# Dynamical Henneaux-Teitelboim Gravity

Emma Albertini<EMAIL_ADDRESS>Imperial College London  Kyle
Barnes<EMAIL_ADDRESS>Brown University  Gabriel Herczeg
<EMAIL_ADDRESS>Brown University

###### Abstract

We consider a modified gravity model which we call “dynamical Henneaux-
Teitelboim gravity” because of its close relationship with the Henneaux-
Teitelboim formulation of unimodular gravity. The latter is a fully
diffeomorphism-invariant formulation of unimodular gravity, where full
diffeomorphism invariance is achieved by introducing two additional non-
dynamical fields: a scalar, which plays the role of a cosmological constant,
and a three-form whose exterior derivative is the spacetime volume element.
Dynamical Henneaux-Teitelboim gravity is a generalization of this model that
includes kinetic terms for both the scalar and the three-form with arbitrary
couplings. We study the field equations for the cases of spherically symmetric
and homogeneous, isotropic configurations. In the spherically symmetric case,
we solve the field equations analytically for small values of the coupling to
obtain an approximate black hole solution. In the homogeneous and isotropic
case, we perturb around de Sitter space to find an approximate cosmological
background for our model.

##  I Introduction

###  I.1 Overview

Cosmic inflation solves a number of fundamental problems in cosmology by
positing an early period of rapid expansion driven by a constant part of the
stress energy tensor—it provides a framework for explaining why the observable
universe is nearly flat and isotropic guth1981inflationary ;
kleban2016inhomogeneous ; east2016beginning , and gives a mechanism for large-
scale structure to form from density perturbations in the early universe
liddle2000cosmological . Meanwhile, the present cosmological epoch is
dominated by dark energy that is usually attributed to a cosmological
constant, albeit a vastly smaller one caldwell1998cosmological ;
ratra1988cosmological . In light of these facts, as well as the recent
controversy over the Hubble tension di2021realm ; knox2020hubble , and the
long-standing cosmological constant problems weinberg1989cosmological ;
adler1995vacuum ; padilla2015lectures , simple modifications of general
relativity where the cosmological constant is promoted to a dynamical field
are worthy of consideration.

One route to accomplishing this is to consider modified gravity models where
the Bianchi identities no longer imply conservation of the stress energy
tensor, but only the weaker condition that the divergence of the stress tensor
is a total derivative. A minimal modification of general relativity in which
this can be achieved is unimodular gravity josset2017dark . In the standard
formulation of unimodular gravity, the spacetime volume element is constrained
to be equal to a prescribed background density, with the result that the
equations of motion become the _trace-free_ Einstein equations

$R_{\mu\nu}-\frac{1}{4}Rg_{\mu\nu}=\kappa(T_{\mu\nu}-\frac{1}{4}Tg_{\mu\nu}),\qquad\kappa=8\pi
G_{N}.$ (1)

Taking the divergence of (1) and using the contracted Bianchi identity
$\nabla^{\mu}R_{\mu\nu}=\frac{1}{2}\nabla_{\nu}R$, leads to

$\nabla^{\mu}T_{\mu\nu}=\frac{1}{4}\nabla_{\nu}(R+\kappa T).$ (2)

If one _imposes_ conservation of the stress tensor, then we learn that
$\Lambda\equiv\frac{1}{4}(R+\kappa T)$ is a constant, which is just the pure-
trace part of the Einstein equations, in which case unimodular gravity is
classically equivalent to general relativity. However, if we are willing to
consider non-conserved stress tensors, then unimodular gravity is physically
distinct from general relativity even at the classical level.111For a recent
review of the sometimes subtle differences between unimodular gravity and
general relativity, see carballo2022unimodular . Some recent proposals for
modified gravity models that draw on ideas from unimodular gravity and
topological field theory lead to the same type of non-conservation of the
stress tensor, and consequent fluctuations in the effective cosmological
constantAlexander:2018tyf ; alexander2020chiral ; alexander2022black . Here,
we promote the cosmological constant to a dynamical field by considering a
minimal modification of a fully diffeomorphism invariant, background
independent reformulation of unimodular gravity due to Henneaux and Teitelboim
henneaux1989cosmological  and we study the implications for spherically
symmetric black holes and homegeneous, isotropic cosmology.

The paper is structured as follows: in section I.2, we give a brief review of
unimodular gravity; in section II, we introduce dynamical Henneaux-Teitelboim
gravity; section III is devoted to the study of spherically symmetric black
holes, and in section IV we take a first look at the implications of dynamical
Henneaux-Teitelboim gravity for a simple homogeneous, isotropic cosmological
model.

###  I.2 Unimodular Gravity

The action for unimodular gravity is simply the Einstein-Hilbert action where
the spacetime volume element $\sqrt{-g}$ has been replaced with a fixed
background density $\epsilon$:

$S_{\mbox{\tiny
UMG}}=\int_{\mathcal{M}}\epsilon\left(\frac{1}{2\kappa}R(g)+\mathcal{L}_{M}(g,\psi)\right)d^{4}x,$
(3)

where $g_{\mu\nu}$ is a metric satisfying $\sqrt{-g}=\epsilon$, and $\psi$
represents any matter fields that are present. Varying (3) with respect to
$g^{\mu\nu}$ leads to the trace-free Einstein equation (1). The introduction
of the background density $\epsilon$ breaks diffeomorphism invariance down to
_transverse_ diffeomorphism invariance—i.e., invariance with respect to
diffemorphisms generated by vector fields $\xi$ satisfying

$\mathscr{L}_{\xi}(\epsilon\,d^{4}x)=0.$ (4)

In order to preserve full diffeomorphism invariance while retaining the spirit
of unimodular gravity, Henneaux and Teitelboim henneaux1989cosmological
considered the action222Smolin has shown in smolin2009quantization  that
certain aspects of the cosmological constant problem can be resolved using
path integral methods based on the action (5).

$S_{\mbox{\tiny
HT}}=\frac{1}{\kappa}\int\mathbf{\mathbf{V\\!ol}}(g)\,\left(\frac{1}{2}R(g)-\varphi+\kappa\mathcal{L}_{M}(g,\psi)\right)+\varphi\,d\mathbf{H},$
(5)

where $\varphi$ is a scalar, $\mathbf{H}$ is a three-form, and
$\mathbf{V\\!ol}(g)=\sqrt{-g}\,d^{4}x$ is the volume form associated with
$g_{\mu\nu}$. Varying with respect to $g^{\mu\nu}$ gives the full set of
Einstein’s equations with $\varphi$ playing the role of the cosmological
constant:

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\varphi g_{\mu\nu}=\kappa T_{\mu\nu}.$ (6)

On the other hand, varying with respect to $\mathbf{H}$ leads to $d\varphi=0$,
so the gravitational equations of motion are equivalent to the usual Einstein
equations. Finally, varying with respect to $\varphi$ gives
$\mathbf{V\\!ol}(g)=d\mathbf{H}$, so while $\mathbf{H}$ is varied in the
action, it can be thought of as a potential for the spacetime volume in much
the same way that $\epsilon$ fixes the spacetime volume in the standard
formulation of unimodular gravity. Since there are no background structures,
the action (5) is invariant under the full group of diffeomorphisms. There is
also a gauge symmetry under $\mathbf{H}\to\mathbf{H}+d\bm{\Omega}$ for any
two-form $\bm{\Omega}$.

