Indeed, one recognizes the first four terms to be related to the homogeneous
piece in the coordinate transformation of the connection, while the last term
is produced by the inhomogeneous piece. Also, one should note that even tough
the connection is not a tensor, its Lie derivative is a $(1,2)$ tensor! This
is also nicely evident from equation (3.51), where the right hand side is
completely constructed from tensorial quantities.

The formulas (3.51) play a role in characterizing symmetries of metric-affine
geometries, as alluded to before. In the works Hohmann:2019 ; DAmbrosio:2021b
; Hohmann:2021 ; DAmbrosio:2021 ; Heisenberg:2022 ; DAmbrosio:2023b ,
symmetries of metric-affine geometries were defined as follows: Let
$\phi_{s}:\mathbb{R}\times\mathcal{M}\to\mathcal{M}$ be a $1$-parameter family
of diffeomorphisms which satisfies $\phi_{0}=\text{id}$,
$\phi_{s}\circ\phi_{t}=\phi_{s+t}$, and which is smooth in the parameter $s$.
This $1$-parameter family is a symmetry of a metric-affine geometry
$\boldsymbol{(\mathcal{M},g,\Gamma)}$ if

$\displaystyle\begin{cases}\phi^{*}_{s}g_{\mu\nu}&\overset{!}{=}\quad
g_{\mu\nu}\\\
\phi^{*}_{s}\Gamma^{\alpha}{}_{\mu\nu}&\overset{!}{=}\quad\Gamma^{\alpha}{}_{\mu\nu}\end{cases}$
(3.53)

This is called the symmetry condition. It demands that neither the metric nor
the connection change under the action of the diffeomorphism. It is important
that the connection appears in this definition. This ensures that all objects
constructed from the connection, such as curvature, torsion, and non-
metricity, respect the symmetry generated by $\phi_{s}$.

Since $\phi_{s}$ is smooth in $s$, we can also consider the infinitesimal
symmetry condition, obtained by expanding the original symmetry condition to
first order in $s$ around $s=0$. It reads

$\displaystyle\begin{cases}\mathcal{L}_{\xi}g_{\mu\nu}&\overset{!}{=}\quad
0\\\ \mathcal{L}_{\xi}\Gamma^{\alpha}{}_{\mu\nu}&\overset{!}{=}\quad
0\end{cases}$ with
$\displaystyle\xi\coloneqq\left.\frac{\mathrm{d}}{\mathrm{d}s}\phi_{s}\right|_{s=0}\,,$
(3.54)

where $\xi$ is the vector field which generates the flow $\phi_{s}$. It is
often called the generating vector field. These symmetry conditions and the
Lie derivatives of metric and connection will reappear when we discuss
cosmology and black holes in subsections 6.1 and 6.2.

###  3.7 Integration in the Presence of Torsion and Non-Metricity: The
Generlized Gauss Theorem

Integration on manifolds $\mathcal{M}$ is a subject usually covered in courses
on calculus of several variables or differential geometry. The theorems of
Gauss and Stokes, into which this subject culminates, can be assumed to be
familiar to all readers due to their widespread use in physics.

What concerns us here, however, is how non-trivial geometric features,
characterized by torsion and non-metricity, affect Gauss’ theorem. The
importance of this investigation lies in the fact that Gauss’ theorem appears
in variational principles or discussions of conserved charges Delhom:2020 .

Let us begin by recalling that on a Riemannian manifold $(\mathcal{M},g)$,
where torsion and non-metricity both vanish, we are uniquely left with the
Levi-Civita connection
$\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}$ and the covariant
derivative operator $\mathcal{D}_{\mu}$ it induces. Using the easy to verify
identity

$\left\\{\genfrac{}{}{0.0pt}{}{\lambda}{\lambda\mu}\right\\}=\partial_{\mu}\log\sqrt{|g|}\,,$
(3.55)

one can re-express the divergence of a vector field $v^{\mu}$ as

$\mathcal{D}_{\mu}v^{\mu}=\frac{1}{\sqrt{|g|}}\partial_{\mu}\left(\sqrt{|g|}v^{\mu}\right)\,.$
(3.56)

Gauss’ theorem, in its familiar form, emerges from this simple identity
PoissonBook

$\displaystyle\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\mathcal{D}_{\mu}v^{\mu}$
$\displaystyle=\int_{\mathcal{M}}\mathrm{d}^{4}x\,\partial_{\mu}\left(\sqrt{|g|}v^{\mu}\right)=\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\varepsilon\,n_{\mu}v^{\mu}\,.$
(3.57)

Here, $\partial\mathcal{M}$ denotes the boundary of $\mathcal{M}$, $n^{\mu}$
is the outward-pointing normal vector to $\partial\mathcal{M}$,
$\varepsilon\coloneqq n_{\mu}n^{\mu}=\pm 1$, and $h$ is the determinant of the
induced metric on $\partial\mathcal{M}$ (obtained by pulling back $g_{\mu\nu}$
to $\partial\mathcal{M}$).

Clearly, this result crucially depends on the identity (3.55) for the Levi-
Civita connection. Hence, if we change the connection, we can no longer expect
to find the same form of Gauss’ theorem as the one given in (3.57).

However, an analogous theorem holds for a generic metric-affine geometry
$(\mathcal{M},g,\Gamma)$. To see that, we make use of the decomposition (3.47)
of the connection which we encountered in the previous subsection:

$\Gamma^{\alpha}{}_{\mu\nu}=\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}+L^{\alpha}{}_{\mu\nu}+K^{\alpha}{}_{\mu\nu}\,.$
(3.58)

This allows us to write the divergence of the vector field $v^{\mu}$ as

$\nabla_{\mu}v^{\mu}=\mathcal{D}_{\mu}v^{\mu}+L^{\lambda}{}_{\lambda\mu}v^{\mu}+K^{\lambda}{}_{\lambda\mu}v^{\mu}\,.$
(3.59)

Next, we use the easy provable relations

$\displaystyle L^{\lambda}{}_{\lambda\mu}$ $\displaystyle=-\frac{1}{2}Q_{\mu}$
and $\displaystyle K^{\lambda}{}_{\lambda\mu}$ $\displaystyle=-T_{\mu}\,,$
(3.60)

and combine these with the identity (3.56) to finally obtain

$\nabla_{\mu}v^{\mu}=\frac{1}{\sqrt{|g|}}\partial_{\mu}\left(\sqrt{|g|}v^{\mu}\right)-\left(\frac{1}{2}Q_{\mu}+T_{\mu}\right)v^{\mu}\,.$
(3.61)

This form of the divergence of a vector field lends itself to formulating the
analogue of Gauss’ theorem for a generic metric-affine geometry. We find that
the generalized Gauss’ theorem for a generic metric-affine geometry
$(\mathcal{M},g,\Gamma)$ takes the form

$\boxed{\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\nabla_{\mu}v^{\mu}=\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\varepsilon\,n_{\mu}v^{\mu}-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\left(\frac{1}{2}Q_{\mu}+T_{\mu}\right)v^{\mu}}$
(3.62)

This can also be re-cast into a slightly different form by noticing that the
covariant derivative of the square root of the metric determinant is given by

$\nabla_{\mu}\sqrt{|g|}=\frac{1}{2}\sqrt{|g|}\,Q_{\mu}\,.$ (3.63)

This allows us to phrase Gauss’ theorem as a statement about the divergence of
a vector density, $\nabla_{\mu}\left(\sqrt{|g|}v^{\mu}\right)$, rather than
about the divergence of a vector field, $\nabla_{\mu}v^{\mu}$. Concretely, the
generalized Gauss theorem can be equivalently stated as

$\boxed{\int_{\mathcal{M}}\mathrm{d}^{4}x\,\nabla_{\mu}\left(\sqrt{|g|}\,v^{\mu}\right)=\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\varepsilon\,n_{\mu}v^{\mu}-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,T_{\mu}v^{\mu}}$
(3.64)

This concludes our discussion of the generalized Gauss theorem and we proceed
with presenting important and useful identities for metric-affine geometries.

###  3.8 Collection of Geometric Identities

In concrete computations involving the covariant derivative $\nabla_{\mu}$
with respect to a generic affine connection $\Gamma^{\alpha}{}_{\mu\nu}$, it
is often necessary to commute two such operators in order to obtain simpler
expressions or expressions with a more transparent geometric meaning. Recall
that the covariant derivative can act on scalar, vector, and tensor fields (or
densities). Thus, the simplest case to consider is the action of the
commutator
$[\nabla_{\mu},\nabla_{\nu}]\coloneqq\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu}$
on a scalar field $f$. Simply by using the basic definitions given in (3.32),
one finds

$\boxed{[\nabla_{\mu},\nabla_{\nu}]f=-T^{\lambda}{}_{\mu\nu}\partial_{\lambda}f}$
(3.65)

Thus, the commutator acting on scalar fields vanishes if and only if the
torsion tensor vanishes. Next, we consider the commutator
$[\nabla_{\mu},\nabla_{\nu}]$ acting on a vector field $v^{\alpha}$. It is
again only necessary to use the basic definitions given in (3.22) and (3.32),
but the computations become longer. What they boil down to is the identity

$\boxed{[\nabla_{\mu},\nabla_{\nu}]v^{\alpha}=R^{\alpha}{}_{\lambda\mu\nu}v^{\lambda}-T^{\lambda}{}_{\mu\nu}\nabla_{\lambda}v^{\alpha}}$
(3.66)

Observe that when torsion vanishes, the above identity reduces to the form
familiar from Riemannian geometry:

$\displaystyle[\nabla_{\mu},\nabla_{\nu}]v^{\alpha}=R^{\alpha}{}_{\lambda\mu\nu}v^{\lambda}\,.$
(3.67)

However, $R^{\alpha}{}_{\lambda\mu\nu}$ is not the curvature tensor with
respect to the Levi-Civita connection, since the connection could be metric-
incompatible, i.e., it could have a non-zero non-metricity tensor.

It is also useful to prove an analogous identity for the commutator acting on
a $1$-form $\omega_{\alpha}$. In this case, one finds

$\boxed{[\nabla_{\mu},\nabla_{\nu}]\omega_{\alpha}=-R_{\alpha}{}^{\lambda}{}_{\mu\nu}\omega_{\lambda}-T^{\lambda}{}_{\mu\nu}\nabla_{\lambda}\omega_{\alpha}}$
(3.68)

Note the appearance of a minus sign in front of the curvature tensor! With the
identities (3.66) and (3.68) at our disposal, we can easily prove that the
commutator $[\nabla_{\mu},\nabla_{\nu}]$ acting on a tensor field of type
$(p,q)$ is given by

$\boxed{\begin{aligned}
S^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}=&\phantom{..-}R^{\mu_{1}}{}_{\lambda\mu\nu}S^{\lambda\dots\mu_{p}}{}_{\lambda\dots\nu_{q}}+\dots+R^{\mu_{p}}{}_{\lambda\mu\nu}S^{\mu_{1}\dots\lambda}{}_{\lambda\dots\nu_{q}}\\\
&-R_{\nu_{1}}{}^{\lambda}{}_{\mu\nu}S^{\mu_{1}\dots\mu_{p}}{}_{\lambda\dots\nu_{q}}-\dots-
R_{\nu_{q}}{}^{\lambda}{}_{\mu\nu}S^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\lambda}\\\
&-T^{\lambda}{}_{\mu\nu}\nabla_{\lambda}S^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}\end{aligned}}$
(3.69)

This identity follows from the previous two mentioned above together with the
fact that a $(p,q)$ tensor lives in the tensor product space
$T\mathcal{M}^{\otimes p}\otimes T^{*}\mathcal{M}^{\otimes q}$.

Identity (3.69) is very general and covers almost every case of interest. The
cases not covered by identity (3.69) only involve tensor densities. To remedy
that, we need to understand how the commutator acts on the metric tensor and,
more importantly, on its determinant. For the metric tensor, it follows from
the definition of the non-metricity tensor and from identity (3.69) that we
can express the commutator as

$\boxed{[\nabla_{\mu},\nabla_{\nu}]g_{\alpha\beta}=2\nabla_{[\mu}Q_{\nu]\alpha\beta}=-2R_{(\alpha\beta)\mu\nu}-T^{\lambda}{}_{\mu\nu}Q_{\lambda\alpha\beta}}$
(3.70)

Using
$\nabla_{\mu}|g|^{\frac{w}{2}}=\frac{w}{2}|g|^{\frac{w}{2}}g^{\alpha\beta}\nabla_{\mu}g_{\alpha\beta}$,
where the integer $w\geq 0$ is the density weight introduced in 2.2, together
with the above identity, one finds that the commutator acting on
$|g|^{\frac{w}{2}}$ is given by

$\displaystyle\boxed{[\nabla_{\mu},\nabla_{\nu}]|g|^{\frac{w}{2}}=\frac{w}{2}|g|^{\frac{w}{2}}g^{\alpha\beta}[\nabla_{\mu},\nabla_{\nu}]g_{\alpha\beta}=w\,|g|^{\frac{w}{2}}\nabla_{[\mu}Q_{\nu]\alpha\beta}=w\,|g|^{\frac{w}{2}}\nabla_{[\mu}Q_{\nu]}}$
(3.71)

In the last step we used $Q_{\nu}\coloneqq g^{\alpha\beta}Q_{\nu\alpha\beta}$.
This finally allows us to determine the action of the commutator on a tensor
density of type $(p,q)$ and weight $w$:

$\displaystyle\boxed{[\nabla_{\mu},\nabla_{\nu}]\left(|g|^{\frac{w}{2}}S^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}\right)=|g|^{\frac{w}{2}}\left([\nabla_{\mu},\nabla_{\nu}]S^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}+w\,\nabla_{[\mu}Q_{\nu]}\right)}$
(3.72)

where the first commutator on the right hand side is of course given by
identity (3.69). Observe that all previous identities follow from this one. By
setting $w=0$, we recover the identities for tensor fields (as opposed to
tensor densities), including the one for scalar fields, which corresponds to
$w=0$ together with $p=q=0$.

Finally, we remark that the covariant derivative $\nabla_{\mu}$ with respect
to a generic affine connection $\Gamma^{\alpha}{}_{\mu\nu}$ satisfies the
Jacobi identity:

$\boxed{[\nabla_{\alpha},[\nabla_{\beta},\nabla_{\gamma}]]+[\nabla_{\beta},[\nabla_{\gamma},\nabla_{\alpha}]]+[\nabla_{\gamma},[\nabla_{\alpha},\nabla_{\beta}]]=0}$
(3.73)

We now turn to important identities involving the curvature tensor. Some of
these identities will play an important role in subsections 4.2 and 4.3, where
they greatly simplify and illuminate the definition of the Teleparallel
Equivalent of GR (TEGR) and the Symmetric Teleparallel Equivalent of GR
(STEGR), respectively.

To begin with, we remark that it follows from the definitions of the curvature
tensor, the torsion tensor, and the covariant derivative, that the following
identities hold

$\boxed{\begin{aligned} R^{\alpha}{}_{\mu(\nu\rho)}&=0\\\
R^{\mu}{}_{[\alpha\beta\gamma]}-\nabla_{[\alpha}T^{\mu}{}_{\beta\gamma]}+T^{\lambda}{}_{[\alpha\beta}T^{\mu}{}_{\gamma]\lambda}&=0\\\
\nabla_{[\alpha}R^{\mu}{}_{|\nu|\beta\gamma]}-T^{\lambda}{}_{[\alpha\beta}R^{\mu}{}_{|\nu|\gamma]\lambda}&=0\end{aligned}}$
(3.74)

When we discuss teleparallel theories of gravity, it will prove to be useful
to know how to relate the curvature tensor $R^{\alpha}{}_{\mu\nu\rho}(\Gamma)$
of the affine connection to the curvature tensor
$\mathcal{R}^{\alpha}{}_{\mu\nu\rho}(g)$ of the Levi-Civita connection. To
establish a relationship between the two, we point out that adding a $(1,2)$
tensor to a given connection $\Gamma^{\alpha}{}_{\mu\nu}$ results in a new and
equally valid connection

$\displaystyle\hat{\Gamma}^{\alpha}{}_{\mu\nu}=\Gamma^{\alpha}{}_{\mu\nu}+\Omega^{\alpha}{}_{\mu\nu}\,.$
(3.75)

This follows directly from the transformation behaviour of a connection under
changes of coordinates (cf. equation (2.64)). We can then compute the
curvature tensor of the connection $\hat{\Gamma}^{\alpha}{}_{\mu\nu}$ and
express it in terms of the curvature of $\Gamma^{\alpha}{}_{\mu\nu}$ as well
as contributions coming from the tensor $\Omega^{\alpha}{}_{\mu\nu}$. One
finds

$\displaystyle\boxed{\hat{R}^{\alpha}{}_{\beta\mu\nu}=R^{\alpha}{}_{\beta\mu\nu}+T^{\lambda}{}_{\mu\nu}\Omega^{\alpha}{}_{\lambda\beta}+2\mathcal{D}_{[\mu}\Omega^{\alpha}{}_{\nu]\beta}+2\Omega^{\alpha}{}_{[\mu|\lambda|}\Omega^{\lambda}{}_{\nu]\beta}}$
(3.76)

where $\mathcal{D}$ is the covariant derivative with respect to the Levi-
Civita connection, as usual. This identity allows us to easily find a relation
between $R^{\alpha}{}_{\mu\nu\rho}(\Gamma)$ and
$\mathcal{R}^{\alpha}{}_{\mu\nu\rho}(g)$. We simply assume that the original
connection was the Levi-Civita one, i.e.,
$\Gamma^{\alpha}{}_{\mu\nu}=\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}$,
while the tensor $\Omega^{\alpha}{}_{\mu\nu}$ is the sum of contortion and
disformation tensor,
$\Omega^{\alpha}{}_{\mu\nu}=K^{\alpha}{}_{\mu\nu}+L^{\alpha}{}_{\mu\nu}$.
Thus, we find that the two curvature tensors are related to each other via

