$\displaystyle\begin{multlined}\sqrt{-g}g^{\mu\lambda}\partial_{\mu}\bar{\Gamma}_{\kappa\lambda}^{\kappa}=\partial_{\mu}\bigl{(}\sqrt{-g}g^{\mu\lambda}\bar{\Gamma}_{\kappa\lambda}^{\kappa}\bigr{)}-\sqrt{-g}\,\bar{\Gamma}_{\kappa\lambda}^{\kappa}\partial_{\mu}g^{\mu\lambda}\\\
+\frac{1}{2}\sqrt{-g}g_{\alpha\beta}g^{\mu\lambda}\bar{\Gamma}_{\kappa\lambda}^{\kappa}\partial_{\mu}g^{\alpha\beta}\,,\end{multlined}\sqrt{-g}g^{\mu\lambda}\partial_{\mu}\bar{\Gamma}_{\kappa\lambda}^{\kappa}=\partial_{\mu}\bigl{(}\sqrt{-g}g^{\mu\lambda}\bar{\Gamma}_{\kappa\lambda}^{\kappa}\bigr{)}-\sqrt{-g}\,\bar{\Gamma}_{\kappa\lambda}^{\kappa}\partial_{\mu}g^{\mu\lambda}\\\
+\frac{1}{2}\sqrt{-g}g_{\alpha\beta}g^{\mu\lambda}\bar{\Gamma}_{\kappa\lambda}^{\kappa}\partial_{\mu}g^{\alpha\beta}\,,$
(4.4.7)

where the first term in each is a boundary term. Note that now we no longer
have a metric compatible connection $\bar{\nabla}_{\mu}g_{\nu\lambda}\neq 0$,
we do not rewrite derivatives of the metric in terms of connection terms. In
fact, doing so would change the fundamental variables of the theory from the
set $\\{g,\bar{\Gamma}\\}$ to $\\{g,T,Q\\}$ with $T$ and $Q$ representing
torsion and non-metricity. Later we will look at formulating the theory with
the latter representation.

Next, we can separate off these boundary terms and arrive at

$\sqrt{-g}\,\bar{R}=\partial_{\kappa}\bigl{(}\sqrt{-g}g^{\mu\lambda}\bar{\Gamma}_{\mu\lambda}^{\kappa}\bigr{)}-\partial_{\mu}\bigl{(}\sqrt{-g}g^{\mu\lambda}\bar{\Gamma}_{\kappa\lambda}^{\kappa}\bigr{)}\\\
+\sqrt{-g}g^{\mu\lambda}\bigl{(}\bar{\Gamma}_{\kappa\rho}^{\kappa}\bar{\Gamma}_{\mu\lambda}^{\rho}-\bar{\Gamma}_{\mu\rho}^{\kappa}\bar{\Gamma}_{\kappa\lambda}^{\rho}\bigr{)}-\sqrt{-g}\,\bar{\Gamma}_{\mu\lambda}^{\kappa}\partial_{\kappa}g^{\mu\lambda}+\sqrt{-g}\,\bar{\Gamma}_{\kappa\lambda}^{\kappa}\partial_{\mu}g^{\mu\lambda}\\\
+\frac{1}{2}\sqrt{-g}g_{\alpha\beta}g^{\mu\lambda}\bar{\Gamma}_{\mu\lambda}^{\kappa}\partial_{\kappa}g^{\alpha\beta}-\frac{1}{2}\sqrt{-g}g_{\alpha\beta}g^{\mu\lambda}\bar{\Gamma}_{\kappa\lambda}^{\kappa}\partial_{\mu}g^{\alpha\beta}\,.$
(4.4.8)

Introducing the notation $\bar{\mathbf{B}}$ to denote the new boundary term
and $\bar{\mathbf{G}}$ to denote the remaining bulk terms, we have the
following decomposition

$\displaystyle\sqrt{-g}\bar{R}=\sqrt{-g}\big{(}\bar{\mathbf{G}}+\bar{\mathbf{B}}\big{)}\,,$
(4.4.9)

with $\bar{\mathbf{G}}$ and $\mathbf{\bar{B}}$ given by

$\displaystyle\bar{\mathbf{G}}$
$\displaystyle:=g^{\mu\lambda}\big{(}\bar{\Gamma}^{\kappa}_{\kappa\rho}\bar{\Gamma}^{\rho}_{\mu\lambda}-\bar{\Gamma}^{\kappa}_{\mu\rho}\bar{\Gamma}^{\rho}_{\kappa\lambda}\big{)}+\big{(}\bar{\Gamma}^{\mu}_{\mu\lambda}\delta^{\nu}_{\kappa}-\bar{\Gamma}^{\nu}_{\kappa\lambda}\big{)}\big{(}\partial_{\nu}g^{\kappa\lambda}-\frac{1}{2}g_{\alpha\beta}g^{\kappa\lambda}\partial_{\nu}g^{\alpha\beta}\big{)}\,,$
(4.4.10) $\displaystyle\mathbf{\bar{B}}$
$\displaystyle:=\frac{1}{\sqrt{-g}}\partial_{\kappa}\big{(}\sqrt{-g}(g^{\mu\lambda}\bar{\Gamma}^{\kappa}_{\mu\lambda}-g^{\kappa\lambda}\bar{\Gamma}^{\mu}_{\mu\lambda})\big{)}\,.$
(4.4.11)

The similarities with the Levi-Civita formulation in (4.1.8) and (4.1.9) are
quite apparent: the boundary term is form equivalent with $\Gamma$ replaced by
$\bar{\Gamma}$; the same is not true for the bulk term. Notice that the index
combination on the quadratic part of $\bar{\mathbf{G}}$ (4.4.10) differs from
$\mathbf{G}$ (4.1.8), and the quadratic parts actually have opposite signs.

The partial derivative terms in $\bar{\mathbf{G}}$ can be written in a more
succinct way which makes the comparison more clear

$\displaystyle\bar{\mathbf{G}}=g^{\mu\lambda}\big{(}\bar{\Gamma}^{\kappa}_{\kappa\rho}\bar{\Gamma}^{\rho}_{\mu\lambda}-\bar{\Gamma}^{\kappa}_{\mu\rho}\bar{\Gamma}^{\rho}_{\kappa\lambda}\big{)}-\frac{1}{2}\partial_{\lambda}g^{\mu\nu}\bar{E}_{\mu\nu}{}^{\lambda}\,,$
(4.4.12)

where the object $\bar{E}{}_{\mu\nu}{}^{\lambda}$ is defined by

$\displaystyle\bar{E}_{(\mu\nu)}{}^{\lambda}:=-2\frac{\partial\mathbf{G}}{\partial_{\lambda}g^{\mu\nu}}=2\bar{\Gamma}^{\lambda}_{(\mu\nu)}-2\delta^{\lambda}_{(\mu}\bar{\Gamma}^{\rho}_{|\rho|\nu)}-g_{\mu\nu}g^{\kappa\rho}\bar{\Gamma}^{\lambda}_{\rho\kappa}+g_{\mu\nu}g^{\kappa\lambda}\bar{\Gamma}^{\rho}_{\rho\kappa}\,.$
(4.4.13)

This is just the metric-affine version of the superpotential used in the Levi-
Civita case, see equations (4.1.1) and (4.1.20). Let us also note that
equation (4.4.12) only determines the symmetric part of
$\bar{E}_{\mu\nu}{}^{\lambda}$, hence the explicit symmeterization brackets on
(4.4.13). Our first field equation is obtained by varying with respect to the
metric and will only contain the symmetric part of $E_{\mu\nu}{}^{\lambda}$.
However, when considering a generalised Einstein-Cartan type theory, we will
be able to also choose its skew-symmetric part in a particular way to make the
equations of motion consistent. This will be made more clear in the following.

For the unique torsion-free, metric-compatible, Levi-Civita connection the
bulk terms simplify to

$\displaystyle\bar{\mathbf{G}}|_{\bar{\Gamma}\rightarrow\Gamma}=g^{\mu\lambda}\bigl{(}\Gamma^{\kappa}_{\mu\rho}\Gamma^{\rho}_{\kappa\lambda}-\Gamma^{\kappa}_{\kappa\rho}\Gamma^{\rho}_{\mu\lambda}\bigr{)}=\mathbf{G}\,,$
(4.4.14)

which is just the Levi-Civita bulk term (4.1.8). In the equation above the
partial derivatives of the metric in (4.4.10) have been rewritten as
connection terms. This result is not immediately obvious but can be seen by
noting that in this limit

$\partial_{\lambda}g^{\mu\nu}\bar{E}_{\mu\nu}{}^{\lambda}|_{\bar{\Gamma}\rightarrow\Gamma}=\partial_{\lambda}g^{\mu\nu}E_{\mu\nu}{}^{\lambda}=-4\mathbf{G}\,,$
(4.4.15)

with the final equality following from equation (4.1.21). This result, along
with the knowledge that the quadratic part of $\bar{\mathbf{G}}$ has an
opposite sign to $\mathbf{G}$, leads to the total limit as being
$-\mathbf{G}+2\mathbf{G}=\mathbf{G}$.

##### The Palatini tensor

The above decomposition of the Ricci scalar is unique in the sense that there
is only one canonical boundary term. However, one can write $\bar{\mathbf{G}}$
in a variety of different equivalent ways, as was already done in (4.4.10) and
(4.4.12). Formulation (4.4.12) turns out to be useful as the field equations
will contain the object $\bar{E}_{\mu\nu}{}^{\lambda}$. Another previously
used form is given by

$\displaystyle\bar{\mathbf{G}}=g^{\mu\lambda}\bigl{(}\bar{\Gamma}^{\kappa}_{\rho\lambda}\bar{\Gamma}^{\rho}_{\mu\kappa}-\bar{\Gamma}^{\kappa}_{\mu\lambda}\bar{\Gamma}^{\rho}_{\rho\kappa}\bigr{)}+\bar{\Gamma}^{\kappa}_{\mu\lambda}P^{\mu\lambda}{}_{\kappa}\,,$
(4.4.16)

where one has to pay special attention to the index positions in the quadratic
part compared to (4.4.10) and (4.4.16). This formulation can be found in many
of the works by Hehl [307, 308, 301, 302, 303]. Here
$P^{\mu\lambda}{}_{\kappa}$ is the Palatini tensor

$P^{\mu\nu}{}_{\lambda}=-Q_{\lambda}{}^{\mu\nu}+\frac{1}{2}g^{\mu\nu}Q_{\lambda\rho}{}^{\rho}+\delta_{\lambda}^{\mu}Q_{\rho}{}^{\rho\nu}-\frac{1}{2}\delta^{\mu}_{\lambda}Q^{\nu}{}_{\rho}{}^{\rho}\\\
+T^{\mu}{}_{\lambda}{}^{\nu}+g^{\mu\nu}T^{\rho}{}_{\rho\lambda}+\delta^{\mu}_{\lambda}T^{\rho\nu}{}_{\rho}\,,$
(4.4.17)

which we originally introduced in equation (3.1.3). This object is a tensor,
as can be seen from its definition in terms of other tensors, but clearly its
contraction with the affine connection leads to a non-covariant term.

Let us show how one arrives at (4.4.16) starting from our original affine bulk
term $\bar{\mathbf{G}}$. From the definitions of torsion and non-metricity we
have the relations

$\partial_{\lambda}g^{\mu\nu}=Q_{\lambda}{}^{\mu\nu}-2\bar{\Gamma}{}^{(\mu}_{\lambda\eta}g^{\nu)\eta}\,,\qquad\bar{\Gamma}{}^{\mu}_{\nu\lambda}=T^{\mu}{}_{\nu\lambda}+\bar{\Gamma}{}^{\mu}_{\lambda\nu}\,,$
(4.4.18)

from which we obtain

$\displaystyle\bar{\mathbf{G}}$
$\displaystyle=g^{\mu\lambda}\big{(}\bar{\Gamma}{}^{\kappa}_{\kappa\rho}\bar{\Gamma}{}^{\rho}_{\mu\lambda}-\bar{\Gamma}{}^{\kappa}_{\mu\rho}\bar{\Gamma}{}^{\rho}_{\kappa\lambda}\big{)}+\big{(}\bar{\Gamma}{}^{\mu}_{\mu\lambda}\delta^{\nu}_{\kappa}-\bar{\Gamma}{}^{\nu}_{\kappa\lambda}\big{)}\big{(}\partial_{\nu}g^{\kappa\lambda}-\frac{1}{2}g_{\alpha\beta}g^{\kappa\lambda}\partial_{\nu}g^{\alpha\beta}\big{)}$
$\displaystyle=g^{\mu\lambda}\big{(}\bar{\Gamma}{}^{\kappa}_{\kappa\rho}\bar{\Gamma}{}^{\rho}_{\mu\lambda}-\bar{\Gamma}{}^{\kappa}_{\mu\rho}\bar{\Gamma}{}^{\rho}_{\kappa\lambda}\big{)}+2g^{\mu\lambda}\big{(}\bar{\Gamma}{}^{\kappa}_{\rho\lambda}\bar{\Gamma}{}^{\rho}_{\mu\kappa}-\bar{\Gamma}{}^{\kappa}_{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\rho\kappa}\big{)}$
$\displaystyle+\bar{\Gamma}{}^{\kappa}_{\mu\lambda}\big{(}\delta^{\mu}_{\kappa}Q_{\rho}{}^{\rho\lambda}-Q_{\kappa}{}^{\mu\lambda}+\frac{1}{2}g^{\mu\lambda}Q_{\kappa\rho}{}^{\rho}-\frac{1}{2}\delta^{\mu}_{\kappa}Q^{\lambda\rho}{}_{\rho}+T^{\mu}{}_{\kappa}{}^{\lambda}-g^{\mu\lambda}T^{\rho}{}_{\kappa\rho}+\delta^{\mu}_{\kappa}T^{\rho\lambda}{}_{\rho}\big{)}$
$\displaystyle=g^{\mu\lambda}\big{(}\bar{\Gamma}{}^{\kappa}_{\rho\lambda}\bar{\Gamma}{}^{\rho}_{\mu\kappa}-\bar{\Gamma}{}^{\kappa}_{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\rho\kappa}\big{)}+\bar{\Gamma}{}^{\kappa}_{\mu\lambda}P^{\mu\lambda}{}_{\kappa}\,.$
(4.4.19)

Note that the quadratic terms have flipped sign in going from the first to the
final line. Throughout this section it is important to be aware of the index
combinations on the quadratic part: in terms of the metric and affine
connection only, such as in (4.4.10) and (4.4.12), the traced connection terms
come first. In terms of the Palatini tensor (4.4.16), the traced parts come
last. For later use, we will define the quadratic part of the final line of
(4.4.19) as

$\bar{\mathbf{G}}_{\textrm{quad}}:=g^{\mu\lambda}\big{(}\bar{\Gamma}{}^{\kappa}_{\rho\lambda}\bar{\Gamma}{}^{\rho}_{\mu\kappa}-\bar{\Gamma}{}^{\kappa}_{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\rho\kappa}\big{)}\,.$
(4.4.20)

It follows that the quadratic part of the original definition (4.4.10) is just
minus $\bar{\mathbf{G}}_{\textrm{quad}}$, but this additional definition will
be useful during the following calculations.

It is also important to be aware that this Palatini form of the bulk term
(4.4.16) should not be used in variation procedures, because the quadratic
part depends on the affine connection whilst the Palatini tensor is written in
terms of torsion and non-metricity. We instead will always work directly with
the set $\\{g,\bar{\Gamma}\\}$ where $g$ and $\bar{\Gamma}$ are independent.
This removes any confusion when making variations of an action that could
include, for example, both the non-metricity tensor $Q_{\lambda\mu\nu}$ and
derivatives of the metric, which are not independent.

It turns out that the Palatini tensor can be defined by the variation of
$\bar{\mathbf{G}}$ with respect to the connection, which we will also show
explicitly. Splitting the calculation into two parts, the variation of the
quadratic part of $\bar{\mathbf{G}}$ and the part containing derivatives of
the metric131313Here we are looking at the variation of $\bar{\mathbf{G}}$ in
the form (4.4.10). We noted that the quadratic part has an opposite sign
compared with the Levi-Civita bulk term $\mathbf{G}$ and the Palatini version
in (4.4.16). This is the reason for the minus sign in equation (4.4.21).,
leads to

$\displaystyle-\delta_{\bar{\Gamma}}\bar{\mathbf{G}}_{\textrm{quad}}$
$\displaystyle=\delta\bar{\Gamma}{}^{\gamma}_{\alpha\beta}\Big{(}\delta^{\alpha}_{\gamma}g^{\mu\lambda}\bar{\Gamma}{}^{\beta}_{\mu\lambda}+g^{\alpha\beta}\bar{\Gamma}{}^{\kappa}_{\kappa\gamma}-g^{\alpha\kappa}\bar{\Gamma}{}^{\beta}_{\gamma\kappa}-g^{\kappa\beta}\bar{\Gamma}{}^{\alpha}_{\kappa\gamma}\Big{)}\,,$
(4.4.21) $\displaystyle\delta_{\bar{\Gamma}}\bar{\mathbf{G}}_{\textrm{other}}$
$\displaystyle=\delta\bar{\Gamma}{}^{\gamma}_{\alpha\beta}\Big{(}\delta^{\alpha}_{\gamma}\partial_{\mu}g^{\mu\beta}-\partial_{\gamma}g^{\alpha\beta}+\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\partial_{\gamma}g^{\mu\nu}-\frac{1}{2}\delta^{\alpha}_{\gamma}g_{\mu\nu}g^{\kappa\beta}\partial_{\kappa}g^{\mu\nu}\Big{)}\,.$
(4.4.22)

Rewriting the partial derivatives of the metric in terms of non-metricity and
connection terms, along with some cancellations, allows us to write the total
variation as

