††institutetext: Department of Physics, University of California, Santa
Barbara, CA 93106, USA

# The Spacetime Geometry of Fixed-Area States in Gravitational Systems

Xi Dong<EMAIL_ADDRESS>Donald Marolf<EMAIL_ADDRESS>Pratik Rath
<EMAIL_ADDRESS>Amirhossein Tajdini and<EMAIL_ADDRESS>Zhencheng Wang
<EMAIL_ADDRESS>
###### Abstract

The concept of fixed-area states has proven useful for recent studies of
quantum gravity, especially in connection with gravitational holography. We
explore the Lorentz-signature spacetime geometry intrinsic to such fixed-area
states in this paper. This contrasts with previous treatments which focused
instead on Euclidean-signature saddles for path integrals that prepare such
states. We analyze general features of fixed-area state geometries and
construct explicit examples. The spacetime metrics are real at real times and
have no conical singularities. With enough symmetry the classical metrics are
in fact smooth, though more generally their curvatures feature power-law
divergences along null congruences launched orthogonally from the fixed-area
surface. While we argue that such divergences are not problematic at the
classical level, quantum fields in fixed-area states feature stronger
divergences. At the quantum level we thus expect fixed-area states to be well-
defined only when the fixed-area surface is appropriately smeared.

##  1 Introduction

The study of entropy and quantum entanglement is a central focus of modern
treatments of the AdS/CFT correspondence and its possible generalizations. In
general, for a given boundary region $R$, the Hubeny-Rangamani-Takayanagi
(HRT) Hubeny:2007xt  generalization of the Ryu-Takayanagi formula Ryu:2006ef
tells us that the entropy of region $R$ in the dual CFT is given by
$\frac{A[\gamma_{R}]}{4G}$ where $G$ is the bulk Newton constant and
$A[\gamma_{R}]$ is the area of the smallest extremal surface $\gamma_{R}$
satisfying both $\partial\gamma_{R}=\partial R$ and the requirement that $R$
and $\gamma_{R}$ be homologous within some Cauchy surface Headrick:2007km ;
Wall:2012uf . The proof of this relation Dong:2016hjy  generalizes the
Lewkowycz-Maldacena argument Lewkowycz:2013nqa  for the time-symmetric case.

As a result, the area $A[\gamma_{R}]$ of the HRT surface $\gamma_{R}$ plays a
critical role in many discussions of AdS/CFT. It is thus natural to study bulk
states in which the distribution of $A[\gamma_{R}]$ is sharply peaked with
only very small fluctuations. Such ‘fixed-area’ states were introduced in
Refs. Akers:2018fow ; Dong:2018seb  to reproduce the entanglement properties
of simple tensor network models of quantum error correction111Though one may
also construct similar tensor network models with more general entanglement
properties by adding additional degrees of freedom to the tensor network
Donnelly:2016qqt . Pastawski:2015qua ; Hayden:2016cfa  and have since proved
to be useful for a variety of constructions and analyses; see e.g. Bao:2018pvs
; Dong:2019piw ; Marolf:2019zoo ; Penington:2019kki ; Marolf:2020vsi ;
Dong:2020iod ; Akers:2020pmf . This is in part due to the fact that the
replica trick is particularly straightforward to apply to fixed-area states,
as there is a sense in which the usual back-reaction associated with replica
numbers $n\neq 1$ vanishes for fixed-area states Akers:2018fow ; Dong:2018seb
.

Our goal here is to explore and elucidate the spacetime geometries associated
with such states. While the original works Akers:2018fow ; Dong:2018seb
observed that saddles for Euclidean path integrals preparing such states will
generally feature conical singularities at the fixed-area surface, the
spacetime geometry intrinsic to fixed-area states has received relatively
little attention. This has led to some confusion in the literature, especially
with regard to the relation between fixed-area states and the microcanonical
thermofield-double in the presence of a time-translation symmetry Goel:2020yxl
. We now discuss this apparent puzzle as an appetizer to our general treatment
of the spacetime geometry of fixed-area states.

### A possible confusion: the Microcanonical TFD vs Fixed-area states

Figure 1: Left: Starting from a TFD state with inverse temperature
$\beta_{0}$, fixing the area of the HRT surface $\gamma_{R}$, corresponding to
the time slice $\tau=0$, results in a Euclidean saddle with a conical
singularity (red) with opening angle $2\pi m$, where
$m=\frac{\beta_{0}}{\beta}$. Right: The same state prepared as a
microcanonical TFD state by imposing asymptotic fixed energy boundary
conditions. This results in a smooth Euclidean saddle with boundary length
$\beta$. The $\mathbb{Z}_{2}$ symmetric Cauchy slice $\Sigma_{sym}$ (green and
blue) on both saddles has identical data and thus, results in identical
Lorentzian spacetimes upon time evolution.

Fixed-area states may be constructed by starting from a seed state
$\ket{\psi}$ and applying a quasi-projection operator that, for a given
boundary region $R$, restricts the probability distribution of the HRT area to
be sharply peaked around a particular value $A_{0}$. From Refs. Akers:2018fow
; Dong:2018seb , it is also known that the entanglement spectrum of a fixed-
area state is quite flat, so that the eigenvalues of the modular Hamiltonian
$\hat{K}_{R}$ are also sharply peaked.

A particularly simple case is one in which the seed-state $\ket{\psi}$ is a
thermofield-double (TFD) state for which the bulk geometry has a static
Killing field with bifurcate horizon; i.e., in the bulk $|\psi\rangle$
describes a standard 2-sided black hole. Since $|\psi\rangle$ is the TFD
state, the norm $\langle\psi|\psi\rangle$ is computed by a Euclidean thermal
path integral with boundary $S^{1}\times X$ for some $X$, where the metric on
this boundary is also a product. It is easiest to visualize the associated
bulk saddles when the bulk is 2-dimensional as in the case of Jackiw-
Teitelboim gravity (where by ‘area’ we mean the value of the dilaton). In that
case, a bulk saddle can be represented as a disk in which there is a preferred
point that represents the Euclidean horizon, see Fig. 1.

We wish to consider the state constructed from $|\psi\rangle$ by fixing the
area of the Euclidean horizon. In this case, the Euclidean horizon coincides
with the HRT surface for the region defined by taking all of $X$ at some point
(which we may call $\tau=0$) on the $S^{1}$. As described in Ref. Dong:2018seb
, the corresponding fixed-area state is defined by the path integral with the
same asymptotically AdS boundary conditions as the one that defines
$|\psi\rangle$, but where the area of this HRT surface is also fixed to some
$A_{0}$ as a boundary condition. Since we do not integrate over that area,
saddles for this path integral need not satisfy the corresponding equation of
motion at the HRT-surface. In particular, such saddles need not be smooth, and
can instead have a conical singularity of arbitrary (constant) strength along
the HRT surface.

If we consider saddles that preserve all symmetries, then in many cases there
will be an analogue of Birkhoff’s theorem which states that, at least locally,
the possible bulk solutions are just the set of appropriately-symmetric
(Euclidean) black holes. Fixing $A[\gamma_{R}]$ to some $A_{0}$ will then
select precisely one such solution. But the period $\beta$ of the smooth
Euclidean black hole with horizon area $A_{0}$ will not generally match the
period $\beta_{0}$ of the $S^{1}$ at infinity. Nevertheless, we can use the
freedom to introduce a conical singularity at $\gamma_{R}$ (in this case with
deficit angle $2\pi\left(1-\frac{\beta_{0}}{\beta}\right)$) to change the
period of this solution to match $\beta_{0}$; see Fig. 1.

On the other hand, as discussed in Ref. Dong:2018seb , one expects that in the
leading semiclassical approximation the above fixed-area state will be
equivalent to the microcanonical thermofield double so long as the area
$A_{0}$ chosen above is not too small (so that the microcanonical ensemble is
dominated by AdS-Schwarzschild black holes). The point here is that the seed
state $|\psi\rangle$ above was chosen to be the usual (canonical) thermofield
double, and so has modular Hamiltonian

$\hat{K}_{R}=\beta H+\log Z,$ (1.1)

where $H$ is the boundary Hamiltonian and the second term makes up the
normalization. Furthermore, for each energy $E$ (again chosen to not be too
small), the entropy is maximized by states that are well-described by an AdS-
Schwarzschild black hole with horizon area $A$ determined by $E.$ As a result,
restricting the canonical TFD to a narrow band of energies is essentially the
same as restricting to a narrow band of areas.

The interesting point then is that the restriction on energies can be
implemented by performing an inverse Laplace transform. This can be done
semiclassically by integrating over boundary length $\beta$ to find a saddle
with definite energy. Unlike the above fixed-area saddle, the corresponding
Euclidean geometry is just a smooth disk with period $\beta$ at infinity
determined as usual by the energy, or equivalently by the horizon area. See
e.g. Marolf:2018ldl ; Marolf:2022jra . Note also that in this case the period
at infinity was not fixed as a boundary condition, but was determined
dynamically by the saddle-point conditions.

Naively, it may appear that the conical defect appearing in the fixed-area
state description might leave some singular imprint on the Lorentzian
spacetime described by that state. In contrast, it is clear that no such issue
arises for the microcanonical TFD. Nevertheless, we will argue that these
indeed lead to the same smooth Lorentzian classical solution.

Note in particular that the fixed-area Euclidean solution has a
$\mathbb{Z}_{2}$ symmetry that leaves invariant a particular slice,
$\Sigma_{sym}$. The $\mathbb{Z}_{2}$ symmetry implies that $\Sigma_{sym}$ has
vanishing extrinsic curvature222Subtleties in this argument associated with
the fact that $\Sigma_{sym}$ passes through the defect will be discussed in
Sec. 2.1 below. $K_{ij}=0$, so the data on $\Sigma_{sym}$ also provides Cauchy
data for a Lorentzian solution. Furthermore, the $U(1)$ symmetry of the
Euclidean solution means that the induced metric $h_{ij}$ on $\Sigma_{sym}$ is
just the usual induced metric on the surface of time-symmetry associated with
the black hole of area $A_{0}$. In particular, the conical singularity leaves
no imprint on either $h_{ij}$ or $K_{ij}$. Thus, the resulting Lorentzian
spacetime is completely smooth until one reaches the usual black hole
singularities. In particular, this Lorentzian spacetime is completely smooth
at the HRT surface of interest. And since the corresponding $\Sigma_{sym}$ in
the microcanonical TFD saddle has precisely the same $h_{ij}$ and $K_{ij}$, it
defines precisely the same Lorentzian solution.

Now, the above setting is not generic, and it turns out that the Lorentzian
solutions generally become singular when the U(1) symmetry is broken. but
these are power law singularities, not conical singularities or strict
shockwaves. We will describe the details of these singularities in Sec. 4.4
below.

### Overview

Our treatment begins in Sec. 2 with a brief review of fixed-area states and
their preparation via path integrals. As mentioned above, a common algorithm
Akers:2018fow ; Dong:2018seb ; Dong:2019piw  for constructing fixed-area
states involves first using a standard Euclidean (or, more generally, complex)
path integral to construct a more familiar semiclassical bulk state and then
modifying this prescription to fix the area of $A[\gamma_{R}]$. This then
allows us to study the spacetime geometry of the fixed-area state in terms of
the boundary conditions imposed on the above Euclidean path integral. Sec. 2
extends previous such discussions by using Schwinger-Keldysh-like
constructions to study the spacetime geometry intrinsic to the fixed-area
state itself and to cleanly separate this geometry from that associated with
sources used to prepare the state.

While fixed-area states can be of use in constructing replica saddles, and
while real-time replica saddles require complex metrics Colin-Ellerin:2020mva
; Colin-Ellerin:2021jev , we will show that the spacetime metrics in fixed-
area states are generally real at real times. Furthermore, they have no
conical singularities.

As emphasized earlier, a $U(1)$ symmetric Euclidean solution results in a
smooth Lorentzian spacetime. We thus analyze various examples where this
$U(1)$ symmetry is broken to demonstrate the features of generic fixed-area
states. Our main analyses will be performed at the classical level, though we
will comment on quantum effects at the end.

We first discuss a warmup example in Sec. 3 where we consider a non-
gravitational scalar field theory in $1+1$ dimensions. This highlights the
prominent features that we expect from fixed-area states such as the existence
of power law divergences in the scalar field on the lightcone of the fixed-
area surface.

We then move on to a general discussion of the structure of the Lorentzian
spacetimes of fixed-area states in gravitational theories. We propose a
general ansatz for the form of the classical solution in Sec. 4.1. We then
construct detailed examples in Jackiw-Teitelboim (JT) gravity and
AdS${}_{3}\times X$ for compact $X$ in Sec. 4.2 and Sec. 4.3 demonstrating the
validity of our ansatz. The general structure of the above-mentioned
singularities on the light cone of the fixed-area surface are analyzed in Sec.
4.4. We close with some final discussion in Sec. 5 including comments on
higher derivative and quantum corrections.

##  2 Schwinger-Keldysh path integrals for fixed-area states

As described in Sec. 1, fixed-area states are simply states of gravitational
systems in which the distribution of some HRT-area operator $A[\gamma_{R}]$ is
sharply peaked, i.e. the width $\Delta A$ is small. Let us first discuss
precisely what we mean by sharply peaked. As anticipated in Ref.
Bousso:2020yxi  and as established in Refs. molly ; xi , in the semiclassical
approximation the action of $\frac{A[\gamma_{R}]}{4G}$ is given by a so-called
boundary-condition-preserving kink transform, which in particular induces a
relative boost between the two entanglement wedges of some rapidity $s$. From
the uncertainty relation, we have

$\Delta A\Delta s\gtrsim O(G),$ (2.1)

where $s$ is the relative boost between the two entanglement wedges on either
side of the HRT surface. Depending on the value of $\Delta A$, we can classify
fixed-area states into two types: pseudo-eigenstates and squeezed states.

Pseudo-eigenstates are very sharply peaked and have $\Delta A\sim
O(G^{\alpha})$ with $\alpha\geq 1$. This leads to $\Delta s\rightarrow\infty$
in the semiclassical limit. As a result, we do not expect a single geometry to
describe such states. On the other hand, squeezed states have $\Delta A\sim
O(G^{\alpha})$ with $\alpha\in(\frac{1}{2},1)$. Such states are expected to
have a semiclassical description, and yet have $\Delta A$ parametrically
smaller than states usually constructed by Euclidean path integrals
Marolf:2018ldl . Here we shall focus on such squeezed states and describe
their associated Lorentzian spacetimes.

