# Quantum-corrected anti-de Sitter space-time

Jacob C. Thompson<EMAIL_ADDRESS>School of Mathematics and
Statistics, The University of Sheffield, Hicks Building, Hounsfield Road,
Sheffield. S3 7RH United Kingdom  Elizabeth Winstanley
<EMAIL_ADDRESS>School of Mathematics and Statistics, The
University of Sheffield, Hicks Building, Hounsfield Road, Sheffield. S3 7RH
United Kingdom

###### Abstract

We study the back-reaction of a quantum scalar field on anti-de Sitter (AdS)
space-time. The renormalized expectation value of the stress-energy tensor
operator for a massless, conformally-coupled quantum scalar field on global
AdS space-time in four space-time dimensions acts as a source term on the
right-hand-side of the Einstein equations for the quantum-corrected metric. We
solve the quantum-corrected Einstein equations numerically and find deviations
from pure AdS which increase as the temperature of the quantum scalar field
state increases. We interpret these quantum-corrected metrics as
asymptotically-AdS solitons, and study the mass of these solitons as a
function of the temperature of the quantum scalar field.

##  I Introduction

The classical Einstein equations govern the interaction between space-time
geometry and matter (we use units in which $8\pi G=c=1$):

$G_{\mu\nu}+\Lambda g_{\mu\nu}=T_{\mu\nu},$ (1)

where $G_{\mu\nu}$ is the Einstein tensor, $g_{\mu\nu}$ the metric tensor with
signature $(-,+,+,+)$ and $\Lambda$ the cosmological constant. The left-hand-
side of (1) describes the geometry of space-time. On the right-hand-side,
$T_{\mu\nu}$ is the classical stress-energy tensor (SET) of matter or
classical fields other than gravity. In a full theory of quantum gravity, both
the left- and right-hand-sides of (1) would be quantized, in other words, all
matter fields, as well as the space-time geometry on which the matter
propagates, would be quantum in nature.

Developing a full theory of quantum gravity is a rather ambitious challenge,
and a more modest undertaking is to consider instead quantum field theory on
curved space-time. In this paradigm, space-time (giving the quantities on the
left-hand-side of (1)) continues to be classical but the matter (and fields
other than gravity) is quantized. This means that the classical SET on the
right-hand-side of (1) is replaced by the (renormalized) expectation of a
quantum stress-energy tensor (RSET) operator
$\langle{\hat{T}}_{\mu\nu}\rangle$, giving the semiclassical Einstein
equations (SCEE).

In four space-time dimensions, the SCEE take the form

$G_{\mu\nu}+\Lambda g_{\mu\nu}+\alpha H_{\mu\nu}^{(1)}+\beta
H_{\mu\nu}^{(2)}=\langle{\hat{T}}_{\mu\nu}\rangle.$ (2)

As a result of the renormalization prescription for the RSET [1, 2, 3], the
left-hand-side includes two additional, classical, conserved geometric
tensors, $H^{(1,2)}_{\mu\nu}$, given by

$\displaystyle H^{(1)}_{\mu\nu}$
$\displaystyle=\dfrac{1}{2}R^{2}g_{\mu\nu}-2RR_{\mu\nu}-2g_{\mu\nu}\Box
R+2\nabla_{\mu}\nabla_{\nu}R,$ $\displaystyle H^{(2)}_{\mu\nu}$
$\displaystyle=\dfrac{1}{2}R^{\sigma\lambda}R_{\sigma\lambda}g_{\mu\nu}-\Box
R_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}R$
$\displaystyle\qquad-\dfrac{1}{2}g_{\mu\nu}\Box
R-2R^{\sigma\lambda}R_{\sigma\mu\lambda\nu},$ (3)

where $R_{\mu\nu\lambda\sigma}$ is the Riemann tensor, $R_{\mu\nu}$ the Ricci
tensor and $R$ the Ricci scalar, and two additional constants $\alpha$ and
$\beta$. The tensors $H^{(1,2)}_{\mu\nu}$ contain four derivatives of the
space-time metric.

A fully-consistent solution of the SCEE would involve the RSET on the right-
hand-side of (2) being computed on the space-time background which gives the
geometric quantities on the left-hand-side. This is, in general, a very
challenging problem, particularly when the fourth-order terms
$H^{(1,2)}_{\mu\nu}$ are included, and, furthermore, the equations (2) are
known to possess runaway solutions [4, 5, 6]. Nonetheless, rigorous work on
solving the SCEE consistently can be found, for example, for a background
Einstein static universe [7], cosmological space-times [8, 9, 10, 11] and
static space-times [12, 13].

Given the difficulties inherent in solving the SCEE, there are various lines
of attack in the literature to simplify the problem. First, one can take a
perturbative (in $\hbar$) approach (see, for example, [14, 15]), in which the
space-time metric takes the form

$g_{\mu\nu}=g_{\mu\nu}^{0}+\hbar g_{\mu\nu}^{(1)}+{\mathcal{O}}(\hbar^{2}),$
(4)

where $g_{\mu\nu}^{0}$ is a given fixed background space-time metric and
$g_{\mu\nu}^{(1)}$ is the first order correction to the metric. Linearizing
the SCEE (2) (with the assumption that the constants $\alpha$ and $\beta$ are
${\mathcal{O}}(\hbar)$) yields a set of linear equations for
$g_{\mu\nu}^{(1)}$ which are second-order in time derivatives [14, 15], and
which involve, as a source term, the RSET computed on the fixed space-time
metric $g_{\mu\nu}^{0}$. Even with the recent development of efficient methods
of computing the RSET on black hole backgrounds (see, for example, [16, 17,
18, 19]), solving these reduced equations is still difficult and often
approximations to the RSET are employed to make progress (see, for example,
[15]).

An alternative approach is to employ an analytic approximation for the RSET
which is valid on a general class of space-time metrics, so that the right-
hand-side of (2) can be written in closed form in terms of one or more unknown
metric functions. The aim is then to solve (2) exactly for these unknown
metric functions. For example, the Polyakov approximation [20, 21] results
from an $s$-wave approximation to the RSET on static, spherically symmetric
space-times. This approximation has the advantage of involving only second-
order derivatives of the metric, and hence it is possible to solve the SCEE
(2) under the simplifying assumption $\alpha=\beta=0$. This strategy been used
to find spherically symmetric semiclassical stars and black holes (see, for
example, [22, 23, 24, 25, 26, 27] for a selection of recent works using this
approach).

In this paper we follow a third, even less ambitious, approach. We fix a
background space-time having metric $g^{0}_{\mu\nu}$ and find the RSET on this
background. This RSET is then used as the source term in the SCEE (2), which
are then solved nonperturbatively to give the quantum corrected (QC) metric
$g_{\mu\nu}^{{\rm{QC}}}$ whose curvature tensors appear on the left-hand-side
of (2). This approach has been applied in three dimensions (where the
additional curvature tensors $H^{(1,2)}_{\mu\nu}$ are absent) to find the QC
BTZ black holes and naked singularities [28, 29, 30], and also more recently
to three-dimensional black holes in massive gravity [31]. In this paper, we
will only be concerned with quantum states for which the constants $\alpha$
and $\beta$ in (2) both vanish, so that the Einstein equations we will be
solving can be written as

$G_{\mu\nu}^{\rm{QC}}+\Lambda
g_{\mu\nu}^{\rm{QC}}=\langle{\hat{T}}_{\mu\nu}[g^{0}]\rangle,$ (5)

where the quantities on the left-hand-side involve the QC metric
$g_{\mu\nu}^{\rm{QC}}$ while the RSET on the right-hand-side is computed on
the fixed background metric $g_{\mu\nu}^{0}$. The terms on the left-hand-side
of (5) are second order in derivatives of the QC metric
$g_{\mu\nu}^{\rm{QC}}$, which facilities finding solutions. This approach is
often referred to as the “back-reaction” problem.

In this paper we address the back-reaction problem for a simple toy model,
taking the initial fixed background metric $g_{\mu\nu}^{0}$ to be that of four
dimensional global anti-de Sitter (AdS) space-time. Since AdS is a maximally
symmetric space-time, this is the simplest nontrivial example of the back-
reaction problem (the back-reaction on Minkowski space-time being trivial for
a quantum field in the global vacuum state). Quantum field theory on four-
dimensional global AdS was initiated many years ago [32, 33] for a massless,
conformally-coupled scalar field $\Phi$ satisfying the Klein-Gordon equation

$0=\left(\nabla_{\mu}\nabla^{\mu}-\dfrac{1}{6}R\right)\Phi.$ (6)

There are two key features which render the back-reaction problem on pure AdS
nontrivial. First, since AdS is not a globally hyperbolic space-time, it is
necessary to apply boundary conditions to the quantum field at null infinity
in order to have a well-defined theory (see, for example [32, 33, 34, 35,
36]). The application of the simplest possible boundary conditions (namely
Dirichlet and Neumann) yields a vacuum state which replicates the maximal
symmetry of the underlying space-time [32, 33, 37, 38], in which case the RSET
in that state is a multiple of the AdS metric. However maximal symmetry is
broken in the vacuum state if more general boundary conditions are considered,
such as Robin boundary conditions [39, 36, 40]. Maximal symmetry is also
broken for thermal states [33], even when the boundary conditions are such
that the corresponding vacuum state is maximally symmetric. The back-reaction
problem becomes nontrivial when the RSET on right-hand-side of the “quantum-
corrected” Einstein equations (QCEE) (5) is evaluated in a nonmaximally-
symmetric quantum state, since then it is no longer the case that the RSET is
proportional to the space-time metric. Solving the back-reaction problem to
find the QC metric $g_{\mu\nu}^{\rm{QC}}$ in such a scenario is the focus of
the present work.

The outline of this paper is as follows. We begin, in Sec. II by reviewing the
(trivial) back-reaction problem for maximally-symmetric quantum states on
maximally symmetric space-times. In Sec. III we review the RSET for a
massless, conformally coupled scalar field in a thermal state on pure AdS
space-time, as computed in [33]. Our analysis of the QC metrics solving the
QCEE (5) with the above RSET as a source on the right-hand-side begins in Sec.
IV, where we employ an ansatz for the QC metric which is particularly useful
for comparing with the pure AdS metric. We derive the QCEE using this metric
ansatz and outline our numerical procedure for solving these equations. The
results of this numerical work are presented in Sec. V, where we consider the
QC metrics resulting from applying Dirichlet, Neumann and transparent boundary
conditions to the quantum scalar field. To aid the physical interpretation of
these QC metrics, in Sec. VI we use an alternative ansatz for the QC metric,
which enables us to interpret the QC metrics as soliton-like space-times which
are asymptotically-AdS. We compute the masses of these solitons and
investigate the dependence of the mass on the temperature of the quantum
field. Our conclusions are presented in Sec. VII.

##  II QC maximally symmetric space-times

We now review some simple examples of QC space-times, for which the RSET is
known exactly in closed form for a massless, conformally coupled, scalar
field. These examples involve background space-times which are maximally
symmetric. We first show that in this situation the constants $\alpha$ and
$\beta$ in the SCEE (2) can be set to zero if the RSET preserves the maximal
symmetry of the background space-time.

###  II.1 RSET for maximally symmetric states

Consider a background space-time which is maximally symmetric. In this case
the Riemann tensor is given in terms of the space-time metric:

$R_{\mu\nu\lambda\sigma}={\mathcal{K}}\left(g_{\mu\lambda}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\lambda}\right)$
(7)

where ${\mathcal{K}}$ is a constant. In four space-time dimensions, the Ricci
tensor and curvature scalar are then:

$R_{\mu\nu}=3{\mathcal{K}}g_{\mu\nu},\qquad R=12{\mathcal{K}}.$ (8)

The geometric tensors $H_{\mu\nu}^{(1,2)}$ (3) are therefore also proportional
to the metric tensor $g_{\mu\nu}$ when evaluated on the background space-time.

Now consider a maximally symmetric quantum state, for example, a global vacuum
state (assuming such a state exists and is Hadamard). In such a state the two-
point function $G(x,x^{\prime})$ is a function only of the world function
$\sigma(x,x^{\prime})$ for the separated space-time points $x$, $x^{\prime}$
[37]. Furthermore, the Hadamard parametrix $G_{\rm{H}}(x,x^{\prime})$ (see,
for example, [3]), which must be subtracted from the two-point function as
part of the renormalization process, is also a function only of
$\sigma(x,x^{\prime})$. The RSET is obtained from the regularized two-point
function

$G_{\rm{R}}(x,x^{\prime})=G(x,x^{\prime})-G_{\rm{H}}(x,x^{\prime})$ (9)

by applying a second-order differential operator ${\mathcal{T}}_{\mu\nu}$ and
then taking the coincidence limit $x^{\prime}\rightarrow x$. In other words,
the RSET in a maximally symmetric vacuum state $|0\rangle$ is

$\langle 0|{\hat{T}}_{\mu\nu}|0\rangle=\lim_{x^{\prime}\rightarrow
x}\left\\{{\mathcal{T}}_{\mu\nu}\left[G_{\rm{R}}(x,x^{\prime})\right]\right\\}-g_{\mu\nu}v_{1}.$
(10)

For a massless, conformally coupled scalar field, the operator is [3]

$\displaystyle{\mathcal{T}}_{\mu\nu}=$
$\displaystyle~{}\dfrac{2}{3}g_{\nu}{}^{\nu^{\prime}}\nabla_{\mu}\nabla_{\nu^{\prime}}+\dfrac{1}{6}g_{\mu\nu}g^{\rho\sigma^{\prime}}\nabla_{\rho}\nabla_{\sigma^{\prime}}$
$\displaystyle~{}-\dfrac{1}{3}g_{\mu}{}^{\mu^{\prime}}g_{\nu}{}^{\nu^{\prime}}\nabla_{\mu^{\prime}}\nabla_{\nu^{\prime}}+\dfrac{1}{3}g_{\mu\nu}\nabla_{\rho}\nabla^{\rho}+\dfrac{1}{6}G_{\mu\nu},$
(11)

where $g_{\nu}{}^{\nu^{\prime}}$ is the bivector of parallel transport and a
prime ′ denotes derivatives with respect to the space-time point $x^{\prime}$.
The quantity $v_{1}$ in the final term in (10) is the coincidence limit of a
state-independent, geometric biscalar which is necessary in order for the RSET
to be conserved [3]. For a maximally symmetric space-time, $v_{1}$ is a
position-independent constant.

