# Quadratic Metric-Affine Gravity: Solving for the Affine-Connection

Damianos Iosifidis  Institute of Theoretical Physics, Department of Physics
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
<EMAIL_ADDRESS>
###### Abstract

We consider the most general 11 parameter parity even quadratic Metric-Affine
Theory whose action consists of the usual Einstein-Hilbert plus the 11
quadratic terms in torsion, non-metricity as well as their mixing. By
following a certain procedure and using a simple trick we are able to find the
unique solution of the affine connection in terms of an arbitrary
hypermomentum. Given a fairly general non-degeneracy condition our result
provides the exact form of the affine connection for all types of matter.
Subsequently we compute the forms of torsion and non-metricity in terms of
their sources (hypermomentum tensor) and also express the metric field
equations in effectively Einstein’s GR with modified source terms that depend
on the hypermomentum and its derivatives. We show that in the absence of
matter the Theory always reduces to GR. Finally we generalize our result and
find the form of the connection for a wider class of quadratic Theories.

###### Contents

  1. I Introduction
  2. II The setup
  3. III The sources of MAG
    1. III.1 Canonical and Metrical Energy Momentum and Hypermomentum Tensors
  4. IV Full Quadratic Theory
  5. V The Quadratic Parity Even MAG Theory
  6. VI Solving for the affine connection
    1. VI.1 Totally Antisymmetric Hypermomentum (Purely Fermionic Matter)
    2. VI.2 Totally Symmetric Hypermomentum
  7. VII Effective GR with hypermomentum contributions
  8. VIII Generalized Quadratic Theory
  9. IX Conclusions
  10. X Acknowledgments
  11. A Relations among parameters
    1. A.1 Elements of matrix $A$ in terms of Theory parameters
    2. A.2 The $\gamma^{\prime}s$
  12. B Post-Riemannian Expansions
    1. B.1 Distortion vectors in terms of torsion and non-metricity vectors
    2. B.2 Post-Riemannian expansions

##  I Introduction

The geometric interpretation of gravity initiated by Einstein on the
development of General Relativity (GR) is indeed an astonishing one. However,
the geometric arena GR lies upon is that of Riemannian geometry
eisenhart2016riemannian  which is built on the assumptions of a symmetric and
metric-compatible connection. That is, a Riemannian space possesses no torsion
(i.e. symmetric connection) and has vanishing non-metricity (metric-
compatibility). Consequently, gravity is descried solely by the curvature and
the only dynamical field is the metric tensor. Interestingly, equivalent
formulations of GR can be formulated by attributing gravity to either solely
torsion (teleparallel gravity aldrovandi2012teleparallel ), solely non-
metricity (symmetric teleparallel nester1998symmetric ;
jimenez2018teleparallel ) or both (general teleparallel jimenez2019general )
by restricting the connection in a certain way. Even more intriguing is the
possibility of leaving the underlying geometry free of restrictions and be in
the realm of non-Riemannian geometry eisenhart2012non . This generalized arena
has all, curvature, torsion and non-metricity non-vanishing and the
gravitational Theories described by non-Riemannian geometry go by the name
Metric-Affine Gravity (MAG) hehl1995metric ; iosifidis2019metric ;
gronwald1997metric 111Some recent developments in MAG and applications include
iosifidis2019metric ; iosifidis2019exactly ; Iosifidis:2020gth ;
iosifidis2021perfect ; percacci2020new ; shimada2019metric ; delhom2019ricci ;
iosifidis2019torsion ; hohmann2020metric ; Mikura:2020qhc ; Jimenez:2020dpn ;
Ariwahjoedi:2020wmo ; klemm2020einstein ; vitagliano2011dynamics ;
iosifidis2019scale ; iosifidis2021cosmology  .

MAG is very interesting to study for many reasons. Firstly, when written in
the language of differential forms it can be seen to represent a gauge Theory
of gravity hehl1995metric . Next, the modifications appearing in MAG (in
comparison to GR) are due to spacetime torsion and non-metricity with both of
them having a clear geometric interpretation (see for instance
iosifidis2019metric ). Furthermore, an essential characteristic of MAG is the
extended geometry-intrinsic structure interrelation. More precisely, in MAG
one also has the notion of hypermomentum source hehl1976hypermomentum ;
hehl1995metric  on top of the usual energy-momentum tensor. The hypermomentum
tensor is formally defined by the variation of the matter sector of the action
with respect to the independent affine connection and in it it encodes the
spin, dilation and shear of matter hehl1976hypermomentum ; hehl1995metric .We
see therefore that MAG represents a generalization of the well known Einstein-
Cartan Theory where now apart from the spin, the dilation and shear222For an
interesting possible relation among the shear current and the hadronic
properties of matter see Hehl:1978cb . of matter (i.e. all intrinsic parts)
are taken into account.

Of course, in order to explore this Metric-Affine land the first thing we
should be concerned with is the choice of the gravitational Lagrangian.
Admittedly, the most obvious possibility for the latter is the generalized
Einstein-Hilbert term (i.e. the full Ricci scalar) which contains non-
Riemannian parts as well. However, one soon realises that this choice
encounters already an issue: The invariance of the Ricci scalar under
projective transformations of the connection (see for instance
iosifidis2020linear ) demands an identically vanishing dilation current. As a
result only matter with no dilation charge for the hypermomentum is allowed by
the Theory. The most straightforward remedy for this problem is to supplement
the Gravitational action with quadratic torsion and non-metricity invariants
that would spoil the aforementioned invariance. In fact, the inclusion of
these quadratic terms is not simply a matter of convenience for breaking the
projective invariance but their presence is indispensable if one wants to keep
generality. Indeed, was already observed in vitagliano2011dynamics  (see also
obukhov1996exact  ), from a field theoretic point of view, the dimension of
the quadratic torsion and non-metricity invariants is the same with that of
the Ricci scalar (being inverse length) and since there is no fundamental
principle to exclude them they must appear in the Gravitational action for MAG
to this order. It is known that there are $3$ pure torsion, $5$ pure non-
metricity and $3$ mixed quadratic (parity even) scalars pagani2015quantum ;
iosifidis2019scale . Therefore, the gravitational part of the parity even
quadratic MAG action consists of the usual Einstein-Hilbert term along with
the $11$ quadratic torsion and non-metricity scalars. Of course, to the
aforementioned gravitational part one must also add a matter sector that will
(in general) possess a non-vanishing hypermomentum333The Cosmology of the full
$11$ parameter quadratic Theory in the presence of a Cosmological hyperfluid
has been recently studied in iosifidis2021cosmology .. Then, after varying
with respect to the metric and the independent affine connection the first
task is to find the form of the affine connection which will subsequently
reveal the expressions for torsion and non-metricity.

Surprisingly, up to now there is no prescribed method allowing one to solve
for the affine connection for the general quadratic Theory when the
hypermomentum is non-vanishing. It is exactly this point we will address in
this work. That is, given some general non-degeneracy conditions we will find
the exact and unique solution of the connection for the full quadratic Theory.
More specifically, the paper is organized as follows. Firstly, we fix notation
and definitions and briefly discuss the geometric arena of MAG along with the
physical content of its sources. We then consider the full $11$ parameter
quadratic Lagrangian and derive the associated metric and connection field
equations. Focusing on the latter and using the recent result of
Iosifidis:2021ili  we find the unique and exact solution of the affine
connection in terms of the hypermomentum and its contractions. Subsequently,
we compute the exact forms of torsion and non-metricity for the quadratic
Theory. Then, turning our attention to the metric field equations, we apply a
post-Riemannian expansion and express the latter as effective Einstein’s GR
field equations with modified source terms due to the intrinsic properties of
matter (i.e. hypermomentum). The resulting Theory serves as certain
generalization of Einstein-Cartan Theory where now apart from spin, the
dilation and shear of matter are taken into account, resulting in a non-
vanishing non-metricity as well. Finally, generalizing the previous
considerations, we consider a collection of matter fields non-minimally
coupled to the quadratic torsion and non-metricity invariants (i.e. promoting
the $11$ parameters to functions) and also for this generalized action we
derive the exact solution for the connection.