Now, fixed-area geometries must of course have a specified value of
$A[\gamma_{R}]$. But as noted above, so long as we consider the squeezed state
case (so that $A[\gamma_{R}]$ is not specified too precisely), we expect that
the state can remain semiclassical. And since there are no other constraints,
one further expects that all other aspects of the semiclassical spacetime
geometry can be chosen arbitrarily (so long as they solve the equations of
motion). However, as described in Refs. Akers:2018fow ; Dong:2018seb ;
Dong:2019piw , one is typically interested in starting with some semiclassical
bulk state $|\psi\rangle$, perhaps constructed using a gravitational path
integral, and then applying a projection-like operator333We use the term
“projection-like operator” to mean a Hermitian operator for which the variance
of $A[\gamma_{R}]$ is small in the state
$|\psi\rangle_{A_{0}}:=\Pi_{R}|\psi\rangle$. We do not require
$\Pi_{R}^{2}=\Pi_{R}$. In particular, we might consider a Gaussian
$\Pi_{R}=e^{-\frac{(A[\gamma_{R}]-A_{0})^{2}}{2\sigma^{2}}}$ with some small
width $\sigma$. $\Pi_{R}$ that restricts this state to a range of
$A[\gamma_{R}]$-eigenvalues of some small width $\Delta A$ about a central
value $A_{0}$. We will thus investigate the spacetime geometries of fixed-area
states that arise from such constructions and in particular their relation to
the path integral boundary conditions used to define $|\psi\rangle$.

Recall that the squeezed state regime $\alpha\in(\frac{1}{2},1)$ described
above suffices to fix the value of $A$ in the semiclassical limit
$G\rightarrow 0$. In particular, in that limit we may use the recipe described
in Refs. Dong:2018seb ; Dong:2019piw  for studying
$|\psi\rangle_{A_{0}}:=\Pi_{R}|\psi\rangle$. The recipe begins by supposing
that we have already constructed a gravitational path integral that computes
the original state $|\psi\rangle$, which in particular means that we are given
boundary conditions for that path integral. From the bulk point of view we can
think of the new state $|\psi\rangle_{A_{0}}$ as being created from
$|\psi\rangle$ by the insertion of additional sources on $\gamma_{R}$, though
of course the location of $\gamma_{R}$ must be determined dynamically in a
manner that takes into account the back-reaction from those sources. As
explained in Refs. Dong:2018seb ; Dong:2019piw , in the saddle-point
approximation this means that saddles for $|\psi\rangle_{A_{0}}$ can be taken
to satisfy the same asymptotic AdS boundary conditions as $|\psi\rangle$ (with
precisely the same sources at the asymptotic boundary), so long as we also 1)
impose the usual equations of motion away from $\gamma_{R}$, 2) allow the bulk
to have a codimension-2 conical singularity of arbitrary strength on a locus
$\gamma_{R}$ homologous to $R$ and satisfying $\partial\gamma_{R}=\partial R$,
3) choose the strength of the conical singularity so that
$A[\gamma_{R}]=A_{0}$, and 4) impose appropriate boundary conditions at
$\gamma_{R}$.

In particular, in Euclidean signature, Appendix A of Ref. Dong:2019piw  shows
that the Euclidean Einstein-Hilbert action (including the delta-function term
in the Ricci scalar associated with the conical singularity at $\gamma_{R}$)
defines a good variational principle for this problem when the metric near
$\gamma_{R}$ takes the following form:

$\displaystyle ds^{2}=dzd\bar{z}+T\frac{(\bar{z}dz-
zd\bar{z})^{2}}{z\bar{z}}+h_{ij}dy^{i}dy^{j}+2iW_{j}dy^{j}(\bar{z}dz-
zd\bar{z}),$ (2.2) $\displaystyle
T=\hat{o}(1),\quad\partial_{r}T=\frac{\hat{o}(1)}{r},\quad\partial_{r}h_{ij}=\frac{\hat{o}(1)}{r},\quad\partial_{r}W_{j}=\frac{\hat{o}(1)}{r},$
(2.3)

where $z$ is defined as $z=re^{im\theta}$ with $\theta\sim\theta+2\pi$, and
$T$, $h_{ij}$, and $W_{j}$ are functions of all coordinates
$(z,\bar{z},y^{i})$. Furthermore, $\hat{o}(1)$ denotes terms that vanish in
the $r\rightarrow 0$ limit at least as fast as some power law $r^{\eta}$ with
$\eta>0$.444This fixes a typo in v1 of Ref. Dong:2019piw . We refer to the
conditions imposed by Eq. (2.2), (2.3) as boundary conditions to be imposed on
Euclidean metrics at $\gamma_{R}$.

We emphasize that the conical singularities on the surfaces $\gamma_{R}$ are
associated with insertions of the operator $\Pi_{R}$ and, as such, they
represent features of the way that the state $|\psi\rangle_{A_{0}}$ is being
prepared rather than a feature intrinsic to the state itself. Indeed, if we
can find another state $|\tilde{\psi}\rangle$ described by a smooth bulk
saddle which yields the same fixed-area state up to quantum corrections

$|\tilde{\psi}\rangle_{A_{0}}=\Pi_{R}|\tilde{\psi}\rangle\approx\Pi_{R}|\psi\rangle=|\psi\rangle_{A_{0}},$
(2.4)

but where the saddle-point value of $A[\gamma_{R}]$ in the state
$|\tilde{\psi}\rangle$ is already $A_{0}$, then the fixed-area saddle with
asymptotically-AdS boundary conditions associated with the state
$|\tilde{\psi}\rangle$ will be smooth regardless of the strength of the
conical singularity in the original saddle defined by the asymptotically-AdS
boundary conditions associated with $|\psi\rangle$.

To study the geometry intrinsic to the state $|\psi\rangle_{A_{0}}$, we should
instead compute correlation functions in this state (which in the
semiclassical limit should then factorize into a product of one-point
functions). We thus consider

${}_{A_{0}}\langle\psi|g_{\mu_{1}\nu_{1}}(x_{1})\dots
g_{\mu_{n}\nu_{n}}(x_{n})|\psi\rangle_{A_{0}}=\langle\psi|\Pi_{R}g_{\mu_{1}\nu_{1}}(x_{1})\dots
g_{\mu_{n}\nu_{n}}(x_{n})\Pi_{R}|\psi\rangle,$ (2.5)

where issues related to the gauge-dependence of the
$g_{\mu_{i}\nu_{i}}(x_{i})$ will not affect our discussion and will thus be
ignored. Note that it is critical that there are two insertions of the
operator $\Pi_{R}$ in Eq. (2.5). In particular, even if $\Pi_{R}$ were an
exact projector, the fact that $\Pi_{R}$ will generally not commute with
$g_{\mu_{i}\nu_{i}}(x_{i})$ would make it difficult to use such a property to
remove either copy of $\Pi_{R}$. Note also that the operators that sample the
desired geometry naturally live between the two projectors.

We may thus construct a path integral that computes Eq. (2.5) by first
constructing path integrals for the bra and ket wavefunctions
${}_{A_{0}}\langle\psi|=\langle\psi|\Pi_{R}$ and
$|\psi\rangle_{A_{0}}=\Pi_{R}|\psi\rangle$ and then using these wavefunctions
as boundary conditions for a path integral that computes correlators of the
$g_{\mu_{i}\nu_{i}}(x_{i})$. The first step above is identical to constructing
path integrals for the unconstrained bra and ket states $\langle\psi|$ and
$|\psi\rangle$, except for the insertion of a constraint on the area of
$\gamma_{R}$. Thus the final path integral involves constraints on two such
surfaces $\gamma_{R}$ (though these surfaces may sometimes coincide).

We should of course add suitable sources $T^{\mu_{i}\nu_{i}}(x_{i})$ to the
action with respect to which we can vary to obtain the desired insertions of
$g_{\mu_{i}\nu_{i}}(x_{i})$. However, we see that in an appropriate sense we
will make such variations only in the region between the two surfaces
$\gamma_{R}$. Saddles for this problem will thus have two codimension-2
conical singularities, and it is only the region of the spacetime that in some
sense lies between those singularities555We will explain the correct sense in
more detail shortly. This sense will be clearest in Lorentz signature where
the Cauchy problem is well-posed. that can be directly interpreted as the
geometry intrinsic to the fixed-area state $|\psi\rangle_{A_{0}}$. In
particular, in the leading saddle-point approximation we can simply set the
sources $T^{\mu_{i}\nu_{i}}(x_{i})$ to zero and take the insertions of
$g_{\mu_{i}\nu_{i}}(x_{i})$ to report the saddle-point value of the metric at
the point $x_{i}$. In that sense it is in fact sufficient to study the path
integral that computes the norm

${}_{A_{0}}\langle\psi|\psi\rangle_{A_{0}}=\langle\psi|\Pi_{R}^{2}|\psi\rangle.$
(2.6)

###  2.1 Saddle points for fixed-area path integrals

We now wish to describe the saddle points of this path integral. The
constraints on the areas of the $\gamma_{R}$ surfaces mean that we do not
integrate over these areas and, as a result, two of the Einstein equations
need not be satisfied by our saddles, one at each of the two $\gamma_{R}$
surfaces. As explained in Refs. Akers:2018fow ; Dong:2018seb ; Dong:2019piw ,
this extra freedom allows conical singularities of arbitrary strength on each
$\gamma_{R}$-surface. While the conical deficit or excess must be constant
along each such surface, the fact that the constraints remove two Einstein
equations means that the strengths of the singularities on the two
$\gamma_{R}$ surfaces may be chosen independently. Furthermore, the idea that
our path integral may be thought of as computing matrix elements associated
with the bra and ket states ${}_{A_{0}}\langle\psi|$, $|\psi\rangle_{A_{0}}$,
and that each contributes one of the conical singularities, suggests that one
should be able to cut the saddle ${\cal M}$ into two pieces ${\cal
M}_{1},{\cal M}_{2}$ along some codimension-1 surface $\Sigma_{cut}$ (so that
$\partial{\cal M}_{1}\supset\Sigma_{cut}\subset\partial{\cal M}_{2}$) such
that ${\cal M}_{1},{\cal M}_{2}$ each contain only one of the $\gamma_{R}$
surfaces, see Fig. 2. We will thus impose this requirement below, with the
understanding that we think of each piece as being closed so that
$\Sigma_{cut}\subset{\cal M}_{1},{\cal M}_{2}$. Thus, this condition can be
satisfied in the degenerate case where the two $\gamma_{R}$ surfaces at least
coincide (in part or in whole) by taking $\Sigma_{cut}$ to pass through the
locus of this coincidence. Note that this restriction requires the tangent
spaces of the surfaces $\gamma_{R}$ to coincide at any point where the two
surfaces intersect.

Figure 2: The saddle point manifold $\mathcal{M}$ for
$\langle\psi|\Pi_{R}^{2}|\psi\rangle$ can be split into two parts
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ along a slice $\Sigma_{cut}$ such that
there are two conical defects (red) at $\gamma_{R}$, with one on the bra side
of the cut and the other on the ket side. On the right, we have added a
regulator $\epsilon$ that moves each of the resulting singularities away from
the cut. The original $\mathcal{M}$ should be understood as the limit
$\epsilon\rightarrow 0$ where the two $\gamma_{R}$ surfaces coincide.

Now, in many cases the state $|\psi\rangle$ will have been constructed by
specifying boundary conditions on a Euclidean asymptotically-AdS boundary. But
we may nevertheless be interested in the metric $g_{\mu_{i}\nu_{i}}(x_{i})$ at
real times $t_{i}$. In this case our path integral will integrate over
spacetimes which follow a Schwinger-Keldysh-like contour through the plane of
complex times. Nevertheless, since $\Pi_{R}$ is Hermitian the expression Eq.
(2.6) is manifestly real. As a result, the path integral will have a
$\mathbb{Z}_{2}$ symmetry that exchanges the parts of the boundary associated
with boundary conditions for $\langle\psi|$ and for $|\psi\rangle$ and which
simultaneously acts by complex-conjugation. Since this symmetry acts as a
reflection (and conjugation) on the asymptotically AdS boundary, any bulk
saddle ${\cal M}$ that preserves this conjugation symmetry must have a
codimension-1 surface $\Sigma_{sym}$ that is invariant under the action of
this $\mathbb{Z}_{2}$ symmetry666The argument uses the fact that our
asymptotically AdS boundary conditions require ${\cal M}$ to have a $C^{1}$
conformal compactification so that at each point on the boundary the
conformally rescaled spactime can be approximated by a Euclidean half-space.
One may then show the existence of $\Sigma_{sym}$ by exhaustively studying the
symmetries of the Euclidean plane. as shown in Fig. 3. We will consider only
such saddles below.

Figure 3: (a) The saddle point for
$\langle\psi|\Pi_{R}(0)O(t_{0})\Pi_{R}(0)|\psi\rangle$ has two conical defects
(red) at $\gamma_{R}$ on the bra and ket side respectively. It has a
$\mathbb{Z}_{2}$ symmetry that leaves invariant the Cauchy slice
$\Sigma_{sym}$ (green). This slice splits the saddle into two parts
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$. The region of the saddle point shaded
light blue can be thought of as the spacetime inherent to the fixed-area
state, while the yellow portion is involved in the preparation of the state.
We may take the blue portion to lie at real Lorentzian times and the yellow
portion to lie at real Euclidean times. In that sense, the two blue portions
each involve the same interval of real Lorentzian times and should perhaps be
drawn as being degenerate with each other, but we have separated the pieces
for ease of visualization. (b) The corresponding Schwinger-Keldysh contour in
the complex-time plane.

