III | $\neq 0$ | $\neq 0$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $\neq 0$ |  $5$
IV | $\neq 0$ | $\neq 0$ | $\neq 0$ | $\neq 0$ |  ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $1$
V (STEGR) | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ |  N/A |  $4$
VI | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  N/A |  $6$
VII | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $\neq 0$ |  $8$
VIII | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | $\neq 0$ |  ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $4$
IX | $\neq 0$ | $\neq 0$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $6$
X | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  N/A |  $10$
XI | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $9$
Table 2: Each primary sector (first column) is defined by the vanishing of
certain combinations of the parameters $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, and
$c_{5}$ (columns two through six). For brevity, and following DAmbrosio:2020 ,
we have defined $\tilde{c}\coloneqq c_{1}+c_{2}+c_{3}+c_{4}+c_{5}$,
$\hat{c}\coloneqq 2c_{1}+c_{2}+c_{4}$, and $c_{35}\coloneqq 2c_{3}+c_{5}$. The
vanishing of these parameter combinations (or the vanishing of combinations
thereof) correspond to the appearance of one or more primary constraints (the
number of independent primary constraints is shown in the last column). STEGR
is contained in sector V.

Notice that in sector V, which harbours STEGR as a special case, the number of
primary constraints matches the one of GR. However, just as in TG, it is
currently unknown in which sector secondary constraints occur and how many
there are. Hence, the exact number of degrees of freedom is not known for most
sectors. At most, we can currently say that sector $0$ propagates ten degrees
of freedom, but it is also a highly pathological theory. Sector X propagates
no degrees of freedom, while sector XI has less degrees of freedom than GR.
Both sectors are therefore uninteresting. Finally, sector V contains STEGR,
which has two degrees of freedom, but it is unclear whether other theories
with a different number of degrees of freedom can co-habitate that sector.

The reason these questions have remained unanswered thus far is because of
challenges posed by the Dirac-Bergmann algorithm, as mentioned in the previous
subsection. These challenges seem to afflict all teleparallel theories of
gravity, as has recently been argued in DAmbrosio:2023 , and the development
of new methods—or at least the exploration of other known methods but applied
to teleparallel theories—seems to be necessary.

Finally, we remark that STEGR distinguishes itself from the other possible
theories within the five-parameter family by having an enhanced set of
symmetries. In BeltranJimenez:2019c , perturbations around Minkowski space
were studied. The perturbative ansatz $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$,
with $|h_{\mu\nu}|\ll 1$, leads to the quadratic Lagrangian

$\displaystyle
L=c_{1}\partial_{\alpha}h_{\mu\nu}\partial^{\alpha}h^{\mu\nu}+\left(c_{2}+c_{4}\right)\partial_{\alpha}h_{\mu\nu}\partial^{\mu}h^{\alpha\nu}+c_{3}\partial_{\alpha}h\partial^{\alpha}h+c_{5}\partial_{\mu}h^{\mu}{}_{\nu}\partial^{\nu}h\,,$
(5.25)

where $h\coloneqq\eta^{\mu\nu}h_{\mu\nu}=h^{\mu}{}_{\mu}$ is the trace of the
perturbations. This is nothing but the most general Lagrangian for a spin-$2$
field. As is well known from the Fierz-Pauli analysis of this Lagrangian,
symmetries have to be imposed in order to remove ghostly degrees of freedom.
Demanding that the theory is invariant under

$\displaystyle h_{\mu\nu}\qquad\mapsto\qquad
h_{\mu\nu}+2\partial_{(\mu}\xi_{\nu)}\qquad\text{ for some vector $\xi^{\mu}$
which satisfies }\partial_{\mu}\xi^{\mu}=0\,,$ (5.26)

so-called transversal diffeomorphisms, leads to the condition

$\displaystyle 2c_{1}+c_{2}+c_{4}=0\,,$ (5.27)

which is indeed satisfied by the STEGR parameters101010This is simply the
condition $\hat{c}=0$, which defines Sector II and which is also a part of
Sector V.. In order to recover the two propagating degrees of freedom of a
massless spin-$2$ field, one can further impose linearized diffeomorphisms,

$\displaystyle h_{\mu\nu}\qquad\mapsto\qquad
h_{\mu\nu}+2\partial_{(\mu}\xi_{\nu)}\,,$ (5.28)

where the vector field $\xi^{\mu}$ is now unrestricted. This leads to

$\displaystyle 2c_{1}=-2c_{3}=c_{5}\,.$ (5.29)

Both conditions taken together then imply

$\displaystyle c_{3}$ $\displaystyle=-c_{1},$ $\displaystyle c_{4}$
$\displaystyle=-2c_{1}-c_{2},$ $\displaystyle c_{5}=2c_{1}\,,$ (5.30)

which is equivalent to

$\displaystyle\tilde{c}$ $\displaystyle=0,$ $\displaystyle\hat{c}$
$\displaystyle=0,$ $\displaystyle c_{35}$ $\displaystyle=0\,.$ (5.31)

These are precisely the defining equations of sector V in Table 2. STEGR,
which inhabits sector V, is therefore distinguish through its symmetries and
the healthy degrees of freedom it propagates. Instead of imposing linearized
diffeomorphisms, one could have also imposed the linearized Weyl symmetry

$\displaystyle h_{\mu\nu}\qquad\mapsto\qquad
h_{\mu\nu}+\phi\,\eta_{\mu\nu}\,,$ (5.32)

for some arbitrary scalar field and in addition to the transverse
diffeomorphisms (5.26). Demanding this symmetry implies

$\displaystyle c_{3}$ $\displaystyle=-\frac{3}{8}c_{1},$ $\displaystyle c_{5}$
$\displaystyle=2c_{1}\,.$ (5.33)

This describes a linearized version of unimodular gravity, which is
essentially GR plus the constraint $\sqrt{|g|}=1$. As a consequence, in
unimodular gravity the cosmological constant emerges from an integration
constant Alvarez:2023 . Notice that Sector V does not respect the linearized
Weyl symmetry. This symmetry only seems to be respected by Sector IX, which
has nothing to do with STEGR or GR. However, it should be pointed out that the
classification was obtained without restricting the metric through the
condition $\sqrt{|g|}=1$.

###  5.3 Non-Linear Extensions: $f(\mathcal{R})$, $f(\mathbb{T})$,
$f(\mathbb{Q})$ and $f(\mathbb{G})$ Theories

As we discussed in section 4, one can set up a geometric trinity to describe
gravity. Einstein’s original formulation based on non-vanishing curvature is
equivalent to TEGR, which is based on non-vanishing torsion, and both theories
are in turn equivalent to STEGR, which is built on a non-vanishing non-
metricity tensor. The General Teleparallel Equivalent of GR unifies the
torsion and non-metricity description of gravity and is also equivalent to GR.

These four formulations are equivalent in the sense that they posses the same
field equations, propagate the same degrees of freedom, and therefore
ultimately possess the same solution space. Each formulation of the trinity
can be derived from an action principle. We recall that these actions are
given by

$\displaystyle\mathcal{S}_{\rm EH}[g]$
$\displaystyle=\phantom{-}\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,\mathcal{R}\,\mathrm{d}^{4}x$
$\displaystyle\mathcal{S}_{\rm TEGR}[\Lambda]$
$\displaystyle=-\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,\mathbb{T}\,\mathrm{d}^{4}x$
$\displaystyle\mathcal{S}_{\rm STEGR}[g,\xi]$
$\displaystyle=-\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,\mathbb{Q}\,\mathrm{d}^{4}x\,.$
$\displaystyle\mathcal{S}_{\rm GTEGR}[g,\Lambda]$
$\displaystyle=-\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,\mathbb{G}\,\mathrm{d}^{4}x\,.$
(5.34)

The actions are equivalent, in the sense spelled out above, but they are not
equal. In fact, they depend on different fields and they all differ by
boundary terms. This opens the door for yet another generalization of the
geometrical trinity of gravity. Namely, we can replace the scalars
$\mathcal{R}$, $\mathbb{T}$, $\mathbb{Q}$, and $\mathbb{G}$ by arbitrary
functions and obtain the following action functionals:

$\displaystyle\mathcal{S}_{f(\mathcal{R})}[g]$
$\displaystyle\coloneqq\phantom{-}\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,f(\mathcal{R})\,\mathrm{d}^{4}x$
$\displaystyle\mathcal{S}_{f(\mathbb{T})}[\Lambda]$
$\displaystyle\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,f(\mathbb{T})\,\mathrm{d}^{4}x$
$\displaystyle\mathcal{S}_{f(\mathbb{Q})}[g,\xi]$
$\displaystyle\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,f(\mathbb{Q})\,\mathrm{d}^{4}x\,$
$\displaystyle\mathcal{S}_{f(\mathbb{G})}[g,\Lambda]$
$\displaystyle\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{|g|}\,f(\mathbb{G})\,\mathrm{d}^{4}x$
(5.35)

The motivation for this non-linear extension is that the added freedom in
choosing a function $f$ may help in explaining the accelerated expansion of
the universe, structure formation, and other phenomena which in the trinity of
GR require the introduction of dark energy and dark matter.

Indeed, given that the original functionals differed by boundary terms, one
has to conclude that the resulting non-linear extensions are no longer
equivalent to each other! In particular this means that each one of the above
functionals gives rise to its own peculiar field equations with its own number
of propagating degrees of freedom.

Probably the most studied and best understood theory among these three is
$f(\mathcal{R})$ gravity, since it was first proposed by Buchdahl in 1970
Buchdahl:1970 . Given the extensive literature and the fact that our focus is
on $f(\mathbb{Q})$ gravity, we shall just discuss some basic aspects of
$f(\mathcal{R})$ gravity and refer the reader to the extensive review articles
Sotiriou:2008 ; Capozziello:2011  and references therein.

####  $\boldsymbol{f(\mathcal{R})}$ Gravity

Following the same route that led to Einstein’s field equations, it is
straightforward to deduce the equations of $f(\mathcal{R})$ gravity. They are

$\displaystyle
f^{\prime}(\mathcal{R})\,\mathcal{R}_{\mu\nu}-\frac{1}{2}f(\mathcal{R})\,g_{\mu\nu}+\left(g_{\mu\nu}\Box-\mathcal{D}_{\mu}\mathcal{D}_{\nu}\right)f^{\prime}(\mathcal{R})=\kappa\,\mathcal{T}_{\mu\nu}\,,$
(5.36)

where we have defined
$f^{\prime}(\mathcal{R})\coloneqq\frac{\mathrm{d}f(\mathcal{R})}{\mathrm{d}\mathcal{R}}$
and $\Box\coloneqq g^{\mu\nu}\mathcal{D}_{\mu}\mathcal{D}_{\nu}$. If we choose
$f(\mathcal{R})=\mathcal{R}$, the equations reproduce to Einstein’s field
equations, as they should. In order to avoid this trivial case, we shall now
assume $f^{\prime\prime}(\mathcal{R})\neq 0$. Then one sees that the above
field equations are actually fourth order non-linear equations for the metric,
due to the second order differential operator
$g_{\mu\nu}\Box-\mathcal{D}_{\mu}\mathcal{D}_{\nu}$ acting on
$f^{\prime}(\mathcal{R})$ (which itself already contains second order
derivatives of the metric).

What may seem alarming at first sight is actually not that troublesome. One
can show Sotiriou:2008 ; Capozziello:2011  that the theory propagates three
healthy degrees of freedom: Two degrees of freedom corresponding to a massless
graviton and one scalar degree of freedom.

####  $\boldsymbol{f(\mathbb{T})}$ Gravity

Starting from the $f(\mathbb{T})$ action coupled to matter fields $\Psi$,

$\displaystyle\mathcal{S}_{f(\mathbb{T})}[g,\Gamma]\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,f(\mathbb{T})+\mathcal{S}_{\rm
matter}[g,\Psi]\,,$ (5.37)

one finds a set of metric and connection field equations

$\displaystyle\left(\nabla_{\alpha}+T_{\alpha}\right)\left[f^{\prime}(\mathbb{T})\,S_{(\mu\nu)}{}^{\alpha}\right]+f^{\prime}(\mathbb{T})\,t_{\mu\nu}-\frac{1}{2}f(\mathbb{T})g_{\mu\nu}$
$\displaystyle=\kappa\,\mathcal{T}_{\mu\nu}$
$\displaystyle\left(\nabla_{\mu}+T_{\mu}\right)\left[f^{\prime}(\mathbb{T})\,S_{[\alpha}{}^{\mu}{}_{\beta]}\right]$
$\displaystyle=0\,.$ (5.38)

It should be noted that in contrast to the $f(\mathcal{R})$ field equations,
the metric field equations of $f(\mathbb{T})$ gravity are second order.
Furthermore, in the case $f(\mathbb{T})=\mathbb{T}$ the field equations reduce
to the equations of TEGR, as had to be expected. In practice it is often
helpful to re-write the metric field equations in the form

$\displaystyle
f^{\prime}(\mathbb{T})\,G_{\mu\nu}-\frac{1}{2}\left(f(\mathbb{T})-f^{\prime}(\mathbb{T})\,\mathbb{T}\right)+f^{\prime\prime}(\mathbb{T})S_{(\mu\nu)}{}^{\alpha}\partial_{\alpha}\mathbb{T}$
$\displaystyle=\kappa\,\mathcal{T}_{\mu\nu}\,.$ (5.39)

In this form it is evident that the case $f^{\prime\prime}(\mathbb{T})=0$ with
$f^{\prime}(\mathbb{T})=1$, which is equivalent to
$f(\mathbb{T})=\mathbb{T}+$const, simply reproduces Einstein’s equations with
a cosmological constant $\Lambda=-\frac{\text{const.}}{2}$. This form of the
equations also highlights that the dynamics will be modified whenever
$f^{\prime\prime}(\mathbb{T})\neq 0$. However, despite some efforts, it has so
far not been possible to determine the precise number of degrees of freedom
propagated by the theory. The number ranges between three Ferraro:2018  and
five Li:2011 ; Blagojevic:2020 .

