# Stability of non-degenerate Ricci-type Palatini theories

Jaakko Annala  and Syksy Räsänen

###### Abstract

We study the stability of theories where the gravitational action has
arbitrary algebraic dependence on the three first traces of the Riemann
tensor: the Ricci tensor, the co-Ricci tensor, and the homothetic curvature
tensor. We collectively call them Ricci-type tensors. We allow arbitrary
coupling to matter. We consider the case when the connection is unconstrained,
and the cases when either torsion or non-metricity is assumed to vanish. We
find which combinations of Ricci-type tensors lead to new degrees of freedom
around Minkowski and FLRW space, and when there are ghosts. None of the
theories with new degrees of freedom are healthy, except the previously known
case when torsion is zero and the action depends only on the Ricci tensor. We
find that projective invariance is not a sufficient condition for a theory to
be ghost-free.

HIP-2022-33/TH

##  1 Introduction

Renormalisation in quantum field theory in curved spacetime requires higher
order curvature terms, and they also appear in proposed ultraviolet
completions of general relativity [1, 2, 3, 4, 5]. In the metric formulation
of gravity, non-linear curvature terms generally lead to higher derivative
equations of motion that have new degrees of freedom that suffer from the
Ostrogradski instability [6, 7, 8, 9, 10, 11]. (It is possible that this
classical instability does not persist in quantum theory [12].) In 4 spacetime
dimensions, non-linear curvature terms always (boundary terms aside) lead to
higher derivative equations of motion and new degrees of freedom, but not all
of them are unstable. In 4 dimensions, the most general healthy action with no
matter and no dependence on derivatives of the Riemann tensor depends on the
curvature only via Ricci scalar. With arbitrary algebraic dependence on the
Ricci scalar, the action has one extra non-ghost scalar degree of freedom
[13]. When matter is added, there are further stable cases, such as the
Horndeski and beyond Horndeski theories that include scalar fields [14].

In the Palatini formulation, the metric and the connection are independent
degrees of freedom [15, 16, 17, 18]. The equations of motion are therefore
second order for any action that is algebraic in the Riemann tensor and where
the connection appears only via the Riemann tensor.111Fermions, unlike scalar
fields and gauge fields, necessarily couple to a connection in the action,
although this can be taken to be the Levi–Civita connection rather than the
full connection [16, 19, 20, 21, 22, 23, 24, 25, 26, 27]. For this reason
higher order curvature terms do not necessarily lead to new degrees of
freedom, and new degrees of freedom do not involve the Ostrogradski
instability. However, the theory can still be unstable because kinetic terms
or gradient terms can have the wrong sign, or potentials can be tachyonic or
unbounded from below [28, 29, 30, 31, 32]. (Fields with a kinetic term of the
wrong sign may not be a problem in the quantum theory [33, 34].) As in the
metric case, requiring the theory to be stable greatly restricts the action.
However, unlike in the metric case, the most general stable theory is not
known even when there is no matter. In addition to the Riemann curvature
tensor, the geometrical properties of the manifold are described by the non-
metricity tensor and the torsion tensor. There are hence many more
gravitational terms than in the metric case, and terms that individually
correspond to an unstable theory can lead to a stable theory when combined
[28, 31, 32].

Many Palatini theories have been considered in the literature. It is known
that a theory with arbitrary algebraic dependence on the symmetric part of the
Ricci tensor does not contain any new degrees of freedom [35, 36, 37, 38]. If
the torsion is taken to be zero, a theory with arbitrary algebraic dependence
on the antisymmetric part of the Ricci tensor is equivalent to normal gravity
with an additional healthy vector degree of freedom [39, 40, 29]. The
stability properties of some parity-invariant theories quadratic in the
curvature, non-metricity, and torsion and are also known [41, 42, 31, 43].

In the Palatini formulation, the Einstein–Hilbert action is invariant under
the projective transformation of the connection [17]. It has been suggested
that requiring projective invariance also for extended actions guarantees
their stability [44, 29, 30]. Stable Palatini theories without projective
symmetry are known, so it is not a necessary condition for stability [45, 29,
30].

We extend previous work by considering the stability of theories where the
connection enters only via the first contractions of the Riemann tensor. There
are three independent first contractions: the Ricci tensor, the co-Ricci
tensor, and the homothetic curvature tensor. We collectively call them Ricci-
type curvature tensors, or just Ricci-type tensors.

Hamiltonian analysis is the definitive way to establish the dynamical content
and stability of a theory [46]. However, with these complicated theories the
analysis can be quite involved. We take a simpler route. We perform a Legendre
transformation and introduce auxiliary fields to make the theory linear in the
Riemann tensor. Expanding around Minkowski space or the
Friedmann–Lemaître–Robertson–Walker (FLRW) universe to first order, we solve
for the connection and insert the solution back into the action. Having
reduced the theory to metric gravity with the Einstein–Hilbert action plus
minimally coupled matter, we then look at the kinetic sector of the new matter
fields.

This method has its limits. We are restricted to non-degenerate theories,
meaning theories for which the Legendre transformation is invertible. We also
cannot find the dynamical content of the theory in general, only around
specific backgrounds. If the theory is found to contain unstable degrees of
freedom, this is sufficient to rule it out, but the reverse does not hold. It
is possible for a theory to be well-behaved around one background but unstable
around another. A theory may also contain new degrees of freedom around one
background but not another. This can indicate a strong coupling problem [47,
48, 49, 50]. We find examples of both situations. We also find a projectively
symmetric theory where perturbations around Minkowski space are unstable,
showing that projective symmetry is not a sufficient condition for stability.

In section 2 we first introduce the relevant geometrical quantities and the
action. We then make the Legendre transformation and introduce the auxiliary
fields. We look at the degrees of freedom and stability in three cases: the
general case when the connection is unconstrained, the zero torsion case, and
the zero non-metricity case. We consider perturbations around the Minkowski
background, and in the general case also around the spatially flat FLRW
background, and find when kinetic terms have the wrong sign. We summarise our
findings and outline open questions in section 3.

##  2 Degrees of freedom

###  2.1 Geometrical quantities and the action

####  2.1.1 Torsion and non-metricity

We work in the Palatini formulation, where the metric and the connection
$\Gamma^{\gamma}_{\alpha\beta}$ are independent degrees of freedom. The
connection is related to the covariant derivative as
$\nabla_{\beta}A^{\alpha}=\partial_{\beta}A^{\alpha}+\Gamma^{\alpha}_{\beta\gamma}A^{\gamma}$,
$\nabla_{\beta}A_{\alpha}=\partial_{\beta}A_{\alpha}-\Gamma^{\gamma}_{\beta\alpha}A_{\gamma}$.
We can decompose the connection as

$\Gamma^{\gamma}_{\alpha\beta}=\overset{g}{\mathring{\Gamma}}{}^{\gamma}_{\alpha\beta}+L\indices{{}^{\gamma}_{\alpha\beta}}\
,$ (2.1)

where $\overset{g}{\mathring{\Gamma}}{}^{\gamma}_{\alpha\beta}$ is the
Levi–Civita connection of the metric $g_{\alpha\beta}$, and
$L\indices{{}^{\gamma}_{\alpha\beta}}$ is the related distortion tensor. In
this decomposition we can define the Levi–Civita connection with respect to
any metric, it does not have to be the physical spacetime metric. Both the
Levi–Civita connection and the distortion tensor depend on the choice of the
metric, while their sum does not.