##  II The setup

We shall consider a generalized $n-dim$ manifold endowed with a metric and an
independent affine connection ($\mathcal{M}$, g, $\nabla$). We will use the
definitions and notations of Iosifidis:2020gth  so we will go through the
basic setup rather briefly. On this non-Riemannian manifold endowed with the
metric $g_{\mu\nu}$ and an independent affine connection with components
$\Gamma^{\lambda}_{\;\;\;\mu\nu}$, we define the curvature, torsion and non-
metricity tensors according to

$R^{\mu}_{\;\;\;\nu\alpha\beta}:=2\partial_{[\alpha}\Gamma^{\mu}_{\;\;\;|\nu|\beta]}+2\Gamma^{\mu}_{\;\;\;\rho[\alpha}\Gamma^{\rho}_{\;\;\;|\nu|\beta]}$
(1) $S_{\mu\nu}^{\;\;\;\lambda}:=\Gamma^{\lambda}_{\;\;\;[\mu\nu]}$ (2)
$Q_{\alpha\mu\nu}:=-\nabla_{\alpha}g_{\mu\nu}$ (3)

and the deviation of the affine connection $\Gamma^{\lambda}_{\;\;\;\mu\nu}$
from the Levi-Civita one defines the so-called distortion tensor
schouten1954ricci

$\displaystyle
N^{\lambda}_{\;\;\;\;\mu\nu}:=\Gamma^{\lambda}_{\;\;\;\mu\nu}-\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}=\frac{1}{2}g^{\alpha\lambda}(Q_{\mu\nu\alpha}+Q_{\nu\alpha\mu}-Q_{\alpha\mu\nu})-g^{\alpha\lambda}(S_{\alpha\mu\nu}+S_{\alpha\nu\mu}-S_{\mu\nu\alpha})$
(4)

where $\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}$ is the usual Levi-Civita
connection calculated only by the metric and its first derivatives. Once the
distortion is given, torsion and non-metricity can be quite easily computed
through (see for instance iosifidis2019metric )

$S_{\mu\nu\alpha}=N_{\alpha[\mu\nu]}\;\;,\;\;\;Q_{\nu\alpha\mu}=2N_{(\alpha\mu)\nu}$
(5)

Out of torsion we can construct a vector as well as a pseudo-vector. Our
definitions for the torsion vector and pseudo-vector are

$S_{\mu}:=S_{\mu\lambda}^{\;\;\;\;\lambda}\;\;,\;\;\;t_{\mu}:=\epsilon_{\mu\alpha\beta\gamma}S^{\alpha\beta\gamma}$
(6)

respectively. Note that the former is defined for any dimension while the
latter only for $n=4$. Continuing with non-metricity, we define the Weyl and
the second non-metricity vector according to

$Q_{\alpha}:=Q_{\alpha\mu\nu}g^{\mu\nu}\;,\;\;q_{\nu}=Q_{\alpha\mu\nu}g^{\alpha\mu}$
(7)

Finally, from the Riemann tensor we can construct the three contractions

$R_{\nu\beta}:=R^{\mu}_{\;\;\nu\mu\beta}$ (8)
$\hat{R}_{\alpha\beta}:=R^{\mu}_{\;\;\mu\alpha\beta}$ (9)
$\breve{R}^{\mu}_{\;\;\beta}:=R^{\mu}_{\;\;\nu\alpha\beta}g^{\nu\alpha}$ (10)

As usual, the first one is the Ricci tensor (which is not symmetric in
general), the second one is the homothetic curvature and the last one is the
the co-Ricci tensor. Note that the first two aforementioned tensors can be
formed without the use of any metric while for the latter a metric is
required. As for the generalized Ricci scalar, the latter is still uniquely
defined since

$R:=g^{\mu\nu}R_{\mu\nu}=-g^{\mu\nu}\breve{R}_{\mu\nu}\;\;,\;\;\;g^{\mu\nu}\hat{R}_{\mu\nu}=0$
(11)

Furthermore, let us mention that by virtue of (4) each quantity can be split
into its Riemannian part (i.e. computed with respect to the Levi-Civita
connection) plus non-Riemannian contributions. For instance inserting the
connection decomposition (4) into the definition (1) we obtain for the Riemann
tensor 444Quantities with $\widetilde{}$ will always denote Riemannian parts
unless otherwise stated.

${R^{\mu}}_{\nu\alpha\beta}=\widetilde{R}^{\mu}_{\phantom{\mu}\nu\alpha\beta}+2\widetilde{\nabla}_{[\alpha}{N^{\mu}}_{|\nu|\beta]}+2{N^{\mu}}_{\lambda|\alpha}{N^{\lambda}}_{|\nu|\beta]}\,,$
(12)

The last decomposition is very useful and we are going to be using it later on
in order to express the metric field equations of our Theory in Einstein-like
form plus modified matter sources. For instance, with the use of the above the
post-Riemannian expansion of the Ricci scalar reads

$\displaystyle
R=\tilde{R}+\frac{1}{4}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}-\frac{1}{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}-\frac{1}{4}Q_{\mu}Q^{\mu}+\frac{1}{2}Q_{\mu}q^{\mu}+S_{\mu\nu\alpha}S^{\mu\nu\alpha}-2S_{\mu\nu\alpha}S^{\alpha\mu\nu}-4S_{\mu}S^{\mu}$
$\displaystyle+2Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+2S_{\mu}(q^{\mu}-Q^{\mu})+\tilde{\nabla}_{\mu}(q^{\mu}-Q^{\mu}-4S^{\mu})$
(13)

We now close this section with a final definition before we turn our attention
into the material content of Metric-Affine Theories.

###### Definition 1

Consider the components $\Psi_{\alpha\mu\nu}$ of a rank-3 tensor field. We
define the $1^{st}$, $2^{nd}$ and $3^{rd}$ contractions of
$\Psi_{\alpha\mu\nu}$ according to

$\Psi^{(1)}_{\mu}:=\Psi_{\alpha\beta\mu}g^{\alpha\beta}\;\;,\;\;\Psi^{(2)}_{\mu}:=\Psi_{\alpha\mu\beta}g^{\alpha\beta}\;\;,\;\;\Psi^{(3)}_{\mu}:=\Psi_{\mu\alpha\beta}g^{\alpha\beta}$
(14)

respectively.

Let us now move on and discuss some basic physical aspects of MAG.

##  III The sources of MAG

###  III.1 Canonical and Metrical Energy Momentum and Hypermomentum Tensors

As we have already pointed out, in the holonomic555In an anholonomic
description we have three independent fields $g_{ab},\vartheta^{b}$ and
$\Gamma^{a}_{\;\;b}$, that is the tangent space metric, the co-frame and the
spin connection. However, the metric field equations are redundant in this
case and therefore we have only two independent fields in this case as well
(see hehl1995metric )  description of MAG one starts with an independent
fields affine connection $\Gamma^{\lambda}_{\;\;\;\mu\nu}$ along with the
metric $g_{\mu\nu}$. The field equations of a given MAG Theory are obtained by
varying the total action independently with respect to those fields. Then, the
variations of the matter part of the actions would be the sources of Gravity.
Let $\mathcal{L}_{m}$ be the matter Lagrangian of the Theory. As usual we
define the energy-momentum tensor of matter by the metric variation of the
matter sector, viz.

$T^{\alpha\beta}:=+\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta
g_{\alpha\beta}}$ (15)

On the other hand, since we now have an independent affine connection on which
the matter part can just as well depend upon, we define the variation of the
latter with respect to the affine connection

$\Delta_{\lambda}^{\;\;\;\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}}$
(16)

Note that the hypermomentum can be split into its three irreducible pieces of
spin, dilation and shear according to666Each of these irreducible pieces forms
an invariant subgroup under rotations.

$\Delta_{\mu\nu\alpha}=\tau_{\mu\nu\alpha}+\frac{1}{n}\Delta_{\alpha}g_{\mu\nu}+\hat{\Delta}_{\mu\nu\alpha}$
(17)

where $\tau_{\mu\nu\alpha}:=\Delta_{[\mu\nu]\alpha}$ is the spin part,
$\Delta_{\alpha}:=\Delta_{\mu\nu\alpha}g^{\mu\nu}$ the dilation (trace) and
$\hat{\Delta}_{\mu\nu\alpha}=\Delta_{(\mu\nu)\alpha}-\frac{1}{n}\Delta_{\alpha}g_{\mu\nu}$
the shear (symmetric traceless part). The aforementioned sources are subject
to the conservation law

$\sqrt{-g}(2\tilde{\nabla}_{\mu}T^{\mu}_{\;\;\alpha}-\Delta^{\lambda\mu\nu}R_{\lambda\mu\nu\alpha})+\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}(\sqrt{-g}\Delta_{\alpha}^{\;\;\mu\nu})+2S_{\mu\alpha}^{\;\;\;\;\lambda}\hat{\nabla}_{\nu}(\sqrt{-g}\Delta_{\lambda}^{\;\;\;\mu\nu})=0$
(18)

which serves as the generalization of the conservation law of energy-momentum
for matter with microstructure (see) Now we have all the ingredients needed to
study model building and more specifically quadratic MAG Theories.