The singular terms ${\mathcal{T}}_{\mu\nu}[G_{\rm{H}}(x,x^{\prime})]$ are
purely geometric and give rise to a renormalization of the constants $8\pi G$
(which we have already set equal to unity), $\Lambda$, $\alpha$ and $\beta$ in
the SCEE (2). Since $G_{\rm{H}}(x,x^{\prime})$ is a function only of
$\sigma(x,x^{\prime})$, using the form of the operator (11) and the expansions
of $g_{\nu}{}^{\nu^{\prime}}\nabla_{\mu}\nabla_{\nu^{\prime}}\sigma$ and
$\nabla_{\mu}\nabla_{\nu}\sigma$ given in [41], it is straightforward to show
that ${\mathcal{T}}_{\mu\nu}[G_{\rm{H}}(x,x^{\prime})]$ is proportional to the
metric tensor $g_{\mu\nu}$. Therefore, it is sufficient to renormalize the
cosmological constant $\Lambda$ in the SCEE (2) and to set the constants
$\alpha=0=\beta$.

For the remainder of this paper, we will mostly be concerned with a QC metric
in the situation where the original background space-time is maximally
symmetric and possesses a maximally symmetric vacuum state. We therefore set
$\alpha=0=\beta$ in (2) and from here on will study the solutions of the
simplified SCEE

$G_{\mu}{}^{\nu}+\Lambda\delta_{\mu}{}^{\nu}=\langle{\hat{T}}_{\mu}{}^{\nu}\rangle,$
(12)

where we have raised the second index for later convenience. Suppose that the
RSET for the maximally symmetric vacuum state is given by

$\langle 0\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert
0\rangle={\mathcal{C}}\delta_{\mu}{}^{\nu},$ (13)

where ${\mathcal{C}}$ is a position-independent constant. Then, for any other
quantum state $\rvert\psi\rangle$ (12) takes the form

$G_{\mu}{}^{\nu}+\Lambda\delta_{\mu}{}^{\nu}=\langle\psi\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert\psi\rangle-\langle
0\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert
0\rangle+{\mathcal{C}}\delta_{\mu}{}^{\nu},$ (14)

giving

$G_{\mu}{}^{\nu}+{\widetilde{\Lambda}}\delta_{\mu}{}^{\nu}=\langle\psi\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert\psi\rangle-\langle
0\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert 0\rangle,$ (15)

where ${\widetilde{\Lambda}}=\Lambda-{\mathcal{C}}$ is the renormalized
cosmological constant. From now on we shall refer to (15) as the “quantum-
corrected Einstein equations” (QCEE).

###  II.2 Examples

Before we study solutions of (15) for a background global AdS space-time, we
review some simple examples of maximally symmetric states on maximally
symmetric space-times.

####  II.2.1 Minkowski space-time

The simplest possible maximally symmetric space-time background is Minkowski
space-time, whose metric in static, spherically symmetric coordinates
$(t,r,\theta,\varphi)$ takes the form

$\mathop{}\\!\mathrm{d}{s}^{2}=-\mathop{}\\!\mathrm{d}{t}^{2}+\mathop{}\\!\mathrm{d}{r}^{2}+r^{2}\mathop{}\\!\mathrm{d}{\Omega}^{2},$
(16)

where

$\mathop{}\\!\mathrm{d}{\Omega}^{2}=\mathop{}\\!\mathrm{d}{\theta}^{2}+\sin^{2}\theta\,\mathop{}\\!\mathrm{d}{\varphi}^{2}$
(17)

is the metric on the two-sphere. In the global Minkowski vacuum
$|0_{{\rm{M}}}\rangle$, the RSET $\langle
0_{{\rm{M}}}|{\hat{T}}_{\mu\nu}|0_{{\rm{M}}}\rangle\equiv 0$. In this case
there are no quantum corrections to the space-time metric and the back-
reaction problem is trivial.

####  II.2.2 De Sitter space-time

De Sitter (dS) space-time is maximally symmetric space-time with positive
scalar curvature. In global coordinates $(\tau,\rho,\theta,\varphi)$ the
metric can be written as

$\mathop{}\\!\mathrm{d}{s}^{2}=a^{-2}\sec^{2}\tau\left[-\mathop{}\\!\mathrm{d}{\tau}^{2}+\mathop{}\\!\mathrm{d}{\rho}^{2}+\sin^{2}\rho\,\mathop{}\\!\mathrm{d}{\Omega}^{2}\right],$
(18)

where the cosmological constant is $\Lambda=3a^{2}>0$ and $a$ is an inverse
length scale. The preferred quantum state on dS is the Bunch-Davies state [42,
43], which is the unique maximally symmetric Hadamard state for a massless,
conformally coupled, scalar field. The RSET in this state is (setting
$\hbar=1$ from henceforth) [42, 44, 45]

$\langle
0_{{\rm{BD}}}|{\hat{T}}_{\mu}{}^{\nu}|0_{{\rm{BD}}}\rangle=\dfrac{a^{2}}{960\pi^{2}}\delta_{\mu}{}^{\nu}.$
(19)

In this case the QCEE (12) give the QC metric to be dS space-time with a
renormalized cosmological constant:

$\Lambda\rightarrow{\widetilde{\Lambda}}_{\rm{dS}}=\Lambda-\dfrac{a^{2}}{960\pi^{2}}.$
(20)

Solutions of the SCEE of dS form have been studied extensively in the
literature: see, for example, [46, 10, 11].

####  II.2.3 Anti-de Sitter space-time

Anti-de Sitter (AdS), like dS, is a maximally symmetric space-time, the
difference being that AdS has constant negative scalar curvature. In global
coordinates $(\tau,\rho,\theta,\varphi)$ the metric is written as

$\mathop{}\\!\mathrm{d}{s}^{2}=a^{-2}\sec^{2}\rho\left[-\mathop{}\\!\mathrm{d}{\tau}^{2}+\mathop{}\\!\mathrm{d}{\rho}^{2}+\sin^{2}\rho\mathop{}\\!\mathrm{d}{\Omega}^{2}\right],$
(21)

where $a$ is the inverse AdS length scale, related to the cosmological
constant by $\Lambda=-3a^{2}<0$. The coordinate ranges are
$\tau\in(-\pi,\pi]$, $\rho\in[0,\pi/2)$, $\theta\in[0,\pi]$ and
$\phi\in[0,2\pi]$.

In order to have a well-defined quantum field theory on AdS, it is necessary
to impose boundary conditions on the massless, conformally coupled, scalar
field [32, 34, 35, 36]. In this work we consider Dirichlet, Neumann and
transparent boundary conditions [32]. In this case the vacuum state for the
scalar field preserves the underlying maximal symmetry of the AdS space-time
[32, 38, 36] and the RSET in the vacuum state, $\langle
0\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert 0\rangle$, is proportional to the metric
[33]

$\langle 0\rvert{\hat{T}}_{\mu}{}^{\nu}\rvert
0\rangle=-\dfrac{a^{2}}{960\pi^{2}}\delta_{\mu}{}^{\nu}.$ (22)

As in the de Sitter case, the QCEE give the QC metric in this case to be AdS
space-time with a renormalized cosmological constant:

$\Lambda\rightarrow{\widetilde{\Lambda}}_{\rm{adS}}=\Lambda+\dfrac{a^{2}}{960\pi^{2}}.$
(23)

The corrections to the cosmological constant due to the vacuum RSET are very
small in both the dS (20) and AdS (23) cases and are such that the magnitude
of the cosmological constant is reduced.

##  III RSET for thermal states on AdS

In the previous section, we have considered only maximally symmetric states,
namely the vacuum state on Minkowski space-time, the Bunch-Davies state on dS,
and the vacuum state on global AdS with either transparent, Dirichlet or
Neumann boundary conditions. We now switch our focus to nonmaximally symmetric
states, in particular thermal states for a massless, conformally coupled,
scalar field. The global thermal state on Minkowski space-time is spatially
homogeneous and isotropic, so the solution of the QCEE (15) is not
particularly interesting in this case. Due to the presence of the cosmological
horizon, there is a preferred temperature on the static patch of dS. The
Bunch-Davies state corresponds, on the static patch of dS, to a state at this
preferred temperature, and therefore the solution of the QCEE (15) is trivial
in this case. For the remainder of this paper we therefore focus on thermal
states on AdS. In this section we review the thermal RSET as computed in [33].
Two aspects make this interesting for our purposes: first, AdS has no
preferred temperature so we can consider the effect of varying the
temperature; and second, the thermal RSET is not maximally symmetric.

The quantum scalar field ${\hat{\Phi}}$ is taken to be in a thermal state at
inverse temperature $\beta$ and for the remainder of this paper we set
$k_{{\rm{B}}}=1$. The difference in the RSET between the thermal
$\rvert\beta\rangle$ and vacuum $\rvert 0\rangle$ states is [33] (with a minor
typographical error corrected):

$\displaystyle{\widetilde{T}}_{\mu}{}^{\nu}$
$\displaystyle=\langle\beta\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert\beta\rangle-\langle
0\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert 0\rangle$
$\displaystyle=\dfrac{a^{2}}{8\pi^{2}}\left\\{{\mathcal{F}}_{1}(\rho)\delta_{\mu}{}^{\nu}+{\mathcal{F}}_{2}(\rho)\tau_{\mu}\tau^{\nu}+{\mathcal{F}}_{3}(\rho)\rho_{\mu}\rho^{\nu}\right\\}$
(24a) where
$\tau_{\bullet}=(a^{-1}\sec\rho,0,0,0),\qquad\rho_{\bullet}=(0,a^{-1}\sec\rho,0,0),$
(24b) are unit length timelike and radial vectors, respectively. The functions
${\mathcal{F}}_{1}(\rho)$, ${\mathcal{F}}_{2}(\rho)$ and
${\mathcal{F}}_{3}(\rho)$ are given by $\displaystyle{\mathcal{F}}_{1}(\rho)=$
$\displaystyle\dfrac{4}{3}\cos^{4}\rho\,f_{3}(a\beta)+\lambda\cot\rho\left[-\dfrac{1}{6}\csc^{2}\rho\cos(2\rho)S_{0}(a\beta,\rho)+\dfrac{1}{3}\cot\rho\,C_{1}(a\beta,\rho)+\dfrac{2}{3}\cos^{2}\rho\,S_{2}(a\beta,\rho)\right],$
$\displaystyle{\mathcal{F}}_{2}(\rho)=$
$\displaystyle\dfrac{16}{3}\cos^{4}\rho\,f_{3}(a\beta)+\lambda\cot\rho\left[\dfrac{1}{6}\left(3-\cot^{2}\rho\right)S_{0}(a\beta,\rho)+\cot\rho\left(1-\dfrac{2}{3}\cos^{2}\rho\right)C_{1}(a\beta,\rho)\right.$
$\displaystyle\qquad\left.+2\cos^{2}\rho\,S_{2}(a\beta,\rho)\right],$
$\displaystyle{\mathcal{F}}_{3}(\rho)=$
$\displaystyle\lambda\cot\rho\left[\dfrac{1}{6}\left(3\csc^{2}\rho-4\right)S_{0}(a\beta,\rho)+\cot\rho\left(\dfrac{2}{3}\sin^{2}\rho-1\right)C_{1}(a\beta,\rho)-\dfrac{2}{3}\cos^{2}\rho\,S_{2}(a\beta,\rho)\right],$
(24c) where we have defined $\displaystyle f_{m}(x)$
$\displaystyle=\sum_{n=1}^{\infty}n^{m}\left(e^{nx}-1\right)^{-1},$ (24d)
$\displaystyle\widetilde{f}_{m}(x)$
$\displaystyle=\sum_{n=1}^{\infty}n^{m}(-1)^{n}\left(e^{nx}-1\right)^{-1},$
(24e) $\displaystyle S_{m}(x,\rho)$
$\displaystyle=\sum_{n=1}^{\infty}n^{m}(-1)^{n}\left(e^{nx}-1\right)^{-1}\sin(2n\rho),$
(24f) $\displaystyle C_{m}(x,\rho)$
$\displaystyle=\sum_{n=1}^{\infty}n^{m}(-1)^{n}\left(e^{nx}-1\right)^{-1}\cos(2n\rho),$
(24g) and the second quantity ${\widetilde{f}}_{m}(x)$ has been defined for
later convenience. In (24c), the constant $\lambda$ depends on the boundary
conditions imposed: $\lambda=\begin{cases}1&{\mbox{Dirichlet boundary
conditions,}}\\\ -1&{\mbox{Neumann boundary conditions,}}\\\
0&{\mbox{transparent boundary conditions.}}\end{cases}$ (24h)

When $\lambda=0$ and we consider transparent boundary conditions, we have
${\mathcal{F}}_{3}(\rho)=0$ and the functions ${\mathcal{F}}_{1,2}(\rho)$
simplify considerably:

$\displaystyle{\mathcal{F}}_{1}(\rho)=$
$\displaystyle\dfrac{4}{3}\cos^{4}\rho\,f_{3}(a\beta),$
$\displaystyle{\mathcal{F}}_{2}(\rho)=$
$\displaystyle\dfrac{16}{3}\cos^{4}\rho\,f_{3}(a\beta).$ (25)

In the limit $\beta\rightarrow\infty$, the functions $f_{m}(a\beta)$,
$S_{m}(a\beta,\rho)$ and $C_{m}(a\beta,\rho)$ all tend to zero and the
difference in RSET expectation values (24) vanishes.

