# Black holes in $f(\mathbb{Q})$ Gravity

Fabio D’Ambrosio<EMAIL_ADDRESS>Shaun D.B. Fell<EMAIL_ADDRESS>Lavinia Heisenberg<EMAIL_ADDRESS>Simon Kuhn
<EMAIL_ADDRESS>Institute for Theoretical Physics, ETH Zurich, Wolfgang-
Pauli-Strasse 27, 8093, Zurich, Switzerland

###### Abstract

We systematically study the field equations of $f(\mathbb{Q})$ gravity for
spherically symmetric and stationary metric-affine spacetimes. Such spacetimes
are described by a metric as well as a flat and torsionless affine connection.
In the Symmetric Teleparallel Equivalent of GR (STEGR), the connection is pure
gauge and hence unphysical. However, in the non-linear extension
$f(\mathbb{Q})$, it is promoted to a dynamical field which changes the
physics. Starting from a general metric-affine geometry, we construct the most
general static and spherically symmetric forms of the metric and the affine
connection. We then use these symmetry reduced geometric objects to prove that
the field equations of $f(\mathbb{Q})$ gravity admit GR solutions as well as
beyond-GR solutions, contrary to what has been claimed in the literature. We
formulate precise criteria, under which conditions it is possible to obtain GR
solutions and under which conditions it is possible to obtain beyond-GR
solutions. We subsequently construct several perturbative corrections to the
Schwarzschild solution for different choices of $f(\mathbb{Q})$, which in
particular include a hair stemming from the now dynamical affine connection.
We also present an exact beyond-GR vacuum solution. Lastly, we apply this
method of constructing spherically symmetric and stationary solutions to
$f(\mathbb{T})$ gravity, which reproduces similar solutions but without a
dynamical connection.

###### Contents

  1. I Introduction
  2. II Symmetric Teleparallelism and $f(\mathbb{Q})$ Gravity
  3. III Symmetry reduction of the metric and the connection
    1. III.1 Symmetry reduction of $\Gamma^{\alpha}{}_{\mu\nu}$ under the assumption that $T^{\alpha}{}_{\mu\nu}=0$
    2. III.2 Implementation of $R^{\alpha}{}_{\mu\nu\rho}=0$
    3. III.3 Solution set 1
    4. III.4 Solution set 2
    5. III.5 Solution set 1 as the double scaling limit of solution set 2
    6. III.6 Diagonalizing the metric, canonical $\mathcal{S}^{2}$ part, and transformation behavior of the solution sets
  4. IV Symmetry reduced field equations for the metric and the connection
    1. IV.1 Off-diagonal metric field equation for solution set 2
    2. IV.2 Off-diagonal metric field equation for solution set 1
    3. IV.3 Self-consistency of the remaining field equations for solution set 2
    4. IV.4 Reduction of the field equations by a constraint on the connection
  5. V Approximate and exact solutions
    1. V.1 Approximate vacuum solution beyond GR for $f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$
    2. V.2 Approximate electro-vacuum solution beyond GR for $f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}-2\Lambda$
    3. V.3 Approximate solutions beyond GR from constraints on the connection
    4. V.4 A comment on isotropic coordinates
    5. V.5 Exact vacuum solutions beyond GR for $f(\mathbb{Q})=\mathbb{Q}^{\kappa}$
  6. VI Application to $f(\mathbb{T})$ gravity
    1. VI.1 Metric Teleparallelism and $f(\mathbb{T})$ gravity
    2. VI.2 Symmetry reduction of the connection and metric
    3. VI.3 Symmetry reduced field equations for the metric and the connection
  7. VII Conclusion
  8. A Connection transformation when diagonalizing the metric
  9. B Approximate solutions for $f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{\kappa}$
    1. B.1 Connection hair solutions
    2. B.2 Constraint solutions

##  I Introduction

General Relativity (GR) is by far the most successful description of gravity
we have. Its predictions for gravitational effects on solar system scales and
on cosmological scales fit the observations very well. However, some slight
tensions, such as the $H_{0}$ tension, have appeared over the past few years.
In addition, theoretical difficulties such as singularities, quantum gravity
and a lacking explanation for the origin of dark matter and dark energy spoil
the beauty of GR. It is thus fruitful to look at generalizations of GR, which
might resolve both observational and theoretical issues.

One such generalization is provided by Symmetric Teleparallelism (ST), which
is rooted in a different set of geometric postulates than GR. The key
difference between ST and GR is the role played by the affine connection,
$\Gamma^{\alpha}{}_{\mu\nu}$. In GR, it is postulated that the connection is
torsionless and metric-compatible, which immediately implies that it is
uniquely given by the Levi-Civita connection. In ST the postulate of metric-
compatibility is dropped and one instead demands that
$\Gamma^{\alpha}{}_{\mu\nu}$ is torsionless and gives rise to a vanishing
Riemann tensor. As long as the connection satisfies these postulates, it can
be chosen arbitrarily and, in particular, it is independent of the metric.
With curvature and torsion of $\Gamma^{\alpha}{}_{\mu\nu}$ postulated to be
zero, the only non-trivial object left in ST which characterizes the affine
geometry is the non-metricity tensor, $Q_{\alpha\mu\nu}$. This tensor can be
used to construct the so-called non-metricity scalar $\mathbb{Q}$, which will
be defined in section II, and which in turn defines the action of ST:
$\mathcal{S}[g,\Gamma]:=\int\mathrm{d}^{4}x\,\sqrt{-g}\,\mathbb{Q}$. It is
well-known BeltranJimenez:2017 ; Jimenez:2018 ; Heisenberg:2018vsk ;
BeltranJimenez:2019 ; DAmbrosio:2020  that this action is equivalent to the
Einstein-Hilbert action of GR up to a boundary term. ST therefore provides a
different geometric description of gravity, which is nevertheless equivalent
to GR. In particular, it can be shown that the affine connection only appears
in a boundary term in the action and it is hence unphysical. More precisely,
the field equations of the metric do not depend on the choice of connection
and the connection field equations are identically satisfied for any choice of
connection which is compatible with the postulates of ST. This means that the
physical degrees of freedom reside in the metric while the connection does not
carry any physical information.

This changes when one considers generalisations, such as generic theories
which are quadratic in the non-metricity tensor Dambrosio:2020b , or
extensions of ST Jimenez:2019 . What is of interest to us in the present paper
is the non-linear extension described by
$\int\mathrm{d}^{4}x\,\sqrt{-g}\,f(\mathbb{Q})$ BeltranJimenez:2017 , where
$f$ is an a priori arbitrary function. Not only is this theory not equivalent
to $f(R)$ gravity, but the theory now harbours degrees of freedom in the
metric and in the affine connection, because the dependence on
$\Gamma^{\alpha}{}_{\mu\nu}$ can no longer be absorbed in a boundary term in
the action.

The connection can thus be expected to influence the metric, which describes
the gravitational field. In this paper we show that this expectation is indeed
realized, contrary to what has been claimed in the literature Zhao:2021 ;
Lin:2021 . We show this by systematically studying the most general stationary
and spherically symmetric spacetimes within $f(\mathbb{Q})$ gravity.

The paper is organized as follows: Section II is dedicated to introducing
symmetric teleparallelism, via the Palatini formalism, as well as
$f(\mathbb{Q})$ gravity. This serves the purpose to recall basic definitions
and fixing notations and conventions. In section III we perform a detailed
symmetry reduction of the metric and the connection. In particular, we show
that there are two ways of giving an explicit parametrization of stationary,
spherically symmetric, torsionless, and flat connections (subsections III.3
and III.4). Moreover, we show that the trivial connection (i.e., the
connection in coincident gauge, $\Gamma^{\alpha}_{\ \mu\nu}=0$) fails to be
spherically symmetric and that the connection used in Zhao:2021 ; Lin:2021
belongs to the second parametrization class studied here (cf. Table 2 for a
definition of this class). In subsection III.5 we then show that the first
parametrization class can be obtained from the second one by a well-defined
double scaling limit. We also show that the metric can be brought into a
diagonal form –with two arbitrary functions of $r$ in the first half of the
diagonal and the standard metric of a $2$-sphere on the second half of the
diagonal– by means of a diffeomorphism which does not alter the structure and
defining properties of the above-mentioned parametrizations of the connection.

Subsequently, in section IV, we use the diagonal metric and the two classes of
connection to perform a symmetry reduction of the metric and connection field
equations of $f(\mathbb{Q})$ gravity. In subsection IV.2, we show that the
first parametrization class cannot produce any solutions which go beyond the
standard Schwarzschild-deSitter-Nordström solution. The symmetry reduced field
equations for the second parametrization class, which we discuss in subsection
IV.1, offer more flexibility. We discuss under which conditions the field
equations can produce solutions beyond the well-known GR solutions for
spherically symmetric and stationary spacetimes – and we explain why the
connection used in Zhao:2021 ; Lin:2021  could only produce GR solutions for
arbitrary choices of the function $f$.

In section V, we use our insights to describe approximate solutions which go
beyond Schwarzschild-deSitter-Nordström for
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$, assuming $\alpha$ is a
small parameter. In section V.5 we also present an exact vacuum solution which
goes beyond GR for the case $f(\mathbb{Q})=\mathbb{Q}^{\kappa}$, for
$\kappa\in\mathbb{R}\setminus\\{0\\}$, demonstrating that such solutions exist
in $f(\mathbb{Q})$ gravity.

Finally, in section VI, we sketch how the approach described in detail for
$f(\mathbb{Q})$ can be transferred to $f(\mathbb{T})$ gravity, a
generalization of Metric Teleparalellism. This theory of gravity is described
again by a metric and a connection, but the latter is now postulated to be
flat and metric-compatible, but with non-vanishing torsion. Since the
construction of stationary and spherically symmetric affine geometries, as
well as the analysis of the equations of motion, work in complete analogy to
$f(\mathbb{Q})$ gravity, one can easily construct the most general stationary,
spherically symmetric, flat, and metric-compatible spacetimes of
$f(\mathbb{T})$ gravity. We report our results, and their relations to
$f(\mathbb{Q})$ gravity, and compare them to the literature Hohmann:2019 ;
Bahamonde:2020vpb ; Hohmann:2019nat ; DeBenedictis:2016aze ; Ruggiero:2015oka
.

We conclude the paper in section VII with a brief discussion of the main
results and an outlook on future research.

##  II Symmetric Teleparallelism and $f(\mathbb{Q})$ Gravity

Let $(\mathcal{M},g_{\mu\nu},\Gamma^{\alpha}_{\ \mu\nu})$ be a metric-affine
geometry, where $\mathcal{M}$ is a four-dimensional manifold, $g_{\mu\nu}$
denotes the components of the metric tensor of signature $(-,+,+,+)$, and
$\Gamma^{\alpha}_{\ \mu\nu}$ represents an affine connection. The connection
defines a notion of covariant differentiation through its action on vectors
and co-vectors,

$\displaystyle\nabla_{\mu}V^{\alpha}$
$\displaystyle=\partial_{\mu}V^{\alpha}+\Gamma^{\alpha}_{\
\mu\lambda}V^{\lambda}$ $\displaystyle\nabla_{\mu}V_{\alpha}$
$\displaystyle=\partial_{\mu}V_{\alpha}-\Gamma^{\lambda}_{\
\mu\alpha}V_{\lambda},$ (2.1)

and it can be used to describe three independent geometric properties of a
spacetime: curvature, torsion, and non-metricity. The first two objects,
curvature and torsion, are defined by

$\displaystyle R^{\alpha}_{\ \beta\mu\nu}$
$\displaystyle:=2\partial_{[\mu}\Gamma^{\alpha}_{\
\nu]\beta}+2\Gamma^{\alpha}_{[\mu|\lambda|}\Gamma^{\lambda}_{\ \nu]\beta}$
$\displaystyle T^{\alpha}_{\ \mu\nu}$ $\displaystyle:=2\Gamma^{\alpha}_{\
[\mu\nu]},$ (2.2)

and symmetric teleparallelism demands that both tensors vanish. We refer to

$R^{\alpha}_{\ \beta\mu\nu}\overset{!}{=}0\quad\text{and}\quad T^{\alpha}_{\
\mu\nu}\overset{!}{=}0$ (2.3)

as the postulates of symmetric teleparallelism. With curvature and torsion set
to zero, the non-metricity tensor is the only remaining non-trivial object. As
it measures the failure of the connection to be metric-compatible, it is
defined by

$Q_{\alpha\mu\nu}:=\nabla_{\alpha}g_{\mu\nu}=\partial_{\alpha}g_{\mu\nu}-2\Gamma^{\lambda}{}_{\alpha(\mu}g_{\nu)\lambda}.$
(2.4)

Notice that the Riemann and torsion tensor depend on the connection only,
while the non-metricity tensor also depends on the metric. Due to the symmetry
of the non-metricity tensor in its last two indices, at quadratic order there
are only five independent scalars that can be built from the non-metricity
tensor. Hence, a natural starting point for defining a Lagrangian which
describes gravity in terms of non-metricity is a linear combination of these
five terms. As it turns out BeltranJimenez:2017 ; Jimenez:2018 ;
BeltranJimenez:2019 , GR is described by a linear combination of only four of
these contractions, which define the so called non-metricity scalar,

$\mathbb{Q}:=-\frac{1}{4}\,Q_{\alpha\beta\gamma}Q^{\alpha\beta\gamma}+\frac{1}{2}\,Q_{\alpha\beta\gamma}Q^{\beta\alpha\gamma}+\frac{1}{4}\,Q_{\alpha}Q^{\alpha}-\frac{1}{2}c_{5}\,Q_{\alpha}\bar{Q}^{\alpha},$
(2.5)

where $Q_{\alpha}:=Q_{\alpha\nu}{}^{\nu}$ and
$\bar{Q}_{\alpha}:=Q^{\nu}{}_{\nu\alpha}$ denote the two independent traces of
the non-metricity tensor. Since any connection can be decomposed into its
torsion-, non-metricity-, and Levi-Civita-parts, it is easy to show that the
non-metricity scalar can be written as

$\mathbb{Q}=\mathcal{D}_{\mu}(Q^{\mu}-\bar{Q}^{\mu})+\mathcal{R},$ (2.6)

where $\mathcal{D}_{\mu}$ denotes the covariant derivative with respect to the
Levi-Civita connection and $\mathcal{R}$ is the Ricci scalar of the Levi-
Civita connection. This identity shows that symmetric teleparallelism, defined
by the action
$\mathcal{S}[g,\Gamma]:=\int\mathrm{d}^{4}x\,\sqrt{-g}\,\mathbb{Q}$, is
equivalent to the Einstein-Hilbert formulation,
$\mathcal{S}[g]:=\int\mathrm{d}^{4}x\,\sqrt{-g}\,\mathcal{R}$, up to a
boundary term.

It makes it also evident that the connection of ST is unphysical, since it is
completely contained in the boundary term and only the Levi-Civita part of
$\Gamma^{\alpha}{}_{\mu\nu}$ contributes to the metric field equations. Hence,
the physical degrees of freedom are all contained in the metric and the
connection can be freely chosen, as long as it satisfies the postulates of
symmetric teleparallelism.

In the present work, however, we are not interested in the theory defined by
$\mathbb{Q}$ alone. Rather, we want to consider non-linear extensions defined
by the action functional BeltranJimenez:2017

$\mathcal{S}[g,\Gamma;\lambda,\rho]:=\int_{\mathcal{M}}\mathrm{d}^{4}x\left(\frac{1}{2}\sqrt{-g}\,f(\mathbb{Q})+\lambda_{\alpha}{}^{\beta\mu\nu}R^{\alpha}{}_{\beta\mu\nu}+\rho_{\alpha}{}^{\mu\nu}T^{\alpha}{}_{\mu\nu}\right)+\mathcal{S}_{\textsf{matter}},$
(2.7)

where the tensor densities $\lambda_{\alpha}{}^{\beta\mu\nu}$ and
$\rho_{\alpha}{}^{\mu\nu}$ act as Lagrange multipliers which enforce the
postulates of symmetric teleparallelism and where $f$ is an arbitrary function
solely subjected to the condition
$f^{\prime}(\mathbb{Q}):=\frac{\mathrm{d}f(\mathbb{Q})}{\mathrm{d}\mathbb{Q}}\neq
0$. This last requirement is necessary in order to obtain non-trivial field
equations.

It is important to notice that for generic $f$, there is no identity analogous
to (2.6). Hence, in $f(\mathbb{Q})$ gravity, the connection can in general not
be absorbed into a boundary term and it has to be expected that
$\Gamma^{\alpha}{}_{\mu\nu}$ carries degrees of freedom, in addition to the
ones contained in the metric. There is evidence for this in the existing
literature Jimenez:2019  and we will show this explicitly in this paper in
section IV and in section V. Moreover, the connection can no longer be
arbitrarily chose, as was done in Lin:2021 ; Zhao:2021 , since it has its own,
non-trivial, field equations which need to be satisfied. Concretely, the field
equations of $f(\mathbb{Q})$ gravity are given by Jimenez:2018

$\displaystyle\mathcal{M}_{\mu\nu}:=\frac{2}{\sqrt{-g}}\nabla_{\alpha}\left[\sqrt{-g}P^{\alpha}{}_{\mu\nu}f^{\prime}(\mathbb{Q})\right]+f^{\prime}(\mathbb{Q})q_{\mu\nu}-\frac{1}{2}f(\mathbb{Q})g_{\mu\nu}-T_{\mu\nu}$
$\displaystyle=0$
$\displaystyle\mathcal{C}_{\alpha}:=\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}\,f^{\prime}(\mathbb{Q})P^{\mu\nu}{}_{\alpha}\right)$
$\displaystyle=0,$ (2.8)

where $T_{\mu\nu}$ denotes the stress-energy tensor (not to be confused with
the torsion tensor which carries an addition contravariant index) and where we
have introduced the non-metricity conjugate $P^{\alpha}{}_{\mu\nu}$ and the
symmetric tensor $q_{\mu\nu}$ defined by

$\displaystyle P^{\alpha}{}_{\mu\nu}$
$\displaystyle:=\frac{1}{2}\frac{\partial\mathbb{Q}}{\partial
Q_{\alpha}{}^{\mu\nu}}=-\frac{1}{4}Q^{\alpha}{}_{\mu\nu}+\frac{1}{2}Q_{(\mu}{}^{\alpha}{}_{\nu)}+\frac{1}{4}g_{\mu\nu}Q^{\alpha}-\frac{1}{4}\left(g_{\mu\nu}\bar{Q}^{\alpha}+\delta^{\alpha}{}_{(\mu}Q_{\nu)}\right)$
$\displaystyle q_{\mu\nu}$ $\displaystyle:=\frac{\partial\mathbb{Q}}{\partial
g^{\mu\nu}}=P_{(\mu|\alpha\beta}Q_{\nu)}{}^{\mu\nu}-2P^{\alpha\beta}{}_{(\nu}Q_{\alpha\beta|\mu)}.$
(2.9)

The metric field equations in (II) can also be re-written in the useful and
more suggestive form Zhao:2021 ; Lin:2021

$f^{\prime}(\mathbb{Q})G_{\mu\nu}-\frac{1}{2}g_{\mu\nu}(f(\mathbb{Q})-f^{\prime}(\mathbb{Q})\mathbb{Q})+2f^{\prime\prime}(\mathbb{Q})P^{\alpha}{}_{\mu\nu}\partial_{\alpha}\mathbb{Q}=T_{\mu\nu},$
(2.10)

where $f^{\prime\prime}$ denotes the second derivative of $f$ with respect to
$\mathbb{Q}$ and $G_{\mu\nu}$ is the Einstein tensor (with respect to the
Levi-Civita connection and with vanishing cosmological constant). In this
form, it becomes obvious that for $f(\mathbb{Q})=\mathbb{Q}+2\Lambda$, the
metric field equations are equivalent to the Einstein field equations with
cosmological constant $\Lambda$.