##  IV Full Quadratic Theory

In order to set up a Metric-Affine Theory, the first thing one has to specify
is the underlying gravitational action that is going to be used in
constructing the Theory. The most obvious possibility is to replace the usual
(Riemannian) Einstein-Hilbert term with the full Ricci scalar, namely
$\tilde{R}\rightarrow R$ and consider the gravitational action

$S_{EH}[g,\Gamma]=\frac{1}{2\kappa}\int d^{n}x\sqrt{-g}R$ (19)

as the representative one for MAG. Even though this is indeed a good choice
once the matter part of the action does not depend on the connection (i.e. for
matter with $\Delta_{\alpha\mu\nu}=0$), in the sense that it gives Einstein’s
field equations, problems set in when matter couples to the connection. It is
known that the problem lies on the projective invariance of the Ricci scalar
(see for instance) which then implies a vanishing dilation current i.e.
$\Delta^{\mu}=0$ once the connection field equations are imposed. It is
therefore necessary, from a physical point of view, to consider actions that
go beyond the mere generalization of the non-Riemannian Einstein-Hilbert term
and break the projective invariance. Interestingly enough, the above reason is
not the only one that demands a modification to the above action. There is yet
another more fundamental reason enforcing us to extend 19 that has to do with
dimensional analysis. Indeed, as it was already noted in obukhov1996exact
(see also vitagliano2011dynamics ) the dimension of the Ricci scalar is
inverse squared length [$L^{-2}$]. However, since the connection has
dimensions of inverse length, quadratic torsion and non-metricity invariants
have the same dimensions [$L^{-2}$] with the Ricci scalar. Therefore, from an
effective field Theory perspective these terms may as well supplement the
Ricci scalar, being of the same dimension. It turns out that to quadratic
order there are $3$ pure torsion, $5$ pure non-metricity and $3$ mixed scalars
(see for instance pagani2015quantum ; iosifidis2019scale ). Therefore, the
most general parity even quadratic MAG action in arbitrary dimension $n$ reads

$\displaystyle S_{quad}=\frac{1}{2\kappa}\int
d^{n}x\sqrt{-g}\Big{[}b_{1}S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}S_{\alpha\mu\nu}S^{\mu\nu\alpha}+b_{3}S_{\mu}S^{\mu}$
$\displaystyle
a_{1}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+a_{3}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+a_{5}Q_{\mu}q^{\mu}$
$\displaystyle+c_{1}Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+c_{2}Q_{\mu}S^{\mu}+c_{3}q_{\mu}S^{\mu}\Big{]}$
(20)

where the $a_{i}^{\prime}s$, $b_{j}^{\prime}s$ and $c_{k}^{\prime}s$ are
dimensionless coupling constants. Of course to the above one could also add
scalars constructed by the covariant derivatives of torsion and non-metricity,
like for instance $g^{\mu\nu}\nabla_{\mu}Q_{\nu}$. However, one can always
apply a post-Riemannian expansion to such terms ending up with total
derivative terms plus some combinations of the above-included quadratic
torsion and non-metricity invariance777For a detailed example see
vitagliano2011dynamics .. As a result the above $11$ parameter action is
indeed the most general quadratic one (and parity even). Note that in
vitagliano2011dynamics  the connection solution of the action was found but
only when the $3$ quadratic torsion terms where included. Our intention here
is to generalize these results and find the connection solution for the full
quadratic action $(\ref{quad})$.

##  V The Quadratic Parity Even MAG Theory

As we already mentioned the main intention of this work is to analyze the full
$11$-parameter quadratic MAG Theory including all possible, parity even terms
in torsion and non-metricity in addition to the Einstein-Hilbert term. Adding
a matter part as well, our full action reads

$\displaystyle S[g,\Gamma,\Phi]=\frac{1}{2\kappa}\int
d^{n}x\sqrt{-g}\Big{[}R+b_{1}S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}S_{\alpha\mu\nu}S^{\mu\nu\alpha}+b_{3}S_{\mu}S^{\mu}$
$\displaystyle
a_{1}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+a_{3}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+a_{5}Q_{\mu}q^{\mu}$
$\displaystyle+c_{1}Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+c_{2}Q_{\mu}S^{\mu}+c_{3}q_{\mu}S^{\mu}\Big{]}+S_{M}[g,\Gamma,\Phi]$
(21)

In the above, the first, second and third lines contain (apart from the usual
Einstein-Hilbert term) the pure torsion, pure non-metricity and mixed
quadratic parity even parts888In obukhov1997irreducible  to this quadratic
Lagrangian, a homothetic curvature squared term was added
$\hat{R}_{\mu\nu}\hat{R}^{\mu\nu}$. This term however is of dimension
[$L^{-4}$] and its inclusion is not along the lines of effective field Theory
approach. We shall therefore consider no such terms in our study here.. Let us
highlight that the matter action we are considering here can (and will) in
general depend on the connection which will result in a non-vanishing
hypermomentum tensor. Varying the above action independently with respect to
the metric and the affine connection, after some long calculations we obtain
the field equations

g-Variation

$R_{(\mu\nu)}-\frac{R}{2}g_{\mu\nu}-\frac{\mathcal{L}_{2}}{2}g_{\mu\nu}+\frac{1}{\sqrt{-g}}\hat{\nabla}_{\alpha}\Big{(}\sqrt{-g}(W^{\alpha}_{\;\;\;(\mu\nu)}+\Pi^{\alpha}_{\;\;\;(\mu\nu)})\Big{)}+A_{(\mu\nu)}+B_{(\mu\nu)}+C_{(\mu\nu)}=\kappa
T_{\mu\nu}$ (22)

where we have abbreviated

$\hat{\nabla}_{\mu}:=2S_{\mu}-\nabla_{\mu}$ (23)
$W^{\alpha}_{\;\;(\mu\nu)}=2a_{1}Q^{\alpha}_{\;\;\mu\nu}+2a_{2}Q_{(\mu\nu)}^{\;\;\;\;\alpha}+(2a_{3}Q^{\alpha}+a_{5}q^{\alpha})g_{\mu\nu}+(2a_{4}q_{(\mu}+a_{5}Q_{(\mu})\delta^{\alpha}_{\nu)}$
(24)
$\Pi^{\alpha\mu\nu}=c_{1}S^{\alpha\mu\nu}+c_{2}g^{\mu\nu}S^{\alpha}+c_{3}g^{\alpha\mu}S^{\nu}$
(25)

$A_{\mu\nu}=a_{1}(Q_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q_{\alpha\beta\mu}Q^{\alpha\beta}_{\;\;\;\;\nu})-a_{2}Q_{\alpha\beta(\mu}Q^{\beta\alpha}_{\;\;\;\;\nu)}+a_{3}(Q_{\mu}Q_{\nu}-2Q^{\alpha}Q_{\alpha\mu\nu})-a_{4}q_{\mu}q_{\nu}-a_{5}q^{\alpha}Q_{\alpha\mu\nu}$
(26)
$B_{\mu\nu}=b_{1}(2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\alpha\beta}-S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\;\nu})-b_{2}S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\beta\alpha}+b_{3}S_{\mu}S_{\nu}$
(27)
$C_{\mu\nu}=\Pi_{\mu\alpha\beta}Q_{\nu}^{\;\;\;\alpha\beta}-(c_{1}S_{\alpha\beta\nu}Q^{\alpha\beta}_{\;\;\;\;\mu}+c_{2}S^{\alpha}Q_{\alpha\mu\nu}+c_{3}S^{\alpha}Q_{\mu\nu\alpha})=c_{1}(Q_{\mu}^{\;\;\;\alpha\beta}S_{\nu\alpha\beta}-S_{\alpha\beta\mu}Q^{\alpha\beta}_{\;\;\;\;\nu})+c_{2}(S_{\mu}Q_{\nu}-S^{\alpha}Q_{\alpha\mu\nu})$
(28)

$\Gamma$-Variation

$\displaystyle\left(\frac{Q_{\lambda}}{2}+2S_{\lambda}\right)g^{\mu\nu}-Q_{\lambda}^{\;\;\mu\nu}-2S_{\lambda}^{\;\;\mu\nu}+\left(q^{\mu}-\frac{Q^{\mu}}{2}-2S^{\mu}\right)\delta_{\lambda}^{\nu}+4a_{1}Q^{\nu\mu}_{\;\;\;\;\lambda}+2a_{2}(Q^{\mu\nu}_{\;\;\;\;\lambda}+Q_{\lambda}^{\;\;\;\mu\nu})+2b_{1}S^{\mu\nu}_{\;\;\;\;\lambda}$
$\displaystyle+2b_{2}S_{\lambda}^{\;\;\;[\mu\nu]}+c_{1}\Big{(}S^{\nu\mu}_{\;\;\;\;\lambda}-S_{\lambda}^{\;\;\;\nu\mu}+Q^{[\mu\nu]}_{\;\;\;\;\;\lambda}\Big{)}+\delta_{\lambda}^{\mu}\Big{(}4a_{3}Q^{\nu}+2a_{5}q^{\nu}+2c_{2}S^{\nu}\Big{)}+\delta_{\lambda}^{\nu}\Big{(}a_{5}Q^{\mu}+2a_{4}q^{\mu}+c_{3}S^{\mu}\Big{)}$
$\displaystyle+g^{\mu\nu}\Big{(}a_{5}Q_{\lambda}+2a_{4}q_{\lambda}+c_{3}S_{\lambda}\Big{)}+\Big{(}c_{2}Q^{[\mu}+c_{3}q^{[\mu}+2b_{3}S^{[\mu}\Big{)}\delta^{\nu]}_{\lambda}=\kappa\Delta_{\lambda}^{\;\;\;\mu\nu}$
(29)

The metric field equations (22) are the analogue of the Einstein field
equations where now apart from the explicit additional terms of torsion and
non-metricity, both the Ricci tensor and scalar contain post-Riemannian pieces
as well. The exact deviation from the usual field equations of GR will be
clear after we solve for the affine connection and substitute all post-
Riemannian expansions back in (22). This will be clearly illustrated in what
follows. Before we move on to solve for the connection let us elaborate a
little bit in the metric field equations. Taking the trace of the latter we
obtain