$\boxed{\begin{aligned}
R^{\alpha}{}_{\mu\nu\rho}(\Gamma)&=\mathcal{R}^{\alpha}{}_{\mu\nu\rho}(g)+T^{\lambda}{}_{\nu\rho}K^{\alpha}{}_{\lambda\mu}+2\mathcal{D}_{[\nu}K^{\alpha}{}_{\rho]\mu}+T^{\lambda}{}_{\nu\rho}L^{\alpha}{}_{\lambda\mu}+2\mathcal{D}_{[\nu}L^{\alpha}{}_{\rho]\mu}\\\
&\phantom{=\mathcal{R}^{\alpha}{}_{\mu\nu\rho}}+2K^{\alpha}{}_{[\nu|\lambda}K^{\lambda}{}_{\rho]\mu}+2L^{\alpha}{}_{[\nu|\lambda}K^{\lambda}{}_{\rho]\mu}+2K^{\alpha}{}_{[\nu|\lambda}L^{\lambda}{}_{\rho]\mu}+2L^{\alpha}{}_{\nu[\lambda}L^{\lambda}{}_{\rho]\mu}\end{aligned}}$
(3.77)

The actual identity which will prove to be useful in teleparallel theories of
gravity is the one which relates the Ricci scalars of the two connections. It
reads

$\displaystyle\boxed{R(\Gamma)=\mathcal{R}(g)+\mathbb{T}+\mathbb{Q}+T^{\rho\mu\nu}Q_{\mu\nu\rho}-T^{\mu}Q_{\mu}+T^{\mu}\bar{Q}_{\mu}+\mathcal{D}_{\alpha}\left(Q^{\alpha}-\bar{Q}^{\alpha}+2T^{\alpha}\right)}$
(3.78)

where we have introduced the torsion scalar and non-metricity scalar,
respectively defined by

$\displaystyle\mathbb{T}$
$\displaystyle\coloneqq\frac{1}{2}\left(\frac{1}{4}T_{\alpha\mu\nu}+\frac{1}{2}T_{\mu\alpha\nu}-g_{\alpha\mu}T_{\nu}\right)T^{\alpha\mu\nu}$
$\displaystyle\mathbb{Q}$
$\displaystyle\coloneqq\frac{1}{4}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}-\frac{1}{2}Q_{\alpha\mu\nu}Q^{\mu\alpha\nu}-\frac{1}{4}Q_{\alpha}Q^{\alpha}+\frac{1}{2}Q_{\alpha}\bar{Q}^{\alpha}\,.$
(3.79)

Two special cases of this identity which play a role in TEGR and STEGR,
respectively, are

$\displaystyle\boxed{R(\Gamma)=\mathcal{R}(g)+\mathbb{T}+2\mathcal{D}_{\alpha}T^{\alpha}}$
(3.80)

where we set non-metricity to zero, and

$\displaystyle\boxed{R(\Gamma)=\mathcal{R}(g)+\mathbb{Q}+\mathcal{D}_{\alpha}\left(Q^{\alpha}-\bar{Q}^{\alpha}\right)}$
(3.81)

where torsion vanishes.

Finally, we recall that for a general connections the curvature tensor is not
antisymmetric in the first two indices, so one can form the non-zero
homothetic tensor $H_{\mu\nu}=R^{\lambda}{}_{\lambda\mu\nu}$. However, by
taking traces of the second Bianchi identity above, one can show

$H_{\mu\nu}=2R_{[\mu\nu]}+2\nabla_{[\mu}T_{\nu]}+\nabla_{\lambda}T^{\lambda}{}_{\mu\nu}+T_{\lambda}T^{\lambda}{}_{\mu\nu}\,.$

It thus follows that the homothetic tensor is not an independent trace of the
curvature tensor. It can be expressed with the help of other, already defined
tensors. Another trace of the curvature tensor is the co-Ricci tensor
$P^{\mu}{}_{\nu}=R^{\mu\lambda}{}_{\nu\lambda}$. However, using the
straightforward identity
$\nabla_{[\mu}Q_{\nu]\rho\sigma}=-R_{(\rho\sigma)\mu\nu}$ one can show

$P^{\mu}{}_{\nu}=R^{\mu}{}_{\nu}-2\nabla_{[\nu}Q_{\lambda]}{}^{\mu\lambda}\,.$

So this trace is also not independent. For the Levi-Civita connection one has
by metric-compatibility that $P_{\mu\nu}=R_{\mu\nu}$ as well as
$H_{\mu\nu}=0$; from the latter it follows then that the Ricci tensor is
symmetric, as one is used to from Riemannian geometry.

## 4 The Geometrical Trinity of General Relativity

In 1915, Einstein completed his General Theory of Relativity and he based it
on Riemannian geometry. He found this at the time relatively new branch of
mathematics to be an adequate language to (a) develop a field theoretic
description of gravity which cures the action-at-a-distance problem of
Newtonian gravity, (b) fully explore the consequences of the equivalence
principle, and (c) implement the idea that the laws of Nature do not depend on
our arbitrary choice of coordinate systems. The latter one was an idea which,
at the time, was unheard of and revolutionary. Today, we call this the
principle of general covariance.

Even though it was never Einstein’s intention to “geometrize gravity”
Lehmkuhl:2014 , as it is sometimes phrased, the theory he developed lends
itself to an interpretation of the phenomena of gravity as the manifestation
of the curvature of spacetime. This has been the prevalent interpretation of
gravity for the past 100 years.

However, as we saw in sections 2 and 3, Riemannian geometry is a special case
of the much more general theory of metric-affine geometry. There is no
physical principle that we know of which unequivocally selects Riemannian
geometry as the only viable description of gravity. In fact, there are three
distinct and yet physically equivalent descriptions of gravity, which are
rooted in the mathematical framework of metric-affine geometry. These
formulations ascribe gravitational phenomena either to non-vanishing
curvature, torsion, or non-metricity. These descriptions form the geometric
trinity of General Relativity Heisenberg:2018 ; BeltranJimenez:2019c .

Figure 12: Three different but equivalent representations of gravity: General
Relativity (based on curvature), the Symmetric Teleparallel Equivalent of GR
(based on non-metricity), and the Teleparallel Equivalent of GR (based on
torsion).

The next three subsections are dedicated to the corners of the triangle shown
in Figure 12: GR, TEGR, and STEGR. We also discuss CGR as a gauge fixed
version of STEGR, as well as GTEGR from which TEGR and STEGR emerge and which
can be thought of as the lower edge of the triangle in Figure 12.

First, we review Einstein’s original formulation in order to establish some
basic facts and notations. This will also facilitate the comparison with the
other two formulations, which we present in subsections 4.2 (TEGR) and 4.3
(STEGR). In all three formulations, the starting point is a generic metric-
affine geometry and we follow a strict structure in order to construct the
theory and work out its main features: a. The Geometric Postulates; b. Form of
the Connection; c. Construction of the Action Functional; d. The Metric and
Connection Field Equations; e. The Palatini Formulation of the Action
Principle; f. The Bianchi Identities; and finally g. Counting Degrees of
Freedom. We deviate from this basic structure when we discuss CGR as the gauge
fixed version of STEGR in subsection 4.4. Similarly, a different approach is
used in subsection 4.5, where the focus is on GTEGR, its special properties,
and how TEGR and STEGR emerge from this more general theory.

###  4.1 Einstein’s Original Formulation of General Relativity

#### The Geometric Postulates

We start with a metric-affine geometry $(\mathcal{M},g,\Gamma)$ and stress
that at this stage, neither $\mathcal{M}$ nor $g$ are fixed. This means that
we do not choose a particular manifold nor do we choose a particular metric on
that manifold. Both entities, $\mathcal{M}$ and $g$, will be determined later
as solutions of Einstein’s field equations. However, we need to select a
connection $\Gamma$ (or, as we will see later, select at least a class of
connections) in order to formulate the theory. Thus, we postulate that
$\Gamma$ satisfies

$\displaystyle T^{\alpha}{}_{\mu\nu}$ $\displaystyle\overset{!}{=}0$ and
$\displaystyle Q_{\alpha\mu\nu}$ $\displaystyle\overset{!}{=}0\,.$ (4.1)

These two postulates leave the curvature tensor
$\mathcal{R}^{\alpha}{}_{\mu\nu\rho}$ as the only non-zero tensor which
characterizes the spacetime geometry. It will be the main building block for
GR.

#### Form of a Torsionless, Metric-Compatible Connection

It is well-known, and it can also be checked using (3.47), that these two
geometric postulates are satisfied if and only if the connection is given by
the Levi-Civita connection,

$\Gamma^{\alpha}{}_{\mu\nu}\equiv\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}=\frac{1}{2}g^{\alpha\lambda}\left(\partial_{\mu}g_{\nu\lambda}+\partial_{\nu}g_{\mu\lambda}-\partial_{\lambda}g_{\mu\nu}\right)\,.$
(4.2)

Because the connection is completely determined by the metric, we will omit
$\Gamma$ from the triple $(\mathcal{M},g,\Gamma)$ and simply say that
spacetime is modelled by the pair $(\mathcal{M},g)$, where it is silently
understood that $\Gamma$ is given by (4.2).

#### Construction of the Action Functional

The action functional which defines GR is the famous Einstein-Hilbert (EH)
action plus the equally famous Gibbons-Hawking-York (GHY) boundary term
York:1972 ; Gibbons:1977 . Including a cosmological term and a matter action
for completeness, we can define GR by the functional

$\displaystyle\mathcal{S}_{\rm GR}[g,\Psi]$
$\displaystyle\coloneqq\mathcal{S}_{\rm EH}[g]+\mathcal{S}_{\rm
GHY}[h]+\mathcal{S}_{\rm matter}[g,\Psi]$
$\displaystyle\coloneqq\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\left(\mathcal{R}-2\Lambda\right)+\frac{1}{\kappa}\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\varepsilon\,\mathcal{K}+\mathcal{S}_{\rm
matter}[g,\Psi]\,,$ (4.3)

with $\kappa\coloneqq 8\pi G$. The fist integral on the second line is the
aforementioned EH action (including a cosmological constant $\Lambda$), the
second integral is the GHY boundary term, and $\mathcal{S}_{\rm
matter}[g,\Psi]$ is the action of (tensorial) matter fields $\Psi$ which are
minimally coupled to the gravitational field $g_{\mu\nu}$. Spinorial fields
are not described by the action (4.1). To describe Fermions, we would have to
describe gravity in terms of a tetrad field, rather than a metric tensor.

As we will discuss shortly, the GHY term is necessary whenever the manifold
$\mathcal{M}$ has a boundary $\partial\mathcal{M}$, otherwise the variational
principle is ill-defined and would not yield any field equations. In the above
boundary integral, $\varepsilon$ is defined as $\varepsilon\coloneqq
n_{\mu}n^{\mu}=\pm 1$, where $n^{\mu}$ is the normal vector to
$\partial\mathcal{M}$. This vector is normalized to either $\varepsilon=+1$
(when $\partial\mathcal{M}$ is timelike) or $\varepsilon=-1$ (when
$\partial\mathcal{M}$ is spacelike). Furthermore, $h$ denotes the determinant
of the metric intrinsic to $\partial\mathcal{M}$, while $\mathcal{K}$ is the
trace of the extrinsic curvature of $\partial\mathcal{M}$ viewed as
hypersurface embedded into $\mathcal{M}$. For a didactical discussion of
hypersurfaces, embeddings, and the concept of extrinsic curvature, we refer
the reader to Poisson’s book PoissonBook .

#### The Field Equations

Our next task is to find field equations for the metric which contain at most
second order derivatives of $g$ and which are of the form

$\displaystyle\mathcal{E}(g)_{\mu\nu}$ $\displaystyle=8\pi
G\,\mathcal{T}_{\mu\nu}$ with
$\displaystyle\mathcal{D}_{\mu}\mathcal{E}^{\mu}{}_{\nu}$ $\displaystyle=0\,,$
(4.4)

and where $\mathcal{T}_{\mu\nu}$ stands for the energy-momentum tensor of
matter fields. This form of the field equations can be motivated by
considering the Newtonian limit. The requirement of second order field
equations is necessary for having a well-posed initial value problem and the
divergence-freeness of the tensor $\mathcal{E}_{\mu\nu}$ ensures that the
covariant conservation of the matter energy-momentum tensor is a consequence
of the field equations.

Given only these requirements, Lovelock showed Lovelock:1969  that the left
hand side of the field equations has to be

$\mathcal{E}(g)_{\mu\nu}=a\,G_{\mu\nu}+\Lambda\,g_{\mu\nu}\,,$ (4.5)

where $a$ and $\Lambda$ are real constants and $G_{\mu\nu}$ is the so-called
Einstein tensor, which is explicitly given by

$G_{\mu\nu}\coloneqq\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}\,g_{\mu\nu}\,.$
(4.6)

This tensor indeed only contains first and second order derivatives of the
metric. Taking again into consideration the Newtonian limit, we find that the
constant $a$ is equal to $1$ and $\Lambda$, the so-called cosmological
constant, has to be very small. Measurements performed by the Planck
collaboration Planck:2015 ; Planck:2018  revealed that the cosmological
constant is positive and of the order of $\Lambda\sim 10^{-52}\unit{m^{-2}}$,
in SI units. Thus, the Einstein field equations take on the form

$\boxed{\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}}$
(4.7)

Given a set of initial data on a three-dimensional Cauchy surface, these
equations determine the metric and the manifold, $(\mathcal{M},g)$, up to
diffeomorphisms HawkingEllisBook ; WaldBook . This is analogous to Maxwell’s
equations, which determine the vector potential $A^{\mu}$ up to gauge
transformations. For more details and a mathematically robust formulation of
the initial value problem of GR, see for instance HawkingEllisBook ; WaldBook
.

The field equations (4.7) follow from the action functional (4.1) by taking a
variation with respect to the inverse metric $g^{\mu\nu}$ and demanding that
this variation vanishes. As mentioned above, if $\mathcal{M}$ has a boundary
the variational principle is ill-defined unless we add the GHY boundary term.
The necessity of this term was first realized by York York:1972  and shortly
afterwards also by Gibbons and Hawking Gibbons:1977 . Its origin is easy to
understand if we consider the variation of the Einstein-Hilbert action with
respect to $g^{\mu\nu}$, which results in

$\displaystyle\delta_{g}\mathcal{S}_{\rm EH}[g]=$
$\displaystyle\phantom{-}\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\left(\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}\right)\delta
g^{\mu\nu}$
$\displaystyle-\frac{1}{2\kappa}\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\varepsilon\,\delta
h^{\mu\nu}n^{\alpha}\partial_{\alpha}\left(\delta g_{\mu\nu}\right)\,.$ (4.8)

Recall that in the calculus of variation it is assumed that the variation
$\delta g^{\mu\nu}$ is fixed at the boundary, $\left.\delta
g^{\mu\nu}\right|_{\partial\mathcal{M}}=0$, but otherwise arbitrary. The
condition $\left.\delta g^{\mu\nu}\right|_{\partial\mathcal{M}}=0$ implies
that derivatives of $\delta g^{\mu\nu}$ in directions tangential to
$\partial\mathcal{M}$ vanish, but it does not imply that derivatives in the
direction normal to $\partial\mathcal{M}$ vanish. In particular, one can
conclude that

$\left.n^{\alpha}\partial_{\alpha}\left(\delta
g^{\mu\nu}\right)\right|_{\partial\mathcal{M}}\neq 0\,,$ (4.9)

which in turn implies that the boundary integral in (4.1) does not vanish in
general. Hence, the variation of the EH action with respect to $g^{\mu\nu}$
under the boundary condition $\left.\delta
g^{\mu\nu}\right|_{\partial\mathcal{M}}=0$ does not imply the Einstein field
equations, unless one also imposes the additional boundary condition
$\left.n^{\alpha}\partial_{\alpha}\left(\delta
g^{\mu\nu}\right)\right|_{\partial\mathcal{M}}=0$.

Gibbons, Hawking, and York realized that this problem can be circumvented by
the introduction of a boundary integral, whose variation precisely cancels the
boundary integral in (4.1). Indeed, the variation of the GHY functional reads

$\displaystyle\delta_{g}\mathcal{S}_{\rm
GHY}[h]=\frac{1}{2\kappa}\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\varepsilon\,\delta
h^{\mu\nu}n^{\alpha}\partial_{\alpha}\left(\delta g_{\mu\nu}\right)\,,$ (4.10)

which then implies that the total variation of the GR action is given by

$\displaystyle\delta_{g}\mathcal{S}_{\rm
GR}[g,\Psi]=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\left(\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}\right)\delta
g^{\mu\nu}-\frac{1}{2}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\mathcal{T}_{\mu\nu}\delta
g^{\mu\nu}\,,$ (4.11)

where we have defined the energy-momentum tensor of matter fields as

$\displaystyle\boxed{\mathcal{T}_{\mu\nu}\coloneqq-\frac{2}{\sqrt{|g|}}\frac{\delta\mathcal{S}_{\rm
matter}}{\delta g^{\mu\nu}}}$ (4.12)

Thus, only once one has supplemented the action by an appropriate boundary
term does one reproduce the celebrated Einstein field equations. As we will
see, neither in TEGR nor STEGR are boundary terms needed.

#### The Palatini Formulation of the Action Principle

Einstein’s General Relativity can also be formulated in the framework of a
general metric-affine geometry $(\mathcal{M},g,\Gamma)$, where $\Gamma$ a
priori possesses non-trivial curvature, torsion, and non-metricity. What is
needed is a slight adaptation of the action principle. In the so-called
Palatini formalism, metric and connection are regarded as two independent
fields and the action is varied with respect to both. As we will see below,
even if we start with a completely general $\Gamma$, the connection field
equations turn out to be purely algebraic equations which fix $\Gamma$ to be
the Levi-Civita connection up to a projective symmetry. This gives us back
Einstein’s original connection and its original field equations for the
metric.