In particular, in this case the theory only propagates two degrees of freedom
of the metric, while the connection is pure gauge. As mentioned above, for a
generic function $f$ it can be expected, and there is also evidence from a
perturbative analysis Jimenez:2019 , that the theory propagates more than two
degrees of freedom and should therefore lead to potentially interesting
deviations from GR. Indeed, we will see later (cf sections IV and V) that
under certain conditions, the connection is no longer pure gauge, but rather,
one component becomes dynamical and this leads to solutions which go beyond
the standard GR solutions.

We conclude this section by recalling that there exists a special gauge choice
in which symmetric teleparallelism, where the connection can be arbitrarily
chosen, can be cast in a particularly simple form: The so called coincident
gauge BeltranJimenez:2017 . In this gauge, the connection is trivial, i.e.
$\Gamma^{\alpha}{}_{\mu\nu}=0$. It is obtained by observing that the first
postulate of symmetric teleparallelism, the vanishing of curvature, implies
that the connection must have the form

$\Gamma^{\alpha}_{\
\mu\nu}=\left(\Lambda^{-1}\right)^{\alpha}{}_{\rho}\partial_{\mu}\Lambda^{\rho}{}_{\nu},$
(2.11)

where $\Lambda^{\alpha}{}_{\beta}\in GL(4,\mathbb{R})$. The requirement of
vanishing torsion further restricts the matrix $\Lambda^{\alpha}{}_{\beta}$ to
have the form $\Lambda^{\alpha}{}_{\beta}=\partial_{\beta}\xi^{\alpha}$, for
arbitrary $\xi^{\alpha}$, and the connection consequently becomes

$\Gamma^{\alpha}_{\ \mu\nu}=\frac{\partial
x^{\alpha}}{\partial\xi^{\lambda}}\partial_{\mu}\partial_{\nu}\xi^{\lambda}.$
(2.12)

Hence, the connection can be set globally to zero by the affine gauge choice
$\xi^{\alpha}=M^{\alpha}_{\ \beta}x^{\beta}+\xi^{\alpha}_{0}$, where
$M^{\alpha}_{\ \beta}$ is a non-degenerate matrix with constant entries and
$\xi^{\alpha}_{0}$ is a constant vector Jimenez:2018 .

##  III Symmetry reduction of the metric and the connection

Our goal is the study of the field equations of $f(\mathbb{Q})$ gravity for
spherically symmetric and stationary spacetimes. To that end, we first assume
that the ten metric components and the $64$ components of the affine
connection can be expressed in the chart
$(t,r,\theta,\phi)\in\mathbb{R}\times\mathbb{R}_{>0}\times[0,\pi]\times[0,2\pi)$.
The next step is to find the most general form of the metric and the
connection which respect the symmetries of the spacetime and, in the case of
the connection, which is compatible with the postulates of symmetric
teleparallelism.

The notion of symmetry we use is the same as the one given in Hohmann:2019
and we recall it here for convenience: Let $G$ be a group,
$\Phi:G\times\mathcal{M}\to\mathcal{M}$ the action of the group on the
spacetime manifold $\mathcal{M}$, and denote by
$\Phi_{u}:\mathcal{M}\to\mathcal{M}$ for $u\in G$ the induced diffeomorphism.
We then say that a metric-affine geometry
$(\mathcal{M},g_{\mu\nu},\Gamma^{\alpha}{}_{\mu\nu})$ is symmetric under the
group action if and only if

$\displaystyle(\Phi_{u}^{*}g)_{\mu\nu}$ $\displaystyle=g_{\mu\nu}$
$\displaystyle(\Phi_{u}^{*}\Gamma)^{\alpha}{}_{\mu\nu}$
$\displaystyle=\Gamma^{\alpha}{}_{\mu\nu},$ (3.1)

for all $u\in G$ and where $\Phi_{u}^{*}$ denotes the pull-back of $\Phi_{u}$.
In our case, the group $G$ will be the group of spatial rotations, $SO(3)$,
and the group of time translations. Moreover, in practice, it is more
convenient to consider infinitesimal actions of $G$ on the metric-affine
geometry. The above symmetry conditions can then easily be re-expressed as

$\displaystyle(\mathcal{L}_{\xi}g)_{\mu\nu}$ $\displaystyle=0$
$\displaystyle(\mathcal{L}_{\xi}\Gamma)^{\alpha}{}_{\mu\nu}$
$\displaystyle=0,$ (3.2)

where $\mathcal{L}_{\xi}$ stands for the Lie derivative along $\xi$, which
representatively stands for the generating vector fields of the Lie algebra
$\mathfrak{g}$ of $G$. Our task is therefore to implement (III) for the
generator of time-translations and the generators of $SO(3)$. Of course, the
most general spherically symmetric and stationary form of the metric is well-
known and we can simply state the result:

$g_{\mu\nu}=\begin{pmatrix}g_{tt}&g_{tr}&0&0\\\ g_{tr}&g_{rr}&0&0\\\
0&0&g_{\theta\theta}&0\\\
0&0&0&g_{\theta\theta}\,\sin^{2}\theta\end{pmatrix},$ (3.3)

where all four independent components
$\\{g_{tt},g_{tr},g_{rr},g_{\theta\theta}\\}$ only depend on $r$. In the case
of the connection, we could refer to the results reported in Hohmann:2019 ,
where the symmetry reduction of a general affine connection under the action
of $G=SO(3)$ has been carried out. However, we will perform the symmetry
reduction of the connection in detail under the additional assumptions that it
is torsionless and stationary. With the former assumption we already achieve
the implementation of one of the two postulates of symmetric teleparallelism.
The assumption of stationarity will play an important role in subsection
III.2, where we implement the second postulate of symmetric teleparallelism.
Note that of the three conditions the connection has to fulfill,
$\mathcal{L}_{\zeta}\Gamma^{\alpha}_{\ \mu\nu}=0$, $T^{\alpha}_{\ \mu\nu}=0$,
$R^{\alpha}_{\ \beta\mu\nu}=0$, the first and second are linear in
$\Gamma^{\alpha}_{\ \mu\nu}$ and thus have unique solutions. But the Riemann
tensor is quadratic in the connection, and one might thus obtain several
solutions for the connection.

###  III.1 Symmetry reduction of $\Gamma^{\alpha}{}_{\mu\nu}$ under the
assumption that $T^{\alpha}{}_{\mu\nu}=0$

The torsionless condition, $T^{\alpha}{}_{\mu\nu}\overset{!}{=}0$, simply
forces the connection to be symmetric in its lower indices,
$\Gamma^{\alpha}{}_{[\mu\nu]}=0$. This reduces the amount of independent
connection components from $64$ to $40$. Next, we implement the condition of
stationarity. Since the generating vector field of time-translations is simply
given by $\mathcal{T}:=\mathcal{T}^{\alpha}\partial_{\alpha}=\partial_{t}$, we
immediately find

$(\mathcal{L}_{\mathcal{T}}\Gamma)^{\alpha}{}_{\mu\nu}=\partial_{t}\Gamma^{\alpha}{}_{\mu\nu}\overset{!}{=}0.$
(3.4)

In words: All $40$ components of the connection are, unsurprisingly, time-
independent. Implementing spherical symmetry requires a little more work. To
begin with, we recall that the generating vector fields of $SO(3)$ are

$\displaystyle\mathcal{R}_{x}$
$\displaystyle:=\mathcal{R}^{\alpha}_{x}\partial_{\alpha}=\sin\phi\,\partial_{\theta}+\frac{\cos\phi}{\tan\theta}\,\partial_{\phi}$
$\displaystyle\mathcal{R}_{y}$
$\displaystyle:=\mathcal{R}^{\alpha}_{y}\partial_{\alpha}=-\cos\phi\,\partial_{\theta}+\frac{\sin\phi}{\tan\theta}\,\partial_{\phi}$
$\displaystyle\mathcal{R}_{z}$
$\displaystyle:=\mathcal{R}^{\alpha}_{z}\partial_{\alpha}=-\partial_{\phi}.$
(3.5)

It is easiest to start with the generator $\mathcal{R}_{z}$ since this one
simply gives us

$(\mathcal{L}_{\mathcal{R}_{z}}\Gamma)^{\alpha}{}_{\mu\nu}=-\partial_{\phi}\Gamma^{\alpha}{}_{\mu\nu}\overset{!}{=}0,$
(3.6)

which means that all connection components are independent of the angular
coordinate $\phi$. To implement the remaining two symmetry conditions, it is
convenient to study linear combinations of Lie derivatives. The first one
reads

$\cos\phi\,(\mathcal{L}_{\mathcal{R}_{x}}\Gamma)^{\alpha}{}_{\mu\nu}+\sin\phi\,(\mathcal{L}_{\mathcal{R}_{y}}\Gamma)^{\alpha}{}_{\mu\nu}\overset{!}{=}0$
(3.7)

and it leads to a set of equations which explicitly determine $28$ of the $40$
connection components. The solutions can be subdivided into three groups. The
first group consists of $20$ components which are forced to be zero,

$\displaystyle\Gamma^{t}{}_{t\theta}$ $\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{t\phi}$ $\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{r\theta}$ $\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{r\phi}$ $\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{\theta\phi}$ $\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{t\theta}$ $\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{t\phi}$ $\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{r\theta}$ $\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{r\phi}$ $\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{\theta\phi}$ $\displaystyle=0$
$\displaystyle\Gamma^{\theta}{}_{tt}$ $\displaystyle=0$
$\displaystyle\Gamma^{\theta}{}_{tr}$ $\displaystyle=0$
$\displaystyle\Gamma^{\theta}{}_{rr}$ $\displaystyle=0$
$\displaystyle\Gamma^{\theta}{}_{\theta\theta}$ $\displaystyle=0$
$\displaystyle\Gamma^{\theta}{}_{\theta\phi}$ $\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{tt}$ $\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{tr}$ $\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{rr}$ $\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{\theta\theta}$ $\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{\phi\phi}$ $\displaystyle=0.$ (3.8)

The second group contains two components which are explicitly given by
trigonometric functions,

$\displaystyle\Gamma^{\theta}{}_{\phi\phi}$ $\displaystyle=-\cos\theta\
\sin\theta$ $\displaystyle\Gamma^{\phi}{}_{\theta\phi}$
$\displaystyle=\cot\theta.$ (3.9)

This is an important result, since it tells us that the coincident gauge,
i.e., the trivial connection $\Gamma^{\alpha}{}_{\mu\nu}=0$, fails to be
spherically symmetric. Any attempt to find spherically symmetric solutions to
the field equations of $f(\mathbb{Q})$ gravity using the coincident gauge is
therefore bound to fail and one should expect inconsistencies, unless we are
in symmetric teleparallelism which is described by the affine function
$f(\mathbb{Q})=a\,\mathbb{Q}+b$.
Finally, we find that in the third group, six components can be expressed
algebraically in terms of other components:

$\displaystyle\Gamma^{t}{}_{\phi\phi}$
$\displaystyle=\Gamma^{t}{}_{\theta\theta}\sin^{2}\theta$
$\displaystyle\Gamma^{r}{}_{\phi\phi}$
$\displaystyle=\Gamma^{r}{}_{\theta\theta}\sin^{2}\theta$
$\displaystyle\Gamma^{\phi}{}_{t\theta}$
$\displaystyle=-\Gamma^{\theta}{}_{t\phi}\csc^{2}\theta$
$\displaystyle\Gamma^{\theta}{}_{t\theta}$
$\displaystyle=\Gamma^{\phi}{}_{t\phi}$
$\displaystyle\Gamma^{\phi}{}_{r\theta}$
$\displaystyle=-\Gamma^{\theta}{}_{r\phi}\csc^{2}\theta$
$\displaystyle\Gamma^{\theta}{}_{r\theta}$
$\displaystyle=\Gamma^{\phi}{}_{r\phi}.$ (3.10)

From the initially 40 independent connection components, we are left with
$40-20-2-6=12$ components. These twelve independent components are

$\left\\{\Gamma^{t}{}_{tt},\Gamma^{t}{}_{tr},\Gamma^{t}{}_{rr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{tt},\Gamma^{r}{}_{tr},\Gamma^{r}{}_{rr},\Gamma^{r}{}_{\theta\theta},\Gamma^{\phi}{}_{t\phi},\Gamma^{\theta}{}_{t\phi},\Gamma^{\phi}{}_{r\phi},\Gamma^{\theta}{}_{r\phi}\right\\}$
(3.11)

and these are functions of $r$ and $\theta$, potentially. We can further
restrict the functional dependence of these components by considering the last
symmetry condition which reads

$\sin\phi\,(\mathcal{L}_{\mathcal{R}_{x}}\Gamma)^{\alpha}{}_{\mu\nu}-\cos\phi\,(\mathcal{L}_{\mathcal{R}_{y}}\Gamma)^{\alpha}{}_{\mu\nu}\overset{!}{=}0.$
(3.12)

This condition leads to a total of twelve first order differential equations
for precisely the twelve independent components given in (3.11). These
equations are explicitly given by

$\displaystyle\partial_{\theta}\Gamma^{\theta}{}_{t\phi}-\Gamma^{\theta}{}_{t\phi}\cot\theta$
$\displaystyle=0$
$\displaystyle\partial_{\theta}\Gamma^{\theta}{}_{r\phi}-\Gamma^{\theta}{}_{r\phi}\cot\theta$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{t}{}_{tt}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{t}{}_{tr}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{t}{}_{rr}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{r}{}_{tt}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{r}{}_{tr}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{r}{}_{rr}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{\phi}{}_{t\phi}$
$\displaystyle=0$ $\displaystyle\partial_{\theta}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=0.$ (3.13)

The first two differential equations in the first line are easily solved and
give us

$\Gamma^{\theta}{}_{t\phi}=\sin\theta\,c_{1}(r)\quad\text{and}\quad\Gamma^{\theta}{}_{r\phi}=\sin\theta\,c_{2}(r),$
(3.14)

where $c_{1}$ and $c_{2}$ are arbitrary functions of $r$, while the remaining
ten equations tell us that the other components are only functions of $r$.
This completes the symmetry reduction of the connection and we are left with
the twelve independent functions

$\left\\{c_{1}(r),c_{2}(r),\Gamma^{t}{}_{tt}(r),\Gamma^{t}{}_{tr}(r),\Gamma^{t}{}_{rr}(r),\Gamma^{t}{}_{\theta\theta}(r),\Gamma^{r}{}_{tt}(r),\Gamma^{r}{}_{tr}(r),\Gamma^{r}{}_{rr}(r),\Gamma^{r}{}_{\theta\theta}(r),\Gamma^{\theta}{}_{t\theta}(r),\Gamma^{\theta}{}_{r\theta}(r)\right\\}.$
(3.15)

We will use the results obtained in this subsection in order to implement
$R^{\alpha}{}_{\mu\nu\rho}\overset{!}{=}0$. This will be the subject of the
next subsection and it will further reduce the amount of independent
connection components.

###  III.2 Implementation of $R^{\alpha}{}_{\mu\nu\rho}=0$

In the previous subsection, we already used the symmetric teleparallelism
postulate that $T^{\alpha}{}_{\mu\nu}=0$ and we imposed the symmetry
conditions (3.4), (3.6), (3.7), and (3.12). From these conditions we learned
that there are only twelve independent connection components, all of which are
solely functions of $r$, and that the remaining $28$ components are given by
equations (III.1), (III.1), and (III.1). We can now use these facts to
simplify the equations which arise from imposing that the Riemann tensor of
$\Gamma^{\alpha}{}_{\mu\nu}$ has to vanish. After rather long and
unenlightening computations, one finds $24$ non-trivial equations. Since the
Riemann tensor is quadratic in $\Gamma$ and linear in $\partial\Gamma$, and
because half the connection components are zero, one can reasonably expect
that these $24$ equations can be separated into non-linear algebraic equations
and first-order differential equations. Indeed, one finds that there are
twelve algebraic equations,

$\displaystyle\Gamma^{t}{}_{\theta\theta}\Gamma^{\theta}{}_{t\phi}$
$\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{\theta\theta}\Gamma^{\theta}{}_{t\phi}$
$\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{\theta\theta}\Gamma^{\theta}{}_{r\phi}$
$\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{\theta\theta}\Gamma^{\theta}{}_{r\phi}$
$\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{\theta\theta}\Gamma^{\theta}{}_{t\phi}+\Gamma^{r}{}_{\theta\theta}\Gamma^{\theta}{}_{r\phi}$
$\displaystyle=0$ $\displaystyle
1+\Gamma^{t}{}_{\theta\theta}\Gamma^{\phi}{}_{t\phi}+\Gamma^{r}{}_{\theta\theta}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{tr}\Gamma^{r}{}_{\theta\theta}+\Gamma^{t}{}_{\theta\theta}\left(\Gamma^{t}{}_{tt}-\Gamma^{\phi}{}_{t\phi}\right)$
$\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{\theta\theta}\Gamma^{r}{}_{tt}+\Gamma^{r}{}_{\theta\theta}\left(\Gamma^{r}{}_{tr}-\Gamma^{\phi}{}_{t\phi}\right)$
$\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{tt}\Gamma^{\theta}{}_{r\phi}+\Gamma^{\theta}{}_{t\phi}\left(\Gamma^{t}{}_{tt}-2\Gamma^{\phi}{}_{t\phi}\right)$
$\displaystyle=0$
$\displaystyle\Gamma^{\theta}{}_{r\phi}\left(\Gamma^{r}{}_{tr}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{\theta}{}_{t\phi}\left(\Gamma^{t}{}_{tr}-\Gamma^{\phi}{}_{r\phi}\right)$
$\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{t\phi}\left(\Gamma^{t}{}_{tt}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{r}{}_{tt}\Gamma^{\phi}{}_{r\phi}+(\Gamma^{\theta}{}_{t\phi})^{2}\csc^{2}\theta$
$\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{tr}\Gamma^{\phi}{}_{t\phi}+\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{r}{}_{tr}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{\theta}{}_{t\phi}\Gamma^{\theta}{}_{r\phi}\csc^{2}\theta$
$\displaystyle=0,$ (3.16)

for only ten of the twelve independent connection components. The remaining
twelve equations are first-order differential equations,