For the moment, let us assume $\Sigma_{sym}$ does not intersect either surface
$\gamma_{R}$ so that it lies in a smooth region of the saddle-point spacetime.
In this case, $\Sigma_{sym}$ has a well-defined induced metric $h_{ij}$ and
extrinsic curvature. We will use the symbol $K^{E}_{ij}$ to denote the
extrinsic curvature defined using Euclidean conventions, and we will use
$K^{L}_{ij}=iK^{E}_{ij}$ to denote the extrinsic curvature defined using
Lorentz-signature conventions. Since all equations of motion are satisfied on
$\Sigma_{sym}$, the data $h_{ij},K^{E}_{ij}$ (or $h_{ij},K^{L}_{ij}$), i.e.,
the metric and extrinsic curvature will clearly satisfy the relevant
constraint equations. Furthermore, the invariance of $\Sigma_{sym}$ under the
conjugation symmetry means that $h_{ij}$ must be real and the real part of
$K^{E}_{ij}$ must vanish. This of course simply means that $K^{E}_{ij}$ is
purely imaginary, or equivalently that $K^{L}_{ij}$ is real. In other words,
just as in the analysis of real-time replica wormholes in Ref. Colin-
Ellerin:2020mva , symmetry requires $\Sigma_{sym}$ to define Cauchy data
appropriate to the Lorentz-signature initial value problem.

Let us now consider the case where $\Sigma_{sym}$ intersects some $\gamma_{R}$
at some point $x$. In this case one might worry that the extrinsic curvature
of $\Sigma_{sym}$ at $x$ is not well-defined. In particular, consider surfaces
$\Sigma_{\pm}$ on either side of $\Sigma_{sym}$, where we take the conjugation
symmetry to interchange $\Sigma_{+}$ and $\Sigma_{-}$. When there is a conical
singularity at $x$, the surfaces $\Sigma_{\pm}$ will have different extrinsic
curvatures at $x$ even in the limit where
$\Sigma_{\pm}\rightarrow\Sigma_{sym}$. Indeed, with Euclidean signature
conventions the real parts of their extrinsic curvatures will have delta-
functions along $\gamma_{R}$ of opposite signs. In contrast, the conjugation
symmetry requires that the imaginary parts of their $K^{E}_{ij}$ tensors
match, so this imaginary part is continuous and unambiguous.

Since the two surfaces $\gamma_{R}$ are in principle independent, the case
where they intersect is just a degenerate limit of the more general case where
they do not. As noted above, when they do not intersect the conjugation
symmetry requires the real part of the extrinsic curvature of $\Sigma_{sym}$
to vanish. We will thus define this to also be the case when the $\gamma_{R}$
intersect. In effect, this is the statement that we should define the
extrinsic curvature on $\Sigma_{sym}$ by first regulating the problem in a
manner that separates the $\gamma_{R}$ surfaces (with $\Sigma_{sym}$ lying
between them) as shown in Fig. 2. We then take a limit where the regulator is
removed.777Similar reasoning tells us that we can always take the conical
singularities to lie inside the Euclidean region. This means that the boundary
conditions described by Eqs. (2.2) and (2.3) suffice to treat them. The case
where the conical singularities lies at the boundary between the Euclidean and
Lorentzian region is to be regarded as a limit of the case where the
singularities lie entirely in the Euclidean region of the contour. But in
practice it suffices to simply compute the (well-defined) imaginary part of
$K^{E}_{ij}$ and to take the real part of $K^{E}_{ij}$ to vanish.

Now, when $\gamma_{R}$ intersects $\Sigma_{sym}$ at $x$, the conical
singularity on $\gamma_{R}$ also means that some equations of motion are not
satisfied at $x$. However, the fact that we required the existence of
$\Sigma_{cut}$ means that the conjugation symmetry must exchange the two
surfaces $\gamma_{R}$. As a result, each point of the fixed-point set
$\Sigma_{sym}$ that intersects one copy of $\gamma_{R}$ must in fact intersect
both copies. Furthermore, as noted in Ref. Dong:2019piw  (see footnote 15),
conical singularities can be thought of as arising from spacelike cosmic-brane
sources that lie along $\gamma_{R}$. The effective stress tensor of such
branes has non-zero components only tangent to $\gamma_{R}$, and in particular
tangent to $\Sigma_{sym}$ at any point of intersection. But the constraints on
initial data on $\Sigma_{sym}$ are associated with components of the equations
of motion normal to $\Sigma_{sym}$, so they receive no contributions from such
sources – i.e., even when $\gamma_{R}$ intersects $\Sigma_{sym}$ the initial
data on $\Sigma_{sym}$ satisfies the constraints that guarantee the initial
value problem of the desired theory to be well-posed.

Figure 4: The saddle can be extended in the real time direction by evolving
using the data on Cauchy slice $\Sigma_{\text{sym}}$. As an example, we depict
the Schwinger-Keldysh contour for forward evolution up to surface $\Sigma_{2}$
and backward evolution up to surface $\Sigma_{1}$.

As a result, given any conjugation-symmetric saddle for the path integral that
computes Eq. (2.6), we are free to extend it in the real time direction by
evolving forward in time from $\Sigma_{sym}$ up to some new Cauchy surface
$\Sigma_{2}$ and then backwards from $\Sigma_{2}$ back to $\Sigma_{sym}$.
Indeed, we could just as well evolve backward in time from $\Sigma_{sym}$ to
some $\Sigma_{1}$ and then forwards again from $\Sigma_{1}$ to $\Sigma_{sym}$
as shown in Fig. 4, or we could even insert additional timefolds and evolve
forwards and backwards in any manner that we like. The new piece of this
spacetime corresponding to our excursion along the real-time axis will be real
and Lorentz signature, and will satisfy all of the equations of motion. In
particular, it will be free of conical singularities. Futhermore, extending
the saddle in this way leaves the action of the saddle unchanged because the
factors of $e^{iS}$ associated with the forward-in-time parts of this
evolution will precisely cancel the factors of $e^{-iS}$ associated with the
backwards-in-time evolution. It is the geometry along this real-time excursion
that we explore in more detail below. As a final remark we note that this part
of the spacetime lies between the two surfaces $\gamma_{R}$ in the sense that
it lies between two copies of the Cauchy surface $\Sigma_{sym}$, while the
copies of $\gamma_{R}$ lie in the two regions between the copies of
$\Sigma_{sym}$ and the Euclidean asymptotically AdS boundary.

##  3 Warmup: Scalar Field Theory in $1+1$ dimensions

We would like to understand fixed-area states in the absence of a $U(1)$
symmetry. In order to do so, we first discuss a related-but-simpler problem
involving a free scalar field $\phi$ in 1+1 dimensions on a fixed conical
background. Due to the fixed background, this example cannot be directly
interpreted as involving a fixed-area state. Nevertheless, the analysis will
highlight the key ideas needed for our discussion of dynamical gravity in Sec.
4. There we will analyze the general structure of fixed-area states and
illustrate it with various examples.

Figure 5: Left: Euclidean preparation of the state of a scalar field on a
conical background of opening angle $2\pi m$. A source for the scalar field
(pink) is present at the asymptotic boundary and which breaks the $U(1)$
symmetry of the geometry. This prepares initial data for Lorentzian evolution
on the $\mathbb{Z}_{2}$ symmetric slice defined by the union of the blue half-
line at $\theta=0$ line and the green half-line at $\theta=\pi$. Note that
there is really only a single blue half-line; the marks on the diagram
indicate that the two copies are to be identified. Right: The initial data
prepared by the Euclidean solution is used to generate the Lorentzian solution
in Minkowski space by solving the equation of motion as described in the main
text. The solution at the small black dot in region II is obtained by
propagating the data along left and right-moving light rays (for a free
massless field).

As discussed in Sec. 2, saddles for Euclidean path integrals preparing fixed-
area states typically contain conical singularities. Here we study a toy model
of the influence of such singularities on solutions to equations of motion by
considering saddles for a scalar field path integral on a fixed conical
background. We can choose the time-contour of the background so that the
saddle is purely Euclidean, or we can choose a contour for which the saddle
contains pieces that evolve in real Lorentzian time.

We take the background Euclidean geometry to be the simple $U(1)$-symmetric
cone. However, we will impose boundary conditions for the scalar field that
break this $U(1)$ symmetry as shown in Fig. 5. In particular, the boundary
conditions will preserve a $\mathbb{Z}_{2}$ conjugation symmetry. Although the
geometry in this example is flat, we will sometimes refer to these boundary
conditions as ‘sources’ using the terminology common in the AdS/CFT context.

Any scalar field saddle on the Euclidean cone determines values of the scalar
field $\phi$ on the $\mathbb{Z}_{2}$ symmetric Cauchy slice. Assuming that the
saddle respects this $\mathbb{Z}_{2}$ symmetry, the real part of the normal
derivative of $\phi$ must vanish.

As a result, the data on this slice gives Cauchy data for a real solution to
the equations of motion for our scalar in $1+1$-dimensional Minkowski
spacetime (with no conical singularity). We will see that such solutions have
power-law singularities on the past and future lightcones of the point $x=0$.
We present the analysis for a massless scalar field in Sec. 3.1 and for a
massive scalar field in Sec. 3.2, both of which have qualitatively similar
properties.

Since gravity is non-dynamical in this example, at some level we have simply
chosen the Lorentzian spacetime by hand to be a singularity-free Minkowski
space. But one may view this choice as arising from a natural analogue of the
discussion surrounding Fig. 3. If a regulator was introduced to split the
original Euclidean conical singularity (say, with conical deficit $\delta$)
into two singularities (both with deficits $\delta/2$) while preserving
$\mathbb{Z}_{2}$ symmetry, then the symmetric slice would have vanishing
extrinsic curvature and induced metric $dx^{2}$. This is precisely the data
for the $t=0$ slice of Minkowski space. One may also think of the Lorentz-
signature Minkowski space as being generated by analytically-continuing the
geometry from the Euclidean region between the singularities.

###  3.1 Massless Field

Consider a free massless real scalar field on a fixed conical background. The
Euclidean action is given by

$\displaystyle S_{\rm massless}=\frac{1}{2}\int
d^{2}x\sqrt{g}(\partial\phi)^{2},$ (3.1)

with the standard metric given by

$\displaystyle ds^{2}=dr^{2}+m^{2}r^{2}d\theta^{2},\qquad\theta\in[0,2\pi).$
(3.2)

In this coordinate system, the time-reflection symmetric slice corresponds to
$\theta=0$ and $\theta=\pi$ as shown in Fig. 5.

In order to have a non-trivial classical solution, we turn on sources for the
scalar field at some large distance cutoff boundary, $r=r_{c}$, and solve the
equations for $r\leq r_{c}$. A simple source we consider which breaks the
$U(1)$ symmetry is the boundary condition

$\displaystyle\phi_{b}(\theta)\equiv\phi(r_{c},\theta)=2r_{c}^{k/m}\cos(k\theta)$
(3.3)

with some positive integer $k$. This source satisfies
$\left.\partial_{\theta}\phi_{b}\right|_{\theta=0,\pi}=0$, as would any real
source that preserves the $\mathbb{Z}_{2}$ symmetry
$\theta\rightarrow-\theta$. As a result, the slice defined by $\theta=0,\pi$
provides initial data for a real Lorentzian solution. Solving the equation of
motion associated with Eq. (3.1) is straightforward. Furthermore, in order for
our action to yield a well-defined variational principle for the scalar field,
we require the solution to be finite at $r=0$. This uniquely determines the
solution to be

$\displaystyle\phi(r,\theta)=2r^{k/m}\cos(k\theta).$ (3.4)

The desired Lorentzian initial data is found simply by setting $\theta=0,\pi$
in Eq. (3.4). Defining the coordinate $X=r$ when $\theta=0$ and $X=-r$ when
$\theta=\pi$, we can write the full initial value of $\phi$ in the form

$\displaystyle\phi_{0}(X)=2X^{k/m}\Theta(X)+2(-1)^{k}(-X)^{k/m}\Theta(-X).$
(3.5)

When combined with the condition $\partial_{T}\phi(T,X)|_{T=0}=0$, the data
$\phi(T=0,X)=\phi_{0}(X)$ determines a unique Lorentzian solution on Minkowski
space (which we take to have the standard metric $ds^{2}=-dT^{2}+dX^{2}$).

While this example is simple enough that we could explicitly solve for
$\phi(T,X)$ everywhere, it will be instructive to first find the corresponding
solutions in the left and right Rindler wedges marked as regions I and III in
Fig. 5. The point here is that, by causality, the analytic continuation
$\theta\to\theta+it$ of Eq. (3.4) is guaranteed to coincide in regions I and
III with the Lorentzian solutions constructed above. This follows from the
fact that the analytic continuation gives a Lorentzian solution with the
correct initial data on both the positive and negative $X$-axes at $t=0$,
together with the fact that such data defines a unique solution in each
Rindler wedge888 Simple attempts to extend the argument to include the origin
$X=T=0$ will fail for the simple reason that the background Euclidean cone is
not analytic at this point due to the delta-function Ricci scalar supported at
the tip of the cone. However, we describe a more sophisticated such extension
in Appendix B.. More explicitly, the above analytic continuation will give
$\phi(X,T)$ in e.g. region I using $X=r\cosh(mt),T=r\sinh(mt)$. The solutions
in regions I, III are thus given by

$\displaystyle\phi(U,V)=V^{k/m}+(-U)^{k/m},$ $\displaystyle(\text{Region
I}):\,V>0\;{\rm and}\;U<0,$ (3.6)
$\displaystyle\phi(U,V)=(-1)^{k}\left((-V)^{k/m}+U^{k/m}\right),$
$\displaystyle(\text{Region III}):V<0\;{\rm and}\;U>0,$ (3.7)

where $U,V$ are the null coordinates

$\displaystyle V=T+X,\qquad U=T-X.$ (3.8)

The key issue is then to find the solutions in regions II and IV. Since the
initial data is real, on general grounds we must find a real solution
everywhere in the Lorentzian manifold. On the other hand, starting from Eq.
(3.6) and Eq. (3.7), one might naively expect to find a complex solution when
$k/m$ is not an integer, as this is certainly the result of e.g. analytically
continuing $V$ across the horizon at $V=0$.