$\Big{(}1-\frac{n}{2}\Big{)}R+\Big{(}1-\frac{n}{2}\Big{)}\mathcal{L}_{2}-\frac{1}{\sqrt{-g}}\tilde{\nabla}_{\alpha}\Big{(}\sqrt{-g}(\Pi^{\alpha}+W^{\alpha})\Big{)}=\kappa
T$ (30)

where

$\Pi^{\alpha}:=\Pi^{\alpha}_{\;\;\mu\nu}g^{\mu\nu}=(c_{1}+nc_{2}+c_{3})S^{\alpha}$
(31)
$W^{\alpha}:=W^{\alpha}_{\;\;\mu\nu}g^{\mu\nu}=(2a_{1}+2na_{3}+a_{5})Q^{\alpha}+(2a_{2}+2a_{4}+na_{5})q^{\alpha}$
(32)

Expanding out the Ricci scalar into its Riemannian part plus non-Riemannian
contributions, equation (30) becomes

$\displaystyle\tilde{R}+\Big{(}a_{1}+\frac{1}{4}\Big{)}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+\Big{(}a_{2}-\frac{1}{2}\Big{)}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+\Big{(}a_{3}-\frac{1}{4}\Big{)}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+\Big{(}a_{5}+\frac{1}{2}\Big{)}Q_{\mu}q^{\mu}$
$\displaystyle+(b_{1}+1)S_{\alpha\mu\nu}S^{\alpha\mu\nu}+(b_{2}-2)S_{\alpha\mu\nu}S^{\mu\nu\alpha}+(b_{3}-4)S_{\mu}S^{\mu}+(c_{1}+2)Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+(c_{2}-2)Q_{\mu}S^{\mu}+(c_{3}+2)q_{\mu}S^{\mu}$
$\displaystyle+\frac{1}{\sqrt{-g}}\widetilde{\nabla}_{\mu}\Big{[}\sqrt{-g}\Big{(}(2a_{2}+2a_{4}+na_{5}+1)q^{\mu}+(2a_{1}+2na_{3}+a_{5}-1)Q^{\mu}+(c_{1}+nc_{2}+c_{3}-4)S^{\mu}\Big{)}\Big{]}=\kappa
T$ (33)

Note that for the parameter choice $a_{1}=-a_{3}=-1/4$, $a_{2}=-a_{5}=1/2$,
$a_{4}=0$, $b_{1}=-1$, $b_{2}=1$, $b_{3}=4$ , $c_{1}=-c_{2}=c_{3}=-2$ all non-
Riemannian pieces of torsion and non-metricity disappear and $(\ref{effEin})$
is the same with the trace of Einstein’s field equations of GR. The same holds
true for the Ricci tensor and essentially the Theory is GR for this parameter
choice. Of course this is no surprise, since choosing the parameters this way
all the additional quadratic torsion and non-metricity terms exactly cancel
the corresponding ones coming from the post Riemannian expansion of the Ricci
scalar yielding just $\tilde{R}$ in the end. However, this is extreme fine
tuning and in order to keep our discussion as general as possible we shall not
impose any a priori relation among the couplings of the Theory.

##  VI Solving for the affine connection

We shall now pursue to solve for the affine connection and derive its exact
form for the quadratic Theory of the previous section. Even though the
connection field equations $(\ref{G})$ are fairly complicated as we show
below, using a certain result and also performing a simple trick the general
solution can be obtained quite easily. We are going to use the following
Theorem which we state here without proof. The detailed proof can be found in
Iosifidis:2021ili .

###### Theorem 1

Consider the $15$ parameter linear tensor equation

$\alpha_{1}N_{\alpha\mu\nu}+\alpha_{2}N_{\nu\alpha\mu}+\alpha_{3}N_{\mu\nu\alpha}+\alpha_{4}N_{\alpha\nu\mu}+\alpha_{5}N_{\nu\mu\alpha}+\alpha_{6}N_{\mu\alpha\nu}+\sum_{i=1}^{3}\Big{(}\alpha_{7i}N^{(i)}_{\mu}g_{\alpha\nu}+\alpha_{8i}N^{(i)}_{\nu}g_{\alpha\mu}+\alpha_{9i}N^{(i)}_{\alpha}g_{\mu\nu}\Big{)}=B_{\alpha\mu\nu}$
(34)

where $N_{\alpha\mu\nu}$ are the components of the unknown tensor field,
$B_{\alpha\mu\nu}$ the components of a given (known) tensor field and
$\alpha_{i}$, $i=1,2,...,6$ are scalars. Define the matrices999The elements
$\gamma_{ij}$ are linear combinations of the parameters $a_{i}$ and their
exact relations are given in the appendix.

$\Gamma:=\begin{pmatrix}\gamma_{11}&\gamma_{12}&\gamma_{13}\\\
\gamma_{21}&\gamma_{22}&\gamma_{23}\\\ \gamma_{31}&\gamma_{32}&\gamma_{33}\\\
\end{pmatrix}$ (35)

and

$A:=\begin{pmatrix}\alpha_{1}&\alpha_{2}&\alpha_{3}&\alpha_{4}&\alpha_{5}&\alpha_{6}\\\
\alpha_{3}&\alpha_{1}&\alpha_{2}&\alpha_{5}&\alpha_{6}&\alpha_{4}\\\
\alpha_{2}&\alpha_{3}&\alpha_{1}&\alpha_{6}&\alpha_{4}&\alpha_{5}\\\
\alpha_{4}&\alpha_{6}&\alpha_{5}&\alpha_{1}&\alpha_{3}&\alpha_{2}\\\
\alpha_{5}&\alpha_{4}&\alpha_{6}&\alpha_{2}&\alpha_{1}&\alpha_{3}\\\
\alpha_{6}&\alpha_{5}&\alpha_{4}&\alpha_{3}&\alpha_{2}&\alpha_{1}\\\
\end{pmatrix}$ (36)

Then, given that both of the above matrices are non-singular, the unique
solution to $(\ref{eq4})$ reads

$N_{\alpha\mu\nu}=\tilde{a}_{11}B_{\alpha\mu\nu}+\tilde{a}_{12}\hat{B}_{\nu\alpha\mu}+\tilde{a}_{13}\hat{B}_{\mu\nu\alpha}+\tilde{a}_{14}\hat{B}_{\alpha\nu\mu}+\tilde{a}_{15}\hat{B}_{\nu\mu\alpha}+\tilde{a}_{16}\hat{B}_{\mu\alpha\nu}$
(37)

where $\tilde{a}_{1j}$ $j=1,2,...,6$ are the first row elements of the inverse
matrix $A^{-1}$ and

$\hat{B}_{\alpha\mu\nu}={B}_{\alpha\mu\nu}-\sum_{i=1}^{3}\sum_{j=1}^{3}\Big{(}\alpha_{7i}\tilde{\gamma}_{ij}B^{(j)}_{\mu}g_{\alpha\nu}+\alpha_{8i}\tilde{\gamma}_{ij}B^{(j)}_{\nu}g_{\alpha\mu}+\alpha_{9i}\tilde{\gamma}_{ij}B^{(j)}_{\alpha}g_{\mu\nu}\Big{)}$
(38)

An immediate consequence of the above result is the following.

###### Corollary 1

If $B_{\alpha\mu\nu}=0$ and both matrices $A$ and $\Gamma$ are not singular
then $N_{\alpha\mu\nu}=0$ is the unique solution of $(\ref{eq4})$.

Let us now go back into the field equations of the quadratic Theory and in
particular consider the connection field equations (29).

A clever trick now will allow us to solve for the connection. The crucial step
is to use equations (5) and also their contractions (see appendix) in order
eliminate torsion and non-metricity and express everything in terms of the
distortion tensor and its associated vectors. After some lengthy but rather
straightforward algebra we finally arrive at

$\displaystyle(4a_{1}+b_{1}-c_{1})N_{\alpha\mu\nu}+\Big{(}-1+2a_{2}+\frac{c_{1}+b_{2}}{2}\Big{)}N_{\nu\alpha\mu}+\Big{(}-1+2a_{2}+\frac{c_{1}+b_{2}}{2}\Big{)}N_{\mu\nu\alpha}$
$\displaystyle+(2a_{2}-b_{1}+c_{1})N_{\alpha\nu\mu}+\Big{(}2a_{2}-\frac{b_{2}}{2}\Big{)}N_{\nu\mu\alpha}+\Big{(}4a_{1}-\frac{b_{2}}{2}-c_{1}\Big{)}N_{\mu\alpha\nu}$
$\displaystyle+\left(2a_{5}+c_{2}-\frac{b_{3}+c_{3}}{2}\right)g_{\nu\alpha}N^{(1)}_{\mu}+\left(2a_{4}+\frac{b_{3}}{2}+c_{3}\right)g_{\nu\alpha}N^{(2)}_{\mu}+\left(1+2a_{4}+\frac{c_{3}}{2}\right)g_{\nu\alpha}N^{(3)}_{\mu}$
$\displaystyle+\left(8a_{3}-2c_{2}+\frac{b_{3}}{2}\right)g_{\mu\alpha}N^{(1)}_{\nu}+\left(2a_{5}+c_{2}-\frac{c_{3}+b_{3}}{2}\right)g_{\mu\alpha}N^{(2)}_{\mu}+\left(2a_{5}-\frac{c_{3}}{2}\right)g_{\mu\alpha}N^{(3)}_{\nu}$
$\displaystyle+\left(2a_{5}-\frac{c_{3}}{2}\right)g_{\mu\nu}N^{(1)}_{\alpha}+\left(2a_{4}+\frac{c_{3}}{2}+1\right)g_{\mu\nu}N^{(2)}_{\alpha}+2a_{4}g_{\mu\nu}N^{(3)}_{\alpha}=\kappa\Delta_{\alpha\mu\nu}$
(39)