In addition to restoring background independence and full diffeomorphism
invariance, the Henneaux-Teitelboim formulation of unimodular gravity
automatically has a conserved stress tensor, unlike the standard formulation.
However, if one introduces additional terms into the Lagrangian that couple
the new fields $\varphi$ and $\mathbf{H}$ to the metric, this may no longer be
the case. As we shall see in the following section, introducing kinetic terms
for both $\varphi$ and $\mathbf{H}$ is sufficient to violate conservation of
the stress tensor just enough to induce fluctuations in the effective
cosmological constant $\varphi$, although including a kinetic term for just
one of them is not.

##  II Dynamical Henneaux-Teitelboim gravity

We now consider “dynamical Henneaux-Teitelboim gravity,” defined by the action

$S_{\mbox{\tiny DHT}}=S_{\mbox{\tiny HT}}-\alpha\\!\int
d\mathbf{H}\wedge*d\mathbf{H}-\beta\\!\int d\varphi\wedge*d\varphi,$ (7)

which enjoys the same symmetries as the Henneaux-Teitelboim formulation of
unimodular gravity—it is background independent and fully diffeomorphism-
invariant, and it retains the gauge symmetry
$\mathbf{H}\to\mathbf{H}+d\bm{\Omega}$ for any two-form $\bm{\Omega}$. Here,
and for the remainder of the article we use geometrized units $c=G_{N}=1$,
which renders $\kappa$ dimensionless, and gives the action dimensions of
$\textrm{(length)}^{2}$. As a result, $\alpha$ acquires dimensions of
$\textrm{(length)}^{-2}$, while $\beta\sim\textrm{(length)}^{4}$.

Varying (7) with respect to $g^{\mu\nu}$ leads to the Einstein field equations
with the _dynamical_ field $\varphi$ playing the role of the cosmological
constant, and where the stress tensor gets contributions from the old matter
Lagrangian $\mathcal{L}_{M}(g,\psi)$, as well as the kinetic terms for
$\mathbf{H}$ and $\varphi$:

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\varphi g_{\mu\nu}=\kappa
T_{\mu\nu},$ (8) $\displaystyle
T_{\mu\nu}=T^{(\psi)}_{\mu\nu}+T^{(\varphi)}_{\mu\nu}+T^{(H)}_{\mu\nu},$ (9)

where

$\displaystyle T^{(\psi)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{2}{\sqrt{-g}}\frac{\partial\mathcal{L}_{M}}{\partial
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L}_{M},$ (10) $\displaystyle
T^{(\varphi)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\beta\left(\nabla_{\mu}\varphi\nabla_{\nu}\varphi-\frac{1}{2}g_{\mu\nu}\nabla^{\rho}\varphi\nabla_{\rho}\varphi\right),$
(11) $\displaystyle T^{(\mathbf{H})}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\alpha\frac{\tilde{Q}^{2}}{|g|}g_{\mu\nu},$ (12)

with $\mathbf{Q}=d\mathbf{H}$, and
$\tilde{Q}=\frac{1}{4!}Q_{abcd}\epsilon^{abcd}$. The trace-free part of the
gravitational equation (8) is just the trace-free Einstein equation (1),
which, as we noted in the introduction, implies that the divergence of the
stress tensor is a total derivative
$\nabla^{\mu}T_{\mu\nu}=\frac{1}{4}\nabla_{\nu}(R+\kappa T)$. On the other
hand, the pure trace part of (8) is simply $\varphi=\frac{1}{4}(R+\kappa T)$.
Putting these together gives

$\nabla_{\nu}\varphi=\nabla^{\mu}T_{\mu\nu}.$ (13)

So we see that fluctuations in $\varphi$ are sourced by non-conservation of
the stress tensor of precisely the type we have alluded to. As for the scalar
and three-form equations of motion, we can vary (7) with respect to $\varphi$
to obtain

$d\mathbf{H}=\mathbf{V\\!ol}(g)-2\beta d*d\varphi,$ (14)

while varying (7) with respect to $\mathbf{H}$ gives

$d\varphi-2\alpha\,d*d\mathbf{H}=0.$ (15)

Taking the Hodge star of (14) and inserting the result into (15) leads to

$d(\varphi+4\alpha\beta\square\varphi)=0,$ (16)

while taking the Hodge star of (15) and plugging into (14) gives

$d(\mathbf{H}+4\alpha\beta d^{\dagger}d\mathbf{H})=\mathbf{V\\!ol}(g),\\\ $
(17)

where $d^{\dagger}\equiv*\,d\,*$. From equations (16) and (17), we see that if
either $\alpha=0$ or $\beta=0$, then we recover
${d\mathbf{H}=\mathbf{V\\!ol}(g)}$ and $d\varphi=0$, and the model is
equivalent to the Henneaux-Teitelboim formulation of unimodular gravity.
However, when $\alpha\beta\neq 0$, we obtain non-trivial dynamics for
$\varphi$.

##  III Black Holes

We would like to investigate the implications of this model for black holes.
Our approach is similar to astorga2019compact , in which metrics for black
holes and other compact objects were studied for the general case of
unimodular gravity with a non-conserved stress tensor, although here we focus
on the black holes solutions of dynamical Henneaux-Teitelboim gravity in
particular. For this, we consider the static, spherically symmetric ansatz

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-A(r)dt^{2}+B(r)dr^{2}+r^{2}d\Omega^{2}_{2},$ (18)
$\displaystyle\mathbf{H}$ $\displaystyle=$ $\displaystyle
h(r)\sin\theta\,dt\wedge d\theta\wedge d\phi,$ (19) $\displaystyle\varphi$
$\displaystyle=$ $\displaystyle\varphi(r),$ (20)

where the ansatz for $\mathbf{H}$ is the most general spherically symmetric,
static three-form up to gauge transformations, and
$d\Omega_{2}^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}~{}$ is the round metric
on the unit two-sphere. Additionally, we will only consider contributions to
the stress tensor from $\mathbf{H}$ and $\varphi$, i.e., we will set
$\mathcal{L}_{M}=0$ in (7).

Since the stress tensor for $\mathbf{H}$ is pure trace, it will be convenient
to divide the Einstein equations into trace-free and pure trace parts. The
trace-free equations are (1), while the pure-trace equation is

$\varphi=\frac{1}{4}(R+\kappa T),$ (21)

where

$\displaystyle R=\frac{1}{2r^{2}A^{2}B^{2}}$ $\displaystyle\Big{[}$
$\displaystyle r^{2}BA^{\prime
2}+r^{2}AA^{\prime}B^{\prime}-2rAB\left(rA^{\prime\prime}+2A^{\prime}\right)$
(22) $\displaystyle\,+\,4A^{2}\left(rB^{\prime}+B^{2}-B\right)\Big{]}$

and

$T=\left(\beta\frac{\varphi^{\prime 2}}{B}-\frac{4\alpha h^{\prime
2}}{ABr^{4}}\right)$ (23)

for the static, spherically symmetric ansatz (18)-(20).
Similarly, the trace-free equations reduce to