Luckily, the massless field is simple enough to allow us to clarify this issue
by finding the explicit solutions in the past and future wedges. As is well-
known, the most general solution to the massless $1+1$ Klein-Gordon equation
is given by

$\displaystyle\phi(U,V)=f(V)+g(U)$ (3.9)

for arbitrary functions $f,g$. Since the data in the left and right Rindler
wedges fully determines both $f$ and $g$, the full solution must be

$\displaystyle\phi(U,V)=V^{k/m}+(-1)^{k}(U)^{k/m},$
$\displaystyle(\text{Region II}):\,V>0\;{\rm and}\;U>0,$ (3.10)
$\displaystyle\phi(U,V)=(-1)^{k}(-V)^{k/m}+(-U)^{k/m},$
$\displaystyle(\text{Region IV}):\,V<0\;{\rm and}\;U<0.$ (3.11)

Eqs. (3.10) and (3.11) are clearly real, solve the equation of motion, and
induce the correct initial data on the surface $T=0$. Thus they give the
desired solutions. We also describe an alternate construction of the same
solution in Appendix A.1 by using a plane-wave basis.

The important feature of Eqs. (3.10), (3.11) is that they display power law
behaviour near the horizons. Furthermore, for generic values of $m$,
sufficiently high derivatives of $\phi(U,V)$ diverge. This can already be seen
from the initial data at $T=0$ in Eq. (3.5), which due to the time-symmetry of
our solutions is closely related to the data on the horizons $U=0$ and $V=0$.
A similar feature will be true of the metric in the fixed-area states studied
in Sec. 4.4, where it will lead to divergent tidal forces.

###  3.2 Massive Field

We now consider the case of a real massive scalar with a mass $\mu$ in a
“fixed-area” state in 1+1 dimensional Minkowski space. This example will play
a key role in understanding higher dimensional examples in the presence of
gravity. In particular, as we will see later, the equations near the horizon
for a scalar field coupled to gravity in $AdS$ behave similar to the example
studied below. Here the various coordinates and both the Euclidean and
Lorentzian metrics are chosen to be the same as in Sec. 3.1.

The equation of motion for the massive field is given by

$\displaystyle(\nabla^{2}-\mu^{2})\phi=0.$ (3.12)

In Euclidean signature, we impose the following boundary conditions at
$r=r_{c}$,

$\displaystyle\phi(r_{c},\theta)=2I_{k/m}(\mu r_{c})\cos(k\theta).$ (3.13)

The regular solution at the origin consistent with this boundary condition is

$\displaystyle\phi(r,\theta)=2I_{k/m}(\mu r)\cos(k\theta).$ (3.14)

In much the same way as in the massless case, by changing coordinates to
$(U,V)$ and performing an analytic continuation one may show the Lorentzian
solution in the Rindler wedges to be given by

$\displaystyle\phi(U,V)=I_{k/m}(\mu\sqrt{-UV})\left((-V/U)^{\frac{k}{2m}}+(-U/V)^{\frac{k}{2m}}\right),$
$\displaystyle V\geq 0,U\leq 0,$
$\displaystyle\phi(U,V)=(-1)^{k}I_{k/m}(\mu\sqrt{-UV})\left((-V/U)^{\frac{k}{2m}}+(-U/V)^{\frac{k}{2m}}\right),$
$\displaystyle U\geq 0,V\leq 0,$ (3.15)

where $I_{\nu}(x)$ is the modified Bessel function of the first kind. Note
that as $\mu\to 0$, these solutions approach the massless solutions Eq. (3.6)
and Eq. (3.7) (up to a multiplicative constant associated with the different
normalizations of Eq. (3.3) and Eq. (3.13)).

Given the solutions in the Rindler wedges, we can solve the equations of
motion to extend the solution into regions II and IV. In fact, this example is
simple enough that we can simply guess the solution. This is what we do below.
But the interested reader may consult Appendix A.2 for a derivation of the
result along the line $X=0$ performed by expanding the solution in terms of
plane waves.

If we guess that solutions in regions II and IV have the form
$(V/U)^{\frac{k}{2m}}G(\sqrt{UV})$, then the equation of motion sets the
function $G$ to be a linear combination of Bessel functions
$J_{k/m}(\mu\sqrt{UV})$ and $Y_{k/m}(\mu\sqrt{UV})$. One can check that only
$J_{k/m}(\mu\sqrt{UV})$ is consistent with continuity of the solution across
the horizons (as $U,V\to 0$), and thus with the absence of delta-function
sources in the equations of motion. The solution is thus given by

$\displaystyle\phi(U,V)=J_{k/m}(\mu\sqrt{UV})\left((V/U)^{\frac{k}{2m}}+(-1)^{k}(U/V)^{\frac{k}{2m}}\right)\qquad
U\geq 0,V\geq 0,$ (3.16)
$\displaystyle\phi(U,V)=J_{k/m}(\mu\sqrt{UV})\left((U/V)^{\frac{k}{2m}}+(-1)^{k}(V/U)^{\frac{k}{2m}}\right)\qquad
U\leq 0,V\leq 0.$ (3.17)

Note that in (for example) region II, the limit $UV\to 0$ gives $\phi(U,V)\sim
V^{k/m}+(-1)^{k}U^{k/m}$ which coincides with Eq. (3.10). So for the types of
sources we have considered, the leading behaviour of the solution near the
lightcone is determined by the massless solution. In particular, the power law
divergences found in Sec. 3.1 are not tied to the massless nature of that
example, but are generic for all masses. We will find a similar feature to be
true in arbitrary theories of gravity coupled to matter.

##  4 Fixed-area states in gravity

We now turn our attention to the Lorentzian geometry of fixed-area states in
the presence of dynamical gravity. We will show how to obtain the Lorentzian
metric from the Euclidean path integral which prepares the fixed-area state.
In Sec. 4.1, we begin with a discussion of the general structure of fields as
power series expansions near the HRT surface in both Euclidean and Lorentzian
signature. Then, we illustrate the prescription for constructing the Lorentz-
signature fixed-area state solutions in two simple examples. In Sec. 4.2, we
construct exact fixed-area solutions in JT gravity coupled to a scalar field.
In Sec. 4.3, we construct fixed-area solutions in AdS3 gravity coupled to a
scalar field by including the effects of backreaction perturbatively. This
example illustrates the generic features of fixed-area states in higher
dimensional gravity since it arises from dimensional reduction. Thus, it
complements the example in JT gravity. Finally, Sec. 4.4 discusses
singularities on the horizons of fixed-area states in the presence of
arbitrary boundary sources.

###  4.1 General Lorentzian solutions

Before discussing the structure of the Lorentzian solutions, let us first
review the structure of Euclidean conical solutions as analyzed in Ref.
Dong:2019piw . The metric near the codimension-2 defect at the HRT surface
$\gamma_{R}$ takes the form

$\displaystyle ds^{2}$ $\displaystyle=dzd\bar{z}+T\frac{(\bar{z}dz-
zd\bar{z})^{2}}{z\bar{z}}+h_{ij}dy^{i}dy^{j}+2iW_{j}dy^{j}(\bar{z}dz-
zd\bar{z}),$ (4.1)

where $y^{i}$ denotes the transverse directions and $(z,\bar{z})$ represent
the normal directions to $\gamma_{R}$. Note $z$ may be written as
$z=re^{im\theta}$ with $\theta\sim\theta+2\pi$. The quantities $T,h_{ij}$ and
$W_{j}$ are functions of all the coordinates $(z,\bar{z},y^{i})$. For Einstein
gravity minimally coupled to scalar matter with standard two-derivative
actions, the metric components in general have the following power series
expansion in terms of powers of $z,\bar{z}$ in a neighbourhood of
$\gamma_{R}$:

$\displaystyle T$ $\displaystyle=\sum_{\begin{subarray}{c}p,q,s=0\\\
pq>0\;\text{or}\;s>0\end{subarray}}^{\infty}T_{pqs}(y^{i})z^{p/m}\bar{z}^{q/m}(z\bar{z})^{s},$
(4.2) $\displaystyle W_{i}$
$\displaystyle=\sum_{\begin{subarray}{c}p,q,s=0\end{subarray}}^{\infty}W_{i,pqs}(y^{i})z^{p/m}\bar{z}^{q/m}(z\bar{z})^{s},$
(4.3) $\displaystyle h_{ij}$
$\displaystyle=\sum_{\begin{subarray}{c}p,q,s=0\end{subarray}}^{\infty}h_{ij,pqs}(y^{i})z^{p/m}\bar{z}^{q/m}(z\bar{z})^{s},$
(4.4)

where $2\pi m$ is the opening angle of the conical defect at $\gamma_{R}$. Any
scalar matter field $\phi$ has a similar series expansion near the conical
defect:

$\displaystyle\phi$
$\displaystyle=\sum_{p,q,s=0}^{\infty}\phi_{pqs}(y^{i})z^{p/m}\bar{z}^{q/m}(z\bar{z})^{s}.$
(4.5)

In particular, it was demonstrated in Ref. Dong:2019piw  that such a power
series expansion near $\gamma_{R}$ provides a good variational ansatz for
finding Euclidean conical solutions with specified asymptotic boundary
conditions.

Figure 6: The Lorentzian spacetime can be divided into four wedges with
respect to the HRT surface $\gamma_{R}$ labelled I-IV. The power series
expansion we propose is valid in the regions shaded pink and blue. The
solution in the pink region is obtained by analytically continuing the
Euclidean solution. The solution in the blue region is obtained by a simple
transformation of the analytically continued solution.

As discussed previously, the Euclidean solution can be analytically continued
to obtain a solution in each of the Rindler wedges defined by the HRT surface
$\gamma_{R}$. We promote the complex coordinates $z,\bar{z}$ to lightcone
coordinates $U,V$ and assuming that the power series expansion is valid in a
tubular neighbourhood $|z\bar{z}|\leq r_{0}^{2}$ around $\gamma_{R}$, we
obtain a Lorentzian solution in a corresponding hyperboloidal region
$-r_{0}^{2}\leq UV<0$ as shown in Fig. 6 (shaded pink). For any field
$\phi(U,V)$, we obtain a power-series expansion in the Rindler wedges of the
form

$\displaystyle\phi(U,V)=\sum_{p,q,s}\phi_{pqs}V^{p/m}(-U)^{q/m}(-UV)^{s},$
$\displaystyle U\leq 0,V\geq 0,$
$\displaystyle\phi(U,V)=\sum_{p,q,s}(-1)^{p+q}\phi_{pqs}(-V)^{p/m}U^{q/m}(-UV)^{s},$
$\displaystyle U\geq 0,V\leq 0,$ (4.6)

In order to construct the solution in the future and past wedges, we first
demonstrate a simple procedure for generating new solutions to the equations
of motion. Given a solution for a field $\phi(U,V)$ in terms of the
coefficients $\phi_{pqs}$, another solution is given by the set of
coefficients $\tilde{\phi}_{pqs}=e^{i\alpha_{1}p}e^{i\alpha_{2}q}\phi_{pqs}$,
where $\alpha_{1,2}$ are arbitrary constants. This can be easily seen in the
case when $m$ is chosen to be irrational, since terms of the form
$V^{p/m}U^{q/m}$ are linearly independent for different values of $p,q$. In
this case, at any given order the equations of motion generate a relation for
the coefficient $\phi_{pqs}$ in terms of a combination of homogenous product
of coefficients such as $\phi_{p_{1}q_{1}s_{1}}\phi_{p_{2}q_{2}s_{2}}$ with
$p_{1}+p_{2}=p$ and $q_{1}+q_{2}=q$. This makes it clear that multiplying the
coefficients by a phase that linearly depends on $p$ or $q$ doesn’t alter the
defining relation arising from the equation of motion. Thus, the new set of
coefficients $\tilde{\phi}_{pqs}$ also solves the equation of motion.

For rational values of $m$, one can take the limit from irrational $m$ and the
limit is well-behaved as demonstrated in Ref. Dong:2019piw . Since the
Lorentzian equations of motion we are analyzing here are Wick-rotated versions
of the Euclidean equations of motion analyzed there, the same arguments go
through in our case.

In general, this new solution would not be as useful since it is complex while
one is usually interested in a real solution given real initial data. However,
in our problem, this transformation will turn out to be useful in generating
real solutions in the past and future wedges. In particular, we propose that
the solution in these regions take the form:

$\displaystyle\phi(U,V)=\sum_{p,q,s}(-1)^{s+q}\phi_{pqs}V^{p/m}U^{q/m}(UV)^{s},$
$\displaystyle U\geq 0,V\geq 0,$
$\displaystyle\phi(U,V)=\sum_{p,q,s}(-1)^{p+s}\phi_{pqs}(-V)^{p/m}(-U)^{q/m}(UV)^{s},$
$\displaystyle U\leq 0,V\leq 0,$ (4.7)

The above power series can be obtained from Eq. (4.1) by applying a phase
rotation to the coefficients that is linear in $p,q$ as described above. This
ensures that it solves the equations of motion. It is also easy to check that
this ansatz agrees with Eq. (4.1) on all horizons, and so satisfies the
desired initial data on these null surfaces. Uniqueness of the null initial
value problem then guarantees that the correct solution will be of this form.
Indeed, one can directly check that the solutions in Sec. 3 take this form. We
now demonstrate this procedure in more detail in various explicit examples in
JT gravity and $AdS_{3}$.

###  4.2 Example 1: JT Gravity + Matter

We begin with an example involving JT gravity minimally coupled to a massless
real scalar field. We use $\Phi$ to represent the dilaton in order to
distinguish it from the matter field $\phi$. This case is sufficiently simple
that the equations of motion can be solved exactly with arbitrary boundary
sources Almheiri:2014cka . We first solve for the dilaton profile in Euclidean
signature, and then obtain the Lorentzian solution using the initial value
formulation.

