\right)\dot{v}^{2}-2r\left(\dot{r}\ddot{v}-\dot{v}\ddot{r}\right)}{2r(2\dot{v}\dot{r}-f(r)\dot{v}^{2})^{3/2}}\,.\end{split}$
(79)

After imposing our gauge-fixing condition (73), this expression reduces to

$K=\frac{1}{\sqrt{-f(r)(r/L)^{2(d-1)}+\dot{r}^{2}}}\left(-\left(\frac{L}{r}\right)^{d-1}r\ddot{r}+\frac{1}{2}\left(2(d-1)f(r)+rf^{\prime}(r)\right)\right)\,,$
(80)

where we have substituted $\dot{v}(\sigma)$ and $\ddot{v}(\sigma)$ (derived by
taking the derivative of eq. (73)) in terms of $\dot{r}(\sigma)$ and
$\ddot{r}(\sigma)$. We note that this expression does not distinguish between
the future and past boundary surfaces, however, this distinction arises in the
final step where we eliminate $\dot{r}$ and $\ddot{r}$ using the radial
equation of motion (75). The final result with any conserved momentum
$P_{v}^{\varepsilon}$ is then

$K_{\Sigma_{\varepsilon}}(P_{v}^{\varepsilon})=-\varepsilon\,\frac{\alpha_{\textrm{\tiny
B}}}{\alpha_{\varepsilon}L}\,.$ (81)

Hence we have precisely recovered eq. (67) and confirmed that the extremal
boundaries are constant mean curvature slices. While we are generally
interested in positive coefficients, i.e., $\alpha_{\textrm{\tiny
B}},\,\alpha_{\pm}>0$, we note that the above expression is invariant under
$\alpha_{\textrm{\tiny B}}/\alpha_{\varepsilon}\to-\alpha_{\textrm{\tiny
B}}/\alpha_{\varepsilon},\ \varepsilon\to-\varepsilon$. That is, the two
surfaces $\Sigma_{\pm}$ are interchanged if, e.g., we flip the sign of
$\alpha_{\textrm{\tiny B}}$.

Turning to the time evolution of the extremal surfaces in the planar black
hole background (15), we introduce a dimensionless radial coordinate and mass
parameter,

$w\equiv\left(\frac{r}{r_{h}}\right)^{d}\,,\qquad
x\equiv\left(\frac{r_{h}}{L}\right)^{d}\propto M\,.$ (82)

Recall the extremal surfaces are described by the classical mechanics problem
in eqs. (75) and (76) and the trajectories of interest come in from
$r=\infty$, reflect of the potential at
$\mathcal{U}(P_{v}^{\varepsilon},r=r_{\pm,\textrm{\tiny min}})=0$ and return
to $r=\infty$. Hence using eq. (76), the turning points $r_{\pm,\textrm{\tiny
min}}$ for the surfaces $\Sigma_{\pm}$ are determined by

$-f(r_{\pm,\textrm{\tiny min}})\left(\frac{r_{\pm,\textrm{\tiny
min}}}{L}\right)^{d}=\left(P_{v}^{\pm}\pm\frac{\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\pm}}\left(\frac{r_{\pm,\textrm{\tiny
min}}}{L}\right)^{d}\right)^{2}\,.$ (83)

Using the dimensionless variables in eq. (82), the latter is simply recast as
the following quadratic equation

$0=\left(1+\frac{\alpha_{\textrm{\tiny
B}}^{2}}{d^{2}\alpha_{\varepsilon}^{2}}\right)x^{2}\,w_{\varepsilon,\rm
min}^{2}-\left(x-\varepsilon\,\frac{2\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\,P_{v}^{\varepsilon}\right)x\,w_{\varepsilon,\rm
min}+P_{v}^{\varepsilon}{}^{2}\,.$ (84)

This corresponds to an ellipse in the ($P^{\varepsilon}_{v},w_{\varepsilon,\rm
min}$)-plane, as illustrated in figure 4. Solving eq. (84) for the turning
points yields

$\begin{split}w_{\varepsilon,\rm
min}=\frac{d\,\alpha_{\varepsilon}}{2x(d^{2}\alpha_{\varepsilon}^{2}+\alpha_{\textrm{\tiny
B}}^{2})}\left[(d\,\alpha_{\varepsilon}x-2\varepsilon\alpha_{\textrm{\tiny
B}}{P_{v}^{\varepsilon}})+\hat{\lambda}\sqrt{d^{2}\alpha_{\varepsilon}^{2}\left(x^{2}-4P_{v}^{\varepsilon}{}^{2}\right)-4\varepsilon
d\,\alpha_{\varepsilon}\alpha_{\textrm{\tiny
B}}\,P_{v}^{\varepsilon}x}\right]\,,\end{split}$ (85)

where $\hat{\lambda}=\pm$.181818A simple check of this result is to consider
the limit $\alpha_{+}\to\infty$ as in eq. (68), which leads to the result
associated with the extremal volume surface, i.e.,
$\lim\limits_{\alpha_{+}\to\infty}w_{+,\mathrm{min}}=\frac{1}{2}\big{(}1+\hat{\lambda}\sqrt{1-{4P^{+\,2}_{v}}/{x^{2}}}\big{)}$.
Alternatively, for a given minimal radius, we can determine the conserved
momentum $P_{v}^{\epsilon}$ as

$P_{v}^{\varepsilon}(\pm)=x\left(-\frac{\varepsilon\alpha_{\textrm{\tiny
B}}}{d\alpha_{\varepsilon}}\,w_{\varepsilon,\rm{min}}\pm\sqrt{w_{\varepsilon,\rm{min}}\left(1-w_{\varepsilon,\rm{min}}\right)}\right)\,.$
(86)

Here the plus/minus signs correspond to moving the boundary time slice towards
positive or negative times. Note that for $P_{v}^{\varepsilon}$ to be real, we
must have $0\leq w_{\varepsilon,\rm min}\leq 1$. That is, the turning point
lies between the singularity (i.e., $w=0$) and the horizon (i.e., $w=1$).
Finally, we remark that in this expression, the conserved momentum
$P_{v}^{\varepsilon}$ is proportional to the dimensionless mass parameter
$x\propto M$.

Figure 4: Illustration of the relation between the conserved momentum
$P_{v}^{\varepsilon}$ and the minimal radius $w_{\varepsilon,\rm min}$ in eq.
(84), for $\alpha_{\textrm{\tiny B}}=1=\alpha_{\pm}$, $x=1$ and $d=2$. The
solutions $P_{v}^{\varepsilon}(+)$ in eq. (86) correspond to the black arcs,
while $P_{v}^{\varepsilon}(-)$ correspond to the red arcs. The critical
momenta $P_{\infty}^{\varepsilon}(\pm)$ in eq. (88) correspond to the maximum
and minimum vertical extrema of the ellipses marked by the horizontal dashed
lines. The corresponding turning points $w_{f}(\pm)$ given in eq. (89) are
marked by the vertical dashed lines.

Recall that we are generally interested in the case where there are two real
solutions for eq. (85), e.g., see figure 3, and further the larger root is the
correct turning point for the trajectories of interest. That is, we must
choose $\hat{\lambda}=+$ in the above expression.191919The choice
$\hat{\lambda}=-$ would correspond to a trajectory that begins at the
singularity $r=0$, reflects off the potential at the corresponding
$w_{\varepsilon,\rm min}(\hat{\lambda}=-)$ (which actually corresponds to the
maximum radius, rather than the minimum), and returns to $r=0$. Hence these
extremal surfaces are anchored to the singularity, rather than the asymptotic
boundary. As illustrated in figure 4, we must also choose the momentum
$P^{\varepsilon}_{v}$ such that the argument of the square-root above is
positive, i.e.,

$P^{\varepsilon}_{\infty}(-)\leq P^{\varepsilon}_{v}\leq
P^{\varepsilon}_{\infty}(+)$ (87)

where

$\begin{split}P^{\varepsilon}_{\infty}(+)&\equiv\frac{x}{2}\left(\sqrt{1+\frac{\alpha_{\textrm{\tiny
B}}^{2}}{d^{2}\alpha^{2}_{\varepsilon}}}-\varepsilon\,\frac{\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\right)\,,\qquad
P^{\varepsilon}_{\infty}(-)\equiv-\frac{x}{2}\left(\sqrt{1+\frac{\alpha_{\textrm{\tiny
B}}^{2}}{d^{2}\alpha^{2}_{\varepsilon}}}+\varepsilon\,\frac{\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\right)\,.\end{split}$ (88)