(a) Energy density

(b) Pressure

(c) Pressure deviator

Figure 1: Energy density $E_{\beta}^{(\rm{D})}(\rho)$, pressure
$P_{\beta}^{(\rm{D})}(\rho)$ and pressure deviator
$\Pi_{\beta}^{(\rm{D})}(\rho)$ (26) for a massless, conformally coupled,
scalar field on global AdS with Dirichlet boundary conditions applied, for
varying inverse temperature $\beta\in(\tfrac{\pi}{12},\tfrac{11\pi}{24})$ and
fixed inverse AdS length scale $a=1$. Figure 2: Ratio of the energy density
$E_{\beta}^{(\rm N)}(\rho)$ with Neumann boundary conditions applied to the
energy density $E_{\beta}^{(\rm D)}(\rho)$ with Dirichlet boundary conditions
applied. The inverse AdS length scale is fixed to be $a=1$ and the inverse
temperature $\beta\in(\tfrac{\pi}{12},\tfrac{11\pi}{24})$.

From the form of the RSET (24), we have
${\widetilde{T}}_{\theta}{}^{\theta}={\widetilde{T}}_{\varphi}{}^{\varphi}$
and therefore it suffices to consider the three components
${\widetilde{T}}_{\tau}{}^{\tau}$, ${\widetilde{T}}_{\rho}{}^{\rho}$ and
${\widetilde{T}}_{\theta}{}^{\theta}$. If we define

$\displaystyle E_{\beta}(\rho)$
$\displaystyle=-\widetilde{T}_{\tau}{}^{\tau},$ (26a) $\displaystyle
P_{\beta}(\rho)$
$\displaystyle=\dfrac{1}{3}\left[\widetilde{T}_{\rho}{}^{\rho}+2\widetilde{T}_{\theta}{}^{\theta}\right],$
(26b) $\displaystyle\Pi_{\beta}(\rho)$
$\displaystyle=\dfrac{2}{3}\left[\widetilde{T}_{\rho}{}^{\rho}-\widetilde{T}_{\theta}{}^{\theta}\right],$
(26c) these are the (local) energy density, pressure and pressure deviator
respectively [47].

In Fig. 1, we show the energy density, pressure and pressure deviator when
Dirichlet boundary conditions are applied. All three quantities (energy
density, pressure and pressure deviator) tend toward zero as the temperature
is decreased. This corresponds to the thermal RSET tending towards the vacuum
expectation value of the RSET in the low temperature limit, as expected. The
energy density and pressure for Neumann or transparent boundary conditions are
qualitatively extremely similar to those shown in Fig. 1.

In Fig. 2, we show the ratio of the energy density $E_{\beta}^{(\rm N)}(\rho)$
when Neumann boundary conditions are applied to the scalar field to the energy
density $E_{\beta}^{(\rm D)}(\rho)$ when Dirichlet boundary conditions are
applied. This ratio is appreciably different from unity close to the boundary.
While both $E_{\beta}^{(\rm N)}(\rho)$ and $E_{\beta}^{(\rm D)}(\rho)$ tend to
zero as $\rho\rightarrow\tfrac{\pi}{2}$, we see from Fig. 2 that the energy
density for Neumann boundary conditions is going to zero more slowly than the
energy density for Dirichlet boundary conditions. It is notable from Fig. 2
that the ratio $E_{\beta}^{(\rm N)}(\rho)/E_{\beta}^{(\rm D)}(\rho)$ increases
as the temperature decreases.

Similar behaviour is observed for the ratio of the energy density
$E_{\beta}^{(\rm T)}(\rho)$ for transparent boundary conditions to the
Dirichlet energy density $E_{\beta}^{(\rm D)}(\rho)$, except that
$E_{\beta}^{(\rm T)}(\rho)/E_{\beta}^{(\rm D)}(\rho)$ is approximately half
$E_{\beta}^{(\rm N)}(\rho)/E_{\beta}^{(\rm D)}(\rho)$. The corresponding
pressure ratios $P_{\beta}^{(\rm N)}(\rho)/P_{\beta}^{(\rm D)}(\rho)$ and
$P_{\beta}^{(\rm T)}(\rho)/P_{\beta}^{(\rm D)}(\rho)$ have similar properties.
The pressure deviator for the case of Neumann boundary conditions is exactly
minus the pressure deviator in the case of Dirichlet boundary conditions, that
is, $\Pi_{\beta}^{(\rm D)}(\rho)=-\Pi_{\beta}^{(\rm N)}(\rho)$. In the case of
transparent boundary conditions the pressure deviator is identically zero.

Unlike the RSET in the vacuum state, we see from Fig. 1 that the thermal RSET
is not maximally symmetric. A maximally symmetric state has constant energy
density which equals the pressure, and vanishing pressure deviator. The energy
density and pressure distributions shown in Fig. 1 tend to ‘clump’ towards the
origin at $\rho=0$ and tend to zero as the space-time boundary at $\rho=\pi/2$
is approached [47]. In contrast to the situations discussed in Sec. II.2, we
therefore expect that the QC metric resulting from the thermal RSET will not
be maximally symmetric. Finding this QC metric is the focus of the remainder
of the paper.

##  IV QC Einstein equations

In this section we derive the explicit form of the QCEE (15) governing QC AdS
space-times and outline our numerical method for solving these equations.

###  IV.1 QCEE for QC AdS

We start with our ansatz for the QC metric, which takes the form

$\mathop{}\\!\mathrm{d}{{s}}_{\rm{QC}}^{2}=-A(\rho)\mathop{}\\!\mathrm{d}{\tau}^{2}+B(\rho)\mathop{}\\!\mathrm{d}{\rho}^{2}+C(\rho)\mathop{}\\!\mathrm{d}{\Omega}^{2}$
(27)

where $A(\rho)$, $B(\rho)$ and $C(\rho)$ are functions of $\rho$, to be
determined by the QCEE (15):

$G_{\mu}{}^{\nu}+{\widetilde{\Lambda}}\delta_{\mu}{}^{\nu}={\widetilde{T}}_{\mu}{}^{\nu},$
(28)

where ${\widetilde{T}}_{\mu}{}^{\nu}$ is the difference between the thermal
and vacuum RSET (24) and
${\widetilde{\Lambda}}={\widetilde{\Lambda}}_{\rm{AdS}}$ is the renormalized
cosmological constant given by (23). With the metric ansatz (27), the QCEE
(28) take the form

$\displaystyle{\widetilde{T}}_{\tau}{}^{\tau}=$
$\displaystyle\dfrac{4B^{2}C({\widetilde{\Lambda}}C-1)-2B^{\prime}CC^{\prime}-B(C^{\prime
2}-4CC^{\prime\prime})}{4B^{2}C^{2}},$ (29a)
$\displaystyle{\widetilde{T}}_{\rho}{}^{\rho}=$
$\displaystyle\dfrac{2A^{\prime}CC^{\prime}+A(4BC[{\widetilde{\Lambda}}C-1]+C^{\prime
2})}{4ABC^{2}},$ (29b) $\displaystyle{\widetilde{T}}_{\theta}{}^{\theta}=$
$\displaystyle-\dfrac{1}{4A^{2}B^{2}C^{2}}\left[A^{\prime 2}BC^{2}\right.$
$\displaystyle\left.-AC(A^{\prime}BC^{\prime}+C\\{2A^{\prime\prime}B-A^{\prime}B^{\prime}\\})\right.$
$\displaystyle-\left.A^{2}(4{\widetilde{\Lambda}}B^{2}C^{2}-B^{\prime}CC^{\prime}-B\\{C^{\prime
2}-2CC^{\prime\prime}\\})\right],$ (29c)

where a prime denotes differentiation with respect to $\rho$, and the
$\varphi$ component has been omitted as
${\widetilde{T}}_{\theta}{}^{\theta}={\widetilde{T}}_{\varphi}{}^{\varphi}$.

Using (29a, 29b), we find that (29c) can be written as

$\left({\widetilde{T}}_{\theta}{}^{\theta}-{\widetilde{T}}_{\rho}{}^{\rho}\right)-\dfrac{C}{C^{\prime}}{\widetilde{T}}_{\rho}{}^{\rho}{}^{\prime}+\dfrac{A^{\prime}C}{2AC^{\prime}}\left({\widetilde{T}}_{\tau}^{\tau}-{\widetilde{T}}_{\rho}{}^{\rho}\right)=0.$
(30)

Taking the covariant derivative of the RSET gives

$\nabla_{\mu}{\widetilde{T}}_{\rho}{}^{\mu}=\\\
\dfrac{1}{2B}\left[\dfrac{2C^{\prime}}{C}\left({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\theta}{}^{\theta}\right)+2{\widetilde{T}}_{\rho}{}^{\rho}{}^{\prime}+\dfrac{A^{\prime}}{A}\left({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\tau}{}^{\tau}\right)\right],$
(31)

which is precisely (30) multiplied by $-\tfrac{C^{\prime}}{BC}$. Hence,
provided the QCEE (29a–29c) hold, conservation of the RSET follows, as is
expected. Thus the QCEE and RSET conservation give three independent equations
for three unknown functions $A(\rho)$, $B(\rho)$ and $C(\rho)$.

To simplify these, we first rearrange (29a, 29b, 30) to give

$\displaystyle 0$
$\displaystyle={\widetilde{\Lambda}}-{\widetilde{T}}_{\tau}{}^{\tau}-\dfrac{C^{\prime
2}}{4BC^{2}}+\dfrac{2BC^{\prime\prime}-2B^{2}-B^{\prime}C^{\prime}}{2B^{2}C},$
(32a) $\displaystyle 0$
$\displaystyle={\widetilde{\Lambda}}-{\widetilde{T}}_{\rho}{}^{\rho}+\dfrac{C^{\prime
2}}{4BC^{2}}+\dfrac{A^{\prime}C^{\prime}-2AB}{2ABC},$ (32b) $\displaystyle 0$
$\displaystyle=A^{\prime}C({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\tau}{}^{\tau})+2A(C{\widetilde{T}}_{\rho}{}^{\rho}{})^{\prime}-2AC^{\prime}{\widetilde{T}}_{\theta}{}^{\theta}.$
(32c)

Eq. (32c) can be rearranged to give $A^{\prime}(\rho)/A(\rho)$ in terms of
$C(\rho)$ and RSET components:

$\dfrac{A^{\prime}}{A}=2\dfrac{(C{\widetilde{T}}_{\rho}{}^{\rho}{})^{\prime}-C^{\prime}{\widetilde{T}}_{\theta}{}^{\theta}}{C({\widetilde{T}}_{\tau}{}^{\tau}-{\widetilde{T}}_{\rho}{}^{\rho})}.$
(33)

We use this equation to eliminate $A^{\prime}/A$ from (32b), to give

$0={\widetilde{\Lambda}}-{\widetilde{T}}_{\rho}{}^{\rho}-\dfrac{1}{C}+\dfrac{C^{\prime
2}}{4BC^{2}}\\\
-\dfrac{C^{\prime}(C{\widetilde{T}}_{\rho}{}^{\rho}{}^{\prime}+C^{\prime}{\widetilde{T}}_{\rho}{}^{\rho}-C^{\prime}{\widetilde{T}}_{\theta}{}^{\theta})}{BC^{2}({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\tau}{}^{\tau})}.$
(34)

We can now rearrange (34) to give an expression for $B(\rho)$ in terms of
$C(\rho)$ (and its derivatives) and RSET components (and derivatives thereof):

$B=\dfrac{C^{\prime
2}({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\tau}{}^{\tau})-4C^{\prime}(C{\widetilde{T}}_{\rho}{}^{\rho}{}^{\prime}+C^{\prime}{\widetilde{T}}_{\rho}{}^{\rho}-C^{\prime}{\widetilde{T}}_{\theta}{}^{\theta})}{4C({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\tau}{}^{\tau})(1-C{\widetilde{\Lambda}}+C{\widetilde{T}}_{\rho}{}^{\rho})}.$
(35)

We then substitute for $B(\rho)$ and $B^{\prime}(\rho)$ from (35) into (32a)
to yield a single equation for $C$:

$\displaystyle 0$
$\displaystyle=\dfrac{1}{CC^{\prime}}\left(2CC^{\prime\prime}+C^{\prime
2}\right)\dfrac{\left({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\tau}{}^{\tau}\right)\Xi_{\rho}}{\Upsilon}$
$\displaystyle\quad+2\dfrac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}{\rho}}\left(\dfrac{\left({\widetilde{T}}_{\rho}{}^{\rho}-{\widetilde{T}}_{\tau}{}^{\tau}\right)\Xi_{\rho}}{\Upsilon}\right)+\dfrac{\Xi_{\tau}}{C},$
(36a) where we have defined
$\Upsilon=C^{\prime}\left({\widetilde{T}}_{\tau}{}^{\tau}+3{\widetilde{T}}_{\rho}{}^{\rho}-4{\widetilde{T}}_{\theta}{}^{\theta}\right)+4C{\widetilde{T}}_{\rho}{}^{\rho}{}^{\prime}$
(36b) and
$\Xi_{\bullet}=C\left({\widetilde{\Lambda}}-{\widetilde{T}}_{\bullet}{}^{\bullet}\right)-1.$
(36c)

We have therefore reduced the QCEE to a single, second order, nonlinear ODE
(ordinary differential equation) for the metric function $C(\rho)$. Once
$C(\rho)$ is known, we can then use (35) to give the metric function $B(\rho)$
and also solve the first order ODE (33) to give the remaining metric function
$A(\rho)$. To solve these ODEs, we require boundary conditions on the metric
functions, which are described next.