$\displaystyle\partial_{r}\Gamma^{\theta}{}_{t\phi}$ $\displaystyle=0$
$\displaystyle\partial_{r}\Gamma^{\phi}{}_{t\phi}$ $\displaystyle=0$
$\displaystyle\partial_{r}\Gamma^{r}{}_{tr}$
$\displaystyle=\Gamma^{t}{}_{rr}\Gamma^{r}{}_{tt}-\Gamma^{t}{}_{tr}\Gamma^{r}{}_{tr}$
$\displaystyle\partial_{r}\Gamma^{t}{}_{tt}$
$\displaystyle=\Gamma^{t}{}_{tr}\Gamma^{r}{}_{tr}-\Gamma^{t}{}_{rr}\Gamma^{r}{}_{tt}$
$\displaystyle\partial_{r}\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=\Gamma^{t}{}_{\theta\theta}\left(\Gamma^{\phi}{}_{r\phi}-\Gamma^{t}{}_{tr}\right)-\Gamma^{t}{}_{rr}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle\partial_{r}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=\Gamma^{r}{}_{\theta\theta}\left(\Gamma^{\phi}{}_{r\phi}-\Gamma^{r}{}_{rr}\right)-\Gamma^{t}{}_{\theta\theta}\Gamma^{r}{}_{tr}$
$\displaystyle\partial_{r}\Gamma^{\theta}{}_{r\phi}$
$\displaystyle=\Gamma^{\theta}{}_{r\phi}\left(\Gamma^{r}{}_{rr}-2\Gamma^{\phi}{}_{r\phi}\right)+\Gamma^{t}{}_{rr}\Gamma^{\theta}{}_{t\phi}$
$\displaystyle\partial_{r}\Gamma^{r}{}_{tt}$
$\displaystyle=\Gamma^{r}{}_{tt}\left(\Gamma^{t}{}_{tr}-\Gamma^{r}{}_{rr}\right)+\Gamma^{r}{}_{tr}\left(1-\Gamma^{t}{}_{tt}\right)$
$\displaystyle\partial_{r}\Gamma^{t}{}_{tr}$
$\displaystyle=\Gamma^{t}{}_{rr}\left(\Gamma^{t}{}_{tt}-\Gamma^{r}{}_{tr}\right)+\Gamma^{t}{}_{tr}\left(\Gamma^{r}{}_{rr}-\Gamma^{t}{}_{tr}\right)$
$\displaystyle\partial_{r}\Gamma^{\theta}{}_{t\phi}$
$\displaystyle=\Gamma^{\theta}{}_{r\phi}\left(\Gamma^{r}{}_{tr}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{\theta}{}_{t\phi}\left(\Gamma^{t}{}_{tr}-\Gamma^{\phi}{}_{r\phi}\right)$
$\displaystyle\partial_{r}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{r}{}_{rr}-\Gamma^{\phi}{}_{r\phi}\right)+\Gamma^{t}{}_{rr}\Gamma^{\phi}{}_{t\phi}+(\Gamma^{\theta}{}_{r\phi})^{2}\csc^{2}\theta$
$\displaystyle\partial_{r}\Gamma^{\phi}{}_{t\phi}$
$\displaystyle=\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{r}{}_{tr}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{t}{}_{tr}\Gamma^{\phi}{}_{t\phi}+\Gamma^{\theta}{}_{t\phi}\Gamma^{\theta}{}_{r\phi}\csc^{2}\theta.$
(3.17)

Let us first have a closer look at the algebraic equations (III.2): Because
there are more equations than independent functions, it is not clear whether
the equations can even be solved and because they are non-linear, one cannot
expect to find unique solutions. As it turns out, the system of equations can
be solved and one finds five distinct sets of solutions. In each solution set,
one can express some connection components in terms of other components in a
highly non-linear fashion. However, what is remarkable, is that all five
solution sets share one particularly simple solution:

$\Gamma^{\theta}{}_{t\phi}=0\quad\text{and}\quad\Gamma^{\theta}{}_{r\phi}=0.$
(3.18)

A quick look at equation (3.14) reveals that this is equivalent to

$c_{1}(r)=0\quad\text{and}\quad c_{2}(r)=0,$ (3.19)

which reduces the amount of independent connection components (3.15) from
twelve to ten. Moreover, if we plug (3.18) back into the algebraic equations,
we find the simpler system

$\displaystyle
1+\Gamma^{t}{}_{\theta\theta}\Gamma^{\phi}{}_{t\phi}+\Gamma^{r}{}_{\theta\theta}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=0$
$\displaystyle\Gamma^{t}{}_{\theta\theta}\left(\Gamma^{t}{}_{tt}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{t}{}_{tr}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{\theta\theta}\left(\Gamma^{r}{}_{tr}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{t}{}_{\theta\theta}\Gamma^{r}{}_{tt}$
$\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{t\phi}\left(\Gamma^{t}{}_{tt}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{r}{}_{tt}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=0$
$\displaystyle\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{r}{}_{tr}-\Gamma^{\phi}{}_{t\phi}\right)+\Gamma^{t}{}_{tr}\Gamma^{\phi}{}_{t\phi}$
$\displaystyle=0.$ (3.20)

Notice that these are five equations for eight connection components,
$\left\\{\Gamma^{t}{}_{tt},\Gamma^{t}{}_{tr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{tt},\Gamma^{r}{}_{tr},\Gamma^{r}{}_{\theta\theta},\Gamma^{\phi}{}_{t\phi},\Gamma^{\phi}{}_{r\phi}\right\\}$.
Hence, it is now obvious that the system is solvable but underdetermined and
clearly we obtain the same five distinct sets of solutions as before. After
all, we used the algebraic equations to obtain the solution (3.18).

Now we turn to the differential equations. Since no matter which of the five
solution sets we use, we always find (3.18), we can use this solution to
simplify the twelve differential equations. Notice that (3.18) implies that
the first, the seventh and the tenth differential equation are trivially
satisfied, while the last two loose their $\csc^{2}\theta$ terms. Furthermore,
the second equation can easily be solved and we find

$\partial_{r}\Gamma^{\phi}{}_{t\phi}=0\quad\Longleftrightarrow\quad\Gamma^{\phi}{}_{t\phi}=c,$
(3.21)

where $c\in\mathbb{R}$ is a constant (since at this stage we already know that
every independent connection component is purely a function of $r$). After
using (3.21), the last differential equation turns into an algebraic equation,

$\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{r}{}_{tr}-c\right)+\Gamma^{t}{}_{tr}\,c=0.$
(3.22)

We can therefore update our system (III.2) of algebraic equations by adding
(3.22) to it. This gives us a total of six algebraic equations for eight
connection components. Moreover, since we solved one differential equation,
three dropped out, and one turned into an algebraic equation, we are now left
with seven differential equations:

$\displaystyle\partial_{r}\Gamma^{r}{}_{tr}$
$\displaystyle=\Gamma^{t}{}_{rr}\Gamma^{r}{}_{tt}-\Gamma^{t}{}_{tr}\Gamma^{r}{}_{tr}$
$\displaystyle\partial_{r}\Gamma^{r}{}_{tt}$
$\displaystyle=\Gamma^{r}{}_{tr}\left(\Gamma^{r}{}_{tr}-\Gamma^{t}{}_{tt}\right)+\Gamma^{r}{}_{tt}\left(\Gamma^{t}{}_{tr}-\Gamma^{r}{}_{rr}\right)$
$\displaystyle\partial_{r}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{r}{}_{rr}-\Gamma^{\phi}{}_{r\phi}\right)+c\,\Gamma^{t}{}_{rr}$
$\displaystyle\partial_{r}\Gamma^{t}{}_{tt}$
$\displaystyle=\Gamma^{t}{}_{tr}\Gamma^{r}{}_{tr}-\Gamma^{t}{}_{rr}\Gamma^{r}{}_{tt}$
$\displaystyle\partial_{r}\Gamma^{t}{}_{tr}$
$\displaystyle=\Gamma^{t}{}_{rr}\left(\Gamma^{t}{}_{tt}-\Gamma^{r}{}_{tr}\right)+\Gamma^{t}{}_{tr}\left(\Gamma^{r}{}_{rr}-\Gamma^{t}{}_{tr}\right)$
$\displaystyle\partial_{r}\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=\Gamma^{t}{}_{\theta\theta}\left(\Gamma^{\phi}{}_{r\phi}-\Gamma^{t}{}_{tr}\right)-\Gamma^{t}{}_{rr}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle\partial_{r}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=\Gamma^{r}{}_{\theta\theta}\left(\Gamma^{\phi}{}_{r\phi}-\Gamma^{r}{}_{rr}\right)-\Gamma^{t}{}_{\theta\theta}\Gamma^{r}{}_{tr}.$
(3.23)

Notice that these equations allow us to re-express the $r$-derivative of seven
of the ten independent connection components. The only components which do not
appear on the left hand side are $\Gamma^{t}{}_{rr}$ and $\Gamma^{r}{}_{rr}$
(and $\Gamma^{\phi}{}_{t\phi}=c$, whose derivative is trivial). Furthermore,
observe that the right hand side of the first equation in (III.2) is equal to
$(-1)$ times the right hand side of the fourth equation. This means we get the
following relation between the left hand sides:

$\partial_{r}\Gamma^{r}{}_{tr}=-\partial_{r}\Gamma^{t}{}_{tt}\quad\Longleftrightarrow\quad\Gamma^{t}{}_{tt}=k-\Gamma^{r}{}_{tr},$
(3.24)

where $k\in\mathbb{R}$ is a constant. This follows again from the fact that
all connection coefficients we are left with are purely functions of $r$. This
is again a useful relation and we are left with six differential equations. As
we will see later, the six differential equations play a crucial role in
determining the propagating degrees of freedom of $f(\mathbb{Q})$ gravity.

This is all the information we can extract from the differential equations at
this point. The next step is to return to the algebraic equations,
supplemented by the new equation (3.22), and study the solution sets which
arise from solving these equations. Before doing so, let us briefly summarize
the situation thus far:

  1. 1.

We started with a general affine connection $\Gamma^{\alpha}{}_{\mu\nu}$,
which has $64$ independent components.

  2. 2.

Implementing $T^{\alpha}{}_{\mu\nu}\overset{!}{=}0$ in subsection III.1
brought this number down to $40$ independent components.

  3. 3.

The first two symmetry conditions, equations (3.4) and (3.6), told us that all
connection components are independent of the coordinates $t$ and $\phi$.

  4. 4.

The third symmetry condition, equation (3.7), told us that the $20$ components
(III.1) are zero. Moreover, we found that two components are given solely by
trigonometric functions, equation (III.1), and we found six algebraic
relations in (III.1). This brought the number of independent components down
to $40-20-2-6=12$ and we learned that the connection in coincident gauge fails
to be spherically symmetric.

  5. 5.

The fourth symmetry condition, equation (3.12), gave use twelve first order
differential equations. These equations tell us that all twelve independent
connection components are functions of $r$ alone, and no other coordinate.
(Minor exception for the components in (3.14), but they turn out to be zero
later on).

  6. 6.

We then proceeded to implement $R^{\mu}{}_{\nu\rho\sigma}\overset{!}{=}0$ in
this subsection and we found that we get twelve non-linear algebraic equations
and twelve first order differential equations. The non-linear equations all
have two solutions in common: $\Gamma^{\theta}{}_{t\phi}=0\text{ and
}\Gamma^{\theta}{}_{r\phi}=0$. These solutions eliminate $c_{1}(r)$ and
$c_{2}(r)$ from the list of independent connection components. Hence, we are
left with the ten independent components
$\left\\{\Gamma^{t}{}_{tt},\Gamma^{t}{}_{tr},\Gamma^{t}{}_{rr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{tt},\Gamma^{r}{}_{tr},\Gamma^{r}{}_{rr},\Gamma^{r}{}_{\theta\theta},\Gamma^{\phi}{}_{t\phi},\Gamma^{\phi}{}_{r\phi}\right\\}$,
which are all functions of $r$ and nothing else.

  7. 7.

Among the differential equations we find
$\partial_{r}\Gamma^{\phi}{}_{t\phi}=0$, which tells us that
$\Gamma^{\phi}{}_{t\phi}$ is a constant. Moreover, we found the relation
$\partial_{r}\Gamma^{r}{}_{tr}=-\partial_{r}\Gamma^{t}{}_{tt}$, which implies
$\Gamma^{t}{}_{tt}=k-\Gamma^{r}{}_{tr}$. This reduces our list of independent
connection components to
$\left\\{c,k,\Gamma^{t}{}_{tr},\Gamma^{t}{}_{rr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{tt},\Gamma^{r}{}_{tr},\Gamma^{r}{}_{rr},\Gamma^{r}{}_{\theta\theta},\Gamma^{\phi}{}_{r\phi}\right\\}$,
where $c$ and $k$ are real constants.

  8. 8.

Finally, there are six differential equations left. These differential
equations allow us to express the $r$-derivative of the connection components
$\left\\{\Gamma^{t}{}_{tr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{tt},\Gamma^{r}{}_{tr},\Gamma^{r}{}_{\theta\theta},\Gamma^{\phi}{}_{r\phi}\right\\}$
in terms of the other connection components. What remains un-determined are
the derivatives $\partial_{r}\Gamma^{t}{}_{rr}$ and
$\partial_{r}\Gamma^{r}{}_{rr}$. This means that in the field equations, only
$\Gamma^{t}{}_{rr}$ and $\Gamma^{r}{}_{rr}$ can become dynamical.

  9. 9.

We also have six non-linear algebraic equations for the eight connection
components
$\left\\{c,k,\Gamma^{t}{}_{tr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{tt},\right.$
$\left.\Gamma^{r}{}_{tr},\Gamma^{r}{}_{\theta\theta},\Gamma^{\phi}{}_{r\phi}\right\\}$.
Notice that these are the same variables as in bullet point 8. (apart from $c$
and $k$). Clearly, the system is underdetermined and we should expect to get
more than one solution to these equations.

Indeed, we find that the remaining algebraic equations now admit two solution
sets. Since these solution sets look independent, we will study them
separately in subsections III.3 and III.4. We will then show in subsection
III.5 that the two sets are actually related to each other by a double scaling
limit. It is nevertheless convenient to distinguish between the two sets and
they both play a crucial role in the study of the symmetry reduced field
equations of $f(\mathbb{Q})$ gravity, which will be discussed in section IV.

###  III.3 Solution set 1

As mentioned in the previous subsection, we are left with six non-linear
algebraic equations, given by (III.2) and (3.22). These equations do not admit
a unique solution. Rather, there are two sets of solutions. For the first
solution set we find that two components are zero and three components can be
expressed in terms of the constants $c,k$, and the function
$\Gamma^{\phi}{}_{r\phi}$:

$\displaystyle\Gamma^{t}{}_{\theta\theta}$ $\displaystyle=-\frac{1}{c}$
$\displaystyle\Gamma^{r}{}_{tr}$ $\displaystyle=k-c$
$\displaystyle\Gamma^{t}{}_{tr}$
$\displaystyle=\frac{2c-k}{c}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle\Gamma^{r}{}_{\theta\theta}$ $\displaystyle=0$
$\displaystyle\Gamma^{r}{}_{tt}$ $\displaystyle=0.$ (3.25)

Clearly, we have to assume that $c\neq 0$ for this solution set to be well-
defined and admissible. Applying this solution set to the differential
equations (III.2) reduces them to four algebraic equations,

$\displaystyle(c-k)\Gamma^{\phi}{}_{r\phi}$ $\displaystyle=0$
$\displaystyle(c-k)(2c-k)$ $\displaystyle=0$
$\displaystyle(c-k)(2c-k)\Gamma^{\phi}{}_{r\phi}$ $\displaystyle=0$
$\displaystyle c-k$ $\displaystyle=0,$ (3.26)

and two differential equations,

$\displaystyle
c\,\Gamma^{t}_{rr}+\Gamma^{\phi}{}_{r\phi}(\Gamma^{r}{}_{rr}-\Gamma^{\phi}{}_{r\phi})-\partial_{r}\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=0$
$\displaystyle(2c-k)\left(c^{2}\,\Gamma^{t}{}_{rr}+c\,\Gamma^{r}{}_{rr}\Gamma^{\phi}{}_{r\phi}+(k-2c)\,(\Gamma^{\phi}{}_{r\phi})^{2}-c\,\partial_{r}\Gamma^{\phi}{}_{r\phi}\right)$
$\displaystyle=0.$ (3.27)

The algebraic equations have obviously a unique solution which is given by

$c=k\neq 0.$ (3.28)

Applying this solution to the two differential equations shows that they are
actually the same and we simply get

$\partial_{r}\Gamma^{\phi}{}_{r\phi}=c\,\Gamma^{t}_{rr}+\Gamma^{\phi}{}_{r\phi}(\Gamma^{r}{}_{rr}-\Gamma^{\phi}{}_{r\phi}).$
(3.29)

This is all the information we can extract from these equations. In
particular, we do not find any further conditions, constraints, or
integrability conditions. What we learn thus is the following: Solution set 1
describes a stationary, spherically symmetric, torsionless, and flat
connection in terms of a real constant $c\neq 0$ and the three arbitrary
functions
$\Gamma^{t}{}_{rr}(r),\Gamma^{r}{}_{rr}(r),\Gamma^{\phi}{}_{r\phi}(r)$. We
refer to these functions as the independent components which define solution
set 1, because every component which belongs to solution set 1 can be
expressed in terms of these functions, the constant $c$, and trigonometric
functions. Moreover, the derivative $\partial_{r}\Gamma^{\phi}{}_{r\phi}$ can
be expressed in terms of the independent functions. The same is not true for
the derivatives $\partial_{r}\Gamma^{t}{}_{rr}$ and
$\partial_{r}\Gamma^{r}{}_{rr}$ of the independent components
$\Gamma^{t}{}_{rr}$, $\Gamma^{r}{}_{rr}$: These derivatives remain
undetermined and potentially render these components dynamical.