####  $\boldsymbol{f(\mathbb{Q})}$ Gravity

The $f(\mathbb{Q})$ action, which includes minimally coupled matter fields
$\Psi$, reads

$\displaystyle\mathcal{S}_{f(\mathbb{Q})}[g,\Gamma]\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,f(\mathbb{Q})+\mathcal{S}_{\rm
matter}[g,\Psi]$ (5.40)

and it gives rise to the following metric and connection field equations:

$\displaystyle\frac{2}{\sqrt{|g|}}\nabla_{\alpha}\left[\sqrt{|g|}\,f^{\prime}(\mathbb{Q})\,P^{\alpha}{}_{\mu\nu}\right]+f^{\prime}(\mathbb{Q})\,q_{\mu\nu}-\frac{1}{2}f(\mathbb{Q})\,g_{\mu\nu}$
$\displaystyle=\kappa\,\mathcal{T}_{\mu\nu}$
$\displaystyle\nabla_{\mu}\nabla_{\nu}\left(\sqrt{|g|}\,f^{\prime}(\mathbb{Q})\,P^{\mu\nu}{}_{\alpha}\right)$
$\displaystyle=0$ (5.41)

These field equations are structurally very similar to the field equations of
STEGR. However, it is possible to re-write the metric field equations as
DAmbrosio:2021 ; DAmbrosio:2021b

$\displaystyle
f^{\prime}(\mathbb{Q})\,G_{\mu\nu}-\frac{1}{2}\left(f(\mathbb{Q})-f^{\prime}(\mathbb{Q})\,\mathbb{Q}\right)g_{\mu\nu}+2f^{\prime\prime}(\mathbb{Q})P^{\alpha}{}_{\mu\nu}\partial_{\alpha}\mathbb{Q}$
$\displaystyle=\kappa\,\mathcal{T}_{\mu\nu}\,.$ (5.42)

In this form, it is evident that $f^{\prime\prime}(\mathbb{Q})=0$ with
$f^{\prime}(\mathbb{Q})=\text{const}.$ reproduces the Einstein field equations
with a cosmological constant. It is also clear that the dynamics will be
considerably modified by the last term on the left hand side. In fact, we will
see in subsection 6.3 that this term has an effect on the counting of primary
constraints and thus also impacts the number of physical degrees of freedom.
It is however important to emphasize that the question how many degrees of
freedom $f(\mathbb{Q})$ propagates has not yet been answered satisfactorily.
The current state will be discussed in more details in subsection 6.3.

Luckily, not knowing the number of physical degrees of freedom does not
constitute an obstacle when it comes to applying the theory to cosmology or
black holes physics. In this context, or more generally whenever we want to
study specific spacetimes, it can be useful to notice that $f(\mathbb{Q})$
gravity can contain the GR solutions as special cases. In fact, if we impose
the condition

$\displaystyle\mathbb{Q}=\mathbb{Q}_{0}=\text{const.}$ (5.43)

and if we assume that we can actually satisfy this condition, then it follows
that the metric field equations take the form

$\displaystyle
G_{\mu\nu}-\frac{1}{2}\frac{f(\mathbb{Q}_{0})-f^{\prime}(\mathbb{Q}_{0})\mathbb{Q}_{0}}{f^{\prime}(\mathbb{Q}_{0})}\,g_{\mu\nu}=\frac{\kappa}{f^{\prime}(\mathbb{Q}_{0})}\,\mathcal{T}_{\mu\nu}\,.$
(5.44)

Formally, this can be read as Einstein’s field equations with an effective
cosmological constant and a re-scaled energy-momentum tensor

$\displaystyle\Lambda_{\rm eff}$
$\displaystyle\coloneqq-\frac{1}{2}\frac{f(\mathbb{Q}_{0})-f^{\prime}(\mathbb{Q}_{0})\mathbb{Q}_{0}}{f^{\prime}(\mathbb{Q}_{0})}$
$\displaystyle\hat{\mathcal{T}}_{\mu\nu}$
$\displaystyle\coloneqq\frac{1}{f^{\prime}(\mathbb{Q}_{0})}\mathcal{T}_{\mu\nu}\,.$
(5.45)

Thus, it is possible to recover certain GR solutions in $f(\mathbb{Q})$
gravity, even when $f^{\prime\prime}(\mathbb{Q})\neq 0$, i.e., even when we
are not in the GR sector of the theory.

For applications and formal considerations it can also be useful to know the
Bianchi identities of $f(\mathbb{Q})$ gravity. Given that the theory is
generally covariant, it is possible to find such Bianchi identities by
following the same reasoning as in GR (or TEGR and STEGR). One finds the
identity

$\displaystyle\mathcal{D}_{\mu}\mathcal{M}^{\mu}{}_{\nu}+\mathcal{C}_{\nu}$
$\displaystyle\equiv 0\,,$ (5.46)

where we have defined

$\displaystyle\mathcal{M}_{\mu\nu}$
$\displaystyle\coloneqq\frac{2}{\sqrt{|g|}}\nabla_{\alpha}\left[\sqrt{|g|}\,f^{\prime}(\mathbb{Q})\,P^{\alpha}{}_{\mu\nu}\right]+f^{\prime}(\mathbb{Q})\,q_{\mu\nu}-\frac{1}{2}f(\mathbb{Q})\,g_{\mu\nu}$
$\displaystyle\mathcal{C}_{\alpha}$
$\displaystyle\coloneqq\nabla_{\mu}\nabla_{\nu}\left(\sqrt{|g|}\,f^{\prime}(\mathbb{Q})\,P^{\mu\nu}{}_{\alpha}\right)\,.$
(5.47)

We emphasize that in contrast to STEGR, $\mathcal{M}_{\mu\nu}$ does not
satisfy the identity $\mathcal{D}_{\mu}\mathcal{M}^{\mu}{}_{\nu}=0$ and thus
the connection field equations are not just trivial identities. Quite on the
contrary, the connection field equations are now dynamical equations for the
connection, which can have physical degrees of freedom.

What one can conclude, however, is that when the metric field equations are
satisfied, i.e., when $\mathcal{M}_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}$
holds, then the connection field equations are also satisfied, due to the
Bianchi identities. In fact, we easily find

$\displaystyle\mathcal{D}_{\mu}\mathcal{M}^{\mu}{}_{\nu}+\mathcal{C}_{\nu}=\kappa\,\underbrace{\mathcal{D}_{\mu}\mathcal{T}^{\mu}{}_{\nu}}_{=0}+\mathcal{C}_{\nu}=\mathcal{C}_{\nu}\equiv
0\,.$ (5.48)

This fact can, for instance, be used to simplify the Hamiltonian analysis of
the theory DAmbrosio:2023 .

####  $\boldsymbol{f(\mathbb{G})}$ Gravity

As discussed in subsection 4.5, the General Teleparallel Equivalent of GR
encompasses TEGR and STEGR at the same time. That is, TEGR and STEGR emerge
from this more general theory as partially gauge-fixed theories. It is
therefore no surprise that one can also consider the non-linear extension
$\mathbb{G}\mapsto f(\mathbb{G})$ and that this modification has some
relations to $f(\mathbb{T})$ and $f(\mathbb{Q})$ gravity. Following
Hohmann:2022 ; Heisenberg:2022 , it is convenient to first introduce the
auxiliary tensors

$\displaystyle M^{\alpha}{}_{\mu\nu}$
$\displaystyle\coloneqq\Gamma^{\alpha}{}_{\mu\nu}-\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}=K^{\alpha}{}_{\mu\nu}+L^{\alpha}{}_{\mu\nu}$
$\displaystyle Z_{\alpha}{}^{\mu\nu}$
$\displaystyle=-M_{\alpha}{}^{\mu\nu}-M^{\nu}{}_{\alpha}{}^{\mu}+M^{\rho}{}_{\alpha\rho}g^{\mu\nu}+M^{\mu\rho}{}_{\rho}\delta^{\nu}{}_{\alpha}\,.$
(5.49)

With their help, one can express the field equations of $f(\mathbb{G})$ in the
relatively compact form

$\displaystyle
f^{\prime}(\mathbb{G})\,G_{\mu\nu}-\frac{1}{2}\left(f(\mathbb{G})-f^{\prime}(\mathbb{G})\,\mathbb{G}\right)g_{\mu\nu}+\mathcal{D}_{(\mu}f^{\prime}(\mathbb{G})\,M^{\sigma}{}_{\nu)\sigma}+f^{\prime\prime}(\mathbb{G})\left(\,M^{[\rho\sigma]}{}_{\sigma}\,g_{\mu\nu}-\,M^{\rho}{}_{(\mu\nu)}\right)\partial_{\rho}\mathbb{G}=\kappa\,\mathcal{T}_{\mu\nu}$
$\displaystyle\nabla_{\rho}\left(f^{\prime}(\mathbb{G})\,Z_{\mu}{}^{\nu\rho}\right)-f^{\prime}(\mathbb{G})\,M^{\lambda}{}_{\rho\lambda}Z_{\mu}{}^{\nu\rho}=0\,.$
(5.50)

It can be verified that by imposing either $Q_{\alpha\mu\nu}=0$ or
$T^{\alpha}{}_{\mu\nu}=0$, which we should read as partial gauge-fixing
conditions for the connection, one recovers the $f(\mathbb{T})$ and
$f(\mathbb{Q})$ field equations, respectively. Moreover, just as before, the
metric field equations reveal that choosing $f^{\prime\prime}(\mathbb{G})=0$
with $f^{\prime}(\mathbb{G})\neq 0$ simply yields Einstein’s field equations.
Finally, if we impose the condition $\mathbb{G}=\mathbb{G}_{0}=\text{const}$,
we find that $f(\mathbb{G})$ can contain some of the GR solutions, since then
the field equations reduce to

$\displaystyle
G_{\mu\nu}-\frac{1}{2}\frac{f(\mathbb{G}_{0})-f^{\prime}(\mathbb{G}_{0})\mathbb{G}_{0}}{f^{\prime}(\mathbb{G}_{0})}g_{\mu\nu}$
$\displaystyle=\frac{\kappa}{f^{\prime}(\mathbb{G}_{0})}\mathcal{T}_{\mu\nu}\,.$
(5.51)

That is, we obtain Einstein’s field equations with an effective cosmological
constant and a rescaled energy-momentum tensor. All of this is unsurprising,
since all these properties hold in $f(\mathbb{T})$ and $f(\mathbb{Q})$
gravity. However, since $f(\mathbb{G})$ does not make use of a gauge-fixing
condition such as $Q_{\alpha\mu\nu}=0$ or $T^{\alpha}{}_{\mu\nu}=0$, it is
possible that it leaves more freedom to find interesting beyond-GR solutions.
A first attempt at finding cosmological solutions has been carried out in
Heisenberg:2022 . The fact that removing gauge-fixing conditions can have
advantages has also been shown in Gomes:2023 , where the so-called canonical
frame has been scrutinized in the context of a general teleparallel cosmology.

## 6 $f(\mathbb{Q})$ Gravity

The non-metricity formulation of gravity, and in particular the non-linear
extension $f(\mathbb{Q})$, have witnessed a flurry of research activities over
the past few years. Most of these activities concern applications to cosmology
and black hole physics. This is natural, considering that one of the
motivations for studying non-linear extensions is the possibility to explain
phenomena which in standard GR require the introduction of dark energy, the
inflaton field, and dark matter. In this section we give an overview over the
most important results which have been obtained in applications of
$f(\mathbb{Q})$ gravity to cosmology and black holes. We also briefly touch
upon the question of how many degrees of freedom the theory propagates.

###  6.1 Cosmology in $f(\mathbb{Q})$

Given that the coincident gauge can always be used in symmetric teleparallel
theories of gravity, the simplest thing to do when working on applications of
the theory is to use this particular gauge plus a fixed background metric.
This was precisely the ansatz in BeltranJimenez:2019 , where $f(\mathbb{Q})$
cosmology was studied for the first time. Specifically, the authors used the
ansatz

$\displaystyle\Gamma^{\alpha}{}_{\mu\nu}$ $\displaystyle=0$ and $\displaystyle
g_{\mu\nu}$ $\displaystyle=\begin{pmatrix}-N(t)^{2}&0&0&0\\\ 0&a(t)^{2}&0&0\\\
0&0&a(t)^{2}&0\\\ 0&0&0&a(t)^{2}\end{pmatrix}\,,$ (6.1)

where $N(t)$ and $a(t)$ are the usual lapse function and scale factor of the
FLRW spacetime. As it turns out, the non-metricity scalar for this ansatz is
simply given by

$\displaystyle\mathbb{Q}=6\frac{H^{2}}{N^{2}}\,,$ (6.2)

where $H\coloneqq\frac{\dot{a}}{a}$ is the usual Hubble function. It is
evident that the symmetry-reduced action

$\displaystyle\mathcal{S}[N,a]\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}t\,\mathrm{d}^{3}\vec{x}\,a(t)^{3}\,N(t)\,f(\mathbb{Q})\,,$
(6.3)

which is the $f(\mathbb{Q})$ action evaluated on the FLRW metric and in
coincident gauge, has a residual time-reparametrization invariance
BeltranJimenez:2017b ; BeltranJimenez:2019 . By exploiting this
reparametrization freedom, we can fix the lapse function to unity, $N(t)=1$.
The idea now is to study the resulting cosmological equations

$\displaystyle 6f^{\prime}\,H^{2}-\frac{1}{2}f$ $\displaystyle=\rho$
$\displaystyle\left(12H^{2}\,f^{\prime\prime}+f^{\prime}\right)\dot{H}$
$\displaystyle=-\frac{1}{2}(\rho+p)\,,$ (6.4)

where $\rho$ and $p$ denote the density and pressure, respectively, and where
we defined $f^{\prime}\coloneqq\frac{\mathrm{d}f}{\mathrm{d}\mathbb{Q}}$ and
$f^{\prime\prime}\coloneqq\frac{\mathrm{d}^{2}f}{\mathrm{d}\mathbb{Q}^{2}}$.
As always, standard matter fields also satisfy the continuity equation

$\displaystyle\dot{\rho}=-3H\left(\rho+p\right)\,.$ (6.5)

A particularly interesting class of theories emerges if we demand that $f$
satisfies the equation

$\displaystyle 6f^{\prime}\,H^{2}-\frac{1}{2}f=\frac{1}{2\kappa}\mathbb{Q}\,,$
(6.6)

with $\kappa=8\pi G$, since this gives the same background evolution as GR,
but the evolution of perturbations is subjected to modifications. Using
equation (6.2), we can rewrite the condition (6.6) equivalently as

$\displaystyle\mathbb{Q}\,f^{\prime}(\mathbb{Q})-\frac{1}{2}f(\mathbb{Q})$
$\displaystyle=\frac{1}{2\kappa}\mathbb{Q}\,,$ (6.7)

which is a simple first order differential equation for $f$, which is solved
by

$\displaystyle
f(\mathbb{Q})=\frac{1}{\kappa}\left(\mathbb{Q}+M\,\sqrt{\mathbb{Q}}\right)\,.$
(6.8)

Here, $M$ is an integration constant and clearly the special case $M=0$
corresponds to STEGR, while $M\neq 0$ leads to a $1$-parameter family of
modified theories. As mentioned before, the background evolution for this
family of theories is the same as in GR. In order to discriminate between
different values of $M$, it is necessary to study perturbations, which exhibit
different behaviour than in GR.