The Palatini action functional for GR in absence of a cosmological constant is
defined as

$\mathcal{S}_{\rm
GR}[g,\Gamma]\coloneqq\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}g^{\mu\nu}R_{\mu\nu}(\Gamma)+\mathcal{S}_{\rm
matter}\,,$ (4.13)

where we recognize the first integral as being the EH action but written in
terms of the Ricci scalar of the general affine connection $\Gamma$, rather
than the Ricci scalar $\mathcal{R}$ of the Levi-Civita connection. Notice also
that it is not necessary to include a boundary term à la Gibbons-Hawking-York,
since there are no second order derivatives. The variational principle is thus
well-defined.

In fact, the metric field equations are determined along the same lines as in
the standard GR case, but without the complication of boundary terms. By
performing the variation of $\mathcal{S}_{\rm GR}[g,\Gamma]$ with respect to
the inverse metric—while keeping the connection fixed—we find

$\displaystyle\left.\delta_{g}\mathcal{S}_{\rm
GR}[g,\Gamma]\right|_{\Gamma}=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left[\delta_{g}\left(\sqrt{|g|}\right)g^{\mu\nu}R_{(\mu\nu)}(\Gamma)+\sqrt{|g|}\,\delta
g^{\mu\nu}R_{\mu\nu}(\Gamma)\right]+\left.\delta_{g}\mathcal{S}_{\rm
matter}\right|_{\Gamma}\,.$ (4.14)

The Ricci tensor is not varied with respect to the metric, since it is
constructed exclusively from the affine connection $\Gamma$. Also, note that
the Ricci tensor is in general not symmetric but that, due to the contraction
with $g^{\mu\nu}$, only its symmetric part contributes. Using the well-known
identity

$\displaystyle\delta_{g}\left(\sqrt{|g|}\right)=-\frac{1}{2}\sqrt{|g|}\,g_{\mu\nu}\delta
g^{\mu\nu}$ (4.15)

together with definition (4.12) we find that the metric field equations can be
written as

$\displaystyle
R_{(\mu\nu)}(\Gamma)-\frac{1}{2}R(\Gamma)\,g_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}\,.$
(4.16)

These equations have the same form as Einstein’s field equations, but the
Ricci tensor and Ricci scalar depend on the affine connection
$\Gamma^{\alpha}{}_{\mu\nu}$, rather than on the Levi-Civita connection
$\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}$.

Next, we turn our attention to the variation with respect to $\Gamma$, while
keeping the metric fixed:

$\displaystyle\left.\delta_{\Gamma}\mathcal{S}_{\rm
GR}[g,\Gamma]\right|_{g}=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,g^{\mu\nu}\delta_{\Gamma}R_{(\mu\nu)}(\Gamma)+\left.\delta_{\Gamma}\mathcal{S}_{\rm
matter}\right|_{g}\,.$ (4.17)

The variation of the Ricci tensor is given by the Palatini identity,

$\displaystyle\delta_{\Gamma}R_{\mu\nu}(\Gamma)=\nabla_{\alpha}\delta\Gamma^{\alpha}{}_{\nu\mu}-\nabla_{\nu}\delta\Gamma^{\alpha}{}_{\alpha\mu}-T^{\alpha}{}_{\beta\nu}\delta\Gamma^{\beta}{}_{\alpha\mu}.$
(4.18)

With the help of the Palatini identity and an integration by parts in order to
move the covariant derivative off $\nabla\delta\Gamma$, we find that the
variation with respect to the connection can be written as

$\displaystyle\left.\mathcal{S}_{\rm GR}[g,\Gamma]\right|_{g}$
$\displaystyle=-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left[\nabla_{\alpha}\left(\sqrt{|g|}\,g^{\mu\nu}\right)\delta\Gamma^{\alpha}{}_{(\mu\nu)}-\nabla_{(\mu}\left(\sqrt{|g|}\,g^{\mu\nu}\right)\delta\Gamma^{\alpha}{}_{\alpha|\nu)}-\sqrt{|g|}\,g^{\mu\nu}T^{\alpha}{}_{\beta(\mu}\delta\Gamma^{\beta}{}_{\alpha|\nu)}\right]$
$\displaystyle\phantom{=}+\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left[\nabla_{\alpha}\left(\sqrt{|g|}\,g^{\mu\nu}\delta\Gamma^{\alpha}{}_{(\mu\nu)}\right)-\nabla_{(\mu}\left(\sqrt{|g|}\,g^{\mu\nu}\,\delta\Gamma^{\alpha}{}_{\alpha|\nu)}\right)\right]+\left.\delta_{\Gamma}\mathcal{S}_{\rm
matter}\right|_{g}\,,$ (4.19)

where we have kept the total divergences. We cannot simply drop these, as we
would usually do, since in a general metric-affine geometry they do not give
rise to pure boundary terms. In fact, the generalized Gauss theorem of
subsection 3.7 tells us that

$\displaystyle\int_{\mathcal{M}}\mathrm{d}^{4}x\nabla_{\alpha}\left(\sqrt{|g|}\,g^{\mu\nu}\delta\Gamma^{\alpha}{}_{(\mu\nu)}\right)$
$\displaystyle=\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\epsilon\,n_{\alpha}g^{\mu\nu}\delta\Gamma^{\alpha}{}_{(\mu\nu)}-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,T_{\alpha}g^{\mu\nu}\delta\Gamma^{\alpha}{}_{(\mu\nu)}$
$\displaystyle\int_{\mathcal{M}}\mathrm{d}^{4}x\nabla_{(\mu}\left(\sqrt{|g|}\,g^{\mu\nu}\,\delta\Gamma^{\alpha}{}_{\alpha|\nu)}\right)$
$\displaystyle=\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{|h|}\,\epsilon\,n_{(\mu}g^{\mu\nu}\,\delta\Gamma^{\alpha}{}_{\alpha|\nu)}-\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{|g|}\,T_{(\mu}g^{\mu\nu}\,\delta\Gamma^{\alpha}{}_{\alpha|\nu)}\,.$
(4.20)

Both boundary integrals vanish because of the standard boundary condition
$\left.\delta\Gamma\right|_{\partial\mathcal{M}}=0$. That is, the boundary
integrals vanish because the variations are being kept fixed at the boundary
of the integration region. However, the bulk integrals on the right side
contribute to the variation of the action and we therefore find

$\displaystyle\left.\mathcal{S}_{\rm GR}[g,\Gamma]\right|_{g}$
$\displaystyle=-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left[\nabla_{\alpha}\left(\sqrt{|g|}\,g^{\mu\nu}\right)\delta\Gamma^{\alpha}{}_{(\mu\nu)}-\nabla_{(\mu}\left(\sqrt{|g|}\,g^{\mu\nu}\right)\delta\Gamma^{\alpha}{}_{\alpha|\nu)}-\sqrt{|g|}\,g^{\mu\nu}T^{\alpha}{}_{\beta(\mu}\delta\Gamma^{\beta}{}_{\alpha|\nu)}\right]$
$\displaystyle\phantom{=}+\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{|g|}\,T_{(\mu}g^{\mu\nu}\,\delta\Gamma^{\alpha}{}_{\alpha|\nu)}-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,T_{\alpha}g^{\mu\nu}\delta\Gamma^{\alpha}{}_{(\mu\nu)}+\left.\delta_{\Gamma}\mathcal{S}_{\rm
matter}\right|_{g}$ (4.21)

After some index reshuffling, we can factor out the common factor
$\delta\Gamma$ and read off the connection field equations

$\displaystyle\nabla_{\alpha}\left(\sqrt{|g|}\,g^{\mu\nu}\right)-\delta^{\mu}{}_{\alpha}\nabla_{\beta}\left(\sqrt{|g|}\,g^{\beta\nu}\right)=\sqrt{|g|}\left[g^{\mu\nu}T_{\alpha}+g^{\beta\nu}T^{\mu}{}_{\alpha\beta}-\delta^{\mu}{}_{\alpha}g^{\beta\nu}T_{\beta}\right]+\tilde{\mathcal{H}}_{\alpha}{}^{\mu\nu}\,,$
(4.22)

where we have also introduced the hypermomentum of matter666Notice that it
follows from the definition that $\tilde{\mathcal{H}}^{\alpha}{}_{\mu\nu}$ is
a tensor density of weight $w=+1$ and that equation (4.22) is thus self-
consistent.,

$\displaystyle\tilde{\mathcal{H}}_{\alpha}{}^{\mu\nu}\coloneqq
2\kappa\frac{\delta\mathcal{S}_{\rm
matter}}{\delta\Gamma^{\alpha}{}_{\mu\nu}}\,.$ (4.23)

These are the connection field equations of GR. Fermionic fields naturally
couple to torsion and do therefore contribute to the hypermomentum. Against
expectation, if torsion is present, bosonic fields also do contribute to the
hypermomentum. A detailed discussion of matter coupling in metric-affine
geometries can be found in BeltranJimenez:2020 .

For simplicity, we shall first assume that torsion and hypermomentum both
vanish. Then, the connection field equations (4.22) reduce to

$\displaystyle\nabla_{\alpha}\left(\sqrt{|g|}\,g^{\mu\nu}\right)-\delta^{\mu}{}_{\alpha}\nabla_{\beta}\left(\sqrt{|g|}\,g^{\beta\nu}\right)=0\,.$
(4.24)

By taking the $\alpha=\mu$ trace of this equation, we obtain

$\displaystyle\nabla_{\beta}\left(\sqrt{|g|}\,g^{\beta\nu}\right)=0\qquad\Longleftrightarrow\qquad
Q^{\nu}-2\bar{Q}^{\nu}=0\,,$ (4.25)

where we have written the equation also in terms of the non-metricity tensor
and its two traces, $Q_{\nu}=g^{\alpha\beta}Q_{\nu\alpha\beta}$ and
$\bar{Q}_{\nu}=g^{\alpha\beta}Q_{\alpha\beta\nu}$. Plugging this result back
into equation (4.24) yields

$\displaystyle\nabla_{\alpha}\left(\sqrt{|g|}\,g^{\mu\nu}\right)=0\qquad\Longleftrightarrow\qquad
g^{\mu\nu}\,Q_{\alpha}-2Q_{\alpha}{}^{\mu\nu}=0\,.$ (4.26)

After contracting this equation with $g_{\mu\nu}$, it follows that

$\displaystyle 4Q_{\alpha}-2g_{\mu\nu}Q_{\alpha}{}^{\mu\nu}=2Q_{\alpha}=0\,.$
(4.27)

Hence, equation (4.26) finally tells us that

$\displaystyle Q_{\alpha}{}^{\mu\nu}=0\,.$ (4.28)

Recall that we assumed $T^{\alpha}{}_{\mu\nu}=0$ and this simplifying
assumption led us to uncover that the connection field equation reduces to
$Q_{\alpha\mu\nu}=0$. We can therefore conclude that the connection is
torsionless and metric-compatible, which uniquely fixes it to be the Levi-
Civita connection. This means that the metric field equations (4.16) become
the standard Einstein equations.

The simplifying assumptions that torsion and hypermomentum vanish can be
lifted and even in full generality it is found that the connection field
equations are purely algebraic equations for the connection which, at the end
of the day, can be completely solved. It is found that the connection is given
by the Levi-Civita connection, up to a projective transformation
$\Gamma^{\alpha}{}_{\mu\nu}\mapsto\Gamma^{\alpha}{}_{\mu\nu}+\delta^{\alpha}{}_{\nu}\xi_{\mu}$.
For the general case, we refer the reader to BeltranJimenez:2017 .

The key lesson here is the following: As long as the action has the same form
as the EH action, changing the geometric framework will not lead to a new
formulation of GR. General Relativity arises naturally if the dynamics is
described by an action of the EH form, even if the variational principle is
formulated à la Palatini. Hence, if we wish to develop teleparallel theories
of gravity, we do not only have to change the geometric framework, we also
have to change the action such that it has a genuinely different form, but is
nevertheless equivalent to the EH action. Equivalent means that the field
equations for the metric are the same and that the theories all propagate the
same number of physical degrees of freedom.

#### The Bianchi Identities

The integrand of the Einstein-Hilbert action is $\sqrt{|g|}\,\mathcal{R}$,
which is a scalar density of weight $w=+1$. Recall that under a change of
coordinates $x^{\mu}\mapsto x^{\prime\mu}(x)$ a scalar density of weight one
transforms as (see equation (2.28))

$\displaystyle\sqrt{|g(x)|}\,\mathcal{R}(x)=\det\left(J\right)\,\sqrt{|g(x^{\prime})|}\,\mathcal{R}(x^{\prime})\,,$
(4.29)

where $J$ is the Jacobian matrix with components
$J^{\mu}{}_{\nu}=\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}$. It
therefore follows from the change of integration variables formula of calculus
that

$\displaystyle\int_{\mathcal{M}}\mathrm{d}^{4}x^{\prime}\,\sqrt{|g(x^{\prime})|}\,\mathcal{R}(x^{\prime})\det\left(J\right)=\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{|g(x)|}\,\mathcal{R}(x)\,.$
(4.30)

In other words, the Einstein-Hilbert action is invariant under
diffeomorphisms. Since this is true for any diffeomorphism, we can just as
well consider a $1$-parameter family of diffeomorphisms: Let
$\phi_{s}:\mathbb{R}\times\mathcal{M}\to\mathcal{M}$ be such a $1$-parameter
family of diffeomorphisms with $\phi_{s=0}=\text{id}$ and with generating
vector field
$v\coloneqq\left.\frac{\mathrm{d}\phi_{s}}{\mathrm{d}s}\right|_{s=0}$. This
family of diffeomorphisms can be read as a family of changes of coordinates,
i.e, $x^{\mu}\mapsto\phi^{\mu}_{s}(x)$ for every value of $s$. The EH action
is invariant under all these diffeomorphisms. Recall from subsection 2.3 that
a $1$-parameter family of diffeomorphisms generates a flow and that the
infinitesimal change of a tensor under a flow is measured by the Lie
derivative. In the case of the metric we have

$\displaystyle\phi^{*}_{s}g_{\mu\nu}=g_{\mu\nu}+s\,\mathcal{L}_{v}g_{\mu\nu}\,,$
(4.31)

where the parameter $|s|\ll 1$ is infinitesimally small and
$\phi^{*}_{s}g_{\mu\nu}$ shall be understood as saying “we applied the
diffeomorphism to the metric”. Applying this $1$-parameter family of
diffeomorphisms to the EH action is tantamount to considering

$\displaystyle\mathcal{S}_{\rm EH}[\phi^{*}_{s}g]\,.$ (4.32)

Due to the invariance we have

$\displaystyle\mathcal{S}_{\rm EH}[\phi^{*}_{s}g]=\mathcal{S}_{\rm EH}[g]\,$
(4.33)

and if we expand this equation in $s$ we find

$\displaystyle\mathcal{S}_{\rm EH}[g]+s\,\delta_{v}\mathcal{S}_{\rm
EH}[g]=\mathcal{S}_{\rm
EH}[g]\qquad\Longrightarrow\qquad\delta_{v}\mathcal{S}_{\rm EH}[g]=0\,,$
(4.34)

where the variation $\delta_{v}\mathcal{S}_{\rm EH}[g]$ is defined as

$\displaystyle 2\kappa\,\delta_{v}\mathcal{S}_{\rm EH}[g]$
$\displaystyle=\int_{\mathcal{M}}\frac{\delta}{\delta
g_{\mu\nu}}\left(\sqrt{|g|}\,\mathcal{R}\right)\,\delta_{v}g_{\mu\nu}\mathrm{d}^{4}x$
$\displaystyle=\int_{\mathcal{M}}\frac{\delta}{\delta
g_{\mu\nu}}\left(\sqrt{|g|}\,\mathcal{R}\right)\,\mathcal{L}_{v}g_{\mu\nu}\mathrm{d}^{4}x\,.$
(4.35)

Of course we already know the variation of $\sqrt{|g|}\,\mathcal{R}$ with
respect to $g_{\mu\nu}$. Up to boundary terms, this is simply the Einstein
tensor with raised indices multiplied by the square root of $|g|$, i.e.,
$\sqrt{|g|}\,G^{\mu\nu}$. By recalling from equation (3.6) that

$\displaystyle\mathcal{L}_{v}g_{\mu\nu}=2\mathcal{D}_{(\mu}v_{\nu)}\,,$ (4.36)

we can further simplify the form of the variation $\delta_{v}\mathcal{S}_{\rm
EH}[g]$ and we find

$\displaystyle
2\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,G^{\mu\nu}\mathcal{D}_{(\mu}v_{\nu)}=2\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\mathcal{D}_{\mu}G^{\mu\nu}v_{\nu}=0\,,$
(4.37)

where we integrated by parts and dropped the boundary term. Since this has to
hold for any diffeomorphism (i.e., for any generating vector field $v^{\mu}$),
we finally find

$\boxed{\begin{aligned} \mathcal{S}_{\rm EH}[g]\text{ invariant under
diffeomorphisms
}&&\Longrightarrow&&\mathcal{D}_{\mu}G^{\mu\nu}=0\end{aligned}}$ (4.38)

These are the Bianchi identities and they are a consequence of the
diffeomorphism invariance of the theory. These equations imply that not all of
Einstein’s field equations are dynamical. This affects the counting of degrees
of freedom, as we will now see.

#### Counting Degrees of Freedom

The basic variable considered in GR is the metric. It has a total of ten
independent components and thus we have an upper bound of ten physical degrees
of freedom for the gravitational field. There is an equal number of second
order partial differential equations, namely
$G_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}$.