$\displaystyle\delta_{\bar{\Gamma}}\bar{\mathbf{G}}$
$\displaystyle=-\delta_{\bar{\Gamma}}\bar{\mathbf{G}}_{\textrm{quad}}+\delta_{\bar{\Gamma}}\bar{\mathbf{G}}_{\textrm{other}}$
$\displaystyle\begin{multlined}=\delta\bar{\Gamma}{}^{\gamma}_{\alpha\beta}\Big{[}\delta^{\alpha}_{\gamma}Q_{\lambda}{}^{\lambda\beta}-Q_{\gamma}{}^{\alpha\beta}+\frac{1}{2}g^{\alpha\beta}Q_{\gamma}{}^{\lambda}{}_{\lambda}-\frac{1}{2}\delta^{\alpha}_{\gamma}Q^{\beta\lambda}{}_{\lambda}\\\
+\delta^{\alpha}_{\gamma}g^{\mu\beta}\big{(}\bar{\Gamma}{}^{\lambda}_{\mu\lambda}-\bar{\Gamma}{}^{\lambda}_{\lambda\mu}\big{)}+g^{\alpha\beta}\big{(}\bar{\Gamma}{}^{\lambda}_{\lambda\gamma}-\bar{\Gamma}{}^{\lambda}_{\gamma\lambda}\big{)}+g^{\mu\beta}\big{(}\bar{\Gamma}{}^{\alpha}_{\gamma\mu}-\bar{\Gamma}{}^{\alpha}_{\mu\gamma}\big{)}\Big{]}\end{multlined}=\delta\bar{\Gamma}{}^{\gamma}_{\alpha\beta}\Big{[}\delta^{\alpha}_{\gamma}Q_{\lambda}{}^{\lambda\beta}-Q_{\gamma}{}^{\alpha\beta}+\frac{1}{2}g^{\alpha\beta}Q_{\gamma}{}^{\lambda}{}_{\lambda}-\frac{1}{2}\delta^{\alpha}_{\gamma}Q^{\beta\lambda}{}_{\lambda}\\\
+\delta^{\alpha}_{\gamma}g^{\mu\beta}\big{(}\bar{\Gamma}{}^{\lambda}_{\mu\lambda}-\bar{\Gamma}{}^{\lambda}_{\lambda\mu}\big{)}+g^{\alpha\beta}\big{(}\bar{\Gamma}{}^{\lambda}_{\lambda\gamma}-\bar{\Gamma}{}^{\lambda}_{\gamma\lambda}\big{)}+g^{\mu\beta}\big{(}\bar{\Gamma}{}^{\alpha}_{\gamma\mu}-\bar{\Gamma}{}^{\alpha}_{\mu\gamma}\big{)}\Big{]}$
(4.4.25)
$\displaystyle\begin{multlined}=\delta\bar{\Gamma}{}^{\gamma}_{\alpha\beta}\Big{[}\delta^{\alpha}_{\gamma}Q_{\lambda}{}^{\lambda\beta}-Q_{\gamma}{}^{\alpha\beta}+\frac{1}{2}g^{\alpha\beta}Q_{\gamma}{}^{\lambda}{}_{\lambda}-\frac{1}{2}\delta^{\alpha}_{\gamma}Q^{\beta\lambda}{}_{\lambda}\\\
+\delta^{\alpha}_{\gamma}T^{\lambda\beta}{}_{\lambda}+g^{\alpha\beta}T^{\lambda}{}_{\lambda\gamma}+T^{\alpha}{}_{\gamma}{}^{\beta}\Big{]}\,,\end{multlined}=\delta\bar{\Gamma}{}^{\gamma}_{\alpha\beta}\Big{[}\delta^{\alpha}_{\gamma}Q_{\lambda}{}^{\lambda\beta}-Q_{\gamma}{}^{\alpha\beta}+\frac{1}{2}g^{\alpha\beta}Q_{\gamma}{}^{\lambda}{}_{\lambda}-\frac{1}{2}\delta^{\alpha}_{\gamma}Q^{\beta\lambda}{}_{\lambda}\\\
+\delta^{\alpha}_{\gamma}T^{\lambda\beta}{}_{\lambda}+g^{\alpha\beta}T^{\lambda}{}_{\lambda\gamma}+T^{\alpha}{}_{\gamma}{}^{\beta}\Big{]}\,,$
(4.4.28)

where in the final line we used the definition of the torsion tensor. We
recognise this object in the brackets to be the Palatini tensor (4.4.17),
hence

$\frac{\partial\bar{\mathbf{G}}}{\partial\bar{\Gamma}_{\mu\nu}^{\lambda}}=P^{\mu\nu}{}_{\lambda}\,.$
(4.4.29)

The fact that this is tensorial, unlike our previously introduced objects,
deserves some comments. It can be understood for a few reasons: firstly, the
metric and connection are treated independently; next, the bulk term is linear
in the connection; lastly, the connection equations of motion coming from
$\delta/\delta\bar{\Gamma}$ (which will be tensorial141414The equations of
motion will be tensorial because the action is (quasi-)diffeomorphism
invariant. This is proven shortly, where we see that $\bar{\mathbf{G}}$ is a
pseudo-scalar.) are derived solely from the bulk term. It therefore follows
that in this case $\partial/\partial\bar{\Gamma}=\delta/\delta\bar{\Gamma}$,
and the resulting equation (4.4.29) is a tensor. In the Levi-Civita case with
$\mathbf{G}$ the connection was not independent
$\delta\bar{\Gamma}\propto\delta\nabla g$, so these arguments did not hold. As
for the affine superpotential $\bar{E}$ in (4.4.13), we varied with respect to
the partial derivative of the metric, holding the undifferentiated metric
fixed. Hence this object also is not a tensor. This short discussion should
clarify some of the differences between the metric-affine and the purely
metric formulations, as well as the importance of the Palatini tensor.

Lastly, let us look at the limit of vanishing non-metricity. In this case we
can swap the indices of the connection terms to introduce torsion tensors that
cancel terms in the Palatini tensor

$\displaystyle\bar{\mathbf{G}}$
$\displaystyle=g^{\mu\lambda}\big{(}\bar{\Gamma}{}^{\kappa}_{\rho\lambda}\bar{\Gamma}{}^{\rho}_{\mu\kappa}-\bar{\Gamma}{}^{\kappa}_{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\rho\kappa}\big{)}+\bar{\Gamma}{}^{\kappa}_{\mu\lambda}P^{\mu\lambda}{}_{\kappa}$
$\displaystyle\stackrel{{\scriptstyle
Q=0}}{{=}}g^{\mu\lambda}\big{(}\bar{\Gamma}{}^{\kappa}_{\rho\lambda}\bar{\Gamma}{}^{\rho}_{\kappa\mu}-\bar{\Gamma}{}^{\kappa}_{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\kappa\rho}\big{)}-\bar{\Gamma}{}^{\lambda}_{\lambda\mu}T^{\rho}{}_{\rho}{}^{\mu}\,.$
(4.4.30)

This is the form of the bulk term in an Einstein-Cartan space, see for example
[290]. When torsion also vanishes the final term is zero and we obtain
$\mathbf{G}$.

####  4.4.2 Coordinate and projective transformations

##### Coordinate transformations

Let us again study the coordinate transformations for these metric-affine bulk
and boundary terms. Here we will go directly to stating the Lie derivatives,
with the calculations detailed in Appendix B.2. For clarity, we restate the
Lie derivatives of the metric and affine connection here

$\displaystyle\mathcal{L}_{\xi}g_{\mu\nu}$
$\displaystyle=\xi^{\lambda}\partial_{\lambda}g_{\mu\nu}+\partial_{\mu}\xi^{\lambda}g_{\lambda\nu}+\partial_{\nu}\xi^{\lambda}g_{\lambda\mu}=2\nabla_{(\mu}\xi_{\nu)}\,,$
(4.4.31) $\displaystyle\mathcal{L}_{\xi}\bar{\Gamma}^{\lambda}_{\mu\nu}$
$\displaystyle=\xi^{\rho}\partial_{\rho}\bar{\Gamma}^{\lambda}_{\mu\nu}-\partial_{\rho}\xi^{\lambda}\bar{\Gamma}^{\rho}_{\mu\nu}+\partial_{\mu}\xi^{\rho}\bar{\Gamma}^{\lambda}_{\rho\nu}+\partial_{\nu}\xi^{\rho}\bar{\Gamma}^{\lambda}_{\mu\rho}+\partial_{\mu}\partial_{\nu}\xi^{\lambda}$
$\displaystyle=\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\xi^{\lambda}+\xi^{\rho}\bar{R}_{\rho\mu\nu}{}^{\lambda}-\bar{\nabla}_{\mu}(T^{\lambda}{}_{\nu\rho}\xi^{\rho})\,.$
(4.4.32)

It is most direct to work with the Palatini form of $\mathbf{G}$ (4.4.16),
which is valid for these calculations. Using the relations above, following
the same methods as in the previous sections, we find

$\displaystyle\mathcal{L}_{\xi}\bar{\mathbf{G}}$
$\displaystyle=\xi^{\mu}\partial_{\mu}\bar{\mathbf{G}}+\partial_{\mu}\partial_{\nu}\xi^{\lambda}\big{(}\bar{M}^{\mu\nu}{}_{\lambda}+P^{\mu\nu}{}_{\lambda}\big{)}\,,$
(4.4.33) $\displaystyle\mathcal{L}_{\xi}\bar{\mathbf{B}}$
$\displaystyle=\xi^{\mu}\partial_{\mu}\bar{\mathbf{B}}-\partial_{\mu}\partial_{\nu}\xi^{\lambda}\big{(}\bar{M}^{\mu\nu}{}_{\lambda}+P^{\mu\nu}{}_{\lambda}\big{)}\,,$
(4.4.34)

where $\bar{M}$ is defined by the variation of the quadratic part of
$\bar{\mathbf{G}}$ (4.4.20) with respect to the connection

$\displaystyle\bar{M}^{\mu\nu}{}_{\lambda}:=\frac{\partial\bar{\mathbf{G}}_{\textrm{quad}}}{\partial\bar{\Gamma}_{\mu\nu}^{\lambda}}=g^{\rho\mu}\bar{\Gamma}^{\nu}_{\lambda\rho}+g^{\rho\nu}\bar{\Gamma}^{\mu}_{\rho\lambda}-g^{\mu\nu}\bar{\Gamma}^{\rho}_{\rho\lambda}-g^{\rho\sigma}\delta^{\mu}_{\lambda}\bar{\Gamma}^{\nu}_{\rho\sigma}\,.$
(4.4.35)

The explicit calculations are given in Appendix B.2.

It turns out that the sum of this new object and the Palatini tensor is
independent from the affine connection altogether. In fact, it is just equal
to the unsymmetrised Levi-Civita object $M^{\mu\nu}{}_{\lambda}$, with all of
the torsion and non-metricity terms in $\bar{\Gamma}$ cancelling

$\bar{M}^{\mu\nu}{}_{\lambda}+P^{\mu\nu}{}_{\lambda}=g^{\rho\mu}\Gamma^{\nu}_{\lambda\rho}+g^{\rho\nu}\Gamma^{\mu}_{\rho\lambda}-g^{\mu\nu}\Gamma^{\rho}_{\rho\lambda}-g^{\rho\sigma}\delta^{\mu}_{\lambda}\Gamma^{\nu}_{\rho\sigma}\,,$
(4.4.36)

again, see Appendix B.2 for the proof. Note that the final term is not
symmetric under $\mu\leftrightarrow\nu$. In the transformations above,
however, this combination appears contracted with the partial derivatives of
$\xi$ and is symmetrised over its first two indices. It follows that we can
rewrite the Lie derivatives in terms of the Levi-Civita version
$M^{\mu\nu}{}_{\lambda}$ defined in (4.1.14)

$\displaystyle\mathcal{L}_{\xi}\bar{\mathbf{G}}$
$\displaystyle=\xi^{\mu}\partial_{\mu}\bar{\mathbf{G}}+M^{\mu\nu}{}_{\lambda}\partial_{\mu}\partial_{\nu}\xi^{\lambda}\,,$
(4.4.37) $\displaystyle\mathcal{L}_{\xi}\bar{\mathbf{B}}$
$\displaystyle=\xi^{\mu}\partial_{\mu}\bar{\mathbf{B}}-M^{\mu\nu}{}_{\lambda}\partial_{\mu}\partial_{\nu}\xi^{\lambda}\,.$
(4.4.38)

The affine connection no longer appears in the inhomogeneous parts of the
pseudo-scalar transformation. This rather odd result reflects the fact that
this decomposition, despite being in the metric-affine framework, solely
depends on the choice of coordinates only. In fact, in Sec. 4.4.4 we decompose
the metric-affine bulk and boundary terms into their Levi-Civita and
contortion parts, and see that $\bar{\mathbf{G}}$ must necessarily transform
in the same way as $\mathbf{G}$. The Lie derivatives given in equations
(4.4.37)-(4.4.38) then immediately follow.

##### Projective transformations

In the previous Levi-Civita cases, the connection depends only on the metric.
Now that we have an independent affine connection, it is important to study
the projective properties of our quantities. In Appendix A we give an
introduction and overview of projective transformations. Moreover, in Sec. 3.1
we saw that a consequence of the projective invariance of the Ricci scalar was
that the Palatini tensor was trace free $P^{\mu\nu}{}_{\nu}=0$. It is
therefore important to consider how the objects $\bar{\mathbf{G}}$ and
$\bar{\mathbf{B}}$ transform.

The generalised projective transformation (A.2) takes the form

$\displaystyle\bar{\Gamma}{}^{\gamma}_{\alpha\beta}\rightarrow\bar{\Gamma}{}^{\gamma}_{\alpha\beta}+c_{1}\delta_{\alpha}^{\gamma}P_{\beta}+c_{2}\delta_{\beta}^{\gamma}P_{\alpha}\,,$
(4.4.39)

with $c_{1}=0$ corresponding to the standard projective transformations
preserving geodesic motion. In Appendix A we show that the Ricci scalar
transforms as

$\bar{R}\rightarrow\bar{R}+2c_{1}\bar{\Gamma}{}^{\gamma}_{[\gamma\mu]}P^{\mu}+c_{1}(1-n)\bar{\nabla}^{\mu}P_{\mu}+c_{1}^{2}(n-1)P^{\mu}P_{\mu}\,,$
(4.4.40)

where $n$ is the number of spacetime dimensions. This is invariant for
projective transformations with $c_{1}=0$.

A direct calculation shows that the affine bulk term transforms as

$\bar{\mathbf{G}}\rightarrow\bar{\mathbf{G}}+c_{1}\Big{[}2\bar{\Gamma}{}^{\kappa}_{[\kappa\mu]}P^{\mu}+(n-1)\big{(}g^{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\mu\lambda}P_{\rho}+P_{\lambda}\partial_{\nu}g^{\nu\lambda}-\frac{1}{2}P^{\nu}g_{\alpha\beta}\partial_{\nu}g^{\alpha\beta}\big{)}\\\
+c_{1}(n-1)P^{\lambda}P_{\lambda}\Big{]}\,.$ (4.4.41)

This can also be rewritten explicitly in terms of non-metricity and
torsion151515Here if one works with the Levi-Civita connection, we see only
the $P^{\lambda}P_{\lambda}$ term remains. This, along with the total
divergence term $\nabla_{\kappa}P^{\kappa}$ in the $\mathbf{B}$ transformation
below, immediately gives the Schouten result for the Levi-Civita Ricci scalar
(A.15).

$\bar{\mathbf{G}}\rightarrow\bar{\mathbf{G}}+c_{1}\Big{[}(2-n)T^{\kappa}{}_{\kappa\mu}P^{\mu}+(n-1)P_{\lambda}\big{(}\bar{\nabla}_{\nu}g^{\nu\lambda}-\frac{1}{2}g_{\alpha\beta}\bar{\nabla}^{\lambda}g^{\alpha\beta}\big{)}\\\
+c_{1}(n-1)P^{\lambda}P_{\lambda}\Big{]}\,.$ (4.4.42)

For the boundary term $\bar{\mathbf{B}}$ we find

$\displaystyle\bar{\mathbf{B}}\rightarrow\bar{\mathbf{B}}+c_{1}(1-n)\frac{1}{\sqrt{-g}}\partial_{\kappa}(\sqrt{-g}P^{\kappa})\,.$
(4.4.43)

Both of these affine quantities share the projective properties of the Ricci
scalar, being invariant for transformations with $c_{1}=0$

$\displaystyle\bar{\mathbf{G}}$ $\displaystyle\rightarrow\bar{\mathbf{G}}\,,$
(4.4.44) $\displaystyle\bar{\mathbf{B}}$
$\displaystyle\rightarrow\bar{\mathbf{B}}\,.$ (4.4.45)

From this we can deduce that any action comprised of these objects will share
this invariance, see [306].

For consistency, we show that the above transformations give the correct
transformation for the Ricci scalar. First, expanding the boundary term
transformation gives

$\displaystyle\bar{\mathbf{B}}\rightarrow\bar{\mathbf{B}}+c_{1}(1-n)\big{(}\partial_{\kappa}P^{\kappa}-\frac{1}{2}g_{\alpha\beta}P^{\kappa}\partial_{\kappa}g^{\alpha\beta})\,,$
(4.4.46)

which then leads to

$\displaystyle\bar{\mathbf{G}}+\bar{\mathbf{B}}$
$\displaystyle\rightarrow\bar{\mathbf{G}}+\bar{\mathbf{B}}+c_{1}\Big{[}2\bar{\Gamma}{}^{\kappa}_{[\kappa\mu]}P^{\mu}+c_{1}(n-1)P^{\lambda}P_{\lambda}$
$\displaystyle+(n-1)\big{(}g^{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\mu\lambda}P_{\rho}+P_{\lambda}\partial_{\nu}g^{\nu\lambda}-\frac{1}{2}P^{\nu}g_{\alpha\beta}\partial_{\nu}g^{\alpha\beta}+\frac{1}{2}g_{\alpha\beta}P^{\kappa}\partial_{\kappa}g^{\alpha\beta}-\partial_{\kappa}P^{\kappa})\Big{]}$
$\displaystyle=\bar{\mathbf{G}}+\bar{\mathbf{B}}+c_{1}\Big{[}2\Gamma^{\kappa}_{[\kappa\mu]}P^{\mu}+c_{1}(n-1)P^{\lambda}P_{\lambda}+(n-1)\big{(}g^{\mu\lambda}\bar{\Gamma}{}^{\rho}_{\mu\lambda}P_{\rho}-\partial^{\kappa}P_{\kappa})\Big{]}$
$\displaystyle=\bar{\mathbf{G}}+\bar{\mathbf{B}}+2c_{1}\bar{\Gamma}{}^{\kappa}_{[\kappa\mu]}P^{\mu}+c_{1}^{2}(n-1)P^{\lambda}P_{\lambda}+c_{1}(1-n)\bar{\nabla}^{\mu}P_{\mu}\,,$
(4.4.47)

which matches the result for the Ricci scalar in equation (4.4.40).

The projective invariance of the action will be important when considering the
field equations in both the unmodified and the modified theories. It also
brings with it some potential problems relating to the uniqueness of solutions
and indeterminacy of the affine connection, which we discussed in the previous
chapter, see [299, 300, 311]. We will see the consequences of this symmetry
later on.

####  4.4.3 Action and diffeomorphism invariance

Beginning with the metric-affine Einstein-Hilbert action $\bar{R}$, we can
once again use our new decomposition
$\bar{R}=\bar{\mathbf{G}}+\bar{\mathbf{B}}$ to extract the bulk part of the
action. We then define the metric-affine Einstein action as

$\displaystyle
S_{\bar{\textrm{E}}}[g,\bar{\Gamma}]=\frac{1}{2\kappa}\int\sqrt{-g}\,\bar{\mathbf{G}}\,d^{4}x\,.$
(4.4.48)

This of course differs from the metric-affine Einstein-Hilbert action (3.1.10)
by the boundary term $\bar{\mathbf{B}}$, and so will lead to equivalent
equations of motion. Variations with respect to the metric and the connection
lead to

$\displaystyle\delta
S_{\bar{\textrm{E}}}=\frac{1}{2\kappa}\int\sqrt{-g}\Big{[}\delta
g^{\mu\nu}\bar{G}_{\mu\nu}+\delta\bar{\Gamma}^{\lambda}_{\mu\nu}P^{\mu\nu}{}_{\lambda}\Big{]}d^{4}x=0\,,$
(4.4.49)

where $\bar{G}_{\mu\nu}$ is the Einstein tensor of a general affine
connection, and $P^{\mu\nu}{}_{\lambda}$ is the Palatini tensor. The full
details of this derivation can be found in Appendix C.1.2. These calculations
will be especially useful in Chapter 5, when we consider modified actions.