The limiting values $P^{\varepsilon}_{\infty}(\pm)$ correspond to the critical
values of the momenta where eq. (85) yields a single solution. The
corresponding turning points $w_{f}=(r_{f}/r_{h})^{d}$ are

$w_{f}(\pm)=\frac{1}{2}\left(1\mp\frac{\varepsilon}{\sqrt{d^{2}\alpha_{\varepsilon}^{2}/\alpha_{\textrm{\tiny
B}}^{2}+1}}\right)\,.$ (89)

Tuning the momenta to the critical values (88) produces the critical
potentials with

$\mathcal{U}(P_{v}^{\varepsilon},w)\big{|}_{w=w_{f}}=0\,,\quad\partial_{w}\mathcal{U}(P_{v}^{\varepsilon},w)\big{|}_{w=w_{f},P_{v}^{\varepsilon}=P_{\infty}^{\varepsilon}}=0\,,$
(90)

as in eq. (32), for which the boundary time diverges. We will find below
(e.g., see eq. (97)) that $P^{\varepsilon}_{v}\to P^{\varepsilon}_{\infty}(+)$
corresponds to $\tau\to+\infty$ and $P^{\varepsilon}_{v}\to
P^{\varepsilon}_{\infty}(-)$, to $\tau\to-\infty$.

####  2.2.2 Time Evolution of Extremal Surfaces

Figure 5: The relation between the boundary time $\tau$ and the conserved
momentum $P_{v}^{\varepsilon}$. We set $\alpha_{\textrm{\tiny
B}}=1=\alpha_{\pm}$, $x=1$ and $d=2$ in both panels and $\varepsilon=+,-$ for
the left and right plot, respectively.

#### Regime A: $P_{\infty}^{\epsilon}(-)\leq P_{v}^{\epsilon}\leq
P_{\infty}^{\epsilon}(+)$

With regards to the time evolution of the boundary time slice $\Sigma_{\tau}$,
we have shown in eq. (31) that the growth rate of the codimension-zero
observable $\mathcal{C}_{\rm gen}(\tau)$ is

$\frac{d}{d\tau}\mathcal{C}_{\rm{gen}}=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\,\left(\alpha_{+}\,P_{v}^{+}(\tau)+\alpha_{-}\,P_{v}^{-}(\tau)\right)\,.$
(91)

Recall that $\alpha_{\pm}$ were absorbed as part of the prefactors in the
previous subsection – see discussion below eq. (71). Hence we would like to
determine the conserved momentum $P_{v}^{\varepsilon}(\tau)$ as a function of
the boundary time. From the definition of the infalling coordinate (15) and
the equations of motion (23), we can obtain the evolution of time coordinate
$t(\sigma)$ along the extremal surfaces as follows

$\dot{t}(\sigma)\equiv\dot{v}-\frac{\dot{r}}{f(r)}=-\frac{1}{f(r)}\left(P_{v}^{\varepsilon}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)\,.$ (92)

Now, if we fix the value of conserved momentum $P_{v}^{\varepsilon}$, we can
integrate the above expression using
$\dot{r}=\sqrt{-\mathcal{U}(P_{v}^{\varepsilon},r)}$ to determine the
corresponding boundary time $t_{\textrm{\tiny R}}$ as

$t_{\textrm{\tiny R}}-t(r_{\pm,\textrm{\tiny
min}})=-\int_{r_{\pm,\textrm{\tiny
min}}}^{\infty}\frac{dr}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\varepsilon},r)}}\,\left(P_{v}^{\varepsilon}+\frac{\varepsilon\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)\,,$ (93)

where $t(r_{\pm,\textrm{\tiny min}})$ denotes the value of the time at the
minimal radius. However, with our symmetric setup (i.e., $t_{\textrm{\tiny
R}}=t_{\textrm{\tiny L}}$), the latter is simply given by
$t(r_{\pm,\textrm{\tiny min}})=0$. An example of $\tau=2t_{\textrm{\tiny R}}$
as a function of $P^{\varepsilon}_{v}$ is explicitly plotted in figure 5.

There are three potentially singular points in the integral: $r=\infty$,
$r=r_{h}$, and $r=r_{\pm,\textrm{\tiny min}}$. We consider each of these in
turn. First, near the asymptotic boundary, the leading behaviour of the
integrand is given by

$\left(P_{v}^{\varepsilon}+\frac{\varepsilon\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)\frac{1}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\varepsilon},r)}}\sim\frac{1}{f(r)}\sim\frac{1}{r^{2}}\,,$
(94)

which yields a finite integral as $r\to\infty$. Second, one may worry about
the horizon $r=r_{h}$ where $f(r)$ vanishes and so the integrand diverges as

$\left(P_{v}^{\varepsilon}+\frac{\varepsilon\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)\frac{1}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\varepsilon},r)}}\sim\left(P_{v}^{\varepsilon}+\frac{\varepsilon\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r_{h}}{L}\right)^{d}\right)\frac{1}{f^{\prime}(r_{h})(r-r_{h})}\,.$
(95)

However, as noted below eq. 28, one can evaluate the integral using the Cauchy
principal value, i.e.,

$\lim_{\epsilon\to 0}\left(\int_{r_{\pm,\textrm{\tiny
min}}}^{r_{h}-\epsilon}+\int_{r_{h}+\epsilon}^{\infty}\right)\,\left(\frac{P_{v}^{\varepsilon}+\frac{\varepsilon\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\varepsilon},r)}}\right)\,dr\,,$
(96)

which is finite due to the cancellation of divergences from the two sides of
the horizon.

Finally, $\mathcal{U}(P_{v}^{\varepsilon},r)$ vanishes at
$r=r_{\pm,\textrm{\tiny min}}$. However, as noted previously, the resulting
divergence is integrable as long as
$\partial_{r}\mathcal{U}(P_{v}^{\varepsilon},r)|_{r=r_{\pm,\textrm{\tiny
min}}}\neq 0$, and the corresponding boundary time is finite. However, in the
special case where the momentum is tuned so that the effective potential has a
double zero at the turning point, the boundary time diverges. As discussed
above and shown figure 4, there are two critical values of the momentum,
$P^{\varepsilon}_{\infty}(\pm)$, for each sign of $\varepsilon$. As shown in
figure 5 then, one finds

$\begin{split}P_{v}^{\varepsilon}&\to
P_{\infty}^{\varepsilon}(+)\,,\qquad\tau\to+\infty\,,\\\
P_{v}^{\varepsilon}&\to
P_{\infty}^{\varepsilon}(-)\,,\qquad\tau\to-\infty\,.\end{split}$ (97)

Combining the late-time limit for the growth rate in eq. (91) with the
critical momenta in (88), we find

$\begin{split}\lim_{\tau\to\infty}\frac{d\mathcal{C}_{\rm{gen}}}{d\tau}&=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\,\left(\alpha_{+}\,P_{\infty}^{+}(+)+\alpha_{-}P_{\infty}^{-}(+)\right)\\\
&=\frac{8\pi M}{d(d-1)}\left(\sqrt{\alpha_{\textrm{\tiny
B}}^{2}+d^{2}\,\alpha_{+}^{2}}+\sqrt{\alpha_{\textrm{\tiny
B}}^{2}+d^{2}\,\alpha_{-}^{2}}\right)\,,\end{split}$ (98)

which is the expected linear growth with a rate proportional to the mass of
the black hole.202020We note that for $\tau\to-\infty$, the result is
identical except for an overall sign. It is interesting to consider the two
limits pointed out in eqs. (68) and (69). First, we take
$\alpha_{+}\to+\infty$ (while keeping $\alpha_{\textrm{\tiny B}}$ and
$\alpha_{-}$ fixed) to recover the standard CV result. Combining eqs. (68) and
(98) then yields

$\begin{split}\lim_{\tau\to\infty}\frac{d\mathcal{C}_{\rm
V}}{d\tau}=\lim_{\alpha_{+}\to+\infty}\lim_{\tau\to\infty}\left(\frac{1}{\alpha_{+}}\,\frac{d\mathcal{C}_{\rm{gen}}}{d\tau}\right)&=\frac{8\pi
M}{d-1}\,,\end{split}$ (99)

which is the expected growth rate for the CV conjecture, e.g.,
Stanford:2014jda ; Carmi:2017jqz . Alternatively, following eq. (69), we take
$\alpha_{\pm}\to 0$ (while keeping $\alpha_{\textrm{\tiny B}}$ fixed) to
recover the CV2.0 proposal. The corresponding late-time linear growth is then
given by

$\begin{split}\lim_{\tau\to\infty}\frac{d\mathcal{C}_{\rm
SV}}{d\tau}=\lim_{\alpha_{\pm}\to
0}\lim_{\tau\to\infty}\left(\frac{1}{\alpha_{\textrm{\tiny
B}}}\,\frac{d\mathcal{C}_{\rm{gen}}}{d\tau}\right)&=\frac{16\pi
M}{d(d-1)}\,,\end{split}$ (100)

which again reproduces the expected growth rate for the CV2.0 proposal, e.g.,
Couch:2016exn  (where the bulk pressure is identified as
$P=-\frac{\Lambda}{8\pi G_{\textrm{\tiny N}}}$).