Another interesting ansatz for studying $f(\mathbb{Q})$ cosmology is a power-
law modification of STEGR:

$\displaystyle
f(\mathbb{Q})=\frac{1}{\kappa}\left[\mathbb{Q}-6\lambda\,M^{2}\left(\frac{\mathbb{Q}}{6M^{2}}\right)^{\alpha}\right]\,,$
(6.9)

where $\lambda$ and $\alpha$ are dimensionless parameters. The modified
Friedmann equation for this ansatz reads

$\displaystyle
H^{2}\left[1+(1-2\alpha)\lambda\left(\frac{H^{2}}{M^{2}}\right)^{\alpha-1}\right]$
$\displaystyle=\frac{\kappa}{3}\rho$ (6.10)

The previous $f$ is contained as a special case for the choice
$\alpha=\frac{1}{2}$, while STEGR emerges from $\alpha=1$. By inspecting the
form of the modified Friedmann equation one can infer that for $\alpha<1$ the
corrections to the GR evolution become important at low curvature, while for
$\alpha>1$ corrections become relevant in the high curvature regime. In other
words, theories with $\alpha>1$ play a role in the early Universe and theories
with $\alpha<1$ provide us with corrections to late-time cosmology. This opens
the possibility for modified inflationary scenarios or a description of the
late-time Universe without dark energy. In fact, various $f(\mathbb{Q})$
cosmology models have been studied and applied to questions pertaining to the
late-time Universe Lymperis:2022 ; Paul:2022 ; Narawade:2023 ; Narawade:2023b
, large scale structures Sokoliuk:2023 , relativistic versions of MOND
Milgrom:2019 ; DAmbrosio:2020b , bouncing cosmologies Bajardi:2020 ;
Agrawal:2021 ; Gadbail:2023 , and quantum cosmology Dimakis:2021 ;
Bajardi:2023 . A lot of effort has also gone into constraining or testing
$f(\mathbb{Q})$ models Dialektopoulos:2019 ; Ayuso:2020 ; Barros:2020 ;
Frusciante:2021 ; Aggarwal:2022 ; De:2022 ; Albuquerque:2022 ; Ferreira:2022 ;
Koussour:2023 ; Najera:2023 ; Bouali:2023 ; Ferreira:2023 ; Subramaniam:2023b
.

The majority of the literature on $f(\mathbb{Q})$ cosmology makes use of the
coincident gauge. However, at this point we would like to recall the
discussion on the $f(\mathbb{Q})$ field equations from subsection 5.3, which
showed that the connection field equations are no longer trivial identities
(this was the case in STEGR). From this it can be expected that
$f(\mathbb{Q})$ propagates more than the two degrees of freedom of GR. When
working in the coincident gauge and using the FLRW metric as an ansatz,
interesting cosmological models can emerge, but we become largely oblivious to
the additional degrees of freedom because of these overly restrictive choices
we made. There are two ways out of this. The first is perturbation theory
around FLRW, while still using the coincident gauge. This avenue was explored
in BeltranJimenez:2019  and it led to the insight that $f(\mathbb{Q})$ gravity
propagates at least two additional degrees of freedom.

The second option is to abandon the coincident gauge and instead work with a
metric and connection which are both compatible with the cosmological
principles of homogeneity and isotropy. The advantage of this method is that
the connection is not completely trivial and it can enrich the phenomenology
to be studied. A systematic study of this approach was undertaken in
Hohmann:2021 ; DAmbrosio:2021  and we shall briefly review the main steps and
results.

#### Symmetries and symmetry-reduction of the metric

Following Hohmann:2019 ; DAmbrosio:2021b ; Hohmann:2021 ; DAmbrosio:2021 ;
Heisenberg:2022 ; DAmbrosio:2023b , we define a (continuous) symmetry of a
metric-affine geometry as follows: Let
$\phi_{s}:\mathbb{R}\times\mathcal{M}\to\mathcal{M}$ be a $1$-parameter family
of diffeomorphisms with $\phi_{s=0}=\text{id}$, which is smooth in $s$ and
which has a generating vector field
$v\coloneqq\left.\frac{\mathrm{d}\phi_{s}}{\mathrm{d}s}\right|_{s=0}$. We say
that $\phi_{s}$ is a continuous symmetry of the metric-affine geometry if and
only if

$\displaystyle\begin{cases}\phi^{*}_{s}g_{\mu\nu}&\overset{!}{=}\quad
g_{\mu\nu}\\\
\phi^{*}_{s}\Gamma^{\alpha}{}_{\mu\nu}&\overset{!}{=}\quad\Gamma^{\alpha}{}_{\mu\nu}\end{cases}\,.$
(6.11)

These are the symmetry conditions. In case there are also tensorial matter
fields $\Psi$ present, we have to impose the additional condition

$\displaystyle\phi^{*}_{s}\Psi\quad\overset{!}{=}\quad\Psi$ (6.12)

because otherwise the field equations would be inconsistent. Heuristically,
this can also be understood as follows: The right hand side of the
$f(\mathbb{Q})$ field equations contain the energy-momentum tensor of the
matter fields. It sources the gravitational field described by
$(g_{\mu\nu},\Gamma^{\alpha}{}_{\mu\nu})$. If the matter sources do not
respect certain symmetries, it is hard to see how they could give rise to a
gravitational field which does respect these symmetries.

Given that the family of diffeomorphisms $\phi_{s}$ is smooth in $s$, we can
re-write the symmetry conditions equivalently as

$\displaystyle\begin{cases}\mathcal{L}_{v}g_{\mu\nu}&\overset{!}{=}\quad 0\\\
\mathcal{L}_{v}\Gamma^{\alpha}{}_{\mu\nu}&\overset{!}{=}\quad 0\\\
\mathcal{L}_{v}\Psi&\overset{!}{=}\quad 0\end{cases}\,,$ (6.13)

where $\mathcal{L}_{v}$ denotes the Lie derivative along the vector field $v$
which generates the symmetry $\phi_{s}$. For a spacetime which is homogeneous
and isotropic, the symmetry generators (written in spherical coordinates) are

$\displaystyle\mathcal{R}_{x}$
$\displaystyle\coloneqq\sin\theta\,\partial_{\theta}+\frac{\cos\phi}{\tan\theta}\,\partial_{\phi}$
$\displaystyle\mathcal{T}_{x}$
$\displaystyle\coloneqq{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\sin\theta\,\cos\phi\,\partial_{r}+\frac{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}{r}\,\cos\theta\,\cos\phi\,\partial_{\theta}-\frac{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}{r}\,\frac{\sin\phi}{\sin\theta}\,\partial_{\phi}$
$\displaystyle\mathcal{R}_{y}$
$\displaystyle\coloneqq-\cos\phi\,\partial_{\theta}+\frac{\sin\phi}{\tan\theta}\,\partial_{\phi}$
$\displaystyle\mathcal{T}_{y}$
$\displaystyle\coloneqq{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\sin\theta\,\sin\phi\,\partial_{r}+\frac{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}{r}\,\cos\theta\,\sin\phi\,\partial_{\theta}+\frac{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}{r}\,\frac{\cos\phi}{\sin\theta}\,\partial_{\phi}$
$\displaystyle\mathcal{R}_{z}$ $\displaystyle\coloneqq-\partial_{\phi}$
$\displaystyle\mathcal{T}_{z}$
$\displaystyle\coloneqq{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\,\cos\theta\,\partial_{r}-\frac{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}{r}\,\sin\theta\,\partial_{\theta}\,,$
(6.14)

where $\mathcal{R}_{i}$ are the generators of spatial rotations,
$\mathcal{T}_{i}$ are the generators of spatial translations, and where we
have introduced

$\displaystyle{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\coloneqq\sqrt{1-k\,r^{2}}\,.$
(6.15)

As explained in Hohmann:2019 ; DAmbrosio:2021 , it actually suffices to only
use $\mathcal{R}_{x}$, $\mathcal{R}_{y}$, $\mathcal{R}_{z}$, and
$\mathcal{T}_{x}$, since the remaining two generators can be obtained by
taking Lie brackets of these four. Moreover, imposing the conditions

$\displaystyle\mathcal{L}_{\mathcal{R}_{i}}g_{\mu\nu}$
$\displaystyle\overset{!}{=}0$ and
$\displaystyle\mathcal{L}_{\mathcal{T}_{i}}g_{\mu\nu}\overset{!}{=}0$ (6.16)

leads to the well-known result

$\displaystyle g_{\mu\nu}=\begin{pmatrix}g_{tt}(t)&0&0&0\\\
0&\frac{g_{rr}(t)}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}&0&0\\\
0&0&g_{rr}(t)\,r^{2}\\\
0&0&0&g_{rr}(t)\,r^{2}\,\sin^{2}\theta\end{pmatrix}\,.$ (6.17)

Thus, the initially ten independent components of the metric are reduced to
merely two independent components, namely $g_{tt}$ and $g_{rr}$, which can
only depend on time. Also, the metric has a simple diagonal form and the
parameter $k\in\mathbb{R}$ famously determines the spatial curvature: If
$k=0$, then the spatial sections are all flat. For $k>0$ one obtains spherical
sections, while $k<0$ describes hyperbolic spatial sections.

#### Symmetry-reduction of the connection

According to our definition of symmetries for metric-affine geometries, we
have to impose the conditions

$\displaystyle\mathcal{L}_{\mathcal{R}_{i}}\Gamma^{\alpha}{}_{\mu\nu}$
$\displaystyle\overset{!}{=}0$ and
$\displaystyle\mathcal{L}_{\mathcal{T}_{i}}\Gamma^{\alpha}{}_{\mu\nu}$
$\displaystyle\overset{!}{=}0$ (6.18)

on the connection. The resulting equations are numerous and long, but
straightforward to solve. One finds Hohmann:2021 ; DAmbrosio:2021

$\displaystyle\Gamma^{t}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}C_{1}&0&0&0\\\
0&\frac{C_{2}}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}}&0&0\\\
0&0&C_{2}\,r^{2}&0\\\ 0&0&0&C_{2}\,r^{2}\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{r}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&C_{3}&0&0\\\
C_{3}&\frac{k\,r}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}}&0&0\\\
0&0&-r\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}&-C_{5}\,r^{2}\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}\,\sin\theta\\\
0&0&-C_{5}\,r^{2}\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}\,\sin\theta&-r\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{\theta}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&C_{3}&0\\\
0&0&\frac{1}{r}&\frac{C_{5}\,\sin\theta}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}\\\
C_{4}&\frac{1}{r}&0&0\\\
0&-\frac{C_{5}\,\sin\theta}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}&0&-\sin\theta\,\cos\theta\end{pmatrix}$
$\displaystyle\Gamma^{\phi}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&C_{3}\\\
0&0&-\frac{C_{5}\,\csc\theta}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}&\frac{1}{r}\\\
0&\frac{C_{5}\,\csc\theta}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}&0&\cot\theta\\\
C_{4}&\frac{1}{r}&\cot\theta&0\end{pmatrix}\,,$ (6.19)

where $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, and $C_{5}$ are arbitrary functions
of time. It should be noted that the initially $4\times 4\times 4=64$
independent components of the connection have been reduced to these five
functions and a few trigonometric functions. However, it should also be noted
that the connection is not symmetric and thus not torsionless. In fact, we
have not yet implemented the postulates of vanishing torsion and vanishing
curvature.