The Euclidean metric for a fixed-area state in AdS2 is

$\displaystyle ds^{2}=\frac{dzd\bar{z}}{(1-z\bar{z}/4)^{2}},$ (4.8)

where $z=re^{im\theta}$ and $\bar{z}=re^{-im\theta}$ with $m$ chosen to
satisfy boundary conditions at Euclidean infinity. On the cutoff surface
$z\bar{z}=r_{c}^{2}=4(1-m\epsilon)^{2}$ where $\epsilon$ is small, we impose a
boundary condition on the matter field $\phi$,

$\displaystyle\phi=2r_{c}^{k/m}\cos(k\theta).$ (4.9)

The equation of motion for the massless scalar field is invariant under
conformal transformations and thus, the solution is the same as the solution
on a flat cone in Sec. 3.1. Thus, we obtain the solution

$\displaystyle\phi=z^{k/m}+\bar{z}^{k/m}.$ (4.10)

The stress tensor for this solution is

$\displaystyle T_{zz}(z)=(k/m)^{2}z^{2(k/m-1)},\quad
T_{\bar{z}\bar{z}}(\bar{z})=(k/m)^{2}\bar{z}^{2(k/m-1)},\quad T_{z\bar{z}}=0,$
(4.11)

and the dilaton equation of motion is Almheiri:2014cka

$\displaystyle-(1-z\bar{z}/4)^{-2}\partial_{\mu}\left((1-z\bar{z}/4)^{2}\partial_{\nu}\Phi\right)=T_{\mu\nu}.$
(4.12)

We thus find

$\displaystyle\Phi$
$\displaystyle=\int^{z}dx\frac{(1-\bar{z}x/4)(x-z)}{(1-z\bar{z}/4)}T_{zz}(x)+\int^{\bar{z}}dx\frac{(1-zx/4)(x-\bar{z})}{(1-z\bar{z}/4)}T_{\bar{z}\bar{z}}(x)+\Phi_{0}$
$\displaystyle=(z^{2k/m}+\bar{z}^{2k/m})\frac{(k/m)\left((1+2k/m)+(1-2k/m)z\bar{z}/4)\right)}{2(1-4k^{2}/m^{2})(1-z\bar{z}/4)}+\Phi_{0},$
(4.13)

where $\Phi_{0}$ can be any solution to pure JT gravity without matter, i.e.,

$\displaystyle\Phi_{0}=\frac{a+dz\bar{z}}{1-z\bar{z}},$ (4.14)

for any constants $a$ and $d$. Note that the ambiguity in choosing the lower
limits of the integrals can be absorbed into $\Phi_{0}$, and possible linear
terms of the form $bz+c\bar{z}$ in the numerator of $\Phi_{0}$ are discarded
since they do not respect the $\mathbb{Z}_{2}$ symmetry and periodicity
$\theta\to\theta+2\pi$.

We choose to study an example where the pure gravity solution is the fixed-
area state defined by a Euclidean black hole at inverse temperature $\beta$,
so that Almheiri:2014cka

$\displaystyle\Phi_{0}=m\Phi_{b}\frac{1+z\bar{z}/4}{1-z\bar{z}/4},$ (4.15)

with $\beta/\epsilon$ the length of the boundary curve and $\Phi_{b}/\epsilon$
the boundary dilaton value.999One may observe that, in our full solution
(4.2), as we approach the cutoff surface $z\bar{z}\to r_{c}$, the dilaton
$\Phi$ does not approach a constant. This may appear inconsistent with the
usual boundary condition $\Phi|_{r=r_{c}}=const$ in JT gravity. However, we
can instead choose to reformulate the boundary conditions by finding a surface
on which the dilaton is in fact constant in our solution and then using the
scalar field profile on the given surface as the boundary condition we impose
for the scalar field. We simply choose the above formulation for convenience.

The next step is to inspect the initial value for the dilaton on the $T=0$
surface, which will then be used to construct the Lorentzian solution. The
$\mathbb{Z}_{2}$ symmetric slice is given by the union of the surfaces
$\theta=0$ and $\theta=\pi$. For $0<X<2$ we take $X=r$, and for $-2<X<0$ we
take $X=-r$. The initial value for $\Phi$ is thus

$\displaystyle\Phi|_{T=0}(X)=$
$\displaystyle\left(X^{2k/m}\Theta(X)+(-X)^{2k/m}\Theta(-X)\right)\frac{(k/m)\left((1+2k/m)+(1-2k/m)X^{2}/4\right)}{(1-4k^{2}/m^{2})(1-X^{2}/4)}$
$\displaystyle+m\Phi_{b}\frac{1+X^{2}/4}{1-X^{2}/4}.$ (4.16)

In the Lorentzian spacetime, the metric may always be written in the form

$\displaystyle ds^{2}=-\frac{dUdV}{(1+UV/4)^{2}},$ (4.17)

with the asymptotic boundaries being located at $UV=-4$.

We start by finding the solution for the scalar field. Since this field
satisfies the massless wave equation, and since this wave equation is
conformally invariant, the result is identical to that in the non-
gravitational example in Sec. 3.1, i.e.,

$\displaystyle\phi(U,V)$ $\displaystyle=V^{k/m}+(-U)^{k/m},$
$\displaystyle\text{ (Region I) }:V>0\text{ and }U<0,$ (4.18)
$\displaystyle\phi(U,V)$ $\displaystyle=V^{k/m}+(-1)^{k}(U)^{k/m},$
$\displaystyle\text{ (Region II) }:V>0\text{ and }U>0,$ (4.19)
$\displaystyle\phi(U,V)$
$\displaystyle=(-1)^{k}\left((-V)^{k/m}+U^{k/m}\right),$ $\displaystyle\text{
(Region III) }:V<0\text{ and }U>0,$ (4.20) $\displaystyle\phi(U,V)$
$\displaystyle=(-1)^{k}(-V)^{k/m}+(-U)^{k/m},$ $\displaystyle\text{ (Region
IV) }:V<0\text{ and }U<0.$ (4.21)

This yields the stress tensor

$\displaystyle T_{UU}(U)$
$\displaystyle=(k/m)^{2}\left(U^{2(k/m-1)}\Theta(U)+(-U)^{2(k/m-1)}\Theta(-U)\right),$
(4.22) $\displaystyle T_{VV}(V)$
$\displaystyle=(k/m)^{2}\left(V^{2(k/m-1)}\Theta(V)+(-V)^{2(k/m-1)}\Theta(-V)\right),$
(4.23) $\displaystyle T_{UV}$ $\displaystyle=0.$ (4.24)

Following Ref. Almheiri:2014cka , the dilaton solution in the Lorentzian
spacetime can then be computed as

$\displaystyle\Phi=\int^{U}dx\frac{(1+Vx/4)(U-x)}{(1+UV/4)}T_{UU}(x)+{\int^{V}}dx\frac{(1+Ux/4)(V-x)}{(1+UV/4)}T_{VV}(x)+\tilde{\Phi}_{0},$
(4.25)

where

$\displaystyle\tilde{\Phi}_{0}=\frac{A+BU+CV+DUV}{1+UV/4}.$ (4.26)

The full Lorentzian solution can then be constructed by performing the
integrals in Eq. (4.25) and choosing integration constants to the prescribed
initial values from Eq. (4.2). Doing so leads us to the result

$\displaystyle\Phi=f(U,V)\frac{(k/m)\left((1+2k/m)-(1-2k/m)UV/4\right)}{2(1-4k^{2}/m^{2})(1+UV/4)}+m\Phi_{b}\frac{1-UV/4}{1+UV/4},$
(4.27)

where

$\displaystyle f(U,V)$ $\displaystyle=V^{2k/m}+(-U)^{2k/m},\quad$
$\displaystyle\text{(Region I):}~{}U<0,V>0$ (4.28) $\displaystyle f(U,V)$
$\displaystyle=V^{2k/m}+U^{2k/m},\quad$ $\displaystyle\text{(Region
II):}~{}U>0,V>0$ (4.29) $\displaystyle f(U,V)$
$\displaystyle=(-V)^{2k/m}+U^{2k/m},\quad$ $\displaystyle\text{(Region
III):}~{}U>0,V<0$ (4.30) $\displaystyle f(U,V)$
$\displaystyle=(-V)^{2k/m}+(-U)^{2k/m},\quad$ $\displaystyle\text{(Region
IV):}~{}U<0,V<0.$ (4.31)

One can then verify that the above expressions solve the equations of motion,
and one can also check that the solution satisfies the general power series
expansion described in Sec. 4.1. Importantly, despite the appearance of
singularities on the lightcone of the HRT surface, the solution is smooth in
the interior of both the past and future wedges.

###  4.3 Example 2: $AdS_{3}$ coupled to a massless scalar field

We now consider an example of a fixed-area state geometry for $AdS_{3}$
gravity minimally coupled to a real scalar field. The corresponding Euclidean
problem was analyzed in detail in Ref. Lewkowycz:2013nqa  for the case $m=1/n$
with $n\in\mathbb{Z^{+}}$ where it was interpreted as a Euclidean replica
geometry. Here we extend the calculation in Ref. Lewkowycz:2013nqa  to conical
defects with arbitrary strength $m$ and focus on the Lorentzian solution whose
initial data agrees with the Euclidean solution on the surface of time-
symmetry.

An interesting feature of this example is the connection to gravity in higher
dimensions. As mentioned in Ref. Lewkowycz:2013nqa , solutions of the form
$AdS_{3}\times S^{3}\times T^{4}$ give rise to three dimensional gravity
coupled to two massless real scalar fields. One obtains scalar fields
$\phi_{1}$ and $\phi_{2}$ from the metric components of $T^{4}$ when
parametrized in the form

$\displaystyle
ds^{2}_{T^{4}}=e^{2\phi_{1}}dy_{1}^{2}+e^{2\phi_{2}}dy_{2}^{2}+e^{-2\phi_{1}}dy_{3}^{2}+e^{-2\phi_{2}}dy_{4}^{2}.$
(4.32)

Studying the fixed-area state solution of the scalar field and metric
components in this example is thus directly related to fixed-area states in
pure gravity in higher dimensions. In particular, as we will discuss, the
Lorentz-signature singularities of the scalar field that arise from the
presence of the conical defect in the Euclidean preparation correspond to
geometric singularities (and generally to Weyl curvature singularities) of the
higher-dimensional gravity theories. For simplicity and clarity, we will set
$\phi_{2}=0$ and denote $\phi=\phi_{1}$ as the only relevant scalar field.

We now turn to the actual preparation of our fixed-area state solution. As
usual, we start in Euclidean signature and use the result to determine the
initial data for the Lorentzian scalar field and metric. Following Ref.
Lewkowycz:2013nqa , we solve the Euclidean equation of motion for the massless
real scalar field and the metric perturbatively by assuming that the source
for the scalar field is small. To zeroth order, the metric of Euclidean
$AdS_{3}$ with a conical defect of strength $m$ is given by

$\displaystyle
ds^{2}=\frac{dr^{2}}{r^{2}+1}+m^{2}r^{2}d\theta^{2}+m^{2}(r^{2}+1)dy^{2},$
(4.33)

where $\theta\in[0,2\pi)$. Here we set $l_{AdS}=1$. The transverse coordinate
$y$ does not play a role for the class of solutions considered here and can be
chosen to be either compact or non-compact. The factor of $m^{2}$ in the last
term in Eq. (4.33) is chosen to ensure that when $y$ is compact with fixed
period, the conformal structure of the boundary torus is independent of $m$.

We turn on a non-trivial source for the scalar field by imposing the following
boundary condition at a cutoff surface $r=r_{c}$,

$\displaystyle\left.\phi\right|_{r=r_{c}}=2\eta\cos(k\theta).$ (4.34)

The equation we are solving is

$\displaystyle\nabla_{\mu}\nabla^{\mu}\phi=0,$ (4.35)

and the solution is then given by

$\displaystyle\phi=2\eta\cos(k\theta)\frac{f_{m,k}(r)}{f_{m,k}(r_{c})},\qquad
f_{m,k}(r)=r^{k/m}{{}_{2}F_{1}}\left(\frac{k}{2m},\frac{k}{2m}+1,\frac{k}{m}+1,-r^{2}\right).$
(4.36)

Using the solution for the scalar field, the stress tensor can be decomposed
in terms of Fourier modes,

$\displaystyle T_{\mu\nu}$
$\displaystyle=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{g_{\mu\nu}}{2}(\nabla\phi)^{2}$
(4.37)
$\displaystyle=T^{0}_{\mu\nu}+T_{\mu\nu}^{+}e^{2ik\theta}+T_{\mu\nu}^{-}e^{-2ik\theta}.$
(4.38)

We can then use the above result to solve the metric equations at leading
order in $\eta$. Following the ansatz in Ref. Lewkowycz:2013nqa , we write the
perturbed metric as

$\displaystyle
ds^{2}=\frac{dr^{2}}{r^{2}+1+g(r,\theta)}+m^{2}r^{2}d\theta^{2}+m^{2}(r^{2}+1)(1+v(r,\theta))dy^{2},$
(4.39)

where $g(r,\theta),v(r,\theta)$ consist of three fourier modes in the angular
direction, e.g.,

$\displaystyle
g(r,\theta)=g_{0}(r)+g^{+}(r)e^{2ik\theta}+g^{-}(r)e^{-2ik\theta}.$ (4.40)

The Einstein equations after dimensional reduction are then given by,

$\displaystyle R_{\mu\nu}-\frac{g_{\mu\nu}}{2}(R+2)=8\pi GT_{\mu\nu},\qquad
T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{g_{\mu\nu}}{2}(\nabla\phi)^{2}.$
(4.41)

By using the ansatz in Eq. (4.39), one can find integral expressions for
$g(r,\theta)$ and $v(r,\theta)$. However, the closed form solutions for
$v(r,\theta)$ or $g(r,\theta)$ will not be important for the discussion of the
Lorentzian evolution. The interested reader may refer to Appendix C for more
details.

So far, we have considered the equations and solution in Euclidean signature.
In order to write down the Lorentzian solution, we follow identical steps to
Sec. 3. Let us first discuss the undeformed metric in the Lorentzian
signature. The initial slice is given by the union of the slices $\theta=0$
and $\theta=\pi$. By continuing $\theta\to\theta+it$, on the $\theta=0,\pi$
slices we find

$\displaystyle
ds_{L}^{2}=-m^{2}r^{2}dt^{2}+\frac{dr^{2}}{r^{2}+1}+m^{2}(r^{2}+1)dy^{2}.$
(4.42)

By rescaling the coordinates, the metric can be written in the more suggestive
form

$\displaystyle
ds_{L}^{2}=-r^{2}{d\tilde{t}}^{2}+\frac{dr^{2}}{r^{2}+1}+(r^{2}+1)d\tilde{y}^{2},\qquad\tilde{t}=mt,\qquad\tilde{y}=my,$
(4.43)

which is just the familiar Lorentz signature $AdS_{3}$ metric with a rescaled
transverse coordinate $y\to my$. In particular, the geometry in Eq. (4.43) is
manifestly smooth in Lorentz signature.