#### Regime B: $P_{v}^{\epsilon}\leq P_{\infty}^{\epsilon}(-)$ or
$P_{v}^{\epsilon}\geq P_{\infty}^{\epsilon}(+)$

Similar to the codimension-one case discussed in Appendix A, we can also
obtain multiple extremal surfaces for $\Sigma_{\pm}$. Besides the extremal
surface joining the two sides at the minimal radius, there is also another
type of extremal surface which moves from $r=\infty$ into the singularity at
$r=0$ and where we allow it to reflect outward to $r=\infty$.212121This type
of extremal surfaces are similar to those discussed in Jorstad:2022mls  for de
Sitter spacetime. However, the volume of the portion along the singularity in
the AdS black hole is zero.. These solutions of the extremality equations
correspond to taking either $P_{v}^{\varepsilon}\geq
P_{\infty}^{\varepsilon}(+)$ or $P_{v}\leq P_{\infty}^{\varepsilon}(-)$. In
other words, we are outside the range in eq. (87), thus the effective
potential never reaches zero, and there is no turning point before the
singularity – see figure 3. The complete connected but non-smooth extremal
surfaces $\Sigma_{\pm}$ are obtained by joining two symmetric left and right
portions at $r=0,t=0$ on the singularity.222222Such solutions could be
considered with the usual CV proposal but are traditionally discarded.
However, within the framework of generalized codimension-one observables
constructed in Belin:2021bga , these non-smooth surfaces arise as a limit of a
class of smooth surfaces when the higher curvature couplings are taken to
zero. But we would like to emphasize that the extremal surfaces with the
maximum volume are still the traditional smooth surfaces, i.e., the non-smooth
surfaces are subdominant saddle points.

Let us consider the asymptotic values of the boundary time with the limit
$|P_{v}^{\varepsilon}|\to\infty$. Applying this limit to eq. (93), we find

$\begin{split}t_{\textrm{\tiny
R}}\left(P^{\varepsilon}_{v}\to\pm\infty\right)&=\mp\int^{\infty}_{0}\frac{dr}{f(r)}=\mp\frac{L^{2}}{r_{h}}\,\frac{\pi}{d}\,\cot\left(\frac{\pi}{d}\right)\,.\\\
\end{split}$ (101)

for $d>1$ 232323Similar to the codimension-one extremal surface we explored
here, the corresponding codimension-$d$ extremal surface, i.e., geodesics
bouncing oﬀ at the singularity, was studied before in Fidkowski:2003nf  and
was found to start from the same boundary time.. In line with the previous
discussion, we have set $t(r_{\pm,\textrm{\tiny min}}=0)=0$ in this
expression. As shown in figure 5, as the conserved momentum
$P_{v}^{\varepsilon}$ decreases from infinity to the critical value
$P_{\infty}^{\varepsilon}(+)$, the corresponding boundary time $\tau$
increases from the above asymptotic value to $+\infty$. Similarly, as
$P_{v}^{\varepsilon}$ increases from minus infinity to
$P_{\infty}^{\varepsilon}(-)$, the boundary time increases from the above
asymptotic value to $-\infty$.

####  2.2.3 Maximization

Although the surfaces above are not smooth, we may still consider them as
candidate saddle points which compete with the smooth extremal surfaces
considered previously for $P_{\infty}^{\epsilon}(-)\leq P_{v}^{\epsilon}\leq
P_{\infty}^{\epsilon}(+)$. Here, we examine the question of which of these
surfaces maximize the value of the observable $\mathcal{C}_{\rm gen}$ given in
eq. (66), and we find that the smooth extremal surfaces always give the
maximal value.

Let us rewrite the codimension-zero observable $\mathcal{C}_{\rm gen}$ as

$\begin{split}\mathcal{C}_{\rm gen}(\tau)&=\frac{2V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\int^{\infty}_{r_{\varepsilon,\rm
min}}\,\frac{\alpha_{\varepsilon}\,dr}{\sqrt{-\mathcal{U}(P_{v}^{\varepsilon},r)}}\left[\left(\frac{r}{L}\right)^{2(d-1)}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\frac{1}{f(r)}\left(P_{v}^{\varepsilon}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)\right]\\\
\end{split},$ (102)

by using the extremality equations (75) and (76) in eq. (29). Furthermore, it
will be useful to recast this expression as a time integral,

$\begin{split}\mathcal{C}_{\rm gen}(\tau)&=-\frac{2V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\int_{0}^{t_{\textrm{\tiny
R}}}\frac{\alpha_{\varepsilon}\,dt}{P_{v}^{\varepsilon}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}}\\\
&\qquad\qquad\qquad\times\
\left[f(r)\left(\frac{r}{L}\right)^{2(d-1)}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\left(P_{v}^{\varepsilon}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)\right]\,.\end{split}$
(103)

If we consider the smooth extremal surfaces with $P_{v}^{\varepsilon}\leq
P_{\infty}^{\varepsilon}(+)$, then in the late-time limit
$\tau=2t_{\textrm{\tiny R}}\to\infty$ (with $P_{v}^{\varepsilon}\to
P_{\infty}^{\varepsilon}(+)$), both extremal surfaces $\Sigma_{\pm}$ approach
a constant radius surface at $r=r_{\pm,f}$. Then after substituting eq. (83)
into eq. (103), $\mathcal{C}_{\rm gen}$ reduces to

$\mathcal{C}_{\rm gen}(\tau\to\infty)=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=+,-}\alpha_{\varepsilon}\,P_{\infty}^{\varepsilon}(+)\,\int_{0}^{\tau\to\infty}d\tau\,.$
(104)

On the other hand, we have found another class of non-smooth extremal surfaces
with $P_{v}^{\varepsilon}\geq P_{\infty}^{\varepsilon}(+)$. In this case, late
boundary time $\tau\to\infty$ is achieved by approaching
$P_{\infty}^{\varepsilon}$ from the above, but the corresponding extremal
surface is not a constant radius surface. This class of extremal surfaces
starts from the asymptotic boundary and extends to the singularity at $r=0$.
Using the inequality $P_{v}^{\varepsilon}\geq P_{\infty}^{\varepsilon}(+)$,
the following always holds along these surfaces

$\mathcal{U}(P_{v}^{\varepsilon},r)=-f(r)\left(\frac{r}{L}\right)^{d}-\left(P_{\infty}^{\varepsilon}+\frac{\varepsilon\alpha_{\textrm{\tiny
B}}}{d\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)^{2}\leq 0\,.$
(105)

As noted above, in the late time limit, we approach the critical momentum,
which we denote as $P_{\infty}^{\varepsilon}(+)\leftarrow
P_{v}^{\varepsilon}$,242424At the infinite time, we have four possible
configurations due to two extremal surfaces., we can decompose the integral
(103) as

$\begin{split}\lim_{P_{\infty}^{\varepsilon}(+)\leftarrow
P_{v}^{\varepsilon}}\mathcal{C}_{\rm gen}&=\frac{V_{x}}{G_{\textrm{\tiny
N}}L}\sum_{\varepsilon=\pm}\int_{0}^{\tau}\frac{\alpha_{\varepsilon}\,d\tau}{P_{\infty}^{\varepsilon}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}}\left[P_{\infty}^{\varepsilon}\left(P_{\infty}^{\varepsilon}+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}\right)+\mathcal{U}(P_{\infty}^{\epsilon},r)\right]\,.\end{split}$
(106)

where the first term is nothing but the result in eq. (104). Firstly, we note
that all spacelike hypersurface anchoring on the infinite boundary time stay
inside the horizon. Noting the various solutions of $P_{\infty}^{\varepsilon}$
given in eq. (88), we will always have