#### Implementing the postulates of vanishing torsion and curvature

The vanishing of torsion is straightforward to implement. We simply have to
demand that the symmetry-reduced connection (6.1) is symmetric, which leads to
the two conditions

$\displaystyle C_{3}-C_{4}$ $\displaystyle=0$ and $\displaystyle C_{5}=0\,.$
(6.20)

This leaves us with $C_{1}$, $C_{2}$, and $C_{3}$ as free functions. Given
that so many connection components are zero and that the free functions only
depend on time, it is not surprising that the condition of vanishing curvature
leaves us algebraic equations and first order differential equations.
Specifically, $R^{\alpha}{}_{\mu\nu\rho}\overset{!}{=}0$ is equivalent to the
set of equations

$\displaystyle C_{1}\,C_{3}-C^{2}_{3}-\dot{C}_{3}$ $\displaystyle=0$
$\displaystyle C_{1}\,C_{2}-C_{2}\,C_{3}+\dot{C}_{2}$ $\displaystyle=0$
$\displaystyle k+C_{2}\,C_{3}$ $\displaystyle=0\,.$ (6.21)

Notice that the spatially flat case is special, since then we have
$C_{2}\,C_{3}=0$, which has three possible solutions:

Case I: $\displaystyle C_{2}=0,\ C_{3}\neq 0\,.$ Case II: $\displaystyle
C_{2}\neq 0,\ C_{3}=0\,.$ Case III: $\displaystyle C_{2}=0,\ C_{3}=0\,.$

If $k\neq 0$, the situation is considerably simpler. Since neither $C_{2}$ nor
$C_{3}$ can be zero, we obtain

$\displaystyle C_{3}=-\frac{k}{C_{2}}\,.$ (6.22)

Using this result, the two differential equations (6.1) reduce to a single
equation:

$\displaystyle k+C_{1}\,C_{2}+\dot{C}_{2}$ $\displaystyle=0\,.$ (6.23)

Given that $C_{2}\neq 0$, we can solve this last equation for $C_{1}$,
obtaining

$\displaystyle C_{1}=-\frac{k+\dot{C}_{2}}{C_{2}}\,.$ (6.24)

We finally arrive at the conclusion that a connection which respects
homogeneity and isotropy, and which is also torsionless and flat under the
assumption that $k\neq 0$ has the form

$\displaystyle\Gamma^{t}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}-\frac{k+\dot{C}_{2}}{C_{2}}&0&0&0\\\
0&\frac{C_{2}}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}}&0&0\\\
0&0&r^{2}\,C_{2}&0\\\ 0&0&0&r^{2}\,C_{2}\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{r}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&-\frac{k}{C_{2}}&0&0\\\
-\frac{k}{C_{2}}&\frac{k\,r}{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}}&0&0\\\
0&0&-r\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}&0\\\
0&0&0&-r\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}^{2}\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{\theta}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&-\frac{k}{C_{2}}&0\\\ 0&0&\frac{1}{r}&0\\\
-\frac{k}{C_{2}}&\frac{1}{r}&0&0\\\
0&0&0&-\sin\theta\,\cos\theta\end{pmatrix}$
$\displaystyle\Gamma^{\phi}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&-\frac{k}{c}\\\ 0&0&0&\frac{1}{r}\\\
0&0&0&\cot\theta\\\ -\frac{k}{C_{2}}&\frac{1}{r}&\cot\theta&0\end{pmatrix}\,.$
(6.25)

We dub this connection $\Gamma^{(k)}$. Now we consider to spatially flat
sections, $k=0$, case by case. For Case I, defined by $C_{2}=0$ under the
assumption that $C_{3}\neq 0$, we obtain the connection $\Gamma^{(\rm I)}$,
which is of the form

$\displaystyle\Gamma^{t}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}C_{3}+\frac{\dot{C}_{3}}{C_{3}}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$
$\displaystyle\Gamma^{r}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&C_{3}&0&0\\\ C_{3}&0&0&0\\\ 0&0&-r&0\\\
0&0&0&-r\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{\theta}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&C_{3}&0\\\ 0&0&\frac{1}{r}&0\\\
C_{3}&\frac{1}{r}&0&0\\\ 0&0&0&-\sin\theta\,\cos\theta\end{pmatrix}$
$\displaystyle\Gamma^{\phi}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&C_{3}\\\ 0&0&0&\frac{1}{r}\\\
0&0&0&\cot\theta\\\ C_{3}&\frac{1}{r}&\cot\theta&0\end{pmatrix}\,,$ (6.26)

This connection depends on the free function $C_{3}(t)$. In the second case,
which is based on the assumption $C_{2}\neq 0$, we obtain the connection
$\Gamma^{(\rm{II})}$, parametrized as

$\displaystyle\Gamma^{t}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}-\frac{\dot{C}_{2}}{C_{2}}&0&0&0\\\
0&C_{2}&0&0\\\ 0&0&r^{2}\,C_{2}&0\\\
0&0&0&r^{2}\,C_{2}\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{r}{}_{\mu\nu}$ $\displaystyle=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&0&-r&0\\\ 0&0&0&-r\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{\theta}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&\frac{1}{r}&0\\\
0&\frac{1}{r}&0&0\\\ 0&0&0&-\sin\theta\,\cos\theta\end{pmatrix}$
$\displaystyle\Gamma^{\phi}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&0&\frac{1}{r}\\\
0&0&0&\cot\theta\\\ 0&\frac{1}{r}&\cot\theta&0\end{pmatrix}\,.$ (6.27)

Finally, the third case, which is clearly the simplest, gives us the
connection $\Gamma^{(\rm{III})}$, which can explicitly be written as

$\displaystyle\Gamma^{t}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}-C_{1}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}$ $\displaystyle\Gamma^{r}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&-r&0\\\
0&0&0&-r\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{\theta}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&\frac{1}{r}&0\\\
0&\frac{1}{r}&0&0\\\ 0&0&0&-\sin\theta\,\cos\theta\end{pmatrix}$
$\displaystyle\Gamma^{\phi}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&0&\frac{1}{r}\\\
0&0&0&\cot\theta\\\ 0&\frac{1}{r}&\cot\theta&0\end{pmatrix}\,.$ (6.28)

In conclusion, we find that a connection which is homogeneous, isotropic,
torsionless, and flat can be parametrized in four distinct ways. The
connections $\Gamma^{(k)}$, $\Gamma^{(\rm{I})}$, $\Gamma^{\rm{II}}$, and
$\Gamma^{(\rm III)}$ could be the source of interesting and rich cosmological
models. Indeed, for the connection $\Gamma^{(\rm{II})}$ with the choice
$f(\mathbb{Q})=\mathbb{Q}^{\kappa}$ (assuming $\kappa\geq 2$), exact vacuum
solutions were obtained DAmbrosio:2021  which can reproduce the scale factor
of a fluid with equation of state $p=w\,\rho$, for some constant $w$. The same
exact vacuum solution can also mimic de Sitter space. This could be of
interest for investigations concerning the early Universe, since this solution
can naturally drive inflation.

The effects of using different connections in $f(\mathbb{Q})$ cosmology have
been studied in DAmbrosio:2021 ; Dimakis:2022 ; Heisenberg:2022 ; Shabani:2023
; Subramaniam:2023 , but a large untapped potential to discover new
interesting solutions remains.

###  6.2 Black Holes in $f(\mathbb{Q})$

It is tempting to start an investigation into black hole solutions in
$f(\mathbb{Q})$ gravity by following the same strategy initially used in
$f(\mathbb{Q})$ cosmology. In fact, the simplest possible strategy one can
think of is to fix the coincident gauge and choose a metric ansatz which is
stationary and spherically symmetric. In formulas:

$\displaystyle\Gamma^{\alpha}{}_{\mu\nu}$ $\displaystyle=0$ and $\displaystyle
g_{\mu\nu}=\begin{pmatrix}-A(r)&0&0&0\\\ 0&B(r)&0&0\\\ 0&0&r^{2}&0\\\
0&0&0&r^{2}\sin^{2}\theta\end{pmatrix}\,,$ (6.29)

where $A$ and $B$ are arbitrary functions of the radial coordinate $r>0$. This
metric ansatz is written in the coordinate system $(t,r,\theta,\phi)$, where
$(r,\theta,\phi)$ are the standard spherical coordinates. The idea is then to
plug the ansatz metric into the $f(\mathbb{Q})$ field equations (5.42) and
work out solutions. As long as $f^{\prime\prime}\neq 0$, one expects to obtain
black hole solutions which deviate from the standard GR solution.

However, as it turns out, this expectation is not only wrong, the entire
strategy is not viable! In fact, one finds without much trouble that the field
equations of $f(\mathbb{Q})$ become inconsistent for the above ansatz metric
in coincident gauge, except in the special case where $f^{\prime\prime}=0$. In
other words, the field equations themselves tell us that the inconsistency
disappears precisely when we are in the regime of the theory which is
equivalent to GR. But in that regime we can only recover GR solutions and
nothing else! After all, the equivalence of GR and STEGR is based on having
the same field equations with the same solution space.

In order to overcome this difficulty, a systematic analysis of stationary and
spherically symmetric spacetimes was carried out in DAmbrosio:2021b . This
means that no special metric nor any special gauge for the connection is
assumed. Rather, the strategy of DAmbrosio:2021b  was to follow three simple
steps:

  1. 1.

Symmetries: Find the most general metric and connection which are stationary
(i.e., time-translation invariant) and spherically symmetric (i.e., invariant
under spatial rotations around the origin);

  2. 2.

Geometric postulates: Use the metric and connection found above and implement
the postulate of vanishing curvature and vanishing torsion,
$R^{\alpha}{}_{\mu\nu\rho}=0$ and $T^{\alpha}{}_{\mu\nu}=0$;

  3. 3.

Field equations: Take the metric and connection which satisfy all symmetries
and geometric postulates and plug them into the $f(\mathbb{Q})$ field
equations.

We provide a brief sketch of the individual steps. The ultimate goal is to
find the simplest representation of a stationary, spherically symmetric
metric-affine geometry $(\mathcal{M},g,\Gamma)$, before studying the field
equations of $f(\mathbb{Q})$ gravity. For details we refer the reader to
DAmbrosio:2021b .

#### Symmetries

In the present context we are interested in finding solutions which are
stationary and spherically symmetric. Given the notion of spacetime symmetry
discussed in the previous subsection, this means that we have to impose

$\displaystyle\begin{cases}\mathcal{L}_{v}g_{\mu\nu}&\overset{!}{=}\quad 0\\\
\mathcal{L}_{v}\Gamma^{\alpha}{}_{\mu\nu}&\overset{!}{=}\quad 0\\\
\mathcal{L}_{v}\Psi&\overset{!}{=}\quad 0\end{cases}\,,$ (6.30)

for the vector fields which generate temporal translations and spatial
rotations around the origin:

$\displaystyle\mathcal{T}$ $\displaystyle\coloneqq\partial_{t}$ (generator of
time-translations) $\displaystyle\mathcal{R}_{x}$
$\displaystyle\coloneqq\sin\phi\,\partial_{\theta}+\frac{\cos\phi}{\tan\theta}\partial_{\phi}$
$\displaystyle\mathcal{R}_{y}$
$\displaystyle\coloneqq-\cos\phi\partial_{\theta}+\frac{\sin\phi}{\tan\theta}\partial_{\phi}$
(generators of rotations) $\displaystyle\mathcal{R}_{z}$
$\displaystyle\coloneqq-\partial_{\phi}$ (6.31)

Obtaining time-translation invariance and invariance with respect to rotations
in the $\phi$-direction is easy: All metric and connection components have to
be independent of the coordinates $t$ and $\phi$. Invariance with respect to
the $\mathcal{R}_{x}$ and $\mathcal{R}_{y}$ generators requires a little more
work. However, the result for the metric is well-known (see for instance
WaldBook ; CarrollBook ): A metric which is time-translation and rotationally
invariant necessarily has the form

$\displaystyle g_{\mu\nu}=\begin{pmatrix}g_{tt}&g_{tr}&0&0\\\
g_{tr}&g_{rr}&0&0\\\ 0&0&g_{\theta\theta}&0\\\
0&0&0&g_{\theta\theta}\sin^{2}\theta\end{pmatrix}$ (6.32)

with respect to the coordinates $(t,r,\theta,\phi)$. In particular, we point
out that $g_{tt},g_{tr},g_{rr}$, and $g_{\theta\theta}$ are only functions of
$r$.

In the case of the connection it is easier to first impose the postulate of
vanishing torsion and then to work out the remaining two symmetry conditions.
Since torsion is the anti-symmetric part of the connection, a torsionless
connection is simply one that is symmetric in its lower two indices. Imposing
this condition also has the effect of reducing the number of independent
components of the connection from $4\times 4\times 4=64$ to
$4\times\frac{4\times(4+1)}{2}=40$.

Furthermore, it is more convenient to consider the following linear
combinations when imposing the remaining symmetry conditions:

$\displaystyle\cos\phi\,\mathcal{L}_{\mathcal{R}_{x}}\Gamma^{\alpha}{}_{\mu\nu}+\sin\phi\,\mathcal{L}_{\mathcal{R}_{y}}\Gamma^{\alpha}{}_{\mu\nu}$
$\displaystyle\overset{!}{=}0$
$\displaystyle\sin\phi\,\mathcal{L}_{\mathcal{R}_{x}}\Gamma^{\alpha}{}_{\mu\nu}-\cos\phi\,\mathcal{L}_{\mathcal{R}_{y}}\Gamma^{\alpha}{}_{\mu\nu}$
$\displaystyle\overset{!}{=}0\,.$ (6.33)

We emphasize that imposing these conditions is strictly equivalent to imposing
$\mathcal{L}_{\mathcal{R}_{x}}\Gamma^{\alpha}{}_{\mu\nu}\overset{!}{=}0$ and
$\mathcal{L}_{\mathcal{R}_{y}}\Gamma^{\alpha}{}_{\mu\nu}\overset{!}{=}0$. By
imposing the first linear combination we learn that

  * a)

Twenty of the $40$ components of $\Gamma^{\alpha}{}_{\mu\nu}$ are exactly
zero;

  * b)

Two components are given in terms of trigonometric functions;

  * c)

Six components are determined through algebraic relations to other components
of the connection.

Hence, out of the initially $64$ independent components of the connection,
three of the symmetry conditions and the postulate of vanishing torsion bring
this number down to only $40-20-2-6=12$ independent components which are
functions of $r$ and $\theta$.

Finally, the second linear combination implements the last symmetry condition.
It leads to a set of twelve first order partial differential equations for
precisely the twelve independent connection components we are left with after
imposing the first three symmetry conditions. These equations can be solved,
but because these are partial differential equations with respect to $\theta$,
the solutions all depend on $r$. Hence, we find that the symmetry conditions
together with the postulate of vanishing torsion leave us with twelve
independent connection components, all of which are purely functions of $r$
and nothing else.