However, since Einstein’s equations are generally covariant, we are free to
perform diffeomorphisms. Each diffeomorphism provides us with four choices,
which in turn grants us the freedom to fix four components of the metric.
Moreover, the Bianchi identities tell us that four of Einstein’s field
equations are actually constraints, rather than dynamical equations. This
follows from expanding the Bianchi identities as

$\displaystyle\mathcal{D}_{\mu}G^{\mu\nu}$
$\displaystyle=\mathcal{D}_{0}G^{0\nu}+\mathcal{D}_{i}G^{i\nu}$
$\displaystyle=\partial_{0}G^{0\nu}+\left\\{\genfrac{}{}{0.0pt}{}{\mu}{\mu\lambda}\right\\}G^{\lambda\nu}+\left\\{\genfrac{}{}{0.0pt}{}{\nu}{\mu\lambda}\right\\}G^{\mu\lambda}+\partial_{i}G^{i\nu}=0\,,$
(4.39)

where the index $i$ stands for spatial derivatives and is summed over the
numbers $1,2,3$. Let us now determine the order of spatial and temporal
derivatives of the metric in each term. To that end, we recall that
$G_{\mu\nu}$ is second order in all derivatives, while the Levi-Civita
connection is only first order in all derivatives. Hence, it follows that

$\displaystyle\left\\{\genfrac{}{}{0.0pt}{}{\mu}{\mu\lambda}\right\\}G^{\lambda\nu}+\left\\{\genfrac{}{}{0.0pt}{}{\nu}{\mu\lambda}\right\\}G^{\mu\lambda}\qquad\leadsto\qquad\text{Highest
derivatives contained: $\partial^{2}_{0}g_{\mu\nu}$,
$\partial_{0}\partial_{i}g_{\mu\nu}$, and
$\partial_{i}\partial_{j}g_{\mu\nu}$}\,.$ (4.40)

Furthermore, since $\partial_{i}$ increases the order of spatial derivatives,
we find that

$\displaystyle\partial_{i}G^{i\nu}\qquad\leadsto\qquad\text{Highest
derivatives contained: $\partial^{2}_{0}\partial_{i}g_{\mu\nu}$,
$\partial_{0}\partial_{i}\partial_{j}g_{\mu\nu}$,
$\partial_{i}\partial_{j}\partial_{k}g_{\mu\nu}$, etc. }$ (4.41)

In other words, these two terms contain third order derivatives. However, the
temporal derivatives are at most second order! This is an important
realization because $\partial_{0}G^{0\nu}$ contains third order time
derivatives, provided that $G^{0\nu}$ contains second order time derivatives.
However, the Bianchi identities tell us that this cannot be the case. None of
the other terms we analyzed contains third order time derivatives. The highest
order is two. Hence, there is nothing which could cancel the presumed third
order time derivatives in $\partial_{0}G^{0\nu}$, which is necessary for the
Bianchi identities to hold.

It follows that the assumption that $G^{0\nu}$ contains second order time
derivatives is wrong! At most, it can contain first order time derivatives
(and indeed it does). The important conclusion is that the four equations
$G^{0\nu}=\kappa\,\mathcal{T}^{0\nu}$ constitute constraints on the initial
data, rather than dynamical equations for the metric. So we are finally left
with

$\displaystyle 10\text{ metric components }-4\text{ diffeomorphisms }-4\text{
constraints }=2\text{ physical degrees of freedom}.$

###  4.2 The Teleparallel Equivalent of General Relativity (TEGR)

In the previous subsection we saw how GR emerges from an action principle in
conjunction with the geometric postulates of vanishing torsion and vanishing
non-metricity. We also saw that the postulates can be dropped and that GR
emerges from a Palatini variational principle, where the connection $\Gamma$
is assumed to be an independent field, but which is ultimately fixed by the
field equations to be precisely the Levi-Civita connection. This fact hinges
on the form of the action: As long as the action has the EH form, GR emerges
naturally. It follows that in order to obtain an equivalent but different
geometric formulation of GR, we need to change the geometric framework as well
as the action principle. In the following, we show how the so-called
Teleparallel Equivalent of GR, or TEGR for short, achieves this.

#### The Geometric Postulates

TEGR attributes the effects of gravity to a non-vanishing torsion tensor. The
starting point is a metric-affine geometry $(\mathcal{M},g,\Gamma)$, where the
connection is postulated to satisfy

$\displaystyle R^{\alpha}{}_{\mu\nu\rho}$ $\displaystyle\overset{!}{=}0$ and
$\displaystyle Q_{\alpha\mu\nu\phantom{\rho}}$
$\displaystyle\overset{!}{=}0\,.$ (4.42)

This may raise the question, in what sense GR and TEGR could even be
equivalent to each other, given that curvature is postulated to vanish in the
latter one. The key observation to resolve this apparent tension is the
following: What is postulated to vanish in TEGR is the curvature tensor
$R^{\alpha}{}_{\mu\nu\rho}$ with respect to the affine connection $\Gamma$,
not the curvature tensor $\mathcal{R}^{\alpha}{}_{\mu\nu\rho}$ with respect to
the Levi-Civita connection. Moreover, recall that the two curvature tensors
are related to each other by the identity (3.77). Starting from this identity,
we have seen that an important special case emerges when non-metricity is set
to zero. Namely equation (3.80), which relates the two Ricci scalars and which
we repeat here for convenience:

$R(\Gamma)=\mathcal{R}(g)+\mathbb{T}+2\mathcal{D}_{\alpha}T^{\alpha}\,.$
(4.43)

The torsion scalar $\mathbb{T}$ is explicitly given by

$\mathbb{T}\coloneqq\frac{1}{2}\left(\frac{1}{4}T_{\alpha\mu\nu}+\frac{1}{2}T_{\mu\alpha\nu}-g_{\alpha\mu}T_{\nu}\right)T^{\alpha\mu\nu}\,,$
(4.44)

and $\mathcal{D}_{\mu}$ still denotes the covariant derivative with respect to
the Levi-Civita connection, not the covariant derivative $\nabla_{\mu}$ with
respect to the more general connection $\Gamma$. This equation will prove to
be the key to formulate GR in terms of torsion, rather than curvature. But
before that, we study the form of the connection more closely.

#### Form of a Flat, Metric-Compatible Connection

The geoemtric postulates demand that the connection be flat and metric-
compatible. These requirements do not completely fix the connection, unlike
the postulates used in GR. Rather, we end up with a whole class of connections
which satisfy the postulates of flatness and metric-compatibility. To see how
this comes about, we start with the observation that the trivial connection,
i.e., the connection $\hat{\Gamma}^{\alpha}{}_{\mu\nu}=0$ is obviously flat.
Now recall that under a change of coordinates a connection transforms
inhomogeneously (cf. equation (2.64)). For our trivial connection, we find
that a change of coordinates from $\hat{x}^{\mu}$ to $x^{\mu}(\hat{x})$ leads
to

$\displaystyle\hat{\Gamma}^{\alpha}{}_{\mu\nu}\qquad\mapsto\qquad\Gamma^{\alpha}{}_{\mu\nu}$
$\displaystyle=\frac{\partial
x^{\alpha}}{\partial\hat{x}^{\beta}}\frac{\partial\hat{x}^{\rho}}{\partial
x^{\mu}}\frac{\partial\hat{x}^{\sigma}}{\partial
x^{\nu}}\underbrace{\hat{\Gamma}^{\beta}{}_{\rho\sigma}}_{=0}+\frac{\partial
x^{\alpha}}{\partial\hat{x}^{\lambda}}\frac{\partial^{2}\hat{x}^{\lambda}}{\partial
x^{\mu}\partial x^{\nu}}=\frac{\partial
x^{\alpha}}{\partial\hat{x}^{\lambda}}\partial_{\mu}\frac{\partial\hat{x}^{\lambda}}{\partial
x^{\nu}}\,.$ (4.45)

If we read $\frac{\partial\hat{x}^{\mu}}{\partial x^{\nu}}$ as the components
of a matrix $\Lambda$, we can write the last equation as

$\displaystyle\boxed{\Gamma^{\alpha}{}_{\mu\nu}=\left(\Lambda^{-1}\right)^{\alpha}{}_{\lambda}\partial_{\mu}\Lambda^{\lambda}{}_{\nu}}$
(4.46)

This is a key equation for all teleparallel theories of gravity for the
following two facts:

  1. 1.

If the curvature tensor is zero in one coordinate system, it is zero in any
other coordinate system. Since it is zero for the trivial connection
$\hat{\Gamma}^{\alpha}{}_{\mu\nu}=0$, it is also zero for the connection
$\Gamma^{\alpha}{}_{\mu\nu}$ in equation (4.46), since this connection has
been obtained by a change of coordinates.

  2. 2.

The change of coordinates is completely arbitrary and the vanishing of
$R^{\alpha}{}_{\mu\nu\rho}$ for the connection in (4.46) does not depend on
the details of the transformation. Thus, we may as well “forget” the origin of
(4.46). That is to say, we can conclude that any connection of the form
(4.46), where $\Lambda^{\mu}{}_{\nu}$ is a matrix belonging to the general
linear group $GL(4,\mathbb{R})$ is flat.

Now we turn to the second postulate, which demands metric-compatibility. By
plugging the flat connection (4.46) into $Q_{\alpha}{}^{\mu\nu}=0$, we find

$g^{\lambda(\mu}\partial_{\alpha}\Lambda^{\nu)}{}_{\rho}(\Lambda^{-1})^{\rho}{}_{\lambda}=\frac{1}{2}\partial_{\alpha}g^{\mu\nu}\,.$
(4.47)

This equation allows us to eliminate the metric and express it in terms of the
$\Lambda^{\mu}{}_{\nu}$. Observe that the metric has ten components while the
matrix $\Lambda^{\mu}{}_{\nu}$ has $4\times 4=16$ components. This redundancy
in the description is well-understood Golovnev:2017 ; Krssak:2018 ;
BeltranJimenez:2019c : Six of the components of $\Lambda^{\mu}{}_{\nu}$
reflect the freedom to perform local Lorentz transformations. This is a
symmetry of the theory.

We reach the following conclusions: Choose a matrix $\Lambda\in
GL(4,\mathbb{R})$ and write the connection as in (4.46). The so defined
connection is guaranteed to be flat. Furthermore, impose (4.47) in order to
obtain a metric-compatible connection. This turns the metric into an auxiliary
field. For completeness, we remark that the torsion tensor is given by

$T^{\alpha}{}_{\mu\nu}=2\left(\Lambda^{-1}\right)^{\alpha}{}_{\lambda}\,\partial_{[\mu}\Lambda^{\lambda}{}_{\nu]}\,.$
(4.48)

for any flat connection parametrized by $\Lambda\in GL(4,\mathbb{R})$.

#### Construction of the Action Functional

The construction of an action functional which is equivalent but not equal to
the one of GR is fairly straightforward. As alluded to above, the key
observation is that the Ricci scalar of a metric-compatible connection is
related to the Ricci scalar of the Levi-Civita connection via equation (4.43).
If we also impose flatness, which amounts to $R(\Gamma)=0$, we find

$\displaystyle\mathcal{R}(g)=-\mathbb{T}(\Lambda)-2\mathcal{D}_{\alpha}T^{\alpha}(\Lambda)\,.$
(4.49)

The notation $\mathbb{T}(\Lambda)$ emphasizes that $\mathbb{T}$ depends on
$\Lambda$ and that the metric has been integrated out from the metricity
condition. This equation now allows us to simply replace the Ricci scalar in
the EH action by the right hand side of equation (4.49). However, such an
action would be strictly equal to the original EH action, because the
connection carries a Levi-Civita piece and a torsion piece. The scalars
$\mathbb{T}$ and $\mathcal{D}_{\alpha}T^{\alpha}$ conspire in such a way, that
the torsion piece drops out and only the Levi-Civita part contributes to the
action.

However, by dropping $\mathcal{D}_{\alpha}T^{\alpha}$, which amounts to a mere
boundary term, we obtain an action which is genuinely different from the EH
action, but which leads to the same field equations. Thus, we define the
action of TEGR as

$\displaystyle\boxed{\mathcal{S}_{\rm
TEGR}[\Lambda]\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\mathbb{T}(\Lambda)+\mathcal{S}_{\rm
matter}}$ (4.50)

It is silently understood that the metric can be expressed in terms of
$\Lambda$. However, observe that $\Lambda$ has $16$ components, while the
metric has only ten. Consequently, we find more field equations than in GR. As
we will see later, ten of these equations are precisely the Einstein field
equations. The remaining six are Bianchi identities related to the local
Lorentz symmetry expressed through six of the components of $\Lambda$.

A further consequence of dropping $\mathcal{D}_{\alpha}\mathbb{T}^{\alpha}$ is
that the action functional only depends on first order derivatives. Thus, the
variational principle is well-defined without having to add boundary terms à
la Gibbons-Hawking-York. This is one of the features we had anticipated in the
previous subsection.

The action for TEGR could also have been constructed using a different
strategy, which does not rely on the geometric identity (4.49). Rather, the
strategy which we shall briefly sketch relies on counting degrees of freedom:
Given the postulates of vanishing curvature and vanishing non-metricity, the
only remaining tensor which can play a fundamental role is the torsion tensor.
Thus, our task is to construct a scalar from the torsion tensor, which is then
used to define the action. Clearly, this scalar cannot be linear in the
torsion tensor. It has to be at least quadratic. As it turns out (we will
discuss this point in more detail in subsection 5.1), there are precisely
three independent scalars one can build from contractions of the torsion
tensor with itself and with the help of the metric. Thus, the most general
scalar assumes the form

$\displaystyle\hat{\mathbb{T}}\coloneqq
c_{1}\,T_{\alpha\mu\nu}T^{\alpha\mu\nu}+c_{2}\,T_{\mu\alpha\nu}T^{\alpha\mu\nu}+c_{3}\,T_{\mu}T^{\mu}\,,$
(4.51)

where $c_{1}$, $c_{2}$, and $c_{3}$ are arbitrary, real constants. Using this
scalar, it is easy to derive field equations and perform a counting of degrees
of freedom around a Minkowski background. In order to obtain precisely two
degrees of freedom, as in GR, one finds that the parameters have to be chosen
as

$\displaystyle c_{2}$ $\displaystyle=2c_{1}$ and $\displaystyle
c_{3}=-4c_{1}\,.$ (4.52)

Up to an over all normalization, this reproduces the torsion scalar (4.44) and
thus the action (4.50).

#### The Palatini Formulation of the Action Principle

This action can also be written in a manifestly covariant form, which
highlights which type of metric-affine geometry is being considered, i.e.,
which geometric postulates are being implemented. This action functional reads

$\mathcal{S}_{\textsf{TEGR}}[g,\Gamma;\tilde{\Pi},\tilde{\chi}]\coloneqq-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left(\frac{1}{2\kappa}\sqrt{|g|}\,\mathbb{T}+\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}\,R^{\alpha}{}_{\mu\nu\rho}+\tilde{\chi}^{\alpha}{}_{\mu\nu}\,Q_{\alpha}{}^{\mu\nu}\right)+\mathcal{S}_{\rm
matter}\,,$ (4.53)

where $\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}$ and
$\tilde{\chi}^{\alpha}{}_{\mu\nu}$ are tensor densities of weight $w=+1$ which
act as Lagrange multipliers. These multipliers enforce the postulates of
vanishing curvature and vanishing non-metricity. It should also be noted that
the Lagrange multipliers possess the symmetries
$\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}=\tilde{\Pi}_{\alpha}{}^{\mu[\nu\rho]}$
and $\tilde{\chi}^{\alpha}{}_{\mu\nu}=\tilde{\chi}^{\alpha}{}_{(\mu\nu)}$,
which they inherit from the curvature tensor and the non-metricity tensor,
respectively. Just as in the Palatini formulation of GR, the connection
$\Gamma$ refers to a generic affine connection. A priori, it has nothing to do
with the previous connection which is parametrized by the matrix $\Lambda$.
When it comes to working out the field equations, the Palatini formalism
offers some advantages.

#### The Metric and Connection Field Equations

Based on the Palatini action given in (4.53), we can perform four independent
variations. These are

$\displaystyle\frac{\delta\mathcal{S}_{\rm TEGR}}{\delta g^{\mu\nu}}$
$\displaystyle\overset{!}{=}0,$ $\displaystyle\frac{\delta\mathcal{S}_{\rm
TEGR}}{\delta\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}}$
$\displaystyle\overset{!}{=}0$ $\displaystyle\frac{\delta\mathcal{S}_{\rm
TEGR}}{\delta\Gamma^{\alpha}{}_{\mu\nu}}$ $\displaystyle\overset{!}{=}0$
$\displaystyle\frac{\delta\mathcal{S}_{\rm
TEGR}}{\delta\tilde{\chi}^{\alpha}{}_{\mu\nu}}$
$\displaystyle\overset{!}{=}0,.$ (4.54)

The variations with respect to the Lagrange multipliers are the most
straightforward ones. The variations with respect to the metric and the
connection require more work. Also, the field equations that follow from the
variational principle are highly coupled in the sense that the Lagrange
multipliers appear with covariant derivatives acting on them. Untangling the
field equations, cleanly implementing the conditions of vanishing curvature
and vanishing non-metricity, and bringing the equations into a simple form
requires some effort. For a detailed derivation we refer the reader to
BeltranJimenez:2018 . The end result is

$\boxed{\begin{aligned} &\textsf{Metric field
equations:}&(\nabla_{\alpha}+T_{\alpha})S_{(\mu\nu)}{}^{\alpha}+t_{\mu\nu}-\frac{1}{2}\mathbb{T}\,g_{\mu\nu}&=\kappa\,\mathcal{T}_{\mu\nu}\\\
&\textsf{Connection field
equations:}&(\nabla_{\alpha}+T_{\alpha})\left[\sqrt{|g|}{S}_{[\mu}{}^{\alpha}{}_{\nu]}\right]&=0\end{aligned}}$

where we have introduced the torsion conjugate
$\boldsymbol{S}_{\alpha}{}^{\mu\nu}$ and the symmetric tensor $t_{\mu\nu}$,

$\displaystyle S_{\alpha}{}^{\mu\nu}$
$\displaystyle\coloneqq\frac{\partial\mathbb{T}}{\partial
T^{\alpha}{}_{\mu\nu}}=\frac{1}{4}\,T_{\alpha}{}^{\mu\nu}+\frac{1}{2}\,T^{[\mu}{}_{\alpha}{}^{\nu]}-\delta_{\alpha}{}^{[\mu}T^{\nu]}$
$\displaystyle t_{\mu\nu}$
$\displaystyle\coloneqq\frac{\partial\mathbb{T}}{\partial
g^{\mu\nu}}=\frac{1}{2}S_{(\mu|}{}^{\lambda\kappa}\,T_{\nu)\lambda\kappa}-T^{\lambda\kappa}{}_{(\mu}\,S_{\lambda\kappa|\nu)}\,.$
(4.55)

It is also assumed that the hypermomentum density, which enters as
$(\nabla_{\alpha}+T_{\alpha})\tilde{\mathcal{H}}_{[\mu}{}^{\alpha}{}_{\nu]}$
in the field equations is either identically zero (i.e., the matter content
and the matter couplings have been chosen such that there is no contribution
to this tensor density) or that it is conserved in the sense that

$\displaystyle\left(\nabla_{\alpha}+T_{\alpha}\right)\tilde{\mathcal{H}}_{\mu}{}^{\alpha}{}_{\nu}=0\,.$
(4.56)

This conservation law holds by virtue of the gauge invariance of the matter
sector and is confirmed in all standard cases where a non-trivial
hypermomentum arises from coupling matter fields to the connection. For more
details, see BeltranJimenez:2018 ; BeltranJimenez:2020 .