$P_{\infty}^{\varepsilon}(+)+\frac{\varepsilon\,\alpha_{\textrm{\tiny
B}}}{d\,\alpha_{\varepsilon}}\left(\frac{r}{L}\right)^{d}>0$ (107)

for $r\leq r_{h}$. Secondly, we obviously have
$\mathcal{U}(P_{\infty}^{\epsilon},r)\leq 0$ for this type of extremal
surface. Assuming $\alpha_{\varepsilon}\geq 0$ for both surfaces
$\Sigma_{\pm}$, we can conclude that

$\lim_{P_{v}^{\varepsilon}\to P_{\infty}^{\varepsilon}(+)}\mathcal{C}_{\rm
gen}(\tau\to\infty)\geq\lim_{P_{\infty}^{\varepsilon}(+)\leftarrow
P_{v}^{\varepsilon}}\mathcal{C}_{\rm gen}(\tau\to\infty)\,.$ (108)

In other words, the late-time extremal surface yielding the maximal value of
the observable is the smooth extremal surface at $r=r_{f}$, i.e., the surface
approached by taking the limit $P_{v}^{\varepsilon}\to
P_{\infty}^{\varepsilon}(+)$ from below. Using the same argument as for the
maximal volume associated with the effective potential (see Appendix A.2.3 for
more details), one can further show that $\mathcal{C}_{\rm gen}$ is maximized
at any boundary time $\tau$ by the surfaces with the smaller momentum, viz,

$\mathcal{C}_{\rm gen}(\tau)\big{|}_{P_{v}^{\varepsilon}\leq
P_{\infty}^{\varepsilon}(+)}>\mathcal{C}_{\rm
gen}(\tau)\big{|}_{P_{v}^{\varepsilon}\geq P_{\infty}^{\varepsilon}(+)}\,.$
(109)

A similar proof also applies for negative boundary times with

$\mathcal{C}_{\rm gen}(\tau)\big{|}_{P_{v}^{\varepsilon}\geq
P_{\infty}^{\varepsilon}(-)}>\mathcal{C}_{\rm
gen}(\tau)\big{|}_{P_{v}^{\varepsilon}\leq P_{\infty}^{\varepsilon}(-)}\,.$
(110)

##  3 Gravitational Observables and the Symplectic Form

In this section we discuss gauge invariant functionals on the phase space of
asymptotically AdS gravity theories. An important concept will be the
Hamiltonian vector field conjugate to such observables. In classical
mechanics, a Hamiltonian vector field $X_{f}$ conjugate to a function $f$ on
phase space is obtained by pulling up the indices of $\mathrm{d}f$ using the
inverse of the symplectic form $\Omega$:

$(X_{f})^{a}=\Omega^{ab}\partial_{b}f.$ (111)

Such vector fields on phase space are to be thought of as linearized
variations of initial data. This is more natural in field theory, so we will
refer to the Hamiltonian vector field as the conjugate variation from now on.

In gravitational field theories, the variation conjugate to an observable can
be found using a construction due to Peierls Peierls:1952cb , which will be
briefly reviewed below in section 3.3 (see Kirklin:2019xug ; Harlow:2019yfa ;
Harlow:2021dfp ; Goeller:2022rsx  for recent uses and a more in-depth review).
The Peierls construction is very general, but a disadvantage is that finding
the conjugate variation requires solving equations of motion for an auxiliary
system. It will therefore be advantageous to directly adapt the naive formula
(111) in gravity, which does not require solving differential equations. When
using the canonical formalism, in section 3.1, we first analyze the conjugate
variation for functionals defined on spacelike codimension-one slices on which
the functional is extremal252525We will only need that the equations of motion
are second order, so the discussion in this section applies also to Lovelock
gravity.. The canonical construction is equivalent with the one of Peierls,
and indeed we will show in section 3.3.1 that observables are required to be
extremized in the Peierls construction.

There are certain gauge invariant observables that do not seem to be in this
class of extremized functionals. As explained in section 2.1.4, we could fix a
gauge invariant slice by extremizing a functional, and then integrate a
different functional on this slice. An example in this class would be to
integrate any covariant density on the maximal volume slice. We will close
this section by explaining how these observables are in fact related to
extremized functionals, and show how to find their conjugate variations.

###  3.1 Canonical formalism for codimension one

In the following, we will focus on pure GR for simplicity. However, it is
straightforward to extend the discussion by including matter. According to the
canonical decomposition, the initial data are the induced metric $h_{ij}$ and
its conjugate momenta $\pi^{ij}$ on a Cauchy slice $\Sigma$. The momenta are
related to the extrinsic curvature of the Cauchy slice $\Sigma$ via

$\pi^{ij}\equiv\sqrt{h}(K^{ij}-h^{ij}K)\,.$ (112)

The symplectic form between two variations $\delta_{1},\delta_{2}$ is defined
by Lee:1990nz

$\Omega(\delta_{1},\delta_{2})=\int_{\Sigma}\left[\delta_{1}\pi^{ij}\delta_{2}h_{ij}-\delta_{1}h_{ij}\delta_{2}\pi^{ij}\right]\,.$
(113)

This quantity is equivalent with the covariant phase space symplectic form up
to boundary terms localized on $\partial\Sigma$. In particular, this
expression is conserved on-shell, in the sense that it is independent of the
slice $\Sigma$ provided that the anchoring $\partial\Sigma$ is fixed.

Let us first consider a functional $W(h_{ij},\pi^{ij})$ associated with a
density functional $w(h_{ij},\pi^{ij})$ which depends on the initial data on
the Cauchy slice $\Sigma$, viz,

$W(h_{ij},\pi^{ij}):=\int_{\Sigma}w(h_{ij},\pi^{ij})\,.$ (114)

We can construct the conjugate variation $\delta_{w}$ by naively applying the
formula (111)

$\delta_{w}\pi^{ij}=-\frac{\delta w}{\delta
h_{ij}},\quad\delta_{w}h_{ij}=+\frac{\delta w}{\delta\pi^{ij}}\,.$ (115)

Using (113), it is obvious that

$\Omega(\delta,\delta_{w})=\delta W\equiv\int_{\Sigma}\delta w\,.$ (116)

However, for $\delta_{w}$ to be a legitimate initial data variation, it must
preserve the constraints of GR. Denoting by $\mathcal{H}_{\perp}$ the
Hamiltonian constraint, and by $\mathcal{H}^{k}$ the momentum constraint, we
must require

$\begin{split}0=\delta_{w}\mathcal{H}_{\perp}&=\frac{\delta\mathcal{H}_{\perp}}{\delta
h_{ij}}\delta_{w}h_{ij}+\frac{\delta\mathcal{H}_{\perp}}{\delta\pi^{ij}}\delta_{w}\pi^{ij}=\frac{\delta\mathcal{H}_{\perp}}{\delta
h_{ij}}\frac{\delta
w}{\delta\pi^{ij}}-\frac{\delta\mathcal{H}_{\perp}}{\delta\pi^{ij}}\frac{\delta
w}{\delta h_{ij}}\equiv\\{\mathcal{H}_{\perp},w\\}_{\rm P.B.},\\\
0=\delta_{w}\mathcal{H}^{k}&=\frac{\delta\mathcal{H}^{k}}{\delta
h_{ij}}\delta_{w}h_{ij}+\frac{\delta\mathcal{H}^{k}}{\delta\pi^{ij}}\delta_{w}\pi^{ij}=\frac{\delta\mathcal{H}^{k}}{\delta
h_{ij}}\frac{\delta
w}{\delta\pi^{ij}}-\frac{\delta\mathcal{H}^{k}}{\delta\pi^{ij}}\frac{\delta
w}{\delta h_{ij}}\equiv\\{\mathcal{H}^{k},w\\}_{\rm P.B.}\,,\end{split}$ (117)

where ${\rm P.B.}$ denotes the Poisson bracket.262626Given two functions $f$
and $g$ defined on the phase space constructed by the canonical coordinates
$(q^{a},p_{a})$, the Poisson bracket can be defined by the following
equivalent expressions:  $\\{f,g\\}_{\rm P.B.}\equiv\frac{\partial f}{\partial
q^{a}}\frac{\partial g}{\partial p_{a}}-\frac{\partial f}{\partial
p_{a}}\frac{\partial g}{\partial
q^{a}}\equiv\Omega(X_{g},X_{f})=X_{g}f=-X_{f}g\,.$ (118) with the symplectic
form taken as $\Omega\equiv dp_{a}\wedge dq^{a}$. This is equivalent to
defining the Poisson bracket as $\\{f,g\\}_{\rm
P.B.}\equiv\Omega^{-1}(\mathrm{d}f,\mathrm{d}g)=X_{g}f$ by using the inverse
of the symplectic form $\Omega^{-1}$ ($\mathrm{d}$ is the exterior derivative
acting on the phase space). This latter definition is the one used for the
Poisson bracket employed throughout this section. The second equation just
entails diffeomorphism invariance in the surface, and it is automatic if we
have constructed $W$ by integrating the volume density $\sqrt{h}$ times a
functional that transforms as a scalar under diffeomorphism fixing the slice
$\Sigma$. The first equation is more interesting. Taking the Hamiltonian
density $\mathcal{H}_{\rm ADM}=N\mathcal{H}_{\perp}+N^{a}\mathcal{H}_{a}$,
Hamilton’s equations read