Now we see that the latter equation falls in the case of $Theorem-1$ with the
obvious identifications among the parameters $\alpha_{i}$ and
$a_{i},b_{j},c_{k}$ etc. Therefore, in applying $Theorem-1$ to the latter and
setting $B_{\alpha\mu\nu}=\kappa\Delta_{\alpha\mu\nu}$ we solve for the
distortion

$N_{\alpha\mu\nu}=\tilde{a}_{11}B_{\alpha\mu\nu}+\tilde{a}_{12}\hat{B}_{\nu\alpha\mu}+\tilde{a}_{13}\hat{B}_{\mu\nu\alpha}+\tilde{a}_{14}\hat{B}_{\alpha\nu\mu}+\tilde{a}_{15}\hat{B}_{\nu\mu\alpha}+\tilde{a}_{16}\hat{B}_{\mu\alpha\nu}$
(40)

where

$\hat{B}_{\alpha\mu\nu}={B}_{\alpha\mu\nu}-\sum_{i=1}^{3}\sum_{j=1}^{3}\Big{(}\alpha_{7i}\tilde{\gamma}_{ij}B^{(j)}_{\mu}g_{\alpha\nu}+\alpha_{8i}\tilde{\gamma}_{ij}B^{(j)}_{\nu}g_{\alpha\mu}+\alpha_{9i}\tilde{\gamma}_{ij}B^{(j)}_{\alpha}g_{\mu\nu}\Big{)}$
(41)

where we have assumed that the parameters of the Quadratic Theory are totally
independent101010In fact, if one wants to keep fair generality this is the way
to go. If there were certain relations among the parameters these would signal
some invariance under connection transformations which, in turn, would then
impose certain constraints on the matter fields (see iosifidis2020linear ). It
is therefore crucial, also from a physical point of view, not to impose any
relations among the parameters of the Theory.  and as a result $det(A)\neq 0$
and $\det(\Gamma)\neq 0$ which imply that indeed (42) is the unique solution
of (39). As a result, in the general non-degenerate case when there are no
relations among the parameters, the general solution of the latter equation
reads

$N_{\alpha\mu\nu}=\kappa\Big{(}\tilde{a}_{11}\hat{\Delta}_{\alpha\mu\nu}+\tilde{a}_{12}\hat{\Delta}_{\nu\alpha\mu}+\tilde{a}_{13}\hat{\Delta}_{\mu\nu\alpha}+\tilde{a}_{14}\hat{\Delta}_{\alpha\nu\mu}+\tilde{a}_{15}\hat{\Delta}_{\nu\mu\alpha}+\tilde{a}_{16}\hat{\Delta}_{\mu\alpha\nu}\Big{)}$
(42)

where

$\hat{\Delta}_{\alpha\mu\nu}=\Delta_{\alpha\mu\nu}-\sum_{i=1}^{3}\sum_{j=1}^{3}\Big{(}\alpha_{7i}\tilde{\gamma}_{ij}\Delta^{(j)}_{\mu}g_{\alpha\nu}+\alpha_{8i}\tilde{\gamma}_{ij}\Delta^{(j)}_{\nu}g_{\alpha\mu}+\alpha_{9i}\tilde{\gamma}_{ij}\Delta^{(j)}_{\alpha}g_{\mu\nu}\Big{)}$
(43)

and $\tilde{\gamma}_{ij}$ are the first row elements of the inverse matrix
$\Gamma^{-1}$. The relations of the $\gamma_{ij}^{\prime}s$ with the
parameters of the quadratic Theory are given in the appendix. Having solved
for the distortion we then substitute this form back in the connection
decomposition and find the exact expression for the affine connection in
Quadratic MAG Theories, which reads

$\Gamma^{\lambda}_{\;\;\;\mu\nu}=\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}+\kappa\Big{(}\tilde{a}_{11}\hat{\Delta}_{\alpha\mu\nu}+\tilde{a}_{12}\hat{\Delta}_{\nu\alpha\mu}+\tilde{a}_{13}\hat{\Delta}_{\mu\nu\alpha}+\tilde{a}_{14}\hat{\Delta}_{\alpha\nu\mu}+\tilde{a}_{15}\hat{\Delta}_{\nu\mu\alpha}+\tilde{a}_{16}\hat{\Delta}_{\mu\alpha\nu}\Big{)}$
(44)

with this we can then easily find the forms of torsion and non-metricity,
sourced by arbitrary hypermomentum, as

$S_{\mu\nu\alpha}=\kappa\Big{[}(\tilde{a}_{11}-\tilde{a}_{14})\hat{\Delta}_{\alpha[\mu\nu]}+(\tilde{a}_{13}-\tilde{a}_{15})\hat{\Delta}_{[\mu\nu]\alpha}+(\tilde{a}_{16}-\tilde{a}_{12})\hat{\Delta}_{[\mu|\alpha|\nu]}\Big{]}$
(45)

and

$Q_{\alpha\mu\nu}=2\kappa\Big{[}(\tilde{a}_{11}+\tilde{a}_{16})\hat{\Delta}_{(\mu\nu)\alpha}+(\tilde{a}_{12}+\tilde{a}_{15})\hat{\Delta}_{\alpha(\mu\nu)}+(\tilde{a}_{13}+\tilde{a}_{14})\hat{\Delta}_{(\mu|\alpha|\nu)}\Big{]}$
(46)

respectively. These last two expressions show how the sources (hypermomentum)
excite the non-Riemannian degrees of freedom. It is interesting to note that
when $\tilde{a}_{11}=\tilde{a}_{14}$, $\tilde{a}_{13}=\tilde{a}_{15}$ and
$\tilde{a}_{16}=\tilde{a}_{12}$, torsion vanishes for arbitrary hypermomentum,
while for $\tilde{a}_{11}=-\tilde{a}_{16}$, $\tilde{a}_{12}=-\tilde{a}_{15}$
and $\tilde{a}_{13}=-\tilde{a}_{14}$ spacetime non-metricity identically
vanishes, again without any assumption about the sources. Note also that there
are no parameter values for which both $S_{\mu\nu\alpha}=0$ and
$Q_{\alpha\mu\nu}=0$ at the same time.111111This is tightly connected with the
fact that there is no transformation of the affine connection that can
simultaneously transform to zero both torsion and non-metricity. For a
discussion of this aspect see iosifidis2020linear . It also immediately
follows from $Corollary$-$1$ that when the hypermomentum sources are switched-
off, the Theory reduces to GR. Therefore, we have:

###### Corollary 2

In vacuum (or more generally when the matter decouples from the connection)
the quadratic Theory is indistinguishable from General Relativity as in
Einstein-Cartan case.

Let us now examine two characteristic cases for the hypermomentum tensor.

###  VI.1 Totally Antisymmetric Hypermomentum (Purely Fermionic Matter)

For purely fermionic matter sectors121212Of course one can always include
additional matter fields that have no dependence on the connection whatsoever.
These, having a vanishing hypermomentum, do not alter the forms of torsion and
non-metricity in any way. the hypermomentum is totally antisymmetric and it is
given by the spin tensor, according to

$\Delta_{\alpha\mu\nu}=\Delta_{[\alpha\mu\nu]}$ (47)

In this case the torsion tensor becomes

$S_{\alpha\mu\nu}=\kappa\Big{(}\tilde{a}_{11}+\tilde{a}_{12}+\tilde{a}_{13}-\tilde{a}_{14}-\tilde{a}_{15}-\tilde{a}_{16}\Big{)}\Delta_{\alpha\mu\nu}$
(48)

while non-metricity identically vanishes

$Q_{\alpha\mu\nu}\equiv 0$ (49)

Therefore, as expected we confirm that fermions cannot excite non-metricity
and only induce spacetime torsion that is totally antisymmetric (i.e. only the
$t^{\mu}$ part of the full tensor is excited).

###  VI.2 Totally Symmetric Hypermomentum

In the case of totally symmetric hypermomentum, that is
$\Delta_{\alpha\mu\nu}=\Delta_{(\alpha\mu\nu)}$, not surprisingly the roles of
torsion and non-metricity are reversed. In particular, for such a matter
configuration, torsion vanishes identically

$S_{\alpha\mu\nu}\equiv 0$ (50)

while non-metricity contains only the totally symmetric part

$Q_{\alpha\mu\nu}=\kappa\Big{(}\tilde{a}_{11}+\tilde{a}_{12}+\tilde{a}_{13}+\tilde{a}_{14}+\tilde{a}_{15}+\tilde{a}_{16}\Big{)}\Delta_{\alpha\mu\nu}$
(51)

We should mention however that even though a totally antisymmetric
hypermomentum tensor possesses some physical substance (i.e fermionic matter),
there are not (as yet) known examples of fundamental particles carrying a
totally symmetric hypermomentum.