These are the defining properties of solution set 1 and they play a crucial
role in simplifying the symmetry reduced field equations in section IV. We
therefore summarize all properties of solution set 1 in the following table
for later convenience and reference:

Independent | All connection components of solution set 1 can be expressed in terms of the three
---|---
components | independent functions $\Gamma^{t}{}_{rr}(r),\Gamma^{r}{}_{rr}(r),\Gamma^{\phi}{}_{r\phi}(r)$, the real constant $c\neq 0$, and trigonometric
| functions.
Non-zero | There are twelve non-zero components in solution set 1 (all other components vanish):
components |  $\Gamma^{t}{}_{rr}$ |  $\Gamma^{r}{}_{rr}$ |  $\Gamma^{\phi}{}_{r\phi}$
|  $\Gamma^{t}{}_{tt}=c$ |  $\Gamma^{t}{}_{tr}=\Gamma^{\phi}{}_{r\phi}$ |  $\Gamma^{t}{}_{\theta\theta}=-\frac{1}{c}$
|  $\Gamma^{t}{}_{\phi\phi}=-\frac{\sin^{2}\theta}{c}$ |  $\Gamma^{\theta}{}_{t\theta}=c$ |  $\Gamma^{\theta}{}_{r\theta}=\Gamma^{\phi}{}_{r\phi}$
|  $\Gamma^{\theta}{}_{\phi\phi}=-\cos\theta\,\sin\theta$ |  $\Gamma^{\phi}{}_{t\phi}=c$ |  $\Gamma^{\phi}{}_{\theta\phi}=\cot\theta$
Derivatives of | Of the three independent functions, the $r$-derivative of $\Gamma^{\phi}{}_{r\phi}$ can be expressed as
independent |  $\partial_{r}\Gamma^{\phi}{}_{r\phi}=c\,\Gamma^{t}_{rr}+\Gamma^{\phi}{}_{r\phi}(\Gamma^{r}{}_{rr}-\Gamma^{\phi}{}_{r\phi})$,
components | while $\partial_{r}\Gamma^{t}{}_{rr}$ and $\partial_{r}\Gamma^{r}{}_{rr}$ cannot be expressed in terms of other components.
Table 1: A concise summary of all the properties which define solution set 1.

Note that alternatively one could also define $\Gamma^{\phi}{}_{r\phi}$ as an
arbitrary function, and in turn fix $\Gamma^{t}{}_{rr}$ by solving (3.29) for
it. This can always be done as $c\neq 0$, and one obtains

$\Gamma^{t}{}_{rr}=\frac{1}{c}\left(\partial_{r}\Gamma^{\phi}{}_{r\phi}-\Gamma^{\phi}{}_{r\phi}(\Gamma^{r}{}_{rr}-\Gamma^{\phi}{}_{r\phi})\right)\
.$ (3.30)

One could thus also choose $\Gamma^{\phi}{}_{r\phi}$ as a possible connection
degree of freedom, which fixes $\Gamma^{t}{}_{rr}$, but this is just an issue
of the freely chosen parametrization of the connection.

###  III.4 Solution set 2

The algebraic equations (III.2) and (3.22) admit a second set of solutions,
which is explicitly given by

$\displaystyle\Gamma^{r}{}_{tr}$
$\displaystyle=c+c\,(2c-k)\,\Gamma^{t}{}_{\theta\theta}$
$\displaystyle\Gamma^{t}{}_{tr}$
$\displaystyle=(2c-k)\frac{\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}\left(1+c\,\Gamma^{t}{}_{\theta\theta}\right)$
$\displaystyle\Gamma^{r}{}_{tt}$
$\displaystyle=-c\,(2c-k)\,\Gamma^{r}{}_{\theta\theta}$
$\displaystyle\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=-\frac{1+c\,\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}},$
(3.31)

where we have to assume $\Gamma^{r}{}_{\theta\theta}\neq 0$ in order to obtain
well-defined expressions. Just as in the previous subsection, we insert the
obtained solutions into the differential equations (III.2). This leads to six
complicated looking equations. None of these equations is algebraic and,
moreover, they are linearly dependent. After some elementary manipulations,
one finally finds the following two independent expressions:

$\displaystyle\partial_{r}\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=-\frac{\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}\left[1+\Gamma^{t}{}_{\theta\theta}\left(3c-k+(2c-k)\,\Gamma^{t}{}_{\theta\theta}\right)\right]-\Gamma^{t}{}_{rr}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle\partial_{r}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=-1-c\,\Gamma^{t}{}_{\theta\theta}\left(2+(2c-k)\,\Gamma^{t}{}_{\theta\theta}\right)-\Gamma^{r}{}_{rr}\Gamma^{r}{}_{\theta\theta}.$
(3.32)

There are now no more equations, constraints, or integrability conditions we
can exploit. Hence, we find that in solution set 2 every component of the
stationary, spherically symmetric, torsionless, and flat connection can be
expressed in terms of two arbitrary constants, $c,k\in\mathbb{R}$, the four
functions
$\Gamma^{t}{}_{rr}(r),\Gamma^{t}{}_{\theta\theta}(r),\Gamma^{r}{}_{rr}(r),\Gamma^{r}{}_{\theta\theta}(r)$,
with $\Gamma^{r}{}_{\theta\theta}\neq 0$, and trigonometric functions. Just as
in the previous subsection, we refer to the above four functions as the
independent connection components which define solution set 2. The derivatives
$\partial_{r}\Gamma^{t}{}_{rr}$ and $\partial_{r}\Gamma^{r}{}_{rr}$ remain
undetermined, while the derivatives $\partial_{r}\Gamma^{t}{}_{\theta\theta}$
and $\partial_{r}\Gamma^{r}{}_{\theta\theta}$ can be expressed in terms of the
independent connection components via (III.4). Table 2 summarizes all
properties of solution set 2 for later convenience and reference.

Independent | All connection components of solution set 2 can be expressed in terms of two
---|---
components | arbitrary constants, $c,k\in\mathbb{R}$, the four functions $\Gamma^{t}{}_{rr}(r),\Gamma^{t}{}_{\theta\theta}(r),\Gamma^{r}{}_{rr}(r),\Gamma^{r}{}_{\theta\theta}(r)$,
| with $\Gamma^{r}{}_{\theta\theta}(r)\neq 0$, and in terms of trigonometric
functions.
Non-zero | There are $16$ non-zero components in solution set 2 (all other components vanish):
components | The four independent functions $\Gamma^{t}{}_{rr}$, $\Gamma^{t}{}_{\theta\theta}$, $\Gamma^{r}{}_{rr}$, $\Gamma^{r}{}_{\theta\theta}$ and
|  $\Gamma^{t}{}_{tt}=-c+k-c(2c-k)\Gamma^{t}{}_{\theta\theta}$ |  $\Gamma^{t}{}_{tr}=\frac{(2c-k)\Gamma^{t}{}_{\theta\theta}(1+c\,\Gamma^{t}{}_{\theta\theta})}{\Gamma^{r}{}_{\theta\theta}}$ |  $\Gamma^{t}{}_{\phi\phi}=\sin^{2}\theta\,\Gamma^{t}{}_{\theta\theta}$
|  $\Gamma^{r}{}_{tt}=-c(2c-k)\Gamma^{r}{}_{\theta\theta}$ |  $\Gamma^{r}{}_{tr}=c+c(2c-k)\Gamma^{t}{}_{\theta\theta}$ |  $\Gamma^{r}{}_{\phi\phi}=\sin^{2}\theta\,\Gamma^{r}{}_{\theta\theta}$
|  $\Gamma^{\theta}{}_{t\theta}=c$ |  $\Gamma^{\theta}{}_{r\theta}=-\frac{1+c\,\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}$ |  $\Gamma^{\theta}{}_{\phi\phi}=-\cos\theta\,\sin\theta$
|  $\Gamma^{\phi}{}_{t\phi}=c$ |  $\Gamma^{\phi}{}_{r\phi}=-\frac{1+c\,\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}$ |  $\Gamma^{\phi}{}_{\theta\phi}=\cot\theta$
Derivatives of | Of the four independent functions, the $r$-derivatives of $\Gamma^{t}{}_{\theta\theta}$ and $\Gamma^{r}{}_{\theta\theta}$ can be expressed as
independent | $\partial_{r}\Gamma^{t}{}_{\theta\theta}=-\frac{\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}\left[1+\Gamma^{t}{}_{\theta\theta}\left(3c-k+(2c-k)\,\Gamma^{t}{}_{\theta\theta}\right)\right]-\Gamma^{t}{}_{rr}\Gamma^{r}{}_{\theta\theta}$
components |  $\partial_{r}\Gamma^{r}{}_{\theta\theta}=-1-c\,\Gamma^{t}{}_{\theta\theta}\left(2+(2c-k)\,\Gamma^{t}{}_{\theta\theta}\right)-\Gamma^{r}{}_{rr}\Gamma^{r}{}_{\theta\theta}$,
| while $\partial_{r}\Gamma^{t}{}_{rr}$ and $\partial_{r}\Gamma^{r}{}_{rr}$
cannot be expressed in terms of other components.
Table 2: A concise summary of all the properties which define solution set 2.

Instead of regarding $\Gamma^{t}{}_{rr}$ and $\Gamma^{r}{}_{rr}$ as being the
free functions, which then fix the first derivatives of
$\Gamma^{t}{}_{\theta\theta}$ and $\Gamma^{r}{}_{\theta\theta}$, one could
define the latter two as free functions, and in turn determine the former by
the equations

$\displaystyle\Gamma^{t}_{\ rr}$ $\displaystyle=\frac{1}{\Gamma^{r}_{\
\theta\theta}}\left(-\frac{\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}\left[1+\Gamma^{t}{}_{\theta\theta}\left(3c-k+(2c-k)\,\Gamma^{t}{}_{\theta\theta}\right)\right]-\partial_{r}\Gamma^{t}{}_{\theta\theta}\right)$
(3.33) $\displaystyle\Gamma^{r}_{\ rr}$ $\displaystyle=\frac{1}{\Gamma^{r}_{\
\theta\theta}}\left(-1-c\,\Gamma^{t}{}_{\theta\theta}\left(2+(2c-k)\,\Gamma^{t}{}_{\theta\theta}\right)-\partial_{r}\Gamma^{r}{}_{\theta\theta}\right).$
(3.34)

Hence the (potential) connection degrees of freedom of $\Gamma^{t}{}_{rr}$ and
$\Gamma^{r}{}_{rr}$ may also be described by $\Gamma^{t}{}_{\theta\theta}$ and
$\Gamma^{r}{}_{\theta\theta}$, but this again just a choice of
parametrization.

Finally, we observe that the spherical connection used in Zhao:2021 ; Lin:2021
falls into solution set 2. Indeed, if we choose
$\\{c,k,\Gamma^{t}{}_{rr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{rr},\Gamma^{r}{}_{\theta\theta}\\}=\\{0,0,0,0,0,-r\\}$,
then the only non-vanishing connection components of solution set 2 are

$\displaystyle\Gamma^{r}{}_{\theta\theta}$ $\displaystyle=-r$
$\displaystyle\Gamma^{r}{}_{\phi\phi}$ $\displaystyle=-r\,\sin^{2}\theta$
$\displaystyle\Gamma^{\theta}{}_{r\theta}$ $\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma^{\theta}{}_{\phi\phi}$
$\displaystyle=-\cos\theta\,\sin\theta$ $\displaystyle\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=\frac{1}{r}$ $\displaystyle\Gamma^{\phi}{}_{\theta\phi}$
$\displaystyle=\cot\theta,$ (3.35)

which agrees precisely with the connection used in Zhao:2021 ; Lin:2021 .

###  III.5 Solution set 1 as the double scaling limit of solution set 2

Solution sets 1 and 2 have certain similarities, even though solution set 2
has more free functions. We will now show that solution set 1 can indeed be
obtained from solution set 2 by a double scaling limit.

The key component to consider is $\Gamma^{\phi}{}_{r\phi}$ of solution set 2,
which can be expressed in terms of $\Gamma^{t}{}_{\theta\theta}$ and
$\Gamma^{r}{}_{\theta\theta}$ (see Table 2). For solution set 1,
$\Gamma^{\phi}{}_{r\phi}$ is an arbitrary free function of $r$. These facts
suggest the following parametrization of $\Gamma^{t}{}_{\theta\theta}$ and
$\Gamma^{r}{}_{\theta\theta}$:

$\displaystyle\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=-\frac{1}{c}+\frac{\lambda}{c}\Phi$ and
$\displaystyle\Gamma^{r}{}_{\theta\theta}$ $\displaystyle=\lambda\Psi,$ (3.36)

where we have to assume $c\neq 0$, $\lambda$ is a parameter, and $\Phi$ and
$\Psi$ are arbitrary functions of $r$. With this parametrization one obtains
in the $\lambda\to 0$ limit and under the assumption that $k=c$

$\lim_{\lambda\to 0}\Gamma^{\phi}{}_{r\phi}=-\frac{\Phi}{\Psi}.$ (3.37)

In words: The component $\Gamma^{\phi}{}_{r\phi}$ of solution set 2 is mapped
to an arbitrary function of $r$, just as required by solution set 1. The
parametrization (3.36) also ensures that the other non-trivial components of
solution set 2 are correctly mapped to their counterparts in solution set 1:

$\displaystyle\left.\lim_{\lambda\to 0}\Gamma^{t}{}_{tt}\right|_{k=c}$
$\displaystyle=c$ $\displaystyle\left.\lim_{\lambda\to
0}\Gamma^{t}{}_{tr}\right|_{k=c}$ $\displaystyle=-\frac{\Phi}{\Psi}$
$\displaystyle\left.\lim_{\lambda\to 0}\Gamma^{t}{}_{\phi\phi}\right|_{k=c}$
$\displaystyle=-\frac{\sin^{2}\theta}{c}$ $\displaystyle\left.\lim_{\lambda\to
0}\Gamma^{r}{}_{tt}\right|_{k=c}$ $\displaystyle=0$
$\displaystyle\left.\lim_{\lambda\to 0}\Gamma^{r}{}_{tr}\right|_{k=c}$
$\displaystyle=0$ $\displaystyle\left.\lim_{\lambda\to
0}\Gamma^{r}{}_{\phi\phi}\right|_{k=c}$ $\displaystyle=0$
$\displaystyle\left.\lim_{\lambda\to
0}\Gamma^{\theta}{}_{r\theta}\right|_{k=c}$
$\displaystyle=-\frac{\Phi}{\Psi}.$ (3.38)

What remains to be checked is whether the double scaling limit also allows us
to recover equation (3.29) from the equations (III.4). Indeed, when plugging
the parametrization (3.36) into (III.4), one finds

$\displaystyle\partial_{r}\Phi$
$\displaystyle=\frac{\Phi^{2}}{\Psi}-c\,\Gamma^{t}{}_{rr}\Psi-\lambda\frac{\Phi^{3}}{\Psi}$
$\displaystyle\partial_{r}\Psi$
$\displaystyle=-\Gamma^{r}{}_{rr}\Psi-\lambda\Phi^{2}.$ (3.39)

In turn, these relations can be used to show that

$\displaystyle\left.\lim_{\lambda\to
0}\partial_{r}\Gamma^{\phi}{}_{r\phi}\right|_{k=c}$
$\displaystyle=-\partial_{r}\left(\frac{\Phi}{\Psi}\right)=c\,\Gamma^{t}{}_{rr}+\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{r}{}_{rr}-\Gamma^{\phi}{}_{r\phi}\right).$
(3.40)

That is to say: We correctly recover equation (3.29) of solution set 1. Thus,
we have succeeded in showing that solution set 1 can be obtained from solution
set 2 via a double scaling limit. We can therefore regard solution set 2 as
the most general parametrization of a connection which is stationary,
spherically symmetric, torsionless, and flat.

There is also another point of view one can take, which regards the choice of
free connection variables. The main difference between solution sets 1 and 2
is the parametrization; solution set 1 is given by the free function
$\Gamma^{\phi}{}_{r\phi}$, while in solution set 2
$\Gamma^{t}{}_{\theta\theta}$ is free and $\Gamma^{\phi}{}_{r\phi}$ is fixed.
In order to see in a different way that solution set 1 can be obtained from 2,
we switch the parametrization in set 2, i.e., we now leave
$\Gamma^{\phi}{}_{r\phi}$ free and set

$\Gamma^{t}{}_{\theta\theta}=-\frac{1+\Gamma^{r}{}_{\theta\theta}\Gamma^{\phi}{}_{r\phi}}{c}.$
(3.41)

With this, one obtains the remaining components of the connection in solution
set 2 as

$\displaystyle\Gamma^{t}{}_{tt}$
$\displaystyle=c+(2c-k)\Gamma^{r}{}_{\theta\theta}\Gamma^{\phi}{}_{r\phi},$
$\displaystyle\Gamma^{t}{}_{tr}$
$\displaystyle=\frac{2c-k}{c}(1+{\Gamma^{r}}_{\theta\theta}{\Gamma^{\phi}}_{r\phi}){\Gamma^{\phi}}_{r\phi},$
$\displaystyle{\Gamma^{r}}_{tt}$
$\displaystyle=-c(2c-k){\Gamma^{r}}_{\theta\theta}$
$\displaystyle{\Gamma^{r}}_{tr}$
$\displaystyle=k-c-(2c-k){\Gamma^{r}}_{\theta\theta}{\Gamma^{\phi}}_{r\phi}$
$\displaystyle{\Gamma^{r}}_{\phi\phi}$
$\displaystyle=\sin^{2}\theta\,{\Gamma^{r}}_{\theta\theta}$
$\displaystyle{\Gamma^{\theta}}_{t\theta}$
$\displaystyle={\Gamma^{\phi}}_{t\phi}=c$
$\displaystyle{\Gamma^{\theta}}_{r\theta}$
$\displaystyle={\Gamma^{\phi}}_{r\phi}$
$\displaystyle{\Gamma^{t}}_{\theta\theta}$
$\displaystyle=-\frac{1+{\Gamma^{\phi}}_{r\phi}{\Gamma^{r}}_{\theta\theta}}{c}$
$\displaystyle{\Gamma^{t}}_{\phi\phi}$
$\displaystyle=-\sin^{2}\theta\,\frac{1+{\Gamma^{\phi}}_{r\phi}{\Gamma^{r}}_{\theta\theta}}{c}$
$\displaystyle{\Gamma^{\theta}}_{\phi\phi}$
$\displaystyle=-\cos\theta\sin\theta$
$\displaystyle{\Gamma^{\phi}}_{\theta\phi}$ $\displaystyle=\cot\theta$ (3.42)

The flatness conditions in this parametrization becomes (3.29) and

$\displaystyle\partial_{r}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=-1+\left(1+\Gamma^{r}_{\ \theta\theta}\Gamma^{\phi}_{\
r\phi}\right)\left(2-\left(2-\frac{k}{c}\right)\left(1+\Gamma^{r}_{\
\theta\theta}\Gamma^{\phi}_{\
r\phi}\right)\right)-\Gamma^{r}{}_{rr}\Gamma^{r}{}_{\theta\theta}.$ (3.43)

Now in this parametrization of solution set 2 we first observe that naivly we
would now want to set $c\neq 0$; but of course this limit can be taken without
a problem, as the parametrization from above shows. This demonstrates that one
has to choose the correct parametrizations for certain limits. We also see in
this parametrization that we can now safely set $\Gamma^{r}_{\
\theta\theta}=0$ and $k=c$ to obtain solution set 1, where we must now, after
taking these limits, demand $c\neq 0$. Note that if one puts $\Gamma^{r}_{\
\theta\theta}=0$ one has to put $k=c$ as well to have still zero curvature.
This shows that solution set 1 is contained in set 2, modulo the
parametrization.