$\displaystyle
u+\frac{1}{2rAB^{2}}(AB^{\prime}+A^{\prime}B)=\frac{-\kappa\beta}{4B}(\varphi^{\prime})^{2}$
(24)
$\displaystyle-u+\frac{1}{2rAB^{2}}(AB^{\prime}+A^{\prime}B)=-3\frac{\kappa\beta}{4B}(\varphi^{\prime})^{2}$
(25) $\displaystyle u=\frac{\beta\kappa}{4B}(\varphi^{\prime})^{2}$ (26)

where we have defined

$\displaystyle u=\frac{1}{8r^{2}A^{2}B^{2}}\\!\\!\\!$ $\displaystyle\Big{(}$
$\displaystyle\\!\\!2ABA^{\prime\prime}r^{2}-AB^{\prime}A^{\prime}r^{2}-B(A^{\prime})^{2}r^{2}$
(27) $\displaystyle\,+\,4A^{2}B^{2}-4A^{2}B\Big{)}.$

In the above, (24), (25) and (26) are the $tt$, $rr$ and $\theta\theta$
components of the trace-free Einstein equations, while the $\phi\phi$
component is identical to the $\theta\theta$ component, and all the off-
diagonal components vanish. Equations (24)-(26) are linearly dependent: by
either subtracting (26) from (24) or adding (26) to (25), we obtain

$\frac{1}{2rAB^{2}}(AB^{\prime}+A^{\prime}B)=-\frac{\kappa\beta}{2B}(\varphi^{\prime})^{2}$
(28)

which can be integrated to give

$AB=e^{-\kappa\beta\int_{r_{0}}^{r}\tilde{r}\varphi^{\prime}(\tilde{r})^{2}d\tilde{r}}\equiv
e^{f(r)}.$ (29)

Inserting the above into (26) and using (27) leads to

$\displaystyle\frac{1}{4}A^{\prime\prime}+\frac{1}{8}\beta\kappa
rA^{\prime}\varphi^{\prime
2}-\frac{A}{2r^{2}}+\frac{1}{2r^{2}}\,e^{f(r)}=\frac{\beta\kappa}{4}(\varphi^{\prime})^{2}A.$
(30)

Doing the same for the trace part of the Einstein equation (21), (22), (23)
and inserting the result into (30) gives

$\displaystyle-\frac{A^{\prime}}{r}-A\left(\frac{1}{r^{2}}-\frac{\kappa\beta\varphi^{\prime
2}}{2}\right)+\left(\frac{1}{r^{2}}-\varphi\right)e^{f(r)}=\frac{\alpha\kappa}{r^{4}}h^{\prime
2},$ (31)

which is a first-order linear equation for $A(r)$. The general solution is

$A(r)=\frac{1}{r}e^{\frac{f(r)}{2}}\left(\int_{\ell}^{r}e^{\frac{1}{2}f(R)}\left(1-G(R)R^{2}\right)dR-2m\right)$
(32)

where the mass $m$ appears as an integration constant, the lower endpoint of
integration $\ell=\sqrt{\frac{3}{\Lambda_{0}}}$, and $\Lambda_{0}$ is fixed by
matching to the (A)dS-Schwarzschild metric in the “decoupling limit,” on which
we expand below, and we have defined

$\displaystyle G(r)=\varphi+\frac{\alpha\kappa}{r^{4}}e^{-f(r)}h^{\prime 2}.$
(33)

From (14) and (15), we notice that we can recover the (A)dS-Schwarzschild
metric in the limit as $\beta\to 0$, $1/\alpha\to 0$ and $1/(\alpha\beta)\to
0$, which we call the decoupling limit. Notice that this limit does not reduce
the order of the equation of motions for the scalar $\varphi$ (16) or for the
three form $\mathbf{H}$ (17). In this case, equation (14) leads to
$d\mathbf{H}=\mathbf{V\\!ol}(g)$ and consequently $d*d\mathbf{H}=0$ in
equation (15). Then equation (17) is redundant, while equation (16) becomes
$d\square\varphi=0$ for which $\varphi=$ constant is a consistent solution.
When $d\mathbf{H}=\mathbf{V\\!ol}(g)$, the stress energy tensor for
$\mathbf{H}$, is just a constant times the metric, which can be absorbed into
the cosmological constant. At the level of the action, we can integrate out
$\mathbf{H}$ by inserting the solution to the equation of motion for $\varphi$
which is $d\mathbf{H}=\mathbf{V\\!ol}(g)$. Then, the gauge kinetic term just
becomes $S_{\mathbf{H}}=\alpha\int\mathbf{V\\!ol}(g)$, which contributes as a
cosmological constant term. We now turn our attention to the equations of
motion (16), (17) for $\mathbf{H}$ and $\varphi$. In order to make progress,
through the decoupling limit, we replace $g_{\mu\nu}$ in equations (16), (17)
with the (A)dS-Schwarzschild metric, $g^{(0)}_{\mu\nu}$. Now the equation of
motion for $\mathbf{H}$ (17) can be integrated once to give

$\displaystyle h(r)-4\alpha\beta
A_{0}(r)\left(h^{\prime\prime}(r)-\frac{2h^{\prime}(r)}{r}\right)=\frac{1}{3}r^{3}+C,$
(34)

where $A_{0}(r)=1-\frac{2m}{r}-\frac{\Lambda}{3}r^{2}$ is the metric function
for an (A)dS-Schwarzschild black hole. This is a linear, inhomogeneous
equation for $h(r)$, so the general solution is of the form

$h(r)=h_{0}(r)+h_{p}(r),$ (35)

where $h_{0}(r)$ is the solution of the homogeneous equation and $h_{p}(r)$ is
a particular solution. One can check that $h_{p}(r)=\frac{1}{3}r^{3}+C$ is a
satisfactory particular solution. The corresponding homogeneous equation can
be solved using the method of Frobenius—i.e., we make a generalized power
series ansatz

$h_{0}(r)=\sum\limits_{n=0}^{\infty}c_{n}r^{-n+q}.$ (36)

The allowed values of $q$ and coefficients $c_{n}$ are calculated in A. Since
the equation is second order, we find two possible values for $q$
corresponding to the two linearly independent solutions of the homogeneous
equation corresponding to (34):

$q=q_{\pm}\equiv\frac{3}{2}\pm\frac{3}{2}\sqrt{1-\frac{1}{3\Lambda\alpha\beta}}.$
(37)

Now the solution to the homogeneous equation becomes

$h_{0}(r)=\sum\limits_{n=0}^{\infty}c^{+}_{n}r^{-n+q_{+}}+\sum\limits_{n=0}^{\infty}c^{-}_{n}r^{-n+q_{-}},$
(38)

where the coefficients are determined by the following base cases and
recursion relation

$\displaystyle c^{\pm}_{1}$ $\displaystyle=$ $\displaystyle 0$ (39)
$\displaystyle c^{\pm}_{2}$ $\displaystyle=$ $\displaystyle\frac{c^{\pm}_{0}\
4\alpha\beta\
q_{\pm}(q_{\pm}-3)}{\frac{4\Lambda\alpha\beta}{3}(q_{\pm}-2)(q_{\pm}-5)+1}$
(40) $\displaystyle c^{\pm}_{n}$ $\displaystyle=$
$\displaystyle\frac{4\alpha\beta}{1+\frac{4\alpha\beta\Lambda}{3}(-n+q_{\pm})(-n+q_{\pm}-3)}\times$
$\displaystyle\Big{[}c^{\pm}_{n-2}(-n+q_{\pm}+2)(-n+q_{\pm}-1)$
$\displaystyle-2m\ c^{\pm}_{n-3}(-n+q_{\pm}+3)(-n+q_{\pm})\Big{]},\quad n>2.$