###  IV.2 Solving the QC Einstein equations

The QCEE (33, 35, 36) are sufficiently complicated that proceeding numerically
is required to find the QC metric functions $A(\rho)$, $B(\rho)$ and
$C(\rho)$. Our assumptions about the form of $C(\rho)$ (see (39) below) mean
that these equations are singular at $\rho=0$. We will start our numerical
integration at $\rho=\epsilon$ for some positive constant $\epsilon\ll 1$ and
integrate for increasing $\rho$ towards the space-time boundary at
$\rho=\tfrac{\pi}{2}$. For this procedure, we require boundary conditions on
the QC metric functions at $\rho=\epsilon$.

We first perform a Taylor series expansion near $\rho=0$ of the three
functions ${\mathcal{F}}_{i}$, $i=1,2,3$ (24c), which take the form:

$\displaystyle{\mathcal{F}}_{1}(\rho)=$
$\displaystyle~{}\dfrac{4}{9}\left[\lambda\widetilde{f}_{1}(a\beta)+3f_{3}(a\beta)+2\lambda\widetilde{f}_{3}(a\beta)\right]-\dfrac{8}{45}\left[\lambda\widetilde{f}_{1}(a\beta)+15f_{3}(a\beta)+10\lambda\widetilde{f}_{3}(a\beta)+4\lambda\widetilde{f}_{5}(a\beta)\right]\rho^{2}+{\mathcal{O}}(\rho^{4}),$
$\displaystyle{\mathcal{F}}_{2}(\rho)=$
$\displaystyle~{}\dfrac{16}{9}\left[\lambda\widetilde{f}_{1}(a\beta)+3f_{3}(a\beta)+2\lambda\widetilde{f}_{3}(a\beta)\right]-\dfrac{16}{45}\left[3\lambda\widetilde{f}_{1}(a\beta)+30f_{3}(a\beta)+20\lambda\widetilde{f}_{3}(a\beta)+7\lambda\widetilde{f}_{5}(a\beta)\right]\rho^{2}+{\mathcal{O}}(\rho^{4}),$
$\displaystyle{\mathcal{F}}_{3}(\rho)=$
$\displaystyle~{}-\dfrac{16\lambda}{45}\left[\widetilde{f}_{1}(a\beta)-\widetilde{f}_{5}(a\beta)\right]\rho^{2}+{\mathcal{O}}(\rho^{4}),$
(37)

where $f_{m}(a\beta)$ and $\widetilde{f}_{m}(a\beta)$ are given in (24d–24e).
Due to the form of the functions in (24c), the above series expansions for
small $\rho$ involve only even powers of $\rho$. Substituting the series (37)
into the expression for the RSET (24a), the RSET components have the following
series expansions for small $\rho$:

$\displaystyle{\widetilde{T}}_{\tau}{}^{\tau}(\rho)$ $\displaystyle=~{}T_{\tau
0}+T_{\tau 2}\rho^{2}+{\mathcal{O}}(\rho^{4}),$
$\displaystyle{\widetilde{T}}_{\rho}{}^{\rho}(\rho)$ $\displaystyle=~{}T_{\rho
0}+T_{\rho 2}\rho^{2}+{\mathcal{O}}(\rho^{4}),$
$\displaystyle{\widetilde{T}}_{\theta}{}^{\theta}(\rho)$
$\displaystyle=~{}T_{\theta 0}+T_{\theta 2}\rho^{2}+{\mathcal{O}}(\rho^{4}),$
(38a) where $\displaystyle T_{\tau 0}$
$\displaystyle=~{}-\dfrac{a^{4}}{6\pi^{2}}\left[\lambda\widetilde{f}_{1}(a\beta)+3f_{3}(a\beta)+2\lambda\widetilde{f}_{3}(a\beta)\right],$
$\displaystyle T_{\tau 2}$
$\displaystyle=~{}\dfrac{a^{4}}{9\pi^{2}}\left[\lambda\widetilde{f}_{1}(a\beta)+9f_{3}(a\beta)+6\lambda\widetilde{f}_{3}(a\beta)\right.$
$\displaystyle\qquad\left.+2\lambda\widetilde{f}_{5}(a\beta)\right],$
$\displaystyle T_{\rho 0}$
$\displaystyle=~{}\dfrac{a^{4}}{18\pi^{2}}\left[\lambda\widetilde{f}_{1}(a\beta)+3f_{3}(a\beta)+2\lambda\widetilde{f}_{3}(a\beta)\right],$
$\displaystyle T_{\rho 2}$
$\displaystyle=~{}-\dfrac{a^{4}}{45\pi^{2}}\left[3\lambda\widetilde{f}_{1}(a\beta)+15f_{3}(a\beta)+10\lambda\widetilde{f}_{3}(a\beta)\right.$
$\displaystyle\qquad\left.+2\lambda\widetilde{f}_{5}(a\beta)\right],$
$\displaystyle T_{\theta 0}$ $\displaystyle=~{}T_{\rho
0}=\dfrac{a^{4}}{18\pi^{2}}\left[\lambda\widetilde{f}_{1}(a\beta)+3f_{3}(a\beta)+2\lambda\widetilde{f}_{3}(a\beta)\right],$
$\displaystyle T_{\theta 2}$
$\displaystyle=~{}-\dfrac{a^{4}}{45\pi^{2}}\left[\lambda\widetilde{f}_{1}(a\beta)+15f_{3}(a\beta)+10\lambda\widetilde{f}_{3}(a\beta)\right.$
$\displaystyle\qquad\left.+4\lambda\widetilde{f}_{5}(a\beta)\right].$ (38b)

Since the components of the RSET have Taylor series expansions (38) for small
$\rho$ which contain only even powers of $\rho$, we assume that the same is
true of the QC metric functions $A(\rho)$, $B(\rho)$ and $C(\rho)$:

$\displaystyle A(\rho)$
$\displaystyle=A_{0}+A_{2}\rho^{2}+{\mathcal{O}}(\rho^{4}),$ (39a)
$\displaystyle B(\rho)$
$\displaystyle=B_{0}+B_{2}\rho^{2}+{\mathcal{O}}(\rho^{4}),$ (39b)
$\displaystyle C(\rho)$
$\displaystyle=C_{2}\rho^{2}+C_{4}\rho^{4}+{\mathcal{O}}(\rho^{6}),$ (39c)

where $A_{i}$, $B_{i}$, $C_{i}$ are constant coefficients. In (39), we have
assumed that the QC metric function $C(\rho)$ is ${\mathcal{O}}(\rho^{2})$ as
$\rho\rightarrow 0$, which is the same behaviour as in the original AdS metric
(21).

We additionally assume that the metric (27) has a regular origin at $\rho=0$.
In order to avoid a curvature singularity at $\rho=0$, it must be the case
that

$B_{0}=C_{2}.$ (40)

Since we have assumed that the QC metric (27) is static, we can rescale the
time coordinate $\tau$, which leads to a rescaling of the metric function
$A(\rho)$. For this reason the coefficient $A_{0}$ in (39a) is not determined
by the QCEE (33). We can therefore set

$A_{0}=1$ (41)

without loss of generality. All the remaining constants in (39) can be fixed
by substituting the series (38, 39) into the QCEE (33, 35, 36a) and ensuring
that these equations are satisfied order-by-order in $\rho$. The lowest order
term in (33) gives

$A_{2}=\dfrac{2\left(2T_{\rho 2}-T_{\theta 2}\right)}{T_{\tau 0}-T_{\theta
0}},$ (42)

which is finite since, from (38), we have $T_{\tau 0}\neq T_{\theta 0}$. From
the lowest-order term in (36a), we find

$C_{2}=\dfrac{12\left(2T_{\theta 2}-T_{\rho 2}\right)}{\left(T_{\tau
0}-T_{\theta 0}\right)\left(T_{\tau 0}-3T_{\theta
0}+2{\widetilde{\Lambda}}\right)},$ (43)

where both terms in the denominator are nonvanishing.

Following the above procedure, we can find series expansions of the RSET and
QC metric functions to any desired order in $\rho$, although the algebraic
expressions for the coefficients in the QC metric functions rapidly become
extremely complicated.

Our numerical method begins by integrating the nonlinear equation for
$C(\rho)$ (36), using Mathematica’s inbuilt NDSolveValue command. We begin the
numerical integration at $\rho=\epsilon=10^{-3}$. Initial conditions for the
numerical integration are given by the series expansion (39c). We perform a
series expansion in the RSET components up to ${\mathcal{O}}(\rho^{8})$, which
is sufficient to give series in $A(\rho)$ and $B(\rho)$ up to
${\mathcal{O}}(\rho^{6})$ and in $C(\rho)$ up to ${\mathcal{O}}(\rho^{8})$.
The coefficients in the series expansions of the RSET components (38) involve
infinite sums which converge extremely rapidly; we included just the first 100
terms in each sum, which was sufficient for our purposes. With series
expansions up to the orders given above, and taking $\epsilon$ in the range
$10^{-2}-10^{-3}$ we estimate the relative error in the solutions of the QCEE
to be approximately $10^{-8}$.

The NDSolveValue command gives $C(\rho)$ as an interpolating function. From
this, the QC metric function $B(\rho)$ is given directly by (35), without any
further integration. To find the remaining QC metric function $A(\rho)$, we
again apply the NDSolveValue command to numerically integrate (33) from
$\rho=\epsilon$, using the series (39a) up to ${\mathcal{O}}(\rho^{6})$ as the
initial condition. In all our numerical integration, we end the integration at
$\rho=\tfrac{\pi}{2}-\epsilon$ since the QC metric functions $A(\rho)$,
$B(\rho)$ and $C(\rho)$ all diverge as $\rho\rightarrow\tfrac{\pi}{2}$ (as is
the case for the original AdS metric functions (21)). In all our numerical
computations we have set the original inverse AdS length scale $a$ to be equal
to unity; this means the the renormalized inverse AdS length scale is

${\widetilde{a}}={\sqrt{-\dfrac{{\widetilde{\Lambda}}}{3}}}=0.9999824\quad{\mbox{to
7 significant figures}},$ (44)

where the renormalized cosmological constant ${\widetilde{\Lambda}}$ is given
by (23).

##  V QC AdS metrics

In this section we present our results for the QC AdS metrics found by solving
the QC Einstein equations, as discussed in the previous section. We first
consider the solutions when Dirichlet or Neumann boundary conditions have been
applied to the quantum scalar field, before turning to the case of transparent
boundary conditions. Our main interest in this section is to explore the
deviation of the QC metrics from pure AdS space-time, as well as considering
the effect the boundary conditions have on the QC metrics.

###  V.1 Dirichlet and Neumann boundary conditions

Figure 3: Ratios between the components of the QC metric (27) and the pure AdS
metric (21) for Neumann boundary conditions. We have fixed the
(unrenormalized) inverse AdS length scale $a=1$ and the inverse temperature
$\beta=\tfrac{\pi}{4}$.

In Fig. 3 we show a typical QC metric solution. We have fixed the inverse
temperature in the RSET to be $\beta=\tfrac{\pi}{4}$ and applied Neumann
boundary conditions to the quantum field. We see that the metric function
$A^{(\rm{N})}(\rho)$ is indistinguishable from that in pure AdS when plotted
on this scale, while the metric functions $B^{(\rm{N})}(\rho)$ and
$C^{(\rm{N})}(\rho)$ are significantly different from the corresponding
functions in pure AdS. At the origin $\rho=0$, we have fixed
$A^{(\rm{N})}(\rho)=1$ as seen in Fig. 3, and we also see that
$B^{(\rm{N})}(\rho)/\sec^{2}\rho$ and $C^{(\rm{N})}(\rho)/\tan^{2}\rho$ have
the same value, in accordance with the series expansions (39b, 39c, 40). The
function $B^{(\rm{N})}(\rho)/\sec^{2}\rho$ is monotonically increasing from
the origin towards the boundary at $\rho=\pi/2$, except for a small region
close to the boundary where it is monotonically decreasing. In contrast, the
function $C^{(\rm{N})}(\rho)/\tan^{2}\rho$ is decreasing close to the origin,
has a local minimum and then increases close to the boundary. Typical
solutions for Dirichlet boundary conditions show similar behaviour.