###  III.6 Diagonalizing the metric, canonical $\mathcal{S}^{2}$ part, and
transformation behavior of the solution sets

Let us recall that our goal is to study the field equations of $f(\mathbb{Q})$
gravity for a stationary and spherically symmetric affine geometry
$(\mathcal{M},g_{\mu\nu},\Gamma^{\alpha}{}_{\mu\nu})$. To that end, we have
performed a symmetry reduction of the connection in subsection III.1 and we
have implemented the postulates of symmetric teleparallelism in the
subsections III.1 and III.2. This led us to the two solution sets studied in
the subsections III.3 and III.4. These two sets arose from demanding that the
Riemann tensor of the affine connection vanishes and both sets represent a
stationary, spherically symmetric, torsionless, and flat connection. Moreover,
we have seen in subsection III.5 that solution set 1 is the double scaling
limit of solution set 2 and this set cannot be further simplified using
symmetries or the postulates of symmetric teleparallelism.

However, what we can try to further simplify is the metric. At the beginning
of section III, we have stated that the most general metric which is
spherically symmetric and stationary takes the form (3.3). This metric can be
brought into a simpler form via the application of a diffeomorphism which
eliminates the off-diagonal component $g_{tr}$ and which brings the
$\mathcal{S}^{2}$ part of the metric into canonical form (i.e., it maps
$g_{\theta\theta}$ to $r^{2}$). Of course, if we apply this diffeomorphism to
the metric, we also need to apply it to the connection. What we will show now,
is that the metric can always be brought into the described form and that the
diffeomorphism which allows us to do so maps the solution sets of the
connection onto themselves. In other words: The diffeomorphism which
simplifies the metric preserves the structure of the solution sets.

To begin with, we notice that the symmetry reduced metric (3.3) describes a
warped geometry. This means the following: Let $(\mathcal{B},h)$ and
$(\mathcal{F},\sigma)$ be (pseudo-) Riemannian manifolds, where $\mathcal{B}$
is called the base space and $\mathcal{F}$ the fiber. Furthermore, let
$f:\mathcal{B}\to\mathbb{R}_{>0}$ be a strictly positive function on the base
space, called the warping factor. A warped geometry is then the manifold
$\mathcal{M}:=\mathcal{B}\times_{f}\mathcal{F}$ which is the topological space
$\mathcal{B}\times\mathcal{F}$ endowed with the metric tensor $g:=h\oplus
f\sigma$. Concretely, this means that the metric on the total space
$\mathcal{M}$ can be written as a metric tensor on $\mathcal{B}$ plus a metric
tensor on $\mathcal{F}$ times a positive function which depends on the
coordinates of $\mathcal{B}$. In our case, we have
$\mathcal{B}=\mathbb{R}\times\mathbb{R}_{>0}$, $\mathcal{F}=\mathcal{S}^{2}$,
and $f(t,r)=g_{\theta\theta}(r)$. Notice that it follows from the fact that
the signature of (3.3) is $(-,+,+,+)$ that $g_{\theta\theta}$ is a strictly
positive function of $r$. We can therefore write the metric (3.3) equivalently
as the warped metric

$g=h_{tt}\,\mathrm{d}t^{2}+2h_{tr}\,\mathrm{d}t\,\mathrm{d}r+h_{rr}\,\mathrm{d}r^{2}+f(r)\,\mathrm{d}\Omega^{2},$
(3.44)

where
$\mathrm{d}\Omega^{2}:=\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi$ is
the canonical metric on the unit $2$-sphere $\mathcal{S}^{2}$. Given the fact
that $f$ is a strictly positive function, we can write it in the suggestive
form $f(r)=\rho(r)^{2}$. This is merely a notational convention and we have
not yet changed anything. But now we choose $\rho$ as a new coordinate and we
perform the change of coordinates on the base space $\mathcal{B}$ from $(t,r)$
to $(t,\rho)$. This is a rather trivial manipulation and the metric tensor in
the new coordinate system reads

$g=h^{\prime}_{tt}\,\mathrm{d}t^{2}+2h^{\prime}_{t\rho}\,\mathrm{d}t\,\mathrm{d}\rho+h^{\prime}_{\rho\rho}\,\mathrm{d}\rho^{2}+\rho^{2}\mathrm{d}\Omega^{2},$
(3.45)

where the transformed metric components are given by

$\displaystyle h^{\prime}_{tt}$ $\displaystyle=h_{tt},$ $\displaystyle
h^{\prime}_{t\rho}$ $\displaystyle=\left(\frac{\partial
r}{\partial\rho}\right)h_{tr},$ $\displaystyle h^{\prime}_{\rho\rho}$
$\displaystyle=\left(\frac{\partial r}{\partial\rho}\right)^{2}h_{rr}.$ (3.46)

Hence, we can always bring the $\mathcal{S}^{2}$ part of the metric into
canonical form as long as $\frac{\partial\rho}{\partial
r}\equiv\frac{1}{2\sqrt{g_{\theta\theta}}}\frac{\partial
g_{\theta\theta}}{\partial r}\neq 0$, which is tantamount to demanding that
$g_{\theta\theta}$ is a monotonous function of $r$.
Let us denote the diffeomorphism which brings the metric into the form (3.45)
by $\phi_{1}:\mathcal{B}\to\mathcal{B}$ and let us now ask if there exists a
second diffeomorphism $\phi_{2}:\mathcal{B}\to\mathcal{B}$ which eliminates
the off-diagonal term $h^{\prime}_{t\rho}$. To that end, we assume that
$\phi_{2}$ only acts on $t$ in the sense that it generates the new coordinate
system $(\tau,\rho)$ from $(t,\rho)$. Under this assumption, the $1$-form
$\mathrm{d}t$ transforms as

$\mathrm{d}t=\left(\frac{\partial
t}{\partial\tau}\right)\mathrm{d}\tau+\left(\frac{\partial
t}{\partial\rho}\right)\mathrm{d}\rho.$ (3.47)

Moreover, we demand that the $\mathcal{B}$-part of the metric tensor in the
new coordinates takes the form

$\left.g\right|_{\mathcal{B}}=h\overset{!}{=}\tilde{h}_{\tau\tau}\,\mathrm{d}\tau^{2}+\tilde{h}_{\rho\rho}\,\mathrm{d}\rho^{2},$
(3.48)

for some (not further specified) functions $\tilde{h}_{\tau\tau}$ and
$\tilde{h}_{\rho\rho}$. By inserting (3.47) into (3.48) we obtain the
following conditions

$\begin{cases}h^{\prime}_{tt}&\overset{!}{=}\ \
\tilde{h}_{\tau\tau}\left(\frac{\partial\tau}{\partial t}\right)^{2}\\\
h^{\prime}_{t\rho}&\overset{!}{=}\ \
\tilde{h}_{\tau\tau}\left(\frac{\partial\tau}{\partial
t}\right)\left(\frac{\partial\tau}{\partial\rho}\right)\\\
h^{\prime}_{\rho\rho}&\overset{!}{=}\ \
\tilde{h}_{\rho\rho}+\tilde{h}_{\tau\tau}\left(\frac{\partial\tau}{\partial\rho}\right)^{2}.\end{cases}$
(3.49)

These are three equations for the three unknown functions $\tau(t,\rho)$,
$\tilde{h}_{\tau\tau}$, and $\tilde{h}_{\rho\rho}$. Hence, the problem is in
principle solvable, provided one specifies initial conditions for
$\tau(t,\rho)$. We can therefore eliminate the off-diagonal term
$h^{\prime}_{t\rho}$.
All in all, we have defined two diffeomorphisms on the base space
$\mathcal{B}$ which diagonalize the metric and bring the $\mathcal{S}^{2}$
part into canonical form. We can also combine these two diffeomorphisms into
$\phi_{2}\circ\phi_{1}:\mathcal{B}\to\mathcal{B}$, with
$(\phi_{2}\circ\phi_{1})(t,r)=(\tau,\rho)$.

In the sequel, we will drop all primes and tildes and denote the metric
components again by $g_{\mu\nu}$, rather than $h_{\mu\nu}$. Also, we change
notation and write $(t,r)$ for $(\tau,\rho)$ (this is for simplicity and
should not cause any confusion). After applying $\phi_{2}\circ\phi_{1}$, the
metric can then be written as

$g_{\mu\nu}=\begin{pmatrix}g_{tt}&0&0&0\\\ 0&g_{rr}&0&0\\\ 0&0&r^{2}&0\\\
0&0&0&r^{2}\sin^{2}\theta\end{pmatrix}.$ (3.50)

This is the form of the metric which we shall use from now on and which will
greatly simplify the field equations to be studied in section IV. However, we
also need to determine how the connection transforms under
$\phi_{2}\circ\phi_{1}$. In general, a connection transforms as

$\tilde{\Gamma}^{\alpha}{}_{\mu\nu}=\frac{\partial\tilde{x}^{\alpha}}{\partial
x^{\beta}}\frac{\partial x^{\rho}}{\partial\tilde{x}^{\mu}}\frac{\partial
x^{\sigma}}{\partial\tilde{x}^{\nu}}\Gamma^{\beta}{}_{\rho\sigma}+\frac{\partial\tilde{x}^{\alpha}}{\partial
x^{\lambda}}\frac{\partial^{2}x^{\lambda}}{\partial\tilde{x}^{\mu}\partial\tilde{x}^{\nu}}.$
(3.51)

under coordinate transformations and this could potentially spoil the
structure of the solution sets. However, notice that $\phi_{2}\circ\phi_{1}$
only acts on the base space $\mathcal{B}$ and that it leaves the fiber
$\mathcal{F}=\mathcal{S}^{2}$ invariant. Intuitively, we would therefore
expect that our diffeomorphism is compatible with spherical symmetry. Indeed,
it is obvious that the metrics (3.3) and (3.50) possess the same Killing
vectors and hence share the same isometry group. This confirms that the
diffeomorphism $\phi_{2}\circ\phi_{1}$ respects spherical symmetry. Moreover,
a diffeomorphism can neither produce curvature nor torsion. But if
$\tilde{\Gamma}^{\alpha}{}_{\mu\nu}$, i.e., the transformed connection, is
invariant under the isometries generated by the Killing vector fields and if
it is torsionless and flat, it gives rise to the same two solution sets we
have discussed in previous subsections. Hence, we see that the diffeomorphism
$\phi_{2}\circ\phi_{1}$ maps the solution sets onto themselves. This can also
be confirmed by a direct computation, see Appendix A.

The importance of this result is that it allows us to use the simpler form of
the metric, given by equation (3.50), together with solution sets 1 and 2 for
the connection. The metric in the form (3.50) together with solution sets 1
and 2 for the connection constitute the simplest and yet most general metric-
affine geometries which are stationary, spherically symmetric, torsionless,
and flat.

##  IV Symmetry reduced field equations for the metric and the connection

In section III we have performed a detailed symmetry reduction of the metric-
affine geometry $(\mathcal{M},g_{\mu\nu},\Gamma^{\alpha}{}_{\mu\nu})$ and we
have found that the simplest –and yet most general form– of a metric-affine
geometry which is stationary, spherically symmetric, torsionless, and flat is
given by the metric (3.50) and the connection has to be chosen such that it
belongs either to solution set 1 (see Table 1) or to solution set 2 (see Table
2).

We now use the metric-affine geometries described above to perform a symmetry
reduction of the metric and connection field equations (II). We first discuss
the symmetry reduced field equations for solution set 2, since, as we have
seen in subsection III.5, solution set 1 can be obtained from solution set 2
by means of a double scaling limit. Hence, once we understand the field
equations for solution set 2, we can immediately derive consequences for the
field equations of solution set 1.

However, before studying the symmetry reduction in detail, we have a brief
look at the structure of the field equations. For both solution sets, the
field equations have the following form:

Structure of metric field equations:
$\displaystyle\begin{pmatrix}\mathcal{M}_{tt}&\mathcal{M}_{tr}&0&0\\\
\mathcal{M}_{tr}&\mathcal{M}_{rr}&0&0\\\ 0&0&\mathcal{M}_{\theta\theta}&0\\\
0&0&0&\mathcal{M}_{\theta\theta}\,\sin^{2}\theta\end{pmatrix}$ Structure of
connection field equations: $\displaystyle\begin{pmatrix}\mathcal{C}_{t}\\\
\mathcal{C}_{r}\\\ 0\\\ 0\end{pmatrix}$ (4.1)

This is true for the vacuum case as well as in the presence of spherically
symmetric matter distributions. What we observe is that there are at most four
independent metric field equations and at most two independent connection
field equations. Moreover, it turns out that we can learn a lot just by
looking at the off-diagonal equation $\mathcal{M}_{tr}$. In the following
subsection we will study $\mathcal{M}_{tr}$ for solution set 2 and we will be
able to derive necessary conditions for the existence of solutions which go
beyond Schwarzschild-deSitter-Nordström. We also note here for completeness
the forms of the scalar $\mathbb{Q}$. For solution set 1 we find

$\mathbb{Q}=\frac{g_{tt}\left(2g_{rr}(3r(c\,r\Gamma^{t}{}_{rr}+\Gamma^{\phi}{}_{r\phi}(r\Gamma^{r}{}_{rr}-r\Gamma^{\phi}{}_{r\phi}+2))-2g_{rr}-2)-3r^{2}\Gamma^{\phi}{}_{r\phi}\partial_{r}g_{rr}\right)+rg_{rr}\partial_{r}g_{tt}(3r\Gamma^{\phi}{}_{r\phi}-4)}{2r^{2}\,(g_{rr}{})^{2}g_{tt}}$
(4.2)

while for solution set 2 we get

$\displaystyle\mathbb{Q}$
$\displaystyle=\frac{1}{2r^{2}g_{rr}{}^{2}g_{tt}{}^{2}\Gamma^{r}{}_{\theta\theta}{}^{2}}\Bigg{(}g_{rr}{}^{2}\Gamma^{r}{}_{\theta\theta}{}^{2}\left(cr^{2}\Gamma^{r}{}_{\theta\theta}\partial_{r}g_{tt}(k-2c)-2crg_{tt}(2c-k)(cr\Gamma^{t}{}_{\theta\theta}(2c\Gamma^{t}{}_{\theta\theta}-k\Gamma^{t}{}_{\theta\theta}+2)+\right.$
$\displaystyle+\left.\Gamma^{r}{}_{\theta\theta}(r\Gamma^{r}{}_{rr}-2)+r)+4g_{tt}{}^{2}(c\Gamma^{t}{}_{\theta\theta}(2c\Gamma^{t}{}_{\theta\theta}-k\Gamma^{t}{}_{\theta\theta}+2)+\Gamma^{r}{}_{\theta\theta}\Gamma^{r}{}_{rr})-2g_{tt}\Gamma^{r}{}_{\theta\theta}\partial_{r}g_{tt}\right)+$
$\displaystyle+g_{rr}g_{tt}\left(2g_{tt}\left(\Gamma^{r}{}_{\theta\theta}{}^{2}\left(r^{2}\Gamma^{t}{}_{rr}(2c\Gamma^{t}{}_{\theta\theta}(k-2c)+k)-2\right)+\right.\right.$
$\displaystyle+\left.\left.r\Gamma^{r}{}_{\theta\theta}(c\Gamma^{t}{}_{\theta\theta}+1)(r\Gamma^{r}{}_{rr}+2)(2c\Gamma^{t}{}_{\theta\theta}-k\Gamma^{t}{}_{\theta\theta}-2)-\left(\Gamma^{t}{}_{\theta\theta}{}^{2}(k-2c)^{2}+2\right)(cr\Gamma^{t}{}_{\theta\theta}+r){}^{2}-\right.\right.$
$\displaystyle-\left.\left.\Gamma^{r}{}_{\theta\theta}{}^{3}\partial_{r}g_{rr}\right)+r\Gamma^{r}{}_{\theta\theta}\left(cr\Gamma^{r}{}_{\theta\theta}{}^{2}\partial_{r}g_{rr}(2c-k)-\partial_{r}g_{tt}(r(\Gamma^{t}{}_{\theta\theta}(c\Gamma^{t}{}_{\theta\theta}(k-2c)+k)+2)+4\Gamma^{r}{}_{\theta\theta})\right)\right)+$
$\displaystyle+r^{2}g_{tt}{}^{2}\Gamma^{r}{}_{\theta\theta}\partial_{r}g_{rr}(c\Gamma^{t}{}_{\theta\theta}+1)(\Gamma^{t}{}_{\theta\theta}(k-2c)+2)\Bigg{)}.$
(4.3)

###  IV.1 Off-diagonal metric field equation for solution set 2

For solution set 2, the off-diagonal metric field equation takes the form

$\mathcal{M}_{tr}=\frac{1}{2}\left(k-2c\,(2c-k)\Gamma^{t}{}_{\theta\theta}\right)\partial_{r}\mathbb{Q}\,f^{\prime\prime}(\mathbb{Q})=0,$
(4.4)

which is valid in the vacuum as well as in the electro-vacuum case. This
equation can be solved in three distinct ways which we will discuss in turn.
The three different possible solutions are

  1. 1.

$\partial_{r}\mathbb{Q}=0$;

  2. 2.

$f^{\prime\prime}(\mathbb{Q})=0$;

  3. 3.

$\frac{1}{2}\left(k-2c\,(2c-k)\Gamma^{t}{}_{\theta\theta}\right)=0$.