In order to make the Lorentzian solution more explicit, let us write the
undeformed metric (4.42) in global coordinates

$\displaystyle
ds_{L}^{2}=-\frac{4dUdV}{(1+UV)^{2}}+m^{2}\left(\frac{1-UV}{1+UV}\right)^{2}dy^{2},$
(4.44)

where

$\displaystyle
U=-\frac{r}{1+\sqrt{1+r^{2}}}e^{-mt},\;V=\frac{r}{1+\sqrt{1+r^{2}}}e^{mt},$
(4.45)

are the global null coordinates. As in the discussion of Sec. 3, the
Lorentzian solution for the scalar field inside the “Rindler” wedges is
straightforward to obtain by analytically continuing Eq. (4.36) to find

$\displaystyle\phi=\frac{\eta}{f_{m,k}(r_{c})}\;\frac{(-2U)^{k/m}+(2V)^{k/m}}{\left(1+UV\right)^{k/m}}{{}_{2}F_{1}}\left(\frac{k}{2m},\frac{k}{2m}+1,\frac{k}{m}+1,\frac{4UV}{(1+UV)^{2}}\right),$
$\displaystyle U<0,\;V>0,$
$\displaystyle\phi=\frac{\eta}{f_{m,k}(r_{c})}\;(-1)^{k}\frac{(2U)^{k/m}+(-2V)^{k/m}}{\left(1+UV\right)^{k/m}}{{}_{2}F_{1}}\left(\frac{k}{2m},\frac{k}{2m}+1,\frac{k}{m}+1,\frac{4UV}{(1+UV)^{2}}\right),$
$\displaystyle U>0,\;V<0.$ (4.46)

We can now obtain the solution in the past and future wedges by solving the
equations of motion and ensuring continuity of the fields across the horizons.
Doing so yields

$\displaystyle\phi=\frac{\eta}{f_{m,k}(r_{c})}\;\frac{(-2U)^{k/m}+(-1)^{k}(-2V)^{k/m}}{\left(1+UV\right)^{k/m}}{{}_{2}F_{1}}\left(\frac{k}{2m},\frac{k}{2m}+1,\frac{k}{m}+1,\frac{4UV}{(1+UV)^{2}}\right),$
$\displaystyle U<0,\;V<0,$
$\displaystyle\phi=\frac{\eta}{f_{m,k}(r_{c})}\;\frac{(-1)^{k}(2U)^{k/m}+(2V)^{k/m}}{\left(1+UV\right)^{k/m}}{{}_{2}F_{1}}\left(\frac{k}{2m},\frac{k}{2m}+1,\frac{k}{m}+1,\frac{4UV}{(1+UV)^{2}}\right),$
$\displaystyle U>0,\;V>0.$ (4.47)

One can check that the above expressions are consistent with the general power
series expansion described in Sec. 4.1. Again, in generic situations, we find
power law divergences on the lightcones.

One can also analyze the behaviour of the Euclidean metric components
$g(r,\theta),v(r,\theta)$ near $r=0$ based on the analysis in Appendix C. The
dependence on $r,\theta$ immediately gives us the dependence on $U,V$ in a
near-horizon region of the Rindler wedges ($UV<0$) by analytic continuation.
We find power-series expansions of the form

$\displaystyle
g^{\pm}(U,V)=(-UV)^{k/m}\sum_{n=1}^{\infty}g^{\pm}_{n}(-UV)^{n},\qquad
g^{0}(U,V)=(-UV)^{k/m}\sum_{n=0}^{\infty}g^{0}_{n}(-UV)^{n}$ (4.48)
$\displaystyle
v^{\pm}(U,V)=(-UV)^{k/m}\sum_{n=0}^{\infty}v^{\pm}_{n}(-UV)^{n},\qquad
v^{0}(U,V)=(-UV)^{k/m}\sum_{n=1}^{\infty}v^{0}_{n}(-UV)^{n}$ (4.49)

where $g^{\pm,0}_{n},v^{\pm,0}_{n}$ are coefficients found by solving the
Euclidean equations of motion, Eq. (C.6) – Eq. (C.11), in a power series
expansion near $r=0$. Given these solutions in the Rindler wedges, the by-now-
familiar ansatz defined by Eq. (4.1) and Eq. (4.1) gives the solutions in the
past and future wedges. In particular, when $UV>0$ we find the solutions have
the following form

$\displaystyle
g^{\pm}(U,V)=(UV)^{k/m}\sum_{n=1}^{\infty}g^{\pm}_{n}(-UV)^{n},\qquad
g^{0}(U,V)=(UV)^{k/m}\sum_{n=0}^{\infty}g^{0}_{n}(-UV)^{n}$ (4.50)
$\displaystyle
v^{\pm}(U,V)=(UV)^{k/m}\sum_{n=0}^{\infty}v^{\pm}_{n}(-UV)^{n},\qquad
v^{0}(U,V)=(UV)^{k/m}\sum_{n=1}^{\infty}v^{0}_{n}(-UV)^{n}$ (4.51)

We can check this by plugging the stress tensor inside the horizon regions and
solving for the metric components directly in Lorentzian signature.

###  4.4 Divergence structure of fixed-area states

As we saw in the examples above, for general sources at the Euclidean
boundary, fixed-area states lead to power-law divergences on the codimension-2
conical defects. Similar divergences then also appear on the lightcones
emanating from the HRT surface in the Lorentzian spacetime. We now generalize
these results to arbitrary dimension and study the divergence structure of the
general solution.

####  4.4.1 Euclidean signature

As usual, we begin by studying the divergence structure in Euclidean
signature. We consider Einstein-Hilbert gravity coupled to classical scalar
fields. We expect other classical matter fields to give similar results.

In quasi-cylindrical coordinates the metric near the codimension-2 defect can
be always be written as Eq. (4.1),

$\displaystyle ds^{2}=dzd\bar{z}+$ $\displaystyle T\frac{(\bar{z}dz-
zd\bar{z})^{2}}{z\bar{z}}+h_{ij}dy^{i}dy^{j}+2iW_{j}dy^{j}(\bar{z}dz-
zd\bar{z}),$ (4.52)

where $T,h_{ij},W_{j}$ are functions of all coordinates $(z,\bar{z},y^{i})$.
The metric components have series expansions in terms of powers of
$z^{1/m},\bar{z}^{1/m}$ and $z\bar{z}$ near $r=0$ as given in Eq. (4.2)-(4.5).

As noted before, in the presence of a $U(1)$ symmetry, all fields are smooth
near $r=0$ and there is no singularity in any derivative of the fields.
Furthermore, when $m=\frac{1}{n}$ for integer $n$ there is a smooth $n$-fold
cover (as in the Lewkowycz-Maldacena discussion of gravitational Renyi
entropies Lewkowycz:2013nqa ), so again there are no power law divergences
(even in the quotient). But more generally we will find divergences. Thus, in
the following discussion, we consider a generic situation where $\frac{1}{m}$
is not an integer and where every coefficient in the power-series expansion of
any metric component and matter field is non-zero.

Analyzing the Christoffel symbols we find

$\displaystyle\Gamma_{\mu\nu}^{\rho}\sim r^{1/m-1},$ (4.53)

where we use the $\sim$ notation to keep track of the leading, non-smooth term
appearing in various quantities like metric components, derivatives of the
matter field, etc. In the case $m>1$, the explicit $r^{1/m-1}$ term in fact
represents the most singular term in the Christoffel symbols. For the Riemann
tensor, there are terms involving the square of Christoffel symbols and terms
involving the second derivatives of the metric. From Eq. (4.53), it is easy to
see that the former terms at most are of order $r^{2/m-2}$. It turns out that
the latter terms give the most singular terms in the Riemann tensor. In order
to see this, let us define

$\displaystyle A_{\mu\nu\rho\sigma}\equiv
R_{\mu\nu\rho\sigma}-\Gamma_{\lambda\mu\sigma}\Gamma^{\lambda}_{\nu\rho}+\Gamma_{\lambda\mu\rho}\Gamma^{\lambda}_{\nu\sigma}.$
(4.54)

We find

$\displaystyle
A_{z\bar{z}z\bar{z}}=T_{,z\bar{z}}+\frac{1}{2}\left(T\frac{z}{\bar{z}}\right)_{,zz}+\frac{1}{2}\left(T\frac{\bar{z}}{z}\right)_{,\bar{z}\bar{z}}\sim
r^{\frac{2}{m}-2},$ (4.55)

where we have used

$\displaystyle
T=T_{110}(z\bar{z})^{1/m}+T_{101}\bar{z}z^{1/m+1}+T_{011}\bar{z}^{1/m+1}z+\cdots.$
(4.56)

Similarly we find

$\displaystyle
A_{z\bar{z}zj}=\frac{1}{2}\left(i(W_{j}\bar{z})_{,z\bar{z}}+i(W_{j}z)_{,zz}-T_{,zi}-\left(T\frac{\bar{z}}{z}\right)_{,\bar{z}i}\right)\sim
r^{1/m-1},$ (4.57)

and

$\displaystyle A_{z\bar{z}ij},A_{ijkl}\sim r^{1/m},$ (4.58)

while

$\displaystyle A_{aijk}\sim r^{1/m-1},$ (4.59)

where $a=z,\bar{z}$. The most singular term comes from $h_{ij,ab}$ and is
given by

$\displaystyle
A_{aibj}=\frac{1}{2}\left(g_{aj,ib}+g_{ib,aj}-g_{ab,ij}-g_{ij,ab}\right)\sim
r^{1/m-2}.$ (4.60)

So in this case the most singular component of the Riemann tensor is

$\displaystyle R_{aibj}\sim\frac{r^{1/m}}{r^{2}}.$ (4.61)

This behavior can be confirmed for the ten-dimensional Riemann tensor in the
example of $AdS_{3}\times S^{3}\times T^{4}$ discussed in Sec. 4.3. In that
case, the coefficient of Eq. (4.61) contains $(\frac{1}{m}-1)$ and therefore
when $m=1$ there is no singularity.

The degree of singularity in Eq. (4.61) naively implies that the Ricci tensor
has similar singularities. For instance $R_{zz}$ contains terms like
$h^{ij}\partial_{z}^{2}h_{ij}$ which naively can be as singular as
$r^{1/m-2}$. However, it was shown in Ref. Dong:2019piw  for pure gravity with
a cosmological constant that solving the Einstein equations sets

$\displaystyle h_{,000}^{ij}h_{ij,100}=h_{,000}^{ij}h_{ij,010}=0.$ (4.62)

This means that due to the equation of motion, the leading term for the Ricci
tensor must be of form $r^{2/m-2}$ which are less singular than terms in
$R_{zizj}$. If the matter coupled to gravity is classical, we expect that
$R_{\mu\nu}\sim T_{\mu\nu}$ and
$T_{\mu\nu}\sim\partial_{\mu}\psi\partial_{\nu}\psi$ where $\psi\sim r^{1/m}$
near $r=0$. Therefore in this case, the most singular terms that scale as
$r^{1/m-2}$ must be in the Weyl tensor and not the Ricci tensor.

####  4.4.2 Lorentzian signature

Generalizing the analysis of divergences to Lorentzian signature is now
straightforward. Continuing $z\rightarrow V,\bar{z}\rightarrow-U$, the
Lorentzian metric takes the form

$\displaystyle ds^{2}=-dUdV-$ $\displaystyle T\frac{(VdU-
UdV)^{2}}{UV}+h_{ij}dy^{i}dy^{j}+2iW_{j}dy^{j}(VdU-UdV),$ (4.63)

where since $T$ vanishes on the horizons $V=0$ and $U=0$, we see that $U$ and
$V$ asymptotically become affine parameters as one approaches either horizon.
We now repeat the analysis of the components of the Riemann tensor. The main
difference from the Euclidean analysis is that $U,V$ are independent. As a
result, we consider the derivative of the metric as $U\to 0$ for a fixed $V$
(and similarly $V\to 0$ for fixed $U$). Note that Eq. (4.1) was originally an
expansion for the metric in $r=\sqrt{z\bar{z}}$. A point with a fixed $V$ and
$U\to 0$ eventually ends up in the small $UV$ region where Eq. (4.1) is a
valid expansion. We use this expansion in the Lorentzian signature to find the
leading singularity as $U\to 0$. However, the dependence on $V$ at fixed $V$
can be arbitrary and we only keep track of powers of $U$.

We find:

$\displaystyle A_{VUVU}\sim U^{1/m-1},\qquad A_{VUVj}\sim U^{1/m},\qquad
A_{VUUj}\sim U^{1/m-1},$ $\displaystyle A_{UVij}\sim U^{1/m},\qquad
A_{ijkl}\sim U^{1/m},\qquad A_{Vijk}\sim U^{1/m},\qquad A_{Uijk}\sim
U^{1/m-1}$ $\displaystyle A_{UiUj}\sim U^{1/m-2},\qquad A_{UiVj}\sim
U^{1/m-1},\qquad A_{ViVj}\sim U^{1/m}.$ (4.64)

Therefore, the most singular components of the Riemann tensor are
$R_{UiUj}\sim U^{1/m-2}$.101010As discussed previously, if $m=1/n$ for integer
$n$, the singular terms in the Riemann tensor vanish.

Although the Riemann tensor itself can be divergent at the horizon (depending
on $m$), the displacement of nearby geodesics passing through the horizon will
be negligible. This can be seen if we integrate the geodesic deviation
equation near $U=0$ twice to find the tidal displacement for nearby geodesics.
The most singular term in the displacement goes as $U^{1/m}$ which vanishes
for $U=0$. Thus, we see that fixed-area states have relatively mild
divergences when working in the classical limit.

##  5 Summary and Discussion

Our work above studied the spacetime geometry intrinsic to fixed-area states
at leading order in the bulk Newton constant $G$. While the saddle point
geometries typically used to prepare such states contain conical
singularities, they represent sources involved in the preparation and are not
part of the fixed-area spacetime itself. Instead, the fixed-area spacetimes
satisfy the usual equations of motion without conical singularities.