The non-metricity tensor and the two non-metricity vectors defined with the
metric $g_{\alpha\beta}$ are

$\displaystyle\overset{g}{Q}_{\gamma\alpha\beta}\equiv\nabla_{\gamma}g_{\alpha\beta}=-2L_{(\alpha|\gamma|\beta)}\
,\quad\overset{g}{Q}_{\gamma}\equiv\overset{g}{Q}_{\gamma\alpha\beta}g^{\alpha\beta}\
,\quad\overset{g}{\hat{Q}}_{\beta}\equiv\overset{g}{Q}_{\gamma\alpha\beta}g^{\gamma\alpha}\
,$ (2.2)

where $L_{\gamma\alpha\beta}\equiv g_{\gamma\mu}L^{\mu}{}_{\alpha\beta}$. The
torsion tensor and the torsion vector are defined independently of the metric
as

$\displaystyle T\indices{{}^{\gamma}_{\alpha\beta}}\equiv
2\Gamma^{\gamma}_{[\alpha\beta]}=2L\indices{{}^{\gamma}_{[\alpha\beta]}}\
,\quad T_{\beta}\equiv T\indices{{}^{\alpha}_{\beta\alpha}}\ .$ (2.3)

The distortion can be written in terms of non-metricity and torsion as

$L\indices{{}^{\gamma}_{\alpha\beta}}=J\indices{{}^{\gamma}_{\alpha\beta}}+K\indices{{}^{\gamma}_{\alpha\beta}}\
,$ (2.4)

where the disformation and the contortion are, respectively,

$J^{\alpha}{}_{\beta\gamma}\equiv\frac{1}{2}g^{\alpha\mu}(\overset{g}{Q}_{\mu\beta\gamma}-\overset{g}{Q}_{\gamma\mu\beta}-\overset{g}{Q}_{\beta\mu\gamma})\
,\quad
K\indices{{}^{\alpha}_{\beta\gamma}}\equiv\frac{1}{2}(T\indices{{}^{\alpha}_{\beta\gamma}}+g_{\gamma\mu}g^{\alpha\nu}T^{\mu}{}_{\nu\beta}+g_{\beta\mu}g^{\alpha\nu}T^{\mu}{}_{\nu\gamma})\
.$ (2.5)

Note that $J_{\alpha\beta\gamma}=J_{\alpha(\beta\gamma)}$ and
$K\indices{{}^{\alpha}_{\beta}{}^{\gamma}}=K\indices{{}^{[\alpha}_{\beta}{}^{\gamma]}}$,
with indices lowered with $g_{\alpha\beta}$ and raised with its inverse. The
tensors $L^{\alpha}{}_{\beta\gamma}$, $J^{\alpha}{}_{\beta\gamma}$, and
$K\indices{{}^{\alpha}_{\beta\gamma}}$ all depend on the choice of the metric,
but to avoid overburdening the notation we do not label them with g, as it is
usually clear from the context which metric is used. Transformations of the
metric shift pieces of the connection between
$\overset{g}{\mathring{\Gamma}}{}^{\gamma}_{\alpha\beta}$,
$J^{\alpha}{}_{\beta\gamma}$, and $K\indices{{}^{\alpha}_{\beta\gamma}}$.

####  2.1.2 The Ricci-type tensors

The Riemann tensor depends only on the connection,

$R^{\alpha}{}_{\beta\gamma\delta}\equiv\partial_{\gamma}\Gamma^{\alpha}_{\delta\beta}-\partial_{\delta}\Gamma^{\alpha}_{\gamma\beta}+\Gamma^{\alpha}_{\gamma\mu}\Gamma^{\mu}_{\delta\beta}-\Gamma^{\alpha}_{\delta\mu}\Gamma^{\mu}_{\gamma\beta}\
.$ (2.6)

With the connection decomposition (2.1), we can write (2.6) in terms of
Levi–Civita and distortion contributions as

$\displaystyle
R^{\alpha}{}_{\beta\gamma\delta}=\overset{g}{\mathring{R}}{}^{\alpha}{}_{\beta\gamma\delta}+2\overset{g}{\mathring{\nabla}}_{[\gamma}L^{\alpha}{}_{\delta]\beta}+2L^{\alpha}{}_{[\gamma|\mu|}L^{\mu}{}_{\delta]\beta}\
,$ (2.7)

where $\overset{g}{\mathring{}}$ denotes a quantity defined with the
Levi–Civita connection
$\overset{g}{\mathring{\Gamma}}{}^{\gamma}_{\alpha\beta}$.

For a general connection the only symmetry of the Riemann tensor is
antisymmetry in the last two indices. There are thus three independent first
contractions of the Riemann tensor: the Ricci tensor, the co-Ricci tensor, and
the homothetic curvature tensor, respectively defined as

$\displaystyle R_{\alpha\beta}$ $\displaystyle\equiv
R\indices{{}^{\mu}_{\alpha\mu\beta}}$
$\displaystyle\hat{R}\indices{{}^{\alpha}_{\beta}}$ $\displaystyle\equiv
g^{\mu\nu}R\indices{{}^{\alpha}_{\mu\nu\beta}}$
$\displaystyle\accentset{\sim}{R}_{\alpha\beta}$ $\displaystyle\equiv
R\indices{{}^{\mu}_{\mu\alpha\beta}}\ .$ (2.8)

We collectively refer to these three quantities (and their linear
combinations) as Ricci-type curvature tensors, or just Ricci-type tensors. The
homothetic curvature tensor $\hat{R}\indices{{}^{\alpha}_{\beta}}$ depends on
the metric, while the other two Ricci-type tensors depend only on the
connection. The full contraction of the Riemann tensor is unique, since
$R\equiv
g^{\alpha\beta}R_{\alpha\beta}=-\hat{R}\indices{{}^{\alpha}_{\alpha}}$ and
$g^{\alpha\beta}\accentset{\sim}{R}_{\alpha\beta}=0$.

Inserting the decomposition of the connection in terms of non-metricity and
torsion (2.1)–(2.5) into the decomposition (2.7) of the Riemann tensor, we can
write the co-Ricci tensor as

$\hat{R}_{\alpha\beta}=-R_{\alpha\beta}+2g^{\mu\nu}\nabla_{[\beta}\overset{g}{Q}_{\mu]\nu\alpha}-T^{\mu\nu}{}_{\beta}\overset{g}{Q}_{\mu\nu\alpha}\
.$ (2.9)

Instead of the co-Ricci tensor, it will turn out to be more convenient to use
the average of the co-Ricci tensor and the Ricci tensor,

$\accentset{\Delta}{R}_{\alpha\beta}\equiv\frac{1}{2}(\hat{R}_{\alpha\beta}+R_{\alpha\beta})=g^{\mu\nu}\nabla_{[\beta}\overset{g}{Q}_{\mu]\nu\alpha}-\frac{1}{2}T^{\mu\nu}{}_{\beta}\overset{g}{Q}_{\mu\nu\alpha}\
.$ (2.10)

Note that $g^{\alpha\beta}\accentset{\Delta}{R}_{\alpha\beta}=0$. Similar
decomposition for the homothetic curvature tensor gives

$\displaystyle\accentset{\sim}{R}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\partial_{[\beta}\overset{g}{Q}_{\alpha]}\ .$ (2.11)

So the homothetic curvature tensor is the exterior derivative of the non-
metricity vector. This decomposition obscures the fact that
$\accentset{\sim}{R}_{\alpha\beta}$ is independent of the metric, as is
transparent from the definition (2.1.2). We can choose to do the decomposition
(2.11) with respect to any metric. So a sufficient (but not necessary)
condition for $\accentset{\sim}{R}_{\alpha\beta}$ to vanish is that the
manifold is such that there exists any symmetric tensor $q_{\alpha\beta}$ that
is non-degenerate ($\det q_{\alpha\beta}\neq 0$) and covariantly constant
($\nabla_{\gamma}q_{\alpha\beta}=0$).