As one can show, the metric field equations are simply the Einstein field
equations in disguise, while the connection field equations arise as Bianchi
identities. As shown in BeltranJimenez:2018 , this is a consequence of the
curvature tensor being the curvature of a $GL(4,\mathbb{R})$ connection. This
has important consequences: The connection field equations carry no dynamical
information. Put differently, these equations do not determine the metric nor
the connection. In fact, they are just trivially satisfied. The dynamics of
the theory is solely determined by the metric field equations, which are the
Einstein equations.

#### The Bianchi Identities

Bianchi identities arise quite naturally whenever an action is invariant under
a certain local symmetry. Since the actions (4.53) and (4.53) are generally
covariant, it is no surprise that Bianchi identities can be found. Starting
from an action of the form $\mathcal{S}[g,\Gamma]$, where $\Gamma$ is a
generic affine connection, and assuming that the action is diffeomorphism
invariant, it was shown in BeltranJimenez:2020  that the Bianchi identities in
a metric-affine geometry take the form

$\displaystyle\boxed{2\mathcal{D}_{\mu}\tilde{\mathcal{M}}^{\mu}{}_{\lambda}-\hat{\nabla}_{\nu}\hat{\nabla}_{\mu}\tilde{\mathcal{C}}_{\lambda}{}^{\mu\nu}+T^{\alpha}{}_{\lambda\nu}\hat{\nabla}_{\mu}\tilde{\mathcal{C}}_{\alpha}{}^{\mu\nu}+\left(R^{\alpha}{}_{\nu\mu\lambda}-T_{\mu}T^{\alpha}{}_{\nu\lambda}\right)\tilde{\mathcal{C}}_{\alpha}{}^{\mu\nu}\equiv
0}$ (4.57)

where

$\displaystyle\tilde{\mathcal{M}}^{\mu\nu}$
$\displaystyle\coloneqq\frac{\delta\mathcal{S}[g,\Gamma]}{\delta g_{\mu\nu}}$
and $\displaystyle\tilde{\mathcal{C}}_{\alpha}{}^{\mu\nu}$
$\displaystyle\coloneqq\frac{\delta\mathcal{S}[g,\Gamma]}{\delta\Gamma^{\alpha}{}_{\mu\nu}}\,.$
(4.58)

are placeholders for the the metric and connection field equations (these are
tensor densities of weight one) and where

$\displaystyle\hat{\nabla}_{\mu}\coloneqq\nabla_{\mu}+T_{\mu}\,.$ (4.59)

The Bianchi identities of GR follow from this general identity as a special
case. Moreover, if we fix the connection to be flat and metric-compatible, we
find

$\displaystyle
2\mathcal{D}_{\mu}\tilde{\mathcal{M}}^{\mu}{}_{\lambda}-\hat{\nabla}_{\nu}\hat{\nabla}_{\mu}\tilde{\mathcal{C}}_{\lambda}{}^{\mu\nu}+T^{\alpha}{}_{\lambda\nu}\hat{\nabla}_{\mu}\tilde{\mathcal{C}}_{\alpha}{}^{\mu\nu}-T_{\mu}T^{\alpha}{}_{\nu\lambda}\tilde{\mathcal{C}}_{\alpha}{}^{\mu\nu}\equiv
0\,.$ (4.60)

Since in TEGR the metric field equations are the same as Einstein’s, i.e.,
since $\tilde{\mathcal{M}}_{\mu\nu}=\sqrt{|g|}\,G_{\mu\nu}$, this simplifies
further to

$\displaystyle\boxed{\hat{\nabla}_{\nu}\hat{\nabla}_{\mu}\tilde{\mathcal{C}}_{\lambda}{}^{\mu\nu}-T^{\alpha}{}_{\lambda\nu}\hat{\nabla}_{\mu}\tilde{\mathcal{C}}_{\alpha}{}^{\mu\nu}+T_{\mu}T^{\alpha}{}_{\nu\lambda}\tilde{\mathcal{C}}_{\alpha}{}^{\mu\nu}\equiv
0}$ (4.61)

due to $\mathcal{D}_{\mu}(\sqrt{|g|}\,G^{\mu\nu})=0$. Thus, only the
connection field equations remain and they have to satisfy the above Bianchi
identity.

#### Counting Degrees of Freedom

If we start with a metric $g_{\mu\nu}$ and a general affine connection
$\Gamma^{\alpha}{}_{\mu\nu}$ we have a total of $10+64$ fields. The flatness
condition $R^{\alpha}{}_{\mu\nu\rho}=0$ drastically reduces this number. Since
any flat connection can be written as
$\Gamma^{\alpha}{}_{\mu\nu}=(\Lambda^{-1})^{\alpha}{}_{\lambda}\partial_{\mu}\Lambda^{\lambda}{}_{\nu}$,
where $\Lambda^{\mu}{}_{\nu}$ is a $GL(4,\mathbb{R})$ matrix, we find that the
connection carries at most $4\times 4=16$ degrees of freedom, rather than
$64$. Finally, we also have to take into account the postulate of vanishing
non-metricity,

$\displaystyle
2(\Lambda^{-1})^{\lambda}{}_{\kappa}\partial_{\alpha}\Lambda^{\kappa}{}_{(\mu}g_{\nu)\lambda}=\partial_{\alpha}g_{\mu\nu}\,,$
(4.62)

which relates the metric and the matrix $\Lambda$. This equation is solved by

$\displaystyle
g_{\mu\nu}=\Lambda^{\alpha}{}_{\mu}\Lambda^{\beta}{}_{\nu}c_{\alpha\beta}\,,$
(4.63)

where $c_{\alpha\beta}$ is a symmetric, constant tensor. It is only natural to
choose $c_{\alpha\beta}=\eta_{\mu\nu}$, where the latter denotes the Minkowski
metric, since we are interested in metrics with Lorentzian signature. Notice
that this also means that instead of potentially $10+16$ degrees of freedom,
we now have at most $16$, since the connection as well as the metric can be
parametrized with the $16$ components of $\Lambda^{\mu}{}_{\nu}$.

At this point it should be noted that flat connections possess a gauge
symmetry. Namely, transformations of the form
$\Lambda\mapsto\Lambda\mathcal{U}$, where $\mathcal{U}\in GL(4,\mathbb{R})$,
leave the curvature tensor and thus the flatness condition invariant. If it
also has to respect the postulate of vanishing non-metricity, then it has to
leave the metric invariant, which means

$\displaystyle\mathcal{U}^{\alpha}{}_{\kappa}\mathcal{U}^{\beta}{}_{\lambda}\Lambda^{\kappa}{}_{\mu}\Lambda^{\lambda}{}_{\nu}\eta_{\alpha\beta}\overset{!}{=}\Lambda^{\alpha}{}_{\mu}\Lambda^{\beta}{}_{\nu}\eta_{\alpha\beta}\,.$
(4.64)

This implies that $\mathcal{U}$ belong to the proper orthochronous Lorentz
group, since this guarantees that the Minkowski metric is left invariant and
it is the part of the Lorentz group which is connected to the identity.

We therefore learn that six components of $\Lambda^{\mu}{}_{\nu}$ simply
represent Lorentz transformations and that these transformations are pure
gauge. That is, they do not change the form of the metric, nor do they affect
the flatness postulate. We have therefore a maximal number of $16-6=10$
degrees of freedom. However, TEGR is a generally covariant theory and
diffeomorphisms remove $2\times 4$ degrees of freedom. Hence, we are finally
left with only two degrees of freedom, as we expected.

###  4.3 The Symmetric Teleparallel Equivalent of General Relativity (STEGR)

#### The Geometric Postulates

We now turn to the third geometric formulation of GR, which ascribes
gravitational phenomena to non-metricity Nester:1998 : The so-called Symmetric
Teleparallel Equivalent of GR (STEGR) BeltranJimenez:2017b . The starting
point is again a metric-affine geometry $(\mathcal{M},g,\Gamma)$, but this
time restricted by the geometric postulates

$\displaystyle R^{\alpha}{}_{\mu\nu\rho}$ $\displaystyle\overset{!}{=}0$ and
$\displaystyle T^{\alpha}{}_{\mu\nu\phantom{\rho}}$
$\displaystyle\overset{!}{=}0\,.$ (4.65)

The postulate of vanishing curvature may, just as in TEGR, raise the question
of how the theory we seek to construct can possibly be equivalent to standard
GR, where curvature plays an essential role. However, the resolution to this
apparent tension is the same as in TEGR: What is postulated to vanish is the
curvature of the affine connection $\Gamma$, not the curvature of the Levi-
Civita connection on which GR is based.

#### Form of a Flat, Torsionless Connection and the Coincident Gauge

Before constructing an action functional for STEGR, let us work out what a
flat and torsionless connection looks like. From the previous subsection, we
recall that a flat connection can always be written as

$\Gamma^{\alpha}{}_{\mu\nu}=\left(\Lambda^{-1}\right)^{\alpha}{}_{\lambda}\,\partial_{\mu}\Lambda^{\lambda}{}_{\nu}\,,$
(4.66)

where $\Lambda^{\mu}{}_{\nu}$ are the components of a matrix belonging to the
general linear group $GL(4,\mathbb{R})$. The postulate of vanishing torsion
can then be rephrased as

$\displaystyle T^{\alpha}{}_{\mu\nu}$ $\displaystyle\overset{!}{=}0$
$\displaystyle\Longrightarrow$
$\displaystyle\partial_{[\mu}\Lambda^{\alpha}{}_{\nu]}$
$\displaystyle\overset{!}{=}0\,.$ (4.67)

This last condition implies that the matrix $\Lambda^{\mu}{}_{\nu}$ can be
written as $\Lambda^{\mu}{}_{\nu}=\partial_{\nu}\xi^{\mu}$, where $\xi^{\mu}$
denotes a collection of four arbitrary functions of the coordinates $x^{\mu}$,
not a vector field! We conclude that a flat, torsionless connection can be
written as

$\Gamma^{\alpha}{}_{\mu\nu}=\frac{\partial
x^{\alpha}}{\partial\xi^{\lambda}}\partial_{\mu}\partial_{\nu}\xi^{\lambda}\,,$
(4.68)

where $\partial x^{\alpha}/\partial\xi^{\lambda}$ should be understood as the
inverse of the Jacobian matrix $\partial\xi^{\lambda}/\partial x^{\alpha}$.
This result means that in any given coordinate system
$\\{x^{0},x^{1},x^{2},x^{3}\\}$ we can choose four independent functions
$\\{\xi^{0},\xi^{1},\xi^{2},\xi^{3}\\}$, such that the Jacobian matrix
$\partial\xi^{\mu}/\partial x^{\nu}$ is invertible (i.e., has a non-zero
determinant) and this allows us then to construct a flat and torsionless
connection via equation (4.68).

Moreover, equation (4.68) reveals that flat and torsionless connections have a
remarkable property: They can be set to zero globally by an appropriate choice
of coordinates. In fact, given any flat and torsionless connection, it
necessarily has the form (4.68) with some functions $\xi^{\mu}$. Therefore, if
we choose our coordinates such that $x^{\mu}=\xi^{\mu}$, the connection is
exactly equal to zero because $\partial_{\mu}\partial_{\nu}\xi^{\lambda}=0$.
This is known as the coincident gauge777More generally, one could also choose
the functions $\xi^{\mu}$ to be of the form
$\xi^{\mu}=M^{\mu}{}_{\nu}x^{\nu}+\xi^{\mu}_{0}$, where $M^{\mu}{}_{\nu}$ is a
non-degenerate matrix with constant entries and $\xi^{\mu}_{0}$ are constants
BeltranJimenez:2018 . This is also known as coincident gauge..

We emphasize that the coincident gauge can always be chosen and that it has
nothing to do with an action principle. It is available as long as the
postulates of vanishing curvature and vanishing torsion are in place. However,
we also stress that there are caveats one has to be aware of when it comes to
working in a fixed coordinate system and wanting to use the coincident gauge.
We will discuss these caveats in subsections 6.1 and 6.2.

#### Construction of the Action Functional

To construct an action functional for STEGR, we follow the same strategy as in
the case of TEGR. The key observation is that the curvature tensor of the
affine connection is related to the curvature tensor of the Levi-Civita
connection by the identity (3.81) we discussed in 3.8. We rewrite this
identity here for convenience:

$\displaystyle
R(\Gamma)=\mathcal{R}(g)+\mathbb{Q}+\mathcal{D}_{\alpha}\left(Q^{\alpha}-\bar{Q}^{\alpha}\right)\,,$
(4.69)

where $\mathcal{D}$ is the covariant derivative with respect to the Levi-
Civita connection, the two traces of the non-metricity tensor are given by

$\displaystyle Q_{\alpha}$ $\displaystyle\coloneqq
Q_{\alpha\lambda}{}^{\lambda}$ and $\displaystyle\bar{Q}_{\alpha}$
$\displaystyle\coloneqq Q^{\lambda}{}_{\lambda\alpha}\,,$ (4.70)

and the non-metricity scalar $\mathbb{Q}$ is defined as

$\displaystyle\mathbb{Q}$
$\displaystyle\coloneqq\frac{1}{4}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}-\frac{1}{2}Q_{\alpha\mu\nu}Q^{\mu\alpha\nu}-\frac{1}{4}Q_{\alpha}Q^{\alpha}+\frac{1}{2}Q_{\alpha}\bar{Q}^{\alpha}\,.$
(4.71)

The latter can also be expressed in terms of the disformation tensor
$L^{\alpha}{}_{\mu\nu}\coloneqq\frac{1}{2}Q^{\alpha}{}_{\mu\nu}-Q_{(\mu}{}^{\alpha}{}_{\nu)}$
as

$\displaystyle\mathbb{Q}=g^{\mu\nu}\left(L^{\alpha}{}_{\alpha\beta}L^{\beta}{}_{\mu\nu}-L^{\alpha}{}_{\beta\mu}L^{\beta}{}_{\nu\alpha}\right)\,.$
(4.72)

Recall that the identity (4.69) is valid only when torsion vanishes. Thus, one
of the geometric postulates is already implemented. The second postulate of
STEGR, which demands that the curvature of the affine connection vanishes,
then implies that

$\mathcal{R}(g)=-\mathbb{Q}-\mathcal{D}_{\alpha}\left(Q^{\alpha}-\bar{Q}^{\alpha}\right)\,.$
(4.73)

In other words, the Ricci scalar of the Levi-Civita connection can be
expressed in terms of the non-metricity scalar and a divergence term. This
allows us to replace $\mathcal{R}(g)$ in the Einstein-Hilbert action by the
right hand side of the identity (4.73). Thus, in the Symmetric Teleparallel
Equivalent of GR gravity is described by the action functional

$\mathcal{S}_{\rm
STEGR}[g,\xi]=-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\mathbb{Q}(g,\xi)+\mathcal{S}_{\textsf{matter}}\,.$
(4.74)

We have dropped the divergence
$\mathcal{D}_{\alpha}\left(Q^{\alpha}-\bar{Q}^{\alpha}\right)$ since, by the
generalized Gauss theorem which we discussed in subsection 3.7, this term
amounts to a mere boundary term which can thus have no influence on the field
equations. Moreover, as we have discussed in the GR and TEGR sections,
changing the action is a necessary step in order to arrive at a genuinely new
formulation.

Notice that the action is a functional of the metric $g_{\mu\nu}$ and the four
functions $\xi^{\alpha}$ which parametrize the flat, torsionless connection
(4.68).

The candidate for the STEGR action could also have been constructed without
knowing the geometric identity (4.73). Just as in TEGR, one can start with the
most general Lagrangian which is quadratic in the non-metricity tensor. Due to
the symmetry of the non-metricity tensor, one finds that there are precisely
five independent scalars one can construct from contractions of the non-
metricity tensor (this will be discussed in more details in subsection 5.2).
The most general Lagrangian then reads

$\displaystyle\hat{\mathbb{Q}}=c_{1}\,Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+c_{2}\,Q_{\mu\alpha\nu}Q^{\alpha\mu\nu}+c_{3}\,Q_{\mu}Q^{\mu}+c_{4}\,\bar{Q}_{\mu}\bar{Q}^{\mu}+c_{5}\,Q_{\mu}\bar{Q}^{\mu}\,,$
(4.75)

where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, and $c_{5}$ are arbitrary, real
constants. By expanding the theory around a Minkowski background and demanding
that it propagates two degrees of freedom, which is tantamount to demanding
that the linearized theory is invariant under linearized diffeomorphisms, one
finds that the parameters $c_{i}$ have to satisfy

$\displaystyle c_{3}$ $\displaystyle=-c_{1},$ $\displaystyle c_{4}$
$\displaystyle=-2c_{1}-c_{2},$ $\displaystyle c_{5}=2c_{1}\,.$ (4.76)

These relations are satisfied by the parameter values which reproduce the
STEGR action and they leave $c_{1}$ and $c_{2}$ free. The linearized theory
cannot fix these parameters, but other considerations can reproduce the STEGR
action up to an overall normalization. For instance, demanding that the full,
non-linear theory satisfies the contracted Bianchi identity
$\mathcal{D}_{\mu}\mathcal{M}^{\mu}{}_{\nu}=0$, where
$\mathcal{M}^{\mu}{}_{\nu}$ stands for the metric field equations, requires
$c_{4}$ to vanish. Thus, we find

$\displaystyle c_{2}$ $\displaystyle=-2c_{1},$ $\displaystyle c_{3}$
$\displaystyle=-c_{1},$ $\displaystyle c_{4}$ $\displaystyle=0,$
$\displaystyle c_{5}$ $\displaystyle=2c_{1}\,,$ (4.77)

which indeed reproduces the action (4.74) up to an overall normalization
constant.