#### Geometric postulates

Since we have already implemented the postulate of vanishing torsion, we are
left with imposing the postulate of vanishing curvature. As can be expected
from the form of the curvature tensor and the fact that $20$ connection
components vanish, this will lead to a set of algebraic equations and a set of
first order partial differential equations. The detailed process of how to
consistently solve all algebraic and differential equations is explained in
DAmbrosio:2021b , where it is found that this ultimately leads to two
different sets of solutions. The first solution set is defined as follows: All
connection components can be expressed in terms of the three arbitrary
functions $\Gamma^{t}{}_{rr}(r)$, $\Gamma^{r}{}_{rr}(r)$,
$\Gamma^{\phi}{}_{r\phi}(r)$, the real constant $c\neq 0$, and trigonometric
functions. Concretely, the connection takes the form

$\displaystyle\Gamma^{t}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}c&\Gamma^{\phi}{}_{r\phi}&0&0\\\
\Gamma^{\phi}{}_{r\phi}&\Gamma^{t}{}_{rr}&0&0\\\ 0&0&-\frac{1}{c}&0\\\
0&0&0&-\frac{\sin^{2}\theta}{c}\end{pmatrix}$
$\displaystyle\Gamma^{r}{}_{\mu\nu}$ $\displaystyle=\begin{pmatrix}0&0&0&0\\\
0&\Gamma^{r}{}_{rr}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$
$\displaystyle\Gamma^{\theta}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&c&0\\\ 0&0&\Gamma^{\phi}{}_{r\phi}&0\\\
c&\Gamma^{\phi}{}_{r\phi}&0&0\\\ 0&0&0&-\sin\theta\,\cos\theta\end{pmatrix}$
$\displaystyle\Gamma^{\phi}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&c\\\ 0&0&0&\Gamma^{\phi}{}_{r\phi}\\\
0&0&0&\cot\theta\\\ c&\Gamma^{\phi}{}_{r\phi}&\cot\theta&0\end{pmatrix}\,.$
(6.34)

Furthermore, the derivative of $\Gamma^{\phi}{}_{r\phi}$ can be written as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\Gamma^{\phi}{}_{r\phi}=c\,\Gamma^{t}{}_{rr}-\Gamma^{\phi}{}_{r\phi}\left(\Gamma^{\phi}{}_{r\phi}+\Gamma^{r}{}_{rr}\right)\,.$
(6.35)

These are all the defining properties of solution set 1. For solution set 2
one finds instead that all connection components can be expressed in terms of
the four arbitrary functions $\Gamma^{t}{}_{rr}(r)$,
$\Gamma^{t}{}_{\theta\theta}(r)$, $\Gamma^{r}{}_{rr}(r)$,
$\Gamma^{r}{}_{\theta\theta}(r)\neq 0$, the two real constants $c$ and $k$,
and trigonometric functions. The connection is explicitly given by

$\displaystyle\Gamma^{t}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}k-c-c\tilde{c}\,\Gamma^{t}{}_{\theta\theta}&\frac{\tilde{c}\,\hat{\Gamma}^{t}{}_{\theta\theta}\,\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}&0&0\\\
\frac{\tilde{c}\,\hat{\Gamma}^{t}{}_{\theta\theta}\,\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}&\Gamma^{t}{}_{rr}&0&0\\\
0&0&\Gamma^{t}{}_{\theta\theta}&0\\\
0&0&0&\Gamma^{t}{}_{\theta\theta}\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{r}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}-c\tilde{c}\,\Gamma^{r}{}_{\theta\theta}&c+c\,\tilde{c}\,\Gamma^{t}{}_{\theta\theta}&0&0\\\
c+c\,\tilde{c}\,\Gamma^{t}{}_{\theta\theta}&\Gamma^{r}{}_{rr}&0&0\\\
0&0&\Gamma^{r}{}_{\theta\theta}&0\\\
0&0&0&\Gamma^{r}{}_{\theta\theta}\,\sin^{2}\theta\end{pmatrix}$
$\displaystyle\Gamma^{\theta}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&c&0\\\
0&0&-\frac{\hat{\Gamma}^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}&0\\\
c&-\frac{\hat{\Gamma}^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}&0&0\\\
0&0&0&-\sin\theta\,\cos\theta\end{pmatrix}$
$\displaystyle\Gamma^{\phi}{}_{\mu\nu}$
$\displaystyle=\begin{pmatrix}0&0&0&c\\\
0&0&0&-\frac{\hat{\Gamma}^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}\\\
0&0&0&\cot\theta\\\
c&-\frac{\hat{\Gamma}^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}&\cot\theta&0\end{pmatrix}\,,$
(6.36)

where we have defined $\tilde{c}\coloneqq 2c-k$ and
$\hat{\Gamma}^{t}{}_{\theta\theta}\coloneqq 1+c\,\Gamma^{t}{}_{\theta\theta}$
in order to compactify the notation. Moreover, the derivatives of
$\Gamma^{t}{}_{\theta\theta}$ and $\Gamma^{r}{}_{\theta\theta}$ can be
expressed in terms of the other free functions. Concretely, one finds

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=-\frac{\left\\{\left[c\,(2c-k)\,\Gamma^{t}{}_{\theta\theta}+3c-k\right]\Gamma^{t}{}_{\theta\theta}+1\right\\}\Gamma^{t}{}_{\theta\theta}}{\Gamma^{r}{}_{\theta\theta}}-\Gamma^{r}{}_{\theta\theta}\Gamma^{t}{}_{rr}$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\Gamma^{r}{}_{\theta\theta}$
$\displaystyle=-c\left((2c-k)\Gamma^{t}{}_{\theta\theta}+2\right)\Gamma^{t}{}_{\theta\theta}-\Gamma^{r}{}_{\theta\theta}\Gamma^{r}{}_{rr}-1\,.$
(6.37)

Observe that in both solution sets the derivatives of $\Gamma^{t}{}_{rr}$ and
$\Gamma^{r}{}_{rr}$ cannot be expressed in terms of other connection
components. Thus, in both cases only these two components should be regarded
as the unknowns to be solved for in the connection field equations.

It was also shown in DAmbrosio:2021b  that the two solution sets are related
to each other by a double scaling limit. However, it should be emphasized that
outside of this particular limit, the two solution sets are genuinely
different and they describe different physics. We elaborate more on this point
further below.

#### Simplest possible form of a stationary, spherically symmetric geometry
$\boldsymbol{(\mathcal{M},g,\Gamma)}$

Recall that our task is not only to find expressions for the metric and the
connection which satisfy the various symmetries and the geometric postulates.
We also wish to find the simplest possible form, as that will hopefully help
in analyzing and solving the field equations. To simplify the form of the
metric, we make use of the diffeomorphism invariance of the theory. This is
possible, since we did not yet fix any particular gauge. As is well-known, it
is possible to find a diffeomorphism which brings the symmetry-reduced metric
(6.32) into the simple diagonal form

$\displaystyle g_{\mu\nu}=\begin{pmatrix}g_{tt}(r)&0&0&0\\\ 0&g_{rr}(r)&0&0\\\
0&0&r^{2}&0\\\ 0&0&0&r^{2}\sin^{2}\theta\end{pmatrix}\,.$ (6.38)

This is of course nothing but the standard form of a metric which is
stationary and spherically symmetric, which can be found in textbooks on GR
HawkingEllisBook ; WaldBook ; CarrollBook . However, in the context of metric-
affine geometries, the diffeomorphism which achieves this transformation has
of course also to be applied to the connection. What is remarkable, is that
even tough this diffeomorphism in general changes the connection, it maps
solution set 1 onto itself and it also maps solution set 2 onto itself!

This means that when we study the field equations of $f(\mathbb{Q})$ gravity,
we can use the metric in its simple symmetry-reduced form (6.38) together with
a connection which either belongs to solution set 1 or solution set 2. This is
the simplest possible form of a stationary and spherically symmetric metric-
affine geometry!

#### A cautionary remark on the coincident gauge

It is worth pausing at this point and discussing why the first approach,
namely the approach based on a metric of the form (6.38) and the coincident
gauge, $\Gamma^{\alpha}{}_{\mu\nu}=0$, fails. This comes simply from the fact
that if the metric has the form (6.38), then the connection cannot be
identically zero if it also has to satisfy the symmetry conditions. This
follows immediately from the two solution sets. Recall that these two solution
sets tell us the possible forms a symmetry-reduced connection can have. Both
sets exclude the possibility $\Gamma^{\alpha}{}_{\mu\nu}=0$, because in both
sets there are components which are purely expressed in terms of trigonometric
function and in both sets there are certain components which are not allowed
to vanish.

Does this mean we cannot use the coincident gauge? No, the coincident gauge
can always be used. But one has to be careful in how one uses it. Our
systematic implementation of symmetries and geometric postulates has shown
what form the metric and the connection are allowed to have in the coordinate
system ${t,r,\theta,\phi}$. What the coincident gauge tells us, is that there
exists a different coordinate system where $\Gamma^{\alpha}{}_{\mu\nu}=0$, but
where the metric will no longer have its simple diagonal form! A
diffeomorphism which trivializes the connection will necessarily complicate
the metric. In a sense, all the information which resided in the symmetry-
reduced connection is “moved” onto the metric by the diffeomorphism. Hence,
nothing is gained by using the coincident gauge, which is why we prefer to
stick to the two solution sets described above. In the context of stationary
and spherically symmetric spacetimes, the transformations which produced the
coincident gauge for both solution sets have been worked out Bahamonde:2022 .

#### Symmetry-reduced form of the field equations

The symmetry-reduced form of the field equations are obtained by plugging the
metric ansatz (6.38) and either the connection from solution set 1 or the
connection from solution set 2 into the $f(\mathbb{Q})$ field equations
(5.42). In both cases we find that the structure of the field equations is

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: $\displaystyle\begin{pmatrix}\mathcal{C}_{t}\\\
\mathcal{C}_{r}\\\ 0\\\ 0\end{pmatrix}$ (6.39)

Of course, the components of these tensors are different for the two different
solution sets of the connection. However, in both cases it turns out to be
highly advantageous to first study the off-diagonal component of the metric
field equations, i.e., $\mathcal{M}_{tr}=0$. This leads to two very similar
and yet still different equations:

  * •

For solution set 1: $\mathcal{M}_{tr}=0\quad\longrightarrow\quad
c\,\partial_{r}\mathbb{Q}\,f^{\prime\prime}(\mathbb{Q})=0$.

  * •

For solution set 2:
$\mathcal{M}_{tr}=0\quad\longrightarrow\quad\left(k-2c(2c-k)\Gamma^{t}{}_{\theta\theta}\right)\partial_{r}\mathbb{Q}\,f^{\prime\prime}(\mathbb{Q})=0$.

We observe that both equations admit $\partial_{r}\mathbb{Q}$ and
$f^{\prime\prime}(\mathbb{Q})=0$ as solutions. The first option amounts to
saying that the non-metricity scalar is constant. In fact, the metric and the
connection for both solution sets only depend on $r$ and $\theta$, but, as was
shown in DAmbrosio:2021b , the non-metricity scalar does not inherit the
$\theta$-dependence. Thus, $\partial_{r}\mathbb{Q}=0$ is really saying that
the non-metricity scalar is a constant. It is then easy to see that this does
not yield any solutions which go beyond GR. In fact, the $f(\mathbb{Q})$ field
equations for $\mathbb{Q}=$const. simply become

$\displaystyle
f^{\prime}(\mathbb{Q}_{0})G_{\mu\nu}+\frac{1}{2}\left(f(\mathbb{Q}_{0})-f^{\prime}(\mathbb{Q}_{0})\,\mathbb{Q}_{0}\right)g_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}\,,$
(6.40)

where $\mathbb{Q}_{0}$ is a constant number. These equations can be re-written
in the more suggestive form

$\displaystyle G_{\mu\nu}+\Lambda_{\rm
eff}g_{\mu\nu}=\kappa\,\bar{\mathcal{T}}_{\mu\nu}\,,$ (6.41)

where we have introduced

$\displaystyle\Lambda_{\rm eff}$
$\displaystyle\coloneqq\frac{1}{2}\frac{f(\mathbb{Q}_{0})-f^{\prime}(\mathbb{Q}_{0})\mathbb{Q}_{0}}{f^{\prime}(\mathbb{Q}_{0})}$
and $\displaystyle\bar{\mathcal{T}}_{\mu\nu}$
$\displaystyle\coloneqq\frac{1}{f^{\prime}(\mathbb{Q}_{0})}\mathcal{T}_{\mu\nu}\,.$
(6.42)

Thus, we obtain the Einstein field equations with an effective cosmological
constant and a re-scaled energy-momentum tensor! Notice that the re-scaling
and the effective cosmological constant are well-defined since we always
assume $f^{\prime}\neq 0$. Otherwise, one would end up with a trivial, non-
dynamical theory. Thus, we conclude that solving the off-diagonal metric field
equation with $\mathbb{Q}=$const. does not yield beyond GR solutions.

The second option is to solve $\mathcal{M}_{tr}=0$ by
$f^{\prime\prime}(\mathbb{Q})=0$. However, we already know that this means
that $f(\mathbb{Q})=a\,\mathbb{Q}+b$, where $a$ and $b$ are two real
constants. In other words, this option just produces STEGR plus a cosmological
constant. Give that STEGR is equivalent to GR, with this option we just
recover GR solutions and nothing else. Hence, also in this case we learn that
we can only obtain GR solutions for both solution sets of the connection.

This leads us to the third option, which is to impose the constraint equations

$\displaystyle\mathcal{M}_{tr}$ $\displaystyle=0$
$\displaystyle\quad\longrightarrow\quad$ $\displaystyle c$ $\displaystyle=0$
(for solution set 1) $\displaystyle\mathcal{M}_{tr}$ $\displaystyle=0$
$\displaystyle\quad\longrightarrow\quad$
$\displaystyle\left(k-2c(2c-k)\Gamma^{t}{}_{\theta\theta}\right)$
$\displaystyle=0$ $\displaystyle\text{(for solution set 2)}\,.$ (6.43)

A quick glace at the defining properties of solution set 1 reveals that $c=0$
is not possible. In fact, solution set 1 is only valid if $c\neq 0$. Hence, we
reach the important conclusion that solution set 1 only contains the GR
solutions! If we wish to find beyond GR solution, our only hope is solution
set 2. Indeed, the constraint equation (6.2) for solution set 2 does have
interesting solutions. As it turns out DAmbrosio:2021b , there are two
branches.