##  VII Effective GR with hypermomentum contributions

Having successfully solved for the affine connection in terms of the
hypermomentum sources we can then consider the full post-Riemannian expansion
of the metric field equations (22). After some lengthy calculations (see
appendix for details) we finally find

$\displaystyle\Big{(}\tilde{R}_{\mu\nu}-\frac{\tilde{R}}{2}g_{\mu\nu}\Big{)}=\kappa
T_{\mu\nu}-\tilde{\nabla}_{\alpha}N^{\alpha}_{\;\;\;(\mu\nu)}+\tilde{\nabla}_{(\nu}N^{(2)}_{\mu)}-N^{(2)}_{\lambda}N^{\lambda}_{\;\;\;(\mu\nu)}+N^{\alpha}_{\;\;\;\lambda(\nu}N^{\lambda}_{\;\;\;\mu)\alpha}$
$\displaystyle+\frac{1}{2}g_{\mu\nu}\Big{[}\tilde{\nabla}_{\alpha}(N^{(3)\alpha}-N^{(2)\alpha})+N^{(3)}_{\alpha}N^{(2)\alpha}-N_{\alpha\beta\gamma}N^{\beta\gamma\alpha}\Big{]}$
$\displaystyle+\frac{g_{\mu\nu}}{2}\Big{[}\Big{(}\frac{b_{1}}{2}+2a_{1}-c_{1}\Big{)}N_{\alpha\beta\gamma}N^{\alpha\beta\gamma}+\Big{(}-\frac{b_{1}}{2}+a_{2}+c_{1}\Big{)}N_{\alpha\beta\gamma}N^{\alpha\gamma\beta}$
$\displaystyle+\Big{(}\frac{b_{2}}{2}+2a_{2}+c_{1}\Big{)}N_{\alpha\beta\gamma}N^{\beta\gamma\alpha}+\Big{(}-\frac{b_{2}}{2}+2a_{1}+a_{2}-c_{1}\Big{)}N_{\mu\nu\alpha}N^{\nu\mu\alpha}$
$\displaystyle+\Big{(}\frac{b_{3}}{4}+4a_{3}-c_{2}\Big{)}N^{(1)}_{\alpha}N^{(1)}_{\beta}g^{\alpha\beta}+\Big{(}\frac{b_{3}}{4}+a_{4}+\frac{c_{3}}{4}\Big{)}N^{(2)}_{\alpha}N^{(2)}_{\beta}g^{\alpha\beta}+a_{4}N^{(3)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
$\displaystyle+\Big{(}2a_{5}-\frac{c_{3}}{2}+c_{2}-\frac{b_{3}}{2}\Big{)}N^{(1)}_{\alpha}N^{(2)}_{\beta}g^{\alpha\beta}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N^{(2)}_{\alpha}N^{(3)}_{\beta}g^{\alpha\beta}+\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N^{(1)}_{\alpha}N^{(3)}_{\beta}g^{\alpha\beta}\Big{]}$
$\displaystyle-\Big{(}4a_{1}-\frac{c_{1}}{2}\Big{)}\frac{1}{\sqrt{-g}}\hat{\nabla}_{\beta}\Big{(}\sqrt{-g}g^{\alpha\beta}N_{(\mu\nu)\alpha}\Big{)}-\Big{(}2a_{2}+\frac{c_{1}}{2}\Big{)}\frac{1}{\sqrt{-g}}\hat{\nabla}_{\beta}\Big{(}\sqrt{-g}g^{\alpha\beta}N_{(\mu|\alpha|\nu)}\Big{)}-2a_{2}\frac{1}{\sqrt{-g}}\hat{\nabla}_{\beta}\Big{(}\sqrt{-g}g^{\alpha\beta}N_{\alpha(\mu\nu)}\Big{)}$
$\displaystyle-\frac{1}{\sqrt{-g}}\hat{\nabla}_{\alpha}\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N^{\alpha(1)}g_{\mu\nu}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N^{\alpha(2)}g_{\mu\nu}+a_{5}N^{\alpha(3)}g_{\mu\nu}\right]$
$\displaystyle-\frac{1}{\sqrt{-g}}\hat{\nabla}_{(\mu}\left[\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(1)}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(2)}+2a_{4}N_{\nu)}^{(3)}\right]$
$\displaystyle-\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\alpha\beta\nu}-\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\beta\alpha\nu}-\frac{c_{1}}{2}N^{\alpha\beta}_{\;\;\;\;(\mu}(N_{\beta|\nu)\alpha}+N_{\alpha|\nu)\beta})$
$\displaystyle+(N^{\alpha\;\;\beta}_{\;\;(\nu}+N_{(\nu}^{\;\;\alpha\beta})\left[2a_{1}N_{\alpha|\mu)\beta}+a_{2}N_{\beta|\mu)\alpha}+\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N_{\mu)\alpha\beta}+\Big{(}a_{2}+\frac{c_{1}}{2}\Big{)}N_{\mu)\beta\alpha}\right]$
$\displaystyle-\frac{1}{4}(N_{\beta(\nu|\alpha}-N_{\beta\alpha(\nu})\Big{(}2b_{1}(N^{\beta\;\;\alpha}_{\;\;\mu)}-b_{2}N^{\alpha\;\;\beta}_{\;\;\mu)}-2b_{1}N^{\beta\alpha}_{\;\;\;\mu)}+b_{2}N^{\alpha\beta}_{\;\;\;\mu)}\Big{)}+b_{1}N_{\mu\alpha\beta}N_{\nu}^{\;\;[\alpha\beta]}$
$\displaystyle+(N_{\mu\nu\alpha}+N_{\nu\mu\alpha})\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N^{(1)\alpha}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N^{(2)\alpha}+a_{5}N^{(3)\alpha}\right]$
$\displaystyle-\Big{(}4a_{3}-\frac{c_{3}}{2}\Big{)}N_{\mu}^{(1)}N_{\nu}^{(1)}-\Big{(}\frac{b_{3}}{4}-a_{4}\Big{)}N_{\mu}^{(2)}N_{\nu}^{(2)}+a_{4}N_{\mu}^{(3)}N_{\nu}^{(3)}-\Big{(}c_{3}-\frac{b_{3}}{2}\Big{)}N_{(\mu}^{(1)}N_{\nu)}^{(2)}+2a_{4}N_{(\mu}^{(2)}N_{\nu)}^{(3)}$
(52)

and recall that

$N_{\alpha\mu\nu}=\kappa\Big{(}\tilde{a}_{11}\hat{\Delta}_{\alpha\mu\nu}+\tilde{a}_{12}\hat{\Delta}_{\nu\alpha\mu}+\tilde{a}_{13}\hat{\Delta}_{\mu\nu\alpha}+\tilde{a}_{14}\hat{\Delta}_{\alpha\nu\mu}+\tilde{a}_{15}\hat{\Delta}_{\nu\mu\alpha}+\tilde{a}_{16}\hat{\Delta}_{\mu\alpha\nu}\Big{)}$
(53)

That is, we see that the Theory is GR with modified/extended matter sources.
Indeed, since the distortion is linear in the hypermomentum the extra terms
appearing on the right-hand side of (52) are quadratic on the hypermomentum
and linear in its derivatives. As a result, the source now is not solely given
by the usual energy-momentum tensor of matter but we also have hypermomentum
contributions. In this sense the microstructure of matter (hypermomentum)
modifies the usual matter content in a highly non-trivial manner. Of course,
given the complexity of (52) when all $11$ parameters are included, no
concrete conclusions can be drawn and the exact effects of these contributions
depend both on the number of parameters included and on the given model for
the hypermomentum source that one considers.

##  VIII Generalized Quadratic Theory

It is worth stressing out that in applying the results of $Theorem-1$ in order
to solve equation (39) for the affine connection, nowhere did we assume that
the coefficients $a_{i}$, $b_{j}$ and $c_{k}$ are constants. Subsequently, the
main result for the affine connection will continue to hold true even if we
promote the aforementioned constants to be spacetime functions. In particular,
we have the following result.