#### The Palatini Formulation of the Action Principle

Just as in TEGR, we can employ the Palatini formalism in order to express the
action principle in a manifestly covariant way which also highlights which
type of metric-affine geometry is being considered. This action is defined as

$\mathcal{S}_{\rm
STEGR}[g,\Gamma]\coloneqq-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left(\frac{1}{2\kappa}\sqrt{|g|}\,\mathbb{Q}(g,\Gamma)+\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}\,R^{\alpha}{}_{\mu\nu\rho}+\tilde{\chi}_{\alpha}{}^{\mu\nu}\,T^{\alpha}{}_{\mu\nu}\right)+\mathcal{S}_{\textsf{matter}}\,,$
(4.78)

where the Lagrange multipliers $\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}$ and
$\tilde{\chi}_{\alpha}{}^{\mu\nu}$ are tensor densities of weight $w=+1$.
These Lagrange multipliers inherit the symmetries
$\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}=\tilde{\Pi}_{\alpha}{}^{\mu[\nu\rho]}$
and $\tilde{\chi}_{\alpha}{}^{\mu\nu}=\tilde{\chi}_{\alpha}{}^{[\mu\nu]}$ from
the curvature and torsion tensor, respectively.

Notice that the action is a functional of the metric $g_{\mu\nu}$ and a
generic affine connection $\Gamma^{\alpha}{}_{\mu\nu}$. By varying the above
action with respect to the Lagrange multiplier densities, one obtains two
constraints on the connection. Namely, the connection is restricted to be flat
and torsionless. Since these constraints do not completely fix the connection,
we still have the freedom to choose four arbitrary functions $\xi^{\alpha}$ in
order to parametrize the connection in agreement with equation (4.68).

As a final comment we add that the curvature tensor measures the change in
direction of a vector which is being parallel transported around a closed loop
(cf. subsection 3.2). Hence, when curvature vanishes, there is no change in
direction and the vector remains, in this sense, parallel to itself. This
justifies the use of the term teleparallel. Moreover, the vanishing of torsion
implies that the connection $\Gamma^{\alpha}{}_{\mu\nu}$ is symmetric in its
lower two indices. Hence the use of the word symmetric in Symmetric
Teleparallel Equivalent of GR.

#### The Metric and Connection Field Equations

To obtain the field equations of STEGR we can either take the action (4.74) as
starting point or the action (4.78). In either case we make the observation
that the non-metricity tensor is linear in first order derivatives of the
metric and that the non-metricity scalar $\mathbb{Q}$ is consequently
quadratic in first order derivatives. Due to the absence of second order
derivatives in either action principle, there is no need to add boundary terms
à la Gibbons-Hawking-York. Both variational principles are well-defined as
they stand.

If we choose to work with the Palatini formalism, we have to vary the action
(4.78) with respect to the metric $g_{\mu\nu}$, the general affine connection
$\Gamma^{\alpha}{}_{\mu\nu}$, as well as the Lagrange multiplier densities
$\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}$ and $\tilde{\chi}_{\alpha}{}^{\mu\nu}$.

If instead we work with the action (4.74), we only need to perform variations
with respect to $g_{\mu\nu}$ and $\xi^{\alpha}$. The first approach turns out
to be simpler, despite the additional variations one has to perform. The
computations have been carried out in great detail in BeltranJimenez:2018 .
For the variation with respect to the inverse metric, one obtains the metric
field equations

$\boxed{\frac{2}{\sqrt{|g|}}\nabla_{\alpha}\left[\sqrt{|g|}\,P^{\alpha}{}_{\mu\nu}\right]+q_{\mu\nu}-\frac{1}{2}\mathbb{Q}\,g_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}}$
(4.79)

As always, $\mathcal{T}_{\mu\nu}$ denotes the energy-momentum tensor of matter
fields and we have introduced the non-metricity conjugate
$\boldsymbol{P^{\alpha}{}_{\mu\nu}}$ and the symmetric tensor $q_{\mu\nu}$,
respectively defined by

$\displaystyle P^{\alpha}{}_{\mu\nu}$
$\displaystyle\coloneqq\frac{1}{2}\frac{\partial\mathbb{Q}}{\partial
Q_{\alpha}{}^{\mu\nu}}=-\frac{1}{4}\,Q^{\alpha}{}_{\mu\nu}+\frac{1}{2}\,Q_{(\mu}{}^{\alpha}{}_{\nu)}+\frac{1}{4}\,g_{\mu\nu}Q^{\alpha}-\frac{1}{4}\,\left(g_{\mu\nu}\bar{Q}^{\alpha}+\delta^{\alpha}{}_{(\mu}Q_{\nu)}\right)$
$\displaystyle q_{\mu\nu}$
$\displaystyle\coloneqq\frac{\partial\mathbb{Q}}{\partial
g^{\mu\nu}}=P_{(\mu|\alpha\beta}Q_{\nu)}{}^{\alpha\beta}-2P^{\alpha\beta}{}_{(\nu}Q_{\alpha\beta|\mu)}\,.$
(4.80)

It should be noted that the non-metricity scalar $\mathbb{Q}$ can be expressed
with the help of the non-metricity tensor and its conjugate as

$\mathbb{Q}=P_{\alpha\mu\nu}Q^{\alpha\mu\nu}\,.$ (4.81)

The variations with respect to the Lagrange multipliers and the general affine
connection boil down to a connection field equation of the form

$\boxed{\nabla_{\mu}\nabla_{\nu}\left(\sqrt{|g|}\,P^{\mu\nu}{}_{\alpha}\right)=0}$
(4.82)

Here, just as in TEGR, we have assumed that the hypermomentum density (4.23)
vanishes. Alternatively, we could have demanded that it is conserved,
$\nabla_{\mu}H_{\alpha}{}^{\mu\nu}=0$.

In both field equations the connection is flat and torsionless, as required by
the geometric postulates. It is thus parametrized by the four functions
$\xi^{\alpha}$. Moreover, observe that we have ten metric field equations and
four connection field equations. These numbers match the number of fields in
the theory, namely ten metric components and four $\xi$’s. However, just as in
the case of GR and TEGR, not all equations are independent due to the
diffeomorphism invariance of the theory.

#### The Bianchi Identities

The action (4.78) is manifestly invariant under diffeomorphisms. This follows
from the fact that $\sqrt{|g|}\,\mathbb{Q}$ is a scalar density of weight
$w=+1$. Also, since the Lagrange multipliers have density weight $w=+1$ and
they are fully contracted, the curvature and torsion constraints are also
scalar densities with the correct weight. Correct means that the integrand
transforms in such a way under diffeomorphisms that the integral remains
invariant. Following the same considerations as in GR, but now also taking the
transformation behaviour of the connection into account, we find the following
identities for STEGR:

$\displaystyle\mathcal{D}_{\mu}\mathcal{M}^{\mu}{}_{\nu}+\mathcal{C}_{\nu}\equiv
0\,,$ (4.83)

where we have defined

$\displaystyle\mathcal{M}_{\mu\nu}$
$\displaystyle\coloneqq\frac{2}{\sqrt{|g|}}\nabla_{\alpha}\left[\sqrt{|g|}\,P^{\alpha}{}_{\mu\nu}\right]+q_{\mu\nu}-\frac{1}{2}\mathbb{Q}\,g_{\mu\nu}$
$\displaystyle\mathcal{C}_{\alpha}$
$\displaystyle\coloneqq\nabla_{\mu}\nabla_{\nu}\left(\sqrt{|g|}\,P^{\mu\nu}{}_{\alpha}\right)$
(4.84)

The tensor $\mathcal{M}_{\mu\nu}$ is simply the expression that appears on the
left side of the metric field equations, while $\mathcal{C}_{\alpha}$
represents the left side of the connection field equations. As a consequence
of these Bianchi identities, it follows that if the metric field equations are
satisfied, i.e., if $\mathcal{M}_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}$, then

$\displaystyle\kappa\,\mathcal{D}_{\mu}\mathcal{T}^{\mu}{}_{\nu}+\mathcal{C}_{\nu}\equiv
0\,.$ (4.85)

By invoking the covariant conservation of energy-momentum of matter fields,
i.e., $\mathcal{D}_{\mu}\mathcal{T}^{\mu}{}_{\nu}$, we find

$\displaystyle\text{If
}\mathcal{M}_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}\text{ is satisfied, then
}\mathcal{C}_{\nu}\equiv 0\,.$ (4.86)

In other words, if the metric field equations are satisfied, then the
connection field equations become mere identities. That is, the connection
field equations are trivially satisfied and carry no dynamical information.

Since one can show that $\mathcal{M}_{\mu\nu}=G_{\mu\nu}$, where the right
hand side is the Einstein tensor without cosmological constant, one can reach
an even stronger conclusion: The Einstein tensor satisfies the Bianchi
identity $\nabla_{\mu}G^{\mu}{}_{\nu}$ also off-shell, i.e., when the Einstein
equations are not satisfied. By combining this fact with the Bianchi identity
of STEGR, one reaches the conclusion that

$\displaystyle\boxed{\mathcal{C}_{\nu}\equiv 0}$ (4.87)

is always true! Since $\mathcal{M}_{\mu\nu}=G_{\mu\nu}$ implies that
$\mathcal{M}_{\mu\nu}$ is independent of $\xi^{\alpha}$ (it only knows about
the Levi-Civita part of the connection and nothing else) it follows that
$\mathcal{M}_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}$ are purely equations for
the metric. Furthermore, since $\mathcal{C}_{\nu}=0$ is always identically
satisfied, there are no equations which determine the four functions
$\xi^{\alpha}$! They remain completely arbitrary. What these considerations
show is the following:

  * •

STEGR is equivalent to GR in the sense that both theories possess the same
field equations and consequently the same solution space. They are
nevertheless rooted in different mathematical frameworks, they used different
fields in their formulation, and this opens the door to conceptual and
philosophical differences between the two theories.

  * •

There is a sense in which STEGR is invariant under two copies of the
diffeomorphism group. First its action is manifestly diffeomorphism invariant
and its field equations are manifestly general covariant. Thus, performing a
diffeomorphism which changes the metric and the connection simultaneously does
not affect the theory. Secondly, we have the freedom to choose the four
functions $\xi^{\alpha}$ at will. The metric field equations do not depend
upon this choice and there are no dynamical equations which determine the
$\xi$’s. Thus, this constitutes a second freedom.

As we will see later, the independence of $\mathcal{M}_{\mu\nu}$ from the
$\xi$’s hinges on a carefully balanced cancellation. If one considers the most
general non-metricity scalar $\hat{\mathbb{Q}}$, as we do in 5.2, this
independence is lost unless one chooses certain parameters in the theory in a
careful way. Also, we will see that in $f(\mathbb{Q})$ gravity the $\xi$’s are
no longer arbitrary. They come with their own dynamical field equations.

#### Counting Degrees of Freedom

After the discussion of the Bianchi identities it comes as no surprise that
STEGR propagates two physical degrees of freedom. Concretely, the counting
goes as follows: The theory is formulated in terms of a metric $g_{\mu\nu}$
with ten components and a general affine connection
$\Gamma^{\alpha}{}_{\mu\nu}$ with $4\times 4\times 4=64$ components. By either
postulating the vanishing of curvature and torsion or by solving the
constraints that arise from the Palatini formulation of the theory, one finds
that the connection carries four potential degrees of freedom (the $\xi$’s).

This leaves us with $10+4=14$ variables and an equal number of field
equations. However, the metric field equations only contain the metric and no
$\xi$’s. Also, there are no dynamical equation for the $\xi$’s. They remain
completely arbitrary and do not constitute anything physical. In fact, as we
will see in the next subsection, they play the role of Stückelberg fields,
which ensure that the theory is generally covariant.

This leaves us with at most ten dynamical variables, namely the metric
components. However, since the metric field equations are simply the Einstein
field equations, the same counting as in GR assures us that only two of these
components represent physical degrees of freedom.

Alternatively, one could also argue as follows: STEGR is a diffeomorphism
invariant theory and it is also invariant under the replacement
$\xi^{\alpha}\mapsto\hat{\xi}^{\alpha}$, where $\hat{\xi}^{\alpha}$ is a new
set of four functions which parametrize the flat, torsionless connection.
Thus, the $\xi$’s play no dynamical role and since diffeomorphisms remove
$2\times 4$ degrees of freedom one finds again $14-4-2\times 4=2$ physical
degrees of freedom.

Thus, in either case, we conclude that STEGR propagates the same two degrees
of freedom as GR, as had to be expected. This will no longer be true when we
consider generalizations of STEGR in subsections 5.2 and 5.3 and in particular
in section 6.

###  4.4 Coincident General Relativity (CGR)

Coincident General Relativity, or CGR for short, refers to a special case of
STEGR. In fact, CGR is simply STEGR in coincident gauge. Even tough this might
seem trivial, CGR has played an important role in applications such as
$f(\mathbb{Q})$ cosmology BeltranJimenez:2019 ; DAmbrosio:2020b ; Bajardi:2020
; Ayuso:2020 ; Frusciante:2021 ; Atayde:2021 ; Anagnostopoulos:2021 ;
Capozziello:2022 . Furthermore, by comparing and contrasting CGR with full
STEGR and what is nowadays called the Einstein action, one is lead to a deeper
understanding of the role played by the flat and torsionless connection (or,
equivalently, by the $\xi$’s).

Let us begin by introducing the action of CGR. As mentioned above, CGR makes
use of the coincident gauge, which means the flat and torsionless connection
$\Gamma^{\alpha}{}_{\mu\nu}$ vanishes globally. Upon using the decomposition
(3.47), we find that this implies

$\displaystyle\Gamma^{\alpha}{}_{\mu\nu}=\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}+L^{\alpha}{}_{\mu\nu}$
$\displaystyle\overset{\text{CG}}{=}0$ $\displaystyle\Longrightarrow$
$\displaystyle
L^{\alpha}{}_{\mu\nu}\overset{*}{=}-\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\mu\nu}\right\\}\,.$
(4.88)

The star on top of the equal sign shall remind us that this relation only
holds in the coincident gauge. This last equality is particularly useful if we
recall that the STEGR action can be written in terms of the disformation
tensor $L^{\alpha}{}_{\mu\nu}$ alone (cf. equation 4.72). Hence, the CGR
action takes the form

$\displaystyle\mathcal{S}_{\text{CGR}}[g]\equiv\mathcal{S}_{\text{STGR}}[g,\Gamma=0]$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,g^{\mu\nu}\left(L^{\alpha}{}_{\alpha\beta}L^{\beta}{}_{\mu\nu}-L^{\alpha}{}_{\beta\mu}L^{\beta}{}_{\nu\alpha}\right)$
$\displaystyle\overset{*}{=}\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,g^{\mu\nu}\left(\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\alpha\beta}\right\\}\left\\{\genfrac{}{}{0.0pt}{}{\beta}{\mu\nu}\right\\}-\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\beta\mu}\right\\}\left\\{\genfrac{}{}{0.0pt}{}{\beta}{\nu\alpha}\right\\}\right)\,.$
(4.89)

The first integral is valid in complete generality, while the second one holds
only in the coincident gauge. Observe further that the action is a functional
of the metric alone. This is one of the features which make CGR attractive for
studying applications: The connection has been dealt with and even globally
trivialized, such that one only has to work with the metric. However, when
using CGR as the starting point for defining non-linear modifications such as
in $f(\mathbb{Q})$ gravity, one is prone to encounter subtleties related to
having fixed the coincident gauge. These subtleties have to do with assuming
some background symmetries (spherical symmetry, homogeneity and isotropy, …)
and we will discuss them in more detail in subsections 6.1 and 6.2.

Here, we wish to highlight an other feature of CGR. Namely that its action is
exactly equal to the so-called Einstein action, which in turn is simply the
Einstein-Hilbert action without the second order derivatives. This can easily
be seen by recalling that the Ricci tensor is given by

$\displaystyle\mathcal{R}_{\mu\nu}=\partial_{\alpha}\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\nu\mu}\right\\}-\partial_{\nu}\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\alpha\mu}\right\\}+\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\alpha\beta}\right\\}\left\\{\genfrac{}{}{0.0pt}{}{\beta}{\mu\nu}\right\\}-\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\beta\mu}\right\\}\left\\{\genfrac{}{}{0.0pt}{}{\beta}{\nu\alpha}\right\\}\,.$
(4.90)

By comparing (4.90) to the action (4.4) it follows that the Einstein action is
given by the Ricci scalar $\mathcal{R}=g^{\mu\nu}\mathcal{R}_{\mu\nu}$ minus
the term

$\displaystyle
g^{\mu\nu}\left(\partial_{\alpha}\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\nu\mu}\right\\}-\partial_{\nu}\left\\{\genfrac{}{}{0.0pt}{}{\alpha}{\alpha\mu}\right\\}\right)\,$
(4.91)

which contains the second order derivatives of the metric. Hence, neither the
CGR nor the Einstein action require the GHY boundary term. Both of them give
rise to a well-defined variational principle.

However, despite looking the same, there is a crucial difference between the
Einstein action and the CGR action. The former is not diffeomorphism
invariant. This follows from the fact that the connection does not transform
like a tensor under coordinate transformations (see equation (2.64)) and one
can check that the Einstein action picks up boundary terms under such
transformations.

The CGR action, on the other hand, is the gauge-fixed version of a perfectly
covariant functional, namely the STEGR action. Hence, we can interpret the
connection as a Stückelberg field which restores the general covariance of the
Einstein action.