Branch 1: $\displaystyle\Gamma^{t}{}_{\theta\theta}$
$\displaystyle=\frac{k}{2c(2c-k)}$ for $c\neq 0$ and $k\neq 2c$
$\displaystyle\Gamma^{t}{}_{rr}$
$\displaystyle=\frac{k(8c^{2}+2ck-k^{2})}{8c^{2}(2c-k)^{2}\left(\Gamma^{r}{}_{\theta\theta}\right)^{2}}$
Branch 2: $\displaystyle\Gamma^{t}{}_{rr}$
$\displaystyle=-\frac{\Gamma^{t}{}_{\theta\theta}}{\left(\Gamma^{r}{}_{\theta\theta}\right)^{2}}$
$\displaystyle c$ $\displaystyle=k=0\,.$ (6.44)

Both branches are viable in the sense that they lead to self-consistent field
equations, as has been shown in DAmbrosio:2021b . Moreover, it has also been
shown that both branches lead to beyond-GR solutions. Some solutions have been
derived explicitly.

#### Overview of different developments and outlook

Let us summarize the situation thus far: We began with a systematic
implementation of stationarity and spherical symmetry. This drastically
restricted the form of the metric and of the connection. Then, we proceeded
with imposing the geometric postulates. In particular, the postulate of
vanishing curvature led to further restrictions on the connection and we found
that there are two possible parametrizations for a symmetry-reduced connection
which also satisfies the geometric postulates. We dubbed these
parametrizations solution set 1 and solution set 2.

Remarkably, it is possible to diagonalize the metric and bring it into the
standard form of a stationary and spherically symmetric metric without
spoiling the solution sets. That is, the diffeomorphism which brings the
metric into its simplest form maps solution set 1 onto itself and solution set
2 onto itself. Thus, the metric (6.38) together with solution sets 1 and 2 for
the connection provide us with the simplest representation of a stationary and
spherically symmetric metric-affine geometry $(\mathcal{M},g,\Gamma)$. The
solution sets also allow us to understand why the coincident gauge leads to
inconsistent field equations, if we simultaneously insist that the metric
ansatz has the form (6.38).

By studying the symmetry-reduced metric field equations, we finally learned
that solution set 1 only contains the standard GR solutions. If one wishes to
find beyond-GR solutions, one has to work with solution set 2. Within this
solution set, one finds that the field equations allow for two branches. That
is, the off-diagonal equation $\mathcal{M}_{tr}=0$ imposes a constraint on the
connection which admits two genuinely different solutions. Both solutions are
fully consistent and can be used to further study the field equations.

This leads us to the question of what can be achieved with these different
branches and modified gravity equations. In DAmbrosio:2021b , different
methods were used to find beyond-GR black hole solutions. Some exact, but
rather unphysical solutions were found. Perturbative techniques led to
approximate solutions of the field equations which are asymptotically flat,
but which lead to multiple horizons and black hole masses which depend on the
connection. Regular black holes, black bounces, and quasi normal modes within
the context of $f(\mathbb{Q})$ gravity were studied in Junior:2023 ;
Gogoi:2023 .

Besides black holes, the stationary and spherically symmetric spacetimes
considered here have inspired a flurry of investigations into wormholes in
$f(\mathbb{Q})$ gravity Banerjee:2021 ; Mustafa:2021 ; Parsaei:2022 ;
Hassan:2022b ; Hassan:2022 ; Hassan:2022c ; Venkatesha:2023 ; Jan:2023 ;
Godani:2023 ; Mustafa:2023 ; Mishra:2023  as well as modified stellar
solutions Wang:2021 ; Maurya:2022b ; Maurya:2022 ; Errehymy:2022 ;
Sokoliuk:2022 ; Calza:2022 ; Chanda:2022 ; Bhar:2023 ; Ditta:2023 ;
Maurya:2023 . Some thought has also been given to the question, how
observational data could be used to constrain $f(\mathbb{Q})$ gravity
DAgostino:2022 . The beyond-GR black hole and stellar solutions could play an
important role in this regard.

###  6.3 Hamiltonian Analysis and Degrees of Freedom of $f(\mathbb{Q})$
Gravity

The question how many degrees of freedom are being propagated in
$f(\mathbb{Q})$ is currently under debate. Findings from cosmological
perturbation theory performed in BeltranJimenez:2019  revealed that
$f(\mathbb{Q})$ possesses at least two additional degrees of freedom compared
to GR. Based on this insight, and the expectation that the primary constraints
of $f(\mathbb{Q})$ gravity are all second class due to its general covariance,
led to the educated guess that the theory propagates six degrees of freedom
DAmbrosio:2020c .

A more systematic approach based on the Hamiltonian analysis, performed in
coincident gauge, was attempted in Hu:2022 . The authors concluded that there
are eight degrees of freedom. However, this conclusion was challenges by
DAmbrosio:2023 , who put an upper bound of seven degrees of freedom using a
kinetic matrix approach. In the same paper, mistakes in the analysis of
Hu:2022  were brought to light and general issues with the Hamiltonian
analysis were discussed. In particular, it was pointed out that the standard
approach due to Dirac DiracBook ; Dirac:1950  and Bergmann Bergmann:1951
encounters severe obstacles and new methods, such as the kinetic matrix
approach, have to be employed. Finally, yet another Hamiltonian analysis was
attempted by Tomonari:2023 , who concluded that there are six degrees of
freedom. This is in agreement with the upper bound of DAmbrosio:2023  and the
authors claim to have overcome the obstacles of the Dirac-Bergmann algorithm
which were pointed out in DAmbrosio:2023 . However, as we will discuss further
below, the resolution is not beyond doubt.

At the moment, only three things seem clear: (a) The theory propagates at
least four degrees of freedom, (b) there are at most seven degrees of freedom,
and (c) there is confusion about what the precise number might be.

To better understand this unsatisfying state of affairs we shall briefly
review the main results on which everyone agrees. Then we discuss the points
where mistakes were made or where opinions drift apart.

#### ADM formulation and primary constraints

In order to perform the Hamiltonian analysis, it is advantageous to employ the
ADM formalism. Under the (weak) assumption that $\mathcal{M}$ has the topology
$\mathcal{M}\simeq\mathbb{R}\times\Sigma$, where $\Sigma$ is a three-
dimensional spacelike hypersurface, we can split the coordinates
$\\{x^{\mu}\\}$ into one temporal and three spatial coordinates,
$\\{t,x^{a}\\}$. The spatial index takes values in $\\{1,2,3\\}$. Moreover,
the metric can be written as

$\displaystyle
g_{\mu\nu}=\begin{pmatrix}-N^{2}+h_{ab}N^{a}N^{b}&h_{ab}N^{b}\\\
h_{ab}N^{b}&h_{ab}\end{pmatrix}\,,$ (6.45)

where $N>0$ is the lapse function, $N^{a}$ is called the shift vector field,
and $h_{ab}$ is the three-dimensional metric intrinsic to $\Sigma$. Spatial
indices are raised and lowered with $h_{ab}$. Also, we refer to
$\\{N,N^{a},h_{ab}\\}$ collectively as ADM variables. From now on, we work
exclusively in coincident gauge. Hence, $\Gamma^{\alpha}{}_{\mu\nu}=0$
globally and consequently covariant derivatives are turned into partial
derivatives, $\nabla_{\mu}=\partial_{\mu}$.

The first step in the Hamiltonian analysis then consists in determining the
momentum densities $\tilde{\pi}_{0}$, $\tilde{\pi}_{a}$, and
$\tilde{\pi}^{ab}$ conjugate to lapse, shift, and intrinsic metric,
respectively. The second step is to determine which of the momentum densities
can be solved for the velocities $\dot{N}$, $\dot{N}^{a}$, and $\dot{h}_{ab}$.
Momenta which are independent of any velocities, i.e., which are of the form
$\tilde{\pi}=\tilde{f}(N,N^{a},h_{ab})$, give rise to primary constraints
$\boldsymbol{\tilde{C}}$ of the form
$\tilde{C}\coloneqq\tilde{\pi}-\tilde{f}$. They put constraints on the
physical field configurations and have thus the effect of lowering the number
of degrees of freedom.

In $f(\mathbb{Q})$, however, one encounters an obstacle in determining primary
constraints if the action functional (5.40) is used: Since the momenta are
defined by taking variations of $\mathcal{S}_{f(\mathbb{Q})}$ with respect to
$\dot{N}$, $\dot{N}^{a}$ and $\dot{h}_{ab}$, one finds that they are all
proportional to $f^{\prime}(\mathbb{Q})$. Thus, it is impossible to solve for
the velocities without specifying a concrete function $f$.

This obstacle is overcome by introducing an auxiliary scalar field $\phi$ and
instead considering the equivalent action functional

$\displaystyle\mathcal{S}[N,N^{a},h_{ab},\phi]\coloneqq\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|h|}\,N\,\left[f(\phi)-f^{\prime}(\phi)\left(\phi-\mathbb{Q}\right)\right]\,.$
(6.46)

The field equations derived from this functional are

$\displaystyle\frac{2}{\sqrt{|g|}}\partial_{\alpha}\left[\sqrt{|g|}P^{\alpha}{}_{\mu\nu}f^{\prime}(\phi)\right]+f^{\prime}(\phi)\,q_{\mu\nu}-\frac{1}{2}\left[f(\phi)-f^{\prime}(\phi)\left(\phi-\mathbb{Q}\right)\right]$
$\displaystyle=0$ $\displaystyle
f^{\prime\prime}(\phi)\left(\phi-\mathbb{Q}\right)$ $\displaystyle=0\,.$
(6.47)

The first equation, which is obtained from varying the action with respect to
the metric variables, has almost the form (5.40), while the second equation is
purely algebraic and admits two solutions:

$\displaystyle f^{\prime\prime}(\phi)$ $\displaystyle=0$ or
$\displaystyle\phi-\mathbb{Q}=0\,.$ (6.48)

In the first case, we can conclude that $f(\phi)=a\,\phi+b$, for some real
constants $a$ and $b$. We can always rescale the action such that $a=1$ and
then we find that the first equation reduces precisely to the metric field
equation of STEGR plus a cosmological constant $\Lambda\propto b$. The second
case is even simpler, since it straightforwardly reproduces the metric field
equations of $f(\mathbb{Q})$ gravity. Thus, we conclude that the field
equations are equivalent to the field equations of $f(\mathbb{Q})$ for any
$f$, after we have solved the equations for $\phi$. The action (6.46) can thus
be regarded as equivalent to the action (5.40).

The benefit of working with this action is that $\mathbb{Q}$ is “pulled out”
of $f$, which allows us to study the momenta more easily. The momentum
densities computed from the action (6.46) are given by Hu:2022 ;
DAmbrosio:2023 ; Tomonari:2023

$\displaystyle\tilde{\pi}_{0}$
$\displaystyle\coloneqq\frac{\delta\mathcal{S}}{\delta\dot{N}}=0,$
$\displaystyle\tilde{\pi}^{ab}$
$\displaystyle\coloneqq\frac{\delta\mathcal{S}}{\delta\dot{h}_{ab}}=\sqrt{h}\,f^{\prime}\left(K_{ab}-K\,h_{ab}\right)$
$\displaystyle\tilde{\pi}_{a}$
$\displaystyle\coloneqq\frac{\delta\mathcal{S}}{\delta\dot{N}^{a}}=-\frac{\sqrt{h}}{N}f^{\prime\prime}\partial_{a}\phi$
$\displaystyle\tilde{\pi}_{\phi}$
$\displaystyle\coloneqq\frac{\delta\mathcal{S}}{\delta\dot{\phi}}=\frac{\sqrt{h}}{N}f^{\prime\prime}\partial_{a}N^{a}\,,$
(6.49)

where $K_{ab}$ and $K$ are the extrinsic curvature and its trace, with the
former defined as

$\displaystyle
K_{ab}\coloneqq\frac{1}{2N}\left(\mathcal{D}_{(a}N_{b)}-\dot{h}_{ab}\right)\,.$
(6.50)

It is important to note that these momenta have been obtained after having
performed a series of partial integrations in order to bring the action (6.46)
into a nicer form, which gives rise to simpler momenta. Performing
integrations by parts and dropping boundary terms is allowed, since this does
not alter the field equations and, consequently, does not alter the number of
degrees of freedom.

Notice that in the special case $f^{\prime\prime}=0$, which corresponds to
STEGR, these momenta reduce precisely to the momenta found in the Hamiltonian
analysis of STEGR in DAmbrosio:2020c  in the coincident gauge. From now on, we
shall always assume $f^{\prime\prime}\neq 0$, since we are only interested in
the degrees of freedom of the modified theory.

From the form of the momenta we can immediately infer that there are five
primary constraints. These are

$\displaystyle\tilde{C}$ $\displaystyle\coloneqq\tilde{\pi}_{0}\approx 0,$
$\displaystyle\tilde{C}_{a}$
$\displaystyle\coloneqq\tilde{\pi}_{a}+\frac{\sqrt{h}}{N}f^{\prime\prime}\partial_{a}\phi\approx
0,$ $\displaystyle\tilde{C}_{\phi}$
$\displaystyle\coloneqq\tilde{\pi}_{\phi}-\frac{\sqrt{h}}{N}f^{\prime\prime}\partial_{a}N^{a}\approx
0\,,$ (6.51)

where $\approx$ stands for “weak equality” in the sense of Dirac and Bergmann
Dirac:1950 ; DiracBook ; Bergmann:1951  (see also SundermeyerBook ;
HenneauxBook ; Wipf:1993 ). Up to this point, there is complete agreement
between Hu:2022 ; DAmbrosio:2023 ; Tomonari:2023 .