####  2.1.3 Projective transformation

An important property of the Einstein–Hilbert action in the Palatini
formulation is invariance under the projective transformation [17]

$\Gamma^{\gamma}_{\alpha\beta}\to\Gamma^{\gamma}_{\alpha\beta}+\delta\indices{{}^{\gamma}_{\beta}}V_{\alpha}\
,$ (2.12)

where $V_{\alpha}$ is an arbitrary vector. Under this transformation non-
metricity and torsion change as

$\displaystyle\overset{g}{Q}_{\gamma\alpha\beta}$
$\displaystyle\to\overset{g}{Q}_{\gamma\alpha\beta}-2V_{\gamma}g_{\alpha\beta}$
$\displaystyle T\indices{{}^{\gamma}_{\alpha\beta}}$ $\displaystyle\to
T\indices{{}^{\gamma}_{\alpha\beta}}+2\delta\indices{{}^{\gamma}_{[\beta}}V_{\alpha]}\
.$ (2.13)

Thus the non-metricity and torsion vectors change as

$\displaystyle\overset{g}{Q}_{\alpha}$ $\displaystyle\to$
$\displaystyle\overset{g}{Q}_{\alpha}-8V_{\alpha}$
$\displaystyle\overset{g}{\hat{Q}}_{\alpha}$ $\displaystyle\to$
$\displaystyle\overset{g}{\hat{Q}}_{\alpha}-2V_{\alpha}$ $\displaystyle
T_{\alpha}$ $\displaystyle\to$ $\displaystyle T_{\alpha}+3V_{\alpha}\ .$
(2.14)

The traceless part of the non-metricity tensor and the part of the torsion
tensor that does not contribute to $T_{\alpha}$ are invariant. Therefore, if
the theory is invariant under the projective transformation, we can completely
exchange torsion for non-metricity if the torsion tensor can be written in
terms of the torsion vector,
$T\indices{{}^{\gamma}_{\alpha\beta}}=-\frac{2}{3}\delta\indices{{}^{\gamma}_{[\alpha}}T_{\beta]}$.
Equivalently, if non-metricity is involved only via the non-metricity vectors
$\overset{g}{Q}_{\alpha}$ and $\overset{g}{\hat{Q}}$, and they are related by
$\overset{g}{Q}_{\alpha}=4\overset{g}{\hat{Q}}$, it can be exchanged for
torsion.222This is the case in $f(R)$ gravity [51].

The Riemann tensor transforms as

$\displaystyle R^{\alpha}{}_{\beta\gamma\delta}\to
R^{\alpha}{}_{\beta\gamma\delta}+2\delta^{\alpha}{}_{\beta}\partial_{[\gamma}V_{\delta]}\
,$ (2.15)

so the Ricci-type tensors (2.1.2) and (2.10) transform as

$\displaystyle R_{\alpha\beta}\to
R_{\alpha\beta}+2\partial_{[\alpha}V_{\beta]}$
$\displaystyle\hat{R}_{\alpha\beta}\to\hat{R}_{\alpha\beta}+2\partial_{[\alpha}V_{\beta]}$
$\displaystyle\accentset{\sim}{R}_{\alpha\beta}\to\accentset{\sim}{R}_{\alpha\beta}+8\partial_{[\alpha}V_{\beta]}$
$\displaystyle\accentset{\Delta}{R}_{\alpha\beta}\to\accentset{\Delta}{R}_{\alpha\beta}+2\partial_{[\alpha}V_{\beta]}\
.$ (2.16)

The symmetric parts of the Ricci-type tensors are invariant under the
projective transformation. Also, the antisymmetric combination

$\displaystyle\accentset{P}{R}_{\alpha\beta}\equiv(\alpha+4\beta)R_{[\alpha\beta]}-\alpha\accentset{\Delta}{R}_{[\alpha\beta]}-\beta\accentset{\sim}{R}_{\alpha\beta}\
,$ (2.17)

is projectively invariant; here $\alpha$ and $\beta$ are any projectively
invariant quantities.

####  2.1.4 Action and Legendre transformation

We consider the most general Palatini theory where the action depends
algebraically on the Ricci-type tensors, and has no other dependence on the
connection. We can take the action to depend on any set of independent
combinations of the Ricci-type tensors. For convenience we choose
$R_{\alpha\beta},\accentset{\Delta}{R}_{(\alpha\beta)},\accentset{P}{R}_{\alpha\beta}$,
and $\accentset{\sim}{R}_{\alpha\beta}$. The action reads

$S=\int\mathrm{d}^{4}x\sqrt{-g}\frac{1}{2}F(g_{\alpha\beta},R_{\alpha\beta},\accentset{\Delta}{R}_{(\alpha\beta)},\accentset{P}{R}_{\alpha\beta},\accentset{\sim}{R}_{\alpha\beta},\Psi,\partial\Psi)\
,$ (2.18)

where $g\equiv\det{g_{\alpha\beta}}$, and $\Psi$ and $\partial\Psi$
collectively denote matter fields and their derivatives (which can be higher
than first order), with arbitrary coupling to the Ricci-type tensors. In order
to have all of the degrees of freedom of the Ricci-type tensors represented,
we must have $\alpha\neq 0$ in $\accentset{P}{R}_{\alpha\beta}$ defined in
(2.17). Also, the terms in $\accentset{P}{R}_{\alpha\beta}$ proportional to
$\beta$ are already separately variables in the action. So without loss of
generality we set $\alpha=1$, $\beta=0$. We perform a Legendre transformation
to make the action linear in the Ricci-type tensors [52, 53, 54, 55, 35]

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\frac{1}{2}\left[F\left(g_{\alpha\beta},\Sigma_{\alpha\beta},\Delta_{\alpha\beta},\Pi_{\alpha\beta},\Theta_{\alpha\beta},\Psi,\partial\Psi\right)+\frac{\partial
F}{\partial\Sigma_{\alpha\beta}}(R_{\alpha\beta}-\Sigma_{\alpha\beta})\right.$
(2.19) $\displaystyle\left.+\frac{\partial
F}{\partial\Delta_{\alpha\beta}}(\accentset{\Delta}{R}_{(\alpha\beta)}-\Delta_{\alpha\beta})+\frac{\partial
F}{\partial\Pi_{\alpha\beta}}(\accentset{P}{R}_{\alpha\beta}-\Pi_{\alpha\beta})+\frac{\partial
F}{\partial\Theta_{\alpha\beta}}(\accentset{\sim}{R}_{\alpha\beta}-\Theta_{\alpha\beta})\right]\
,$

where $\Sigma_{\alpha\beta}$, $\Delta_{\alpha\beta}=\Delta_{(\alpha\beta)}$,
$\Pi_{\alpha\beta}=\Pi_{[\alpha\beta]}$, and
$\Theta_{\alpha\beta}=\Theta_{[\alpha\beta]}$ are auxiliary fields. Let us
check when the action (2.19) is equivalent to the original action (2.18). Let
us first consider $\Sigma_{\alpha\beta}$. If the action is linear in
$\Sigma_{\alpha\beta}$, equivalence follows trivially. Let us then assume that
this is not the case, so
$\partial^{2}F/(\partial\Sigma_{\alpha\beta}\partial\Sigma_{\gamma\delta})\neq
0$. Variation with respect to $\Sigma_{\alpha\beta}$ gives