###  4.5 The General Teleparallel Equivalent of General Relativity (GTEGR)

So far we have seen that gravity can be described from three different
perspectives: Following Einstein’s original path, we can encode gravity in the
curvature of spacetime while setting torsion and non-metricity to zero. Or we
can describe gravity using torsion in a flat and metric-compatible spacetime.
The third option is to work in a flat and torsionless spacetime, but with non-
zero non-metricity.

We can think of these three descriptions as the three corners of a triangle,
as illustrated in Figure 12. We can also give meaning to the edges of the
triangle. Of particular interest to us is the lower edge which connects TEGR
and STEGR. In fact, there exists yet another teleparallel theory of gravity,
called the General Teleparallel Equivalent of GR (GTEGR) BeltranJimenez:2019d
; Hohmann:2022 ; Heisenberg:2022 , which subsumes TEGR and STEGR in the sense
that these two theories are the gauge-fixed offsprings of a more general
parent theory.

To construct the theory, we start again with a general metric-affine geometry
$(\mathcal{M},g,\Gamma)$ and a single geometric postulate,

$\displaystyle R^{\alpha}{}_{\mu\nu\rho}\overset{!}{=}0\,.$ (4.92)

The geometric identity (3.78) we encountered in subsection 3.8 then allows us
to write the Ricci scalar of the Levi-Civita connection as

$\displaystyle-\mathcal{R}(g)$
$\displaystyle=\mathbb{T}+\mathbb{Q}+T^{\rho\mu\nu}Q_{\mu\nu\rho}-T^{\mu}Q_{\mu}+T^{\mu}\bar{Q}_{\mu}+\mathcal{D}_{\alpha}\left(Q^{\alpha}-\bar{Q}^{\alpha}+2T^{\alpha}\right)$
$\displaystyle=\mathbb{G}+\mathcal{D}_{\alpha}\left(Q^{\alpha}-\bar{Q}^{\alpha}+2T^{\alpha}\right)\,,$
(4.93)

where in the last equation we have introduced the scalar

$\displaystyle\mathbb{G}\coloneqq\mathbb{T}+\mathbb{Q}+T^{\rho\mu\nu}Q_{\mu\nu\rho}-T^{\mu}Q_{\mu}+T^{\mu}\bar{Q}_{\mu}\,.$
(4.94)

We then define the following action

$\displaystyle\mathcal{S}_{\rm
GTEGR}[g,\Lambda]\coloneqq-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\,\mathbb{G}(g,\Lambda)+\mathcal{S}_{\rm
matter}\,,$ (4.95)

where $\Lambda\in GL(4,\mathbb{R})$ is the matrix used to parametrize the flat
connection. Since the EH action and the action of GTEGR only differ by a total
derivative, it comes as no surprise that both actions describe the same
theory. However, notice that while GR only makes use of a metric, GTEGR also
involves the matrix $\Lambda\in GL(4,\mathbb{R})$ in its definition. This
mismatch in the number of fields is no reason for concern, since GTEGR enjoys
an additional symmetry. In fact, $\delta_{\Lambda}\mathcal{S}_{\rm GTEGR}=0$
is satisfied off-shell, which means that the connection is not dynamical
BeltranJimenez:2019d . Put differently, this means that only the metric
carries physical degrees of freedom while the connection is pure gauge.
Furthermore, since the metric field equations obtained from (4.95) have to be
the Einstein field equations, the metric propagates exactly the same two
degrees of freedom as GR.

Observe further that in the absence of non-metricity the scalar $\mathbb{G}$
reduces to $\mathbb{T}$ and TEGR is recovered. Similarly, when torsion is
absent, $\mathbb{G}$ reduces to the non-metricity scalar $\mathbb{Q}$ and
STEGR emerges. Demanding either the vanishing of torsion or the vanishing of
non-metricity amounts to imposing additional conditions on the connection, as
we have seen in previous subsections. It is in this sense that we can think of
TEGR and STEGR as partially gauge-fixed versions of the more general theory
GTEGR: The pure gauge connection of GTEGR can be partially fixed by either
imposing $Q_{\alpha\mu\nu}=0$ or $T^{\alpha}{}_{\mu\nu}=0$, which simply
amounts to working with TEGR or STEGR, respectively.

###  4.6 Non-flat combinations in the edges and in the dot

So far we have discussed the three corners of Figure 12 as well as one edge.
This corresponds to four different formulations of General Relativity:
Standard GR based on curvature and the teleparallel theories TEGR, STEGR, and
GTEGR which are all based on the postulate of vanishing curvature. It is only
natural to ask whether other equivalent formulations are possible. In
particular, there are two more edges present in Figure 12. These would
correspond to non-flat geometries with either torsion or non-metricity (but
not both at the same time). Finally, we can also imagine a dot in the center
of the triangle, which represents a theory based on non-vanishing curvature,
torsion, and non-metricity. A modified version of Figure 12 could look like
13.

Figure 13: Of the metric-affine theories of gravity which live on the edges
and in the center of the triangle, only the General Teleparallel theory is
equivalent to GR.

Notice that in all three new cases we wish to discuss, the postulate of
teleparallelism (i.e., the condition $R^{\alpha}{}_{\mu\nu\rho}=0$) is not
imposed. This has far-reaching consequences.

Recall that in TEGR, STEGR, and GTEGR the crucial step was to impose
$R^{\alpha}{}_{\mu\nu\rho}=0$, which immediately implies that the connection
has the form $\Gamma=(\Lambda)^{-1}\partial\Lambda$. Since the Lagrangians
which define these three theories are all quadratic in $T$ and $Q$, we find
that they all possess something akin to a “kinetic term”,
$T^{2}\sim(\partial\Lambda)^{2}$ and $Q^{2}\sim(\partial\Lambda)^{2}$.
However, if the flatness condition is not imposed, we lose this “kinetic
term”. In particular, the actions

$\displaystyle\mathcal{S}_{\rm Einstein-Cartan}[g,\Gamma]$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\left(R+\mathbb{T}\right)+\mathcal{S}_{\rm
matter}[g,\Gamma,\Psi]$ $\displaystyle\mathcal{S}[g,\Gamma]$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\left(R+\mathbb{Q}\right)+\mathcal{S}_{\rm
matter}[g,\Gamma,\Psi]$ $\displaystyle\mathcal{S}[g,\Gamma]$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{|g|}\left(R+\mathbb{T}+\mathbb{Q}\right)+\mathcal{S}_{\rm
matter}[g,\Gamma,\Psi]$ (4.96)

are all deprived of this “kinetic term”. The first one corresponds to the left
edge in Figure 13 and is also known as the Einstein-Cartan action. The action
in the middle represents the theory living on the right edge, while the action
on the bottom corresponds to the dot in Figure 13. As it turns out, in all
three cases the connection is a mere auxiliary field which can be integrated
out. After integrating out the connection, the resulting actions are not
equivalent to GR! Rather, one obtains three modified gravity theories.

Furthermore, one can also show that integrating out the connection changes the
way matter fields couple in these theories, leading again to non-GR behaviour.
This is in the same spirit as what was shown in BeltranJimenez:2017  for more
general Lagrangians based on the Palatini formalism.

###  4.7 Matter Coupling

Our discussion of the geometric trinity and the equivalence between
teleparallel theories of gravity and GR was so far limited to the pure gravity
sector. Does the equivalence between teleparallel theories and GR also hold in
the presence of matter fields?

In GR, the coupling of the gravitational field to matter fields follows the
so-called minimal coupling principle. It states that a matter theory
formulated in Minkowski space is promoted to a matter theory coupled to the
gravitational field $g_{\mu\nu}$ by replacing $\eta_{\mu\nu}\mapsto
g_{\mu\nu}$ and $\partial_{\mu}\mapsto\mathcal{D}_{\mu}$, provided that the
matter fields only couple to $g_{\mu\nu}$, $g^{\mu\nu}$, and $\sqrt{|g|}$, but
not derivatives of the metric. Is the minimal coupling principle preserved in
TEGR and STEGR?

Let us first consider TEGR and naively apply the minimal coupling principle in
the form $\eta_{\mu\nu}\mapsto g_{\mu\nu}$ and
$\partial_{\mu}\mapsto\nabla_{\mu}$, where $\nabla_{\mu}$ is the covariant
derivative operator with respect to the connection
$\Gamma^{\alpha}{}_{\mu\nu}$. As a specific example, we consider the
electromagnetic potential $A_{\mu}$ and its associated Maxwell $2$-form
$F_{\mu\nu}\coloneqq\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. According to
the minimal coupling principle, the Maxwell $2$-form becomes

$\displaystyle
F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-T^{\alpha}{}_{\mu\nu}A_{\alpha}\,.$
(4.97)

We immediately conclude that the minimal coupling principle fails, since the
Maxwell action picks up terms proportional to the torsion tensor, thus
spoiling the equivalence between TEGR and GR. For fermionic fields, one
obtains a similar failure of the minimal coupling principle. The Dirac
Lagrangian is directly affected by the connection in the presence of an axial
torsion.

In STEGR the situation is quite different: The minimal coupling principle is
preserved even in the presence of non-metricity. In the case of the
electromagnetic field $A_{\mu}$ it is straightforward to verify that non-
metricity does not contribute to $F_{\mu\nu}$ due to its symmetry and thus one
finds

$\displaystyle
F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,,$
(4.98)

just as in GR. For fermions, this property remains unchanged. The non-
metricity drops out completely from the Dirac Lagrangian due to the symmetry
of the non-metricity tensor. For a more detailed analysis of matter couplings
in TEGR and STEGR we refer the reader to BeltranJimenez:2020 .

The key message here is that in the presence of matter fields, the equivalence
is only maintained between STEGR and GR. TEGR coupled to matter fields is no
longer equivalent to GR.

## 5 The Geometrical Trinity of Modified Gravity Theories

In section 4 we introduced three different geometric approaches to formulate
the theory of General Relativity. This so-called geometric trinity of GR has
conceptual advantages. For instance, the teleparallel theories TEGR and STEGR
possess well-defined variational principles BeltranJimenez:2018 ;
BeltranJimenez:2019c  without the need of adding a GHY boundary term.
Furthermore, STEGR and CGR have inspired new approaches to define the elusive
gravitational energy-momentum BeltranJimenez:2017b ; BeltranJimenez:2019b ;
BeltranJimenez:2021 ; Koivisto:2022 , it is possible to compute black hole
entropy without adding counter terms to the action BeltranJimenez:2018 ;
BeltranJimenez:2019b ; Heisenberg:2022b , and the coincident gauge might open
a new avenue toward the quantization of the gravitational field.

However, since the field equations of TEGR and STEGR are identical to the
Einstein field equations, these theories cannot address any phenomenological
questions which elude GR—such as the accelerated expansion of the Universe or
the shape of galactic rotation curves.

Such questions are typically addressed by theories of modified gravity and the
geometric trinity of GR presented in the previous section can be used as
starting point for developing such modifications. There are two approaches
which are commonly considered in the literature:

  1. 1.

The actions of TEGR and STEGR are quadratic in the torsion and the non-
metricity tensors, respectively. One can thus try to construct the most
general scalar which is quadratic in the torsion or non-metricity tensor and
take this scalar to define an action functional. In the case of torsion, one
finds a three-parameter family of theories described by an action which is
quadratic in the torsion tensor. In the case of non-metricity, one finds a
five-parameter family of quadratic Lagrangians. These generalizations are
discussed in subsections 5.1 and 5.2, respectively.

  2. 2.

An other popular direction is to consider non-linear extensions of the form
$f(\mathbb{T})$ and $f(\mathbb{Q})$, where $f$ is some function which is only
subjected to the condition that its first derivative does not vanish. Non-
linear extensions of this type are the subject of subsection 5.3. In section 6
we will have a closer look at $f(\mathbb{Q})$, its application to cosmology,
black hole physics, and the question of how many degrees of freedom the theory
propagates.

Since these modifications of GR are based on the framework of metric-affine
geometry, we will sometimes refer to them as the geometrical trinity of
modified gravity theories.

###  5.1 Quadratic Actions for Torsion Theories

Recall from subsection 4.2 that the action of TEGR is constructed solely from
quadratic contractions of the torsion tensor. Concretely, we defined the so-
called torsion scalar as

$\displaystyle\mathbb{T}\coloneqq\frac{1}{2}\left(\frac{1}{4}T_{\alpha\mu\nu}+\frac{1}{2}T_{\mu\alpha\nu}-g_{\alpha\mu}T_{\nu}\right)T^{\alpha\mu\nu}\,.$
(5.1)

Now we are interested in constructing the most general scalar which is
quadratic in the torsion tensor. To that end, we need to consider the
symmetries of $T^{\alpha}{}_{\mu\nu}$. A priori, a tensor with three indices
can be contracted in six different ways with itself; one just has to perform
all possible permutations of indices. However, because $T^{\alpha}{}_{\mu\nu}$
is antisymmetric in its lower indices, only two of these contractions are
independent:

$\displaystyle T_{\alpha\mu\nu}T^{\alpha\mu\nu}$ and $\displaystyle
T_{\mu\alpha\nu}T^{\alpha\mu\nu}\,.$ (5.2)

The next thing to consider is the trace of the torsion tensor. Due to its
antisymmetry, the torsion tensor possesses only one trace; $T_{\mu}\coloneqq
T^{\alpha}{}_{\mu\alpha}$. Thus, the only other quadratic contraction we can
build out of the torsion tensor is

$T_{\mu}T^{\mu}\,.$ (5.3)

With this we have exhausted all options and we conclude that the most general
scalar which is quadratic in the torsion tensor is a linear combination of the
three terms discussed above:

$\displaystyle\hat{\mathbb{T}}\coloneqq
c_{1}\,T_{\alpha\mu\nu}T^{\alpha\mu\nu}+c_{2}\,T_{\mu\alpha\nu}T^{\alpha\mu\nu}+c_{3}\,T_{\mu}T^{\mu}\,,$
(5.4)

where $c_{1}$, $c_{2}$, and $c_{3}$ are arbitrary, real constants. It is easy
to see that the scalar $\hat{\mathbb{T}}$ reduces to $\mathbb{T}$ for the
parameter choice $c_{1}=\frac{1}{4}$, $c_{2}=\frac{1}{2}$, $c_{3}=-1$. Using
the general torsion scalar defined in (5.4), we can now write the action
functional of Teleparallel Gravity (TG)888The theory defined by this action is
sometimes referred to as New General Relativity in the literature (for
instance in Hayashi:1979 ; BeltranJimenez:2017b ; Blixt:2018 ). as

$\mathcal{S}_{\textsf{TG}}[g,\Gamma,\Psi]\coloneqq-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left(\frac{1}{\kappa}\sqrt{|g|}\,\hat{\mathbb{T}}+\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}\,R^{\alpha}{}_{\mu\nu\rho}+\tilde{\chi}^{\alpha}{}_{\mu\nu}\,Q_{\alpha}{}^{\mu\nu}\right)+\mathcal{S}_{\textsf{matter}}[g,\Psi]\,,$
(5.5)

where the matter fields $\Psi$ are assumed to be minimally coupled and having
a vanishing hypermomentum. This action looks deceptively similar to the action
of TEGR, since we have only replaced $\mathbb{T}$ by the more general
$\hat{\mathbb{T}}$. Indeed, even the field equations look very similar:

Metric field equations:
$\displaystyle(\nabla_{\alpha}+T_{\alpha})\hat{S}_{(\mu\nu)}{}^{\alpha}+\hat{t}_{\mu\nu}-\frac{1}{2}\hat{\mathbb{T}}\,g_{\mu\nu}$
$\displaystyle=\kappa\,\mathcal{T}_{\mu\nu}$ Connection field equations:
$\displaystyle(\nabla_{\alpha}+T_{\alpha})\left[\sqrt{|g|}\hat{S}_{[\mu}{}^{\alpha}{}_{\nu]}\right]$
$\displaystyle=0\,,$ (5.6)

where the hatted torsion conjugate $\hat{S}_{\alpha}{}^{\mu\nu}$ and the
symmetric tensor $\hat{t}_{\mu\nu}$ are defined as

$\displaystyle\hat{S}_{\alpha}{}^{\mu\nu}$
$\displaystyle\coloneqq\frac{\partial\hat{\mathbb{T}}}{\partial
T^{\alpha}{}_{\mu\nu}}=c_{1}\,T_{\alpha}{}^{\mu\nu}+c_{2}\,T^{[\mu}{}_{\alpha}{}^{\nu]}+c_{3}\,\delta_{\alpha}{}^{[\mu}T^{\nu]}$
$\displaystyle\hat{t}_{\mu\nu}$
$\displaystyle\coloneqq\frac{\partial\hat{\mathbb{T}}}{\partial
g^{\mu\nu}}=\frac{1}{2}\hat{S}_{(\mu|}{}^{\lambda\kappa}\,T_{\nu)\lambda\kappa}-T^{\lambda\kappa}{}_{(\mu}\,\hat{S}_{\lambda\kappa|\nu)}\,.$
(5.7)

Unsurprisingly, also the Bianchi identities posses the same form we
encountered before and it is still possible to parametrize the connection as
described in subsection 4.2, since the latter one has nothing to do with an
action principle. Thus, TEGR and TG are theories which look very similar.
However, there are important differences when it comes to the number of
physical degrees of freedom. As we have seen in subsection 4.2, TEGR
possesses, as expected, two degrees of freedom.

For TG, the situation can look quite different in this regard. Even though the
field equations have a similar form and are still second order partial
differential equations, the number of degrees of freedom depends on how one
chooses the parameters $c_{1}$, $c_{2}$, and $c_{3}$. That is because these
parameters appear in certain combinations in the field equations and there are
choices which can make some second order time derivatives of the metric
disappear. Thus, some equations can be turned into constraints, rather than
dynamical equations, which has an impact on the number of degrees of freedom.
Similarly, equations which are constraints in TEGR can be turned into
dynamical equations by detuning the parameters $c_{1}$, $c_{2}$, and $c_{3}$.
Moreover, it is possible that for certain parameter combinations further
constraint equations appear as integrability conditions, thus affecting the
number of degrees of freedom even more.