As was discussed in detail in Sec 3.1.1, the Palatini tensor is trace-free
over $P^{\mu\nu}{}_{\nu}=0$. This was a consequence of the projective
invariance of the affine Ricci scalar $\bar{R}$. The invariance of the bulk
term $\bar{\mathbf{G}}$ leads to the same conclusion, and the connection
equation of motion therefore only fixes 60 of the 64 components of the affine
connection. In the case of vanishing hypermomentum we obtain the solutions for
the connection given in (3.1.7), and we return to GR in the Levi-Civita
geometry. The general field equations are thus

$\displaystyle\bar{G}_{(\mu\nu)}$
$\displaystyle=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}\,,$ (4.4.50) $\displaystyle
P^{\mu\nu}{}_{\lambda}$
$\displaystyle=2\kappa\Delta^{\mu\nu}{}_{\lambda}\,,\,$ (4.4.51)

with the symmetric energy-momentum tensor and hypermomentum defined in
equations (3.2.1) and (3.2.3). The dynamics of this theory are then completely
equivalent to the metric-affine formulation of the Einstein-Hilbert action
studied in the previous chapter.

Let us now discuss the invariances of the metric-affine Einstein action.
Firstly, the action is clearly invariant under local Lorentz transformations,
as we have made no reference to the tangent space and tetrads whatsoever.
Similarly, we showed the projective invariance of the affine bulk term
$\bar{\mathbf{G}}$ and hence the invariance of the action as a whole. With
regards to diffeomorphisms, the metric-affine bulk term transforms
infinitesimally as

$\mathcal{L}_{\xi}\bar{\mathbf{G}}=\xi^{\mu}\partial_{\mu}\bar{\mathbf{G}}+M^{\mu\nu}{}_{\lambda}\partial_{\mu}\partial_{\nu}\xi^{\lambda}\,.$
(4.4.52)

This transformation is exactly the same as in the Levi-Civita case, and the
final term can be written as a total derivative. Applying this to the action
leads to

$\displaystyle\delta_{\xi}S_{\bar{\textrm{E}}}[g,\bar{\Gamma}]$
$\displaystyle=\frac{1}{2\kappa}\int\delta_{\xi}(\sqrt{-g}\,\bar{\mathbf{G}})\,d^{4}x$
$\displaystyle=\frac{1}{2\kappa}\int\partial_{\mu}\partial_{\nu}\big{(}\sqrt{-g}M^{\mu\nu}{}_{\gamma}\big{)}\xi^{\gamma}d^{4}x=0\,,$
(4.4.53)

where we have discarded boundary terms depending on $\xi$ and its derivatives
and made use of the contracted Bianchi identity (4.1.63) in the final
equality.

Again, this result should not be surprising given that the action differs from
a diffeomorphism invariant one by a boundary term. However, it is perhaps
unexpected that this method does not give rise to any new identities besides
the standard (Levi-Civita) contracted Bianchi identity. To highlight this, let
us take the traditional approach of taking our standard field variations and
letting them be generated by an infinitesimal diffeomorphism

$\displaystyle\delta_{\xi}S_{\bar{\textrm{E}}}[g,\bar{\Gamma}]$
$\displaystyle=\frac{1}{2\kappa}\int\Big{[}\delta_{\xi}g^{\mu\nu}\bar{G}_{\mu\nu}+\delta_{\xi}\bar{\Gamma}{}^{\lambda}_{\mu\nu}P^{\mu\nu}{}_{\lambda}\Big{]}\sqrt{-g}d^{4}x$
$\displaystyle=\frac{1}{2\kappa}\int\Big{[}-2$
$\displaystyle\nabla_{(\mu}\xi_{\nu)}\bar{G}{}^{\mu\nu}+P^{\mu\nu}{}_{\lambda}\Big{(}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\xi^{\lambda}+\xi^{\rho}\bar{R}_{\rho\mu\nu}{}^{\lambda}-\bar{\nabla}_{\mu}(T^{\lambda}{}_{\nu\rho}\xi^{\rho})\Big{)}\Big{]}\sqrt{-g}d^{4}x$
$\displaystyle=$ $\displaystyle\ \textrm{`surface
terms'}+\frac{1}{2\kappa}\int\xi^{\mu}\Big{[}2\nabla_{\nu}\bar{G}{}^{(\nu}{}_{\mu)}+\frac{1}{\sqrt{-g}}\bar{\nabla}_{\nu}\bar{\nabla}_{\lambda}(\sqrt{-g}P^{\lambda\nu}{}_{\mu})$
$\displaystyle+$
$\displaystyle\frac{1}{\sqrt{-g}}\bar{\nabla}_{\nu}(\sqrt{-g}P^{\nu\lambda}{}_{\rho})T^{\rho}{}_{\lambda\mu}+\bar{R}_{\mu\nu\lambda}{}^{\rho}P^{\nu\lambda}{}_{\rho}\Big{]}\sqrt{-g}d^{4}x=0\,,$
(4.4.54)

where we have simply inserted the Lie derivative definitions in (4.4.31) and
(4.4.32). Discarding boundary terms, we are led to the identity

$2\nabla_{\nu}\bar{G}{}^{(\nu}{}_{\mu)}+\frac{1}{\sqrt{-g}}\bar{\nabla}_{\nu}\bar{\nabla}_{\lambda}(\sqrt{-g}P^{\lambda\nu}{}_{\mu})+\frac{1}{\sqrt{-g}}\bar{\nabla}_{\nu}(\sqrt{-g}P^{\nu\lambda}{}_{\rho})T^{\rho}{}_{\lambda\mu}\\\
+\bar{R}_{\mu\nu\lambda}{}^{\rho}P^{\nu\lambda}{}_{\rho}\equiv 0\,,$ (4.4.55)

which is manifestly covariant161616We hazard the guess that this final
identity (4.4.55) could be equated with the twice-contracted affine Bianchi
identity (2.1.41), such that Noether’s theorem again leads to a geometric
identity as it did in the Levi-Civita case. However, the required algebraic
manipulations to verify this equivalence appear tediously long.. A similar
result is obtained in the teleparallel case in [148].

As a last note, we mention that the above analysis could be performed in the
orthonormal basis working instead with the tetrad and affine spin connection
$\\{e,\omega\\}$, see for instance [167]. In that case, we would discuss local
Lorentz pseudo-invariance. The resulting equations of motion would still be
the same, due to the action differing from the metric-affine Einstein-Hilbert
action by a boundary term. However, the boundary term would of course be
different to the $\bar{\mathbf{B}}$ studied here, just as the Levi-Civita
boundary terms $\mathbf{B}$ and $\mathfrak{B}$ had a non-trivial difference.

####  4.4.4 Relation to teleparallel gravity

The affine Riemann tensor and its contractions can be decomposed into a sum of
Levi-Civita terms and contortion terms, see equations (2.1.54)–(2.1.56). For
the affine Ricci scalar, these contracted contortion terms can be expressed in
terms of the geometric scalars of torsion and non-metricity. We made use of
this, along with the decomposition of the Ricci scalar into its bulk and
boundary terms, in equations (4.3.1)-(4.3.2). Let us restate the first of
those fundamental equations (in the coordinate basis) here for clarity

$\displaystyle\bar{R}=\overbrace{\mathbf{G}+\mathbf{B}}^{R}+\,T-B_{T}+Q+B_{Q}+\mathbf{C}\,.$
(4.4.56)

Recall that $T$ and $Q$ are the torsion and non-metricity scalars, given in
(3.3.11) and (3.3.25) respectively. The torsion and non-metricity boundary
terms $B_{T}$ and $B_{Q}$ are given in (3.3.12) and (3.3.26). The cross terms
$\mathbf{C}$ can be found in [222].

One can then show that the affine quantities $\bar{\mathbf{G}}$ and
$\bar{\mathbf{B}}$ are related to the geometric scalars by [173]

$\displaystyle\bar{\mathbf{G}}$ $\displaystyle=\mathbf{G}+T+Q+\mathbf{C}\,,$
(4.4.57) $\displaystyle\bar{\mathbf{B}}$
$\displaystyle=\mathbf{B}-B_{T}+B_{Q}\,.$ (4.4.58)

The affine bulk term is simply the sum of the Levi-Civita, torsion, and non-
metricity bulk terms, and similarly for the affine boundary term.

It is important to remember that the equivalence between the scalars $Q$ and
$T$ and the bulk terms $\mathbf{G}$ and $\mathfrak{G}$ only holds in the
teleparallel framework, where vanishing curvature
$\bar{R}_{\lambda\mu\nu}{}^{\gamma}(\bar{\Gamma})=0$ has been enforced. Here
we allow for connections with affine curvature
$\bar{R}_{\lambda\mu\nu}{}^{\gamma}\neq 0$ and so this equivalence is broken.
Hence, a combination like $\mathbf{G}+Q$ does not vanish in general in any
specific set of coordinates or in any gauge. This marks a significant
departure from the analysis in Sec. 4.3.

We also remark that even in the teleparallel case, the affine bulk and
boundary terms are not forced to vanish individually. Instead, only their sum
needs to vanish $\bar{\mathbf{G}}+\bar{\mathbf{B}}=0$. Moreover, they will
actually reduce to the gauge boundary terms identified in (3.3.19) and
(3.3.29). Let us show this in the simple case of a symmetric teleparallel
geometry $T=\mathbf{C}=0$, in which case equations (4.4.57) and (4.4.58) can
be written as

$\displaystyle\bar{\mathbf{G}}$ $\displaystyle=\mathbf{G}+Q=-b_{\xi}\,,$
(4.4.59) $\displaystyle\bar{\mathbf{B}}$
$\displaystyle=\mathbf{B}+B_{Q}=b_{\xi}\,,$ (4.4.60)

where we have made use of the teleparallel relations (4.3.9). We therefore see
that the requirement for $\bar{\mathbf{G}}$ or $\bar{\mathbf{B}}$ to vanish is
exactly the same as the conditions for $b_{\xi}$ to vanish. This is satisfied
in the coincident gauge.

When we study the modified metric-affine theories based on $\bar{\mathbf{G}}$
and $\bar{\mathbf{B}}$, we will be working in a geometric setting with non-
vanishing curvature, and so the torsion and non-metricity scalars are no
longer teleparallel. This opens up a new avenue of gravitational theories to
study that will differ from GR more significantly. Instead, these theories
will more closely resemble the metric-affine theories with a truly independent
and dynamical connection, see Sec. 3.2.

This concludes this section and this chapter, where we have focussed on the
different possible decompositions of the Ricci scalar within different
geometric frameworks. All of the resulting actions were shown to be
dynamically equivalent to their Einstein-Hilbert counterparts and exhibit the
same invariant properties. In the Levi-Civita cases, we saw an interesting
relationship with the teleparallel equivalents of General Relativity. In the
next chapter, we look at modified theories of gravity based on non-linear
modifications of these pseudo-invariant actions.

[75mm] There are some scientists who go about preaching that Nature always
takes on the simplest solutions. Yet the simplest solution by far would be
nothing at all in the Universe. Nature is far more inventive than that, so I
refuse to go along thinking it always has to be simple. ‘Lectures on
Gravitation’ (1963)
Richard P. Feynman

##  Chapter 5 Modified theories of gravity

In this chapter we study modified theories of gravity based on the non-
covariant decompositions of the Einstein-Hilbert action. The form of these
modifications follows the route taken in $f(R)$ gravity (1.2.2), allowing our
generalised action to depend on an arbitrary function of our decomposed terms.
We do not introduce any additional fields and work in $n=4$ spacetime
dimensions. Moreover, we only consider minimally coupled matter. However,
diffeomorphism invariance or local Lorentz invariance is explicitly broken in
these models, leading to a number of interesting consequences.

In the first section we work in the standard Riemannian framework of General
Relativity. This is based on our work [222]. We take a detailed look at the
role of diffeomorphism invariance and the conservation of energy-momentum in
these theories. Our modifications can also be shown to be equivalent to the
modified teleparallel theories, which we prove in Sec. 5.2. We then propose a
unified framework from which one can study all of the previously introduced
modified theories. In the final section we move on to considering our non-
covariant modifications in the more general metric-affine setting, taking a
close look at a modified Einstein-Cartan type theory, following our work
[173].

###  5.1 Modifications of the Einstein action

Inspired by the modifications of $f(R)$, $f(T)$ and $f(Q)$ gravity, we now
consider non-linear modifications of the Einstein action (4.1.54). These take
the form $f(\mathbf{G})$, with $\mathbf{G}$ corresponding to the bulk part of
the Ricci scalar defined in (4.1.8), and $f$ representing an arbitrary
function. To be even more general, and to make contact with limiting cases
such as $f(R)$ gravity, let us propose our action take the following form
[222]

$S_{\textrm{grav}}[g]=\frac{1}{2\kappa}\int
f(\mathbf{G},\mathbf{B})\sqrt{-g}d^{4}x\,,$ (5.1.1)

where $\mathbf{B}$ is the boundary term (4.1.9) in the decomposition
$R=\mathbf{G}+\mathbf{B}$. These more general modifications are motivated by
the analogous treatment in teleparallel $f(T,B_{T})$ gravity [375], where the
similar relation $R=-T+B_{T}$ holds.

The $f(\mathbf{G},\mathbf{B})$ class of theories encompass both General
Relativity and $f(R)$ gravity for specific forms of the function $f$. However,
they also give room for deviations by breaking the diffeomorphism symmetry
present in both of those two cases. As shown in the previous chapter, the
Einstein action given by $f=\mathbf{G}$ was diffeomorphism invariant up to
boundary terms (4.1.61). Similarly, the bulk and boundary terms transformed as
pseudo-scalars, see (4.1.48) and (4.1.49). However, a non-linear function of a
boundary term is no longer a boundary term. Hence, non-linear modifications
will not share the pseudo-diffeomorphism invariance of the bulk and boundary
terms $\mathbf{G}$ and $\mathbf{B}$.

Moving on to the dynamical equations of motion, variations of the action
(5.1.1) lead to

$\delta S_{\textrm{grav}}=\frac{1}{2\kappa}\int\Big{(}\delta
f(\mathbf{G},\mathbf{B})-\frac{1}{2}f(\mathbf{G},\mathbf{B})g_{\rho\sigma}\delta
g^{\rho\sigma}\Big{)}\sqrt{-g}d^{4}x\,,$ (5.1.2)

where

$\delta f(\mathbf{G},\mathbf{B})=\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{G}}\delta\mathbf{G}+\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{B}}\delta\mathbf{B}\,.$ (5.1.3)

Following our work in [222], the field equations are calculated to be

$H_{\rho\sigma}=\kappa T_{\rho\sigma}\,,$ (5.1.4)

where we define

$H_{\rho\sigma}:=\frac{\partial
f}{\partial\mathbf{G}}\Big{[}G_{\rho\sigma}+\frac{1}{2}g_{\rho\sigma}\mathbf{G}\Big{]}+\frac{1}{2}E_{\rho\sigma}{}^{\gamma}\partial_{\gamma}\Big{(}\frac{\partial
f}{\partial\mathbf{G}}\Big{)}-\frac{1}{2}g_{\rho\sigma}f(\mathbf{G},\mathbf{B})\
\\\ +\frac{1}{2}\frac{\partial f}{\partial\mathbf{B}}\
g_{\rho\sigma}\mathbf{B}+g_{\rho\sigma}\partial^{\mu}\partial_{\mu}\Big{(}\frac{\partial
f}{\partial\mathbf{B}}\Big{)}-\partial_{\rho}\partial_{\sigma}\Big{(}\frac{\partial
f}{\partial\mathbf{B}}\Big{)}\\\
+\frac{1}{2}g_{\rho\sigma}\partial_{\mu}(g^{\mu\nu})\partial_{\nu}\Big{(}\frac{\partial
f}{\partial\mathbf{B}}\Big{)}+\frac{1}{\sqrt{-g}}\partial_{(\rho}(\sqrt{-g})\partial_{\sigma)}\Big{(}\frac{\partial
f}{\partial\mathbf{B}}\Big{)}\,.$ (5.1.5)

The superpotential $E_{\rho\sigma}{}^{\gamma}$ is defined in equation (4.1.1)
and $T_{\rho\sigma}$ is the energy-momentum tensor of a minimally coupled
matter action $S_{\textrm{M}}[g,\varPhi^{A}]$. For the full details of the
derivation of these equations of motion, see Appendix C.2.1. Also observe that
for $f=\mathbf{G}+\mathbf{B}$ or $f=\mathbf{G}$, the field equations reduce to
$G_{\rho\sigma}=\kappa T_{\rho\sigma}$ and GR is recovered as expected.

Let us make a few preliminary comments on these field equations. Firstly, they
are of course symmetric $H_{[\rho\sigma]}=0$ because they were computed by
varying with respect to the symmetric metric tensor. They are also generically
fourth-order in derivatives of the metric, as there are terms with second
derivatives of $\mathbf{B}$. A key departure from theories such as $f(R)$
gravity is that the gravitational equations (5.1.5) are no longer covariant,
following from the transformation properties111Note that Levi-Civita
connection and all of the non-covariant objects $\mathbf{G}$, $\mathbf{B}$ and
$E_{\rho\sigma}{}^{\gamma}$ are still invariant under global Poincaré
transformations (2.2.2). of $\mathbf{G}$, $\mathbf{B}$ and
$E_{\rho\sigma}{}^{\gamma}$. We also do not make any assumptions about the
invariant properties of the matter action, which we will return to shortly.

The lack of diffeomorphism invariance at the level of the action (5.1.1)
implies that we do not automatically have a Bianchi-like identity. This is in
contrast to the Einstein action (4.1.54), and covariant theories such as
$f(R)$ gravity [376] or even $f(T)$ gravity [377]. As we do not (yet) assume
our matter action $S_{\textrm{M}}[g,\varPhi^{A}]$ to be invariant under
diffeomorphisms, we cannot infer the usual covariant conservation law
$\nabla^{\mu}T_{\mu\nu}=0$. Furthermore, if $S_{\textrm{M}}[g,\varPhi^{A}]$ is
not a scalar then the energy-momentum $T_{\mu\nu}$ may not be a true rank-two
tensor, and the equation $\nabla^{\mu}T_{\mu\nu}$ would not be well-defined.

Having said this, let us now make the following observation. If we divide
(5.1.4) by $\partial f/\partial\mathbf{G}$ we can isolate the Einstein tensor
$G_{\rho\sigma}$ and move all remaining terms to the right-hand side. This
would yield an equation of the form

$\displaystyle
G_{\rho\sigma}=\frac{\kappa}{f_{,\mathbf{G}}}T_{\rho\sigma}+T^{(f)}_{\rho\sigma}\,,$
(5.1.6)

where $f_{,\mathbf{G}}=\partial f/\partial\mathbf{G}$ and
$T^{(f)}_{\rho\sigma}$ stands for the collection of all the remaining terms.
In this sense, the right-hand side can be thought of as modifications to the
effective energy-momentum tensor. This formulation has two implications:
first, mathematical consistency requires that the right-hand side of (5.1.6)
must behave as a rank 2 tensor, possibly under some additional conditions;
second, the twice contracted Bianchi identities imply that the right-hand side
must also be covariantly conserved

$\displaystyle\nabla^{\rho}G_{\rho\sigma}=0\qquad\Rightarrow\qquad\nabla^{\rho}\Bigl{[}\frac{\kappa}{f_{,\mathbf{G}}}T_{\rho\sigma}+T^{(f)}_{\rho\sigma}\Bigr{]}=0\,.$
(5.1.7)

Perhaps unexpectedly, we arrive at a conservation equation.

Note that we did not explicitly assume the diffeomorphism invariance of either
of the actions, but in the next section we will show how such a condition has
in fact been applied a posteriori. We will also clarify, in a more fundamental
way, how this conservation equation has emerged.