$\displaystyle\frac{\partial^{2}F}{\partial\Sigma_{\alpha\beta}\partial\Sigma_{\gamma\delta}}\left(R_{\gamma\delta}-\Sigma_{\gamma\delta}\right)=0\
.$ (2.20)

If $\partial^{2}F/(\partial\Sigma_{\alpha\beta}\partial\Sigma_{\gamma\delta})$
is invertible with respect to the indices $\alpha\beta$ (or equivalently
$\gamma\delta$), (2.20) gives $\Sigma_{\alpha\beta}=R_{\alpha\beta}$. The
invertibility requirement imposes a constraint on the form of $F$. We need the
same assumption also for the second partial derivatives of $F$ with respect to
$\Delta_{\alpha\beta}$, $\Pi_{\alpha\beta}$, and $\Theta_{\alpha\beta}$. We
call theories where $F$ satisfies these constraints non-degenerate. For such
theories we also get
$\Delta_{\alpha\beta}=\accentset{\Delta}{R}_{(\alpha\beta)}$,
$\Pi_{\alpha\beta}=\accentset{P}{R}_{\alpha\beta}$, and
$\Theta_{\alpha\beta}=\accentset{\sim}{R}_{\alpha\beta}$. Inserting these
solutions back into (2.19) recovers the action (2.18).

We now introduce the field redefinitions (we use units where the Planck mass
is unity)

$\displaystyle\sqrt{-g}\frac{\partial F}{\partial\Sigma_{(\alpha\beta)}}$
$\displaystyle\equiv$ $\displaystyle\sqrt{-q}q^{\alpha\beta}\
,\quad\sqrt{-g}\frac{\partial
F}{\partial\Sigma_{[\alpha\beta]}}\equiv\sqrt{-q}B^{\alpha\beta}$ (2.21)
$\displaystyle\sqrt{-g}\frac{\partial F}{\partial\Delta_{\alpha\beta}}$
$\displaystyle\equiv$ $\displaystyle\sqrt{-q}S^{\alpha\beta}\
,\quad\sqrt{-g}\frac{\partial
F}{\partial\Pi_{\alpha\beta}}\equiv\sqrt{-q}P^{\alpha\beta}$ (2.22)
$\displaystyle\sqrt{-g}\frac{\partial F}{\partial\Theta_{\alpha\beta}}$
$\displaystyle\equiv$ $\displaystyle\sqrt{-q}H^{\alpha\beta}\ ,$ (2.23)

where $q\equiv 1/\det q^{\alpha\beta}=\det q_{\alpha\beta}$, with
$q_{\alpha\beta}$ being the inverse of $q^{\alpha\beta}$. (The inverse has to
exist for the field redefinition (2.21) to be consistent.) In terms of these
new fields, the action (2.19) reads

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}\frac{1}{2}\Bigg{[}q^{\alpha\beta}R_{(\alpha\beta)}+B^{\alpha\beta}R_{[\alpha\beta]}+S^{\alpha\beta}\accentset{\Delta}{R}_{(\alpha\beta)}+\Pi^{\alpha\beta}\accentset{P}{R}_{\alpha\beta}+H^{\alpha\beta}\accentset{\sim}{R}_{\alpha\beta}$
(2.24)
$\displaystyle+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+B^{\alpha\beta})\Sigma_{\alpha\beta}-S^{\alpha\beta}\Delta_{\alpha\beta}-P^{\alpha\beta}\Pi_{\alpha\beta}-H^{\alpha\beta}\Theta_{\alpha\beta}\Bigg{]}$
$\displaystyle\equiv$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}\frac{1}{2}\Bigg{[}R^{\alpha}{}_{\beta\gamma\delta}f_{\alpha}{}^{\beta\gamma\delta}$
$\displaystyle+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+B^{\alpha\beta})\Sigma_{\alpha\beta}-S^{\alpha\beta}\Delta_{\alpha\beta}-P^{\alpha\beta}\Pi_{\alpha\beta}-H^{\alpha\beta}\Theta_{\alpha\beta}\Bigg{]}\
,$

where we have used the definitions (2.1.2) and (2.10) of the Ricci-type
tensors to gather the factors multiplying the Riemann tensor into the term

$\displaystyle f_{\alpha}{}^{\beta\gamma\delta}$ $\displaystyle\equiv$
$\displaystyle\left(q^{\beta[\delta}+B^{\beta[\delta}+\frac{1}{2}S^{\beta[\delta}+\frac{1}{2}P^{\beta[\delta}\right)\delta^{\gamma]}{}_{\alpha}$
(2.25)
$\displaystyle+\frac{1}{2}S^{\mu[\delta}g^{\gamma]\beta}g_{\mu\alpha}-\frac{1}{2}P^{\mu[\delta}g^{\gamma]\beta}g_{\mu\alpha}+H^{\gamma\delta}\delta^{\beta}{}_{\alpha}\
.$

The field $q_{\alpha\beta}$ plays the role of the metric, and we have the
symmetric field $S^{\alpha\beta}$, and the antisymmetric fields
$B^{\alpha\beta}$, $P^{\alpha\beta}$, and $H^{\alpha\beta}$. The old fields
$g_{\alpha\beta},\Sigma_{\alpha\beta},\Delta_{\alpha\beta},\Pi_{\alpha\beta}$,
and $\Theta_{\alpha\beta}$ are solved algebraically in terms of the new fields
$q_{\alpha\beta},B^{\alpha\beta},S^{\alpha\beta},P^{\alpha\beta}$,
$H^{\alpha\beta}$, and the matter fields $\Psi$ (and their derivatives) from
the field redefinitions (2.21)–(2.23).

The action is linear in the Riemann tensor, which is the only place where the
connection appears. So the action is quadratic in the connection. Variation
with respect to the connection hence gives a linear equation of motion, which
we can solve, and insert the solution back into the action. The kinetic terms
for the remaining fields then indicate the number of degrees of freedom and
whether they are ghosts or healthy. We look at the degrees of freedom in three
cases: when the connection is unconstrained, when torsion is assumed to
vanish, and when non-metricity is assumed to vanish.

###  2.2 General case

####  2.2.1 Equation of motion

We start with the case when no constraints are imposed on the connection.
Projective transformation of the new Legendre transformed action then plays an
important role. Projective invariance of the new action (2.24) is distinct
from possible projective invariance of the original action (2.18). Whether or
not the original formulation of the theory is projectively invariant depends
on which combinations of Ricci-type tensors the action contains. In the new
action, part of the physics related to the connection in the original action
has been transferred to the auxiliary fields, which are taken to be
projectively invariant. Under a projective transformation, the Riemann tensor
transforms as (2.15), so the new action (2.24) changes as

$\displaystyle\delta_{P}S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}f_{\alpha}{}^{\alpha\gamma\delta}\partial_{\gamma}V_{\delta}$
(2.26) $\displaystyle=$
$\displaystyle-\int\mathrm{d}^{4}x\sqrt{-q}V_{\delta}\overset{q}{\mathring{\nabla}}{}_{\gamma}f_{\alpha}{}^{\alpha\gamma\delta}\
,$

where on the last line we have switched $\partial_{\alpha}V_{\beta}$ to
$\overset{q}{\mathring{\nabla}}_{\alpha}V_{\beta}$ using the fact that the
contribution of the Levi–Civita connection vanishes due to the antisymmetry
$f_{\alpha}{}^{\beta\gamma\delta}=f_{\alpha}{}^{\beta[\gamma\delta]}$, done a
partial integration, and dropped a boundary term. The Legendre transformed
action is thus projectively invariant if

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\overset{q}{\mathring{\nabla}}_{\gamma}f_{\alpha}{}^{\alpha\gamma\delta}=\overset{q}{\mathring{\nabla}}_{\gamma}(B^{\gamma\delta}+4H^{\gamma\delta})\
.$ (2.27)

The field $S^{\alpha\beta}$ does not appear in the constraint, corresponding
to the fact that only the antisymmetric parts of the Ricci-type tensors change
under the projective transformation (2.1.3).