###### Theorem 2

Let $\Phi$ be a collection of matter fields (collectively denoted) and $d\Phi$
their exterior derivatives. Furthermore, let
$a_{i}(\Phi,d\Phi)=a_{i}(\Phi,\partial\Phi)$, $b_{j}(\Phi,\partial\Phi)$ and
$c_{j}(\Phi,\partial\Phi)$ $i=1,2,...,5$, $j=1,2,3$ be arbitrary functions
continuous and differentiable in their arguments131313These functions can just
as well depend also on the metric tensor and its derivatives, however not on
the connection.. Then, for the general class of Theories given by the action

$\displaystyle S[g,\Gamma,\Phi]=\frac{1}{2\kappa}\int
d^{n}x\sqrt{-g}\Big{[}R+b_{1}(\Phi,\partial\Phi)S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}(\Phi,\partial\Phi)S_{\alpha\mu\nu}S^{\mu\nu\alpha}+b_{3}(\Phi,\partial\Phi)S_{\mu}S^{\mu}$
$\displaystyle
a_{1}(\Phi,\partial\Phi)Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}(\Phi,\partial\Phi)Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+a_{3}(\Phi,\partial\Phi)Q_{\mu}Q^{\mu}+a_{4}(\Phi,\partial\Phi)q_{\mu}q^{\mu}+a_{5}(\Phi,\partial\Phi)Q_{\mu}q^{\mu}$
$\displaystyle+c_{1}(\Phi,\partial\Phi)Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+c_{2}(\Phi,\partial\Phi)Q_{\mu}S^{\mu}+c_{3}(\Phi,\partial\Phi)q_{\mu}S^{\mu}\Big{]}+S_{M}[g,\Gamma,\Phi]$
(54)

the affine connection is given by

$\Gamma^{\lambda}_{\;\;\;\mu\nu}=\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}+\kappa\Big{(}\tilde{a}_{11}\hat{\Delta}_{\alpha\mu\nu}+\tilde{a}_{12}\hat{\Delta}_{\nu\alpha\mu}+\tilde{a}_{13}\hat{\Delta}_{\mu\nu\alpha}+\tilde{a}_{14}\hat{\Delta}_{\alpha\nu\mu}+\tilde{a}_{15}\hat{\Delta}_{\nu\mu\alpha}+\tilde{a}_{16}\hat{\Delta}_{\mu\alpha\nu}\Big{)}$
(55)

where now $\tilde{\alpha}_{ij}=\tilde{\alpha}_{ij}(\Phi,\partial\Phi)$ and
$\hat{\Delta}_{\mu\nu\alpha}$ is given by (43).

Proof. Following an identical procedure with that of section $VI$ and by
simply promoting the $a_{i}^{\prime}s,b_{j}^{\prime}s$ and $c_{j}^{\prime}s$
to be functions of $\Phi$ and $d\Phi$ we arrive at the stated result.

Comment. Note that in the action (54) are included Scalar-Tensor and Vector-
Tensor Theories as special cases. We have therefore provided the exact form of
the affine connection for Scalar-Tensor and Vector-Tensor Theories whose
actions are of the same form as $(\ref{genquad})$.

##  IX Conclusions

We have considered the most general parity even quadratic MAG Theory in $n$
dimensions. The aforementioned Theory consists of the usual Einstein-Hilbert
term along with the $11$ parity even quadratic terms in torsion and non-
metricity. Additionally, in order to keep things fairly general, we imposed no
restriction on the matter sector, allowing it to have an arbitrary
hypermomentum. Then, after deriving the field equations by varying with
respect to the metric and the independent affine connection, we first focused
on the latter connection field equations. By using a recent result on how to
solve such linear but complicated tensor equations (see cf. Iosifidis:2021ili
) we were able to find the exact solution for the affine connection in terms
of the (arbitrary) hypermomentum and its traces. Given two fairly general non-
degeneracy conditions, the uniqueness of our exact solution is also
guaranteed. With this result we also extracted the exact forms of torsion and
non-metricity for the general Theory.

Having obtained the exact form of the connection, we then used a post-
Riemannian expansion on the metric-field equations and expressed the latter in
an effective GR-like form with modified/extended source terms that depend
quadratically on the hypermomentum and linearly on its derivative. These extra
terms come on top of the usual energy-momentum tensor of matter and modify the
matter content in a highly non-trivial manner, as seen from equation (52).
Furthermore, the non-vanishing of the hypermomentum has a close relation with
the intrinsic properties of matter Puetzfeld:2007ye . With this in mind, we
see that the additional modifications compared to GR have a very nice and
clear physical interpretation, namely, the microstructure of matter modifies
the gravitational field. However, the exact effect of this microstructure in
the general case where all $11$ parameters are included is not so easy to
decode given the complexity of (52). Nevertheless, more manageable expressions
and subsequently a better understanding of these additional hypermomentum
terms can be acquired once a specific choice for the matter sector is made. It
is also important to note that, just like in the Einstein-Cartan Theory, once
the sources are switched off (i.e. vacuum case) the complicated quadratic
Theory reduces to GR. In other words, the new interactions are tied to matter
(i.e. they do not propagate) and as a result outside matter the quadratic
Theory is indistinguishable from GR but changes considerably when matter with
intrinsic structure is considered.

Going one step further we considered the general class of quadratic Metric-
Affine Theories as given by (54) where now the coupling constants between the
torsion and non-metricity invariants have been promoted to functions (of some
given matter fields). Employing the same procedure with the case of constant
coefficients we were able to solve again for the affine connection. In this
generalization specific Scalar-Tensor and Vector-Tensor Metric-Affine Theories
are also included. Therefore, our results here will be of prominent importance
in analysing such Theories. Another very interesting application would be to
compute the right hand-side of (52) for specific matter types (i.e given
$T_{\mu\nu}$ and $\Delta_{\alpha\mu\nu}$ tensors) and see what the new
interactions look like especially in comparison with the Einstein-Cartan
Theory.

##  X Acknowledgments

This research is co-financed by Greece and the European Union (European Social
Fund- ESF) through the Operational Programme ’Human Resources Development,
Education and Lifelong Learning’ in the context of the project “Reinforcement
of Postdoctoral Researchers - 2 nd Cycle” (MIS-5033021), implemented by the
State Scholarships Foundation (IKY).

##  Appendix A Relations among parameters

$\displaystyle\alpha_{1}=4a_{1}+b_{1}-c_{1}\;\;,\;\;\alpha_{2}=\alpha_{3}=-1+2a_{2}+\frac{b_{2}+c_{1}}{2}$
$\displaystyle\alpha_{4}=2a_{2}-b_{1}+c_{1}\;\;,\;\;\alpha_{5}=2a_{2}-\frac{b_{2}}{2}\;\;,\;\;\alpha_{6}=4a_{1}-\frac{b_{2}}{2}-c_{1}$
$\displaystyle\alpha_{71}=2a_{5}+c_{2}-\frac{b_{3}+c_{3}}{2}\;\;,\;\;\alpha_{72}=2a_{4}+c_{3}+\frac{b_{3}}{2}\;\;,\;\;\alpha_{73}=1+2a_{4}+\frac{c_{3}}{2}$
$\displaystyle\alpha_{81}=8a_{3}-2c_{2}+\frac{b_{3}}{2}\;\;,\;\;\alpha_{82}=2a_{5}+c_{2}-\frac{c_{3}+b_{3}}{2}\;\;,\;\;\alpha_{83}=2a_{5}-\frac{c_{3}}{2}$
$\displaystyle\alpha_{91}=2a_{5}-\frac{c_{3}}{2}\;\;,\;\;\alpha_{92}=2a_{4}+\frac{c_{3}}{2}+1\;\;,\;\;\alpha_{93}=2a_{4}$
(56)

###  A.1 Elements of matrix $A$ in terms of Theory parameters

###  A.2 The $\gamma^{\prime}s$

The relations between the elements of $\Gamma$ and the parameters $a_{i}$ read

$\displaystyle\gamma_{11}=a_{1}+a_{3}+a_{71}+na_{81}+a_{91}\;\;,\;\;\gamma_{12}=a_{2}+a_{4}+a_{72}+na_{82}+a_{92}\;\;,\;\;\gamma_{13}=a_{5}+a_{6}+a_{73}+na_{83}+a_{93}$
$\displaystyle\gamma_{21}=a_{2}+a_{5}+na_{71}+a_{81}+a_{91}\;\;,\;\;\gamma_{22}=a_{1}+a_{6}+na_{72}+a_{82}+a_{92}\;\;,\;\;\gamma_{23}=a_{3}+a_{4}+na_{73}+a_{83}+a_{93}$
$\displaystyle\gamma_{31}=a_{5}+a_{6}+a_{71}+a_{81}+na_{91}\;\;,\;\;\gamma_{32}=a_{3}+a_{4}+a_{72}+a_{82}+na_{92}\;\;,\;\;\gamma_{31}=a_{1}+a_{2}+a_{73}+a_{83}+na_{93}$

##  Appendix B Post-Riemannian Expansions

###  B.1 Distortion vectors in terms of torsion and non-metricity vectors

Contracting (5) in all possible ways with the metric we find

$N_{(1)}^{\mu}=\frac{1}{2}Q^{\mu}\;\;,\;\;N_{(2)}^{\mu}=\frac{1}{2}Q^{\mu}+2S^{\mu}\;\;,\;\;N_{(3)}^{\mu}=q^{\mu}-\frac{1}{2}Q^{\mu}-2S^{\mu}$
(57)

which relate the distortion vectors in terms of those of torsion and non-
metricity. Inverting them we have

$Q^{\mu}=2N_{(1)}^{\mu}\;\;,\;\;q^{\mu}=N_{(2)}^{\mu}+N_{(3)}^{\mu}\;\;,\;\;S^{\mu}=\frac{1}{2}\Big{(}N_{(2)}^{\mu}-N_{(1)}^{\mu}\Big{)}$
(58)