####  5.1.1 Diffeomorphism invariance and conservation equations

Let us take a closer look at the diffeomorphisms of action (5.1.1). As
previously noted, the Einstein action was invariant under diffeomorphisms,
owing to the fact that the action differed from a coordinate scalar by a
boundary term. Clearly the modified action (5.1.1) does not share this
property, but it is illustrative to see this explicitly.

Let us proceed to calculate how the modified action transforms under an
infinitesimal symmetry transformation. We make use of our previous expressions
for the Lie derivatives of the bulk and boundary terms, (4.1.48) and (4.1.49),
and find

$\displaystyle\delta_{\xi}S_{\textrm{grav}}$
$\displaystyle=\frac{1}{2\kappa}\int\mathcal{L}_{\xi}\big{(}\sqrt{-g}f(\mathbf{G},\mathbf{B})\big{)}d^{4}x$
$\displaystyle\begin{multlined}=\frac{1}{2\kappa}\int\biggl{[}\nabla_{\mu}\xi^{\mu}f(\mathbf{G},\mathbf{B})\\\
+\Bigl{(}\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{G}}\mathcal{L}_{\xi}\mathbf{G}+\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{B}}\mathcal{L}_{\xi}\mathbf{B}\Bigr{)}\biggr{]}\sqrt{-g}\,d^{4}x\,.\end{multlined}=\frac{1}{2\kappa}\int\biggl{[}\nabla_{\mu}\xi^{\mu}f(\mathbf{G},\mathbf{B})\\\
+\Bigl{(}\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{G}}\mathcal{L}_{\xi}\mathbf{G}+\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{B}}\mathcal{L}_{\xi}\mathbf{B}\Bigr{)}\biggr{]}\sqrt{-g}\,d^{4}x\,.$
(5.1.10)

This leads to

$\delta_{\xi}S_{\textrm{grav}}=\frac{1}{2\kappa}\int\biggl{[}\nabla_{\mu}\xi^{\mu}f(\mathbf{G},\mathbf{B})+\Bigl{(}\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{G}}\partial_{\mu}\mathbf{G}\xi^{\mu}+\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{B}}\partial_{\mu}\mathbf{B}\xi^{\mu}\Bigr{)}\\\
+M^{\alpha\beta}{}_{\gamma}\partial_{\alpha}\partial_{\beta}\xi^{\gamma}\Big{(}\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{G}}-\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{B}}\Big{)}\biggr{]}\sqrt{-g}\,d^{4}x\\\
=\textrm{`surface term'}+\frac{1}{2\kappa}\int
M^{\alpha\beta}{}_{\gamma}\partial_{\alpha}\partial_{\beta}\xi^{\gamma}\bigg{(}\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{G}}-\frac{\partial
f(\mathbf{G},\mathbf{B})}{\partial\mathbf{B}}\bigg{)}\sqrt{-g}\,d^{4}x\,,$
(5.1.11)

which is analogous to equation (4.1.61). Discarding the surface term leaves
the following non-trivial contribution

$\delta_{\xi}S_{\textrm{grav}}=\frac{1}{2\kappa}\int\sqrt{-g}\big{(}f_{,\mathbf{G}}-f_{,\mathbf{B}}\big{)}M^{\alpha\beta}{}_{\gamma}\partial_{\alpha}\partial_{\beta}\xi^{\gamma}\,d^{4}x\,.$
(5.1.12)

As before, we integrate by parts twice and discard the boundary terms, which
leads to

$\delta_{\xi}S_{\textrm{grav}}=\frac{1}{2\kappa}\int\partial_{\alpha}\partial_{\beta}\Bigl{(}\sqrt{-g}M^{\alpha\beta}{}_{\gamma}(f_{,\mathbf{G}}-f_{,\mathbf{B}})\Bigr{)}\xi^{\gamma}\,d^{4}x\,.$
(5.1.13)

If we also include the matter action $S_{\textrm{M}}$, with matter fields
assumed to be on-shell, the total action transforms as

$\displaystyle\delta_{\xi}S_{\textrm{ total}}$
$\displaystyle=\delta_{\xi}S_{\textrm{grav}}+\delta_{\xi}S_{\textrm{M}}$
$\displaystyle=\int\biggl{[}\frac{1}{2\kappa}\partial_{\alpha}\partial_{\beta}\Bigl{(}\sqrt{-g}M^{\alpha\beta}{}_{\gamma}(f_{,\mathbf{G}}-f_{,\mathbf{B}})\Bigr{)}-\nabla_{\alpha}(\sqrt{-g}T^{\alpha}{}_{\gamma})\biggr{]}\xi^{\gamma}\,d^{4}x\,,$
(5.1.14)

where the covariant derivative of the energy-momentum ‘tensor’ should be
understood as implying its usual relation in terms of partial derivatives and
connection terms
$\partial_{\alpha}T^{\alpha}{}_{\gamma}+\Gamma^{\alpha}_{\alpha\beta}T^{\beta}{}_{\gamma}-\Gamma_{\alpha\gamma}^{\beta}T^{\alpha}{}_{\beta}$.
This follows directly from the definition of the Lie derivative of the metric,
but we use the notation $\nabla_{\alpha}$ as shorthand.

If at this point one requires the total action to be invariant under
coordinate transformations, one finds a conservation equation, namely the
vanishing of the integrand of (5.1.14). In the GR case, the integrand vanished
identically (4.1.63). One can again regard this equation as a consequence of
Noether’s theorem, as it relates to the invariance of the action under
transformations of coordinates.

Let us emphasise an important mathematical point that is required for the
consistency of the theory. For the variational formulation to be well-defined,
the total action must be a pseudo-scalar (a coordinate scalar up to boundary
terms). Without this requirement, it seems difficult to construct a well-
defined Lagrangian field theory222For a more formal and geometric discussion
regarding the Lagrangian formulation see [378]. For our purposes, we simply
require that the Euler-Lagrange equations be self-consistent.. We will
therefore assume that (5.1.14) is forced to vanish $\delta_{\xi}S_{\textrm{
total}}=0$ and show that this is in fact required by consistency.

The implementation of the vanishing of the conservation equation (5.1.14)
leads to various interesting points to be addressed. First, let us assume that
the matter action is itself based on a true scalar Lagrangian, which then
implies that the matter energy-momentum tensor is independently conserved
$\nabla_{\alpha}T^{\alpha}{}_{\gamma}=0$. Requiring
$\delta_{\xi}S_{\textrm{t}otal}=0$ then implies the additional equation

$\displaystyle\partial_{\alpha}\partial_{\beta}\Bigl{(}\sqrt{-g}M^{\alpha\beta}{}_{\gamma}(f_{,\mathbf{G}}-f_{,\mathbf{B}})\Bigr{)}=0\,.$
(5.1.15)

It is possible to show that this conservation equation (5.1.15) is in fact
completely equivalent to the vanishing ‘covariant derivative’ of the
gravitational field equations

$\nabla_{\alpha}H^{\alpha}{}_{\gamma}:=\partial_{\alpha}H^{\alpha}{}_{\gamma}+\Gamma^{\alpha}_{\alpha\beta}H^{\beta}{}_{\gamma}-\Gamma^{\beta}_{\alpha\gamma}H^{\alpha}{}_{\beta}=0\,,$
(5.1.16)

where $H_{\rho\sigma}$ is defined in (5.1.5). A direct but long-winded
calculation and expansion of both equations (5.1.15) and (5.1.16) verifies
this fact. Alternatively, one can work with the metric variations of the
action directly and let this variation be generated by an infinitesimal
diffeomorphism to arrive at equation (5.1.16). (This is exactly how we first
derived the contracted Bianchi identity for the Einstein-Hilbert action
(2.3.2)).

Once the equivalence between both forms of conservation equation has been
established, it is straightforward to see that these equations are exactly
what we found in (5.1.6) following from
$\nabla_{\alpha}G^{\alpha}{}_{\beta}=0$. In other words, for our field
equations to be mathematically consistent we require the vanishing
diffeomorphism transformation of the total action $\delta_{\xi}S_{\textrm{
total}}=0$. This can also be seen from the fact that the variations with
respect to the dynamical field variables $g_{\mu\nu}$ are required to vanish
$\delta S_{\textrm{ total}}=0$, and the fields themselves are covariant.

This is a similar conclusion to what is found in many other symmetry-breaking
models. In Sec. 1.2.4, when looking at unimodular gravity, the action becomes
diffeomorphism invariant on-shell when applying the field equations. In the
non-covariant Chern-Simons modifications studied by Jackiw [210, 211], it is
found that diffeomorphism invariance is dynamically reinstated as well. This
is reflected in our modifications here also.

Moving on to the solutions of (5.1.15), we first note that the constraint is
satisfied for all possible geometries if $f_{,\mathbf{G}}-f_{,\mathbf{B}}=0$.
This equation is easily integrated and one finds that all functions of the
form $f(\mathbf{G}+\mathbf{B})$ are general solutions, assuming a sufficiently
regular function $f$. Recalling that this combination is in fact the Ricci
scalar, the coordinate scalar constructed from $\mathbf{G}$ and $\mathbf{B}$,
we find the expected result that within the family of
$f(\mathbf{G},\mathbf{B})$ theories $f(R)$ gravity is the unique
diffeomorphism invariant theory. We will show this result explicitly in Sec.
5.2.1.

Alternatively, let us now consider the case when $f(\mathbf{G},\mathbf{B})\neq
f(\mathbf{G}+\mathbf{B})$, i.e., non-$f(R)$ and non-covariant theories.
Moreover, we will maintain the assumption that the matter action yields an
independent conservation equation $\nabla_{\alpha}T^{\alpha}{}_{\gamma}=0$. As
before, for the total action to be invariant, equation (5.1.15) has to be
satisfied. It is now important that we note that the conservation equation
(5.1.15) in fact depends on the coordinates used. As such, solutions for a
given geometry $g_{\mu\nu}(x)$ will depend explicitly on the coordinate system
$\\{x^{\mu}\\}$. The simplest non-trivial example is when $\mathbf{G}$ and
$\mathbf{B}$ are both constants with respect to some set of coordinates.

In the case when the conservation equation vanishes, the gravitational action
will naively _appear_ to be invariant under diffeomorphisms. This is therefore
associated with a specific set of preferred coordinates. This situation is
similar to that in $f(T)$ gravity when so-called ‘good tetrads’ are considered
[379, 169, 375]. This is now understood as a manifestation of the Weitzenböck
gauge in $f(T)$ gravity [152]. In our case here, these ‘good’ coordinates are
in fact just those associated with coincident gauge in symmetric teleparallel
gravity. Moreover, the guaranteed existence of the coincident gauge in the
symmetric teleparallel theories [344] also guarantees the existence of non-
trivial solutions to (5.1.15). We will return to this point in Sec. 5.2.2,
when we extend the analysis between the bulk and boundary terms $\mathbf{G}$
and $\mathbf{B}$ and the symmetric teleparallel scalars $Q$ and $B_{Q}$ to the
modified theories $f(\mathbf{G},\mathbf{B})$ and $f(Q,B_{Q})$.

Lastly, if we drop the assumption that matter be described by an invariant
action, and hence the covariant conservation of the energy-momentum tensor,
only the integrand of (5.1.13) has to vanish to achieve diffeomorphism
invariance of the total action. We then simply have an equation of the form

$\nabla_{\alpha}\big{(}H^{\alpha}{}_{\gamma}-T^{\alpha}{}_{\gamma}\big{)}=0\,,$
(5.1.17)

where we again understand the covariant derivative here to be shorthand for
the expression in equation (5.1.16). This would allow us to study more exotic
theories where energy-momentum is not conserved. An interesting example is
models where the dark matter energy density can decrease over time whilst the
dark energy density, here modelled through the contributions of
$f(\mathbf{G},\mathbf{B})$, can increase over time. For more on these types of
models, see the review [380].

####  5.1.2 Second-order $f(\mathbf{G})$ gravity

Now that we have introduced the fourth-order $f(\mathbf{G},\mathbf{B})$
gravity class of theories, let us focus on the second-order models. This is
achieved by simply considering functions of the bulk term only, or more
specifically, $f(\mathbf{G},\mathbf{B})=f(\mathbf{G})+c_{1}\mathbf{B}$ for
some constant $c_{1}$. Clearly the linear boundary term does not contribute to
the classical dynamics, but we mention it briefly here for
completeness333Recall in Sec. 2.2.2 we made reference to the role of the
Gibbons-Hawking-York counter term in relation to quantum gravity and black
hole entropy. This same counter term can be included alongside the boundary
term $\mathbf{B}$ if necessary.. In what follows, we set $c_{1}=0$ as it plays
no role in the discussion.

In general, these theories will be non-covariant and differ from GR when
$f(\mathbf{G})\neq c_{2}\mathbf{G}$. They also possess the attractive quality
that they remain second-order, introduce no additional field content, are
four-dimensional, and still reside in the standard Levi-Civita framework.
Lovelock’s theorem is instead violated due to the non-covariance of the
theory.

Defining the second-order modified action as

$S_{\textrm{grav}}[g]=\frac{1}{2\kappa}\int f(\mathbf{G})\sqrt{-g}d^{4}x\,,$
(5.1.18)

the equations of motion follow directly from (5.1.5)

$\displaystyle
f^{\prime}(\mathbf{G})\Big{[}G_{\rho\sigma}+\frac{1}{2}g_{\rho\sigma}\mathbf{G}\Big{]}+\frac{1}{2}f^{\prime\prime}(\mathbf{G})E_{\rho\sigma}{}^{\gamma}\partial_{\gamma}\mathbf{G}-\frac{1}{2}g_{\rho\sigma}f(\mathbf{G})=\kappa
T_{\rho\sigma}\,.$ (5.1.19)

The conservation equation for the gravitational sector (5.1.15) becomes

$\partial_{\alpha}\partial_{\beta}\big{(}\sqrt{-g}M^{\alpha\beta}{}_{\gamma}f^{\prime}(\mathbf{G})\big{)}=0\,.$
(5.1.20)

Clearly this equation is only zero (for all geometries) in the case that
$f^{\prime\prime}(\mathbf{G})=0$, which leads back to GR as expected. We will
again require coordinates to satisfy this constraint for all of the solutions
that we study.

In the introduction we mentioned that diffeomorphism-breaking theories with a
minimal length scale can be motivated by appeals to quantum gravity. We also
mentioned that this had been studied in $f(T)$ gravity (in relation to
breaking local Lorentz invariance) by using the Born-Infeld approach [381,
382]. The Born-Infeld Lagrangian for our $f(\mathbf{G})$ theory reads

$L_{BI}=f(\mathbf{G})=\lambda\Big{(}\sqrt{1+\frac{2\mathbf{G}}{\lambda}}-1\Big{)}\
.$ (5.1.21)

This is simply $\mathbf{G}$ to leading order as $\lambda$ tends to infinity.
In natural units, the parameter $\lambda$ has dimensions of inverse length
squared. Taking $1/\sqrt{|\lambda|}\propto l$ to be the associated length
scale of the theory, we recover GR in the limit that $l\rightarrow 0$ and
$|\lambda|\rightarrow\infty$. One can then think of $l$ as the length scale
related to the non-invariance of the theory under general coordinate
transformations. Alternatively, this can be interpreted as some fundamental
length scale of spacetime (e.g., the Planck length).

An interesting motivation for the Born-Infeld scheme comes from transitioning
from the classical free particle Lagrangian to the relativistic one, see [179,
381]. The free particle Lagrangian of Newtonian mechanics is
$L_{p}=m\dot{x}^{2}/2$. In the Born-Infeld scheme this becomes
$L_{BI}=\sqrt{1+L_{p}/(mc^{2})}-1$, where the new scale $mc^{2}$ has been
introduced. This new Lagrangian is simply that of a relativistic particle, and
the low energy, non-relativistic limit $|\dot{x}|\ll c$ reduces to the
Newtonian mechanics action. We will study these modified Born-Infeld models in
the cosmological setting in Sec. 6.1.1.

A different choice of function $f$ leads to an important comparison with the
theories known as modified Newtonian dynamics (MOND) [217]. It was recently
shown that the $f(\mathbf{G})$ equations of motion (5.1.19) can give rise to
the MOND phenomenology [218], see also [383, 384]. The so-called ‘deep-MOND
limit’ is obtained for the choice of function $f\propto\mathbf{G}^{2/3}$. In
fact, Milgrom recently proposed the theories given by
$\mathcal{F}(\ell_{M}^{2}\mathcal{R})$, where $\ell_{M}$ is the MOND length
scale and $\mathcal{R}$ is the quadratic part of the Ricci scalar [218]. This
turns out to be exactly the $f(\mathbf{G})$ theory presented here444We thank
Moti Milgrom for pointing this out in response to our work [222].. We will
return to this topic when discussing covariantisation in Sec. 5.2. This shows
yet another example of how the $f(\mathbf{G})$ models can be chosen to lead to
physically interesting scenarios.

Returning to the equations of motion, let us rewrite the field equations by
dividing through by $f^{\prime}$ and isolating the divergence-free Einstein
tensor

$\displaystyle
G_{\rho\sigma}=\frac{\kappa}{f^{\prime}}T_{\rho\sigma}+\frac{1}{2}\frac{1}{f^{\prime}}\Bigl{(}(f-f^{\prime}\mathbf{G})g_{\rho\sigma}-f^{\prime\prime}E_{\rho\sigma}{}^{\gamma}\partial_{\gamma}\mathbf{G}\Bigr{)}\,.$
(5.1.22)

The vanishing covariant derivative of the right-hand side can be formally
implemented by using the gravitational constraint equation (5.1.20), which we
can write as

$\displaystyle\frac{1}{2\kappa}\partial_{\alpha}\partial_{\beta}\bigl{(}\sqrt{-g}M^{\alpha\beta}{}_{\gamma}f^{\prime}(\mathbf{G})\bigr{)}-\nabla_{\alpha}T^{\alpha}{}_{\gamma}\sqrt{-g}=0\,.$
(5.1.23)

For conventional matter based on a scalar action, the covariant divergence of
the energy-momentum tensor vanishes and only the initial term survives. As we
mentioned, the non-covariance of the gravitational theory means that certain
systems of coordinates are picked out by the equation above. To illustrate
what this means in practice, let us briefly consider the metrics for
spherically symmetric and FLRW spacetimes, the latter of which will be studied
in greater detail in Chapter 6.

The static and spherically symmetric metric is often written in spherical
coordinates $(t,r,\theta,\phi)$, taking the form

$ds^{2}=-A(r)^{2}dt^{2}+B(r)^{2}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\,.$ (5.1.24)

However, if one uses this metric in this form in the conservation equation
(5.1.20), one is led to the constraint $f^{\prime\prime}(\mathbf{G})=0$. This
constraint also comes from the off-diagonal field equations. This then
trivially leads back to GR. However, this is not a reflection of the theory or
the physical symmetries of the spacetime, but instead a reflection of the
choice of coordinates. In isotropic coordinates $(t,x,y,z)$ the metric takes
the form

$\displaystyle
ds^{2}=-{A(R)}^{2}dt^{2}+{B(R)}^{2}\bigl{(}dx^{2}+dy^{2}+dz^{2}\bigr{)}\,,\qquad
R=\sqrt{x^{2}+y^{2}+z^{2}}\,,$ (5.1.25)

and it is then found that (5.1.20) vanishes independently of the function $f$.
Hence, these are the appropriate coordinates to study models beyond GR. The
same results hold in the more general $f(\mathbf{G},\mathbf{B})$ theories.
Further discussion of spherically symmetric spacetimes is given in our work
[222].