With either fine-tuning or enough symmetry, the fixed-area spacetimes can be
completely smooth at leading order in $G$. More generally, however,
derivatives of fields may diverge on null congruences fired orthogonally from
the fixed-area surface. In particular, as described in Sec. 4.4, for states
defined by cutting open Euclidean path integrals without a $U(1)$ symmetry,
one typically finds the curvature tensor to diverge as $U^{1/m-2}$ as these
null congruences are approached, where $U$ is the affine null parameter
orthogonal to the null congruence and $2\pi m$ is the opening angle of the
Euclidean saddle that prepares the fixed-area state. The singularities are
integrable, meaning that the total tidal distortion experienced by freely-
falling particles crossing the null congruence is finite. Thus such
singularities need not necessarily destroy infalling observers and, in fact,
so long as the coefficients of such singularities are small the effect on such
observers can be negligible.

Importantly, in our example in Sec. 4.2 we found that the equations of motion
could be solved in a manner that continues the solution beyond the power-law
divergences on the lightcones of the HRT surface. The resulting solution then
had a large smooth region in both the past and future wedges. While these
regions are harder to analyze in higher dimensional contexts, as in the three-
dimensional analysis of section 4.3 they will remain amenable to study via
both standard perturbation theory and a near-horizon power series expansion.
This provides strong evidence that a large smooth region will continue to
exist in both the past and future wedges.111111Though a spacelike singularity
may develop after some proper time since, even before fixing the area, we
might consider a state that describes a black hole. Furthermore, while the
singularities we find at the horizons do in principle raise concerns regarding
our control over the effect of any UV corrections on solutions in the past and
future wedges, these concerns can be tamed by smearing out the HRT surface in
the transverse directions and thus effectively introducing a UV cutoff.

Such singularities can be strong enough that they remove the spacetime from
the realm in which the initial value problem for the Einstein-Hilbert gravity
is well-posed. For example, in 3+1 dimensions standard such results require
the curvature to be appropriately square-integrable Klainerman:2012wt ;
Klainerman:2012wy . This is clearly violated for sufficiently large $m$.
However, in our context this may not be a problem as we impose additional
boundary conditions at the fixed-area surface (and, in effect, on the
orthogonal null congruences) adapted from the Euclidean analysis of Ref.
Dong:2019piw . These conditions are chosen to make the Einstein-Hilbert
variational principle well-defined, and one may hope that they similarly
repair the initial value problem. However, we leave the detailed study of such
issues for future work.

Additional singularities will arise once we consider quantum corrections. One
way to see this is to recall the example of a bifurcate Killing horizon in
which the Euclidean saddle had a $U(1)$ symmetry. Such saddles were just
familiar Euclidean black holes with conical singularities inserted at the
horizon so that they could match boundary conditions with some period
$\beta_{0}$ unrelated to the usual inverse temperature $\beta$ of the Lorentz-
signature fixed-area geometry. At the quantum level, this clearly prepares
quantum fields around this black hole in a state of inverse temperature
$\beta_{0}$ which differs from the inverse Hawking temperature $\beta$. This
is well-known to give a singular stress tensor at the black hole horizon, and
in fact the special case $\beta_{0}=\infty$ corresponds to the Boulware vacuum
state for the black hole. The Boulware vacuum is much like the Rindler vacuum
on Rindler space, for which the stress tensor features a quadratic divergence
at the horizon Birrell:1982ix . The associated back-reaction on the metric
would then force the Ricci-tensor to be quadratically divergent as well, so
that general integrated tidal distortions diverge logarithmically. This
suggests that using fixed-area states beyond leading order in $G$ will require
taming this divergence by smearing out the fixed-area surface along the
orthogonal two spacetime dimensions; see also related comments in Ref.
Dong:2019piw . This may also be related to issues regarding quantum
corrections to HRT-areas seen in Ref. Witten:2021unn . We hope to return to
further study of such quantum corrections in the future.

It would also be useful to generalize our results to include perturbative
higher-derivative corrections and non-minimal couplings. In this context, the
area is replaced by a more general geometric entropy functional Dong:2013qoa ;
Camps:2013zua . Nevertheless, in the leading semiclassical approximation,
states of fixed geometric-entropy states are again constructed by using
Euclidean saddles with conical defects Dong:2019piw . The general arguments
described here should thus go through in a similar fashion. In particular,
there is a similar power-series expansion for metric quantities in a conical
defect spacetime in higher-derivative theories Dong:2019piw . This will again
give power-series solutions in the past and future wedges just as described in
Sec. 4.1. However, an important difference in this case is that the power
series expansion involves more singular terms. To resolve this, one may again
consider a smeared version of the fixed geometric-entropy state in order to
obtain reasonable initial data for Lorentzian evolution. It would be
interesting to understand such solutions in greater detail in future.

A final open question involves the states where we fix the area of an HRT
surface $\gamma_{R}$ that is anchored to an asymptotically AdS boundary (say,
at the edges of a boundary subregion $R$). In this case, the area is
divergent. While one can renormalize the HRT-area by subtracting its
expectation value, fluctuations of this renormalized area remain divergent;
see e.g. Marolf:2020vsi . As a result, projecting onto a small window of HRT-
area eigenvalues would remove the state from the CFT Hilbert space.121212The
situation is much like that for the operator $:\phi^{2}(x):$ in the theory of
a free scalar field. A useful notion of fixed-area state in this context will
thus require the introduction of an appropriate boundary UV cutoff.

This then raises the question of how such UV issues will manifest themselves
in the boundary-anchored versions of the calculations described in this work.
One possibility is that, in the absence of a UV regulator, a singular shock
will arise at the boundary anchors and will propagate into the bulk toward
both past and future. On the other hand, related UV concerns arise in the
study of the flow by taking Poisson brackets with the HRT area operator
Bousso:2020yxi ; molly ; xi . But in that context, at least in AdS3, the
behavior turns out to be milder. Indeed, in that context Ref. molly  showed
that the bulk itself remains smooth, and that the CFT singularity is dual only
to a singularity in the manner that the bulk and boundary are attached. It
thus seems likely that boundary-anchored fixed-area geometries will be
similar. A better understanding of the area operator from the CFT perspective,
perhaps along the lines of Ref. Belin:2021htw , may also be useful for
understanding such issues.

###### Acknowledgements.

This material is based upon work by XD, DM, PR, and ZW supported by the Air
Force Office of Scientific Research under award number FA9550-19-1-0360. The
work of DM, PR, and AT was also supported by a grant from the Simons
foundation. Our work was also supported in part by funds from the University
of California.

##  Appendix A Scalar field solution via Fourier expansion

###  A.1 Massless Field

Let us now take a moment to construct the solutions in Eqs. (3.10), (3.11)
directly, without recourse to analytic continuation. As is well-known, the
space of solutions to the Klein-Gordon equation in 1+1 dimensional Minkowski
space has a basis given by plane waves. A general solution $\phi(T,X)$ may
thus be written in the form

$\displaystyle\phi(T,X)=\int_{-\infty}^{+\infty}a(\zeta)e^{i(|\zeta|T-\zeta
X)}d\zeta+\int_{-\infty}^{+\infty}a^{\ast}(\zeta)e^{-i(|\zeta|T-\zeta
X)}d\zeta.$ (A.1)

The initial condition is given by Eq. (3.5):

$\displaystyle\phi_{0}(X)=2X^{k/m}\Theta(X)+2(-1)^{k}(-X)^{k/m}\Theta(-X),\qquad\left.\partial_{T}\phi\right|_{T=0}=0,$
(A.2)

which yields

$\displaystyle a^{\ast}(\zeta)=a(-\zeta),$ (A.3)

and

$\displaystyle
a(\zeta)=\frac{\Gamma\left(\frac{k+m}{m}\right)\left|\zeta\right|^{-\frac{k+m}{m}}\left(-\left((-1)^{k}+1\right)\sin\left(\frac{\pi
k}{2m}\right)+(-i)\left((-1)^{k}-1\right)\text{sgn}(\zeta)\cos\left(\frac{\pi
k}{2m}\right)\right)}{2\pi}.$ (A.4)

This result simplifies for even and odd $k$:

$\displaystyle a(\zeta)=-\frac{\Gamma(1+k/m)\sin(\frac{\pi
k}{2m})}{\pi}|\zeta|^{-k/m-1},$ $\displaystyle k\in 2\mathbb{Z}$ (A.5)
$\displaystyle a(\zeta)=i\frac{\Gamma(1+k/m)\cos(\frac{\pi k}{2m})}{\pi}{\rm
sign}(\zeta)|\zeta|^{-k/m-1},$ $\displaystyle k\in 2\mathbb{Z}+1.$ (A.6)

Here in finding Eq. (A.5) and Eq. (A.6), we rotated the contour of integration
to make the integrals convergent. For $k\in 2\mathbb{Z}$, the field
$\phi(U,V)$ is given by

$\displaystyle\phi(U,V)$
$\displaystyle=\int_{-\infty}^{+\infty}\left(-\frac{\Gamma(1+k/m)\sin(\frac{\pi
k}{2m})}{\pi}|\zeta|^{-k/m-1}\right)\left(e^{i(|\zeta|T-\zeta
X)}+e^{-i(|\zeta|T-\zeta X)}\right)d\zeta$
$\displaystyle=|V|^{k/m}+|U|^{k/m},$ (A.7)

where we used

$\displaystyle\int_{0}^{+\infty}d\zeta\zeta^{-k/m-1}(e^{i\zeta V}+e^{-i\zeta
V})$ $\displaystyle=2|V|^{k/m}\cos(\frac{\pi k}{2m})\Gamma(-k/m).$ (A.8)

In Eq. (A.8), an analytic continuation in $k/m$ is needed to make sense of the
integral. Similarly for odd $k$ we find

$\displaystyle\phi(T,X)$
$\displaystyle=\int_{-\infty}^{+\infty}\left(i\frac{\Gamma(1+k/m)\cos(\frac{\pi
k}{2m})}{\pi}{\rm sign}(\zeta)|\zeta|^{-k/m-1}\right)\left(e^{i(|\zeta|T-\zeta
X)}-e^{-i(|\zeta|T-\zeta X)}\right)d\zeta$ $\displaystyle={\rm
sign}(V)|V|^{k/m}-{\rm sign}(U)|U|^{k/m}.$ (A.9)

Thus, we see that both solutions agree with the solution we found previously
by using appropriate analytic continuations.

###  A.2 Massive Field

In Sec. 3.2, we guessed the solution in the future wedge for the massive
scalar field theory using the separation of variables and continuity of the
solution. Here we derive the same solution directly by performing a Fourier
transform.

Doing the Fourier transform directly on the initial data is not easy in this
case. One way to go around this is to use the integral representation of the
Bessel function,

$\displaystyle
I_{\nu}(x)=-i2^{-\nu-1}x^{\nu}\int\frac{\Gamma(s)}{\Gamma(s+1/2)\Gamma(1/2-s)\Gamma(1+\nu-s)}\left(\frac{x^{2}}{4}\right)^{-s}ds,$
(A.10)

where the integral runs over the imaginary line with a positive real part.

Using this representation, the calculation is almost the same as the massless
case. For simplicity, let’s set $k$ to be even, although the analysis is quite
similar for odd $k$. Defining $V=T+X,U=T-X,\nu\equiv k/m$, we have

$\displaystyle\phi(T=0,X)=2I_{k/m}(\mu|X|)=-i\int\frac{\Gamma(s)}{\Gamma(s+1/2)\Gamma(1/2-s)\Gamma(1+\nu-s)}2^{2s-\nu}|\mu
X|^{\nu-2s}ds.$ (A.11)

As a result,the Fourier coefficients are given by

$\displaystyle
a(\zeta)=\frac{1}{4\pi}\int_{-\infty}^{+\infty}\phi(T=0,X)e^{-i\zeta X}dX$
$\displaystyle=-\frac{i}{2\pi}\int\frac{\Gamma(s)}{\Gamma(s+1/2)\Gamma(1/2-s)\Gamma(1+\nu-s)}2^{2s-\nu}\mu^{\nu-2s}\left(-\Gamma(1+\nu-2s)\sin(\frac{\pi(\nu-2s)}{2})|\zeta|^{-\nu+2s-1}\right)$
$\displaystyle=\int b(s)|\zeta|^{2s-\nu-1},$ (A.12)

where $b(s)\equiv
i\frac{\Gamma(1+\nu-2s)2^{2s-\nu}\sin(\frac{\pi(\nu-2s)}{2})\Gamma(s)}{2\pi\Gamma(s+1/2)\Gamma(1/2-s)\Gamma(1+\nu-s)}\mu^{\nu-2s}$.
Therefore, the field is given by

$\displaystyle\phi(T,X)=\int
ds\,b(s)\int|\zeta|^{2s-\nu-1}\left(e^{i\left(\sqrt{\zeta^{2}+\mu^{2}}T-\zeta
X\right)}+e^{-i\left(\sqrt{\zeta^{2}+\mu^{2}}T-\zeta X\right)}\right)d\zeta.$
(A.13)

This integral is hard to do in general, so for a simpler case, we instead
check that Eq. (A.13) reduces to Eq. (3.16) when $X=0$. In this case we have

$\displaystyle\phi(T,0)$ $\displaystyle=\int
b(s)ds\int|\zeta|^{2s-\nu-1}\left(e^{i\left(\sqrt{\zeta^{2}+\mu^{2}}T\right)}+e^{-i\left(\sqrt{\zeta^{2}+\mu^{2}}T\right)}\right)d\zeta$
$\displaystyle=\int b(s)ds\int\mu^{2s-\nu}|\sinh(y)|^{2s-\nu-1}\left(e^{i\mu
T\cosh(y)}+e^{-i\mu T\cosh(y)}\right)\cosh(y)dy$
$\displaystyle\begin{multlined}=2\int
b(s)ds\int\mu^{2s-\nu}2^{s-\nu/2-1/2}\frac{\Gamma(s-\nu/2)}{\sqrt{\pi}}\times\\\
\qquad\qquad\times\left(-K_{s-\nu/2+1/2}(-i\mu T)(-i\mu
T)^{-s+\nu/2+1/2}-K_{s-\nu/2+1/2}(i\mu T)(i\mu
T)^{-s+\nu/2+1/2}\right)\end{multlined}=2\int
b(s)ds\int\mu^{2s-\nu}2^{s-\nu/2-1/2}\frac{\Gamma(s-\nu/2)}{\sqrt{\pi}}\times\\\
\qquad\qquad\times\left(-K_{s-\nu/2+1/2}(-i\mu T)(-i\mu
T)^{-s+\nu/2+1/2}-K_{s-\nu/2+1/2}(i\mu T)(i\mu T)^{-s+\nu/2+1/2}\right)$
(A.16) $\displaystyle=-2\int
b(s)ds\int\mu^{2s-\nu}\frac{\Gamma(s-\nu/2)}{\sqrt{\pi}}\int
d\tilde{s}\frac{1}{4\pi
i}\Gamma(\tilde{s})\Gamma(\tilde{s}-s+\nu/2-1/2)\left[\left(\frac{i\mu
T}{2}\right)^{1-2\tilde{s}}+\left(\frac{-i\mu
T}{2}\right)^{1-2\tilde{s}}\right].$ (A.17)