Figure 4: Ratios between the components of the QC metric (27) and the pure AdS
metric (21) for Dirichlet boundary conditions. The top plot shows
$A^{(\rm{D})}(\rho)/\sec^{2}\rho$, the middle plot
$B^{(\rm{D})}(\rho)/\sec^{2}\rho$ and the bottom plot
$C^{(\rm{D})}(\rho)/\tan^{2}\rho$. Keeping the inverse AdS length scale $a=1$
fixed, the inverse temperature lies in the range
$\beta\in(\frac{\pi}{12},\frac{11\pi}{24})$.

We now study how the QC metrics change as we vary the inverse temperature
$\beta$ in the RSET. In Fig. 4 we plot the ratios
$A^{(\rm{D})}(\rho)/\sec^{2}\rho$ (top plot),
$B^{(\rm{D})}(\rho)/\sec^{2}\rho$ (middle plot) and
$C^{(\rm{D})}(\rho)/\tan^{2}(\rho)$ (bottom plot) for Dirichlet boundary
conditions. We have found QC metrics for a range of inverse temperatures, with
$\beta=\tfrac{\pi}{12}$ corresponding to the highest temperature studied. The
zero-temperature limit corresponds to $\beta\rightarrow\infty$; in this case
the QC metric is just pure AdS with a renormalized cosmological constant (23).

First consider the ratio $A^{(\rm{D})}(\rho)/\sec^{2}\rho$ (top plot in Fig.
4). Looking at the vertical scale, it can be seen that this ratio is very
close to unity for all values of the temperature considered and all
$\rho\in[0,\tfrac{\pi}{2})$. The only significant deviation from unity is
close to the boundary as $\rho\rightarrow\tfrac{\pi}{2}$, with the deviations
from unity increasing as the temperature increases.

Turning now to the ratio $B^{(\rm{D})}(\rho)/\sec^{2}\rho$ (middle plot in
Fig. 4), we find much more significant variations compared to the pure AdS
case. At the origin, the value of the ratio $B^{(\rm{D})}(\rho)/\sec^{2}\rho$
decreases rapidly as the temperature $\beta^{-1}$ increases, and is close to
zero for the large temperature $\beta^{-1}=\tfrac{12}{\pi}$. For all values of
the inverse temperature studied, we find that the ratio
$B^{(\rm{D})}(\rho)/\sec^{2}\rho$ initially increases as $\rho$ increases from
zero. The ratio $B^{(\rm{D})}(\rho)/\sec^{2}\rho$ then has a maximum for
$\rho$ close to the boundary. For sufficiently high temperature, the maximum
value of the ratio $B^{(\rm{D})}(\rho)/\sec^{2}\rho$ is greater than unity. To
within the accuracy of our numerical integration, it appears to be the case
that $B^{(\rm{D})}(\rho)/\sec^{2}\rho$ is approaches unity as
$\rho\rightarrow\tfrac{\pi}{2}$ and the space-time boundary is approached.

Finally we consider the ratio $C^{(\rm{D})}(\rho)/\tan^{2}\rho$ (bottom plot
in Fig. 4). Like the ratio $B^{(\rm{D})}(\rho)/\sec^{2}\rho$, at the origin
$\rho=0$ the ratio $C^{(\rm{D})}(\rho)/\tan^{2}\rho$ decreases rapidly as the
temperature increases, and is close to zero for the highest temperature
studied (corresponding to $\beta=\tfrac{\pi}{12}$). Unlike the ratio
$B^{(\rm{D})}(\rho)/\sec^{2}\rho$, we see that
$C^{(\rm{D})}(\rho)/\tan^{2}\rho$ does not change greatly as the radial
coordinate $\rho$ increases. The ratio generally has a local minimum for some
value of $\rho$ and is slightly increasing close to the space-time boundary.
The value that the ratio $C^{(\rm{D})}(\rho)/\tan^{2}\rho$ approaches as
$\rho\rightarrow\tfrac{\pi}{2}$ decreases with increasing temperature.

Figure 5: Ratios between the components of the QC metric (27) when Neumann
boundary conditions are applied to the scalar field and the corresponding QC
metric components when Dirichlet boundary conditions are applied. The top plot
shows $A^{(\rm N)}(\rho)/A^{(\rm D)}(\rho)$, the middle plot $B^{(\rm
N)}(\rho)/B^{(\rm D)}(\rho)$ and the bottom plot $C^{(\rm N)}(\rho)/C^{(\rm
D)}(\rho)$. Keeping the inverse AdS length scale $a=1$ fixed, the inverse
temperature lies in the range $\beta\in(\frac{\pi}{12},\frac{11\pi}{24})$.

When Neumann boundary conditions are applied to the scalar field, the plots of
the ratios of the QC metric components to the AdS metric components $B^{(\rm
N)}(\rho)/\sec^{2}\rho$ and $C^{(\rm N)}(\rho)/\tan^{2}(\rho)$ are virtually
indistinguishable by eye from the corresponding plots for Dirichlet boundary
conditions, shown in Fig. 4. In Fig. 5 we therefore show the ratios of the QC
metric components when Neumann boundary conditions are applied to the
corresponding QC metric components when Dirichlet boundary conditions are
applied. From these ratios, we see that the difference between the QC metric
functions for the different boundary conditions is very small, typically less
than one percent.

When we examine the data for the ratio $A^{(\rm{N})}(\rho)/\sec^{2}(\rho)$,
while this is less than unity close to the boundary when Dirichlet boundary
conditions are applied, it is greater than unity for Neumann boundary
conditions. Furthermore, the deviations from unity are larger for Neumann
boundary conditions compared with Dirichlet boundary conditions, and become
significant at smaller values of $\rho$ for Neumann boundary conditions
compared with Dirichlet.

From the plots of the ratios of the QC metric functions $B(\rho)$ and
$C(\rho)$ shown in Fig. 5, close to the origin at $\rho=0$ these ratios are
closer to unity for higher temperatures. Thus the effect of the boundary
conditions on the QC metric is less significant at higher temperatures. This
is in accordance with the properties of the RSET [40], whose components at
high temperature show little dependence on the boundary conditions applied to
the scalar field. Close to the space-time boundary, the QC metric ratios in
Fig. 5 are significantly greater than unity, and increase with increasing
temperature, mimicking the behaviour of the RSET close to the boundary in Fig.
2.

###  V.2 Transparent boundary conditions

When transparent boundary conditions are applied to the quantum scalar field,
the RSET (24) simplifies considerably:

${\widetilde{T}}_{\mu}{}^{\nu}=\dfrac{a^{2}}{6\pi^{2}}\cos^{4}\rho\,f_{3}(a\beta)\,{\rm{Diag}}\,\\{-3,1,1,1\\},$
(45)

where $f_{3}(a\beta)$ is given in (24d). In this case the equation for
$A(\rho)$ (33) reduces to

$\dfrac{A^{\prime}}{A}=2\tan\rho,$ (46)

which can be directly integrated to give

$A(\rho)=\sec^{2}\rho,$ (47)

where we have chosen a constant of integration such that $A(0)=1$, as in (41).
Therefore the QC metric function $A^{(\rm{T})}(\rho)$ takes exactly the same
form as in pure AdS.

The QC metric functions $B^{(\rm{T})}(\rho)$ and $C^{(\rm{T})}(\rho)$ are,
however, not the same as for pure AdS. The equations governing these (35, 36a)
do however simplify, giving the following equation for $C(\rho)$:

$\displaystyle 0$
$\displaystyle=\frac{\Xi_{\tau}}{C}+\dfrac{4}{CC^{\prime}}\left(2CC^{\prime\prime}+C^{\prime
2}\right)\dfrac{\Xi_{\rho}{\widetilde{T}}_{\rho}{}^{\rho}}{\Upsilon}+8\dfrac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}{\rho}}\left(\dfrac{\Xi_{\rho}{\widetilde{T}}_{\rho}{}^{\rho}}{\Upsilon}\right),$
where $\Upsilon$ (36b) is now
$\Upsilon=4\left(C{\widetilde{T}}_{\rho}{}^{\rho}{}^{\prime}-C^{\prime}{\widetilde{T}}_{\rho}{}^{\rho}\right),$
(48a) and $\Xi_{\bullet}$ (36c) is unchanged.

The equation for $B(\rho)$ in terms of $C(\rho)$ (35) also simplifies
considerably:

$B(\rho)=-\dfrac{C^{\prime}}{4C\Xi_{\rho}}\left(C^{\prime}+4C\tan\rho\right).$
(49)

Figure 6: Ratios between the components of the QC metric (27) when transparent
boundary conditions are applied to the scalar field and the corresponding QC
metric components when Dirichlet boundary conditions are applied. The upper
plot shows $B^{(\rm T)}(\rho)/B^{(\rm D)}(\rho)$ and the lower plot $C^{(\rm
T)}(\rho)/C^{(\rm D)}(\rho)$. Keeping the inverse AdS length scale $a=1$
fixed, the inverse temperature lies in the range
$\beta\in(\frac{\pi}{12},\frac{11\pi}{24})$.

Although the equations governing $C(\rho)$ (48) and $B(\rho)$ (49) are much
simpler than for either Dirchlet or Neumann boundary conditions, the resulting
metric functions $B^{(\rm{T})}(\rho)$ and $C^{(\rm{T})}(\rho)$ are very
similar to those shown in Fig. 4. In particular, we find that
$B^{(\rm{T})}(\rho)$ and $C^{(\rm{T})}(\rho)$ for transparent boundary
conditions are not the same as for pure AdS, even though the metric function
$A^{(\rm{T})}(\rho)$ (47) does take the AdS form. Fig. 6 shows the ratios of
the QC metric functions when transparent boundary conditions are applied to
those for Dirichlet boundary conditions, namely $B^{(\rm T)}(\rho)/B^{(\rm
D)}(\rho)$ (upper plot) and $C^{(\rm T)}(\rho)/C^{(\rm D)}(\rho)$ (lower
plot). At low temperatures, the QC metric functions $B^{(\rm T)}(\rho)$ and
$C^{(\rm T)}(\rho)$ for transparent boundary conditions are indistinguistable
from the corresponding QC metric functions for Dirichlet boundary conditions.
At higher temperatures, the ratios $B^{(\rm T)}(\rho)/B^{(\rm D)}(\rho)$ and
$C^{(\rm T)}(\rho)/C^{(\rm D)}(\rho)$ are significantly different from unity
(more so than the ratios $B^{(\rm N)}(\rho)/B^{(\rm D)}(\rho)$ and $C^{(\rm
N)}(\rho)/C^{(\rm D)}(\rho)$ shown in Fig. 5).

To derive the vacuum (and hence thermal) Green’s function with transparent
boundary conditions applied to the conformally-coupled scalar field on AdS, we
first make a conformal transformation to the Einstein static universe (ESU)
with metric

$\mathop{}\\!\mathrm{d}{{\widetilde{s}}}^{2}=a^{-2}\left[-\mathop{}\\!\mathrm{d}{\tau}^{2}+\mathop{}\\!\mathrm{d}{\rho}^{2}+\sin^{2}\rho\mathop{}\\!\mathrm{d}{\Omega}^{2}\right].$
(50)

Since ESU is a globally hyperbolic space-time, it is not necessary to apply
boundary conditions to the field on ESU. Having found the vacuum Green’s
function on ESU, the corresponding Green’s function on AdS will satisfy
transparent boundary conditions [32]. The resulting vacuum Green’s function on
AdS is then used to construct the corresponding thermal Green’s function on
AdS, from which the RSET (45) is then obtained [33]. It is therefore
informative to compare our results in this section for the QC AdS metric with
the corresponding QC ESU metric. ESU solutions of the SCEE (2) are studied in
detail in [7]. Our analysis below is rather simpler, as we are considering a
massless, conformally coupled scalar field and are only interested in the QC
metric rather than solving the full SCEE.

Unlike AdS, the ESU is not a maximally symmetric space-time, and so the
general approach in Sec. II is not immediately applicable. The ESU is a
solution of the classical Einstein equations (1) with a cosmological constant
$\Lambda=a^{2}$ and a nonzero classical stress-energy tensor

${}^{\rm{dust}}T_{\mu}{}^{\nu}=-2a^{2}\,{\rm{Diag}}\\{1,0,0,0\\},$ (51)

corresponding to pressureless dust. Following [7], for the rest of this
section only we reinstate Newton’s constant and consider the modified QCEE

$G_{\mu}{}^{\nu}+\Lambda\delta_{\mu}{}^{\nu}=8\pi
G\left[{}^{\rm{dust}}T_{\mu}{}^{\nu}+\langle{\hat{T}}_{\mu}{}^{\nu}\rangle^{\rm{ESU}}\right]$
(52)

where $\langle{\hat{T}}_{\mu}{}^{\nu}\rangle^{\rm{ESU}}$ is the RSET for the
quantum scalar field on ESU and we have raised the second index for later
convenience. In (52), we have also set the constants $\alpha$ and $\beta$ in
(2) equal to zero; this will be justified a posteriori later in this section.