The first option is tantamount to saying that the non-metricity scalar, when
evaluated on a solution of the field equations, is constant. But this cannot
yield a solution to the field equations which goes beyond Schwarzschild-
deSitter-Nordström. This can be readily seen from the alternative form (2.10)
of the metric field equations. Let us re-write these equations here for
convenience:

$f^{\prime}(\mathbb{Q})G_{\mu\nu}+\frac{1}{2}g_{\mu\nu}(f(\mathbb{Q})-f^{\prime}(\mathbb{Q})\mathbb{Q})+2f^{\prime\prime}(\mathbb{Q})P^{\alpha}{}_{\mu\nu}\partial_{\alpha}\mathbb{Q}=T_{\mu\nu}.$
(4.5)

If we assume that a solution to the connection and metric field equations
exists such that $\mathbb{Q}_{\textsf{sol}}=\textsf{const}.$ \- we will see in
subsection V that such solutions indeed exist, but one can also use that
$\mathbb{Q}$ is linear in $\Gamma^{r}_{\ rr}$, hence one can solve the
equations $\mathbb{Q}=\mathbb{Q}_{\text{sol}}$ for $\Gamma^{r}_{\ rr}$ for any
$\mathbb{Q}_{\text{sol}}\in\mathbb{R}$ to obtain such solutions by a choice of
the connection - then the last term on the left hand side of (4.5) vanishes
and we are left with

$G_{\mu\nu}+\Lambda_{\textsf{eff}}\,g_{\mu\nu}=\bar{T}_{\mu\nu},$ (4.6)

where, as we recall, $G_{\mu\nu}$ is the standard Einstein tensor with respect
to the Levi-Civita connection and where we have defined

$\displaystyle\Lambda_{\textsf{eff}}$
$\displaystyle:=\frac{1}{2}\frac{f(\mathbb{Q}_{\textsf{sol}})-f^{\prime}(\mathbb{Q}_{\textsf{sol}})\mathbb{Q}_{\textsf{sol}}}{f^{\prime}(\mathbb{Q}_{\textsf{sol}})}$
$\displaystyle\bar{T}_{\mu\nu}$
$\displaystyle:=\frac{1}{f^{\prime}(\mathbb{Q}_{\textsf{sol}})}T_{\mu\nu}.$
(4.7)

The first term, $\Lambda_{\textsf{eff}}$, is clearly constant and simply
represents an effective cosmological constant, while $\bar{T}_{\mu\nu}$ is
just a re-scaled energy-momentum tensor. Notice that dividing by $f^{\prime}$
is allowed since we need to assume $f^{\prime}\neq 0$ in order to obtain non-
trivial and self-consistent field equations. We therefore reach the conclusion
that assuming $\mathbb{Q}_{\textsf{sol}}=\text{const.}$ can only produce the
Schwarzschild-deSitter-Nordström solution for arbitrary $f$.

Let us now consider the second option, namely that the off-diagonal field
equation $\mathcal{M}_{tr}$ is solved by $f^{\prime\prime}(\mathbb{Q})=0$. It
is important to notice that this equation has to hold on a solution of all
field equations, i.e., we should write
$f^{\prime\prime}(\mathbb{Q}_{\textsf{sol}})=0$. We can now easily show that
when $f^{\prime\prime}(\mathbb{Q}_{\textsf{sol}})=0$ and
$\mathbb{Q}_{\textsf{sol}}\neq$ const., then it follows that $f$ is
necessarily of the form $f(\mathbb{Q})=a\,\mathbb{Q}+b$, where $a$ and $b$ are
real constants and this form of $f$ holds everywhere, not just on solutions.
To see this, let us first assume that $f(\mathbb{Q})$ does not have the above
form. Then it follows that $f^{\prime\prime}(\mathbb{Q}_{\textsf{sol}})=0$ is
an equation which will be satisfied for at least one
$\mathbb{Q}_{\textsf{sol}}$ (at least one solution needs to exist, otherwise
$f^{\prime\prime}(\mathbb{Q}_{\textsf{sol}})\neq 0$, but the solution needs
not be unique). But if we fix $\mathbb{Q}_{\textsf{sol}}$ via this equation,
we find that $\mathbb{Q}_{\textsf{sol}}$ is a constant! Hence, we fall back
into the GR regime for arbitrary $f$, which we have already discussed above.
If, on the other hand side, we have $f(\mathbb{Q})=a\,\mathbb{Q}+b$, then
$f^{\prime\prime}(\mathbb{Q}_{\textsf{sol}})=0$ is trivially satisfied and
$\mathbb{Q}_{\textsf{sol}}$ is not a constant. This is what we wanted to
prove.

We reach the conclusion that solving $\mathcal{M}_{tr}=0$ via
$f^{\prime\prime}(\mathbb{Q})=0$ automatically leads us into the symmetric
teleparallelism sector of $f(\mathbb{Q})$ gravity. In particular, this means
that we can only get the Schwarzschild-deSitter-Nordström solution and nothing
else. There is only one little caveat, which also applies to solving
$\mathcal{M}_{tr}$ via $\partial_{r}\mathbb{Q}=0$: We have only used the
metric field equations but completely neglected the connection field
equations. However, it is easy to see that these equations will not give rise
to any constraints or inconsistencies which would alter the conclusions we
have reached thus far. In fact, the connection field equations can be written
in the schematic form

$\displaystyle\mathcal{C}_{t}$
$\displaystyle=A\,\left(\partial_{r}\mathbb{Q}\right)\,f^{\prime\prime}(\mathbb{Q})+B\,\left(\partial^{2}_{r}\mathbb{Q}\right)\,f^{\prime\prime}(\mathbb{Q})+C\,\left(\partial_{r}\mathbb{Q}\right)^{2}\,f^{(3)}(\mathbb{Q})=0$
$\displaystyle\mathcal{C}_{r}$
$\displaystyle=D\,\left(\partial_{r}\mathbb{Q}\right)\,f^{\prime\prime}(\mathbb{Q})+E\,\left(\partial^{2}_{r}\mathbb{Q}\right)\,f^{\prime\prime}(\mathbb{Q})+F\,\left(\partial_{r}\mathbb{Q}\right)^{2}\,f^{(3)}(\mathbb{Q})=0,$
(4.8)

where $A$, $B$, $C$, $D$, $E$, and $F$ are complicated functions of the metric
components, their derivatives, and connection components. What matters is that
the connection equations can be written as polynomials in
$\partial_{r}\mathbb{Q}$, $f^{\prime\prime}(\mathbb{Q})$, and their higher
order derivatives, as indicated by equation (IV.1). This makes it obvious that
the solutions $\partial_{r}\mathbb{Q}=0$ or $f^{\prime\prime}(\mathbb{Q})=0$
to $\mathcal{M}_{tr}$ automatically also satisfy the connection field
equations. Hence, no further constraints appear and the equations are self-
consistent. We can therefore confirm that solving $\mathcal{M}_{tr}=0$ via
$\partial_{r}\mathbb{Q}=0$ or $f^{\prime\prime}(\mathbb{Q})=0$ only gives the
Schwarzschild-deSitter-Nordström solution (for arbitrary $f$ in the case of
$\partial_{r}\mathbb{Q}=\textsf{const}$, while $f^{\prime\prime}(\mathbb{Q})$
simply reduces $f(\mathbb{Q})$ to standard symmetric teleparallelism, as one
would expect).

What remains to be examined is the third option. Namely, that
$\mathcal{M}_{tr}=0$ is solved by

$k-2c\,(2c-k)\Gamma^{t}{}_{\theta\theta}=0.$ (4.9)

This is a constraint equation for the connection and we will see shortly that
it does indeed eliminate one of two potential degrees of freedom of the
connection. To that end, we solve (4.9) for $\Gamma^{t}{}_{\theta\theta}$,
which gives us

$\displaystyle\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=\frac{k}{2c(2c-k)}\quad\text{for }c\neq 0\text{ and }k\neq 2c.$
(4.10)

Notice that since $c$ and $k$ are constants, $\Gamma^{t}{}_{\theta\theta}$ is
now constrained to be a constant. In particular, this means that its
derivative vanishes. But its derivative can also be written in terms of other
connection components (cf. last row of Table 2). We therefore obtain the
condition

$0=-\frac{k(8c^{2}+2ck-k^{2})}{8c^{2}(2c-k)^{2}\Gamma^{r}{}_{\theta\theta}}-\Gamma^{t}{}_{rr}\Gamma^{r}{}_{\theta\theta}\quad\Longleftrightarrow\quad\Gamma^{t}{}_{rr}=-\frac{k(8c^{2}+2ck-k^{2})}{8c^{2}(2c-k)^{2}(\Gamma^{r}{}_{\theta\theta})^{2}}.$
(4.11)

This expression for $\Gamma^{t}{}_{rr}$ is well defined since solution set 2
requires $\Gamma^{r}{}_{\theta\theta}\neq 0$ for its very definition and we
already had to assume $c\neq 0$ and $k\neq 2c$ in order to obtain (4.10).
Hence, what we find is that we can fix two of the four free functions of
solution set 2 in terms of other connection components. More importantly, the
constraint equation (4.9) allowed us to fix the component $\Gamma^{t}{}_{rr}$,
whose derivative was up to now not known. This means that up to now,
$\Gamma^{t}{}_{rr}$ was a potential degree of freedom since its derivative
could in principle be determined by one of the field equations. But since
$\Gamma^{t}{}_{rr}$ is now given by (4.11), whose derivatives can all be
determined without using the field equations, it cannot become dynamical and
it is therefore not a physical degree of freedom. Hence, the constraint
equation (4.9) has effectively removed a potential degree of freedom. The only
connection component which can now become dynamical is $\Gamma^{r}{}_{rr}$ \-
or alternatively $\Gamma^{r}_{\ \theta\theta}$ \- because its derivative is
undetermined and can therefore not be eliminated from the field equations.

There is just a little caveat: Our considerations only hold as long as $c\neq
0$ and $k\neq 2c$. To remedy this shortcoming, i.e., to see what happens for
$c=0$, we can solve the constraint equation (4.9) for $k$:

$k=\frac{4c^{2}\,\Gamma^{t}{}_{\theta\theta}}{1+2c\,\Gamma^{t}{}_{\theta\theta}}\quad\text{for
}1+2c\,\Gamma^{t}{}_{\theta\theta}\neq 0.$ (4.12)

Just as before, we can compute the derivative of (4.12) and express the
derivative of $\Gamma^{t}{}_{\theta\theta}$ in terms of other connection
components, using the relation shown in Table 2. This gives us again an
equation which we can solve for a connection component and we obtain

$\Gamma^{t}{}_{rr}=-\frac{\Gamma^{t}{}_{\theta\theta}+5c\,(\Gamma^{t}{}_{\theta\theta})^{2}+4c^{2}\,(\Gamma^{t}{}_{\theta\theta})^{3}}{(1+2c\,\Gamma^{t}{}_{\theta\theta})(\Gamma^{r}{}_{\theta\theta})^{2}},$
(4.13)

which is again well-defined under the assumptions for which (4.12) is valid.
The equations (4.12) and (4.13) have the advantage that they hold for $c=0$
and for $k=2c$. Both options imply

$c=k=0\quad\text{and}\quad\Gamma^{t}{}_{rr}=-\frac{\Gamma^{t}{}_{\theta\theta}}{(\Gamma^{r}{}_{\theta\theta})^{2}}.$
(4.14)

Hence, it is true in full generality111Notice that (4.12) demands
$1+2\,c\,\Gamma^{t}{}_{\theta\theta}\neq 0$ and it seems that we have to treat
this case separately. But this is not true: If
$\Gamma^{t}{}_{\theta\theta}=-\frac{1}{2c}$, then the constraint equation
(4.9) reduces to $c=0$, which leads to an inconsistency. Hence,
$1+2\,c\,\Gamma^{t}{}_{\theta\theta}=0$ is not admissible and we have
therefore already found the most general solution to the constraint equation.
that the constraint equation (4.9) removes one degree of freedom and only
$\Gamma^{r}{}_{rr}$ is left as a candidate for a propagating degree of freedom
stemming from the connection.

Before concluding this subsection, we remark that the connection field
equations can be simplified by using the constraint equation (4.9). In fact,
one can show that the equation $\mathcal{C}_{t}$ can be written as

$\mathcal{C}_{t}=A\,\left(k-2c(2c-k)\Gamma^{t}{}_{\theta\theta}\right)-B\,\left(2c(2c-k)\partial_{r}\Gamma^{t}{}_{\theta\theta}\right)=0,$
(4.15)

where $A$ and $B$ are functions of metric components and their derivatives.
The coefficient of $A$ is simply the constraint equation (4.9), while the
coefficient of $B$ is the derivative of the constraint equation. This means
that once we have choosen a connection which satisfies the constraint equation
(4.9), the connection field equation $\mathcal{C}_{t}$ is automatically
satisfied. The equation $\mathcal{C}_{r}$ is not trivially satisfied and it
contains derivatives of $\Gamma^{r}{}_{rr}$. We therefore find that after
imposing (4.9), the field equations have the following structure:

New structure of metric field equations:
$\displaystyle\begin{pmatrix}\mathcal{M}_{tt}&0&0&0\\\
0&\mathcal{M}_{rr}&0&0\\\ 0&0&\mathcal{M}_{\theta\theta}&0\\\
0&0&0&\mathcal{M}_{\theta\theta}\,\sin^{2}\theta\end{pmatrix}$ New structure
of connection field equations: $\displaystyle\begin{pmatrix}0\\\
\mathcal{C}_{r}\\\ 0\\\ 0\end{pmatrix}$ (4.16)

This means there are at most three independent metric field equations and one
connection field equation for the dynamical variables
$\\{g_{tt},g_{rr},\Gamma^{r}{}_{rr}\\}$. Alternatively, one can trade
$\Gamma^{r}_{\ rr}$ for $\Gamma^{r}_{\ \theta\theta}$ and regard the latter as
the dynamical degree of freedom stemming from the connection.

The next natural step is to check the internal consistency of these equations.
This is necessary since the number of field equations is larger than the
number of dynamical variables and it is therefore not clear whether these
equations can be solved consistently. In subsection IV.3 we show the self-
consistency of the field equations and we will see on an abstract level that
they produce solutions which go beyond Schwarzschild-deSitter-Nordström.

However, before doing so, in subsection IV.2 we briefly discuss the
implications of $\mathcal{M}_{tr}=0$ for solution set 1. In particular, we
show that solution set 1 is not viable when looking for solutions to
$f(\mathbb{Q})$ gravity which go beyond GR.

###  IV.2 Off-diagonal metric field equation for solution set 1

In the previous subsection we have found that the off-diagonal metric field
equation can be solved in three distinct ways. The different options which
solve $\mathcal{M}_{tr}=0$ can be summarized as follows:

  1. 1.

$\partial_{r}\mathbb{Q}_{\textsf{sol}}=0$: If $\mathcal{M}_{tr}=0$ is solved
by $\partial_{r}\mathbb{Q}_{\textsf{sol}}=0$, or, in other words, if
$\mathbb{Q}$ is constant when evaluated on a solution of the field equations -
again one can see from the form of $\mathbb{Q}$ that this can be achieved by
solving $\mathbb{Q}=\mathbb{Q}_{\text{sol}}$ for $\Gamma^{r}_{\ rr}$ \- then
it follows that one can only find the Schwarzschild-deSitter-Nordström
solution for arbitrary $f$ (as long as $f$ satisfies $f^{\prime}\neq 0$, which
is required in order to obtain non-trivial field equations).

  2. 2.

$f^{\prime\prime}(\mathbb{Q}_{\textsf{sol}})=0$: If the second derivative of
$f$ is zero when evaluated on a solution of the field equations, and assuming
that $\partial_{r}\mathbb{Q}_{\textsf{sol}}\neq 0,$ then it follows that $f$
is an affine function; $f(\mathbb{Q})=a\,\mathbb{Q}+b$. This means we are in
the GR sector of $f(\mathbb{Q})$ gravity and it naturally follows that the
only solution is Schwarzschild-deSitter-Nordström.

  3. 3.

Constraint (4.9): The constraint equation eliminates two of the four free
functions of solution set $2$ and we are left with one potential degree of
freedom stemming from the connection; $\Gamma^{r}{}_{rr}$. This is the only
option which allows us to get solutions to $f(\mathbb{Q})$ gravity which go
beyond GR. But from section III.5 we know how to obtain solution set 1 from
set 2, in particular we have to set $\Gamma^{t}_{\ \theta\theta}=-1/c$ and
$k=c$ with now $c\neq 0$. (4.9) then becomes the equation $3c=0$, which cannot
be fulfilled. This is thus not an option.

Only the first two options are viable. Hence, solution set 1 can only give
rise to the Schwarzschild-deSitter-Nordström solution.

###  IV.3 Self-consistency of the remaining field equations for solution set 2

In subsection IV.1 we have seen that the constraint equation (4.9) is the only
possibility to solve $\mathcal{M}_{tr}=0$ which does not immediately force the
Schwarzschild-deSitter-Nordström solution on us. In particular, we have seen
that there are two options to solve the constraint (4.9). Both options and the
assumptions which go into them are summarized in Table 3 for convenience.

| The most general solution to equation (4.9)
---|---
| $\Gamma^{t}{}_{\theta\theta}=\frac{k}{2c(2c-k)}$
Option 1 |  |  |  $\text{for }c\neq 0\text{ and }k\neq 2c$
|
$\Gamma^{t}{}_{rr}=-\frac{k(8c^{2}+2ck-k^{2})}{8c^{2}(2c-k)^{2}(\Gamma^{r}{}_{\theta\theta})^{2}}$
| $c=k=0$
Option 2 |  |  |
|
$\Gamma^{t}{}_{rr}=-\frac{\Gamma^{t}{}_{\theta\theta}}{(\Gamma^{r}{}_{\theta\theta})^{2}}$
Table 3: A summary of the different choices of connection which satisfy the
constraint equation (4.9).

What we need to do now is to show that the remaining field equations are self-
consistent and that they can indeed produce solutions which go beyond GR. The
consistency of the equations is not immediately obvious since we have
potentially four independent equations for the three degrees of freedom
$\\{g_{tt},g_{rr},\Gamma^{r}{}_{rr}\\}$. Moreover, we wish to re-write the
equations in the simplest possible form so that they can, at least in
principle, be integrated. This needs to be done for each option in Table 3
individually, but the strategy to get to the final results is always the same.
This method works in particular when including the electro-vacuum.

  1. 1.

Choose one of the options from Table 3 and simplify all equations and the non-
metricity scalar $\mathbb{Q}$ using the chosen relations.

  2. 2.

Solve the $\mathbb{Q}$ scalar for $\Gamma^{r}{}_{rr}$. (This is the only weak
point of the strategy as there are scenarios in which this step fails. We
discuss the ramifications of this failure in the next subsection.)

  3. 3.

Replace every $\Gamma^{r}{}_{rr}$ in the metric and connection field equations
by the expression obtained in 2. In other words: We trade $\Gamma^{r}{}_{rr}$
in each equation for $\mathbb{Q}$ and we treat $\mathbb{Q}$ as a degree of
freedom. This step will produce much more compact expressions.

  4. 4.

For each option, the metric field equations $\mathcal{M}_{tt}$ and
$\mathcal{M}_{rr}$ only contain first order derivatives of the metric. Solve
these equations for $\partial_{r}g_{tt}$ and $\partial_{r}g_{rr}$. This will
produce expressions of the form222Strictly speaking, the functions $1$ and $2$
also depend on the matter fields as well as $f$ and its first two derivatives.
However, we will suppress these dependencies for all appearing functions here
and in the sequel for the sake of readability.

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=\text{function}_{1}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},CC)$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=\text{function}_{2}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},CC),$
(4.17)

where $CC$ stands for “Connection Components”.

  5. 5.