The first simplification is that

$\displaystyle\sqrt{\pi}\Gamma(1+\nu-2s)2^{2s-\nu}=\Gamma(\nu/2+1/2-s)\Gamma(\nu/2+1-s),$
(A.18)

and

$\displaystyle\Gamma(\nu/2+1-s)\Gamma(s-\nu/2)=\pi/\sin(\pi(s-\nu/2)).$ (A.19)

Using these identities and integrating over $s$ (and wrapping poles to the
left), we find

$\displaystyle-\int
b(s)\mu^{2s-\nu}\Gamma(s-\nu/2)\Gamma(\tilde{s}-s+\nu/2-1/2)=-i\frac{\Gamma(1-\tilde{s})\Gamma(\tilde{s}+\nu/2-1/2)}{\Gamma(3/2-\tilde{s}+\nu/2)}.$

Defining $s_{1}=\tilde{s}-1/2$, and using
$\Gamma(1/2-s_{1})\Gamma(1/2+s_{1})=\pi/\cos(\pi s_{1})$ we have

$\displaystyle\phi(T,0)$ $\displaystyle=-\int
ds_{1}\frac{i}{2\pi}\frac{\Gamma(1/2-s_{1})\Gamma(1/2+s_{1})\Gamma(s_{1}+\nu/2)}{\Gamma(1+\nu/2-s_{1})}2\cos(\pi
s_{1})\left(\frac{\mu T}{2}\right)^{-2s_{1}}$ $\displaystyle=-\int
ds_{1}\frac{i}{\pi}\frac{\Gamma(s_{1}+\nu/2)}{\Gamma(1+\nu/2-s_{1})}\left(\frac{\mu
T}{2}\right)^{-2s_{1}}=2J_{k/m}(\mu T),$ (A.20)

as expected from by Eq. (3.16) by setting $V=T,U=T$ and considering even $k$.

##  Appendix B Finding Solutions by Analytic Continuation in JT gravity

Figure 7: Left: We take solutions corresponding to a conical defect of opening
angle $2\pi m$ and cut pieces of it, denoted I and III, slightly greater than
half so as to avoid the defect. We then patch it onto a portion of the smooth
$\text{AdS}_{2}$ solution, labelled II, by imposing matching conditions on the
geodesic and at the asymptotic boundary. This leads to a regularized split
solution which can be analytically continued to a Lorentzian spacetime. In the
limit $\delta\rightarrow 0$, we obtain the same solution that we expect from
initial data evolution.

This appendix illustrates how the Lorentz signature solutions can be found by
first regularizing the Euclidean conical singularities in a way that splits it
(symmetrically) into two such singularities as in figure 2, and then
analytically continuing the region between them.

For a Euclidean JT gravity solution with a conical defect, we have

$ds^{2}=m^{2}(r^{2}-r_{s}^{2})d\tau^{2}+\frac{dr^{2}}{r^{2}-r_{s}^{2}},$ (B.1)
$\Phi=m\Phi_{b}r,$ (B.2)

where the periodicity of $\tau$ is $2\pi/r_{s}$. When $m=1$, the solution is
smooth, so $r_{s}=\frac{2\pi}{\beta}$.

The boundary condition is that the boundary length is
$\frac{\beta}{\epsilon}$. This means that we should put the cutoff at
$r=r_{c}=\frac{1}{m\epsilon}$, and the dilaton value there is
$\frac{\Phi_{b}}{\epsilon}$.

We cut the solution at the following geodesic:

$\displaystyle r(\lambda)$
$\displaystyle=\frac{1}{m}\sqrt{\delta^{2}+(mr_{s})^{2}}\cosh\lambda,$ (B.3)
$\displaystyle\tau(\lambda)$
$\displaystyle=\frac{1}{r_{s}}\arctan\left(\frac{mr_{s}}{\delta}\tanh\lambda\right),$

where $\delta$ indicates how far the geodesic is from the center of the disk.
At $r=\frac{1}{m\epsilon}$, the affine parameter is

$\lambda_{c}=\cosh^{-1}\frac{1/\epsilon}{\sqrt{\delta^{2}+(mr_{s})^{2}}}.$
(B.4)

In the middle region, we have a JT solution without a defect:

$ds^{2}=(\bar{r}^{2}-\bar{r}_{s}^{2})d\bar{\tau}^{2}+\frac{d\bar{r}^{2}}{\bar{r}^{2}-\bar{r}_{s}^{2}},$
(B.5) $\Phi=\bar{\Phi}_{b}\bar{r},$ (B.6)

where the periodicity of $\bar{\tau}$ is $2\pi/\bar{r}_{s}$. We cut it at the
geodesic

$\displaystyle\bar{r}(\bar{\lambda})$
$\displaystyle=\sqrt{\bar{\delta}^{2}+\bar{r}_{s}^{2}}\cosh\bar{\lambda},$
(B.7) $\displaystyle\bar{\tau}(\bar{\lambda})$
$\displaystyle=\frac{1}{\bar{r}_{s}}\arctan\left(\frac{\bar{r}_{s}}{\bar{\delta}}\tanh\bar{\lambda}\right).$

At the cutoff $\bar{r}=\bar{r}_{c}=\frac{1}{\epsilon}$, the affine parameter
is

$\bar{\lambda}_{c}=\cosh^{-1}\frac{1/\epsilon}{\sqrt{\bar{\delta}^{2}+\bar{r}_{s}^{2}}}.$
(B.8)

We now glue these solutions together to regularize the conical defect solution
to have a neighbourhood of a smooth solution near the $\mathbb{Z}_{2}$
symmetric slice. The first matching condition is that the affine parameters
are the same:

$\lambda_{c}=\bar{\lambda}_{c},$ (B.9)

from which we get

$\bar{\delta}=\sqrt{m^{2}r_{s}^{2}-\bar{r}_{s}^{2}+\delta^{2}}.$ (B.10)

The second matching condition is that the total boundary length is
$\frac{\beta}{\epsilon}$:

$4\left(\frac{\pi}{r_{s}}-\tau(\lambda_{c})\right)+4\left(\frac{\pi}{2\bar{r}_{s}}-\bar{\tau}(\bar{\lambda}_{c})\right)=\beta.$
(B.11)

To leading order in $\epsilon$, we know that the relation between $r_{s}$ and
$\bar{r}_{s}$ can be solved from

$\pi(r_{s}+\bar{r}_{s})-2\bar{r}_{s}\tan^{-1}\left(\frac{mr_{s}}{\delta}\right)-2r_{s}\tan^{-1}\left(\frac{\bar{r}_{s}}{\bar{\delta}}\right)=0.$
(B.12)

This is in general hard to solve, but we can see that in the limit $\delta\to
0$, we have $\bar{r}_{s}=mr_{s}$.

The last matching condition is that, the dilaton values should be the same at
the boundary. Thus, we have

$\bar{\Phi}_{b}=\Phi_{b}.$ (B.13)

If we now analytically continue the middle region to Lorentzian signature, we
obtain a smooth black hole with $\bar{r}_{s}=mr_{s}$, which means
$\bar{\beta}=\frac{\beta}{m}$. The boundary dilaton is
$\bar{\Phi}_{b}=\Phi_{b}$. Note that for this kind of spacetime that we
constructed, the dilaton field only matches along the geodesic when we take
$\delta\to 0$ limit. This is of course, as we expect from fixed-area state in
pure JT gravity with $U(1)$ symmetry, identical to the microcanonical TFD at
the temperature $\bar{\beta}$.

More generally, this technique of regularizing the conical defect solution and
analytically continuing from the neighbourhood of the $\mathbb{Z}_{2}$
symmetric slice might be useful. In certain situations, it may be simpler than
solving the initial value problem.

##  Appendix C Solving the AdS3 metric perturbatively

In Sec. 4.3, we solved for the scalar field in AdS3 and described the solution
for the backreacted metric. In this appendix, we write down all the equations
for the metric components explicitly and show that they can be solved
perturbatively as claimed in Sec. 4.3.

The solution for the scalar field to leading order is

$\displaystyle\phi=2\eta\cos(k\theta)\frac{f_{m,k}(r)}{f_{m,k}(r_{c})},\qquad
f_{m,k}(r)=r^{k/m}{{}_{2}F_{1}}\left(\frac{k}{2m},\frac{k}{2m}+1,\frac{k}{m}+1,-r^{2}\right).$
(C.1)

From this solution, the stress tensor is decomposed into Fourier modes as

$\displaystyle
T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{g_{\mu\nu}}{2}(\nabla\phi)^{2}=T_{\mu\nu}^{-}e^{-2ik\theta}+T_{\mu\nu}^{+}e^{2ik\theta}+T_{\mu\nu}^{0}.$
(C.2)

Following Lewkowycz:2013nqa , the ansatz for the metric to first order in
$\eta^{2}$ is

$\displaystyle
ds^{2}=\frac{dr^{2}}{1+r^{2}+g(r,\theta)}+m^{2}r^{2}d\theta^{2}+m^{2}(1+r^{2})\left(1+v(r,\theta)\right)dy^{2},$
(C.3)

where $g(r,\theta),v(r,\theta)$ are the metric perturbation, and have the
following Fourier expansion,

$\displaystyle
g(r,\theta)=g^{+}(r)e^{2ik\theta}+g(r)^{0}+g^{-}(r)e^{-2ik\theta},\qquad
v(r,\theta)=v^{+}(r)e^{2ik\theta}+v^{0}(r)+v^{-}(r)e^{-2ik\theta}$ (C.4)

Plugging the stress tensor in the Einstein’s equation (we set $8\pi G=1$)

$\displaystyle R_{\mu\nu}-\frac{g_{\mu\nu}}{2}\left(R+2\right)=T_{\mu\nu},$
(C.5)

one finds that Fourier modes decouple to the first order. For the
$yy$-components, we have

$\displaystyle
4k^{2}g^{-}(r)+m^{2}r(1+r^{2})g^{-\prime}(r)=2r^{2}T^{-}_{yy}=\frac{\eta^{2}}{f_{m,k}^{2}(r_{c})}(1+r^{2})\left(k^{2}f_{m,k}(r)^{2}-m^{2}r^{2}(1+r^{2})f_{m,k}^{\prime\,2}\right),$
(C.6) $\displaystyle
g^{0\prime}(r)=\frac{2r}{m^{2}(1+r^{2})}T^{0}_{yy}=\frac{\eta^{2}}{f_{m,k}^{2}(r_{c})}\partial_{r}\left(r(1+r^{2})f_{m,k}(r)f^{\prime}_{m,k}(r)\right),$
(C.7) $\displaystyle
4k^{2}g^{+}(r)+m^{2}r(1+r^{2})g^{+\prime}(r)=2r^{2}T^{+}_{yy}=\frac{\eta^{2}}{f_{m,k}^{2}(r_{c})}(1+r^{2})\left(k^{2}f_{m,k}(r)^{2}-m^{2}r^{2}(1+r^{2})f_{m,k}^{\prime\,2}\right),$
(C.8)

where in Eq. (C.7), the equation of motion for the scalar field is used. These
equations are all first order differential equations and therefore, the
solutions can be written in terms of integrals involving $f_{m,k}(r)$ and its
derivatives. Using the solutions for $g^{\pm}(r),g^{0}(r)$, the $rr$
components of Einstein’s equations give the differential equations for
$v^{\pm},v^{0}$,

$\displaystyle-4k^{2}(1+r^{2})v^{-}+2m^{2}r^{2}g^{-}(r)+m^{2}r(1+r^{2})^{2}v^{-\prime}(r)=2m^{2}r^{2}(1+r^{2})^{2}T_{rr}^{-},$
(C.9) $\displaystyle
2rg^{0}(r)+(1+r^{2})^{2}v^{0\prime}(r)=2T^{0}_{rr}r(1+r^{2})^{2},$ (C.10)
$\displaystyle-4k^{2}(1+r^{2})v^{+}+2m^{2}r^{2}g^{+}(r)+m^{2}r(1+r^{2})^{2}v^{+\prime}(r)=2m^{2}r^{2}(1+r^{2})^{2}T_{rr}^{+},$
(C.11)

where

$\displaystyle
T^{0}_{rr}=\frac{\eta^{2}}{m^{2}r^{2}(1+r^{2})f_{m,k}^{2}(r_{c})}(-k^{2}f_{m,k}(r)^{2}+m^{2}r^{2}(1+r^{2})f^{\prime}_{m,k}(r)^{2}),$
(C.12) $\displaystyle
T_{rr}^{-}=\frac{\eta^{2}}{2m^{2}r^{2}(1+r^{2})f_{m,k}^{2}(r_{c})}(k^{2}f_{m,k}(r)^{2}+m^{2}r^{2}(1+r^{2})f^{\prime}_{m,k}(r)^{2}),$
(C.13) $\displaystyle
T_{rr}^{+}=\frac{\eta^{2}}{2m^{2}r^{2}(1+r^{2})f_{m,k}^{2}(r_{c})}(k^{2}f_{m,k}(r)^{2}+m^{2}r^{2}(1+r^{2})f^{\prime}_{m,k}(r)^{2}).$
(C.14)

Therefore, the equations for $v^{\pm}(r),v^{0}(r)$ are also first order and
the solutions can be written in terms of integrals of the stress tensor or
power series expansions. We also checked that the power series solutions of
$g^{\pm,0},v^{\pm,0}$ satisfy other components of Einstein’s equations and
therefore, the verified the consistency of the metric ansatz. The form of the
power series expansions have been listed in Eq. (4.50).

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