The scalar field equation of motion is given by

$\varphi-4\alpha\beta\left(A^{\prime}+\frac{2}{r}A\right)\varphi^{\prime}-4\alpha\beta
A\varphi^{\prime\prime}=\Lambda,$ (42)

where $A$ and $\varphi$ both depend on the radius $r$. Again, we work around
the Schwarzschild-(A)dS metric $A(r)=A_{0}(r)$, and apply the method of
Frobenius, with our ansatz given by

$\varphi-\Lambda=\sum_{n=0}^{\infty}a_{n}r^{-n+s}.$ (43)

We once again refer the reader to appendix A for the computation of the
allowed values of $s$ and the coefficients $a_{n}$. The roots of the indicial
equation are

$s=s_{\pm}\equiv-\frac{3}{2}\pm\frac{3}{2}\sqrt{1-\frac{1}{3\Lambda\alpha\beta}},$
(44)

so again, we have two independent solutions, as we would expect for a second
order equation. Now the solution becomes

$\varphi=\Lambda+\sum\limits_{n=0}^{\infty}a^{+}_{n}r^{-n+s_{+}}+\sum\limits_{n=0}^{\infty}a^{-}_{n}r^{-n+s_{-}},$
(45)

where the coefficients are

$\displaystyle a^{\pm}_{1}$ $\displaystyle=$ $\displaystyle 0$ (46)
$\displaystyle a^{\pm}_{2}$ $\displaystyle=$
$\displaystyle\frac{a^{\pm}_{0}\left(2s_{\pm}+\frac{3}{4\Lambda\alpha\beta}\right)}{\frac{2\Lambda}{3}(2s_{\pm}+1)}$
(47) $\displaystyle a_{n}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{\frac{\Lambda}{3}(-n+s_{\pm})(-n+s_{\pm}+3)+\frac{1}{4\alpha\beta}}\times$
$\displaystyle\Big{[}a^{\pm}_{n-2}(-n+s_{\pm}+2)(-n+s_{\pm}+3)$
$\displaystyle-2m\ a^{\pm}_{n-3}(-n+s_{\pm}+3)^{2}\Big{]},\quad n>2.$

Note that $\Re(s_{\pm})<0$ for all values of $\Lambda$, $\alpha$ and $\beta$,
so that $\lim_{r\to\infty}\varphi(r)=\Lambda$, with $s_{\pm}$ being real for
$\Lambda<0$ or $\Lambda\geq\frac{1}{3\alpha\beta}$. On the interval
$0<\Lambda<\frac{1}{3\alpha\beta}$, $s_{\pm}$ become complex, including the
physically realistic case when $\Lambda$ is small and positive. One might
worry that such a situation precludes a physically meaningful solution for the
most relevant region of parameter space, however this is not the case. So far,
we have been assuming that the integration constants $a^{+}_{0}$ and
$a^{-}_{0}$ are real and independent, but they still define a solution of the
homogeneous equation even when they are complex, albeit a generically complex-
valued one. If we instead take $a^{-}_{0}=\bar{a}^{+}_{0}$, then $\varphi$ is
real when $s_{\pm}$ are complex conjugates. After some algebra, one finds the
explicitly real form

$\displaystyle\varphi=\Lambda+\sum_{n=0}^{\infty}2r^{-n+\Re(s_{+})}$
$\displaystyle\\!\\!\\!\big{[}$
$\displaystyle\\!\\!\\!w_{n}\cos\big{(}\Im(s_{+})\,\,\textrm{ln}(r)\,\big{)}$
(49)
$\displaystyle\,-\,z_{n}\sin\big{(}\Im(s_{+})\,\textrm{ln}(r)\big{)}\big{]}.$

where $w_{n}=\Re(a^{+}_{n})$ and $z_{n}=\Im(a^{+}_{n})$. The series expansions
above completely characterize the solutions for $\varphi$ and $\mathbf{H}$ in
the asymptotic region. However, the series will not converge for all values of
$r$. We will provide expressions for the radii of convergence of all of our
series solutions at the end of this section. With this in mind, we now search
for a solution to the equations of motion for small $r$.
The equation of motion for $h(r)$ is still given by (34), but now we guess a
power series of the form
$h_{0}(r)=\sum\limits_{n=0}^{\infty}\tilde{c}_{n}r^{n+\tilde{q}}$. The
indicial equation is simply $\tilde{q}(\tilde{q}-3)=0$, giving the two roots
$\tilde{q}_{1}=3$ and $\tilde{q}_{2}=0$. In general, the full solution for the
case of roots $\tilde{q}_{1}>\tilde{q_{2}}$ differing by an integer is given
by

$h_{0}(r)=\sum\limits_{n=0}^{\infty}\tilde{c}_{n}r^{n+\tilde{q}_{1}}+k\ln(r)\
\sum\limits_{n=0}^{\infty}\tilde{c}_{n}r^{n+\tilde{q}_{1}}+\sum\limits_{n=0}^{\infty}\tilde{d}_{n}r^{n+\tilde{q}_{2}},$
(50)

where the procedure for determining the coefficients $\tilde{d}_{n}$ depends
on whether or not $k$ vanishes. If $k$ does vanish, the $\tilde{d}_{n}$ can be
obtained using the usual recurrence relation, and otherwise they must be
computed using a more complicated approach. In general, the value of $k$ is
determined by

$k=\lim_{\tilde{q}\to\tilde{q}_{2}}(\tilde{q}-\tilde{q}_{2})\tilde{c}_{\tilde{q}_{1}-\tilde{q}_{2}}(\tilde{q}).$
(51)

For this particular differential equation, the recurrence relation for general
$\tilde{q}$ is given by

$\displaystyle\tilde{c}_{1}(\tilde{q})$ $\displaystyle=$
$\displaystyle\tilde{c}_{2}(\tilde{q})=0$ (52)
$\displaystyle\tilde{c}_{n+3}(\tilde{q})$ $\displaystyle=$
$\displaystyle\frac{1}{{8\alpha\beta m(n+3+\tilde{q})(n+\tilde{q})}}\times$
$\displaystyle\Big{[}4\alpha\beta\tilde{c}_{n+2}(n+2+\tilde{q})(n-1+\tilde{q})$
$\displaystyle-\tilde{c}_{n}(1+\frac{4\alpha\beta\Lambda}{3}(n+\tilde{q})(n-3+\tilde{q}))\Big{]},\quad
n\geq 0,$

from which we conclude that

$k=\lim_{\tilde{q}\to 0}\tilde{q}\,\tilde{c}_{3}(\tilde{q})=0.$ (54)

Hence, the general solution becomes

$h_{0}(r)=\sum\limits_{n=0}^{\infty}\tilde{c}_{n}r^{n+3}+\sum\limits_{n=0}^{\infty}\tilde{d}_{n}r^{n},$
(55)

where the coefficients are given by:

$\displaystyle\tilde{c}_{1}$ $\displaystyle=$ $\displaystyle\tilde{c}_{2}=0$
(56) $\displaystyle\tilde{c}_{n+3}$ $\displaystyle=$
$\displaystyle\frac{4\alpha\beta\tilde{c}_{n+2}(n+5)(n+2)-\tilde{c}_{n}(1+\frac{4\alpha\beta\Lambda}{3}(n+3)n)}{8\alpha\beta
m(n+6)(n+3)},$

and

$\displaystyle\tilde{d}_{1}$ $\displaystyle=$ $\displaystyle\tilde{d}_{2}=0$
(58) $\displaystyle\tilde{d}_{n+3}$ $\displaystyle=$
$\displaystyle\frac{4\alpha\beta\tilde{d}_{n+2}(n+2)(n-1)-\tilde{d}_{n}(1+\frac{4\alpha\beta\Lambda}{3}n(n-3))}{8\alpha\beta
m(n+3)n},$

with $n\geq 0$. The equation of motion for $\varphi(r)$ is given by (42), but
now we guess a power series of the form
$\varphi(r)=\Lambda+\sum\limits_{n=0}^{\infty}\tilde{a}_{n}r^{n+\tilde{s}}$.
The indicial equation is