The metric field equation $\mathcal{M}_{\theta\theta}$ contains first order
derivatives of the metric and the second order derivative
$\partial^{2}_{r}g_{tt}$. Use (4) to eliminate the first oder derivatives and
then solve $\mathcal{M}_{\theta\theta}$ for $\partial^{2}_{r}g_{tt}$. This
leads to an expression of the form

$\displaystyle\partial^{2}_{r}g_{tt}$
$\displaystyle=\text{function}_{3}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},CC).$
(4.18)

  6. 6.

Since we solved $\mathbb{Q}$ for $\Gamma^{r}{}_{rr}$ and used the resulting
expression to simplify all field equations, the connection equation
$\mathcal{C}_{r}$ is a second order differential equation for $\mathbb{Q}$.
Moreover, it contains first order derivatives of the metric. Use (4) to get
rid of the first order derivatives of the metric and solve the connection
equation for $\partial^{2}_{r}\mathbb{Q}$.

  7. 7.

We can derive a consistency condition for the field equations by looking at
the equations (4) and (4.18). In fact, the $r$-derivative of
$\partial_{r}g_{tt}$ should be equal to $\partial^{2}_{r}g_{tt}$ and this
leads to the consistency condition

$\partial_{r}\text{function}_{1}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},CC)\overset{!}{=}\text{function}_{3}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},CC).$
(4.19)

Notice that $\text{function}_{1}$ and $\text{function}_{3}$ depend on the same
arguments, but the derivative operator on the left hand side generates an
expression of the form

$\partial_{r}\text{function}_{1}=\text{function}_{4}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},CC,\partial_{r}g_{tt},\partial_{r}g_{rr},\partial^{2}_{r}\mathbb{Q},\partial_{r}CC).$
(4.20)

The first order derivatives of the metric can be eliminated using (4) and the
derivatives of the connection components are all known: A quick look at Table
2 reveals that we know how to express all derivatives of connection
components, except $\partial_{r}\Gamma^{t}{}_{rr}$ and
$\partial_{r}\Gamma^{r}{}_{rr}$. But the options described in Table 3 all
allow us to express $\Gamma^{t}{}_{rr}$ in terms of other connection
components, and hence we also know how to express its derivative. Moreover,
$\Gamma^{r}{}_{rr}$ has been traded for $\mathbb{Q}$ in all equations, hence
there are no derivatives of $\Gamma^{r}{}_{rr}$ which we need to worry about
and the consistency condition (4.19) becomes

$\text{function}_{4}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},\partial^{2}_{r}\mathbb{Q},CC)\overset{!}{=}\text{function}_{3}(g_{tt},g_{rr},\mathbb{Q},\partial_{r}\mathbb{Q},CC).$
(4.21)

  8. 8.

Observe that the left hand side of the consistency condition (4.21) depends on
$\partial^{2}_{r}\mathbb{Q}$, while this term does not appear on the right
hand side. At this point, we need to use the connection field equation, which
we solved for $\partial^{2}_{r}\mathbb{Q}$ in step 6. Once we eliminate
$\partial^{2}_{r}\mathbb{Q}$ by the expression found in 6., we find that the
consistency condition (4.21) is satisfied for both options given in Table 3.

This strategy uses all field equations and it makes extensive use of the
properties of solution set 2. What it shows is that the field equations are
self-consistent and that there are only three independent field equations, not
four. The relevant field equations are $\mathcal{M}_{tt}$, $\mathcal{M}_{rr}$
and $\mathcal{C}_{r}$ , while $\mathcal{M}_{\theta\theta}$ is trivially
satisfied - as in GR - when the other three equations are satisfied. Moreover,
these equations determine the dynamical variables
$\\{g_{tt},g_{rr},\Gamma^{r}{}_{rr}\\}$, or, alternatively,
$\\{g_{tt},g_{rr},\mathbb{Q}\\}$ or
$\\{g_{tt},g_{rr},\Gamma^{r}{}_{\theta\theta}\\}$, as we will see shortly.
What remains to be done is to write out explicitly the equations for
$\partial_{r}g_{tt}$, $\partial_{r}g_{rr}$ and $\partial^{2}_{r}\mathbb{Q}$
derived from $\mathcal{M}_{tt}$, $\mathcal{M}_{rr}$ and $\mathcal{C}_{r}$ for
both options in Table 3. For option 1 we find

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=-\frac{g_{tt}\left(2+g_{rr}\left(\mathbb{Q}\,r^{2}-2-\frac{f(\mathbb{Q})\,r^{2}}{f^{\prime}(\mathbb{Q})}\right)\right)}{2r}$
$\displaystyle\phantom{=}+\frac{\left(\frac{(k-4c)^{2}g_{tt}\,r^{2}}{c(2c-k)\Gamma^{r}{}_{\theta\theta}}-4\Gamma^{r}{}_{\theta\theta}\,g_{rr}(2g_{tt}-c(2c-k)r^{2})\right)}{8\,r\,f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\,f^{\prime\prime}(\mathbb{Q})$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=\phantom{+}\frac{g_{rr}\left(2+g_{rr}\left(\mathbb{Q}\,r^{2}-2-\frac{f(\mathbb{Q})\,r^{2}}{f^{\prime}(\mathbb{Q})}\right)\right)}{2\,r}$
$\displaystyle\phantom{=}+\frac{g_{rr}\left(16r+\frac{(k-4c)^{2}\,r^{2}}{c(2c-k)\Gamma^{r}{}_{\theta\theta}}+\frac{4\Gamma^{r}{}_{\theta\theta}\,g_{rr}\,(2g_{tt}+c(2c-k)r^{2})}{g_{tt}}\right)}{8\,r\,f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\,f^{\prime\prime}(\mathbb{Q})$
(4.22)

$\displaystyle\partial^{2}_{r}\mathbb{Q}$
$\displaystyle=\frac{(\partial_{r}\mathbb{Q})}{2(4c(2c-k)\Gamma^{r}{}_{\theta\theta}^{2}g_{rr}(cr^{2}(2c-k)-2g_{tt})+r^{2}(k-4c)^{2}g_{tt})^{2}}\times$
$\displaystyle\times\bigg{(}\bigg{\\{}128c^{2}(k-2c)^{2}(\mathbb{Q}r^{2}-2)\Gamma^{r}{}_{\theta\theta}^{3}g_{tt}g_{rr}^{2}(cr^{2}(2c-k)-2g_{tt})$
$\displaystyle+16c^{3}r(2c-k)^{3}\Gamma^{r}{}_{\theta\theta}^{4}g_{rr}^{2}((\mathbb{Q}r^{2}-2)g_{rr}+6)(cr^{2}(2c-k)-2g_{tt})$
$\displaystyle+8cr(2c-k)(k-4c)^{2}\Gamma^{r}{}_{\theta\theta}^{2}g_{tt}g_{rr}(cr^{2}(2c-k)((\mathbb{Q}r^{2}-2)g_{rr}-6)+$
$\displaystyle
g_{tt}((6-3\mathbb{Q}r^{2})g_{rr}+6))+r^{3}(k-4c)^{4}g_{tt}^{2}((\mathbb{Q}r^{2}-2)g_{rr}-2)\bigg{\\}}$
$\displaystyle-
g_{rr}\,r^{3}\,\left((4c-k)^{2}g_{tt}-4c^{2}(2c-k)^{2}(\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}\right)\frac{f(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}$
$\displaystyle\phantom{=}+2\left(2(4c-k)^{2}\Gamma^{r}{}_{\theta\theta}\,g_{tt}\,g_{rr}\,r+(4c-k)^{2}\,g_{tt}\,r^{2}+4c(2c-k)(\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}\left(2g_{tt}-c(2c-k)\,r^{2}\right)\right)\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})$
$\displaystyle\phantom{=}-2\left((4c-k)^{2}g_{tt}\,r^{2}-4c(2c-k)(\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}\left(2g_{tt}-c(2c-k)r^{2}\right)\right)\frac{f^{(3)}(\mathbb{Q})}{f^{\prime\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\bigg{)}.$
(4.23)

Observe that the only connection coefficients which appear in these equations
are $c$, $k$, and $\Gamma^{r}{}_{\theta\theta}$. The constants $c$ and $k$ are
freely specifiable, but the function $\Gamma^{r}{}_{\theta\theta}$ is fixed,
up to an integration constant, by the differential equation

$\partial_{r}\Gamma^{r}{}_{\theta\theta}=\frac{1}{2}+\frac{k}{4c}-\frac{3}{2c-k}-\Gamma^{r}{}_{rr}\Gamma^{r}{}_{\theta\theta}.$
(4.24)

This differential equation follows from the last row of Table 2 when
specialized to option 1 of Table 3. Also, recall that we traded
$\Gamma^{r}{}_{rr}$ for $\mathbb{Q}$. Hence, if we replace $\Gamma^{r}{}_{rr}$
by its expression in terms of $\mathbb{Q}$ we obtain a highly non-linear
differential equation, which nonetheless determines
$\Gamma^{r}{}_{\theta\theta}$ (at least in principle) in terms of the
dynamical variables $\\{g_{tt},g_{rr},\mathbb{Q}\\}$ and in terms of an
integration constant. Hence, for the field equations (IV.3) to produce a
solution it is necessary to choose three constants, $c$, $k$, and the
integration constant in (4.24), as well as initial conditions for the
dynamical degrees of freedom. Since the metric field equations are first
order, we need a total of two initial conditions for the metric and we need
two initial conditions for $\mathbb{Q}$, given that its differential equation
is second order.

The field equations derived from option 2 have a simpler and more compact form
and they require less specifications in order to be solved:

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=-\frac{g_{tt}\left(2+g_{rr}\left(\mathbb{Q}\,r^{2}-2-\frac{f(\mathbb{Q})\,r^{2}}{f^{\prime}(\mathbb{Q})}\right)\right)}{2r}+\frac{g_{tt}\left(r^{2}-g_{rr}(\Gamma^{r}{}_{\theta\theta})^{2}\right)}{\Gamma^{r}{}_{\theta\theta}\,r\,f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\,f^{\prime\prime}(\mathbb{Q})$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=\phantom{+}\frac{g_{rr}\left(2+g_{rr}\left(\mathbb{Q}\,r^{2}-2-\frac{f(\mathbb{Q})\,r^{2}}{f^{\prime}(\mathbb{Q})}\right)\right)}{2\,r}+\frac{g_{rr}\left((\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}+2\,r\,\Gamma^{r}{}_{\theta\theta}+r^{2}\right)}{\Gamma^{r}{}_{\theta\theta}\,r\,f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\,f^{\prime\prime}(\mathbb{Q})$
(4.25)

$\displaystyle\partial^{2}_{r}\mathbb{Q}$
$\displaystyle=\frac{(\partial_{r}\mathbb{Q})}{2\left[(\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}-r^{2}\right]}\left(2r+g_{rr}\left(4\Gamma^{r}{}_{\theta\theta}\left(1+\Gamma^{r}{}_{rr}\Gamma^{r}{}_{\theta\theta}\right)+2r-\mathbb{Q}\,r^{3}\right)+g_{rr}\,r^{3}\frac{f(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}\right.$
$\displaystyle\phantom{=}\left.-2\left((\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}+2\Gamma^{r}{}_{\theta\theta}\,g_{rr}\,r+r^{2}\right)\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})-\left((\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}-r^{2}\right)\frac{f^{(3)}(\mathbb{Q})}{f^{\prime\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\right)$
(4.26)

Since option 2 demands $c=k=0$, the only connection component which now
appears in the field equations is $\Gamma^{r}{}_{\theta\theta}$, which is
implicitly fixed by the differential equation

$\partial_{r}\Gamma^{r}{}_{\theta\theta}=-1-\Gamma^{r}{}_{rr}\Gamma^{r}{}_{\theta\theta}.$
(4.27)

This equation is again derived from the last row of Table 2 when specialized
to option 2 of Table 3. After replacing $\Gamma^{r}{}_{rr}$ by its expression
in terms of $\mathbb{Q}$ we find the highly non-linear differential equation

$\partial_{r}\Gamma^{r}{}_{\theta\theta}=-\frac{\Gamma^{r}{}_{\theta\theta}\left(f(\mathbb{Q})\,g_{rr}\,r^{2}(\Gamma^{r}{}_{\theta\theta}+r)+(2r-g_{rr}(2\Gamma^{r}{}_{\theta\theta}+r)(\mathbb{Q}\,r^{2}-1))f^{\prime}(\mathbb{Q})\right)}{2\left((\Gamma^{r}{}_{\theta\theta})^{2}\,g_{rr}-r^{2}\right)f^{\prime}(\mathbb{Q})}$
(4.28)

In principle, this equation determines $\Gamma^{r}{}_{\theta\theta}$ in terms
of $\\{g_{tt},g_{rr},\mathbb{Q}\\}$ and an integration constant. The origin of
the integration constant can also be understood in a different way: Instead of
treating $\Gamma^{r}{}_{rr}$ or $\mathbb{Q}$ as a degree of freedom, we can
regard $\Gamma^{r}{}_{\theta\theta}$ as the degree of freedom stemming from
the connection. By solving the above differential equation for $\mathbb{Q}$
and plugging the result into the connection field equation for option 2, we
obtain a differential equation for $\Gamma^{r}{}_{\theta\theta}$. This
differential equation is now third order, and hence we need to specify three
initial conditions for $\Gamma^{r}{}_{\theta\theta}$.

We therefore find that for option 2 we need to specify initial conditions for
$\\{g_{tt},g_{rr},\mathbb{Q}\\}$ ($1+1+2$) and an integration constant for
(4.28), or, alternatively, provide initial conditions for
$\\{g_{tt},g_{rr},\Gamma^{r}{}_{\theta\theta}\\}$ ($1+1+3$). This latter point
of view with a third order differential equation for
$\Gamma^{r}{}_{\theta\theta}$ will be particularly useful in section V, where
we derive approximate and exact solutions for $f(\mathbb{Q})$ gravity.

Before doing so, however, we show that there is also a different strategy to
tackle the symmetry reduced field equations. More precisely, we show that the
caveat alluded to in step 2 of the above strategy opens up a rout to solve the
field equations in a different way. Namely, the connection can be determined
through a constraint equation, while the metric remains dynamical. This will
nevertheless lead to solutions which go beyond GR.

###  IV.4 Reduction of the field equations by a constraint on the connection

In the previous subsection we saw that there are two dynamical equations for
the metric and one dynamical equation for the connection. However, this
conclusion hinges crucially on the validity of step 2, which requires us to
solve $\mathbb{Q}$ for $\Gamma^{r}{}_{rr}$. A closer examination of the non-
metricity scalar $\mathbb{Q}$ reveals that this is only possible if the
coefficient in front of $\Gamma^{r}{}_{rr}$ is different from zero.
Concretely, one finds that $\mathbb{Q}$ takes the schematic form

$\mathbb{Q}=\begin{cases}\frac{1}{4}\left(\frac{(4c-k)^{2}}{c\,(2c-k)\,g_{rr}}-\frac{4(\Gamma^{r}{}_{\theta\theta})^{2}\left(2g_{tt}-c\,(2c-k)\,r^{2}\right)}{g_{tt}\,r^{2}}\right)\frac{\Gamma^{r}{}_{rr}}{\Gamma^{r}{}_{\theta\theta}}+\textsf{other
terms}&\text{for option 1}\\\
2\left(\frac{1}{g_{rr}}-\frac{(\Gamma^{r}{}_{\theta\theta})^{2}}{r^{2}}\right)\frac{\Gamma^{r}{}_{rr}}{\Gamma^{r}{}_{\theta\theta}}+\textsf{other
terms}&\textsf{for option 2}.\end{cases}$ (4.29)

If we now impose the constraint that the factor in front of
$\Gamma^{r}{}_{rr}$ vanishes, i.e., if we impose the condition that
$\Gamma^{r}{}_{\theta\theta}$ is of the form333The signature of the metric
demands that $g_{tt}<0$ and hence $\sqrt{-g_{tt}}$ is real for option 1.

$\Gamma^{r}{}_{\theta\theta}=\begin{cases}\pm\frac{|4c-k|\,\sqrt{-g_{tt}}\,r}{\sqrt{4c\,(2c-k)\,g_{rr}\,(c\,(2c-k)\,r^{2}-2g_{tt})}}&\textsf{for
option 1}\\\ \pm\frac{r}{\sqrt{g_{rr}}}&\textsf{for option 2}\end{cases}$
(4.30)

then it is no longer possible to trade $\Gamma^{r}{}_{rr}$ for $\mathbb{Q}$
and the strategy presented in the previous subsection does not work anymore.
However, this does not mean that the field equations become inconsistent. In
fact, if $\Gamma^{r}{}_{\theta\theta}$ has been chosen to have one of the
forms of (4.30), then it can be shown that the connection field equation
$\mathcal{C}_{r}$ is trivially satisfied. We are thus left with the metric
field equations and they turn out to be self-consistent. The strategy to show
self-consistency is as follows, which again holds when including the electro-
vacuum.

  1. 1.

Choose either option 1 or option 2 from Table 3 and fix
$\Gamma^{r}{}_{\theta\theta}$ by choosing the appropriate expression from
(4.30) for any choice of sign.

  2. 2.

Solve $\mathcal{M}_{tt}$ and $\mathcal{M}_{rr}$ for $\partial_{r}g_{tt}$ and
$\partial_{r}g_{rr}$. This gives rise to expressions of the form

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=\textsf{function}_{1}(g_{tt},g_{rr},c,k)$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=\textsf{function}_{2}(g_{tt},g_{rr},c,k,\partial_{r}\mathbb{Q}),$
(4.31)

where $c$ and $k$ only appear for option 1 and where we have suppressed the
dependence of the two functions on matter fields as well as the function $f$
and its derivatives. Observe that only $\partial_{r}g_{rr}$ depends on
$\partial_{r}\mathbb{Q}$.

  3. 3.

Take the expression for $\mathbb{Q}$ and simplify it using the relations of
option 1/option 2, the corresponding form of $\Gamma^{r}{}_{\theta\theta}$
from (4.30) as well as the expressions obtained in step 2. This leads to an
expression of the form

$\mathbb{Q}=\textsf{function}_{3}(g_{tt},g_{rr},\partial_{r}g_{tt},c,k).$
(4.32)

  4. 4.

Observe that only $\partial_{r}g_{tt}$ appears in $\textsf{function}_{3}$.
Hence, if we replace $\partial_{r}g_{tt}$ by $\textsf{function}_{1}$ and then
take the $r$-derivative of $\mathbb{Q}$, we obtain an equation which we can
solve for $\partial_{r}\mathbb{Q}$. This leads to the schematic expression

$\partial_{r}\mathbb{Q}=\textsf{function}_{4}(g_{tt},g_{rr},c,k),$ (4.33)

which in turn can be used to replace $\partial_{r}\mathbb{Q}$ in
$\textsf{function}_{2}$. This gives us

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=\textsf{function}_{1}(g_{tt},g_{rr},c,k)$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=\widetilde{\textsf{function}}_{2}(g_{tt},g_{rr},c,k).$ (4.34)

  5. 5.