$\frac{d\pi^{ij}}{dt}=-\frac{\delta\mathcal{H}_{\rm ADM}}{\delta
h_{ij}}\,,\quad\quad\frac{dh_{ij}}{dt}=\frac{\delta\mathcal{H}_{\rm{ADM}}}{\delta\pi^{ij}}\,,$
(119)

where $t$ is the time in an ADM $d+1$ decomposition.272727We will use this
notation throughout the section, but we should emphasize that $t$ is the ADM
bulk time, and not the CFT time. We have that $\delta_{w}\mathcal{H}_{\rm
ADM}=N\delta_{w}\mathcal{H}_{\perp}$, because
$\mathcal{H}_{\perp}=\mathcal{H}_{a}=0$ in the background and we have already
argued for $\delta_{w}\mathcal{H}_{a}=0$ from the diffeomorphism invariance of
the density $w$. Using this, it follows that the condition that the
Hamiltonian constraint is satisfied by this initial data deformation
translates to

$\delta_{w}\mathcal{H}_{\perp}=\frac{1}{N}\\{\mathcal{H}_{\rm ADM},w\\}_{\rm
P.B.}=\frac{1}{N}\frac{dW}{dt}=0\,.$ (120)

Since this must hold for an arbitrary choice of lapse, it then implies that
$W$ must be extremal under any shape variation of the Cauchy slice
$\Sigma$.282828The Hamiltonian constraint will generate time evolution that is
trivial at the boundary.

We learned therefore that for extremal $W$, (115) corresponds to an allowed
deformation of initial data. Equivalently, the initial data $(h_{ij}$,
$\pi^{ij})$ defines a complete time dependent solution $g_{ab}$ of Einstein’s
equations, while (115) defines a particular linearized solution
$\delta_{w}g_{ab}$ around this. These two ways of thinking about $\delta_{w}$
are equivalent in globally hyperbolic spacetimes. In the next subsection, we
review the Peierls construction, which obtains the spacetime variation
$\delta_{w}g_{ab}$ directly by solving an auxiliary problem. Before doing
this, we make some further comments about the conjugate variations given in
eq. (115).

Let us discuss the simplest example, where $W$ is the volume of the Cauchy
slice, i.e., $W=\int_{\Sigma}\sqrt{h}$. Gauge invariance requires us to
extremize $W$ with respect to $\Sigma$. On such a $\Sigma$, the conjugate
variation (115) reads as

$\delta_{w}\pi^{ij}=-\frac{1}{2}\sqrt{h}h^{ij}\,,\qquad\delta_{w}h_{ij}=0\,.$
(121)

This initial data variation was studied in detail in Belin:2018bpg  and was
called new York transformation, because of its relation to evolution in York
time.

It is useful to have the form of the general variation (115) in terms of the
extrinsic curvature $K_{ij}$ of the slice $\Sigma$ in GR. The final result is
given by

$\begin{split}\delta_{w}h_{ij}&=A_{ij}-\frac{{A^{k}}_{k}}{d-1}h_{ij}\,,\\\
\delta_{w}K_{ij}&=B_{ij}-\frac{{A^{k}}_{k}}{2(d-1)}K_{ij}-\frac{h_{ij}}{2(d-1)}\left(2{B^{k}}_{k}-K_{kl}A^{kl}\right)\,,\end{split}$
(122)

where we have defined

$A^{ij}=\frac{1}{\sqrt{h}}\frac{\delta w}{\delta K_{ij}}\,,\qquad
B_{ij}=\frac{1}{\sqrt{h}}\frac{\delta w}{\delta h^{ij}}\,.$ (123)

The derivations for those conjugate variations
$\delta_{w}h_{ij},\delta_{w}K_{ij}$ are presented in Appendix E with more
detail.

###  3.2 Holographic dual of the conjugate variation

In the AdS/CFT context, the bulk symplectic form
$\Omega(\delta_{1},\delta_{2})$ has a remarkably simple expression in the
boundary CFT Belin:2018fxe :

$\Omega(\delta_{1},\delta_{2})=i\int_{S^{+}}dx^{+}\int_{S^{-}}dx^{-}\langle\mathcal{O}^{i}(x^{+})^{\dagger}\mathcal{O}^{j}(x^{-})\rangle_{\lambda,\rm{con}}[\delta_{1}\lambda_{i}(x^{+})\delta_{2}\lambda_{j}(x^{-})-\delta_{2}\lambda_{i}(x^{+})\delta_{1}\lambda_{j}(x^{-})].$
(124)

In this expression, $S=S^{+}\cup S^{-}$ is the Euclidean space that the CFT is
defined on,
$\langle\mathcal{O}^{i}(x^{+})^{\dagger}\mathcal{O}^{j}(x^{-})\rangle_{\lambda,\rm{con}}$
denotes the connected correlator of two single trace operators
$\mathcal{O}^{i}$ on $S$ with (potentially complex) sources $\lambda$ turned
on, and $\delta_{1,2}\lambda$ are deformations of these sources. The sources
are such that if we use them to prepare a state $|\lambda\rangle$ by doing a
path integral over $S^{-}$, then the path integral over $S^{+}$ prepares the
conjugate state $\langle\lambda|$, i.e.,

$\begin{split}|\lambda\rangle=T\exp\left({-\int_{S^{-}}dx^{-}\lambda(x^{-})\mathcal{O}(x^{-})}\right)|0\rangle\,,\quad\langle\lambda|=\langle
0|T\exp\left({-\int_{S^{+}}dx^{+}\lambda^{\ast}(x^{+})\mathcal{O}(x^{+})}\right)\,.\end{split}$
(125)

In other words, the manifold $S$ with the sources $\lambda$ must have a time
reflection plus complex conjugation symmetry.

After the normalization of the CFT states $|\lambda\rangle$, equation (124) is
nothing but the Berry curvature two-form. This Berry curvature agrees with the
bulk symplectic form in the bulk geometry dual to the state $|\lambda\rangle$
Belin:2018fxe ; Belin:2018bpg ; Kirklin:2019ror . The corresponding dual
spacetime can be obtained by finding the (generally complex) Euclidean bulk
geometry where the bulk fields have boundary conditions set by the $\lambda$
sources, and then Wick rotating over an appropriate bulk slice to a real
Lorentzian spacetime. The variations $\delta_{1,2}\phi$ entering the bulk
symplectic form are linearized solutions around this background whose
Euclidean boundary conditions are set by $\delta_{1,2}\lambda$.

Supposing we have an extremal functional $W$ in the bulk and have found the
conjugate variation $\delta_{w}\phi$ around a solution that is dual to a state
$|\lambda\rangle$, one can find a source deformation $\delta_{w}\lambda$, that
gives rise to the linearized solution $\delta_{w}\phi$.292929The inverse
problem to this is not well posed, see Belin:2020zjb . This source deformation
$\delta_{w}\lambda$ is thus an intrinsically boundary object and satisfies
$\delta W=\Omega(\delta,\delta_{w})$, where $\Omega$ is the boundary Berry
curvature two form defined in eq. (124), and $\delta$ is an arbitrary
variation of the state.