For the FLRW metric a similar situation occurs. Here we will consider the
spatially flat $k=0$ case only555Spatially curved cases can also be
considered, but the explicit form of the metric in the ‘good coordinates’ is
extremely complex.. Working in spherical coordinates with the metric

$ds^{2}=-N(t)^{2}dt^{2}+a(t)^{2}\big{(}dr^{2}+r^{2}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\big{)}\,,$ (5.1.26)

one again is lead to an equation of the form $f^{\prime\prime}(\mathbf{G})=0$,
forcing the theory back to GR. Working instead in Cartesian coordinates

$ds^{2}=-N(t)^{2}dt^{2}+a(t)^{2}\big{(}dx^{2}+dy^{2}+dz^{2}\big{)}\,,$
(5.1.27)

the constraint equation is satisfied for all choices of $f$. This again
represents the correct choice of coordinates if studying theories beyond GR.
As we pointed out previously, this is similar to the ‘good’ and ‘bad’ tetrads
in modified teleparallel $f(T)$ gravity [169]. It should now be clear from the
previous discussions that these are related to the coincident and Weitzenböck
gauges. We examine this more closely in Sec. 5.2.2. This will explain this
unusual, seemingly coordinate-dependent behaviour of the theory presented
here, showing that it is in fact just an artefact of an implicit gauge choice.

An interesting and related topic, which we unfortunately will not explore
here, is the issue of the number of physical degrees of freedom of these
theories. Without the full diffeomorphism symmetry, one would expect the
number to be greater than the two polarisations of the gravitational field in
GR. But to determine the precise number requires a full Hamiltonian analysis,
which is far from straightforward. Moreover, though second-order theories are
free from the Ostrogradsky instabilities described in Sec. 1.2, it still
remains to be checked whether the additional degrees of freedom are healthy.
For example, if additional propagating degrees of freedom exist but do not
show up at lower perturbation orders, this may indicate a possible strong
coupling problem666This is when a given background solution becomes strongly
coupled at very low (IR) energies. At the classical level, the predictions
from linear perturbations about this background then become unreliable. This
is often hinted at when known degrees of freedom are absent at lower
perturbation orders around specific background, e.g., around Minkowski and
FLRW backgrounds in $f(T)$ gravity [385, 386, 387]., see [203, 204, 207]. For
details on these issues in $f(T)$ and $f(Q)$ gravity, see [387, 202, 388,
389].

##### Tetradic modifications

We have focused on the modifications stemming from the coordinate basis
decomposition of the Ricci scalar $R=\mathbf{G}+\mathbf{B}$, but we could
instead work in the orthonormal basis $R(e)=\mathfrak{G}+\mathfrak{B}$.
Modifications based on these tetradic terms, such as $f(\mathfrak{G})$ and
$f(\mathfrak{G},\mathfrak{B})$, would lead to a different theory than those
studied above. We will not go through the explicit details of the calculation,
but the formulation follows analogously to the $f(\mathbf{G},\mathbf{B})$
theories. The main computational difference is that we instead vary with
respect to the tetrad, and a conservation equation would follow from requiring
the total action to be locally Lorentz invariant. In Section 5.2.2 we show the
resulting equations of motion, by making use of the aforementioned equivalence
with the teleparallel theories. See equation (5.2.8) for the explicit
equations of motion of the $f(\mathfrak{G})$ theories.

We also note that $\mathfrak{G}$ can be obtained by making use of the boundary
term $\mathbb{B}$ in equation (4.2.36). There, we showed it could be related
to the metric bulk term by $\mathfrak{G}=\mathbf{G}+\mathbb{B}$. One could
instead derive the $f(\mathfrak{G})$ theories by taking the tetradic
variations of $f(\mathbf{G}+\mathbb{B})$. To be even more general, the mixed
Einstein action (4.2.44) could be used as the starting point for
modifications. In Sec. 5.2.3 we explore this possibility, showing how this
leads to a unified approach.

###  5.2 Relation to other modified theories

####  5.2.1 $f(R)$ gravity

If we take $f(\mathbf{G},\mathbf{B})$ to be a function of the Ricci scalar
only, that is $f(\mathbf{G},\mathbf{B})=f(\mathbf{G}+\mathbf{B})=f(R)$, we
readily recover the $f(R)$ field equations. This also serves as a good
consistency check. First we use that

$\frac{\partial f(\mathbf{G}+\mathbf{B})}{\partial\mathbf{G}}=\frac{\partial
f(\mathbf{G}+\mathbf{B})}{\partial\mathbf{B}}\ =\frac{\partial f(R)}{\partial
R}\,.$ (5.2.1)

With this simplification we can rewrite the left-hand side of equation (5.1.5)
as follows

$\frac{\partial f}{\partial
R}\Big{[}G_{\rho\sigma}+\frac{1}{2}g_{\rho\sigma}\mathbf{G}+\frac{1}{2}g_{\rho\sigma}\mathbf{B}\Big{]}-\frac{1}{2}g_{\rho\sigma}f(R)+g_{\rho\sigma}\partial^{\mu}\partial_{\mu}\Big{(}\frac{\partial
f}{\partial R}\Big{)}-\partial_{\rho}\partial_{\sigma}\Big{(}\frac{\partial
f}{\partial R}\Big{)}\\\ +\partial_{\gamma}\Big{(}\frac{\partial f}{\partial
R}\Big{)}\Big{[}\frac{1}{2}E_{\rho\sigma}{}^{\gamma}+\frac{1}{2}g_{\rho\sigma}\partial_{\mu}g^{\mu\gamma}+\frac{1}{\sqrt{-g}}\partial_{(\rho}(\sqrt{-g})\delta^{\gamma}_{\sigma)}\Big{]}\,.$
(5.2.2)

Expanding the definition of $E_{\rho\sigma}{}^{\gamma}$ (4.1.1), writing the
partial derivatives of the metric in terms of Christoffel symbols and using
$\mathbf{G}+\mathbf{B}=R$ gives

$\frac{\partial f}{\partial
R}\Big{[}G_{\rho\sigma}+\frac{1}{2}g_{\rho\sigma}R\Big{]}-\frac{1}{2}g_{\rho\sigma}f(R)+g_{\rho\sigma}\partial^{\mu}\partial_{\mu}\Big{(}\frac{\partial
f}{\partial R}\Big{)}-\partial_{\rho}\partial_{\sigma}\Big{(}\frac{\partial
f}{\partial R}\Big{)}\\\ +\partial_{\gamma}\Big{(}\frac{\partial f}{\partial
R}\Big{)}\Big{[}\Gamma^{\gamma}_{\rho\sigma}-g_{\rho\sigma}g^{\mu\nu}\Gamma^{\gamma}_{\mu\nu}\Big{]}\,.$
(5.2.3)

The term in the first square bracket is the Ricci tensor $R_{\rho\sigma}$. The
other connection and partial derivative terms are simply the covariant
derivatives acting on $\partial f/\partial R$, which, unlike $\mathbf{G}$ and
$\mathbf{B}$, is a true coordinate scalar.

To see the equivalence more clearly, take the $f(R)$ field equations (1.2.3)

$\frac{\partial f}{\partial
R}R_{\rho\sigma}+\big{[}g_{\rho\sigma}\Box-\nabla_{\rho}\nabla_{\sigma}\big{]}\frac{\partial
f}{\partial R}-\frac{1}{2}g_{\rho\sigma}f(R)\,,$ (5.2.4)

and expand the derivative terms

$\displaystyle\big{[}g_{\rho\sigma}\Box-\nabla_{\rho}\nabla_{\sigma}\big{]}f_{,R}$
$\displaystyle=g_{\rho\sigma}g^{\mu\nu}\big{(}\partial_{\mu}\nabla_{\nu}-\Gamma^{\gamma}_{\mu\nu}\nabla_{\gamma}\big{)}f_{,R}-\nabla_{\rho}\partial_{\sigma}f_{,R}$
(5.2.5)
$\displaystyle=\Big{(}g_{\rho\sigma}g^{\mu\nu}\partial_{\mu}\partial_{\nu}-g_{\rho\sigma}g^{\mu\nu}\Gamma^{\gamma}_{\mu\nu}\partial_{\gamma}-\partial_{\rho}\partial_{\sigma}+\Gamma^{\gamma}_{\rho\sigma}\partial_{\gamma}\Big{)}f_{,R}\,.$
(5.2.6)

Comparing (5.2.3) with the expressions above shows that these are indeed the
same, such that $f(\mathbf{G}+\mathbf{B})$ yields the standard $f(R)$ field
equations. These are generally fourth-order, due to the second-order terms in
the Ricci scalar (which are confined to the boundary in GR) contributing to
the equations of motion when $f(R)$ is a non-linear function.

In Fig. 5.1 we illustrate the relationship between $f(R)$ gravity and our
$f(\mathbf{G},\mathbf{B})$ theories. The most general model is
$f(\mathbf{G},\mathbf{B})$, which reduces to $f(R)$ by requiring covariance.
This is represented by the vanishing of the conservation equation (5.1.15).
Alternatively, the most general second-order theory is $f(\mathbf{G})$
gravity. Requiring the field equations to be both covariant and second-order
leads back to General Relativity.

${f(\mathbf{G},\mathbf{B})}$${f(\mathbf{G})}$${f(\mathbf{G}+\mathbf{B})}$GRsecond-
orderdiff. invariantsecond-orderdiff. invariant Figure 5.1: Relationship
between the modified theories of gravity and General Relativity.

####  5.2.2 Teleparallel $f(T)$ and $f(Q)$ gravity

The modified teleparallel theories $f(T)$ and $f(Q)$ gravity take a very
similar form to our $f(\mathbf{G})$ theories [222]. They too are both second-
order in derivatives of the metric or tetrad, though they are based in
different geometric settings. In Section 4.3 we showed the equivalence of the
Einstein actions and the teleparallel equivalents of General Relativity, TEGR
and STEGR. Here we extend this discussion to their modified versions as well.
One may then wonder if the equivalence between $f(\mathbf{G})$ and $f(Q)$, and
between $f(\mathfrak{G})$ and $f(T)$, is broken for these modified theories.
Such an equivalence between the Ricci scalar and the bulk terms is clearly
broken when transitioning to non-linear modifications. However, quite
remarkably, the theories are still completely equivalent.

The variations with respect to the inertial connections for the teleparallel
theories lead to non-trivial equations, and the fields that parameterise the
connection become dynamical. Evidence for this can be seen in the increased
number of degrees of freedom777We note that this issue is not fully settled,
but for $f(Q)$ gravity it is clear there are at least two additional
propagating degrees of freedom [390]. in $f(T)$ or $f(Q)$ gravity [391, 392].
But these equations are exactly the same as the ones imposed by requiring the
non-covariant theories $f(\mathbf{G})$ and $f(\mathfrak{G})$ to be invariant
under diffeomorphisms and local Lorentz transformations. And, as we saw in the
previous sections, these constraints were required by mathematical
consistency. Hence, the modified theories are in fact dynamically equivalent,
which we show below.

##### Metric teleparallel gravity

The action of modified metric teleparallel gravity is given by

$S_{f(T)}[e,...]=-\frac{1}{2\kappa}\int f(T)ed^{4}x\,.$ (5.2.7)

We now have the choice of what additional fields to consider as dynamical,
indicated by the ellipsis in $S_{f(T)}[e,...]$. Recall that in TEGR, the
purely inertial spin connection did not play a physical role, and the fields
that parameterised the connection $\Lambda$ were not dynamical. In fact, they
could be separated off in a term that we labelled the ‘purely gauge boundary
term’, see (3.3.18). For non-linear modifications this is no longer the case.

For a consistent variational procedure in the modified action (5.2.7), we have
the following three choices: work in the fixed Weitzenböck gauge
$\omega(\Lambda)=0$; use the purely inertial spin connection parameterised by
the fields $\Lambda$; or vary with respect to an independent spin connection
$\omega$ with Lagrange multipliers enforcing vanishing curvature and non-
metricity. The former of these approaches breaks local Lorentz invariance,
which will be discussed below. The latter two methods are covariant but
introduce additional dynamical fields. These various approaches are compared
in detail in [336], so we refer to that work, as well as [146], for the
explicit calculations.

In the fixed Weitzenböck gauge, known as ‘pure $f(T)$ gravity’, the equations
of motion are

$f^{\prime}(T)\Big{[}G_{\rho\sigma}-\frac{1}{2}g_{\rho\sigma}T\Big{]}+2f^{\prime\prime}(T)\mathscr{S}_{\rho\sigma}{}^{\gamma}\partial_{\gamma}T+\frac{1}{2}g_{\rho\sigma}f(T)=\kappa\Theta_{\rho\sigma}\,,$
(5.2.8)

where $\mathscr{S}^{\mu\nu\lambda}$ is the teleparallel superpotential
(3.3.10) and $\Theta_{\rho\sigma}$ is the symmetric888We assume the matter
action to be a local Lorentz scalar and to only couple to the Levi-Civita spin
connection, hence $S_{\textrm{M}}=S_{\textrm{M}}[e,\varPhi^{A}]$ and
$\Theta_{[\rho\sigma]}=0$. See the discussion at the end of Chapter 2.
(tetradic) energy-momentum tensor (2.3.41), see [393, 394]. Importantly, the
antisymmetric part of the field equations do not vanish in general, leading to
an additional set of equations that must be satisfied

$f^{\prime\prime}(T)\mathscr{S}_{[\rho\sigma]}{}^{\gamma}\partial_{\gamma}T=0\,.$
(5.2.9)

This equation is a consequence of the non-invariance under local Lorentz
transformations. For TEGR, $f(T)=T$ and $f^{\prime\prime}(T)=0$ and this
equation vanishes identically, representing the local Lorentz
(pseudo-)invariance. A similar situation arises in the $f(T,B_{T})$ theories,
where the teleparallel boundary term is also included. The combination
$f(-T+B_{T})$ leads to the locally Lorentz invariant $f(R)$ theories, and the
antisymmetric field equation also vanishes identically [375].

In the fixed, Weitzenböck framework, the torsion tensor is not covariant under
LLTs, and hence the field equations take different values in different choices
of frame $\\{\mathbf{e}_{a}\\}$. Frames in which the antisymmetric equation
(5.2.9) forces $f^{\prime\prime}(T)=0$ were known as ‘bad tetrads’, because
this trivially leads back to GR. On the other hand, if the equation is
satisfied for all $f$, then these were known as ‘good tetrads’ [169]. A
typical example is when the torsion scalar is a constant. The similarities
with the choices of coordinates that lead to $f^{\prime\prime}(\mathbf{G})=0$
in our theories is immediately apparent.

Let us now consider the approach with the inclusion of the purely inertial
spin connection. The action then takes the form $f(T(e,\omega(\Lambda)))$, and
is invariant under local Lorentz transformations performed on the tetrad and
spin connection simultaneously. The explicit form of the field equations can
be found in [170], but the conclusion is that the tetradic field equations are
simply (5.2.8) in an arbitrary gauge. Transforming to the Weitzenböck gauge
$\omega=0$ exactly gives the pure $f(T)$ field equations.

The equations of motion obtained from the variations of the spin connection
end up being identical to the skew-symmetric part (5.2.9), again in an
arbitrary gauge. Therefore the covariant $f(T)$ equations are simply
equivalent to the Weitzenböck ones with gauge-invariance restored.
Importantly, this means that all possible solutions of the theory in either
approach are the same. Any solution of the covariant $f(T)$ equations can be
transformed via a suitable local Lorentz transformation to the Weitzenböck
gauge, and hence exists in the pure approach. For more discussions on this
topic, see [166, 164, 152, 338].

There have been many studies on $f(T)$ gravity, working with a plethora of
different models, see the reviews [162, 145] and references therein. A recent
focus has been on constraining such models with cosmological observations, for
example [395, 396, 397]. Notably, the EFT techniques mentioned in Sec. 1.1.2
have been applied to $f(T)$ gravity in this cosmological context [398, 399,
400]. This appears to be a promising avenue to study these theories in a
model-independent way. Moreover, problems such as strong coupling have been
approached using an EFT perspective [389].

Moving on to the comparison with our theories, recall that $\mathfrak{G}$ and
$T(e,0)$ were equivalent (4.3.4), modulo sign. It follows immediately that
their modifications are also equivalent, and so $f(\mathfrak{G})$ gravity is
simply pure $f(T)$ gravity999Again, we assume that matter couples only to the
Levi-Civita connection, as is standard in teleparallel gravity. In our
$f(\mathfrak{G})$ theories, this is the only natural coupling.. All of the
previous discussions on the equivalence of the covariant vs pure approach now
apply to the non-covariant $f(\mathfrak{G})$ models. This seems quite natural,
as $f(\mathfrak{G})$ gravity is not invariant under local Lorentz
transformations, while the presence of additional degrees of freedom can be
seen explicitly in its covariant form. It is the $\Lambda$ matrices
parameterising the purely inertial spin connection $\omega(\Lambda)$ that
restore the covariance of the theory, and these fields become dynamical when
we consider non-linear modifications. In the next part, when we study
symmetric teleparallel gravity, we will show this equivalence explicitly.

Lastly, we note all of these discussions extend to the modified theories
including boundary terms, namely $f(\mathfrak{G},\mathfrak{B})$ and the
teleparallel $f(T,B_{T})$ models [375]. The equivalence follows from the
discussion of the boundary terms in Sec. 4.2.3, and the relation
$B_{T}=\mathfrak{B}+b_{\Lambda}$. In Sec. 5.2.3 we look closer at these
possible extensions.

##### Symmetric teleparallel gravity

Moving on to modified symmetric teleparallel gravity, the action is given
by101010Note that in some of the $f(Q)$ literature, the non-metricity scalar
is defined with an overall minus sign different from ours. To match up with
these formulations, appropriate field redefinitions are needed. Similar care
must also be taken with non-equivalent definitions of the non-metricity
tensor. This will be discussed below when comparing with $f(\mathbf{G})$
gravity. [158]

$S_{f(Q)}[g,...]=-\frac{1}{2\kappa}\int f(Q)\sqrt{-g}d^{4}x\,.$ (5.2.10)

Again, we must decide which fields to consider independent and dynamical.
Fixing the coincident gauge breaks the diffeomorphism invariance of the action
and immediately gives the $f(\mathbf{G})$ action (5.1.18). In this case, only
the metric tensor enters the action. Alternatively, a covariant formulation
can be used by considering the fields $\xi$ that parameterise the symmetric
teleparallel connection (3.3.21), or using the Lagrange multiplier approach
with an affine connection $\bar{\Gamma}^{\lambda}_{\mu\nu}$. Either of these
latter two approaches lead to the same equations of motion, and we will show
that they are dynamically equivalent to the gauge-fixed formulation shortly.