Plugging the above expressions for $\partial_{r}g_{tt}$ and
$\partial_{r}g_{rr}$ into the remaining metric field equation,
$\mathcal{M}_{\theta\theta}$, does not yield anything new. The equation
$\mathcal{M}_{\theta\theta}$ is trivially satisfied. With this, we have
exhausted all field equations and we have shown their self-consistency even in
the case where the connection component $\Gamma^{r}{}_{\theta\theta}$ is fixed
by a constraint equations, rather than by a dynamical field equation.

In summary, we find that we can fix the connection component
$\Gamma^{r}{}_{\theta\theta}$ through one of the constraints in (4.30) and we
are then left with two dynamical equations for the metric. To solve these
equations, we need to specify initial data for $g_{tt}$ and $g_{rr}$ and, in
the case of option 1, we also need to specify $c$ and $k$.

Also, notice that if we choose to solve the field equations using the
constraints (4.30), we need less initial data than when we let the connection
be dynamical.

In the next section we will derive approximate and exact solutions for both, a
dynamically determined connection and one fixed by the constraints. We will
focus on option 2 because the equations are more compact and simpler444We have
not found any (approximate) solutions when including $c$ or $k$, but it would
be interesting to see what role these constants play and how they affect
solutions. due to the absence of $c$ and $k$.

##  V Approximate and exact solutions

Having shown that we have a self-consistent set of field equations, it is now
time to look for solutions. To that end it is sensible to choose option 2 from
Table 3 since the field equations (IV.3) have a simpler form than (IV.3) and
they do not depend on an arbitrary choice of $c$ and $k$ – they only need the
specification of initial conditions in order to produce well-defined
solutions. Moreover, we want to use the spherical connection considered in
Zhao:2021 ; Lin:2021  as a partial guide line toward finding solutions beyond
the GR solutions. The idea is that the spherical connection of Zhao:2021 ;
Lin:2021  is already known to produce the GR solutions for arbitrary $f$ and a
controlled deformation of that connection could therefore lead to deformations
of the GR solutions for a given choice of $f$. We will explain how this can be
achieved in a perturbative fashion in the next subsection.

However, before doing so, we want to consider the exact spherical connection
used in Zhao:2021 ; Lin:2021  in order to explicitly show that it can only
produce GR solutions for arbitrary $f$ and because this provides a concrete
example for the fact that solutions with
$\mathbb{Q}_{\textsf{sol}}=\textsf{const}$. do exist.

As already noted in subsection III.4, the spherical connection is obtained by
setting
$\\{c,k,\Gamma^{t}{}_{rr},\Gamma^{t}{}_{\theta\theta},\Gamma^{r}{}_{rr},\Gamma^{r}{}_{\theta\theta}\\}=\\{0,0,0,0,0,-r\\}$.
The only non-zero connection coefficients are then explicitly given by

$\displaystyle\Gamma^{r}{}_{\theta\theta}$ $\displaystyle=-r$
$\displaystyle\Gamma^{r}{}_{\phi\phi}$ $\displaystyle=-r\,\sin^{2}\theta$
$\displaystyle\Gamma^{\theta}{}_{r\theta}$ $\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma^{\theta}{}_{\phi\phi}$
$\displaystyle=-\cos\theta\,\sin\theta$ $\displaystyle\Gamma^{\phi}{}_{r\phi}$
$\displaystyle=\frac{1}{r}$ $\displaystyle\Gamma^{\phi}{}_{\theta\phi}$
$\displaystyle=\cot\theta.$ (5.1)

This means that the spherical connection falls into solution set 2 and that it
corresponds to option 2 in Table 3. Hence, we can simply insert this
connection into the field equations (IV.3) and we obtain

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=-\frac{g_{tt}\left(2+g_{rr}\left(\mathbb{Q}\,r^{2}-2-\frac{f(\mathbb{Q})\,r^{2}}{f^{\prime}(\mathbb{Q})}\right)\right)}{2r}-g_{tt}\left(1-g_{rr}\right)(\partial_{r}\mathbb{Q})\,\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=\phantom{+}\frac{g_{rr}\left(2+g_{rr}\left(\mathbb{Q}\,r^{2}-2-\frac{f(\mathbb{Q})\,r^{2}}{f^{\prime}(\mathbb{Q})}\right)\right)}{2\,r}+g_{rr}\left(1-g_{rr}\right)(\partial_{r}\mathbb{Q})\,\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}$
$\displaystyle\partial^{2}_{r}\mathbb{Q}$
$\displaystyle=\frac{1}{2}(\partial_{r}\mathbb{Q})\left(\frac{g_{rr}^{2}(\mathbb{Q}\,r^{2}-2)+g_{rr}(4+\mathbb{Q}\,r^{2})-2}{(g_{rr}-1)^{2}\,r}+\frac{g_{rr}\,r}{g_{rr}-1}\frac{f(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}+2\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})-2\frac{f^{(3)}(\mathbb{Q})}{f^{\prime\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\right).$
(5.2)

If we multiply the first equation by $g_{rr}$ and the second one by $g_{tt}$
and then add the two multiples together we obviously obtain

$\displaystyle g_{rr}\partial_{r}g_{tt}+g_{tt}\partial_{r}g_{rr}$
$\displaystyle=0$ $\displaystyle\Longleftrightarrow$ $\displaystyle
g_{tt}g_{rr}$ $\displaystyle=\textsf{const}.$ (5.3)

This is already a first indication that we will obtain the GR solution. If we
now use the spherical connection to compute the non-metricity scalar
$\mathbb{Q}$ we find

$\mathbb{Q}=-\frac{g_{rr}-1}{g_{tt}\,g_{rr}^{2}\,r}\left(g_{rr}\partial_{r}g_{tt}+g_{tt}\partial_{r}g_{rr}\right).$
(5.4)

Hence, if we evaluate $\mathbb{Q}$ on a solution of the field equations, which
imply that $g_{rr}\partial_{r}g_{tt}+g_{tt}\partial_{r}g_{rr}=0$, we obtain
$\mathbb{Q}_{\textsf{sol}}=0$ from (5.4). In turn this implies that the
connection equation in (V) is trivially satisfied. Moreover, the metric field
equations in (V) reduce to

$\displaystyle\partial_{r}g_{tt}$
$\displaystyle=-\frac{g_{tt}\left(1-g_{rr}\left(1+\frac{f(0)\,r^{2}}{2f^{\prime}(0)}\right)\right)}{r}$
$\displaystyle\partial_{r}g_{rr}$
$\displaystyle=\phantom{+}\frac{g_{rr}\left(1-g_{rr}\left(1+\frac{f(0)\,r^{2}}{2f^{\prime}(0)}\right)\right)}{r}.$
(5.5)

As anticipated, this shows explicitly that solutions with
$\mathbb{Q}_{\textsf{sol}}=\textsf{const}.$ exist and that they can only
produce the Schwarzschild-deSitter-Nordström solution555Our considerations
also hold for the electro-vacuum but we stuck to the pure vacuum case for
simplicity. for an arbitrary choice of $f$. Indeed, upon integrating the above
differential equations one finds

$\displaystyle g_{tt}$
$\displaystyle=c_{2}+\frac{c_{1}\,c_{2}}{r}+\frac{c_{2}}{6}\frac{f(0)}{f^{\prime}(0)}r^{2}\equiv
c_{2}+\frac{c_{1}\,c_{2}}{r}+\frac{c_{2}\,\Lambda_{\textsf{eff}}}{3}r^{2}$
$\displaystyle g_{rr}$ $\displaystyle=\frac{1}{c_{2}\,g_{tt}},$ (5.6)

where we have used that
$\frac{1}{2}\frac{f(0)}{f^{\prime}(0)}=:\Lambda_{\textsf{eff}}$ acts as an
effective cosmological constant, as we have explained in subsection IV.1. This
example will also be the point of departure to construct approximate solutions
to $f(\mathbb{Q})$ gravity which go beyond GR but which reduce to the GR
solutions in an appropriate limit. The key observation is that the metric
field equations for option 2 (cf. equation (IV.3)) imply that

$\displaystyle\frac{g_{rr}\partial_{r}g_{tt}+g_{tt}\partial_{r}g_{rr}}{g_{tt}g_{rr}}=\frac{\partial_{r}(g_{tt}g_{rr})}{g_{tt}g_{rr}}=2\frac{\Gamma^{r}{}_{\theta\theta}+r}{\Gamma^{r}{}_{\theta\theta}}\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q}).$
(5.7)

This equation can easily be integrated and one obtains

$\displaystyle
g_{tt}g_{rr}=c_{1}\,\exp\left(2\int\mathrm{d}r\,\frac{\Gamma^{r}{}_{\theta\theta}+r}{\Gamma^{r}{}_{\theta\theta}}\frac{f^{\prime\prime}(\mathbb{Q})}{f^{\prime}(\mathbb{Q})}(\partial_{r}\mathbb{Q})\right),$
(5.8)

where $c_{1}$ is an integration constant chosen666It follows from $\det g\neq
0$ and the signature of the metric that $g_{tt}g_{rr}$ is strictly negative.
such that $g_{tt}g_{rr}<0$. This equation shows again that the spherical
connection, which imposes $\Gamma^{r}{}_{\theta\theta}=-r$, reproduces the GR
relation $g_{tt}\propto\frac{1}{g_{rr}}$. Unsurprisingly, it also shows that
$\partial_{r}\mathbb{Q}=0$ or $f^{\prime\prime}(\mathbb{Q})$ produce this
relation. However, what is more important to us, is that this equation
suggests that we can consider a deformation of the spherical connection where
$\Gamma^{r}{}_{\theta\theta}=-r+\gamma(r)$. This would lead to a deviation of
the typical $g_{tt}\propto\frac{1}{g_{rr}}$ behavior of GR (although we will
not actually see this in all of the approximate solution derived in the
following subsections because the difference between $g_{tt}$ and $-1/g_{rr}$
are sometimes hidden in higher order perturbations). In the next subsection we
will make this idea more precise and show that it is possible to obtain an
approximate solution for the ansatz
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$, where $\alpha$ is assumed
to be a small parameter.

We also report some results for
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{\kappa}$ for integer $\kappa\geq
2$ in Appendix B, which generalize some of the results derived below.

###  V.1 Approximate vacuum solution beyond GR for
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$

Since we wish to consider the deformation
$\Gamma^{r}{}_{\theta\theta}=-r+\gamma(r)$, it is convenient to solve the
differential equation (4.28) for $\mathbb{Q}$ and plugging the resulting
expression into the differential equation for $\partial^{2}_{r}\mathbb{Q}$.
This results in a third order differential equation for $\gamma(r)$. Moreover,
this equation also contains terms proportional to $f^{(3)}(\mathbb{Q})$. Given
the complexity of this equation, it is sensible to first consider the ansatz
$f(\mathbb{Q})=\mathbb{Q}+\alpha\,\mathbb{Q}^{2}$, which gets rid of terms
proportional to $f^{(3)}(\mathbb{Q})$. Furthermore, we consider the pure
vacuum case with vanishing cosmological constant.

In the sequel we wish to consider $\alpha$ as being a small parameter, i.e.,
$|\alpha|\ll 1$. We can therefore expect that this ansatz will lead to small
deviations from the GR solutions and we choose the ansätze

$\displaystyle g_{tt}$
$\displaystyle=g_{tt}^{(0)}+\alpha\,g_{tt}^{(1)}+\alpha^{2}\,g_{tt}^{(2)}$
$\displaystyle g_{rr}$
$\displaystyle=g_{rr}^{(0)}+\alpha\,g_{rr}^{(1)}+\alpha^{2}\,g_{rr}^{(2)}$
$\displaystyle\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=-r+\alpha\,\gamma^{(1)}+\alpha^{2}\,\gamma^{(2)},$ (5.9)

where $g_{tt}^{(0)}$ and $g_{rr}^{(0)}$ are given by the Schwarzschild
solution:

$\displaystyle g_{tt}^{(0)}$ $\displaystyle=-\left(1-\frac{2M}{r}\right)$ and
$\displaystyle g_{rr}^{(0)}$ $\displaystyle=-\frac{1}{g_{tt}^{(0)}}.$ (5.10)

We include second order terms in $\alpha$ because, as we will see shortly, in
the zeroth and first order equations, the metric and the connection equations
decouple from each other. Only at second order do we find coupled equations
which imply that the connection influences the metric and leads to what we dub
a “connection hair”.

The next step is to plug these ansätze into the field equations and to solve
them order by order. At zeroth order we find, unsurprisingly, that the metric
as well as the connection field equations are trivially satisfied. At first
order in $\alpha$ we find that the metric field equations reduce to

$\displaystyle\partial_{r}g_{tt}^{(1)}$
$\displaystyle=\frac{2M\,r\,g_{tt}^{(1)}-(r-2M)^{2}\,g_{rr}^{(1)}}{(r-2M)\,r^{2}}$
$\displaystyle\partial_{r}g_{rr}^{(1)}$
$\displaystyle=\frac{(2M+r)\,g_{rr}^{(1)}}{(2M-r)\,r}$ (5.11)

while the connection equation is identically fulfilled777Notice that at zeroth
order in $\alpha$ we have symmetric teleparallelism, where the connection and
the metric or not only independent, but the connection is completely
arbitrary. That is why at zeroth order there is no equation for the
connection, there is only an identity. At first order we do get an equation
for the connection, but this equation is identically satisfied for
$\gamma^{(0)}=-r$. Only at second order can we expect something interesting
for the connection.. As anticipated, the metric field equations do not depend
on the connection and the connection field equations do not depend on the
metric. The above equations can easily be integrated and one finds

$\displaystyle g_{tt}^{(1)}$ $\displaystyle=\frac{c_{2}+c_{1}\,(r-2M)}{r}$
$\displaystyle g_{rr}^{(1)}$ $\displaystyle=\frac{c_{2}\,r}{(r-2M)^{2}},$
(5.12)

where $c_{i}$ are real integration constants. We set $c_{1}=0$ in order to
obtain an asymptotically flat solution in the sense that
$\lim_{r\to\infty}g_{tt}=-1$ and $\lim_{r\to\infty}g_{rr}=1$. At this point
one notices that the only effect of the perturbations is to renormalize the
mass, in the sense that we can write the full metric at first order in
$\alpha$ as

$\displaystyle g_{tt}$ $\displaystyle=-1+\frac{2M_{\text{ren}}}{r}$
$\displaystyle g_{rr}$ $\displaystyle=-\frac{1}{g_{rr}},$ (5.13)

where the renormalized mass is defined as

$2M_{\text{ren}}:=2M+\alpha\,c_{2}.$ (5.14)

Since the mass $M$ appears in the Schwarzschild solution only as an
integration constant, the only physically observable mass is $M_{\text{ren}}$.
We now move to the second order and we find that the metric equations now do
depend on $\gamma^{(1)}$. After inserting the solutions for $g_{tt}^{(1)}$,
and $g_{rr}^{(1)}$ into the second order equations we find that they can be
written as

$\displaystyle\partial_{r}g_{tt}^{(2)}$
$\displaystyle=-\frac{-\frac{2Mr^{5}g_{tt}^{(2)}}{r-2M}+g_{rr}^{(2)}r^{4}(r-2M)-\frac{c_{2}r^{5}(c_{2}+c_{2}(r-2M))}{(r-2M)^{2}}+16M^{2}(4\gamma^{(1)}-4r\partial_{r}{\gamma^{(1)}}+r^{2}\partial_{r}^{2}{\gamma^{(1)}})}{r^{6}}$
$\displaystyle\partial_{r}g_{rr}^{(2)}$
$\displaystyle=\frac{-c_{2}^{2}r^{5}-g_{rr}^{(2)}(r+2M)r^{3}(r-2M)^{3}-16M^{2}(r-2M)^{2}(4\gamma^{(1)}-4r\partial_{r}{\gamma^{(1)}}+r^{2}\partial_{r}^{2}{\gamma^{(1)}})}{r^{4}(r-2M)^{4}},$
(5.15)

while for the connection field equations at second order we find the non-
trivial equation

$16\gamma^{(1)}-16r\,\partial_{r}\gamma^{(1)}+6r^{2}\,\partial^{2}_{r}\gamma^{(1)}-r^{3}\,\partial^{3}_{r}\gamma^{(1)}=0.$
(5.16)

The connection field equation can indeed be integrated and we find the
solution

$\gamma^{(1)}=r\,\left(c_{5}+c_{6}\,r^{3}+c_{7}\,r^{3}\ln(r)\right).$ (5.17)

After plugging the above solution for $\gamma^{(1)}$ into the metric field
equations, we are able to integrate them and we find the second order
contributions to the metric:

$\displaystyle g_{tt}^{(2)}$
$\displaystyle=\frac{-c_{2}c_{1}+c_{3}-2Mc_{4}-48M^{2}c_{6}-16M^{2}c_{7}-48M^{2}c_{7}\ln(r)+c_{4}r}{r}$
$\displaystyle g_{rr}^{(2)}$
$\displaystyle=\frac{r(c_{2}^{2}+(c_{3}-16M^{2}(3c_{6}+c_{7}))(r-2M)-48M^{2}c_{7}\ln(r)(r-2M)}{(r-2M)^{3}}.$
(5.18)

Notice that $c_{5}$ does not appear in the metric. In order to maintain
asymptotic flatness, i.e., $\lim_{r\to\infty}g_{tt}=-1$ and
$\lim_{r\to\infty}g_{rr}=1$, we need to set $c_{4}=0$. In particular, we
obtain a new beyond GR logarithmic correction coming from the connection
$\gamma^{(1)}$ – i.e., we obatin a “connection hair”. The full metric
components can be written as

$\displaystyle g_{tt}$
$\displaystyle=-\left(1-\frac{2M_{\textsf{ren}}}{r}\right)+\alpha^{2}\frac{\mu}{r}\ln\left(\frac{r}{r^{*}}\right)$
$\displaystyle g_{rr}$ $\displaystyle=-\frac{1}{g_{tt}}.$ (5.19)

The renormalized mass $M_{\text{ren}}$ is now given by

$2M_{\textsf{ren}}:=2M+\alpha\,c_{2}+\alpha^{2}\left(c_{3}-16M^{2}(3c_{6}+c_{7})\right),$
(5.20)

where the second equation holds up to order $\alpha^{2}$. The scale $r^{*}$
can be introduced by a shift in the constant $c_{6}\rightarrow
c_{6}-48M^{2}c_{7}\ln(r^{*})$, in order to have a dimensionless argument in
the logarithm. We have also defined a new scale

$\displaystyle\mu$ $\displaystyle:=48\,M^{2}\,c_{7},$ (5.21)

which characterizes the strength of the beyond GR correction – a new “black