Variation of the new action (2.24) with respect to the connection gives

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}\frac{1}{2}\delta
R^{\alpha}{}_{\beta\gamma\delta}f_{\alpha}{}^{\beta\gamma\delta}$ (2.28)
$\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}\left(\nabla_{\gamma}\delta\Gamma^{\alpha}_{\delta\beta}+\frac{1}{2}T^{\mu}{}_{\gamma\delta}\delta\Gamma^{\alpha}_{\mu\beta}\right)f_{\alpha}{}^{\beta\gamma\delta}$
$\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}\left[-\left(\nabla_{\gamma}+\frac{1}{2}\overset{q}{Q}_{\gamma}+T_{\gamma}\right)\delta^{\delta}{}_{\mu}+\frac{1}{2}T^{\delta}{}_{\gamma\mu}\right]f_{\alpha}{}^{\beta\gamma\mu}\delta\Gamma^{\alpha}_{\delta\beta}\
,$

where we have used the definition (2.6) of the Riemann tensor, applied the
identity $\nabla_{\alpha}\sqrt{-q}=\frac{1}{2}\sqrt{-q}Q_{\alpha}$, and done a
partial integration. Putting the variation to zero gives the connection
equation of motion

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left[(\nabla_{\gamma}+\frac{1}{2}\overset{q}{Q}_{\gamma}+T_{\gamma})\delta^{\delta}{}_{\mu}-\frac{1}{2}T^{\delta}{}_{\gamma\mu}\right]f_{\alpha}{}^{\beta\gamma\mu}$
(2.29) $\displaystyle=$
$\displaystyle\overset{q}{\mathring{\nabla}}_{\gamma}f_{\alpha}{}^{\beta\gamma\delta}-L^{\mu}{}_{\gamma\alpha}f_{\mu}{}^{\beta\gamma\delta}+L^{\beta}{}_{\gamma\mu}f_{\alpha}{}^{\mu\gamma\delta}\
.$

The $\delta^{\alpha}{}_{\beta}$ trace of this equation gives the constraint
(2.27). So the equation of motion imposes a constraint on the matter fields
that makes the action projectively invariant. Correspondingly, the projective
symmetry leaves undetermined one linear combination of the traces of the
distortion $L\indices{{}^{\alpha}_{\beta\gamma}}$, corresponding to a linear
combination of $\overset{q}{Q}_{\alpha}$, ${\overset{q}{\hat{Q}}}_{\alpha}$,
and $T_{\alpha}$.

As equation (2.29) is linear, it is possible to solve
$L\indices{{}^{\alpha}_{\beta\gamma}}$ exactly. However, the general solution
is rather messy. We consider perturbations around a background, and solve the
equation of motion to first order.

####  2.2.2 Minkowski space

We expand the original metric as $g_{\alpha\beta}=\eta_{\alpha\beta}+\delta
g_{\alpha\beta}$, where $\eta_{\alpha\beta}$ is the Minkowski metric and
$\delta g_{\alpha\beta}$ is a perturbation. The new metric is similarly
expanded as $q_{\alpha\beta}=\eta_{\alpha\beta}+\delta q_{\alpha\beta}$. The
connection $\Gamma^{\gamma}_{\alpha\beta}$ and the fields
$B^{\alpha\beta},S^{\alpha\beta},P^{\alpha\beta}$, and $H^{\alpha\beta}$ are
first order small. To first order in perturbations,
$f_{\alpha}{}^{\beta\gamma\delta}$ defined in (2.25) reads

$f_{\alpha}{}^{\beta\gamma\delta}=q^{\beta[\delta}\delta^{\gamma]}{}_{\alpha}+\delta\tilde{f}_{\alpha}{}^{\beta\gamma\delta}\
,$ (2.30)

where

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\delta\tilde{f}_{\alpha}{}^{\beta\gamma\delta}$
$\displaystyle\equiv$
$\displaystyle\left(B^{\beta[\delta}+\frac{1}{2}S^{\beta[\delta}+\frac{1}{2}P^{\beta[\delta}\right)\delta^{\gamma]}{}_{\alpha}+\frac{1}{2}S^{\mu[\delta}\eta^{\gamma]\beta}\eta_{\mu\alpha}-\frac{1}{2}P^{\mu[\delta}\eta^{\gamma]\beta}\eta_{\mu\alpha}+H^{\gamma\delta}\delta^{\beta}{}_{\alpha}\
.$ (2.31)

To first order, the connection equation of motion (2.29) simplifies to

$-\overset{q}{Q}_{\gamma\alpha\beta}+\eta_{\alpha\gamma}\overset{q}{\hat{Q}}_{\beta}+\eta_{\alpha[\beta}(\overset{q}{Q}_{\gamma]}+2T_{\gamma]})+T_{\alpha\beta\gamma}=2\partial_{\delta}\delta\tilde{f}_{\gamma\beta\alpha}{}^{\delta}\
,$ (2.32)

where indices are raised and lowered with the Minkowski metric. Solving (2.32)
and choosing the torsion vector $T_{\alpha}$ as the part of the connection
left undetermined due to projective invariance, we get

$\displaystyle\overset{q}{Q}_{\gamma\alpha\beta}$ $\displaystyle=$
$\displaystyle\eta_{\alpha\beta}\left(-\frac{2}{3}T_{\gamma}+\partial_{\delta}\delta\tilde{f}_{\gamma\mu}{}^{\mu\delta}+\frac{1}{3}\partial_{\delta}\delta\tilde{f}_{\mu\gamma}{}^{\mu\delta}\right)+2\partial_{\delta}\delta\tilde{f}_{(\alpha\beta)\gamma}{}^{\delta}-2\partial_{\delta}\delta\tilde{f}_{(\alpha|\gamma|\beta)}{}^{\delta}-2\partial_{\delta}\delta\tilde{f}_{\gamma(\alpha\beta)}{}^{\delta}$
$\displaystyle T_{\gamma\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\eta_{\gamma[\beta}T_{\alpha]}-2\partial_{\delta}\delta\tilde{f}_{[\alpha\beta]\gamma}{}^{\delta}+2\partial_{\delta}\delta\tilde{f}_{[\alpha|\gamma|\beta]}{}^{\delta}+2\partial_{\delta}\delta\tilde{f}_{\gamma[\alpha\beta]}{}^{\delta}+\frac{4}{3}\eta_{\gamma[\alpha}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{\beta]\delta\mu}\
,$ (2.33)

with the constraint

$\displaystyle\partial_{\gamma}\delta\tilde{f}_{\alpha}{}^{\alpha\gamma\delta}=\partial_{\gamma}(B^{\gamma\delta}+4H^{\gamma\delta})=0\
.$ (2.34)

Inserting the solution (2.2.2) into the action (2.24), expanding to second
order, and using the constraint (2.34), we obtain the action