The RSET for a massless, conformally coupled scalar field on ESU
$\langle{\hat{T}}_{\mu}{}^{\nu}\rangle^{\rm{ESU}}$ is related to that on AdS
$\langle{\hat{T}}_{\mu}{}^{\nu}\rangle^{\rm{AdS}}$ in the corresponding state
by [2]

$\langle{\hat{T}}_{\mu}{}^{\nu}\rangle^{\rm{AdS}}=\langle{\hat{T}}_{\mu}{}^{\nu}\rangle^{\rm{ESU}}\dfrac{{\sqrt{-{\widetilde{g}}}}}{{\sqrt{-g}}}-\dfrac{1}{2880\pi^{2}}H_{\mu}^{(3)\nu},$
(53)

where $g$, ${\widetilde{g}}$ are, respectively, the determinants of the
metrics (21, 50), and

$\displaystyle H_{\mu\nu}^{(3)}=$
$\displaystyle\frac{1}{3}\nabla_{\mu}\nabla_{\nu}R-\frac{1}{3}g_{\mu\nu}\nabla_{\sigma}\nabla^{\sigma}R-\frac{1}{6}g_{\mu\nu}R^{2}+\frac{1}{3}RR_{\mu\nu}$
$\displaystyle\qquad+R^{\rho\sigma}R_{\rho\mu\sigma\nu}.$ (54)

Since the second term in (53) is independent of the quantum state, we have

${}^{\rm{AdS}}{\widetilde{T}}_{\mu}{}^{\nu}={}^{\rm{ESU}}{\widetilde{T}}_{\mu}{}^{\nu}\cos^{4}\rho,$
(55)

where ${}^{\rm{AdS}}{\widetilde{T}}_{\mu}{}^{\nu}$ is the difference between
the thermal and vacuum expectation values on AdS with transparent boundary
conditions (45) and

${}^{\rm{ESU}}{\widetilde{T}}_{\mu}{}^{\nu}=\langle\beta\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert\beta\rangle^{\rm{ESU}}-\langle
0\lvert{\hat{T}}_{\mu}{}^{\nu}\rvert 0\rangle^{\rm{ESU}}.$ (56)

Therefore the difference between the thermal and vacuum expectation values of
the RSET on ESU takes a particularly simple form:

${}^{\rm{ESU}}{\widetilde{T}}_{\mu}{}^{\nu}=\dfrac{a^{2}}{6\pi^{2}}\,f_{3}(a\beta)\,{\rm{Diag}}\,\\{-3,1,1,1\\}.$
(57)

It is striking that (57) is proportional to the RSET in the vacuum state on
ESU [48]:

$\langle
0_{{\rm{ESU}}}|{\hat{T}}_{\mu}{}^{\nu}|0_{{\rm{ESU}}}\rangle=\dfrac{a^{2}}{480\pi^{2}}{\rm{Diag}}\\{-3,1,1,1\\}.$
(58)

Hence, to find the QC ESU metric, we consider the RSET

$\langle{\hat{T}}_{\mu}{}^{\nu}\rangle^{\rm{ESU}}=a^{2}{\mathcal{P}}\,{\rm{Diag}}\,\\{-3,1,1,1\\},$
(59)

where the constant ${\mathcal{P}}$ is given by

${\mathcal{P}}=\frac{1}{480\pi^{2}}\left[1+80f_{3}(a\beta)\right].$ (60)

To solve (52) with (59) on the right-hand-side, we first define a renormalized
cosmological constant ${\widetilde{\Lambda}}$ (and hence a renormalized
inverse length scale ${\widetilde{a}}$) using the spatial parts of the total
RSET:

${\widetilde{\Lambda}}={\widetilde{a}}^{2}=\Lambda-8\pi
Ga^{2}{\mathcal{P}}=a^{2}\left(1-8\pi G{\mathcal{P}}\right).$ (61)

The modified SCEE (52) now read

$G_{\mu}{}^{\nu}+{\widetilde{\Lambda}}\delta_{\mu}{}^{\nu}=-16\pi
Ga^{2}\,{\rm{Diag}}\,\\{1+2{\mathcal{P}},0,0,0\\}.$ (62)

The right-hand-side of (62) now has the pressure dust form (51) with
renormalized inverse length scale ${\widetilde{a}}$ if we also define a
renormalized Newton constant ${\widetilde{G}}$ (as in [7]) by

$8\pi{\widetilde{G}}=8\pi
G\dfrac{a^{2}}{{\widetilde{a}}^{2}}\left(1+2{\mathcal{P}}\right)=\dfrac{8\pi
G(1+2{\mathcal{P}})}{1-8\pi
G{\mathcal{P}}}=\dfrac{1+2{\mathcal{P}}}{1-{\mathcal{P}}},$ (63)

where in the last equality we have set $8\pi G=1$, as elsewhere in this paper.
We therefore obtain the QCEE

$G_{\mu\nu}^{\rm{QC}}+{\widetilde{\Lambda}}g_{\mu\nu}^{\rm{QC}}=-2{\widetilde{a}}^{2}{\rm{Diag}}\\{1,0,0,0\\},$
(64)

and hence the QC metric takes the ESU form (50), with renormalized inverse
length scale ${\widetilde{a}}$.

Therefore, providing we renormalize both the cosmological constant
${\widetilde{\Lambda}}$ and Newton’s constant ${\widetilde{G}}$, we are
justified in setting the constants $\alpha$ and $\beta$ equal to zero in (52).
While the AdS and ESU space-times are conformally related, and we are dealing
with a conformal quantum field, it is not the case that the resulting QC
metrics are also conformally related, due to the nonlinearity of the QCEE. If
the scalar field is in a thermal state on ESU, the resulting QC metric has ESU
form, but the QC metric sourced by the corresponding thermal state on AdS is
not of the AdS form.

##  VI QC AdS solitons

In the previous section we have constructed the QC AdS metrics by integrating
the QCEE (15). We have found that, in the low temperature limit, the QC AdS
metrics approach pure AdS space-time, with deviations from AdS increasing as
the temperature of the thermal quantum state increases. In this section we
consider an alternative metric ansatz for the QC AdS metrics, which will aid
their interpretation as solitons sourced by the quantum scalar field.

###  VI.1 Alternative QCEE form

To transform the QC AdS metric ansatz (27) into the more familiar form for a
static, spherically symmetric space-time, we first make a change of radial
coordinate, defining a new coordinate $r$ by

$r^{2}=C(\rho).$ (65)

From the results of the previous section, we see that $r\rightarrow 0$ as
$\rho\rightarrow 0$ from (39c). Since $C(\rho)/\tan^{2}\rho$ tends to a finite
nonzero limit as $\rho\rightarrow\pi/2$ (see Fig. 4), we also have
$r\rightarrow\infty$ as $\rho\rightarrow\pi/2$, so that the coordinate $r$ has
range $r\in[0,\infty)$ as is usually the case for the radial coordinate in
spherical polars.

We then make the following metric ansatz:

$\mathop{}\\!\mathrm{d}{s}^{2}=-f(r)e^{\psi(r)}\mathop{}\\!\mathrm{d}{t}^{2}+f(r)^{-1}\mathop{}\\!\mathrm{d}{r}^{2}+r^{2}\mathop{}\\!\mathrm{d}{\Omega}^{2},$
(66)

in terms of functions $f(r)$ and $\psi(r)$ which are to be determined from the
QCEE. We also write $f(r)$ as follows:

$f(r)=1-\dfrac{2m(r)}{r}-\dfrac{{\widetilde{\Lambda}}}{3}r^{2},$ (67)

where ${\widetilde{\Lambda}}$ is the renormalized cosmological constant (23)
and $m(r)$ is to be determined from the QCEE. Pure AdS space-time has
$m(r)\equiv 0$. Differentiating (65) and comparing the two forms (27, 66) of
the QC AdS metric, we have

$\displaystyle f(r)$
$\displaystyle=\dfrac{C^{\prime}(\rho)^{2}}{4B(\rho)C(\rho)},$ $\displaystyle
m(r)$
$\displaystyle=\dfrac{\sqrt{C(\rho)}}{2}\left(1+{\widetilde{a}}^{2}C(\rho)-\dfrac{C^{\prime}(\rho)^{2}}{4B(\rho)C(\rho)}\right),$
$\displaystyle e^{\psi(r)}$
$\displaystyle=\dfrac{4A(\rho)B(\rho)C(\rho)}{C^{\prime}(\rho)^{2}},$ (68)

where ${\widetilde{a}}$ is the renormalized inverse AdS length scale. If we
attempt to find $f(r)$, $m(r)$ and $\psi(r)$ using the expressions (68), we
find that numerical errors in the metric functions $A(\rho)$, $B(\rho)$ and
$C(\rho)$ accumulate rapidly. To obtain numerically satisfactory results for
$f(r)$, $m(r)$ and $\psi(r)$ we instead integrate the QCEE resulting from the
metric ansatz (66).

The QCEE for the metric functions $m(r)$ and $\psi(r)$ take a particularly
simple form:

$\displaystyle\dfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$
$\displaystyle=~{}-\dfrac{r^{2}}{2}{\widetilde{T}}_{\tau}{}^{\tau},$ (69a)
$\displaystyle\dfrac{\mathop{}\\!\mathrm{d}{\psi}}{\mathop{}\\!\mathrm{d}{r}}$
$\displaystyle=~{}\dfrac{r}{f(r)}\left({\widetilde{T}}_{\tau}{}^{\tau}+{\widetilde{T}}_{\rho}{}^{\rho}\right).$
(69b)

Our primary interest lies in the metric function $m(r)$, which we consider in
the following subsection.

###  VI.2 QC AdS soliton masses

(a) Dirichlet boundary conditions

(b) Neumann boundary conditions

(c) Transparent boundary conditions

Figure 7: Derivative
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}$ (70) for
Dirichlet (top), Neumann (middle) and transparent (bottom) boundary
conditions. The inverse temperature lies in the range
$\beta\in(\frac{\pi}{12},\frac{11\pi}{24})$. The inverse AdS length scale is
$a=1$.

In this subsection we study the metric function $m(r)$ (67). Its governing
equation (69a) is readily integrated numerically, by writing it in the form

$\dfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}=-\dfrac{1}{4}C^{\prime}(\rho){\sqrt{C(\rho)}}{\widetilde{T}}_{\tau}{}^{\tau}(\rho),$
(70)

where all the quantities on the right-hand-side are now functions of the
original radial coordinate $\rho$. The derivative (70) is shown in Fig. 7 for
Dirichlet, Neumann and transparent boundary conditions. We see that
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}$ vanishes at
the origin $\rho=0$ for all three boundary conditions. Since the energy
density $-{\widetilde{T}}_{\tau}{}^{\tau}$ is positive for all $\rho$ (see,
for example, Fig. 1), it is also the case that
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}$ is positive
for all $\rho$ and all boundary conditions. However, the behaviour of
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}$ as
$\rho\rightarrow\tfrac{\pi}{2}$ and the boundary is approached depends
strongly on the boundary conditions. For Dirichlet boundary conditions (Fig.
7(a)), the derivative
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}$ has a
maximum which lies closer to the boundary as the temperature increases, and is
monotonically decreasing close to the boundary. This is not the case for
either Neumann (Fig. 7(b)) or transparent (Fig. 7(c)) boundary conditions, for
which it appears that
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}$ is
monotonically increasing towards a finite nonzero value as
$\rho\rightarrow\tfrac{\pi}{2}$, and the value on the boundary increases as
the temperature increases.

Figure 8: Derivative
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ of the metric
function $m(r)$ for Dirichlet boundary conditions. The inverse temperature
lies in the range $\beta\in(\frac{\pi}{12},\frac{11\pi}{24})$, and the inverse
AdS length scale is fixed to be $a=1$. Figure 9: Derivative
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ of the metric
function $m(r)$ for Dirichlet, Neumann and transparent boundary conditions at
a fixed inverse temperature, $\beta=\frac{\pi}{4}$. The inverse AdS length
scale is fixed to be $a=1$.

We also consider the derivative
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ (69a), as shown
in Figs. 8 and 9. In Figs. 8 and 9 we have evaluated the right-hand-side of
(69a) using (65) and transformed the resulting function of $\rho$ to a
function of $r$, again using (65). In Fig. 8 we present
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ for Dirichlet
boundary conditions and a selection of values of the inverse temperature
$\beta$. As expected,
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ is zero when
$r=0$, and tends to zero as $r\rightarrow\infty$. The maximum of
$m^{\prime}(r)$ occurs at a value of $r$ which is temperature-dependent,
occurring at smaller values of $r$ for higher temperatures.

Qualitatively similar behaviour is seen in
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ for Neumann and
transparent boundary conditions, namely
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ vanishes at
$r=0$ and tends to zero as $r\rightarrow\infty$. We find that the rate at
which $\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ as
$r\rightarrow\infty$ depends on the boundary conditions, as can be seen in
Fig. 9 for inverse temperature $\beta=\tfrac{\pi}{4}$ (we find similar
behaviour for other values of the inverse temperature). For larger values of
$r$, the derivative
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ takes the
smallest values for Dirichlet boundary conditions, and the largest values for
Neumann boundary conditions. This is in accordance with the behaviour of
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{\rho}}$ in Fig. 7.