$\displaystyle\tilde{s}^{2}=0,$ (60)

giving repeated root $\tilde{s}_{1}=\tilde{s}_{2}=0$, while the recurrence
relation for general $\tilde{s}$ is given by

$\displaystyle\tilde{a}_{1}(\tilde{s})$ $\displaystyle=$
$\displaystyle\tilde{a}_{2}(\tilde{s})=0$ (61)
$\displaystyle\tilde{a}_{n+3}(\tilde{s})$ $\displaystyle=$
$\displaystyle\frac{1}{2m(n+\tilde{s}+3)^{2}}\times$
$\displaystyle\Big{[}\tilde{a}_{n+2}(n+\tilde{s}+2)(n+\tilde{s}+3)$
$\displaystyle-\tilde{a}_{n}\big{(}\tfrac{1}{4\alpha\beta}+\tfrac{\Lambda}{3}(n+\tilde{s})(n+\tilde{s}+3)\big{)}\Big{]},\quad
n\geq 0.$

In the special case of a repeated root, the solution takes the general form

$\displaystyle\\!\\!\\!\\!\\!\varphi(r)=\Lambda$ $\displaystyle+$
$\displaystyle\sum\limits_{n=0}^{\infty}\tilde{a}_{n}r^{n+\tilde{s}}$ (63)
$\displaystyle+$
$\displaystyle\tilde{K}\Big{(}\ln(r)\sum\limits_{n=0}^{\infty}\tilde{a}_{n}r^{n+\tilde{s}}+\sum\limits_{n=1}^{\infty}\tilde{b}_{n}r^{n+\tilde{s}}\Big{)}$

where $\tilde{K}$ is an integration constant. Here the coefficients $a_{n}$
are given by

$\displaystyle\tilde{a}_{1}$ $\displaystyle=$ $\displaystyle\tilde{a}_{2}=0$
(64) $\displaystyle\tilde{a}_{n+3}$ $\displaystyle=$
$\displaystyle\frac{\tilde{a}_{n+2}(n+2)(n+3)-\tilde{a}_{n}(\frac{1}{4\alpha\beta}+\frac{\Lambda}{3}n(n+3))}{2m(n+3)^{2}},$

and the coefficients $b_{n}$ are related to the $a_{n}$ by

$\displaystyle\tilde{b}_{n}$ $\displaystyle=$
$\displaystyle\frac{d}{d\tilde{s}}\tilde{a}_{n}(\tilde{s})\Big{|}_{\tilde{s}=0},$
(66)

with $n\geq 0$.

The radius of convergence of each power series solution is the distance in the
complex plane between the point around which the series is expanded and the
nearest other regular singular point of the differential equation. For both
(34) and (42), the regular singular points are located at $r=0$, $r=\infty$,
and the zeros of $A_{0}(r)$. Thus, the radii of convergence are determined by
the roots of a general depressed cubic equation

$\displaystyle r^{3}-\frac{3}{\Lambda}r+\frac{6m}{\Lambda}=0$ (67)

with discriminant

$\displaystyle\Delta=\frac{1}{\Lambda^{3}}-\frac{9m^{2}}{\Lambda^{2}}.$ (68)

The roots can be written as

$\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle S_{+}+S_{-}$ (69)
$\displaystyle R_{\pm}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}R_{1}\pm\frac{1}{2}i\sqrt{3}(S_{+}-S_{-})$ (70)

where

$S_{\pm}=\sqrt[3]{-\frac{3m}{\Lambda}\pm\left(-\frac{1}{\Lambda^{3}}+\frac{9m^{2}}{\Lambda^{2}}\right)^{1/2}}.$

If $\Lambda<0$, the discriminant is always negative and $R_{1}$ will be real,
while $R_{\pm}$ are complex conjugates. The moduli are
$|R_{1}|^{2}=(S_{+}+S_{-})^{2}$ while
$|R_{\pm}|^{2}={(S_{+}+S_{-})^{2}-3S_{+}S_{-}}$. Then, $S_{+}>0$ and
$S_{-}<0$, leading to $|R_{\pm}|>|R_{1}|$. So the radius of convergence of the
$r$-series is given by $0<r<|R_{1}|$ while the radius of convergence of the
$\frac{1}{r}$-series is given by $0<\frac{1}{r}<\frac{1}{|R_{\pm}|}$.

If $\Lambda>0$, we have two cases. If $\Lambda>\frac{1}{9m^{2}}$ then the
discriminant is negative and we still have $R_{1}$ real with $R_{\pm}$ being
complex conjugates. Since $S_{+}\\!<0$ and $S_{-}\\!<0$, $|R_{1}|>|R_{\pm}|$.
Then the radius of convergence of the $r$-series is given by $0<r<|R_{\pm}|$
while the radius of convergence of the $\frac{1}{r}$-series is given by
$0<\frac{1}{r}<\frac{1}{|R_{1}|}$.

Finally, if $0<\Lambda<\frac{1}{9m^{2}}$, then $S_{+}$ and $S_{-}$ become
complex conjugates and we have three real roots. This can be seen by writing
$S_{\pm}=(a\pm ib)^{1/3}=\rho^{1/3}e^{\pm i\theta/3}$, where
$a=-\frac{3m}{\Lambda}$,
$b=\left(\frac{1}{\Lambda^{3}}-\frac{9m^{2}}{\Lambda^{2}}\right)^{1/2}$,
$\rho=\sqrt{a^{2}+b^{2}}$ and $\theta=\arctan^{-1}(b/a)$. Then,
$|R_{1}|=2\rho^{1/3}\cos(\theta/3)$, and
$|R_{\pm}|=\rho^{1/3}(\cos(\theta/3)\pm\sqrt{3}\sin(\theta/3))$. Since
$b/a<0$, i.e. $-\pi/2\leq\theta\leq 0$, we have
$|R_{1}|\geq|R_{-}|\geq|R_{+}|$ with equalities occurring only at the
endpoints of the interval. Hence, the radius of convergence of the $r$-series
is given by $0<r<|R_{-}|$ , while the radius of convergence of the
$\frac{1}{r}$-series is given by $0<\frac{1}{r}<\frac{1}{|R_{1}|}$.

With the series solutions for $\varphi$ and $\mathbf{H}$ in hand, one can now,
take into account the approximate back-reaction on the (A)dS-Schwarzschild
background by substituting the solutions for the fields into the general
solution for the metric function (32).