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\frac{1}{2}\Bigg{[}\mathring{R}+\frac{1}{2}\partial_{\gamma}\delta\tilde{f}_{\alpha}{}^{\beta\alpha\gamma}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{\beta\delta\mu}+\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial^{\mu}\delta\tilde{f}_{\alpha\beta\delta\mu}$
(2.35)
$\displaystyle-2\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial^{\mu}\delta\tilde{f}_{\beta\delta\alpha\mu}+\frac{1}{2}\partial_{\gamma}\delta\tilde{f}^{\alpha\beta}{}_{\beta}{}^{\gamma}\partial^{\mu}\delta\tilde{f}_{\alpha}{}^{\delta}{}_{\delta\mu}-\partial_{\gamma}\delta\tilde{f}_{\beta}{}^{\alpha\beta\gamma}\partial_{\mu}\delta\tilde{f}_{\alpha\delta}{}^{\delta\mu}$
$\displaystyle+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+B^{\alpha\beta})\Sigma_{\alpha\beta}-S^{\alpha\beta}\Delta_{\alpha\beta}-P^{\alpha\beta}\Pi_{\alpha\beta}-H^{\alpha\beta}\Theta_{\alpha\beta}\Bigg{]}\
.$

Matter fields end up being minimally coupled because we use
$\accentset{\Delta}{R}_{\alpha\beta}=\frac{1}{2}(\hat{R}_{\alpha\beta}+R_{\alpha\beta})$
rather than $\hat{R}_{\alpha\beta}$ as a variable. Note that the undetermined
vector $T_{\alpha}$ does not appear in the action. Plugging in (2.31) and
using the constraint (2.34) to eliminate $\partial_{\alpha}H^{\alpha\beta}$,
we get

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\frac{1}{2}\Bigg{[}\mathring{R}-\frac{1}{8}\partial_{\alpha}{B}^{\alpha\beta}\partial^{\gamma}{B}_{\gamma\beta}-\frac{1}{4}\partial_{\gamma}{B}_{\alpha\beta}\partial^{\gamma}{B}^{\alpha\beta}$
(2.36)
$\displaystyle+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+B^{\alpha\beta})\Sigma_{\alpha\beta}-S^{\alpha\beta}\Delta_{\alpha\beta}-P^{\alpha\beta}\Pi_{\alpha\beta}-H^{\alpha\beta}\Theta_{\alpha\beta}\Bigg{]}\
.$

The kinetic terms of ${B}^{\alpha\beta}$ (corresponding to
$R_{[\alpha\beta]}$) give an unstable theory [56]. As there is no kinetic term
for $H^{\alpha\beta}$, which corresponds to
$\accentset{\sim}{R}_{\alpha\beta}$, it makes no difference for this
instability whether the action depends on $R_{[\alpha\beta]}$ or the
projectively invariant combination
$4R_{[\alpha\beta]}-\accentset{\sim}{R}_{\alpha\beta}$. (This can also be
shown by explicit calculation.) So projective invariance does not guarantee
stability. On the contrary, the tensor
$\accentset{P}{R}_{\alpha\beta}=R_{[\alpha\beta]}-\accentset{\Delta}{R}_{[\alpha\beta]}$
does not lead to a kinetic term, so the theory may be stable if the action
contains $R_{[\alpha\beta]}$ only in this combination.

We see that the kinetic terms corresponding to different Ricci-type tensors
can cancel each other, so it is not possible to conclude that one of them
cannot be present in a stable theory without considering the others. As we do
not get kinetic terms for $S^{\alpha\beta}$, $P^{\alpha\beta}$, and
$H^{\alpha\beta}$, we cannot draw conclusions about the stability of an action
that contains the corresponding tensors
$\accentset{\Delta}{R}_{(\alpha\beta)}$, $\accentset{P}{R}_{\alpha\beta}$, and
$\accentset{\sim}{R}_{\alpha\beta}$. However, we next show that for
$\accentset{\Delta}{R}_{(\alpha\beta)}$ this feature is peculiar to Minkowski
space, as around a cosmological background it leads to news degrees of
freedom, which may imply a strong coupling problem [47, 48, 49, 50].

####  2.2.3 FLRW universe

We look at the kinetic terms of $S^{\alpha\beta}$, $P^{\alpha\beta}$, and
$H^{\alpha\beta}$ (corresponding to $\accentset{\Delta}{R}_{\alpha\beta}$,
$\accentset{P}{R}_{\alpha\beta}$, and $\accentset{\sim}{R}_{\alpha\beta}$,
respectively) around a cosmological background. It is cumbersome to expand the
action around a FLRW background in the case when we keep dependence on all
Ricci-type tensors. We therefore consider a simplified case, where only
$g^{\alpha\beta}R_{\alpha\beta}$, $\accentset{\Delta}{R}_{(\alpha\beta)}$,
$\accentset{P}{R}_{\alpha\beta}$, and $\accentset{\sim}{R}_{\alpha\beta}$
appear,

$\displaystyle
S=\int\mathrm{d}^{4}x\sqrt{-g}\frac{1}{2}F(g_{\alpha\beta},g^{\alpha\beta}R_{\alpha\beta},\accentset{\Delta}{R}_{(\alpha\beta)},\accentset{P}{R}_{\alpha\beta},\accentset{\sim}{R}_{\alpha\beta},\Psi,\partial\Psi)\
.$ (2.37)

With this simplification the metrics $g_{\alpha\beta}$ and $q_{\alpha\beta}$
are conformal to each other, which makes expansion around a FLRW background
more tractable. Were we to allow $F$ to depend on $R_{(\alpha\beta)}$, the
metrics would be related by a disformal transformation [35, 36, 37].
Performing the Legendre transformation and introducing field redefinitions as
before, the auxiliary fields are defined in (2.22), (2.23), and

$\displaystyle\sqrt{-g}\frac{\partial F}{\partial\Sigma}g^{\alpha\beta}$
$\displaystyle\equiv$ $\displaystyle\sqrt{-q}q^{\alpha\beta}\ ,$ (2.38)

where $\Sigma$ is the auxiliary field corresponding to
$g^{\alpha\beta}R_{\alpha\beta}$. Now the action becomes

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}\frac{1}{2}\Bigg{[}R^{\alpha}{}_{\beta\gamma\delta}f_{\alpha}{}^{\beta\gamma\delta}+\frac{\sqrt{-g}}{\sqrt{-q}}F-\Sigma-S^{\alpha\beta}\Delta_{\alpha\beta}-P^{\alpha\beta}\Pi_{\alpha\beta}-H^{\alpha\beta}\Theta_{\alpha\beta}\Bigg{]}\
,$ (2.39)

with

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!f_{\alpha}{}^{\beta\gamma\delta}$
$\displaystyle=$
$\displaystyle\left(q^{\beta[\delta}+\frac{1}{2}S^{\beta[\delta}+\frac{1}{2}P^{\beta[\delta}\right)\delta^{\gamma]}{}_{\alpha}+\frac{1}{2}S^{\mu[\delta}g^{\gamma]\beta}g_{\mu\alpha}-\frac{1}{2}P^{\mu[\delta}g^{\gamma]\beta}g_{\mu\alpha}+H^{\gamma\delta}\delta^{\beta}{}_{\alpha}$
(2.40) $\displaystyle=$
$\displaystyle\left(q^{\beta[\delta}+\frac{1}{2}S^{\beta[\delta}+\frac{1}{2}P^{\beta[\delta}\right)\delta^{\gamma]}{}_{\alpha}+\frac{1}{2}S^{\mu[\delta}q^{\gamma]\beta}q_{\mu\alpha}-\frac{1}{2}P^{\mu[\delta}q^{\gamma]\beta}q_{\mu\alpha}+H^{\gamma\delta}\delta^{\beta}{}_{\alpha}\
,$

where on the second line we have taken into account that $g_{\alpha\beta}$ is
conformal to $q_{\alpha\beta}$. Because of this, the gravity sector of the
action, where the kinetic terms can arise, does not explicitly depend on
$g_{\alpha\beta}$.