These considerations become slightly more transparent through the lens of a
Hamiltonian analysis. In Blixt:2018 ; Blixt:2019 , the first step of such an
analysis was carried out (see also Blixt:2020  for a detailed review of the
results obtained so far). More precisely, it has been investigated how many
independent so-called primary constraints appear through the vanishing of
certain combinations of the parameters $c_{1}$, $c_{2}$, $c_{3}$. The analysis
of these primary constraints revealed that the three-parameter family of
theories described by the action (5.5) compartmentalizes into nine different
sectors (which we dub the primary sectors, following the nomenclature of
DAmbrosio:2020 ). Each sector is characterized by a different number of
primary constraints (cf. Table 1).

Primary | Parameter combinations | # of primary
---|---|---
sector |  $\mathbf{2c_{1}+c_{2}+c_{3}}$ |  $\mathbf{2c_{1}-c_{2}}$ |  $\mathbf{2c_{1}+c_{2}}$ |  $\mathbf{2c_{1}+c_{2}+3c_{3}}$ | constraints
$\mathbf{0}$ | $\neq 0$ | $\neq 0$ | $\neq 0$ |  $\neq 0$ |  $0$
I | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | $\neq 0$ |  $\neq 0$ |  $3$
II | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ |  $\neq 0$ |  $3$
III | $\neq 0$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $\neq 0$ |  $5$
IV | $\neq 0$ | $\neq 0$ | $\neq 0$ |  ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $1$
V (TEGR) | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ |  $\neq 0$ |  $6$
VI | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $\neq 0$ |  $8$
VII | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ |  ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $4$
VIII | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ |  $9$
Table 1: Each primary sector is defined by the vanishing of certain parameter
combinations. The vanishing of one or more of these parameter combinations
leads to the appearance of primary constraints. The number of independent
primary constraints is listed in the last column. TEGR is contained in sector
V. This table was inspired by Blixt:2020 , but we adopt a slightly different
nomenclature in order to be consistent with DAmbrosio:2020 .

Primary constraints reduce the number of degrees of freedom. Hence, the more
primary constraints, the fewer degrees of freedom there are. However, the
exact number of physical degrees of freedom within each sector has not yet
been determined. In fact, the Hamiltonian analysis could not be carried out to
completion and it is not even known whether there are secondary constraints.
In DAmbrosio:2023  it was argued that the standard Hamiltonian method for
constrained systems (the so-called Dirac-Bergmann algorithm) is in general not
applicable to teleparallel theories of gravity. We briefly touch upon this
subject in subsection 6.3 and refer the reader to DAmbrosio:2023  for more
details on this important open question.

Before concluding this subsection, we emphasize that TEGR has a special place
among the theories described by the action (5.5). In fact, one has to ask what
distinguishes the particular choice of parameters which turns TG into TEGR
from all other possible choices. The answer: Enhanced symmetries. Perturbation
theory around Minkowski space shows Koivisto:2018 ; BeltranJimenez:2019c  that
a self-consistent theory requires $2c_{1}+c_{2}+c_{3}=0$. If this condition is
satisfied, one is left with a $1$-parameter family of theories (up to an
overall normalization) which propagate one additional degrees of freedom
besides the graviton. Among this $1$-parameter family of theories, the one
which satisfies $2c_{1}-c_{2}$ enjoys an additional symmetry and it looses the
additional degree of freedom. One is then left with TEGR. Removing either one
of these parameter conditions leads to a loss of symmetry accompanied by an
increase in degrees of freedom, not all of which are healthy.

###  5.2 Quadratic Actions for Non-Metricity Theories

In subsection 4.3 we constructed STEGR’s action functional from the non-
metricity scalar

$\mathbb{Q}=\frac{1}{4}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}-\frac{1}{2}Q_{\alpha\mu\nu}Q^{\mu\alpha\nu}-\frac{1}{4}Q_{\alpha}Q^{\alpha}+\frac{1}{2}Q_{\alpha}\bar{Q}^{\alpha}\,.$
(5.8)

Now we want to define a new class of theories, which we subsume under the
umbrella term Symmetric Teleparallel Gravity (STG), by using the most general
scalar which is quadratic in the non-metricity tensor. To that end, we need to
consider all possible independent contractions of $Q_{\alpha\mu\nu}$ with
itself. There are six contractions one can build this way, since we have six
possible index permutations. However, because $Q_{\alpha\mu\nu}$ is symmetric
in its last two indices, this cuts down the number to three. Using again the
symmetry of $Q_{\alpha\mu\nu}$, one can then show that only the contractions

$\displaystyle Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}$ and $\displaystyle
Q_{\mu\alpha\nu}Q^{\alpha\mu\nu}$ (5.9)

are independent. Next, we consider the traces of the non-metricity tensor.
Because of its symmetry, there are two such traces:

$\displaystyle Q_{\alpha}$ $\displaystyle\coloneqq
Q_{\alpha\lambda}{}^{\lambda}$ and $\displaystyle\bar{Q}_{\alpha}$
$\displaystyle\coloneqq Q^{\lambda}{}_{\lambda\alpha}\,.$ (5.10)

Using these traces, we can build three more contractions which are quadratic
in the non-metricity tensor:

$\displaystyle Q_{\mu}Q^{\mu}\,,$ $\displaystyle\bar{Q}_{\mu}\bar{Q}^{\mu}\,,$
and $\displaystyle Q_{\mu}\bar{Q}^{\mu}\,.$ (5.11)

With this we have exhausted all possibilities and we finally conclude that the
most general scalar which is quadratic in the non-metricity tensor is

$\displaystyle\hat{\mathbb{Q}}\coloneqq
c_{1}\,Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+c_{2}\,Q_{\mu\alpha\nu}Q^{\alpha\mu\nu}+c_{3}\,Q_{\mu}Q^{\mu}+c_{4}\,\bar{Q}_{\mu}\bar{Q}^{\mu}+c_{5}\,Q_{\mu}\bar{Q}^{\mu}\,,$
(5.12)

where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, and $c_{5}$ are arbitrary, real
constants. Just as in the case of TG, the action of STG is obtained by
replacing $\mathbb{Q}$ with $\hat{\mathbb{Q}}$ in the action of STEGR. This
results in the functional999The theory described by this action is sometimes
referred to as Newer General Relativity. See for instance BeltranJimenez:2017b
.

$\mathcal{S}_{\textsf{STG}}[g,\Gamma,\Psi]\coloneqq-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\left(\frac{1}{\kappa}\sqrt{|g|}\,\hat{\mathbb{Q}}+\tilde{\Pi}_{\alpha}{}^{\mu\nu\rho}\,R^{\alpha}{}_{\mu\nu\rho}+\tilde{\chi}_{\alpha}{}^{\mu\nu}\,T^{\alpha}{}_{\mu\nu}\right)+\mathcal{S}_{\textsf{matter}}[g,\Psi]\,,$
(5.13)

where eventual matter fields $\Psi$ are assumed to be minimally coupled and
having a vanishing hypermomentum $\mathcal{H}_{\alpha}{}^{\mu\nu}$. Not only
the action looks deceptively similar to the one of STEGR, also the field
equations have virtually the same form:

Metric field equations:
$\displaystyle\frac{2}{\sqrt{|g|}}\nabla_{\alpha}\left[\sqrt{|g|}\hat{P}^{\alpha}{}_{\mu\nu}\right]+\hat{q}_{\mu\nu}-\frac{1}{2}\hat{\mathbb{Q}}\,g_{\mu\nu}$
$\displaystyle=\kappa\,\mathcal{T}_{\mu\nu}$ Connection field equations:
$\displaystyle\nabla_{\mu}\nabla_{\nu}\left(\sqrt{|g|}\hat{P}^{\mu\nu}{}_{\alpha}\right)$
$\displaystyle=0\,.$ (5.14)

The hatted non-metricity conjugate $\hat{P}^{\alpha}{}_{\mu\nu}$ and the
symmetric tensor $\hat{q}_{\mu\nu}$ are defined as

$\displaystyle\hat{P}^{\alpha}{}_{\mu\nu}$
$\displaystyle\coloneqq\frac{1}{2}\frac{\partial\hat{\mathbb{Q}}}{\partial
Q_{\alpha}{}^{\mu\nu}}=c_{1}\,Q^{\alpha}{}_{\mu\nu}+c_{2}\,Q_{(\mu}{}^{\alpha}{}_{\nu)}+c_{3}\,g_{\mu\nu}Q^{\alpha}+c_{4}\,\delta^{\alpha}{}_{(\mu}\bar{Q}_{\nu)}+\frac{1}{2}c_{5}\,\left(g_{\mu\nu}\bar{Q}^{\alpha}+\delta^{\alpha}{}_{(\mu}Q_{\nu)}\right)$
$\displaystyle\hat{q}_{\mu\nu}$
$\displaystyle\coloneqq\frac{\partial\hat{\mathbb{Q}}}{\partial
g^{\mu\nu}}=P_{(\mu|\lambda\kappa}Q_{\nu)}{}^{\lambda\kappa}-2P^{\lambda\kappa}{}_{(\mu}Q_{\lambda\kappa|\nu)}$
(5.15)

and it is still true that

$\displaystyle\hat{\mathbb{Q}}=\hat{P}_{\alpha\mu\nu}Q^{\alpha\mu\nu}\,.$
(5.16)

Moreover, the Bianchi identities, which derive from the diffeomorphism
invariance of the action (5.13), read

$\displaystyle\mathcal{D}_{\nu}\hat{\mathcal{M}}^{\nu}{}_{\mu}+\hat{\mathcal{C}}_{\mu}\equiv
0\,,$ (5.17)

where $\hat{\mathcal{M}}_{\mu\nu}$ and $\hat{\mathcal{C}}_{\alpha}$ represent
the left hand side of the field equations (5.2), i.e.,

$\displaystyle\hat{\mathcal{M}}_{\mu\nu}$
$\displaystyle\coloneqq\frac{2}{\sqrt{|g|}}\nabla_{\alpha}\left[\sqrt{|g|}\hat{P}^{\alpha}{}_{\mu\nu}\right]+\hat{q}_{\mu\nu}-\frac{1}{2}\hat{\mathbb{Q}}\,g_{\mu\nu}$
$\displaystyle\hat{\mathcal{C}}_{\alpha}$
$\displaystyle\coloneqq\nabla_{\mu}\nabla_{\nu}\left(\sqrt{|g|}\hat{P}^{\mu\nu}{}_{\alpha}\right)\,.$
(5.18)

Thus, it follows that when the metric field equations are satisfied, the
connection field equations are identically satisfied as a consequence of the
Bianchi identities:

$\displaystyle\hat{\mathcal{M}}_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}\,\text{
satisfied}\quad\Longrightarrow\quad\mathcal{D}_{\nu}\hat{\mathcal{M}}^{\nu}{}_{\mu}=0\quad\Longrightarrow\quad\hat{\mathcal{C}}_{\mu}\equiv
0\,.$ (5.19)

What distinguishes STG from STEGR is the number of physical degrees of
freedom. As we know, STEGR propagates the same two degrees of freedom as GR.
When it comes to STG, the number of degrees of freedom depends on how one
chooses the parameters $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, and $c_{5}$. The
reason is the same as in the case of TG: It is possible to tune the parameters
such that certain second order time-derivatives of the metric drop out from
the field equations, thus turning some of the equations into constraints. The
more independent constraints there are, the lower the number of degrees of
freedom. Conversely, it is also possible that equations which appear as
constraints in STEGR are turned into dynamical equations because the
parameters are no longer finely tuned to lead to certain cancellations. This
has the effect of increasing the number of degrees of freedom and it can also
lead to pathologies.

In DAmbrosio:2020 , the first steps of a Hamiltonian analysis were carried
out. After performing an ADM decomposition of the metric and after applying
the coincident gauge, the momenta conjugate to lapse, shift, and intrinsic
metric were studied. The full expressions, which can be found in
DAmbrosio:2020  are quite long. However, if we only consider the kinetic part
of the Lagrangian of STG, which reads

$\displaystyle\mathcal{L}_{\rm kinetic}$
$\displaystyle=-\sqrt{h}\left(\frac{2\tilde{c}}{N^{3}}\dot{N}^{2}+\frac{c_{35}}{N^{2}}\,\dot{N}\,h^{ab}\dot{h}_{ab}-\frac{\hat{c}}{2N^{3}}h_{ab}\dot{N}^{a}\dot{N}^{b}+\frac{1}{2N}\left\\{c_{1}h^{ac}h^{db}\dot{h}_{ad}\dot{h}_{cb}+c_{3}h^{ac}h^{bd}\dot{h}_{ac}\dot{h}_{bd}\right\\}\right)$
with $\displaystyle\tilde{c}\coloneqq
c_{1}+c_{2}+c_{3}+c_{4}+c_{5},\qquad\hat{c}\coloneqq
2c_{1}+c_{2}+c_{4},\qquad\text{and}\qquad c_{35}\coloneqq 2c_{3}+c_{5}\,,$
(5.20)

we can already gain important insights. In fact, the momenta conjugate to
lapse, shift, and intrinsic metric have the form

$\displaystyle\tilde{\pi}$
$\displaystyle=-\frac{\sqrt{h}}{N^{2}}\left(4\,\tilde{c}\,\dot{N}+c_{35}\,N\,h^{ab}\dot{h}_{ab}\right)+\text{terms
without time derivatives}$ $\displaystyle\tilde{\pi}_{a}$
$\displaystyle=\frac{\sqrt{h}}{N^{3}}\,\hat{c}\,h_{ab}\dot{N}^{b}+\text{terms
without time derivatives}$ $\displaystyle\tilde{\pi}^{ab}$
$\displaystyle=-\frac{\sqrt{h}}{N^{2}}\left(c_{1}h^{ac}h^{bd}\dot{h}_{cd}N+h^{ab}\left\\{c_{2}h^{cd}\dot{h}_{cd}N+c_{35}\dot{N}\right\\}\,\right)+\text{terms
without time derivatives}\,.$ (5.21)

Evidently, the momentum conjugate to lapse is turned into a so-called primary
constraint if the parameters are chosen such that $\tilde{c}=$ and $c_{35}=0$.
Similarly, the momentum conjugate to shift becomes a constraint when
$\hat{c}=0$. These choices correspond precisely to the so-called primary
sectors I and II shown in Table 2. More constraints can be identified through
a systematic analysis based on the kinetic matrix, which is composed of the
following submatrices:

$\displaystyle\frac{\delta^{2}\mathcal{L}}{\delta\dot{N}\delta\dot{N}}$
$\displaystyle=-4\frac{\sqrt{h}}{N^{3}}\tilde{c}$
$\displaystyle\frac{\delta^{2}\mathcal{L}}{\delta\dot{N}^{a}\delta\dot{N}^{b}}$
$\displaystyle=\frac{\sqrt{h}}{N^{3}}\hat{c}\,h_{ab}$
$\displaystyle\frac{\delta^{2}\mathcal{L}}{\delta\dot{N}^{a}\delta\dot{N}}$
$\displaystyle=0$
$\displaystyle\frac{\delta^{2}\mathcal{L}}{\delta\dot{h}_{bc}\delta\dot{N}^{a}}$
$\displaystyle=0$
$\displaystyle\frac{\delta^{2}\mathcal{L}}{\delta\dot{h}_{ab}\delta\dot{N}}$
$\displaystyle=-\frac{\sqrt{h}}{N^{2}}c_{35}h^{ab}$
$\displaystyle\frac{\delta^{2}\mathcal{L}}{\delta\dot{h}_{cd}\delta\dot{h}_{ab}}$
$\displaystyle=-\frac{\sqrt{h}}{2N}\left(c_{1}\,h^{ad}h^{bc}+c_{1}h^{ac}h^{bd}+2c_{3}h^{ab}h^{cd}\right)\,.$
(5.22)

It is found that the determinant of the kinetic matrix $\mathcal{K}$ is given
by

$\displaystyle\det{\mathcal{K}}=8\frac{h^{2}}{N^{18}}\,c^{5}_{1}\,\hat{c}^{3}\left(3c^{2}_{35}-4\left(c_{1}+3c_{3}\right)\tilde{c}\right)\,.$
(5.23)

By demanding that the determinant vanishes, i.e., demanding that the matrix is
degenerate, one finds additional primary sectors. In fact, one finds that
there are four independent solutions to the above equations. These solutions
are

Sector I: $\displaystyle\tilde{c}$ $\displaystyle=0\text{ and }c_{35}=0$
Sector II: $\displaystyle\hat{c}$ $\displaystyle=0$ Sector III: $\displaystyle
c_{1}$ $\displaystyle=0$ Sector IV: $\displaystyle c_{3}$
$\displaystyle=-\frac{c_{1}}{3}+\frac{c^{2}_{35}}{4\tilde{c}}$ (5.24)

To determine the number of constraints in each sector, we only need to compute
$10-\text{rank}(\mathcal{K})$ in each sector. For the first four, we find $1$,
$3$, $5$, and again $1$ primary constraints, respectively.

Even more sectors can be identified by combining the different parameter
conditions in the different sectors, so as to create new and independent
sectors with more constraints. This process is described in detail in
DAmbrosio:2020  and ultimately leads to Table 2.

Primary | Parameter combinations | # of primary
---|---|---
sector |  $\mathbf{\tilde{c}}$ |  $\mathbf{\hat{c}}$ |  $\mathbf{c_{35}}$ |  $\mathbf{c_{1}}$ |  $\mathbf{\frac{c_{1}}{3}+c_{3}-\frac{c^{2}_{35}}{\tilde{c}}}$ | constraints
$\mathbf{0}$ | $\neq 0$ | $\neq 0$ | $\neq 0$ | $\neq 0$ |  $\neq 0$ |  $0$
I | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ |  N/A |  $1$
II | $\neq 0$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}=0}$ | $\neq 0$ | $\neq 0$ |  $\neq 0$ |  $3$