Then, using again (5) along with the above relations and the definition of the
Palatini tensor we can express the latter solely in terms of the distortion as

$P^{\alpha\mu\nu}=g^{\mu\nu}N_{(2)}^{\alpha}+g^{\nu\alpha}N_{(3)}^{\mu}-(N^{\mu\nu\alpha}+N^{\nu\alpha\mu})$
(59)

###  B.2 Post-Riemannian expansions

The Ricci tensor is expanded as

$R_{\nu\beta}=\tilde{R}_{\nu\beta}+\tilde{\nabla}_{\mu}N^{\mu}_{\;\;\;\nu\beta}-\tilde{\nabla}_{\beta}N^{(2)}_{\nu}+N^{(2)}_{\lambda}N^{\lambda}_{\;\;\;\nu\beta}-N^{\mu}_{\;\;\;\rho\beta}N^{\rho}_{\;\;\;\nu\mu}$
(60)

and the Ricci scalar as

$R=\tilde{R}+\tilde{\nabla}_{\mu}(N^{(3)\mu}-N^{(2)\mu})+N^{(3)}_{\mu}N^{(2)\mu}-N_{\alpha\mu\nu}N^{\mu\nu\alpha}$
(61)

As a result for the Einstein-like combination we find

$\displaystyle
R_{(\mu\nu)}-\frac{R}{2}g_{\mu\nu}=\Big{(}\tilde{R}_{(\mu\nu)}-\frac{\tilde{R}}{2}g_{\mu\nu}\Big{)}+\tilde{\nabla}_{\alpha}N^{\alpha}_{\;\;\;(\mu\nu)}-\tilde{\nabla}_{(\nu}N^{(2)}_{\mu)}+N^{(2)}_{\lambda}N^{\lambda}_{\;\;\;(\mu\nu)}-N^{\alpha}_{\;\;\;\lambda(\nu}N^{\lambda}_{\;\;\;\mu)\alpha}$
$\displaystyle-\frac{1}{2}g_{\mu\nu}\Big{[}\tilde{\nabla}_{\alpha}(N^{(3)\alpha}-N^{(2)\alpha})+N^{(3)}_{\alpha}N^{(2)\alpha}-N_{\alpha\beta\gamma}N^{\beta\gamma\alpha}\Big{]}$
(62)

Continuing we calculate

$\displaystyle
b_{1}S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}S_{\alpha\mu\nu}S^{\mu\nu\alpha}+a_{1}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}++c_{1}Q_{\alpha\mu\nu}S^{\alpha\mu\nu}=$
$\displaystyle\Big{(}\frac{b_{1}}{2}+2a_{1}-c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\mu\nu}+\Big{(}-\frac{b_{1}}{2}+a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\nu\mu}$
$\displaystyle+\Big{(}\frac{b_{2}}{2}+2a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\mu\nu\alpha}+\Big{(}-\frac{b_{2}}{2}+2a_{1}+a_{2}-c_{1}\Big{)}N_{\mu\nu\alpha}N^{\nu\mu\alpha}$
(63)

and also

$\displaystyle
b_{3}S_{\mu}S^{\mu}+a_{3}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+a_{5}Q_{\mu}q^{\mu}+c_{2}Q_{\mu}S^{\mu}+c_{3}q_{\mu}S^{\mu}=$
$\displaystyle=\Big{(}\frac{b_{3}}{4}+4a_{3}-c_{2}\Big{)}N^{(1)}_{\mu}N^{(1)}_{\nu}g^{\mu\nu}+\Big{(}\frac{b_{3}}{4}+a_{4}+\frac{c_{3}}{4}\Big{)}N^{(2)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+a_{4}N^{(3)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
$\displaystyle+\Big{(}2a_{5}-\frac{c_{3}}{2}+c_{2}-\frac{b_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N^{(2)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}+\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
(64)

such that

$\displaystyle\mathcal{L}_{2}=\Big{(}\frac{b_{1}}{2}+2a_{1}-c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\mu\nu}+\Big{(}-\frac{b_{1}}{2}+a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\nu\mu}$
$\displaystyle+\Big{(}\frac{b_{2}}{2}+2a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\mu\nu\alpha}+\Big{(}-\frac{b_{2}}{2}+2a_{1}+a_{2}-c_{1}\Big{)}N_{\mu\nu\alpha}N^{\nu\mu\alpha}$
$\displaystyle=\Big{(}\frac{b_{3}}{4}+4a_{3}-c_{2}\Big{)}N^{(1)}_{\mu}N^{(1)}_{\nu}g^{\mu\nu}+\Big{(}\frac{b_{3}}{4}+a_{4}+\frac{c_{3}}{4}\Big{)}N^{(2)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+a_{4}N^{(3)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
$\displaystyle+\Big{(}2a_{5}-\frac{c_{3}}{2}+c_{2}-\frac{b_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N^{(2)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}+\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
(65)

Finally, we calculate

$\displaystyle
W_{\alpha(\mu\nu)}+\Pi_{\alpha(\mu\nu)}=\Big{(}4a_{1}-\frac{c_{1}}{2}\Big{)}N_{(\mu\nu)\alpha}+\Big{(}2a_{2}+\frac{c_{1}}{2}\Big{)}N_{(\mu|\alpha|\nu)}+2a_{2}N_{\alpha(\mu\nu)}$
$\displaystyle\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N_{\alpha}^{(1)}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N_{\alpha}^{(2)}+a_{5}N_{\alpha}^{(3)}\right]g_{\mu\nu}$
$\displaystyle+g_{\alpha(\mu}\left[\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(1)}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(2)}+2a_{4}N_{\nu)}^{(3)}\right]$
(66)

as well as

$\displaystyle
A_{(\mu\nu)}+B_{(\mu\nu)}+C_{(\mu\nu)}=Q_{(\mu}^{\;\;\;\;\alpha\beta}\Big{(}a_{1}Q_{\nu)\alpha\beta}+c_{1}S_{\nu)\alpha\beta}\Big{)}-\Big{(}2a_{1}Q_{\alpha\beta(\mu}+c_{1}S_{\alpha\beta(\mu}+a_{2}Q_{\beta\alpha(\mu}\Big{)}Q^{\alpha\beta}_{\;\;\;\;\nu)}$
$\displaystyle+\Big{(}2b_{1}S_{(\mu}^{\;\;\;\alpha\beta}-b_{2}S_{(\mu}^{\;\;\;\beta\alpha}\Big{)}S_{\nu)\alpha\beta}-b_{1}S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\nu}-Q_{\alpha\mu\nu}\Big{(}2a_{3}Q^{\alpha}+a_{5}q^{\alpha}+c_{2}S^{\alpha}\Big{)}$
$\displaystyle+a_{3}Q_{\mu}Q_{\nu}-a_{4}q_{\mu}q_{\nu}+b_{3}S_{\mu}S_{\nu}+c_{2}S_{(\mu}Q_{\nu)}=$
$\displaystyle=\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\alpha\beta\nu}+\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\beta\alpha\nu}+\frac{c_{1}}{2}N^{\alpha\beta}_{\;\;\;\;(\mu}(N_{\beta|\nu)\alpha}+N_{\alpha|\nu)\beta})$
$\displaystyle-(N^{\alpha\;\;\beta}_{\;\;(\nu}+N_{(\nu}^{\;\;\alpha\beta})\left[2a_{1}N_{\alpha|\mu)\beta}+a_{2}N_{\beta|\mu)\alpha}+\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N_{\mu)\alpha\beta}+\Big{(}a_{2}+\frac{c_{1}}{2}\Big{)}N_{\mu)\beta\alpha}\right]$
$\displaystyle+\frac{1}{4}(N_{\beta(\nu|\alpha}-N_{\beta\alpha(\nu})\Big{(}2b_{1}(N^{\beta\;\;\alpha}_{\;\;\mu)}-b_{2}N^{\alpha\;\;\beta}_{\;\;\mu)}-2b_{1}N^{\beta\alpha}_{\;\;\;\mu)}+b_{2}N^{\alpha\beta}_{\;\;\;\mu)}\Big{)}-b_{1}N_{\mu\alpha\beta}N_{\nu}^{\;\;[\alpha\beta]}$
$\displaystyle-(N_{\mu\nu\alpha}+N_{\nu\mu\alpha})\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N^{(1)\alpha}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N^{(2)\alpha}+a_{5}N^{(3)\alpha}\right]$
$\displaystyle+\Big{(}4a_{3}-\frac{c_{3}}{2}\Big{)}N_{\mu}^{(1)}N_{\nu}^{(1)}+\Big{(}\frac{b_{3}}{4}-a_{4}\Big{)}N_{\mu}^{(2)}N_{\nu}^{(2)}-a_{4}N_{\mu}^{(3)}N_{\nu}^{(3)}$
$\displaystyle+\Big{(}c_{3}-\frac{b_{3}}{2}\Big{)}N_{(\mu}^{(1)}N_{\nu)}^{(2)}-2a_{4}N_{(\mu}^{(2)}N_{\nu)}^{(3)}$
(67)

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