Note that $\delta_{w}\lambda$ may depend on the state $|\lambda\rangle$ in a
complicated way. In certain cases, it could have a natural meaning and display
some form of universality. For example, when $W$ is defined as the area of a
space-like codimension-two surface, $\delta_{w}\lambda$ inserts a small
conical deficit angle, corresponding to changing the Rényi index Dong:2017xht
. When $W$ is the volume of a codimension-one slice, $\delta_{w}\lambda$ was
explored in Belin:2018bpg , but it is still unknown how universal the
variation should be in general states. We will comment on this further in the
section 4.

Instead of studying the meaning of $\delta_{w}\lambda$ for a given functional
$W$, our main point here is that the boundary object $\delta_{w}\lambda$
exists for each diff-invariant functional $W$.303030A subtlety here is that as
we have seen before, the slice or region extremizing $W$ may not exist, or may
not be unique. When the region does not exist, there is also no
$\delta_{w}\lambda$ that could be defined. When it is not unique, there is a
distinct $\delta_{w}\lambda$ corresponding to all the different extrema. The
recipe to construct it is to find the conjugate variation in the bulk either
using eq. (115) from the canonical formalism or the Peierls construction, then
Wick rotate it to Euclidean, and finally follow it to the Euclidean boundary.
In section 3.3, we will provide another example of such a calculation for the
case where the functional is the spacetime volume discussed in section 2.2.

###  3.3 Peierls construction

To construct the variation $\delta_{w}$ conjugate to an arbitrary
diffeomorphism invariant functional $W$ (codimension-zero or codimension-one),
we first briefly review the so-called Peierls construction. It is a covariant
way of “pulling the indices up” of $\delta W$ with the symplectic form. The
Peierls construction is equivalent to the simple formula (115) for
codimension-one functionals, but it applies to a more general class of
functionals. For example, we can apply it to the functionals defined as
integrals between two non-intersecting Cauchy slices $\Sigma_{-}$ and
$\Sigma_{+}$, as shown in Figure 6.

Let us consider a diffeomorphism invariant gravitational theory described by a
given action $S(\phi)$ where we have denotes all dynamical fields as $\phi$.
Then, Peierls’ prescription for generating $\delta_{w}\phi$ is given by the
following Peierls:1952cb :

  1. 1.

Deform the gravitational action by adding the functional $W$ with a small
coefficient $\varrho$, viz, $S\mapsto S-\varrho W$

  2. 2.

Pick a background solution $\phi_{0}$ which solves the equations of motion
from the undeformed action $S$. Then, one can construct linearized solutions
$\phi_{\textrm{\tiny R}/\textrm{\tiny
A}}=\phi_{0}+\varrho\delta_{\textrm{\tiny R}/\textrm{\tiny A}}\phi$ associated
with the deformed action. The subscript $R/A$ is referred to as the
retarded/advanced solution. Obviously, we have that $\delta_{\textrm{\tiny
R}}\phi=0$ in the past of $\Sigma_{-}$ while $\delta_{\textrm{\tiny A}}\phi=0$
in the future of $\Sigma_{+}$. Physically speaking, we obtain
$\phi_{\textrm{\tiny R}}$ by starting from the unperturbed solution $\phi_{0}$
and following how it changes under time evolution due to the perturbation
$\varrho W$. $\phi_{\textrm{\tiny A}}$ is the time reversed picture of this,
where we look for the precursor solution that “lands” on $\phi_{0}$ after the
perturbation has been switched off. See Figure 6 for an illustration.

  3. 3.

The conjugate variation dual to the functional $W$ is then given by

$\delta_{w}\phi=\delta_{\textrm{\tiny R}}\phi-\delta_{\textrm{\tiny
A}}\phi\,.$ (126)

This is a solution of the EOM without the deformation, since acting on either
$\delta_{\textrm{\tiny R/A}}\phi$ with the undeformed linearized equation of
motion gives the same inhomogeneous source term $\varrho\frac{\delta
W}{\delta\phi}|_{\phi_{0}}$, so it cancels in the difference. Following this
Peierls construction, the corresponding conjugate variation $\delta_{w}\phi$
satisfies our target, i.e., $\delta W=\Omega(\delta,\delta_{w})$.

Figure 6: Illustration of the retarded and advanced solutions in the Peierls
construction. The action is deformed by the functional $W$ in the bulk region
$\mathcal{M}$, which lies between the two surfaces denoted $\Sigma_{\pm}$
(i.e., the two orange curves). The blue shading shows the support of the
retarded solution $\delta_{\textrm{\tiny R}}\phi$ in the left panel, and that
of the advanced solution $\delta_{\textrm{\tiny A}}\phi$ in the right panel.

A modern proof of these statements can be found in Harlow:2019yfa , which we
will not repeat here. In the rest of this paper, we will restrict to the case
where $\Sigma_{\pm}$ hit the boundary on the same boundary Cauchy slice, that
is, $\partial\Sigma_{+}=\partial\Sigma_{-}$.313131An even more general class
of observables would be to have $\partial\Sigma_{+}$ be in the future of
$\partial\Sigma_{-}$, hence having a time interval in the boundary, where
possible boundary deformations could be turned on as well. We will briefly
comment on the possible uses of such observables in the discussion section.

####  3.3.1 Extremality in the Peierls construction

In the above construction, it seems like we have obtained the conjugate
variation to an arbitrary functional $W$, while in the canonical discussion of
section 3.1, we have seen that for $\delta_{w}$ to exist, the functional must
be extremized. In this section, we discuss why this extremization is
automatically enforced in the covariant Peierls picture. In particular, we
emphasize that even if $W$ is defined in a codimension-zero region between two
non-intersecting Cauchy slices $\Sigma_{\pm}$, one has to extremize $W$ with
respect to the shapes of $\Sigma_{\pm}$.

The basic point is that the equations of motion for the deformed gravitational
action, i.e.,

$S_{\varrho}=S-\varrho W\,,$ (127)

enforce this extremization. This may seem at first a bit surprising since we
didn’t include the embedding coordinates of the Cauchy slices as dynamical
variables. However, one can consider the deformed action (127) as having two
branes at $\Sigma_{\pm}$ acting as interfaces between regions with $W$ being
turned on or off, and it is a well-known fact that the junction conditions and
Einstein equation fully determine the locations of a brane Freivogel:2005qh .

More formally, the requirement that the deformed action $S_{\varrho}$ is
diffeomorphism invariant means that

$\delta_{\xi}S_{\varrho}=\int\left[\frac{\delta
S_{\varrho}}{\delta\phi}\mathcal{L}_{\xi}\phi+\frac{\delta S_{\varrho}}{\delta
X^{I}}\mathcal{L}_{\xi}X^{I}\right]=0\,,$ (128)

where $\phi$ denotes all dynamical fields including the metric, $X^{I}$ are
embedding fields specifying the location of the surfaces $\Sigma_{\pm}$, and
$\xi$ is an arbitrary diffeomorphism. Evaluate this variation around a
solution of the equations of motion with $\frac{\delta
S_{\varrho}}{\delta\phi}=0$, one can obtain

$\frac{\delta S_{\varrho}}{\delta X^{I}}\partial_{\mu}X^{I}=0\,.$ (129)

When the gradients $dX^{I}$ are linearly independent, this implies that the
embedding fields satisfy their equations of motion $\frac{\delta
S_{\varrho}}{\delta X^{I}}=0$, which reduces to the extremization condition
for the surfaces $\Sigma_{\pm}$ in the limit $\varrho\to 0$. We can specify
the location of two codimension-one surfaces with two scalar fields, so the
$dX^{I}$, $I=\pm$ will in general be linearly independent for spacetimes that
are at least two dimensional. This argument was made in Jacobson:2015mra ,
where it was used for a rather different purpose.v The argument only relies on
the diffeomorphism invariance of the original action, and hence should apply
for any higher curvature theory (Lovelock or otherwise) with any matter
content.

####  3.3.2 Examples of conjugate variations

It is educational to see how the machinery of the Peierls construction
reviewed in section 3.3 reproduces the conjugate variation for a particular
functional $W$. In this subsection, we consider two explicit examples.