Figure 1: Plots of the metric function $A(r)$ are shown for various choices of
the initial values
$(\bar{a}^{+}_{0},\bar{a}^{-}_{0})=\\{0,0.002,0.004,0.008,0.01\\}\times(m_{\tiny{\astrosun}}^{-(s_{+}+2)},m_{\tiny{\astrosun}}^{-(s_{-}+2)})$,
where the label $A_{i}(r)$ coincides with the position of the value in the
list, $m_{\tiny{\astrosun}}$ is the mass of the sun, and $s_{+}\approx 0$,
$s_{-}\approx-3$. Here we have fixed $m=50\,m_{\tiny{\astrosun}}$,
$\Lambda_{0}=10^{-5}m_{\tiny{\astrosun}}^{-2}$,
$r_{0}=10^{5}m_{\tiny{\astrosun}}$, $\beta=10^{-3}m_{\tiny{\astrosun}}^{4}$,
$\alpha=\frac{1}{8}\times 10^{6}m_{\tiny{\astrosun}}^{-2}$ and chosen the
particular solution $h(r)=\frac{1}{3}r^{3}.$ As $a^{\pm}_{0}\to 0$, the
solution approaches the dS-Schwarzschild metric $A_{0}(r)$, while for larger
values of $a^{\pm}_{0}$ the event horizon is pushed outward while the
cosmological horizon is pushed inward. At some critical values of
$a^{\pm}_{0}$ the horizons degenerate and the black hole becomes extremal,
while for still larger values of $a^{\pm}_{0}$ the solution has a naked
singularity.

Figure 2: Plots of the metric function $A(r)$ are shown for various values of
$\Lambda_{0}=\\{1.00,0.25,0.50,2.00,4.00\\}\times
10^{-5}m_{\tiny{\astrosun}}^{-2}$, where the label $A_{i}(r)$ coincides with
the position of the value in the list$,m_{\tiny{\astrosun}}$ is the mass of
the sun, and we have fixed
$a_{0}^{+}=.005m_{\tiny{\astrosun}}^{-(s_{+}+2)},a_{0}^{-}=.005m_{\tiny{\astrosun}}^{-(s_{-}+2)}$,
with $s_{+}\approx 0$, $s_{-}\approx-3$, $m=50m_{\tiny{\astrosun}}$,
$r_{0}=10^{5}m_{\tiny{\astrosun}}$, $\beta=10^{-3}m_{\tiny{\astrosun}}^{4}$,
$\alpha=\frac{1}{8}\times 10^{6}m_{\tiny{\astrosun}}^{-2}$ and chosen the
particular solution $h(r)=\frac{1}{3}r^{3}.$ We see that for small values of
$\Lambda_{0}$ both horizons occur for larger values of $r$, while for larger
values of $\Lambda_{0}$ the horizons are pushed inward and closer together
until at some critical value, the horizons degenerate and the black hole
becomes extremal. For even larger values of $\Lambda_{0}$, no horizons exist
and the solution has a naked singularity.

Using the series solutions for large $r$, one can perform a numerical
integration to plot the profile function $A(r)$ whose zeros are the black hole
and cosmological horizons for different values of $a^{-}_{0}$ and
$\bar{a}^{+}_{0}$, as shown in figure 2. As expected, for appropriate values
of the couplings and integration constants, we find a pair of horizons which
tend to the dS-Schwarzschild ones as $a^{\pm}_{0}\to 0$. Note that the
discrepancy from $A_{0}(r)$ of the inner horizon is greater than the outer
horizon. However, as $\varphi$ is allowed to fluctuate more strongly, i.e., as
$a^{\pm}_{0}$ becomes large, the solution does not have a horizon. In figure 2
we show the results for different values of $\Lambda_{0}$. In order for the
solution to approach dS-Schwarzschild in the decoupling limit, we fix
$\Lambda_{0}=\Lambda+\alpha\kappa$. For $\Lambda_{0}<0$, the lower endpoint of
integration in (32) becomes imaginary and the solution becomes complex and
unphysical. Hence, we see that the model has a dynamical preference for black
holes with $\Lambda_{0}\geq 0$.

This completes our analysis of spherically symmetric black holes in dynamical
Henneaux-Teitelboim gravity. Another very natural question to consider is how
the dynamics of the effective cosmological constant modify the standard
cosmological background of general relativity, which we study in the following
section.

##  IV Cosmological analysis

We will now consider homogeneous, isotropic configurations sourced only by the
fields $\varphi$ and $\mathbf{H}$ with no additional matter content. Our
analysis runs along the same lines as other work on cosmology in unimodular
gravity with a non-conserved stress tensor, e.g. garcia2019cosmic , where the
model we consider here serves as a concrete realization of the general
proposal.

We consider the FRW metric

$ds^{2}=-dt^{2}+a(t)^{2}(dx^{2}+dy^{2}+dz^{2})$ (71)

along with the homogeneous and isotropic fields

$\displaystyle\mathbf{H}$ $\displaystyle=$ $\displaystyle h(t)\,dx\wedge
dy\wedge dz,$ (72) $\displaystyle\varphi$ $\displaystyle=$
$\displaystyle\varphi(t).$ (73)

Again, we are only considering contributions to the stress tensor from
$\mathbf{H}$ and $\varphi$, so we set $\mathcal{L}_{M}=0$ in (7).

We decompose the Einstein equations into trace-free and pure trace parts,
recalling that the stress tensor for $\mathbf{H}$ is pure trace. The trace-
free equation is given by (1), and plugging in our symmetry-reduced
expressions for the metric, three-form, and scalar, we find a single
independent equation:

$\frac{\kappa}{2}\beta\dot{\varphi}^{2}=-\frac{\dot{a}^{2}-a\ddot{a}}{a^{2}}.$
(74)

The trace equation (21) gives

$\varphi=\frac{3}{2}\left(\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\right)-\frac{\kappa}{4}\left(\beta\dot{\varphi}^{2}+\alpha\frac{\dot{h}^{2}}{6a^{6}}\right),$
(75)

which can be combined with (74) to give

$\varphi=\frac{\ddot{a}}{a}+2\frac{\dot{a}^{2}}{a^{2}}-\frac{\kappa\alpha}{24}\frac{\dot{h}^{2}}{a^{6}}.$
(76)

The equations of motion for the scalar and three-form imply (16) and (17),
which upon symmetry reduction become

$\displaystyle\dot{h}-4\alpha\beta\frac{d}{dt}\left(a^{3}\frac{d}{dt}\Big{(}\frac{\dot{h}}{a^{3}}\Big{)}\right)=a^{3}$
(77)
$\displaystyle\varphi-4\alpha\beta\left(3\frac{\dot{a}}{a}\dot{\varphi}+\ddot{\varphi}\right)=\Lambda.$
(78)

Making use of the Hubble parameter $H=\frac{\dot{a}}{a}$, defining
$u=\frac{\varphi}{\Lambda}$, $v=\frac{\dot{h}}{a^{3}}$, and adimensionalizing,
we simplify our problem to three equations333It is worth noting the close
similarity between equations (79), (80) and (81) and the slow roll equations.
See e.g. liddle1994formalizing .:

$\displaystyle u-1=3\tilde{H}\dot{u}+\ddot{u}$ (79) $\displaystyle
v-1=3\dot{v}\tilde{H}+\ddot{v}$ (80)
$\displaystyle\tilde{\Lambda}u=\dot{\tilde{H}}+3\tilde{H}^{2}+pv^{2},$ (81)

where $\tilde{\Lambda}=4\alpha\beta\Lambda$, $p=-\kappa\alpha^{2}\beta/24$,
$\tilde{H}=\sqrt{4\alpha\beta}H$, and $\tilde{t}=t/\sqrt{4\alpha\beta}$. Thus,
we search for a fixed point, and find one at $u=1$, $v=1$, and
$\tilde{H}=H_{0}\equiv\sqrt{\frac{\tilde{\Lambda}-p}{3}}$. Perturbing around
this fixed point decouples two of the equations, yielding