Using the form of the RSET (24), it is straightforward to show that, for all
boundary conditions,

${\widetilde{T}}_{\tau}{}^{\tau}\sim-{\mathcal{T}}_{\infty}^{\lambda}\left(\dfrac{\pi}{2}-\rho\right)^{4}+{\mathcal{O}}\left(\dfrac{\pi}{2}-\rho\right)^{6}$
(71a) as $\rho\rightarrow\tfrac{\pi}{2}$, where
${\mathcal{T}}_{\infty}^{\lambda}=\dfrac{a^{2}}{2\pi^{2}}\sum_{n=1}^{\infty}\left[n^{3}-\dfrac{\lambda}{9}n\left(1+8n^{2}\right)\right]\left(e^{na\beta}-1\right)^{-1}.$
(71b) From the plots in Sec. V, we see that $C(\rho)/\tan^{2}\rho$ approaches
a finite nonzero limit ${\mathcal{C}}_{\infty}^{\lambda}(\beta)$ as
$\rho\rightarrow\infty$, so that
$r^{2}=C(\rho)\sim{\mathcal{C}}_{\infty}^{\lambda}(\beta)\left(\dfrac{\pi}{2}-\rho\right)^{-2}+{\mathcal{O}}\left(\dfrac{\pi}{2}-\rho\right)^{0}$
(71c)

as $\rho\rightarrow\tfrac{\pi}{2}$, where
${\mathcal{C}}_{\infty}^{\lambda}(\beta)$ will depend on the boundary
conditions applied and the inverse temperature $\beta$. Combining the results
in (71), we have

$\dfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}\sim\frac{1}{2}{\mathcal{C}}_{\infty}^{\lambda}(\beta){\mathcal{T}}_{\infty}^{\lambda}\left(\dfrac{\pi}{2}-\rho\right)^{2}\sim{\mathcal{O}}(r^{-2})$
(72)

as $r\rightarrow\infty$ ($\rho\rightarrow\tfrac{\pi}{2}$). From (71b), we have
$0<{\mathcal{T}}_{\infty}^{1}<{\mathcal{T}}_{\infty}^{0}<{\mathcal{T}}_{\infty}^{-1}$,
which is in agreement with the results in Fig. 9, since $\lambda$ is given by
(24h).

Integrating (70) using Mathematica’s inbuilt NIntegrate command gives the
metric function $m$ as a function of $\rho$, which can then be transformed to
a function of $r$ using the relationship (65) and the previously computed
metric function $C(\rho)$. For a regular origin at $r=0$, we must have
$m(0)=0$, which fixes the constant of integration. Similarly, we find the
metric function $\psi(r)$ by integrating the equation for
$\tfrac{\mathop{}\\!\mathrm{d}{\psi}}{\mathop{}\\!\mathrm{d}{\rho}}$ derived
from (69b):

$\dfrac{\mathop{}\\!\mathrm{d}{\psi}}{\mathop{}\\!\mathrm{d}{\rho}}=\dfrac{2B(\rho)C(\rho)}{C^{\prime}(\rho)}\left({\widetilde{T}}_{\tau}{}^{\tau}+{\widetilde{T}}_{\rho}{}^{\rho}\right).$
(73)

We fix $\psi(0)=0$ without loss of generality, since changing the value of
$\psi(0)$ corresponds to a rescaling of the time coordinate $\tau$.

For all values of the temperature considered, we integrate (70, 73) from
$\rho=\epsilon$ to $\rho=\pi/2-\epsilon$, taking $\epsilon=10^{-3}$ (changing
the value of $\epsilon$ in the range $10^{-3}<\epsilon<10^{-2}$ results in an
estimated relative error of no more than $10^{-6}$ in our results). At
$\rho=\epsilon$, we use a series expansion for $m$ constructed using the
corresponding expansions for the metric function $C(\rho)$ (39c) and the RSET
component ${\widetilde{T}}_{\tau}{}^{\tau}$ (38), which give

$m=m_{3}\rho^{3}+m_{5}\rho^{5}+{\mathcal{O}}(\rho^{7}),$ (74a) where the
$m_{i}$ are constants, the first of which is given by
$m_{3}=-\dfrac{1}{6}C_{2}^{\frac{3}{2}}T_{\tau 0},$ (74b)

and the constants $C_{2}$ and $T_{\tau 0}$ are given in (43) and (38b)
respectively. A similar expansion is used for the metric function $\psi$:

$\psi=\psi_{2}\rho^{2}+\psi_{4}\rho^{4}+{\mathcal{O}}(\rho^{6}),$ (75)

where the $\psi_{i}$ are constants and the lowest order term is given by

$\psi_{2}=\dfrac{1}{3}C_{2}T_{\tau 0}.$ (76)

Using (39c, 65), the initial value of $\rho=\epsilon$ corresponds to a very
small value $r={\sqrt{C_{2}}}\epsilon$. The upper limit of our numerical
integration, $\rho=\tfrac{\pi}{2}-\epsilon$, corresponds to
$r={\sqrt{C(\tfrac{\pi}{2}-\epsilon)}}$. As can be seen from Fig. 4, the value
of $C(\rho)/\tan^{2}\rho$ as $\rho\rightarrow\tfrac{\pi}{2}$ decreases with
increasing temperature, and, accordingly, the largest value of $r$ in our
numerical evaluation of the metric functions $m(r)$ and $\psi(r)$ also
decreases as the temperature increases.

Figure 10: Metric function $\psi(r)$ for transparent boundary conditions. The
inverse temperature lies in the range
$\beta\in(\frac{\pi}{12},\frac{11\pi}{24})$ and the inverse AdS length scale
is fixed to be $a=1$.

Our main focus in this section is the metric function $m(r)$, but we first
present, in Fig. 10, some results for the metric function $\psi(r)$ when
transparent boundary conditions are applied (the corresponding results for
Dirichlet and Neumann boundary conditions are qualitatively very similar). We
see that $\psi(r)$ is a monontonically decreasing function of $r$, and that,
as $r$ increases, $\psi(r)$ very rapidly converges to a finite limit. The
magnitude of this limit increases with increasing temperature. The rapid
convergence of $\psi(r)$ as $r\rightarrow\infty$ is predicted by the behaviour
of $\tfrac{\mathop{}\\!\mathrm{d}{\psi}}{\mathop{}\\!\mathrm{d}{r}}$ as
$r\rightarrow\infty$, from (67, 69b, 71):

$\dfrac{\mathop{}\\!\mathrm{d}{\psi}}{\mathop{}\\!\mathrm{d}{r}}\sim{\mathcal{O}}(r^{-5}),$
(77)

where we have used the fact that ${\widetilde{T}}_{\rho}{}^{\rho}$, like
${\widetilde{T}}_{\tau}{}^{\tau}$, is ${\mathcal{O}}(r^{-4})$ as
$r\rightarrow\infty$.

(a) Dirichlet boundary conditions

(b) Neumann boundary conditions

(c) Transparent boundary conditions

Figure 11: Metric function $m(r)$ for Dirichlet (top), Neumann (middle) and
transparent (bottom) boundary conditions. The inverse temperature lies in the
range $\beta\in(\frac{\pi}{12},\frac{11\pi}{24})$. The range of values of $r$
shown decreases as the temperature $\beta^{-1}$ increases. The inverse AdS
length scale is fixed to be $a=1$.

We now turn to Fig. 11, where we show the metric function $m(r)$ as a function
of the new radial coordinate $r$ when the boundary conditions satisfied by the
quantum scalar field are Dirichlet (top), Neumann (middle) or transparent
(bottom). Since the energy density $-{\widetilde{T}}_{\tau}{}^{\tau}$ is
always positive, it follows from (69a) that $m(r)$ is a monotonically
increasing function of $r$. From Fig. 11, we also see that $m(r)$ tends to a
finite limit as $r\rightarrow\infty$, as expected from the behaviour (72) of
$\tfrac{\mathop{}\\!\mathrm{d}{m}}{\mathop{}\\!\mathrm{d}{r}}$ for large $r$.
For fixed temperature, convergence to this limit is quickest when Dirichlet
boundary conditions are considered. Increasing the temperature results in the
function $m(r)$ converging more slowly as $r$ increases.

Figure 12: QC soliton mass $M=\lim_{r\to\infty}m(r)$ as a function of the
inverse temperature $\beta$, for Dirichlet (solid line), Neumann (dashed line)
and transparent (dotted line) boundary conditions. The inverse AdS length
scale is fixed to be $a=1$.

The finite limit $M=\lim_{r\rightarrow\infty}m(r)$ is the mass of the QC
solitons. To see how this depends on the temperature, in Fig. 12 we show $M$
as a function of the inverse temperature $\beta$ for Dirichlet (solid line),
Neumann (dashed line) and transparent (dotted line) boundary conditions. For
all fixed values of the temperature considered, we find that the mass of the
soliton is smallest when Dirichlet boundary conditions are applied, and
greatest when Neumann boundary conditions are applied, with the mass for
transparent boundary conditions lying between the Dirichlet and Neumann
masses. At low temperatures (larger values of $\beta$), the mass $M$ increases
as the temperature increases ($\beta$ decreases). However, this trend does not
continue for ever-increasing temperature. For all boundary conditions, we find
a maximum mass $M$ at very similar values of the inverse temperature
$\beta\sim 0.5$. For larger temperatures, the mass of the QC solitons
decreases as the temperature increases further ($\beta$ decreases further).

This result is somewhat unexpected. For all boundary conditions, the energy
density in the quantum field increases with increasing temperature
monotonically at all values of $\rho$. The total energy in the quantum field
can be found by integrating the energy density with respect to $\rho$. This
gives a finite quantity which is monotonically increasing with temperature for
fixed boundary conditions. Naively one would expect this to be proportional to
the mass of the resulting QC soliton. As a result, one would anticipate that
the QC soliton masses also increase monotonically with increasing temperature,
contrary to our results in Fig. 12.

To understand this, we write the mass $M$ as follows:

$M=-\dfrac{1}{4}\int_{\rho=0}^{\frac{\pi}{2}}C^{\prime}(\rho){\sqrt{C(\rho)}}{\widetilde{T}}_{\tau}{}^{\tau}(\rho)\,d\rho.$
(78)

From this, we see that the behaviour of the soliton mass $M$ as the
temperature increases depends on the behaviour of both the energy density
$-{\widetilde{T}}_{\tau}{}^{\tau}$ and the QC AdS metric function $C(\rho)$
with increasing temperature. While the former (the energy density)
monotonically increases with temperature, it can be seen from Fig. 4 that for
fixed $\rho$, the QC AdS metric function $C(\rho)$ decreases monotonically
with increasing temperature. The behaviour of the QC soliton mass $M$ as a
function of temperature is therefore the result of a complicated interplay
between the behaviours of the energy density in the quantum field and the
resulting QC AdS metric.

##  VII Conclusions

We have explored the back-reaction effect of a quantum scalar field on AdS
space-time. We considered a massless, conformally-coupled scalar field on a
fixed, global AdS space-time background. The RSET for this set-up, with the
scalar field in either the global AdS vacuum or a global thermal state, can be
found in [33] when the field is subject to either Dirichlet, Neumann or
transparent boundary conditions imposed at the space-time boundary.

We used the RSET from [33] as the source term on the right-hand-side of the
QCEE (15). The left-hand-side of the QCEE is purely classical, and describes
the QC metric. When the quantum scalar field is in the global vacuum state,
the RSET is a multiple of the metric and the QCEE are trivially solved by a
renormalization of the cosmological constant. In this case the QC metric is
simply pure AdS space-time with the inverse AdS radius having a renormalized
value.

To find nontrivial QC metrics, we therefore considered the scalar field to be
in a global thermal state, with the right-hand-side of the QCEE being the
difference between the thermal and vacuum expectation values of the RSET. We
first solved the QCEE using a metric ansatz (27) which facilitates comparison
between the QC metric and pure AdS space-time. We find that the QC metrics
deviate from pure AdS space-time, with the deviations increasing as the
temperature of the quantum scalar field state increases.

We then employ an alternative metric ansatz to aid the interpretation of these
QC metrics as asymptotically-AdS solitons. These solitons have a finite mass
which depends on the temperature of the quantum scalar field state. For low
temperatures, the soliton mass increases as the temperature increases, but it
then reaches a maximum. The value of the temperature at which the soliton mass
has a maximum is very similar for the three boundary conditions studied. On
increasing the temperature further, the mass of the soliton decreases. This
surprising result arises from a complicated interplay between the energy
density of the quantum field (which increases as the temperature increases)
and the behaviour of one of the QC metric functions (which decreases as the
temperature increases).