#### Primary Hamiltonian and consistency conditions

The authors of Hu:2022 ; DAmbrosio:2023 ; Tomonari:2023  also agree on the
form of the primary Hamiltonian, which is

$\displaystyle H_{\rm
P}(\Sigma_{t})=H_{0}(\Sigma_{t})+\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\left(\lambda^{0}\tilde{C}_{0}+\lambda^{a}\tilde{C}_{a}+\lambda^{\phi}\tilde{C}_{\phi}\right)\,,$
(6.52)

where $\lambda^{0}$, $\lambda^{a}$, and $\lambda^{\phi}$ are Lagrange
multipliers which enforce the primary constraints and where
$H_{0}(\Sigma_{t})$ is defined as

$\displaystyle H_{0}(\Sigma_{t})$
$\displaystyle\coloneqq\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\left(\dot{N}\tilde{\pi}_{0}+\dot{N}^{a}\tilde{\pi}_{a}+\dot{h}_{ab}\tilde{\pi}^{ab}-\mathcal{L}\right)\,.$
(6.53)

Here, $\Sigma_{t}$ refers to a Cauchy surface, which is simply a leaf in the
foliation of $\mathcal{M}$, i.e., a section of $\mathbb{R}\times\Sigma$. In
yet other words, $\Sigma_{t}$ corresponds to a $t=$const. spacelike
hypersurface.

The Dirac-Bergmann algorithm demands that the primary constraints be preserved
under the time evolution generated by the primary Hamiltonian. This means that
the following Poisson brackets have to vanish when the constraints are
satisfied:

$\displaystyle\\{H_{\rm
P},C_{I}\\}=\\{H_{0},C_{I}\\}+\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\\{C_{J},C_{I}\\}\lambda^{J}\overset{!}{\approx}0\,,$
(6.54)

where the Poisson brackets are defined as

$\displaystyle\\{F(\Psi^{a},\tilde{\pi}_{a}),G(\Psi^{A},\tilde{\Pi}_{A})\\}\coloneqq\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\left(\frac{\delta
F}{\delta\Psi^{A}}\frac{\delta G}{\delta\tilde{\Pi}_{A}}-\frac{\delta
F}{\delta\tilde{\Pi}_{A}}\frac{\delta G}{\delta\Psi^{A}}\right)\,,$ (6.55)

for some fields $\Psi^{A}$ and their conjugate momentum densities
$\tilde{\Pi}_{A}$. Equation (6.54), also called consistency condition, can
give rise to secondary constraints. That is, it can put additional constraints
on the physical field configurations and thus reduce the number of degrees of
freedom even further. It is also possible that it determines the Lagrange
multipliers. This is precisely the point where differences in the works of
Hu:2022 ; DAmbrosio:2023 ; Tomonari:2023  start to emerge.

In Hu:2022  it was argued that (6.54) leads to one secondary constraint and a
system of linear equations for the Lagrange multipliers. It was further argued
that these equations possess unique solutions, hence preventing the appearance
of further constraints. It thus follows that there are $22-6=16$ phase space
degrees of freedom or, equivalently, eight configuration space degrees of
freedom.

This conclusion was challenged by DAmbrosio:2023 ; Tomonari:2023 . It was
first realised in DAmbrosio:2023  that the analysis of Hu:2022  contains an
error. Namely, the equations for the Lagrange multipliers are first order
partial differential equations (PDEs), rather than linear algebraic equations.
This fact was overlooked in Hu:2022  and it drastically changes the situation.
First of all, the original Dirac-Bergmann algorithm for counting degrees of
freedom does not foresee the possibility that the Lagrange multipliers are
constrained by PDEs. It is silently assumed that the equations are always
linear algebraic equations.

That PDEs can arise has been observed also by other authors (see in particular
SundermeyerBook ) and it is understood that this problem is due to the
presence of spatial derivatives of field variables in the primary constraints.
The partial integrations necessary for computing the Poisson brackets in the
consistency conditions (6.54) can move partial derivatives from the field
variables onto the Lagrange multipliers. Unfortunately, the issue has received
relatively little attention and no general procedure is known for how to deal
with this scenario. In certain simple cases it is possible to solve the PDEs
and to reach sensible conclusions from a modified version of the Dirac-
Bergmann algorithm. But the general case is far from under control.

Moreover, it was shown in DAmbrosio:2023  that the PDEs for the Lagrange
multipliers are not all independent, thus leading potentially to further
complications. Several other issues were pointed out in the same work, which
is why a different route was ultimately selected to give at least an upper
bound on the degrees of freedom. Before discussing these issues and the upper
bound in more detail, we turn our attention to Tomonari:2023 . The authors of
Tomonari:2023  propose a method to avoid having to deal with PDEs for the
Lagrange multipliers. We quote directly from their text:

> “For some field $A(x)$ on a $(n+1)$-dimensional spacetime, the term
> $\sqrt{h}A(x)\partial^{(x)}_{I}\delta^{(n)}(\vec{x}-\vec{y})$, where $I$
> runs from $1$ to the dimension of the hypersurface $n$, in PB-algebras can
> be neglected by setting properly spatial boundary conditions of $A(x)$ in
> the variational principle, where $h$ is the determinant of the metric of the
> $n$-dimensional hypersurface.”

The hypersurface the authors refer to is $\Sigma_{t}$ and the term
$\sqrt{h}A(x)\partial^{(x)_{I}}\delta^{(n)}(\vec{x}-\vec{y})$ has the generic
form of the terms which lead to the aforementioned issue. That is, terms of
this form lead to PDEs for the Lagrange multipliers. By dropping all terms of
this form from the constraint algebra, the authors find indeed a linear system
of equations for the Lagrange multipliers. Their analysis leads them to
uncover three secondary and two tertiary constraints. They also conclude that
all constraints are second class, eventually leading to
$\frac{1}{2}(22-5-3-2)=6$ degrees of freedom for $f(\mathbb{Q})$.

However, as we mentioned above, this procedure is not beyond doubt. Shortly
before the quoted passage, the authors of Tomonari:2023  assert that

> “[…] when taking into account that the spatially boundary terms can always
> be neglected by imposing appropriate spatial boundary conditions in the
> variational principle and it never affects the dynamics (time evolution).”

It is correct that, given an action functional, one is allowed to drop or
neglect boundary terms because such terms do not change the field equations.
In this sense, boundary terms do indeed not affect the dynamics. However, it
is not true that spatial boundary conditions in the variational principle do
not affect the dynamics. In fact, boundary conditions constrain the solution
space of a theory! This can readily be seen from the following example: Take
one of the actions of the trinity and derive the field equations without any
further assumptions. One obtains Einstein’s field equations which, in
particular and among many others, admit the Schwarzschild and FLRW spacetimes
as solutions. Now, take the same action but demand that the fields are
asymptotically flat. This is a boundary condition and it has the effect of
eliminating certain solutions. The equations one obtains are still Einstein’s
field equations, but the FLRW spacetime is no longer in the solution space
because it does not satisfy the boundary condition (i.e., it is not
asymptotically flat). Thus, the solution space has been changed by the
imposition of boundary conditions.

Moreover, the term
$\sqrt{h}A(x)\partial^{(x)}_{I}\delta^{(n)}(\vec{x}-\vec{y})$ is being dropped
from the Poisson bracket algebra, rather than from the action. It is not clear
that such a modification does not affect the dynamics. In particular, since
the integrals in questions are integrals over Cauchy surfaces $\Sigma_{t}$,
rather than actual boundary integrals. There is nothing which prevents a
Cauchy surface to cross through the bulk of a spacetime through regions of
intense field strength. In other words, Cauchy surfaces have nothing to do
with the boundary surfaces of spacetimes, where fields are generically assumed
to be weak and thus negligible.

In conclusion, the approach of Tomonari:2023  does indeed allow one to carry
out the Dirac-Bergmann analysis of $f(\mathbb{Q})$ gravity to completion and
count degrees of freedom. However, the method used to achieve this feat is not
beyond all doubts.

#### Issues of the Dirac-Bergmann algorithm

We have mentioned issues with the Dirac-Bergmann algorithm already several
times. Specifically, what was point out in DAmbrosio:2023  is that the
standard algorithm does not foresee consistency conditions involving PDEs for
the Lagrange multipliers. Rather, it only foresees systems of linear equations
of the form

$\displaystyle M\,\vec{\lambda}+\vec{v}\overset{!}{\approx}0\,,$ (6.56)

where $\vec{\lambda}$ contains all $r$ Lagrange multipliers coming from $r$
primary constraints, $\vec{v}$ is a vector built from the fields, their
conjugate momenta, and their derivatives, and $M$ is a $r\times r$ matrix. The
symbol $\overset{!}{\approx}$ means that this equations has to be imposed and
that it only has to hold if the primary constraints hold. Three scenarios can
now emerge111111For more details on the Hamiltonian analysis of constrained
systems and the Dirac-Bergmann algorithm see, for instance, SundermeyerBook ;
HenneauxBook ; Wipf:1993 . See also the more recent Blixt:2020 ;
DAmbrosio:2023 .:

  1. 1.

If $\det{M}\not\approx 0$, the matrix $M$ is invertible and we can solve for
all Lagrange multipliers, $\vec{\lambda}=-M^{-1}\vec{v}$.

  2. 2.

If $\det{M}\approx 0$, it is not possible to solve for all Lagrange
multipliers. If $\text{rank}(M)=m<r$, there are $r-m$ vectors $\vec{u}_{D}$,
with $D\in\\{1,\dots,r-m\\}$ which are null vectors of $M$. That is, these
vectors satisfy $M\vec{u}_{D}=0$. One can show that one can consistently solve
for some of the Lagrange multipliers if and only if
$\vec{u}_{D}^{\intercal}\vec{v}\approx 0$. If this last equation does not
hold, one has to impose it. This leads to additional, so-called secondary
constraints.

  3. 3.

If $\det{M}\approx 0$ and $\vec{u}_{D}^{\intercal}\vec{v}\approx 0$, it is
possible that the consistency condition is trivially satisfied or that it
leads to secondary constraints.

It should be noted that in the cases 2. and 3., some of the Lagrange
multipliers inevitably remain undetermined. Since these multipliers appear in
the primary Hamiltonian, which generates the dynamics, it means that there is
some indeterminacy in the time evolution of the system. This indeterminacy is
well-understood to be related to gauge symmetries. Thus, because of this
connection to gauge symmetry, it is not alarming when the primary Hamiltonian
depends on some arbitrary fields.

This brings us now to the case of PDEs for Lagrange multipliers. These PDEs
can arise when the constraints contain spatial derivatives of field variables.
Because one has to perform an integration by parts in order to compute the
second Poisson bracket in (6.54), one ends up with terms of the form
$\partial\lambda$. We emphasize that the presence of partial derivatives in
the constraints is only a necessary but not a sufficient condition. After all,
also the constraints of electromagnetism and GR possess spatial derivatives,
but they do not cause any problems. This has also been discussed in
DAmbrosio:2023  .

However, if it happens that the partial derivative has been moved onto the
Lagrange multiplier, the system of PDEs has generically the form

$\displaystyle\sum_{i=1}^{d}M^{(i)}\partial_{i}\vec{\lambda}+N\vec{\lambda}+\vec{v}\overset{!}{\approx}0\,.$
(6.57)

We have assumed that there are $d$ spatial dimensions and consequently there
are $d$ matrices $M^{(i)}$ of dimensions $r\times r$ which multiply the $d$
different first order spatial derivatives $\partial_{i}\vec{\lambda}$. We have
also introduced a $r\times r$ matrix $N$ and a $r$-dimensional vector
$\vec{v}$. The $r$ Lagrange multipliers $\vec{\lambda}$ all depend on the $d$
spatial coordinates and time.

As is well-known, in order to obtain a unique solution to a PDE one has to
impose boundary conditions or initial value conditions. But this raises the
question: How do these initial value or boundary conditions affect the primary
Hamiltonian? To be more explicit: We are completely free in choosing these
conditions. But no matter what we choose, this choice will affect the primary
Hamiltonian and it will depend on the field values we arbitrarily chose for
$\vec{\lambda}$. In turn, these field values will show up in the time
evolution of the system. Is there a relation to gauge transformation, as there
is one in the standard case discussed above?

If there is, it is not completely clear how it will manifest. Observe that
there is a difference between $\vec{\lambda}$ not being completely determined
by the linear equations and $\vec{\lambda}$ depending on arbitrary choices for
its initial values or boundary values: In the first case, we are forced to
introduce arbitrary fields which depend on space and time. In the second case,
we arbitrarily fix for instance the $x^{1}$ axis as “initial surface” and
specify initial values on that surface. This amounts to specifying functions
of time and $d-1$ spatial coordinates, since one coordinate is fixed. Albeit,
the fixation of $x^{1}$ was arbitrary.

Nevertheless, we are confronted with open questions and the answers are not
clear. This means that we have no reliable way of dealing with these PDEs such
that we can count the degrees of freedom in a way we can trust.

There is a further problem, also brought to light through the analysis of
DAmbrosio:2023 . Namely, the PDEs for the Lagrange multipliers that emerge in
$f(\mathbb{Q})$ do not give rise to a well-posed initial value formulation.
This means that the PDEs are under-determined. Or, in yet other words, even if
we prescribe initial values for $\vec{\lambda}$, it is not possible to find a
unique solution. We can only determine some of the components of
$\vec{\lambda}$ and they will depend on the un-determined components.

It is known that this happens in gauge theories and that this issue is related
to the freedom of performing gauge transformations. For instance, in
electromagnetism formulated in terms of a vector potential $A^{\mu}$, the
field equations are under-determined. This is tantamount to saying that the
initial value problem is not well-posed. The resolution is to realize that the
field equations determine all components of $A^{\mu}$, except one. Thus, by
imposing a gauge fixing, this issue is resolved and one obtains a unique
solution. Furthermore, as is well-known, this arbitrary gauge fixing does not
affect physical observables.