$\displaystyle\delta u=3H_{0}\delta\dot{u}+\delta\ddot{u}$ (82)
$\displaystyle\delta v=3H_{0}\delta\dot{v}+\delta\ddot{v}$ (83)
$\displaystyle\tilde{\Lambda}\delta u-2p\delta
v=\delta\dot{\tilde{H}}+6H_{0}\delta\tilde{H}.$ (84)

These equations are solvable. The first two equations simply give

$\displaystyle\delta
u=A_{u}e^{\omega_{+}\tilde{t}}+B_{u}e^{\omega_{-}\tilde{t}}$ (85)
$\displaystyle\delta
v=A_{v}e^{\omega_{+}\tilde{t}}+B_{v}e^{\omega_{-}\tilde{t}}$ (86)

where $\omega_{\pm}=-\frac{3H_{0}}{2}\pm\sqrt{\frac{9}{4}H_{0}^{2}+1}$. The
last equation has a homogeneous solution given by

$\delta\tilde{H}^{(h)}=C_{H}e^{-6H_{0}\tilde{t}}$ (87)

and a particular solution given by

$\delta\tilde{H}^{(p)}=A_{H}e^{\omega_{+}\tilde{t}}+B_{H}e^{\omega_{-}\tilde{t}},$
(88)

where the coefficients are

$\displaystyle A_{H}=\frac{\tilde{\Lambda}A_{u}-2pA_{v}}{\omega_{+}+6H_{0}}$
(89) $\displaystyle
B_{H}=\frac{\tilde{\Lambda}B_{u}-2pB_{v}}{\omega_{-}+6H_{0}}.$ (90)

If we impose initial conditions such that $A_{u}=A_{v}=0$, which imply that
$A_{H}=0$, then $\varphi$ relaxes to a constant at late times, allowing for a
de Sitter-like phase of accelerated expansion in the late universe. However,
in general both the exponentially growing and decaying modes will be present,
so we expect that the solution that is compatible with observations of our
current cosmological epoch will be unstable to small perturbations around this
background. Still, having a solution of the linearized equations of motion
which approaches de Sitter at late times may be useful for more sophisticated
analyses that include matter other than $\mathbf{H}$ and $\varphi$.

##  V Discussion

In this work we have introduced dynamical Henneaux-Teitelboim gravity, a
minimal modification of general relativity based closely on the Henneaux-
Teitelboim formulation of unimodular gravity that realizes fluctuations in the
cosmological constant sourced by non-conservation of the energy-momentum
tensor. Furthermore, we have considered the implications of this model for
spherically symmetric black holes and homogeneous, isotropic configurations.
In order to make progress analytically, we have made a number of simplifying
assumptions that should ultimately be lifted in order to probe more realistic
physical scenarios. For the analysis of black holes, we have only presented an
approximate solution of all the field equations by perturbing around the
Schwarzschild-de Sitter metric and using series methods. In order to establish
whether such an approximate solution is physically viable we would need, at a
minimum, to study the geodesic motion of a massive particle orbiting such a
black hole around the equator and compare with astrophysical observations. The
assumption of spherical symmetry dramatically simplifies the equations of
motion, reducing them to ordinary differential equations, but of course, in
order to model rotating black holes, this restriction must be lifted. Studies
of rotating black holes in similar modified gravity models such as dynamical
Chern-Simons gravity alexander2009chern ; yagi2012slowly ; yunes2009dynamical
; alexander2021chern ; alexander2022black1  suggest that going to the slowly-
rotating limit and making use of numerical methods are likely to be necessary.
Finally, the series solutions we have presented for small and large $r$ have
finite radii of convergence, leaving the possibility of intermediate regions,
possibly containing the event horizon, which are not covered by either series
solution. Here again, numerical methods for interpolating between the
solutions for small and large $r$ might be useful. We leave such numerical
studies for the future, but note that for the solutions we have studied in the
decoupling limit that are close to Schwarzschild-de Sitter, the radius of
convergence is well below the scale of the event horizon. On the cosmology
side, we have only considered the contribution to the stress tensor from the
fields $\mathbf{H}$ and $\varphi$ for simplicity. In order to make realistic
predictions, we would need, at a minimum, to include contributions from
general baryonic matter and radiation as well, consider the effects of
cosmological perturbations, and, if they are stable, study the matter-power
spectrum for deviations from $\Lambda$CDM. We have also seen that while a
solution of the linearized equations of motion that approaches de Sitter at
late times exists, this solution is apparently unstable. It would be
interesting to see whether this instability persists in a more realistic
analysis including additional matter and radiation, or whether, perhaps, the
exponentially growing modes might be dynamically suppressed by the inclusion
of additional matter degrees of freedom. The model we have considered in this
article is a natural extension of the Henneaux-Teitelboim formulation of
unimodular gravity because the kinetic terms for the scalar and three-form
fields are invariant under the local symmetries of the latter—spacetime
diffeomorphisms and two-form-valued gauge transformations. Consequently, the
kinetic terms are forced upon us by “Gell-Mann’s totalitarian principle,”
which dictates that “everything that is not forbidden (by local symmetries) is
compulsory.” To put it plainly, renormalization group flow will generate every
term that is compatible with the local symmetries of the action (5), and that
includes kinetic terms for the scalar and three-form fields. In this sense,
the model we have studied in this article follows straightforwardly from (5)
and general principles of effective field theory.

##  VI Acknowledgements

We are grateful to Tucker Manton and Roman Marcarelli for discussions and
helpful feedback on an early version of this manuscript. EA was supported by
the STFC Consolidated Grant ST/W507519/1.

##  Appendix A Coefficients and Indices for Series Solutions

In this section we perform detailed calculations of the coefficients and
indices for the series solutions of the equations of motion for $\mathbf{H}$
and $\varphi$ in the spherically symmetric case. Let us first focus on the
homogeneous equation

$h_{0}(r)-4\alpha\beta
A_{0}(r)\left(h_{0}^{\prime\prime}(r)-\frac{2h_{0}^{\prime}(r)}{r}\right)=0.$
(91)

Putting the series ansatz $h_{0}(r)=\sum\limits_{n=0}^{\infty}c_{n}r^{-n+q}$
into (91) gives

$\displaystyle\sum\limits^{\infty}_{n=0}\Big{[}c_{n}r^{-n+q}-4\alpha\beta
c_{n}(-n+q)(-n+q-3)$ $\displaystyle\left(r^{-n+q-2}-2m\
r^{-n+q-3}-\tfrac{\Lambda}{3}r^{-n+q}\right)\Big{]}=0.$ (92)

Now, reindexing

$\displaystyle\sum\limits^{\infty}_{n=0}$
$\displaystyle\\!\\!c_{n}(1+\tfrac{4\alpha\beta\Lambda}{3}(-n+q)(-n+q-3))r^{-n+q}$
$\displaystyle-4\alpha\beta\\!\\!\\!$
$\displaystyle\sum\limits^{\infty}_{n=2}$
$\displaystyle\\!\\!c_{n-2}(-n+q+2)(-n+q-1)r^{-n+q}$ (93)
$\displaystyle-4\alpha\beta\\!\\!\\!$
$\displaystyle\sum\limits^{\infty}_{n=3}$
$\displaystyle\\!\\!(-2m)c_{n-3}(-n+q+3)(-n+q)r^{-n+q}=0$