The field equations from varying the action $S_{f(Q)}[g,\xi]$ with respect to
the metric are

$f^{\prime}(Q)\big{(}G_{\rho\sigma}-\frac{1}{2}g_{\rho\sigma}Q\big{)}-2f^{\prime\prime}(Q)\mathscr{P}^{\lambda}{}_{\rho\sigma}\partial_{\lambda}Q+\frac{1}{2}g_{\rho\sigma}f(Q)=\kappa
T_{\rho\sigma}\,,$ (5.2.11)

see [202, 401, 402]. The non-metricity conjugate, or superpotential, is
defined as

$\mathscr{P}^{\lambda\rho\sigma}:=\frac{1}{2}\frac{\partial Q}{\partial
Q_{\lambda\rho\sigma}}=\frac{1}{4}Q^{\lambda\rho\sigma}-\frac{1}{2}Q^{(\rho\sigma)\lambda}-\frac{1}{4}g^{\rho\sigma}(Q^{\lambda\mu}{}_{\mu}-Q^{\mu}{}_{\mu}{}^{\lambda})+\frac{1}{4}g^{\lambda(\rho}Q^{\sigma)\mu}{}_{\mu}\,,$
(5.2.12)

from which we can obtain the non-metricity scalar via the contraction
$\mathscr{P}^{\mu\nu\lambda}Q_{\mu\nu\lambda}=Q$. The connection equation of
motion is given by

$\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\big{(}\sqrt{-g}\mathscr{P}^{\mu\nu}{}_{\lambda}f^{\prime}(Q)\big{)}=0\,,$
(5.2.13)

where $\bar{\nabla}_{\mu}$ is the covariant derivative associated with the
symmetric teleparallel connection. Note that we have assumed that
hypermomentum vanishes, as is standard in these teleparallel models. For
further details on these equations of motion we refer to [271, 202].

Let us now show that these equations in the coincident gauge are exactly
equivalent to our $f(\mathbf{G})$ ones. Firstly, we of course have that
$\mathbf{G}=-Q$. The superpotentials of both theories are also easily shown to
coincide

$\displaystyle\mathscr{P}_{\lambda\rho\sigma}(g,0)$
$\displaystyle=\frac{1}{4}\Big{(}-\partial_{\lambda}g_{\rho\sigma}+2\partial_{(\rho}g_{\sigma)\lambda}+2g_{\rho\sigma}g^{\alpha\beta}\partial_{[\lambda}g_{\alpha]\beta}-g^{\alpha\beta}g_{\lambda(\rho}\partial_{\sigma)}g_{\alpha\beta}\Big{)}$
$\displaystyle=\frac{1}{4}\Big{(}2g_{\gamma\lambda}\Gamma_{\rho\sigma}^{\gamma}-2g_{\lambda(\rho}\Gamma_{\sigma)\gamma}^{\gamma}+g_{\rho\sigma}\Gamma_{\lambda\gamma}^{\gamma}-g_{\rho\sigma}g_{\gamma\lambda}g^{\alpha\beta}\Gamma_{\alpha\beta}^{\gamma}\Big{)}$
$\displaystyle=\frac{1}{4}E_{\rho\sigma\lambda}\,,$ (5.2.14)

where we have used that the non-metricity tensor in the coincident gauge is
simply the partial derivative of the metric (3.3.2). We have then expressed
the derivatives of the metric in terms of the Levi-Civita Christoffel symbols,
which gives an equivalent expression to the superpotential
$E^{\rho\sigma\lambda}$ (4.1.1).

Lastly, we must make appropriate field redefinitions such that the original
$f(Q)$ action (5.2.10) aligns with our $f(\mathbf{G})$ action (5.1.18). These
redefinitions111111Alternatively, we can begin with the $f(\mathbf{G})$
equations and make the transformation
$f(\mathbf{G})\rightarrow-f(-\mathbf{G})$. Note again that there are different
definitions of $Q_{\mu\nu\lambda}$, $Q$ and $P^{mu\nu\lambda}$ in the
literature, see footnote 10. Conveniently, in works where the non-metricity
scalar differs from ours by a minus sign, the equivalence with our
$f(\mathbf{G})$ equations is present immediately without any redefinitions
needed, see equation (18) of [401]. are
$f(Q)\rightarrow-f(-Q)=-f(\mathbf{G})$, from which it follows that
$f^{\prime}(Q)\rightarrow f^{\prime}(\mathbf{G})$ and
$f^{\prime\prime}(Q)\rightarrow-f^{\prime\prime}(\mathbf{G})$. Along with
equation (5.2.2), we see that the coincident gauge $f(Q)$ metric field
equations are the same as the $f(\mathbf{G})$ ones (5.1.19).

For the connection equation of motion (5.2.13), in the coincident gauge we
have $\bar{\nabla}_{\mu}\rightarrow\partial_{\mu}$. Using the equivalence of
the superpotentials (5.2.2), we obtain

$\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\big{(}\sqrt{-g}\mathscr{P}^{\mu\nu}{}_{\lambda}f^{\prime}(Q)\big{)}=\frac{1}{4}\partial_{\mu}\partial_{\nu}\big{(}\sqrt{-g}E_{\lambda}{}^{\mu\nu}f^{\prime}(\mathbf{G})\big{)}=0\,.$
(5.2.15)

With the identity $E^{\lambda(\mu\nu)}=M^{\mu\nu\lambda}$ (4.1.22), we see the
above equation is exactly our constraint coming from diffeomorphism invariance
(5.1.20). Further details on the relationship between the connection
variations and infinitesimal diffeomorphisms can be found in [202, 271].

Just like in the $f(T)$ case, the fields $\xi^{\mu}$ that parameterise the
connection act to restore the invariance of the theory: their equations of
motion are exactly those found from the symmetry transformations of the gauge-
fixed theories, as we expected. Performing the covariantisation á la
Stueckelberg on the non-covariant $f(\mathbf{G})$ theories leads uniquely to
$f(Q)$ gravity. This has also been shown by Milgrom in [218].

The discussion can again be extended to arbitrary functions of the boundary
terms as well. In our work [222] (Sec. 5.1), we gave the first formulation of
the $f(\mathbf{G},\mathbf{B})$ theories along with their equations of motion.
More recently, the extended $f(Q,B_{Q})$ modified theories have been
considered [403, 404]. However, it is clear that the dynamics, and hence the
theories themselves, are completely equivalent to our non-covariant theories.
The symmetric teleparallel $f(Q,B_{Q})$ theories are simply the covariantised
versions of our $f(\mathbf{G},\mathbf{B})$ models. Likewise, it is not
difficult to show that fixing the coincident gauge for the $f(Q,B_{Q})$
theories leads exactly to $f(\mathbf{G},\mathbf{B})$ gravity.

As a final observation, recall our brief discussion of Hořava gravity in Sec.
1.2.4. A further comparison can be made between the situation there and the
$f(\mathbf{G})$ and covariant $f(Q)$ theories. In Hořava gravity, when working
in the (gauge-fixed) ADM formalism, one finds the dynamical equations do not
in fact provide enough information to determine the (dynamical) free lapse
function $N$. This is then fixed by imposing the Hamiltonian constraint [208].
However, on certain backgrounds the Hamiltonian constraint vanishes and leaves
the lapse undetermined. In other words, it exhibits some residual
reparameterisation invariance. When covariance is restored via the
Stueckelberg trick, this constraint equation becomes the equation for the
Stueckelberg fields. On these backgrounds, this equation constrains an
otherwise propagating degree of freedom, and so the additional mode becomes
non-dynamical. This is then related to the aforementioned strong-coupling
issue [208].

The same phenomena occur in $f(\mathbf{G})$ and $f(Q)$. The diffeomorphism
conservation constraint
$\partial_{\mu}\partial_{\nu}\big{(}\sqrt{-g}M^{\mu\nu}{}_{\lambda}f^{\prime}(\mathbf{G})\big{)}=0$
becomes the equation of motion for the Stueckelberg fields (the $\xi$’s
parameterising the symmetric teleparallel connection). Similarly, for some
backgrounds, like FLRW in Cartesian coordinates (5.1.27), the theory retains a
residual reparameterisation invariance [147, 202]. This allows us to
generically set the lapse to one $N(t)=1$. The authors of [147, 202] confirm
that in the minisuperspace action, the lapse indeed becomes non-dynamical, and
the theory remains time-reparameterisation invariant.

In comparison to other symmetry-breaking models, the situation in the
$f(\mathbf{G})$, $f(Q)$, $f(\mathfrak{G})$ and $f(T)$ modified theories is
even more complicated, with the full diffeomorphism or local Lorentz
symmetries effectively being broken. In $f(T)$ gravity, this has resulted in
many investigations into the so-called ‘remnant symmetry’ [405, 406, 407,
408]. This is the residual local Lorentz symmetry of the tetrad about a given
solution. The transformation properties of $\mathfrak{G}$, given in (4.2.27),
are no doubt related to this residual symmetry. However, we will not explore
this issue further here.

##### Matter and symmetry

In the modified teleparallel theories, we have assumed that the matter action
does not depend on the teleparallel connection
$S_{\textrm{M}}[e,\varPhi^{A}]$. This is the usual approach when working with
these theories, but some further comments on this topic are warranted. In the
metric teleparallel framework, it is well known that the use of the
Weitzenböck connection is inconsistent with Dirac spinor matter [409]. This is
because the Dirac action couples to the connection, but the Weitzenböck
connection is totally inertial and can be set to zero via a frame
transformation. In this case, one finds that the matter energy-momentum tensor
is not necessarily conserved, leading to inconsistencies. To solve this issue,
matter must instead couple to the Levi-Civita connection of GR rather than the
teleparallel one, which is slightly at odds with the standard gauge theory
procedure.

On the other hand, it has been pointed out that in the symmetric teleparallel
geometry, the Dirac Lagrangian only couples to the Levi-Civita part of the
connection [271, 341]. Therefore none of the problems of the metric
teleparallel framework are present in the symmetric case. Moreover, the
natural coupling procedure of promoting partial derivatives to covariant
derivatives is less unambiguous than in theories with torsion. For further
details on matter couplings, see [410, 411, 271, 341].

Another important topic is spacetime symmetries. In teleparallel theories
beyond the equivalents of General Relativity, one needs to also take account
of the additional geometric structure. This means ensuring that the metric and
affine connection pair, or tetrad and spin connection pair, all respect the
symmetries of the spacetime. In metric teleparallel gravity, this can be
achieved by generalising isometries of the metric to affine frame symmetries,
where the Lie derivatives of the tetrad $\mathcal{L}_{\xi}\mathbf{e}_{a}$ and
spin connection $\mathcal{L}_{\xi}\omega_{ab}$ must both vanish in a given
frame [276]. Alternatively, a frame independent notion of spacetime symmetry
can be adapted from Cartan geometries, which has been employed in [277, 275].
The Weitzenböck gauge can then also be enforced, see for example [412]. This
makes the method applicable to the pure $f(T)$ formulation as well, and hence
also $f(\mathfrak{G})$ gravity.

In symmetric teleparallel gravity, one requires the Lie derivative of both the
metric (2.3.1) and affine connection (2.3.14) to vanish [371]. In this natural
coordinate basis, these concepts are easier to implement. This is a purely
covariant approach, where the affine connection is not assumed to vanish.
However, we are also still free to transform our system to the coincident
gauge via a suitable coordinate transformation. For the spatially flat FLRW
metric (5.1.27), this happens to take the simple form of Cartesian
coordinates. For spatially curved cosmologies it has been noted that the
coincident gauge is not ideal, due to the inability to retain the usual
spherical coordinate form of the metric tensor [401, 371]. However, coincident
gauge coordinates can still be found and used, leading to the exact same
equations of motion, albeit in a more complicated form. This is the choice of
coordinates that would be appropriate for the gauge-fixed $f(\mathbf{G})$
theories.

####  5.2.3 A unified framework

Putting together all of the modifications studied so far, we can construct a
unified framework that encompasses all of these theories. To do so, we make
use of the relationship between the metric bulk and boundary terms
$\mathbf{G}$, $\mathbf{B}$, and the tetradic bulk and boundary terms
$\mathfrak{G}$, $\mathfrak{B}$. These were given by

$\mathfrak{G}-\mathbf{G}=\mathbb{B}\,,\qquad\qquad\mathbf{B}-\mathfrak{B}=\mathbb{B}\,,$
(5.2.16)

with $\mathbb{B}$ defined in (4.2.36). We also use that the metrical
decomposed terms are equivalent to the gauge-fixed symmetric teleparallel
scalars, and the tetradic decomposed terms are equivalent to the gauge-fixed
metric teleparallel scalars. And as emphasised in the previous sections, the
covariant theories are dynamically and physically equivalent.

We then propose the following generalised modified theory in the purely
Riemannian framework, described by the action

$S_{\textrm{grav}}[g,e]=\frac{1}{2\kappa}\int
f\big{(}\mathbf{G}(g),\mathbb{B}(e),\mathbf{B}(g)\big{)}ed^{4}x\,,$ (5.2.17)

where the dynamical variables are the metric and tetrad121212We could have
written this just in terms of the tetrad, but we include the metric to
emphasise that $\mathbf{G}$ and $\mathbf{B}$ can be written just in terms of
the metric.. This can be seen as a modification and generalisation of the
mixed Einstein action in equation (4.2.44). Crucially, the most general
function $f(\mathbf{G},\mathbb{B},\mathbf{B})$ is neither locally Lorentz
invariant nor diffeomorphism invariant, and gives rise to fourth-order
equations of motion.

In Fig. 5.2 we give a representation of the different theories based on this
unified $f(\mathbf{G},\mathbb{B},\mathbf{B})$ action. Requiring Lorentz
invariance leads to the metric modifications $f(\mathbf{G},\mathbf{B})$, which
can be expressed in terms of the natural coordinate basis. On the other hand,
requiring diffeomorphism invariance leads to the tetradic theories
$f(\mathbf{G}+\mathbb{B},\mathbf{B}-\mathbb{B})=f(\mathfrak{G},\mathfrak{B})$
in the orthonormal basis. We retrieve $f(R)$ gravity for
$f(\mathbf{G}+\mathbf{B})=f(\mathfrak{G}+\mathfrak{B})$, which is invariant
under both coordinate and local Lorentz transformations.

${f(\mathbf{G},\mathbb{B},\mathbf{B})}$${f(Q,B_{Q})}$${f(\mathbf{G},\mathbf{B})}$${f(R)}$${f(\mathfrak{G},\mathfrak{B})}$${f(T,B_{T})}$${f(Q)}$${f(\mathbf{G})}$GR${f(\mathfrak{G})}$${f(T)}$
Lorentz invar. (metrical) diff. invar. (tetradic) diff. invarLorentz invar2nd-
order2nd-order2nd-order2nd-order2nd-orderdiff. invar.Lorentz
invar.covariantise gauge fix $\omega=0$ covariantise gauge fix
$\bar{\Gamma}=0$ gauge fix $\bar{\Gamma}=0$ covariantisecovariantise gauge fix
$\omega=0$ Figure 5.2: Relationships between the Riemannian
$f(\mathbf{G},\mathbb{B},\mathbf{B})$ class of theories. The most general
setup is $f(\mathbf{G},\mathbb{B},\mathbf{B})$, from which we obtain
$f(\mathbf{G},\mathbf{B})$ or $f(\mathfrak{G},\mathfrak{B})$ in the metric and
tetradic representations respectively. The former is locally Lorentz invariant
while the latter is diffeomorphism invariant. Covariantisation of these models
leads to the modified symmetric and metric teleparallel theories. Considering
only the bulk terms leads to the second-order modifications. From
$f(\mathbf{G})$ and $f(\mathfrak{G})$ we obtain GR by requiring invariance
under diffeomorphisms and local Lorentz transformations.

As we have discussed, covariantisation of these theories gives rise to the
teleparallel modifications. Specifically, we retrieve $f(Q,B_{Q})$ gravity
from the $f(\mathbf{G},\mathbf{B})$ theories, and $f(T,B_{T})$ from the
$f(\mathfrak{G},\mathfrak{B})$ theories. Despite the teleparallel theories
being fully covariant, the $f(R)$ limit still represents a special case
possessing an ‘enhanced’ symmetry: invariance under coordinate transformations
of the metric tensor and local Lorentz transformations of the tetrad alone
(without considering the affine connection). This is also special because the
degrees of freedom in $f(R)$ gravity are fewer than in the modified
teleparallel theories [391]. This can be understood as the (dynamical) fields
parameterising the teleparallel connection dropping out from the action
entirely in the $f(R)$ limit. We then are led to GR in the second-order, fully
invariant limit, as expected.

On the other hand, if one wanted to work in the other direction, one can begin
in the teleparallel framework and simply fix the gauge using the formulae
developed in this work. This could be accomplished by introducing Lagrange
multipliers to the teleparallel actions. We will briefly comment on this
approach in the next section, when working in the metric-affine geometries.

We should also point out the similar modifications in general teleparallel
gravity [160, 333], described by $f(\mathring{G})$. There, $\mathring{G}$
represents the quadratic contortion terms of a teleparallel geometry with both
torsion and non-metricity. This is the generalisation of the teleparallel
action given in (3.3.8). Clearly this is closely related to our non-covariant
theory here. However, the contractions of the contortion tensor include
torsion and non-metricity cross terms, which are not immediately present in
our $f(\mathbf{G},\mathbb{B},\mathbf{B})$ model. The simplest realisation of
this theory into our non-covariant scheme would be to work in a general
anholonomic frame which is not orthonormal [373]. Then, the gauge-fixed
theories would correspond to the vanishing affine connection in the mixed
basis. However, this is beyond the scope of this thesis.

This unified framework also serves as a useful tool to compare and contrast
the differences between the metric teleparallel and the symmetric teleparallel
theories. It is curious that the equations of motion for these theories often
coincide, leading to equivalent solutions131313Of course, when perturbations
are taken into account, metric vs tetrad perturbations may lead to
differences.. The case of FLRW cosmology is well-known and well-studied, and
it can be attributed to the high amount of symmetry that these spacetimes
possess. The same also occurs in spherically symmetric spacetimes, which we
first pointed out in [222], and has later been studied in more detail [413].

Clearly a necessary condition for the equivalence of these theories is for the
boundary term $\mathbb{B}=\mathfrak{G}-\mathbf{G}$ to vanish. As we noted,
$\mathbb{B}$ is neither a coordinate scalar nor a Lorentz scalar, hence
depends on both the coordinates and frames. One would hope that studying a
theory of the form $f(\mathbf{G}+\alpha\mathbb{B})$, and using the additional
constraints that come from diffeomorphism invariance and local Lorentz
invariance, would be useful in further investigating the relationship between
the $f(\mathbf{G})$ and $f(\mathfrak{G})$ theories. It would be interesting to
prove (or disprove) whether a background solution of $f(Q)$ gravity is always
a solution of $f(T)$ gravity, and vice-versa. Our gauge-fixed theories seem to
be a useful approach for these types of questions, where we only have the
metric and tetrad fields to deal with.

###  5.3 Metric-affine generalisations

Here we return to the metric-affine setting with an independent metric tensor
and affine connection. This allows us to make contact with the other metric-
affine theories directly, such as Einstein-Cartan theory. In particular,
beginning with the affine decomposition studied in Sec. 4.3, we will assume
that the total curvature $\bar{R}_{\mu\nu\gamma}{}^{\lambda}$ is non-
vanishing. To set either non-metricity or torsion to zero, we will use the
Lagrange multiplier method employed throughout the previous chapters.

Setting both torsion and non-metricity to zero will return to the Levi-Civita
framework. But we will see that in some cases, such as when non-linear
functions of boundary terms are considered, the modified theories do not
reduce to their metric counterparts studied in the previous sections. This
will be explained in more detail below.