We have taken a very simple approach in this paper. Instead of attempting to
solve the full SCEE (12), in which the RSET on the right-hand-side is computed
on the same background space-time metric which gives the geometric quantities
on the left-hand-side, we have fixed the background metric for the RSET, while
allowing the QC metric on the left-hand-side to vary (giving the QCEE (15)).
We have however solved the full nonlinear QCEE, rather than considering linear
perturbations about the initial fixed background metric (which, in our case,
was pure AdS space-time), following the approach in [28, 29, 30]. The QCEE can
be considered as an approximation to the SCEE, which will be valid when
quantum corrections to the space-time are small. In our situation, this
corresponds to low temperatures for the quantum scalar field state. As the
temperature of the quantum scalar field increases, we find that quantum
corrections to the space-time also increase, and therefore for sufficiently
large temperature the approximation used in this work will breakdown. A full
nonlinear solution of the SCEE would be required to quantitively estimate the
temperature at which this breakdown occurs. However, one may speculate that
this is may be at a temperature below that at which the QC solitons we find
have a maximum mass. If this were the case, then our result that the QC
soliton mass is a decreasing function of temperature for sufficiently large
temperature may not persist for solutions of the full SCEE.

In this work we have considered only a massless, conformally coupled scalar
field and have worked in four space-time dimensions, applying either
Dirichlet, Neumann or transparent boundary conditions to the quantum field.
The advantage of this is that we have been able to use the simple expressions
in [33] for the RSET expectation values. It would be interesting to explore
the properties of the QC metric resulting from a quantum scalar field with
different mass or coupling to the scalar curvature, with different boundary
conditions applied, or in dimensions other than four. For example, considering
a massless, conformally coupled scalar field on global AdS in four dimensions,
but applying Robin boundary conditions, results in a vacuum state which is no
longer maximally symmetric [36, 40], and hence we anticipate that the QC
metric even for the vacuum state will not be pure AdS. In all these situations
the RSET expectation values are more complicated to compute numerically (see,
for example, [40, 49]). We therefore leave these investigations for future
work.

###### Acknowledgements.

The work of J.T. is supported by an EPSRC studentship. The work of E.W. is
supported by STFC grant number ST/X000621/1. E.W. also acknowledges support of
the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université), and LabEx
CARMIN (ANR-10-LABX-59-01). This work makes use of the Black Hole Perturbation
Toolkit [50], in particular the General Relativity Tensors package by Seth
Hopper and Barry Wardell was employed for tensor computations.

## References

  * Christensen [1978] S. M. Christensen, Regularization, renormalization, and covariant geodesic point separation, Phys. Rev. D 17, 946 (1978).
  * Birrell and Davies [1984] N. D. Birrell and P. C. W. Davies,  _Quantum Fields in Curved Space_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, UK, 1984).
  * Decanini and Folacci [2008] Y. Decanini and A. Folacci, Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension, Phys. Rev. D 78, 044025 (2008), arXiv:gr-qc/0512118 .
  * Simon [1990] J. Z. Simon, Higher derivative Lagrangians, nonlocality, problems and solutions, Phys. Rev. D 41, 3720 (1990).
  * Parker and Simon [1993] L. Parker and J. Z. Simon, Einstein equation with quantum corrections reduced to second order, Phys. Rev. D 47, 1339 (1993), arXiv:gr-qc/9211002 .
  * Suen [1989] W.-M. Suen, The stability of the semiclassical Einstein equation, Phys. Rev. D 40, 315 (1989).
  * Sanders [2022] K. Sanders, Static symmetric solutions of the semi-classical Einstein–Klein–Gordon system, Annales Henri Poincare 23, 1321 (2022), arXiv:2007.14311 [math-ph] .
  * Pinamonti [2011] N. Pinamonti, On the initial conditions and solutions of the semiclassical Einstein equations in a cosmological scenario, Commun. Math. Phys. 305, 563 (2011), arXiv:1001.0864 [gr-qc] .
  * Pinamonti and Siemssen [2015] N. Pinamonti and D. Siemssen, Global existence of solutions of the semiclassical Einstein equation for cosmological spacetimes, Commun. Math. Phys. 334, 171 (2015), arXiv:1309.6303 [math-ph] .
  * Juárez-Aubry [2019] B. A. Juárez-Aubry, Semi-classical gravity in de Sitter spacetime and the cosmological constant, Phys. Lett. B 797, 134912 (2019), arXiv:1903.03924 [gr-qc] .
  * Gottschalk  _et al._ [2023] H. Gottschalk, N. R. Rothe, and D. Siemssen, Cosmological de Sitter solutions of the semiclassical Einstein equation, Annales Henri Poincare 24, 2949 (2023), arXiv:2206.07774 [gr-qc] .
  * Juárez-Aubry [2022] B. A. Juárez-Aubry, Semiclassical gravity in static spacetimes as a constrained initial value problem, Annales Henri Poincare 23, 1451 (2022), arXiv:2011.05947 [gr-qc] .
  * Juárez-Aubry and Modak [2022] B. A. Juárez-Aubry and S. K. Modak, Semiclassical gravity with a conformally covariant field in globally hyperbolic spacetimes, J. Math. Phys. 63, 092303 (2022), arXiv:2110.01719 [math-ph] .
  * Flanagan and Wald [1996] E. E. Flanagan and R. M. Wald, Does back reaction enforce the averaged null energy condition in semiclassical gravity?, Phys. Rev. D 54, 6233 (1996), arXiv:gr-qc/9602052 .
  * Taylor and Breen [2021] P. Taylor and C. Breen, Semiclassical backreaction on asymptotically anti–de Sitter black holes, Phys. Rev. D 103, 025006 (2021), arXiv:2007.09405 [gr-qc] .
  * Levi and Ori [2016] A. Levi and A. Ori, Versatile method for renormalized stress-energy computation in black-hole spacetimes, Phys. Rev. Lett. 117, 231101 (2016), arXiv:1608.03806 [gr-qc] .
  * Levi  _et al._ [2017] A. Levi, E. Eilon, A. Ori, and M. van de Meent, Renormalized stress-energy tensor of an evaporating spinning black hole, Phys. Rev. Lett. 118, 141102 (2017), arXiv:1610.04848 [gr-qc] .
  * Taylor  _et al._ [2022] P. Taylor, C. Breen, and A. Ottewill, Mode-sum prescription for the renormalized stress energy tensor on black hole spacetimes, Phys. Rev. D 106, 065023 (2022), arXiv:2201.05174 [gr-qc] .
  * Arrechea  _et al._ [2023] J. Arrechea, C. Breen, A. Ottewill, and P. Taylor, Renormalized stress-energy tensor for scalar fields in Hartle-Hawking, Boulware, and Unruh states in the Reissner-Nordström spacetime, Phys. Rev. D 108, 125004 (2023), arXiv:2307.10307 [gr-qc] .
  * Parentani and Piran [1994] R. Parentani and T. Piran, The internal geometry of an evaporating black hole, Phys. Rev. Lett. 73, 2805 (1994), arXiv:hep-th/9405007 .
  * Fabbri and Navarro-Salas [2005] A. Fabbri and J. Navarro-Salas,  _Modeling Black Hole Evaporation_ (Imperial College Press, 2005).
  * Carballo-Rubio [2018] R. Carballo-Rubio, Stellar equilibrium in semiclassical gravity, Phys. Rev. Lett. 120, 061102 (2018), arXiv:1706.05379 [gr-qc] .
  * Arrechea  _et al._ [2020] J. Arrechea, C. Barceló, R. Carballo-Rubio, and L. J. Garay, Schwarzschild geometry counterpart in semiclassical gravity, Phys. Rev. D 101, 064059 (2020), arXiv:1911.03213 [gr-qc] .
  * Arrechea  _et al._ [2021] J. Arrechea, C. Barceló, R. Carballo-Rubio, and L. J. Garay, Reissner–Nordström geometry counterpart in semiclassical gravity, Class. Quant. Grav. 38, 115014 (2021), arXiv:2102.03544 [gr-qc] .
  * Arrechea  _et al._ [2022] J. Arrechea, C. Barceló, R. Carballo-Rubio, and L. J. Garay, Semiclassical relativistic stars, Sci. Rep. 12, 15958 (2022), arXiv:2110.15808 [gr-qc] .
  * Arrechea  _et al._ [2024] J. Arrechea, C. Barceló, R. Carballo-Rubio, and L. J. Garay, Ultracompact horizonless objects in order-reduced semiclassical gravity, Phys. Rev. D 109, 104056 (2024), arXiv:2310.12668 [gr-qc] .
  * Beltrán-Palau  _et al._ [2023] P. Beltrán-Palau, A. del Río, and J. Navarro-Salas, Quantum corrections to the Schwarzschild metric from vacuum polarization, Phys. Rev. D 107, 085023 (2023), arXiv:2212.08089 [gr-qc] .
  * Casals  _et al._ [2017] M. Casals, A. Fabbri, C. Martínez, and J. Zanelli, Quantum Backreaction on Three-Dimensional Black Holes and Naked Singularities, Phys. Rev. Lett. 118, 131102 (2017), arXiv:1608.05366 [gr-qc] .
  * Casals  _et al._ [2018] M. Casals, A. Fabbri, C. Martínez, and J. Zanelli, Quantum fields as cosmic censors in $(2+1)$-dimensions, Int. J. Mod. Phys. D 27, 1843011 (2018).
  * Casals  _et al._ [2019] M. Casals, A. Fabbri, C. Martínez, and J. Zanelli, Quantum-corrected rotating black holes and naked singularities in ( 2+1 ) dimensions, Phys. Rev. D 99, 104023 (2019), arXiv:1902.01583 [hep-th] .
  * Chernicoff  _et al._ [2024] M. Chernicoff, G. Giribet, J. Moreno, J. Oliva, R. Rojas, and C. R. d. A. Torres, Quantum backreactions in (A)dS3 massive gravity and logarithmic asymptotic behavior, (2024), arXiv:2404.10127 [hep-th] .
  * Avis  _et al._ [1978] S. J. Avis, C. J. Isham, and D. Storey, Quantum field theory in anti-de Sitter space-time, Phys. Rev. D 18, 3565 (1978).
  * Allen  _et al._ [1987] B. Allen, A. Folacci, and G. W. Gibbons, Anti-de Sitter space at finite temperature, Phys. Lett. B 189, 304 (1987).
  * Ishibashi and Wald [2004] A. Ishibashi and R. M. Wald, Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time, Class. Quant. Grav. 21, 2981 (2004), arXiv:hep-th/0402184 .
  * Dappiaggi  _et al._ [2018] C. Dappiaggi, H. Ferreira, and A. Marta, Ground states of a Klein-Gordon field with Robin boundary conditions in global anti-de Sitter spacetime, Phys. Rev. D 98, 025005 (2018), arXiv:1805.03135 [hep-th] .
  * Morley  _et al._ [2021] T. Morley, P. Taylor, and E. Winstanley, Quantum field theory on global anti-de Sitter space-time with Robin boundary conditions, Class. Quant. Grav. 38, 035009 (2021), arXiv:2004.02704 [gr-qc] .
  * Allen and Jacobson [1986] B. Allen and T. Jacobson, Vector two point functions in maximally symmetric spaces, Commun. Math. Phys. 103, 669 (1986).
  * Kent and Winstanley [2015] C. Kent and E. Winstanley, Hadamard renormalized scalar field theory on anti–de Sitter spacetime, Phys. Rev. D 91, 044044 (2015), arXiv:1408.6738 [gr-qc] .
  * Barroso and Pitelli [2020] V. S. Barroso and J. P. M. Pitelli, Boundary conditions and vacuum fluctuations in ${\mathrm{AdS}}_{4}$, Gen. Rel. Grav. 52, 29 (2020), arXiv:1904.10920 [gr-qc] .
  * Morley  _et al._ [2024] T. Morley, S. Namasivayam, and E. Winstanley, Renormalized stress-energy tensor on global anti-de Sitter space-time with Robin boundary conditions, Gen. Rel. Grav. 56, 38 (2024), arXiv:2308.05623 [hep-th] .
  * Decanini and Folacci [2006] Y. Decanini and A. Folacci, Off-diagonal coefficients of the Dewitt-Schwinger and Hadamard representations of the Feynman propagator, Phys. Rev. D 73, 044027 (2006), arXiv:gr-qc/0511115 .
  * Bunch and Davies [1978] T. S. Bunch and P. C. W. Davies, Quantum field theory in de Sitter space: renormalization by point splitting, Proc. Roy. Soc. Lond. A 360, 117 (1978).
  * Allen [1985] B. Allen, Vacuum states in de Sitter space, Phys. Rev. D 32, 3136 (1985).
  * Page [1982] D. N. Page, Thermal stress tensors in static Einstein spaces, Phys. Rev. D 25, 1499 (1982).
  * Tadaki [1988] S. Tadaki, Stress tensor in de Sitter space, Prog. Theor. Phys. 80, 654 (1988).
  * Dappiaggi  _et al._ [2008] C. Dappiaggi, K. Fredenhagen, and N. Pinamonti, Stable cosmological models driven by a free quantum scalar field, Phys. Rev. D 77, 104015 (2008), arXiv:0801.2850 [gr-qc] .
  * Ambrus  _et al._ [2018] V. E. Ambrus, C. Kent, and E. Winstanley, Analysis of scalar and fermion quantum field theory on anti-de Sitter spacetime, Int. J. Mod. Phys. D 27, 1843014 (2018), arXiv:1801.10366 [hep-th] .
  * Candelas and Dowker [1979] P. Candelas and J. S. Dowker, Field theories on conformally related space-times: some global considerations, Phys. Rev. D 19, 2902 (1979).
  * [49] S. Namasivayam and E. Winstanley, Work in progress.
  * [50] Black Hole Perturbation Toolkit, (bhptoolkit.org).