We now expand the action (2.39) around a spatially flat FLRW background. The
fields are split into background plus perturbations as

$\displaystyle q_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\bar{q}_{\alpha\beta}+\delta q_{\alpha\beta}\ ,\qquad\ \
\bar{q}_{\alpha\beta}=a(\eta)^{2}\eta_{\alpha\beta}$ $\displaystyle
S^{\alpha\beta}$ $\displaystyle=$ $\displaystyle\bar{S}^{\alpha\beta}+\delta
S^{\alpha\beta}\ ,\qquad
L^{\gamma}{}_{\alpha\beta}=\bar{L}^{\gamma}{}_{\alpha\beta}+\delta
L^{\gamma}{}_{\alpha\beta}\ ,$ (2.41)

where overbar denotes a background quantity, and $\eta$ is conformal time. We
decompose $\bar{S}^{\alpha\beta}$ as

$\displaystyle\bar{S}^{\alpha\beta}=S_{1}(\eta)\bar{q}^{\alpha\beta}+S_{2}(\eta)u^{\alpha}u^{\beta}\
,$ (2.42)

where $u^{\alpha}=(a^{-1},0,0,0)$. As the fields $P^{\alpha\beta}$ and
$H^{\alpha\beta}$ are antisymmetric, they are zero on the background, so
$P^{\alpha\beta}=\delta P^{\alpha\beta}$ and $H^{\alpha\beta}=\delta
H^{\alpha\beta}$. When we substitute (2.42) into the action (2.39), the action
turns out not to depend on $S_{1}$.333Therefore, for a maximally symmetric
background, where there is no preferred time direction and hence no
$u^{\alpha}$, we will not get a kinetic term for $S^{\alpha\beta}$, same as in
Minkowski space.

We decompose the background of the distortion tensor as

$\displaystyle\bar{L}^{\gamma}{}_{\alpha\beta}=L_{1}(\eta)u^{\gamma}\bar{q}_{\alpha\beta}+L_{2}(\eta)u_{\alpha}\delta^{\gamma}{}_{\beta}+L_{3}(\eta)u_{\beta}\delta^{\gamma}{}_{\alpha}+L_{4}(\eta)u^{\gamma}u_{\alpha}u_{\beta}+L_{5}(\eta)\epsilon^{\gamma}{}_{\alpha\beta\delta}u^{\delta}\
,$ (2.43)

where $\epsilon_{\alpha\beta\gamma\delta}$ is the Levi–Civita tensor. We
substitute this into the action (2.39), vary the action with respect to the
functions $L_{i}$, and solve the resulting equations to find the solution
$\bar{L}^{\gamma}{}_{\alpha\beta}=\bar{L}^{\gamma}{}_{\alpha\beta}(a,S_{2})$.
The details are given in appendix A. Inserting the background solution back
into the action (2.39), the Riemann tensor term becomes

$\displaystyle S$ $\displaystyle\supset$
$\displaystyle\int\mathrm{d}^{4}xa^{4}\left[\frac{1}{2}{\mathring{R}}(\bar{q})+\frac{3}{4}H^{2}S_{2}^{2}+\frac{3S_{2}^{2}}{16a^{2}(S_{2}^{2}-4)}S_{2}^{\prime}S_{2}^{\prime}+\frac{3HS_{2}}{4a}S_{2}^{\prime}\right]\
,$ (2.44)

where ${}^{\prime}\equiv\mathrm{d}/\mathrm{d}\eta$, $H\equiv
a^{\prime}/a^{2}$, and ${\mathring{R}}(\bar{q})$ is the Levi–Civita Ricci
scalar of the background metric $\bar{q}_{\alpha\beta}$. The kinetic term has
the wrong sign when $|S_{2}|<2$, so the value $S_{2}=0$ is unstable, and the
field runs to $S_{2}=\pm 2$. This is transparent if we write the action in
terms of a scalar field with a canonical kinetic term,
$\chi=\sqrt{3|S_{2}^{2}-4|/8}$.

Let us now consider perturbations. We have to solve the connection to first
order from (2.29). We sketch here the main points of the calculation and
summarise the results, with details given in appendix A. The equation of
motion for the connection to first order in perturbations has the form

$\displaystyle\delta
L^{\sigma}{}_{\mu\nu}M_{\sigma}{}^{\mu\nu}{}_{\alpha}{}^{\beta\gamma}(a,S_{2})+\rho_{\alpha}{}^{\beta\gamma}(a,S_{2},\delta
q_{\mu\nu},\delta S^{\mu\nu},\delta P^{\mu\nu},\delta H^{\mu\nu})=0\ ,$ (2.45)

where $M_{\sigma}{}^{\mu\nu}{}_{\alpha}{}^{\beta\gamma}$ depends only on the
background quantities, while the source term $\rho_{\alpha}{}^{\beta\gamma}$
also depends on the perturbations and their derivatives. The field $\delta
H^{\alpha\beta}$ satisfies the constraint
$\overset{q}{\mathring{\nabla}}_{\alpha}\delta H^{\alpha\beta}=0$ due to
projective invariance, where $\overset{q}{\mathring{\nabla}}$ is the
background covariant derivative. Solving the connection is straightforward but
tedious. Substituting the solution back into the action again gives the
kinetic terms. The kinetic part of the action expanded to second order has the
form

$\displaystyle S$ $\displaystyle\supset$
$\displaystyle\int\mathrm{d}^{4}xa^{4}\Big{(}K_{(1)}^{\mu\alpha\beta\nu\gamma\delta}\partial_{\mu}\delta
q_{\alpha\beta}\partial_{\nu}\delta
q_{\gamma\delta}+K_{(2)}^{\mu\alpha\beta\nu\gamma\delta}\partial_{\mu}\delta
S_{\alpha\beta}\partial_{\nu}\delta
S_{\gamma\delta}+K_{(3)}^{\mu\alpha\beta\nu\gamma\delta}\partial_{\mu}\delta
q_{\alpha\beta}\partial_{\nu}\delta S_{\gamma\delta}\Big{)}\ ,$

where $K_{(i)}^{\mu\alpha\beta\nu\gamma\delta}$ depend on the background
quantities. The fields $P^{\alpha\beta}$ and $H^{\alpha\beta}$ do not get
kinetic terms.