#### A. Re-deriving the new York transformation for the volume

First of all, let us take the codimension-one functional
$W=\int_{\Sigma}\sqrt{h}$ which is used in the CV conjecture (1). Adding this
to the usual Einstein-Hilbert action, one get the deformed action $S-\varrho
W$. This new term $W$ acts as a stress tensor in the equations of motion,
namely

$T^{\mu\nu}=+\varrho\frac{2}{\sqrt{g}}\frac{\delta w}{\delta g_{\mu\nu}}\,.$
(130)

Using the ADM Arnowitt:1959ah  variables $(h_{ij}$, $K_{ij})$, and working in
Gaussian normal coordinates around $\Sigma$ with $N=1$, $N^{i}=0$, one gets

$T^{00}=T^{0i}=0\,,\qquad T^{ij}=\varrho h^{ij}\delta(t)\,.$ (131)

It is clear that the variation $\varrho W$ results in a delta function on the
slice $\Sigma$ (corresponding to ADM time $t=0$) since $W$ is an integral over
this time slice only. Let us think about the equations of motion in ADM
formulation. The fact that $T^{00}=T^{0i}=0$ means that the Hamiltonian and
momentum constraints are unchanged. There is also no change in $\dot{h}_{ij}$
since the stress tensor does not enter here. The only change happens in the
equation for $\dot{K}_{ij}$:

$\dot{K}_{ij}=(\text{r.h.s. in Einstein gravity})+\frac{1}{2}\varrho
h_{ij}\delta(t)\,.$ (132)

We may now write the retarded solution as

$K^{\textrm{\tiny R}}_{ij}=K^{0}_{ij}+\varrho\Theta(t)\delta_{\textrm{\tiny
R}}K_{ij},\quad h^{\textrm{\tiny
R}}_{ij}=h^{0}_{ij}+\varrho\Theta(t)\delta_{\textrm{\tiny R}}h_{ij}\,,$ (133)

where $h^{0}$, $K^{0}$ denote the background solution. Putting this in the ADM
equations for $\dot{h}_{ij}$ and $\dot{K}_{ij}$ gives

$\displaystyle\varrho\delta(t)\delta_{\textrm{\tiny
R}}h_{ij}+\varrho\Theta(t)\frac{d}{dt}\delta_{\textrm{\tiny R}}h_{ij}$
$\displaystyle=-2\varrho\Theta(t)\delta_{\textrm{\tiny R}}K_{ij},$ (134)
$\displaystyle\varrho\delta(t)\delta_{\textrm{\tiny
R}}K_{ij}+\varrho\Theta(t)\frac{d}{dt}\delta_{\textrm{\tiny R}}K_{ij}$
$\displaystyle=\varrho\Theta(t)(\text{linearized r.h.s. for
}\delta_{\textrm{\tiny R}}K_{ij})+\frac{1}{2}\varrho\delta(t)h_{ij}.$

Matching the coefficients of $\Theta(t)$ terms tells us that
$\delta_{\textrm{\tiny R}}h_{ij}$ and $\delta_{\textrm{\tiny R}}K_{ij}$ solve
the undeformed linearized EOM around $(h^{0}_{ij},K^{0}_{ij})$, for $t>0$.
Matching the coefficients of $\delta(t)$ gives:

$\delta_{\textrm{\tiny R}}h_{ij}(t=0)=0\,,\quad\delta_{\textrm{\tiny
R}}K_{ij}(t=0)=\frac{1}{2}h_{ij}(t=0).$ (135)

We can construct the advanced solution in the same way by taking

$K^{\textrm{\tiny A}}_{ij}=K^{0}_{ij}+\varrho\Theta(-t)\delta_{\textrm{\tiny
A}}K_{ij},\quad h^{\textrm{\tiny
A}}_{ij}=h^{0}_{ij}+\varrho\Theta(-t)\delta_{\textrm{\tiny A}}h_{ij}.$ (136)

The difference then is that on the l.h.s of the EOMs the delta functions
appear with an extra sign due to $\Theta(-t)$. This results in initial data

$\delta_{\textrm{\tiny A}}h_{ij}(t=0)=0,\quad\delta_{\textrm{\tiny
A}}K_{ij}(t=0)=-\frac{1}{2}h_{ij}(t=0).$ (137)

According to Peierls’ recipe, the new York transformation is

$\displaystyle\delta_{w}h_{ij}$ $\displaystyle=\Theta(t)\delta_{\textrm{\tiny
R}}h_{ij}-\Theta(-t)\delta_{\textrm{\tiny A}}h_{ij}\,,$ (138)
$\displaystyle\delta_{w}K_{ij}$ $\displaystyle=\Theta(t)\delta_{\textrm{\tiny
R}}K_{ij}-\Theta(-t)\delta_{\textrm{\tiny A}}K_{ij}\,.$

It is clear that the delta functions now cancel from
$\frac{d}{dt}\delta_{w}h_{ij}$ and $\frac{d}{dt}\delta_{w}K_{ij}$, so these
are linearized solutions to the undeformed EOMs. They correspond to initial
data323232There is a $\Theta(0)$ formally in these expressions, but the
correct prescription is to thicken the functional to be between $\Sigma_{\pm}$
with $\Sigma_{-}$ at $t=-\epsilon$ and $\Sigma_{+}$ at $t=\epsilon$. For this
thickened functional, the retarded and advanced solutions now have support
that overlaps in the region $-\epsilon<t<\epsilon$. Therefore, this sends
$\Theta(t)\mapsto\Theta(t+\epsilon)$ and
$\Theta(-t)\mapsto\Theta(-t+\epsilon)$ in the formulas, so we must formally
put $\Theta(0)=1$.

$\delta_{w}h_{ij}(t=0)=0,\quad\delta_{w}K_{ij}(t=0)=h_{ij}(t=0)\,,$ (139)

so we indeed recover the canonical results (121) from the Peierls
construction.

#### B. Spacetime volume in vacuum AdS

As a second example, we define the functional $W$ as that given in eq. (66).
Following the Peierls recipe, we consider the bulk part of the deformed action

$\frac{1}{16\pi G_{\textrm{\tiny
N}}}\int\sqrt{-g}(\mathcal{R}+\frac{d(d-1)}{L^{2}})-\varrho W\,.$ (140)

Between $\Sigma_{-}$ and $\Sigma_{+}$ the solution must look like

$ds^{2}=-d\tau^{2}+L_{\rm def}^{2}\cos^{2}\frac{\tau}{L_{\rm
def}}d\Sigma^{2}\,,$ (141)

with

$L_{\rm def}=L\sqrt{\frac{1}{1-\frac{16\pi\varrho\alpha_{\textrm{\tiny
B}}}{d(d-1)}}}=L\left(1+\frac{8\pi\varrho\alpha_{\textrm{\tiny
B}}}{d(d-1)}\right)+O(\varrho^{2})\,.$ (142)

We consider the retarded ansatz

$\displaystyle ds^{2}_{\textrm{\tiny R}}$
$\displaystyle=-d\tau^{2}+[\Theta(-\tau_{-}-\tau)L^{2}\cos^{2}\frac{\tau}{L}+\Theta(\tau+\tau_{-})\Theta(\tau_{+}-\tau)L_{\rm
def}^{2}\cos^{2}\frac{\tau-\xi_{1}^{\textrm{\tiny R}}}{L_{\rm def}}$ (143)
$\displaystyle+\Theta(\tau-\tau_{+})L^{2}\cos^{2}\frac{\tau-\xi_{2}^{\textrm{\tiny
R}}}{L}]d\Sigma^{2}.$

Here, we have chosen the location of the surfaces $\Sigma_{\pm}$ to be at
$\tau=\pm\tau_{\pm}$, while $\xi^{\textrm{\tiny R}}_{1,2}$ are time shifts
that are further parameters of the ansatz. We determine these four parameters
by solving the equations of motions derived from the deformed action (140).
The ansatz above is a piecewise solution to eqs. (140) and (66). Taking
account of the contributions coming from the codimension-one boundary terms in
eq. (66), we merely need to enforce the Israel junction conditions, namely

$\begin{split}\Delta h_{ij}&=0\,,\\\ \Delta K_{ij}-h_{ij}\Delta
K&=-\frac{8\pi\varrho\,\alpha_{+}}{L}h_{ij}\,,\end{split}$ (144)

where $\Delta h_{ij}$ and $\Delta K_{ij}$ are the jump in the induced metric
and extrinsic curvature across the surface $\Sigma_{+}$. Matching the induced
metric across the interfaces results in