####  5.3.1 $f(\bar{\mathbf{G}},\bar{\mathbf{B}})$ gravity

Let us now study the modifications of the metric-affine decomposition of the
Ricci scalar $\bar{R}=\bar{\mathbf{G}}+\bar{\mathbf{B}}$, where we remind the
reader that $\bar{\mathbf{G}}$ and $\bar{\mathbf{B}}$ are defined as

$\displaystyle\bar{\mathbf{G}}$
$\displaystyle:=g^{\mu\lambda}\big{(}\bar{\Gamma}^{\kappa}_{\kappa\rho}\bar{\Gamma}^{\rho}_{\mu\lambda}-\bar{\Gamma}^{\kappa}_{\mu\rho}\bar{\Gamma}^{\rho}_{\kappa\lambda}\big{)}+\big{(}\bar{\Gamma}^{\mu}_{\mu\lambda}\delta^{\nu}_{\kappa}-\bar{\Gamma}^{\nu}_{\kappa\lambda}\big{)}\big{(}\partial_{\nu}g^{\kappa\lambda}-\frac{1}{2}g_{\alpha\beta}g^{\kappa\lambda}\partial_{\nu}g^{\alpha\beta}\big{)}\,,$
$\displaystyle\mathbf{\bar{B}}$
$\displaystyle:=\frac{1}{\sqrt{-g}}\partial_{\kappa}\big{(}\sqrt{-g}(g^{\mu\lambda}\bar{\Gamma}^{\kappa}_{\mu\lambda}-g^{\kappa\lambda}\bar{\Gamma}^{\mu}_{\mu\lambda})\big{)}\,.$

Recall that all terms with a bar now refer to their metric-affine versions,
and we work in the natural coordinate basis.

We propose the modified gravitational action to be an arbitrary function of
the affine bulk and boundary terms [173]

$\displaystyle S_{\textrm{ grav}}[g,\bar{\Gamma}]=\frac{1}{2\kappa}\int
f(\bar{\mathbf{G}},\bar{\mathbf{B}})\sqrt{-g}\,d^{4}x\,.$ (5.3.1)

Variations with respect to the independent metric and connection lead to

$\delta S_{\textrm{grav}}=\frac{1}{2\kappa}\int\biggl{\\{}\delta
g^{\mu\nu}\Big{[}-\frac{1}{2}g_{\mu\nu}f+f_{,\bar{\mathbf{G}}}\big{(}\bar{G}_{\mu\nu}+\frac{1}{2}g_{\mu\nu}\bar{\mathbf{G}}\big{)}+\frac{1}{2}\bar{E}_{\mu\nu}{}^{\lambda}\partial_{\lambda}f_{,\bar{\mathbf{G}}}\\\
+\frac{1}{2}g_{\mu\nu}f_{,\bar{\mathbf{B}}}\bar{\mathbf{B}}-\frac{1}{2}\bar{E}_{\mu\nu}{}^{\lambda}\partial_{\lambda}f_{,\bar{\mathbf{B}}}\Big{]}+\delta\bar{\Gamma}{}^{\lambda}_{\mu\nu}\Big{[}P^{\mu\nu}{}_{\lambda}f_{,\bar{\mathbf{G}}}+2\partial_{\sigma}f_{,\bar{\mathbf{B}}}\delta_{\lambda}^{[\mu}g^{\sigma]\nu}\Big{]}\biggr{\\}}\sqrt{-g}\,d^{4}x\,,$
(5.3.2)

where the superpotential $\bar{E}_{\mu\nu}{}^{\lambda}$ is defined in
(4.4.13). For the derivation of the field equations, see Appendix C.2.2. Let
us also include the matter action
$S_{\textrm{M}}[g,\bar{\Gamma},\varPhi^{A}]$, which we assume to be minimally
coupled to gravity. The full field equations are

$-\frac{1}{2}g_{\mu\nu}f+f_{,\bar{\mathbf{G}}}\big{(}\bar{G}_{(\mu\nu)}+\frac{1}{2}g_{\mu\nu}\bar{\mathbf{G}}\big{)}+\frac{1}{2}\bar{E}_{(\mu\nu)}{}^{\lambda}\partial_{\lambda}(f_{,\bar{\mathbf{G}}}-f_{,\bar{\mathbf{B}}})+\frac{1}{2}g_{\mu\nu}f_{,\bar{\mathbf{B}}}\bar{\mathbf{B}}=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}\,,$
(5.3.3)
$P^{\mu\nu}{}_{\lambda}f_{,\bar{\mathbf{G}}}+2\partial_{\rho}f_{,\bar{\mathbf{B}}}\delta_{\lambda}^{[\mu}g^{\rho]\nu}=2\kappa\Delta^{\mu\nu}{}_{\lambda}\,,$
(5.3.4)

where the metric (Hilbert) energy-momentum tensor is defined in (3.2.1) and
the hypermomentum defined in (3.2.3).

In the case that the action depends only on the bulk term
$f(\bar{\mathbf{G}})$, all $f_{,\bar{\mathbf{B}}}$ terms vanish. Then, the
connection equation is proportional to Palatini tensor
$P^{\mu\nu}{}_{\lambda}$ (4.4.17). Recall that $\bar{\mathbf{G}}$ is also
algebraic in torsion and non-metricity (4.4.10), therefore in the
$f(\mathbf{G})$ theories the connection field equation is only algebraic. It
is well known that in Einstein-Cartan theory, torsion does not propagate,
which implies the absence of torsional gravitational waves in vacuum. This is
in stark contrast to metric perturbations, which can propagate through vacuum
regions of spacetime. In order to obtain dynamical behaviour in either of
these geometric quantities, we must include the boundary term
$\bar{\mathbf{B}}$. It is precisely those boundary terms that contain the
derivatives of torsion and non-metricity.

It is interesting to note that making the Palatini variations of
$f(\bar{\mathbf{G}})$ and then choosing the Levi-Civita connection (consistent
with solving the connection field equation with vanishing hypermomentum) leads
back to the metric $f(\mathbf{G})$ gravity theory (5.1.19). However,
performing the same procedure with $f(\bar{\mathbf{G}},\bar{\mathbf{B}})$ does
not yield the metric $f(\mathbf{G},\mathbf{B})$ model. This is perhaps
unsurprising as the same situation occurs in Palatini $f(R)$ gravity [306, 92,
93]. The reason is clear from the form of (5.3.2): if one assumes the Levi-
Civita connection, the Palatini tensor $P^{\mu\nu}{}_{\lambda}$ vanishes
identically. Therefore none of the content of the $f(\mathbf{G})$ model is
lost in the connection equations of motion. In contrast, the fourth-order
terms coming from the $f(\ldots,\mathbf{B})$ field equation are lost in the
connection equation of motion. Specifically, the Levi-Civita connection
includes derivatives of the metric, which means an integration by parts would
be performed on the $\partial_{\lambda}f_{,\mathbf{B}}$ term, leading to the
fourth-order terms in the Levi-Civita variation of the
$f(\mathbf{G},\mathbf{B})$ action.

In order to make direct contact with the teleparallel theories, one would need
to consider a more general action than (5.3.1). Firstly, looking back at Fig.
5.2, one can imagine a slightly more general setting with
$f(\bar{\mathbf{G}},\mathbb{B},\bar{\mathbf{B}})$, where $\mathbb{B}$ relates
the coordinate basis and orthonormal basis bulk and boundary terms. This
extension follows readily from the work in this chapter and the previous one.
In the Levi-Civita limit, we would obtain all of Fig. 5.2. Decomposing the
affine quantities into their different geometric parts (4.4.56) yields an even
more general type of theory
$f(\mathbf{G},T,Q,\mathbf{C},\mathbb{B},\mathbf{B},B_{T},B_{Q})$, where again
$\mathbb{B}$ is used to relate the coordinate and orthonormal bases.
Teleparallel theories are then directly included in this action. Moreover, one
easily obtains the limit
$f(\mathbf{G}+T+Q+\mathbf{C},\mathbf{B}-B_{T}+B_{Q})=f(\bar{\mathbf{G}},\bar{\mathbf{B}})$.
In this section we will continue to work the
$f(\bar{\mathbf{G}},\bar{\mathbf{B}})$ theories, but this discussion should
show how more general theories can be constructed, and how these relate with
one another.

####  5.3.2 Diffeomorphism and projective invariance

Let us again return to the topic of symmetry and invariance for our modified
action. In Sec. 4.4.2 it was shown that the affine bulk and boundary terms
were projectively invariant under the transformation

$\bar{\Gamma}^{\gamma}_{\alpha\beta}\rightarrow\bar{\Gamma}^{\gamma}_{\alpha\beta}+\delta^{\gamma}_{\beta}P_{\alpha}\,.$

The same was the case for the affine Ricci scalar $\bar{R}$, and in Sec. 3.1.1
we showed that the consequence was that the Palatini tensor is trace-free
$P^{\mu\nu}{}_{\nu}=0$. Similarly, our modified action
$f(\bar{\mathbf{G}},\bar{\mathbf{B}})$ is then invariant under projective
transformations, and this is reflected in the connection equation also being
trace-free over the same indices. This is easily verified

$P^{\mu\nu}{}_{\nu}f_{,\bar{\mathbf{G}}}+2\partial_{\lambda}f_{,\bar{\mathbf{B}}}g^{\nu[\lambda}\delta^{\mu]}_{\nu}=\partial_{\lambda}f_{,\bar{\mathbf{B}}}g^{\mu\lambda}-\partial_{\lambda}f_{,\bar{\mathbf{B}}}g^{\lambda\mu}=0\,.$
(5.3.5)

The situation here is the same as in the standard metric-affine GR case, or in
modifications such as metric-affine $f(\bar{R})$ gravity. There is little else
to say on this topic, as these modifications neither help nor hinder any
issues associated with projective invariance.

Moving on to diffeomorphisms, the action transforms under an infinitesimal
symmetry transformation as

$\displaystyle\delta_{\xi}S_{\textrm{grav}}$
$\displaystyle=\frac{1}{2\kappa}\int\mathcal{L}_{\xi}\big{(}\sqrt{-g}f(\bar{\mathbf{G}},\bar{\mathbf{B}})\big{)}d^{4}x$
$\displaystyle=\frac{1}{2\kappa}\int\xi^{\lambda}\partial_{\mu}\partial_{\nu}\Big{[}\sqrt{-g}M^{\mu\nu}{}_{\lambda}\big{(}f_{,\bar{\mathbf{G}}}-f_{,\bar{\mathbf{B}}}\big{)}\Big{]}d^{4}x\,.$
(5.3.6)

Here we have performed the same calculations shown explicitly in Sec. 5.1.1,
and discarded any surface terms. We have also made use of the formulae for the
Lie derivatives of these affine terms, see equations (4.4.37)-(4.4.38). For
arbitrary $\xi$, this leaves the expression

$\displaystyle\partial_{\mu}\partial_{\nu}\Big{[}\sqrt{-g}M^{\mu\nu}{}_{\lambda}\big{(}f_{,\bar{\mathbf{G}}}-f_{,\bar{\mathbf{B}}}\big{)}\Big{]}\,.$
(5.3.7)

This expression could also be easily obtained by using the decompositions of
the affine objects

$\bar{\mathbf{G}}=\mathbf{G}+T+Q+C\,,\qquad\bar{\mathbf{B}}=\mathbf{B}-B_{T}+B_{Q}\,,$

which we gave in equations (4.4.57) and (4.4.58). All terms on the right-hand
side, except the bulk and boundary terms, are coordinate scalars. Hence, under
a diffeomorphism, it is only the $\mathbf{G}$ and $\mathbf{B}$ term that
transforms non-covariantly, which directly leads to the conservation equation
(5.3.7).

Unlike for an invariant scalar action, the above transformation need not
vanish identically. The expression is also non-covariant and depends on the
choice of coordinates, due to the non-tensorial nature of the bulk and
boundary terms. One immediately sees that for the choice of function
$f(\bar{\mathbf{G}}+\bar{\mathbf{B}})=f(\bar{R})$ the expression vanishes,
reflecting the diffeomorphism invariance of the Palatini $f(\bar{R})$
theories.

If we insist that our gravitational action remain invariant under the above
transformations, we obtain the conservation law

$\displaystyle\partial_{\mu}\partial_{\nu}\Big{[}\sqrt{-g}M^{\mu\nu}{}_{\lambda}\big{(}f_{,\bar{\mathbf{G}}}-f_{,\bar{\mathbf{B}}}\big{)}\Big{]}=0\,.$
(5.3.8)

We discussed this topic in detail in Sec. 5.1.1, so we refer to that section
for further analysis. If we also include minimally coupled matter in our total
action, which we remind the reader depends on the metric and the connection,
we have the following (on-shell) conservation law

$\frac{1}{2\kappa}\partial_{\mu}\partial_{\nu}\Big{[}\sqrt{-g}M^{\mu\nu}{}_{\lambda}\big{(}f_{,\bar{\mathbf{G}}}-f_{,\bar{\mathbf{B}}}\big{)}\Big{]}-\sqrt{-g}\nabla_{\mu}{}^{(\bar{\Gamma})}T^{\mu}{}_{\lambda}\\\
-\bar{\nabla}_{\nu}\bar{\nabla}_{\mu}(\sqrt{-g}\Delta^{\mu\nu}{}_{\lambda})-\sqrt{-g}\bar{R}_{\lambda\mu\nu}{}^{\rho}\Delta^{\mu\nu}{}_{\rho}-\bar{\nabla}_{\mu}(\sqrt{-g}\Delta^{\mu\nu}{}_{\rho})T^{\rho}{}_{\nu\lambda}=0\,.$
(5.3.9)

For a consistent variation principle of the total action
$S_{\textrm{total}}=S_{\textrm{grav}}+S_{\textrm{M}}$ we require the above
equation (5.3.9) to be satisfied. This is the most general case, as we do not
consider non-minimal couplings here. One sees that the equation (5.3.9)
reduces to the metric $f(\mathbf{G},\mathbf{B})$ conservation law given in
equation (5.1.14) when the connection is Levi-Civita and hypermomentum is
zero.

If the matter action $S_{\textrm{M}}$ is a coordinate scalar then we must
impose that the first of these equations (5.3.8) is satisfied. In that case,
one obtains the usual metric-affine matter-hypermomentum conservation laws

$\sqrt{-g}\nabla_{\mu}{}^{(\bar{\Gamma})}T^{\mu}{}_{\lambda}+\bar{\nabla}_{\nu}\bar{\nabla}_{\mu}(\sqrt{-g}\Delta^{\mu\nu}{}_{\lambda})+\sqrt{-g}\bar{R}_{\lambda\mu\nu}{}^{\rho}\Delta^{\mu\nu}{}_{\rho}\\\
+\bar{\nabla}_{\mu}(\sqrt{-g}\Delta^{\mu\nu}{}_{\rho})T^{\rho}{}_{\nu\lambda}=0\,,$
(5.3.10)

see for example [148]. Vanishing hypermomentum leads to the standard energy-
momentum conservation law $\nabla_{\mu}T^{\mu}{}_{\lambda}=0$. In Chapter 6,
when working in the cosmological setting, we will work with the standard
perfect fluid matter representations, where hypermomentum vanishes and the
matter action is a scalar. We will see that this forces the choice of
coordinates to satisfy the gravitational conservation constraint (5.3.8).

In our work [173], where we first proposed this theory, we conjectured on the
covariantisation of the $f(\bar{\mathbf{G}},\bar{\mathbf{B}})$ models. Our
main questions were the following: how would such a procedure be implemented,
and what would the resulting theory look like? We now feel to be in a better
position to answer such questions. In Section 4.3 we showed how the
Stueckelberg covariantisation could be applied to the Levi-Civita bulk and
boundary terms in both the coordinate and orthonormal bases. Importantly, this
procedure made no reference to geometry, even though it turned out that the
compensating Stueckelberg fields could be given a geometric interpretation. It
follows that the same procedure could be applied here, i.e., using the
decomposition of $\bar{\mathbf{G}}$ into $\mathbf{G}$ and the other affine
scalars (4.4.57), and then applying the Stueckelberg trick to the (metric)
bulk part $\mathbf{G}$. This appears to give a satisfactory answer to the
first question.

With regards to the second question, on the interpretation of the resulting
theory, we must note that because the connection is not flat
$\bar{R}_{\mu\nu\lambda}{}^{\gamma}\neq 0$, the Stueckelberg fields cannot be
related to the affine connection of the spacetime. In other words, we do not
obtain a neat geometric picture like we did when performing the
covariantisation of the metric $f(\mathbf{G},\mathbf{B})$ theories.

However, one answer to this second question is that they could be related to a
different, independent teleparallel connection. This seems reminiscent of the
modified theories with two independent connections [414], as well as the so-
called ‘hybrid-Palatini’ theories [415, 416]. In our case, the second
(auxiliary) connection will be in terms of the Stueckelberg fields, taking the
form of the ‘purely gauge’ symmetric teleparallel connection (3.3.21). This
was actually shown to be consistent by Milgrom (in the context of the bimetric
MOND theories [383]), where the connection associated with the auxiliary
metric was fixed to be the symmetric teleparallel one [218]. This led to the
covariant $f(Q)$ theories. A similar approach was used for the non-covariant
Einstein action by introducing an additional reference connection and metric
[359]. This would be an interesting avenue to explore further in the metric-
affine context, which we believe has not been looked at before.

####  5.3.3 Equivalence with metric-affine $f(R)$ gravity

Metric-affine $f(\bar{R})$ gravity is of course also included in our approach
and achieved by simply setting
$f(\bar{\mathbf{G}},\bar{\mathbf{B}})=f(\bar{\mathbf{G}}+\bar{\mathbf{B}})$.
This again follows from the decomposition of the affine Ricci scalar (4.4.9).
For the metric field equations (5.3.3), the $\bar{E}$ terms cancel to give

$\displaystyle-\frac{1}{2}g_{\mu\nu}f+f^{\prime}(R)\bigl{(}\bar{G}_{(\mu\nu)}+\frac{1}{2}g_{\mu\nu}\bar{\mathbf{G}}+\frac{1}{2}g_{\mu\nu}\bar{\mathbf{B}}\bigr{)}=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}\,,$
(5.3.11)

which can be written as

$\displaystyle
f^{\prime}(\bar{R})\bar{R}_{(\mu\nu)}-\frac{1}{2}g_{\mu\nu}f(\bar{R})=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}\,,$
(5.3.12)

where we have used the relations
$\bar{G}_{\mu\nu}+g_{\mu\nu}\bar{R}/2=\bar{R}_{\mu\nu}$ and
$\bar{\mathbf{G}}+\bar{\mathbf{B}}=\bar{R}$. The connection field equation is

$\displaystyle
P^{\mu\nu}{}_{\lambda}f^{\prime}(R)+2\partial_{\rho}f^{\prime}(\bar{R})\delta_{\lambda}^{[\mu}g^{\rho]\nu}=2\kappa\Delta^{\mu\nu}{}_{\lambda}\,.$
(5.3.13)

This can be written in a more familiar form by noting that

$\frac{2}{\sqrt{-g}}\bar{\nabla}_{\rho}(f^{\prime}(R)\sqrt{-g}\delta_{\lambda}^{[\mu}g^{\rho]\nu})=2\partial_{\rho}f^{\prime}(\bar{R})\delta_{\lambda}^{[\mu}g^{\rho]\nu}\\\
+f^{\prime}(\bar{R})\Big{(}\frac{1}{2}Q_{\lambda}{}^{\rho}{}_{\rho}g^{\mu\nu}-\frac{1}{2}Q^{\nu\rho}{}_{\rho}\delta^{\mu}_{\lambda}+Q_{\rho}{}^{\rho\nu}\delta^{\mu}_{\lambda}-Q_{\lambda}{}^{\mu\nu}\Big{)}\\\
=2\partial_{\rho}f^{\prime}(\bar{R})\delta_{\lambda}^{[\mu}g^{\rho]\nu}+f^{\prime}(\bar{R})P^{\mu\nu}{}_{\lambda}-f^{\prime}(\bar{R})(T^{\mu}{}_{\lambda}{}^{\nu}+g^{\mu\nu}T^{\rho}{}_{\rho\lambda}+\delta^{\mu}_{\lambda}T^{\rho\nu}{}_{\rho})\,,$
(5.3.14)

where in the first line we expanded out the covariant derivatives of the
metric in terms of the non-metricity tensor, and in the second line we
inserted the definition of $P^{\mu\nu}{}_{\lambda}$. Putting together (5.3.13)
and (5.3.14) we arrive at

$\displaystyle\frac{2}{\sqrt{-g}}\bar{\nabla}_{\rho}(f^{\prime}(R)\sqrt{-g}\delta_{\lambda}^{[\mu}g^{\rho]\nu})+f^{\prime}(\bar{R})(T^{\mu}{}_{\lambda}{}^{\nu}+g^{\mu\nu}T^{\rho}{}_{\rho\lambda}+\delta^{\mu}_{\lambda}T^{\rho\nu}{}_{\rho})=2\kappa\Delta^{\mu\nu}{}_{\lambda}\,.$
(5.3.15)

This is the familiar form of the $f(\bar{R})$ connection equation of motion
[306].