Now, let us find the recurrence relation and the allowed values of $q$.
Firstly, consider the base cases. For $n=0$ we have only terms of order
$r^{q}$, which yields the indicial equation

$\displaystyle c_{0}\left(1+\frac{4\Lambda\alpha\beta}{3}q(q-3)\right)=0.$
(94)

Hence,

$\displaystyle c_{0}=0\quad\textrm{or}\quad
q=q_{\pm}\equiv\frac{3}{2}\pm\frac{3}{2}\sqrt{1-\frac{1}{3\Lambda\alpha\beta}}.$
(95)

We can discard $c_{0}=0$ without loss of generality, since this amounts to
reindexing the sum to start from $n=1$. For $n=1$, we have only terms of order
$r^{q-1}$, so

$\displaystyle c_{1}\left(1+\frac{4\Lambda\alpha\beta}{3}(q-1)(q-4)\right)=0,$
(96)

which tells us that either

$\displaystyle c_{1}=0\quad\textrm{or}\quad
q=\frac{5}{2}\pm\frac{3}{2}\sqrt{1-\frac{1}{3\Lambda\alpha\beta}}.$ (97)

We accept the first solution since the second one is inconsistent with
$c_{0}\neq 0$. For $n=2$, we have only terms of order $r^{q-2}$, so

$\displaystyle
c_{2}\left(1+\frac{4\alpha\beta\Lambda}{3}(q-5)(q-2)+\right)-c_{0}4\alpha\beta
q(q-3)=0,$ (98)

which tells us that

$\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle\frac{c_{0}\
4\alpha\beta\ q(q-3)}{\frac{4\Lambda\alpha\beta}{3}(q-2)(q-5)+1}$ (99)

Then, for $n\geq 3$ we have only terms of order $r^{q-n}$, so

$\displaystyle
c_{n}=\frac{4\alpha\beta}{1+\frac{4\alpha\beta\Lambda}{3}(-n+q)(-n+q-3)}\times$
$\displaystyle\Big{[}c_{n-2}(-n+q+2)(-n+q-1)$ (100) $\displaystyle-2m\
c_{n-3}(-n+q+3)(-n+q)\Big{]}$

We now work with the scalar field equation

$\varphi-4\alpha\beta\left(A^{\prime}\varphi^{\prime}+\frac{2}{r}A\varphi^{\prime}+A\varphi^{\prime\prime}\right)=\Lambda$
(101)

Subtracting $\Lambda$ from both sides and letting $y=\varphi-\Lambda$:

$A(r)y^{\prime\prime}(r)+\left(\frac{2}{r}A(r)+A^{\prime}(r)\right)y^{\prime}(r)-\frac{1}{4\alpha\beta}y(r)=0$
(102)

Noting the Schwarzschild-AdS metric
$A(r)=1-\frac{2m}{r}-\frac{\Lambda}{3}r^{2}$, we have:

$\left(1-\frac{2m}{r}-\frac{\Lambda}{3}r^{2}\right)y^{\prime\prime}+\left(\frac{2}{r}-\frac{2m}{r^{2}}-\frac{2\Lambda}{3}r\right)y^{\prime}-\frac{1}{4\alpha\beta}y=0$
(103)

Plugging in the series ansatz $y=\sum\limits_{n=0}^{\infty}a_{n}r^{-n+s}$
yields:

$\displaystyle\sum\limits^{\infty}_{n=0}$ $\displaystyle
a_{n}(-n+s)(-n+s+1)r^{-n+s-2}$ $\displaystyle-\\!\\!\\!$
$\displaystyle\sum\limits^{\infty}_{n=0}$ $\displaystyle\ 2m\
a_{n}(-n+s)^{2}r^{-n+s-3}$ (104) $\displaystyle-\\!\\!\\!$
$\displaystyle\sum\limits^{\infty}_{n=0}$ $\displaystyle
a_{n}\left((-n+s)(-n+s+3)\frac{\Lambda}{3}+\frac{1}{4\alpha\beta}\right)r^{-n+s}=0$

Reindexing gives:

$\displaystyle\sum\limits^{\infty}_{n=2}$ $\displaystyle
a_{n-2}(-n+s+2)(-n+s+3)r^{-n+s}$ $\displaystyle-\\!\\!\\!$
$\displaystyle\sum\limits^{\infty}_{n=3}$ $\displaystyle\ 2m\
a_{n-3}(-n+s+3)^{2}r^{-n+s}$ (105) $\displaystyle-\\!\\!\\!$
$\displaystyle\sum\limits^{\infty}_{n=0}$ $\displaystyle
a_{n}\left((-n+s)(-n+s+3)\frac{\Lambda}{3}+\frac{1}{4\alpha\beta}\right)r^{-n+s}=0$

Now we focus on the recurrence relation. First consider the base case $n=0$.
Here we only have terms of order $r^{s}$, so we have:

$a_{0}\left(\frac{\Lambda}{3}s(s+3)+\frac{1}{4\alpha\beta}\right)=0.$ (106)

Hence

$a_{0}=0\quad\textrm{or}\quad
s=s_{\pm}\equiv-\frac{3}{2}\pm\frac{3}{2}\sqrt{1-\frac{1}{3\Lambda\alpha\beta}}.$
(107)

We can discard $a_{0}=0$ without loss of generality since this amounts to
reindexing the sum to start from $n=1$. For $n=1,$ we have only terms of order
$r^{s-1}$, so we have

$a_{1}\left(\frac{\Lambda}{3}(s-1)(s+2)+\frac{1}{4\alpha\beta}\right)=0,$
(108)

Thus we have

$a_{1}=0\quad\textrm{or}\quad s=s_{\pm}+1.$ (109)

The second solution is inconsistent with $a_{0}\neq 0$. For $n=2,$ we have
only terms of order $r^{s-2}$, giving us:

$-a_{2}\left(\frac{\Lambda}{3}(s+1)(s-2)+\frac{1}{4\alpha\beta}\right)+a_{0}(s+1)s=0,$
(110)

and so we find

$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{a_{0}(s+1)s}{\frac{\Lambda}{3}(s+1)(s-2)+\frac{1}{4\alpha\beta}}$
(111) $\displaystyle=$
$\displaystyle\frac{a_{0}\left(2s+\frac{3}{4\Lambda\alpha\beta}\right)}{\frac{2\Lambda}{3}(2s+1)},$

where we used $s^{2}+3s=-\frac{3}{4\Lambda\alpha\beta}$. Finally, for $n\geq
3$, we have terms of order $r^{s-n}$, so we have:

$\displaystyle-
a_{n}\left(\tfrac{\Lambda}{3}(-n+s)(-n+s+3)+\tfrac{1}{4\alpha\beta}\right)$
$\displaystyle+a_{n-2}(-n+s+2)(-n+s+3)$ (112)
$\displaystyle-2m\,a_{n-3}(-n+s+3)^{3}=0$

which gives us the recurrence relation

$\displaystyle a_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{\frac{\Lambda}{3}(-n+s)(-n+s+3)+\frac{1}{4\alpha\beta}}\times$
$\displaystyle\Big{[}a^{\pm}_{n-2}(-n+s+2)(-n+s+3)$ $\displaystyle-2m\
a_{n-3}(-n+s+3)^{2}\Big{]},\quad n>2.$

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