However, what does this under-determination of the PDEs mean in the context of
Lagrange multipliers? Is there a connection to gauge symmetries of the theory?
All these questions deserve more attention and a detailed analysis, so that we
can trust the results obtained from a modified version of the Dirac-Bergmann
algorithm.

#### Upper and lower bound on the degrees of freedom

Given the obstacles mentioned above, which emerge from applying the Dirac-
Bergmann algorithm to $f(\mathbb{Q})$ gravity, the authors of DAmbrosio:2023
opted for a different approach. Using the so-called kinetic matrix, it was
shown that $f(\mathbb{Q})$ propagates at most seven degrees of freedom.
Together with the four degrees of freedom found through cosmological
perturbation theory in BeltranJimenez:2019 , we have a clear lower and upper
bound.
The kinetic matrix approach sidesteps the issues discussed so far since it is
independent of the Hamiltonian analysis and it is directly concerned with the
field equations. The basic idea can easily be explained with a simple example:
Consider a field theory in $1+1$ dimensions with second order field equations.
Let’s say that the field $\Psi$ in question has two components,
$\Psi=(\Psi_{1},\Psi_{2})$, which are functions of the coordinates
$x^{\mu}=(x^{0},x^{1})$. The coordinate $x^{0}$ plays the role of a time
coordinate, while $x^{1}$ is the spatial coordinate. Then, the second order
field equations can be written as

$\displaystyle\underbrace{\begin{pmatrix}\mathcal{K}_{11}&\mathcal{K}_{12}\\\
\mathcal{K}_{21}&\mathcal{K}_{22}\end{pmatrix}}_{\eqqcolon\mathcal{K}}\partial^{2}_{0}\begin{pmatrix}\Psi_{1}\\\
\Psi_{2}\end{pmatrix}+\underbrace{\begin{pmatrix}\mathcal{M}^{(1)}_{11}&\mathcal{M}^{(1)}_{12}\\\
\mathcal{M}^{(1)}_{21}&\mathcal{M}^{(1)}_{22}\end{pmatrix}}_{\eqqcolon\mathcal{M}^{(1)}}\partial_{0}\partial_{1}\begin{pmatrix}\Psi_{1}\\\
\Psi_{2}\end{pmatrix}+\underbrace{\begin{pmatrix}\mathcal{P}^{(11)}_{11}&\mathcal{P}^{(11)}_{12}\\\
\mathcal{P}^{(11)}_{21}&\mathcal{P}^{(11)}_{22}\end{pmatrix}}_{\eqqcolon\mathcal{P}^{(11)}}\partial^{2}_{1}\begin{pmatrix}\Psi_{1}\\\
\Psi_{2}\end{pmatrix}$ $\displaystyle+\text{ lower order derivatives }=0\,.$
(6.58)

We have introduced three different matrices which multiply the three different
second order derivatives, $\mathcal{K}$, $\mathcal{M}^{(1)}$, and
$\mathcal{P}^{(11)}$. The notation will become clear later on.

Now, what does it mean to solve this system of PDEs? First of all, since the
system is second order, we have to prescribe two initial value conditions, if
we hope to find a unique solution. These conditions are

$\displaystyle\left.\Psi\right|_{t=t_{0}}$ $\displaystyle=F(x^{1})$ and
$\displaystyle\left.\partial_{0}\Psi\right|_{t=t_{0}}$
$\displaystyle=G(x^{1})\,.$ (6.59)

In other words, we prescribe $\Psi$ on a $t=t_{0}$ hypersurface and we
prescribe what its time derivative is on that surface. Observe that if we
evaluate the above equation on that particular surface, we know every term,
except the first one. In fact, we find

$\displaystyle\left.\mathcal{K}\,\partial_{0}^{2}\Psi\right|_{t=t_{0}}+\mathcal{M}^{(1)}\partial_{1}G(x^{1})+\mathcal{P}^{(11)}\partial^{2}_{1}F(x^{1})+\text{
other terms which we know on $t=t_{0}$ }=0\,.$ (6.60)

Notice that since $F(x^{1})$ and $G(x^{1})$ are known functions of $x^{1}$, we
also know what their derivatives with respect to $x^{1}$ are. What we do not
know, is what $\partial^{2}_{2}\Psi$ equals to on the $t=t_{0}$ surface. That
is where the field equations come into play. We can find out what
$\partial^{2}_{2}\Psi$ is, if we can solve the above equations for the second
order time derivatives. That is, if we can write

$\displaystyle\left.\partial^{2}_{0}\Psi\right|_{t=t_{0}}=-\mathcal{K}^{-1}\left(\mathcal{M}^{(1)}\partial_{1}G(x^{1})+\mathcal{P}^{(11)}\partial^{2}_{1}F(x^{1})+\text{
the lower order terms}\right)\,,$ (6.61)

where $\mathcal{K}^{-1}$ is the inverse of $\mathcal{K}$. Hence, if we can
invert $\mathcal{K}$, we can formally integrate the PDE and find out what
$\Psi$ is away from the $t=t_{0}$ surface (see also DAmbrosio:2023  for a more
technical and detailed explanation of this point). What happens if we can not
invert the matrix $\mathcal{K}$? To answer the question, consider the
following case:

$\displaystyle\begin{pmatrix}\mathcal{K}_{11}&\mathcal{K}_{12}\\\
0&0\end{pmatrix}\partial^{2}_{0}\Psi+\mathcal{M}^{(1)}\partial_{0}\partial_{1}\Psi+\mathcal{P}^{(11)}\partial^{2}_{1}\Psi+\text{
lower order derivatives }=0\,.$ (6.62)

Clearly, $\mathcal{K}$ has only rank one and is therefore not invertible.
Observe that this has two implications:

  1. 1.

If we explicitly write out the vector-matrix product, we see that the second
equation has no second order time derivatives. It is thus just a constraint
equation, rather than a dynamical equation.

  2. 2.

The first equation can still be solved for, say, $\partial^{2}_{0}\Psi_{1}$,
but $\partial^{2}_{0}\Psi_{2}$ then appears on the right hand side. Since
there is no equation which determines $\partial^{2}_{0}\Psi_{2}$, we have to
prescribe it by hand. Otherwise we cannot integrate the equation for
$\partial^{2}_{0}\Psi_{1}$. This is what generically happens in gauge
theories.

We learn an important lesson from this simple example: Whether a given second
order PDE can be solved or not is determined by the matrix $\mathcal{K}$ which
multiplies the second order time derivatives. We can generalize this insight
in the following way.

Let spacetime be $d+1$ dimensional and let $\Psi$ be a vector which contains
the $n$ components of a tensor field (that could be a vector, or a metric, or
any other tensor). Then we can write the second order PDE for the field in
question as

$\displaystyle\mathcal{K}\,\partial^{2}_{0}\Psi+\sum_{i=1}^{d}\mathcal{M}^{(i)}\partial_{0}\partial_{i}\Psi+\sum_{i=1,i\leq
j}^{d}\sum_{j=1}^{d}\mathcal{P}^{(ij)}\partial_{i}\partial_{j}\Psi+\text{
lower order terms }=0\,,$ (6.63)

where we have introduced the $n\times n$ kinetic matrix
$\boldsymbol{\mathcal{K}}$, $d$ so-called mixing matrices $\mathcal{M}^{(i)}$,
each of dimension $n\times n$, and $\frac{d(d+1)}{2}$ potential matrices
$\boldsymbol{\mathcal{P}^{(ij)}}$, also each of dimension $n\times n$. If the
kinetic matrix is invertible, we obtain a unique solution for the PDE.
However, if $\mathcal{K}$ is not invertible, we find constraint equations.
From our simple example it is clear that the number of constraint equation is
the same as the number of rows of $\mathcal{K}$ which are zero, in some sense.
Of course, a matrix $\mathcal{K}$ which is degenerate does not always have
rows filled with zero. Rather, it has rows which are linear combinations of
other rows. Thus, the mathematically precise statement is this121212Further
mathematical details can be found in MiersemannBook ; DAmbrosio:2023  and in
the Appendix of DAmbrosio:2022 , which also provides ample illustrations and
examples.:

$\displaystyle\text{If rank}(\mathcal{K})=r\leq
n\quad\Longrightarrow\quad\text{There are $n-r$ constraint equations.}$ (6.64)

It thus follows that by determining the rank of the kinetic matrix, we can
infer how many constraints there are at least. There can be more constraints
than just the $n-r$ which follow from the rank of $\mathcal{K}$ since
integrability conditions can occur. Given that each constraint reduces the
number of degrees of freedom by one, we finally reach the conclusion

$\displaystyle\text{If rank}(\mathcal{K})=r\leq
n\quad\Longrightarrow\quad\text{There are at most $r$ degrees of freedom.}$
(6.65)

In DAmbrosio:2023 , this insight was used to show that the number of degrees
of freedom in $f(\mathbb{Q})$ is at most seven. The argument is as follows:
The basic variables to consider are the ten metric components $g_{\mu\nu}$ and
the four functions $\xi^{\alpha}$, which parametrize the flat and torsionless
connection. Furthermore, if the metric field equations are satisfied, then the
Bianchi identities imply that the connection field equations are automatically
satisfied as well. Thus, we can completely focus on the metric field
equations. If we work in coincident gauge, we remove the four functions
$\xi^{\alpha}$ from our considerations without accidentally killing degrees of
freedom. In fact, all degrees of freedom are now encoded in the ten metric
components, whose dynamics is described by the metric field equations. Thus,
from the originally $10+4$ potential degrees of freedom, we are left with only
ten.

It was then shown that the rank of the kinetic matrix of the metric field
equations is seven, provided that $f^{\prime\prime}\neq 0$. If
$f^{\prime\prime}=0$, the rank is six, which had to be expected since the
Einstein equations contain $10-6=4$ constraints. This is a nice consistency
check.

As a final remark, we point out that the kinetic matrix approach can in
principle also be used to figure out the precise number of degrees of freedom.
This involves also considerations regarding the mixing matrices and the
potential matrices with highly involved computations. For more details on this
outlook we refer the reader to DAmbrosio:2023 .

## 7 Summary

Gravitational phenomena arise from curved spacetime, a concept made possible
by the equivalence principle. This implies that gravity is independent of
matter type. Within the framework of geometry, curvature is just one aspect of
a manifold’s affine properties. In addition to curvature, there are two other
fundamental objects associated with the connection of a metric space: torsion
and non-metricity. In standard General Relativity following Einstein, both
non-metricity and torsion are absent. Embracing the geometric nature of
gravity as advocated by the equivalence principle prompts us to explore
different ways to represent gravity. In one equivalent description of General
Relativity, we envision a flat spacetime with a metric but an asymmetric
connection, where gravity is solely attributed to torsion. Alternatively, we
can construct a third equivalent representation of GR on a flat spacetime
without torsion, attributing gravity to non-metricity. Thus, the same
fundamental physical theory, GR, can be articulated through the Einstein-
Hilbert action, the Teleparallel Equivalent of GR action, or the Symmetric
Teleparallel Equivalent of GR action Heisenberg:2018 ; BeltranJimenez:2019c .

The fundamental foundation of these geometric interpretations paves the way
for innovative approaches to modified gravity. These equivalent descriptions
of General Relativity involving curvature, torsion, and non-metricity provide
diverse starting points for modified gravity theories when scalar quantities
are transformed into arbitrary functions. It’s worth noting that quadratic
non-metricity and torsion Lagrangian with detuned arbitrary 5 and 3
parameters, respectively, can also be considered, albeit with anticipated
complexities. In this review, our primary focus lay on $f(\mathbb{Q})$
theories BeltranJimenez:2017b .

We began by establishing the foundational elements of geometry. Starting with
the basic manifold, we incorporated coordinates, points, and curves. Tensor
fields, including scalars and vectors, were introduced on this manifold. To
facilitate the comparison of vector fields at different points, we introduced
the affine connection, delving into its general properties and the associated
tensor quantities: curvature and torsion tensors. To incorporate the concept
of distance, we introduced the metric, which in turn allowed us to define the
non-metricity tensor. With these components in place, we were well-prepared to
delve into the core principles of General Relativity. We’ve clearly
demonstrated that the theory of General Relativity can be formulated in three
distinct ways: as a curvature theory, a torsion theory, or a non-metricity
theory. We’ve examined the key distinctions, addressed subtle nuances, and
explored the consistent coupling of matter fields within these frameworks. In
doing so, we’ve identified cases where the minimal coupling principle proves
inadequate.

Next, we examined strategies for departing from the principles of General
Relativity in a consistent manner. We explored two complementary approaches:
one involving generic quadratic Lagrangians with arbitrary parameters, and the
other by transforming GR scalars into nonlinear functions. These approaches
led us to derive various theories of modified gravity. Given our primary focus
on $f(Q)$ theories, we provided an overview of the fundamental characteristics
of various modifications before returning our attention to $f(\mathbb{Q})$
theories. Specifically, we introduced the defining Lagrangian, derived the
corresponding field equations, and delved into discussions regarding its
symmetries and Bianchi identities. Having gained a solid grasp of the
overarching principles of the covariant theory, our focus shifted towards
practical applications in cosmology and astrophysics. We specifically examined
both cosmological and spherically symmetric backgrounds, utilizing symmetry
reduction principles to establish the necessary conditions for the metric and
the connection that align with the background symmetries. This systematic
approach enabled us to explore the potential derivation of novel cosmological
and black hole solutions within the framework of $f(\mathbb{Q})$ theories.

Our motto is Qravity: Gravity with $\mathbb{Q}$. We firmly believe that the
intricate structure inherent in the geometric framework of gravity can unlock
fresh and captivating perspectives, leading us into uncharted realms and
confronting the challenges of conventional formulations. Let us wholeheartedly
embrace this captivating new geometry.

## Acknowledgements

LH is supported by funding from the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme grant agreement
No 801781. LH further acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES
Excellence Cluster).

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