$\begin{split}L^{2}\cos^{2}\frac{\tau_{-}}{L}&=L_{\rm
def}^{2}\cos^{2}\frac{\tau_{-}+\xi_{1}^{\textrm{\tiny R}}}{L_{\rm def}}\,,\\\
L_{\rm def}^{2}\cos^{2}\frac{\tau_{+}-\xi_{1}^{\textrm{\tiny R}}}{L_{\rm
def}}&=L^{2}\cos^{2}\frac{\tau_{+}-\xi_{2}^{\textrm{\tiny R}}}{L}\,,\\\
\end{split}$ (145)

while the junction conditions for the extrinsic curvatures impose

$\begin{split}\frac{1}{L}\tan\frac{\tau_{-}}{L}-\frac{1}{L_{\rm
def}}\tan\frac{(\tau_{-}+\xi_{1}^{\textrm{\tiny R}})}{L_{\rm
def}}&=-\frac{8\pi\varrho\alpha_{-}}{(d-1)L}\,,\\\
\frac{1}{L}\tan\frac{(\tau_{+}-\xi_{2}^{\textrm{\tiny R}})}{L}-\frac{1}{L_{\rm
def}}\tan\frac{(\tau_{+}-\xi_{1}^{\textrm{\tiny R}})}{L_{\rm
def}}&=-\frac{8\pi\varrho\alpha_{+}}{(d-1)L}\,.\\\ \end{split}$ (146)

Performing the expansion $L_{\rm def}=L+\varrho\delta L$ with $\delta
L=\frac{8\pi\alpha_{\textrm{\tiny B}}}{d(d-1)}L$, these equations then reduce
to

$\begin{split}\xi_{1}^{\textrm{\tiny R}}&=\varrho\,\delta
L\left(\frac{\tau_{-}}{L}+\cot\frac{\tau_{-}}{L}\right)\,,\\\
\xi_{2}^{\textrm{\tiny R}}&=\varrho\,\delta
L\left(\frac{\tau_{-}+\tau_{+}}{L}+\cot\frac{\tau_{-}}{L}+\cot\frac{\tau_{+}}{L}\right)\,,\\\
\tan\frac{\tau_{-}}{L}&=\frac{(d-1)\delta
L}{8\pi\alpha_{-}L}=\frac{\alpha_{\textrm{\tiny B}}}{d\alpha_{-}}\,,\\\
\tan\frac{\tau_{+}}{L}&=\frac{(d-1)\delta
L}{8\pi\alpha_{+}L}=\frac{\alpha_{\textrm{\tiny B}}}{d\alpha_{+}}\,.\\\
\end{split}$ (147)

Noting that the last two equations give not only $\tau_{\pm}$ but also the
extremality conditions. Indeed, on constant $\tau$ slices, we have
$K=-\frac{d}{L}\tan\frac{\tau}{L}$. Evaluated on the two hypersurface
$\Sigma_{\pm}$ located at $\tau=\tau_{+}$ and $\tau=-\tau_{-}$, the two
constraint equations are nothing but the same extremality conditions shown in
eq. (67).

For the advanced solution, we may consider the ansatz

$\displaystyle ds^{2}_{\textrm{\tiny A}}$
$\displaystyle=-d\tau^{2}+[\Theta(-\tau_{-}-\tau)L^{2}\cos^{2}\frac{\tau-\xi^{\textrm{\tiny
A}}_{2}}{L}+\Theta(\tau+\tau_{-})\Theta(\tau_{+}-\tau)L_{\rm
def}\cos^{2}\frac{\tau-\xi_{1}^{\textrm{\tiny A}}}{L}$ (148)
$\displaystyle+\Theta(\tau-\tau_{+})L^{2}\cos^{2}\frac{\tau}{L}]d\Sigma^{2}\,,$

which leads to the solution to the junction conditions,
$\xi_{1}^{\textrm{\tiny A}}=-\xi_{1}^{\textrm{\tiny
R}}|_{\tau_{-}\mapsto\tau_{+}}$ and $\xi_{2}^{\textrm{\tiny
A}}=-\xi_{2}^{\textrm{\tiny R}}$. Taking the difference between the retarded
and advanced solutions, we find a continuous solution which is a simple shift
in York time

$\delta_{w}:\tau\mapsto\tau+2\delta
L\left(\frac{\tau_{-}+\tau_{+}}{L}+\cot\frac{\tau_{-}}{L}+\cot\frac{\tau_{+}}{L}\right),$
(149)

where $\tau_{\pm}$ are understood to be functions of $\alpha_{\pm}$. Since
this is a diffeomorphism, it gives rise to a boundary term in the symplectic
form. As a result, all variations of $W$ around the vacuum solution are given
by boundary terms, just like for codimension-one functionals Belin:2018bpg .
Moreover, because $\delta_{w}$ is only a shift in York time, the variation is
the same as that for the volume, and the entire $\alpha_{\pm}$ dependence is
encoded in a prefactor of the symplectic form (i.e., as a prefactor in the
conjugate variation). These comments carry over to the boundary dual of the
conjugate variation, that is, the boundary background metric $\gamma_{ij}$ is
deformed as $\delta_{w}\gamma_{ij}\propto\text{sign}(t_{\textrm{\tiny
E}})\gamma_{ij}$ Belin:2018bpg , where $t_{\textrm{\tiny E}}$ is Euclidean
time on the boundary, and the $\alpha_{\pm}$ dependence is purely in the
prefactor of the deformation.

###  3.4 Complexity = Anything2

As discussed in the introduction, not all diffeomorphism invariant observables
take the form of a single extremized functional. Another class of quantities
consists of specifying the slice by extremizing a first functional, and
evaluating a different functional on it. Since both the slice specification
and the functional are diffeomorphism invariant, one would expect these
quantities to be accessible through the CFT. This seems to suggest that the
space of functionals is in fact “squared”, where the slice and quantity to
integrate can be evaluated separately. We will now explain how these
quantities fit in the general framework of Peierls and in particular how to
construct conjugate variations by using the canonical formalism of section
3.1.

Considering the general codimension-one functional defined in eq. (5), viz,

$O_{F_{1},\Sigma_{F_{2}}}(\Sigma_{\textrm{\tiny
CFT}})=\frac{1}{G_{\textrm{\tiny
N}}L}\int_{\begin{subarray}{c}\Sigma_{F_{2}}\end{subarray}}\\!\\!\\!\\!\\!d^{d}\sigma\,\sqrt{h}\,F_{1}(g_{\mu\nu};X^{\mu})\,,$
(150)

we can relate it to an auxiliary one with

$F_{\epsilon}=F_{2}+\epsilon F_{1}\,.$ (151)

For simplicity, we will focus on the case where $F_{2}=1$, i.e., we integrate
the density $F_{1}$ on the maximal volume slice. In this case, the auxiliary
density is $F_{\epsilon}=1+\epsilon F_{1}$ and our auxiliary observable is
given by

$O_{\epsilon}=\int_{\Sigma_{\epsilon}}d^{d}\sigma\sqrt{h}(1+\epsilon
F_{1})\,,$ (152)

where $\Sigma_{\epsilon}$ denotes the hypersurface which extremizes the
functional $O_{\epsilon}$. If we think of this problem perturbatively in
$\epsilon$, we have

$\Sigma_{\epsilon}=\Sigma_{V}+\epsilon\Sigma^{(1)}+\mathcal{O}(\epsilon^{2})\,,$
(153)

where $\Sigma_{V}$ is referred to as the maximal volume slice. In terms of
this expansion, one finds

$O_{\epsilon}=\int_{\Sigma_{V}}d^{d}\sigma\sqrt{h}(1+\epsilon
F_{1})+\mathcal{O}(\epsilon^{2})\,.$ (154)

Because the surface $\Sigma_{V}$ extremizes the volume, there is no leading
order contribution from $\Sigma^{(1)}$. As a consequence, one can then relate
$O_{F_{1},\Sigma_{V}}$ to a family of extremized observables $O_{\epsilon}$
by333333In fact, $\frac{d}{d\epsilon}O_{\epsilon}$ at finite $\epsilon$ is
just $F_{1}$ evaluated on the surface $\Sigma_{\epsilon}$, since the
contribution due to changing the location of the surface is a second order
effect due to extremization of $O_{\epsilon}$.

$O_{F_{1},\Sigma_{V}}=\frac{d}{d\epsilon}O_{\epsilon}\bigg{|}_{\epsilon=0}\,.$
(155)

Supposing $\delta_{O_{\epsilon}}$ is the transformation that generates
variations of $O_{\epsilon}$ through the symplectic form, we have

$\delta