In the case that matter does not couple to the connection, à la Palatini, it
is known that this theory is equivalent to the scalar-tensor theory of Brans-
Dicke (1.2.10) with parameter $\omega=-3/2$ [156]. Interestingly, this differs
from its metric $f(R)$ counterpart with $\omega=0$. As discussed earlier, this
inequivalence is related to the boundary terms present in the affine Ricci
scalar.

Taking the trace of the metric equation (5.3.12) leads to the algebraic
relation $-2f(\bar{R})+f^{\prime}(\bar{R})\bar{R}=\kappa T$. This implies an
algebraic relation between the Ricci scalar $\bar{R}$ and the trace of the
energy-momentum tensor $T$ [156]. Contrast this with the
$f(\bar{\mathbf{G}},\bar{\mathbf{B}})$ equations (5.3.3), where the trace
leads to a dynamical equation as opposed to an algebraic one. This leads to a
number of interesting possibilities beyond what can be found in the metric-
affine $f(\bar{R})$ theories.

####  5.3.4 Modified Einstein-Cartan gravity

Einstein-Cartan theory is derived in the metric-affine setting by imposing
metric-compatibility of the connection. This was achieved in Sec. 3.2.2 by
using the Lagrange multiplier action (3.2.13). Once the Lagrange multiplier
was eliminated from the equations of motion, they read

$\displaystyle\tilde{G}_{(\mu\nu)}$
$\displaystyle=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}+\kappa(\tilde{\nabla}_{\rho}+T^{\sigma}{}_{\rho\sigma})\Delta^{\rho}{}_{(\mu\nu)}\,,$
$\displaystyle\tilde{P}_{\mu\nu\rho}$
$\displaystyle=2\kappa\Delta_{\mu[\nu\rho]}\,,$

see (3.2.17)-(3.2.18). Also recall that we were able to make use of geometric
identities in order to rewrite the first field equation in such a way that the
symmetry brackets could be removed

$\tilde{G}_{\mu\nu}=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}+\kappa(\tilde{\nabla}_{\rho}+T^{\sigma}{}_{\rho\sigma})\Delta^{\rho}{}_{\nu\mu}\,,$

Its symmetric part gives the original first field equation, whilst its skew
part coincides with the second equation. Moreover, the right-hand side was
shown to be equivalent with the canonical energy-momentum tensor (3.2.21).

Following our work [173], and using the same Lagrange multiplier term

$\displaystyle
S_{Q}=-\int\frac{1}{2}\lambda^{\mu\nu\rho}Q_{\mu\nu\rho}\sqrt{-g}\,d^{4}x=\int\frac{1}{2}\lambda^{\mu\nu\rho}\bar{\nabla}_{\mu}g_{\nu\rho}\sqrt{-g}\,d^{4}x\,.$

let us apply this to our modified theory. The complete modified metric field
equation (5.3.3) now reads

$-\frac{1}{2}g_{\mu\nu}f+f_{,\tilde{\mathbf{G}}}\big{(}\tilde{G}_{(\mu\nu)}+\frac{1}{2}g_{\mu\nu}\tilde{\mathbf{G}}\big{)}+\frac{1}{2}\tilde{E}_{(\mu\nu)}{}^{\sigma}\partial_{\sigma}f_{,\tilde{\mathbf{G}}}\\\
+\frac{1}{2}g_{\mu\nu}f_{,\tilde{\mathbf{B}}}\tilde{\mathbf{B}}-\frac{1}{2}\tilde{E}_{(\mu\nu)}{}^{\sigma}\partial_{\sigma}f_{,\tilde{\mathbf{B}}}+\kappa\tilde{\nabla}_{\sigma}\lambda^{\sigma}{}_{\mu\nu}+\kappa
T^{\rho}{}_{\sigma\rho}\lambda^{\sigma}{}_{\mu\nu}=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}\,,$
(5.3.16)

while the connection (5.3.4) and constraint equation are given by

$\displaystyle\tilde{P}^{\mu\nu}{}_{\rho}f_{,\tilde{\mathbf{G}}}+2\partial_{\sigma}f_{,\tilde{\mathbf{B}}}\delta_{\rho}^{[\mu}g^{\sigma]\nu}-2\kappa\lambda^{\mu\nu}{}_{\rho}$
$\displaystyle=2\kappa\Delta^{\mu\nu}{}_{\rho}\,,$ (5.3.17)
$\displaystyle\bar{\nabla}_{\mu}g_{\nu\rho}$ $\displaystyle=0\,.$ (5.3.18)

Note that we have already implemented the constraint equation to remove any
non-metricity terms which would otherwise be present in the field equations.
For this reason, affine quantities appearing in the field equations now are
denoted with a tilde, which implies non-metricity has been set to zero.

We will now follow the same method as shown for the standard Einstein-Cartan
theory of eliminating the Lagrange multipliers and writing the equations in a
unified way. We begin with the $f(\tilde{\mathbf{G}})$ case, without boundary
terms, and then look at the more general
$f(\tilde{\mathbf{G}},\tilde{\mathbf{B}})$ theories.

##### Modified Einstein-Cartan theory without boundary terms

Let us now return to the field equations (5.3.16)–(5.3.18) and assume
$f(\tilde{\mathbf{G}},\tilde{\mathbf{B}})=f(\tilde{\mathbf{G}})$ so that
$f_{,\tilde{\mathbf{B}}}=0$. Moreover, we assume that non-metricity vanishes
via the introduction of the Lagrange multiplier, in which case the field
equations simplify to

$-\frac{1}{2}g_{\mu\nu}f+f_{,\tilde{\mathbf{G}}}\big{(}\tilde{G}_{(\mu\nu)}+\frac{1}{2}g_{\mu\nu}\tilde{\mathbf{G}}\big{)}+\frac{1}{2}\tilde{E}_{(\mu\nu)}{}^{\rho}\partial_{\rho}f_{,\tilde{\mathbf{G}}}\\\
+\kappa\tilde{\nabla}_{\rho}\lambda^{\rho}{}_{\mu\nu}+\kappa
T^{\sigma}{}_{\rho\sigma}\lambda^{\rho}{}_{\mu\nu}=\kappa{}^{(\bar{\Gamma})}T_{\mu\nu}\,,$
(5.3.19)
$\tilde{P}^{\mu\nu}{}_{\rho}f_{,\tilde{\mathbf{G}}}-2\kappa\lambda^{\mu\nu}{}_{\rho}=2\kappa\Delta^{\mu\nu}{}_{\rho}\,.$
(5.3.20)

As before, the Palatini tensor is now skew-symmetric in the last pair of
indices and we find $-\lambda_{\mu\nu\rho}=\Delta_{\mu(\nu\rho)}$ and
$\tilde{P}_{\mu\nu\rho}f_{,\tilde{\mathbf{G}}}=2\kappa\Delta_{\mu[\nu\rho]}$.
Hence, we can write the first and second field equations without any Lagrange
multipliers in the form

$-\frac{1}{2}g_{\mu\nu}f+f_{,\tilde{\mathbf{G}}}\big{(}\tilde{G}_{(\mu\nu)}+\frac{1}{2}g_{\mu\nu}\tilde{\mathbf{G}}\big{)}+\frac{1}{2}\tilde{E}_{(\mu\nu)}{}^{\rho}\partial_{\rho}f_{,\tilde{\mathbf{G}}}\\\
=\kappa\Bigl{(}{}^{(\bar{\Gamma})}T_{\mu\nu}+(\tilde{\nabla}_{\rho}+T^{\sigma}{}_{\rho\sigma})\Delta^{\rho}{}_{(\nu\mu)}\Bigr{)}\,,$
(5.3.21)
$\tilde{P}_{\mu\nu\rho}f_{,\tilde{\mathbf{G}}}=2\kappa\Delta_{\mu[\nu\rho]}\,.$
(5.3.22)

The right-hand side of equation (5.3.21) now takes the familiar form of the
symmetrised canonical energy-momentum tensor.

We now wish to follow the same approach that we used in the standard Einstein-
Cartan theory, of removing the symmetry brackets and considering the full
equation instead. Taking the skew-symmetric part of (5.3.21), we find

$\displaystyle
f_{,\tilde{\mathbf{G}}}\tilde{G}_{[\mu\nu]}+\frac{1}{2}\tilde{E}_{[\mu\nu]}{}^{\rho}\partial_{\rho}f_{,\tilde{\mathbf{G}}}$
$\displaystyle=\kappa(\tilde{\nabla}_{\rho}+T^{\sigma}{}_{\rho\sigma})\Delta^{\rho}{}_{[\nu\mu]}$
(5.3.23)
$\displaystyle=\frac{1}{2}(\tilde{\nabla}_{\rho}+T^{\sigma}{}_{\rho\sigma})(\tilde{P}^{\rho}{}_{\nu\mu}f_{,\tilde{\mathbf{G}}})\,.$
(5.3.24)

After taking into account the geometric identity
$\tilde{G}_{[\mu\nu]}=(\tilde{\nabla}_{\rho}+T^{\sigma}{}_{\rho\sigma})\tilde{P}^{\rho}{}_{\nu\mu}/2$
we arrive at the interesting equation141414Note that here we have somewhat
abused notation by assuming that $f_{,\tilde{\mathbf{G}}}$ acts as a scalar
when acted on by the covariant derivative. This is of course not well defined.
However, our purpose here is only to rewrite the field equations in a more
suggestive form. We will use the original forms (5.3.21)-(5.3.22) when
performing calculations.

$\displaystyle\frac{1}{2}\tilde{E}_{[\mu\nu]}{}^{\rho}\partial_{\rho}f_{,\tilde{\mathbf{G}}}=\frac{1}{2}\tilde{P}^{\rho}{}_{\nu\mu}\partial_{\rho}f_{,\tilde{\mathbf{G}}}\,,$
(5.3.25)

which immediately gives
$\tilde{E}_{[\mu\nu]}{}^{\rho}=\tilde{P}^{\rho}{}_{\nu\mu}$. This equation can
be seen as determining the skew-symmetric part of
$\tilde{E}_{[\mu\nu]}{}^{\rho}$ such that, in complete analogy with Einstein-
Cartan theory, the full first field equation contains the second field
equation. This allows us to define a new object as follows

$\displaystyle\mathcal{E}_{\mu\nu}{}^{\rho}$
$\displaystyle:=\tilde{E}_{(\mu\nu)}{}^{\rho}+\tilde{E}_{[\mu\nu]}{}^{\rho}$
$\displaystyle:=2\tilde{\Gamma}^{\rho}_{(\mu\nu)}-2\delta^{\rho}_{(\mu}\tilde{\Gamma}^{\lambda}_{|\lambda|\nu)}-g_{\mu\nu}(g^{\kappa\lambda}\tilde{\Gamma}^{\rho}_{\lambda\kappa}-g^{\kappa\rho}\tilde{\Gamma}^{\lambda}_{\lambda\kappa})+\tilde{P}^{\rho}{}_{\nu\mu}$
$\displaystyle=2\tilde{\Gamma}{}^{\rho}_{\mu\nu}+2\delta^{\rho}_{[\mu}\tilde{\Gamma}^{\lambda}_{\nu]\lambda}-2\delta^{\rho}_{\mu}\tilde{\Gamma}{}^{\lambda}_{\lambda\nu}-g_{\mu\nu}(g^{\kappa\lambda}\tilde{\Gamma}^{\rho}_{\lambda\kappa}-g^{\kappa\rho}\tilde{\Gamma}^{\lambda}_{\lambda\kappa})\,.$
(5.3.26)

We are now able to re-write the complete first field equation

$\displaystyle-\frac{1}{2}g_{\mu\nu}f+f_{,\tilde{\mathbf{G}}}\big{(}\tilde{G}_{\mu\nu}+\frac{1}{2}g_{\mu\nu}\tilde{\mathbf{G}}\big{)}+\frac{1}{2}\mathcal{E}_{\mu\nu}{}^{\lambda}\partial_{\lambda}f_{,\tilde{\mathbf{G}}}=\kappa\Bigl{(}{}^{(\tilde{\Gamma})}T_{\mu\nu}+(\tilde{\nabla}_{\lambda}+T^{\sigma}{}_{\lambda\sigma})\Delta^{\lambda}{}_{\nu\mu}\Bigr{)}\,,$
(5.3.27)

so that its skew-symmetric part coincides with the second field equation. This
result is somewhat surprising but illustrates the intricate relationship
between the two field equations. It also reduces to standard Einstein-Cartan
when $f(\bar{\mathbf{G}})=\bar{\mathbf{G}}$ as expected.

##### Modified Einstein-Cartan theory with boundary terms

Let us now work with the full $f(\tilde{\mathbf{G}},\tilde{\mathbf{B}})$
theory, given by equations (5.3.16)-(5.3.18). The connection equation with
lowered indices is

$\displaystyle\tilde{P}_{\mu\nu\rho}f_{,\tilde{\mathbf{G}}}+2g_{\mu[\rho}\partial_{\nu]}f_{,\tilde{\mathbf{B}}}-2\kappa\lambda_{\mu\nu\rho}=2\kappa\Delta_{\mu\nu\rho}\,,$
(5.3.28)

from which we immediately see the symmetric part
$\lambda_{\mu\nu\rho}=-\Delta_{\mu(\nu\rho)}$ isolates the Lagrange
multiplier, just like before. The skew-symmetric part now includes an
additional dynamical term

$\displaystyle
P_{\mu\nu\rho}f_{,\tilde{\mathbf{G}}}+2g_{\mu[\rho}\partial_{\nu]}f_{,\tilde{\mathbf{B}}}=2\kappa\Delta_{\mu[\nu\rho]}\,.$
(5.3.29)

The metric field equation is then

$-\frac{1}{2}g_{\mu\nu}f+f_{,\tilde{\mathbf{G}}}\big{(}\tilde{G}_{(\mu\nu)}+\frac{1}{2}g_{\mu\nu}\tilde{\mathbf{G}}\big{)}+\frac{1}{2}\tilde{E}_{(\mu\nu)}{}^{\lambda}\partial_{\lambda}(f_{,\tilde{\mathbf{G}}}-f_{,\tilde{\mathbf{B}}})+\frac{1}{2}g_{\mu\nu}f_{,\tilde{\mathbf{B}}}\tilde{\mathbf{B}}\\\
=\kappa\Big{(}{}^{(\bar{\Gamma})}T_{\mu\nu}+(\tilde{\nabla}_{\lambda}+T^{\sigma}{}_{\lambda\sigma})\Delta^{\lambda}{}_{(\nu\mu)}\Big{)}\,.$
(5.3.30)

Just as before, we now remove the symmetry brackets and consider the full
field equation. First, the antisymmetric part can be written as

$\displaystyle
f_{,\tilde{\mathbf{G}}}\tilde{G}_{[\mu\nu]}+\frac{1}{2}\tilde{E}_{[\mu\nu]}{}^{\lambda}\partial_{\lambda}(f_{,\tilde{\mathbf{G}}}-f_{,\tilde{\mathbf{B}}})=\frac{1}{2}(\tilde{\nabla}_{\lambda}+T^{\sigma}{}_{\lambda\sigma})\Big{(}\tilde{P}^{\lambda}{}_{\nu\mu}f_{,\tilde{\mathbf{G}}}+2\delta^{\lambda}_{[\mu}\partial_{\nu]}f_{,\tilde{\mathbf{B}}}\Big{)}\,,$
(5.3.31)

where on the right-hand side we have used the skew-symmetric part of the
connection equation. By again using the geometric identity
$\tilde{G}_{[\mu\nu]}=(\tilde{\nabla}_{\lambda}+T^{\sigma}{}_{\lambda\sigma})\tilde{P}^{\lambda}{}_{\nu\mu}/2$
we find

$\displaystyle\frac{1}{2}\partial_{\lambda}f_{,\tilde{\mathbf{G}}}\Big{(}\tilde{E}_{[\mu\nu]}{}^{\lambda}-\tilde{P}^{\lambda}{}_{\nu\mu}\Big{)}-\frac{1}{2}\tilde{E}_{[\mu\nu]}{}^{\lambda}\partial_{\lambda}f_{,\tilde{\mathbf{B}}}=(\tilde{\nabla}_{\lambda}+T^{\sigma}{}_{\lambda\sigma})\delta^{\lambda}_{[\mu}\partial_{\nu]}f_{,\tilde{\mathbf{B}}}\,.$
(5.3.32)

The part in the first bracket was derived in the $f(\tilde{\mathbf{G}})$ case
(5.3.25). We can expand the right-hand side, cancelling the second partial
derivatives, to obtain

$\displaystyle(\tilde{\nabla}_{\lambda}+T^{\sigma}{}_{\lambda\sigma})\delta^{\lambda}_{[\mu}\partial_{\nu]}f_{,\tilde{\mathbf{B}}}$
$\displaystyle=\frac{1}{2}\Big{(}T^{\lambda}{}_{\nu\mu}\partial_{\lambda}f_{,\tilde{\mathbf{B}}}+T^{\sigma}{}_{\mu\sigma}\partial_{\nu}f_{,\tilde{\mathbf{B}}}-T^{\sigma}{}_{\nu\sigma}\partial_{\mu}f_{,\tilde{\mathbf{B}}}\Big{)}$
$\displaystyle=-\frac{1}{2}\tilde{P}^{\lambda}{}_{\nu\mu}\partial_{\lambda}f_{,\tilde{\mathbf{B}}}\,,$
(5.3.33)

where the first torsion term comes from commutator of covariant derivatives.
Putting this back into our skew-symmetric field equation (5.3.32) we find

$\displaystyle\Big{(}\tilde{E}_{[\mu\nu]}{}^{\lambda}-\tilde{P}^{\lambda}{}_{\nu\mu}\Big{)}\Big{(}\partial_{\lambda}f_{,\tilde{\mathbf{G}}}-\partial_{\lambda}f_{,\tilde{\mathbf{B}}}\Big{)}=0\,,$
(5.3.34)

where we again have
$\tilde{E}_{[\mu\nu]}{}^{\lambda}=\tilde{P}^{\lambda}{}_{\nu\mu}$. If we use