To determine whether or not there are ghosts we have to diagonalise the
kinetic terms. To facilitate this, we decompose the fields $\delta
q_{\alpha\beta}$ and $\delta S^{\alpha\beta}$ into scalar, vector, and tensor
modes. To second order, these decouple from each other, and we can look at
each of them individually. If any are unstable, the theory is unhealthy. The
tensor modes are the simplest: we have $\delta q_{ij}=h_{ij}$, with
$h{}^{i}{}_{i}=0,\ \partial^{i}h{}_{ij}=0$, and $\delta S^{ij}=s^{ij}$, with
$s{}^{i}{}_{i}=0,\ \partial_{i}s{}^{ij}=0$. Applying this decomposition in
(2.2.3), we get the tensor sector

$\displaystyle S$ $\displaystyle\supset$
$\displaystyle\int\mathrm{d}^{4}xa^{4}\left[\frac{S_{2}^{2}+4}{16a^{2}}\partial_{0}h_{ij}\partial_{0}h^{ij}-\frac{1}{4}\partial_{k}h_{ij}\partial^{k}h^{ij}\right.\left.+\frac{S_{2}^{2}}{4a^{2}(S_{2}^{2}-4)}\partial_{0}s_{ij}\partial_{0}s^{ij}-\frac{S_{2}}{4a^{2}}\partial_{0}h_{ij}\partial_{0}s^{ij}\right]\
,$

where the spatial indices are lowered with $\bar{q}_{ij}$ and raised with its
inverse. Diagonalising, we obtain

$\displaystyle S$ $\displaystyle\supset$
$\displaystyle\int\mathrm{d}^{4}xa^{4}\left[\frac{1}{2a^{2}}\partial_{0}h_{ij}\partial_{0}h^{ij}-\frac{1}{4}\partial_{k}h_{ij}\partial^{k}h^{ij}+\frac{S_{2}^{2}}{4a^{2}(S_{2}^{2}-4)}\partial_{0}s_{ij}\partial_{0}s^{ij}\right]\
.$ (2.48)

The factor in front of the kinetic term of $s^{ij}$ has the same form as the
background solution, and has the wrong sign when $|S_{2}|<2$. In section 2.2.2
we found that $S^{\alpha\beta}$ does not have a kinetic term around Minkowski
space. So whether $S^{\alpha\beta}$ leads to new degrees of freedom depends on
the background, and a theory that contains it is unstable around some FLRW
backgrounds. As $S^{\alpha\beta}$ corresponds to the projectively invariant
tensor $\accentset{\Delta}{R}_{(\alpha\beta)}$, we again see that projective
invariance does not guarantee stability.

As in Minkowski space, we do not obtain kinetic terms for $P^{\alpha\beta}$
and $H^{\alpha\beta}$. It thus remains an open question whether the
corresponding tensors $\accentset{P}{R}_{\alpha\beta}$ and
$\accentset{\sim}{R}_{\alpha\beta}$ lead to new degrees of freedom, and if so,
whether they are healthy.

###  2.3 Zero torsion

Let us now consider the case when the connection is taken to be symmetric a
priori, i.e. torsion vanishes, as often assumed in the Palatini formulation.
This condition does not depend on the choice of the metric. Any projective
transformation generates torsion (as (2.1.3) shows), so this assumption breaks
projective invariance.444It would be possible to consider a symmetrised
version of the projective transformation that maintains the zero torsion
condition. With zero torsion,
$\accentset{\sim}{R}_{\alpha\beta}=2R_{[\alpha\beta]}$, so we can drop the
dependence on $\accentset{\sim}{R}_{\alpha\beta}$, setting
$H^{\alpha\beta}=0$.

When varying the action, we have to take into account that the connection is
symmetric. From (2.28) we see that this amounts to symmetrising
$f_{\alpha}{}^{\beta\gamma\delta}$, and thus the equation of motion (2.29),
with respect to $\beta$ and $\delta$. To first order in the perturbations
around Minkowski space, the equation of motion is

$\displaystyle-\overset{q}{Q}_{\gamma\alpha\beta}+\eta_{\gamma(\alpha}\overset{q}{\hat{Q}}_{\beta)}+\frac{1}{2}\eta_{\alpha\beta}\overset{q}{Q}_{\gamma}-\frac{1}{2}\eta_{\gamma(\alpha}\overset{q}{Q}_{\beta)}=2\partial_{\delta}\delta\tilde{f}_{\gamma(\alpha\beta)}{}^{\delta}\
,$ (2.49)

with the solution

$\displaystyle\\!\\!\\!\\!\\!\\!\overset{q}{Q}_{\gamma\alpha\beta}$
$\displaystyle=$
$\displaystyle-2\partial_{\delta}\delta\tilde{f}_{\gamma(\alpha\beta)}{}^{\delta}+\eta_{\alpha\beta}\partial_{\mu}\delta\tilde{f}_{\gamma\delta}{}^{\delta\mu}+\frac{2}{3}\partial_{\mu}\delta\tilde{f}_{\delta}{}_{(\alpha}{}^{\delta\mu}\eta_{\beta)\gamma}-\frac{1}{3}\eta_{\alpha\beta}\partial_{\mu}\delta\tilde{f}^{\delta}{}_{\delta\gamma}{}^{\mu}$
(2.50)
$\displaystyle+\frac{2}{3}\partial_{\mu}\delta\tilde{f}_{\delta}{}^{\delta}{}_{(\alpha}{}^{\mu}\eta_{\beta)\gamma}-\frac{1}{3}\eta_{\alpha\beta}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{\gamma\delta\mu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{(\alpha}S_{\beta)\gamma}+\frac{1}{2}\partial_{\gamma}S_{\alpha\beta}-\frac{1}{6}\eta_{\gamma(\alpha}\partial_{\beta)}S^{\mu}{}_{\mu}-\frac{1}{6}\eta_{\alpha\beta}S^{\mu}{}_{\mu}+\frac{1}{6}\eta_{\gamma(\alpha}\partial^{\mu}S_{\beta)\mu}+\frac{1}{6}\eta_{\alpha\beta}\partial^{\mu}S_{\gamma\mu}$
$\displaystyle+\frac{1}{2}\partial_{(\alpha}P_{\beta)\gamma}+-\frac{1}{6}\eta_{\gamma(\alpha}\partial^{\mu}P_{\beta)\mu}-\frac{1}{6}\eta_{\alpha\beta}\partial^{\mu}P_{\gamma\mu}+\frac{2}{3}\eta_{\gamma(\alpha}\partial^{\mu}B_{\beta)\mu}-\frac{1}{3}\eta_{\alpha\beta}\partial^{\mu}B_{\gamma\mu}\
.$

Substituting the solution (2.50) into the action gives

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\frac{1}{2}\left[\vphantom{\frac{1}{2}}\right.\mathring{R}(q)+\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\alpha(\beta\delta)}{}^{\mu}+\frac{1}{2}\partial^{\gamma}\delta\tilde{f}^{\alpha\beta}{}_{\beta\gamma}\partial_{\mu}\delta\tilde{f}_{\alpha\delta}{}^{\delta\mu}-\frac{1}{2}\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\beta\alpha\delta}{}^{\mu}$
(2.51)
$\displaystyle-\frac{1}{2}\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\delta\beta\alpha}{}^{\mu}-\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\beta\delta\alpha}{}^{\mu}-\partial_{\gamma}\delta\tilde{f}_{\alpha}{}^{\beta\alpha\gamma}\partial_{\mu}\delta\tilde{f}_{\beta\delta}{}^{\delta\mu}+\partial_{\beta}\delta\tilde{f}_{\alpha}{}^{\alpha\beta\gamma}\partial_{\mu}\delta\tilde{f}_{\gamma\delta}{}^{\delta\mu}$
$\displaystyle+\frac{1}{6}\partial_{\gamma}\delta\tilde{f}_{\alpha}{}^{\beta\alpha\gamma}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{\beta\delta\mu}-\frac{1}{3}\partial_{\beta}\delta\tilde{f}_{\alpha}{}^{\alpha\beta\gamma}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{\gamma\delta\mu}-\frac{1}{6}\partial_{\beta}\delta\tilde{f}_{\alpha}{}^{\alpha\beta\gamma}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{\delta\gamma\mu}$
$\displaystyle+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+B^{\alpha\beta})\Sigma_{\alpha\beta}-S^{\alpha\beta}\Delta_{\alpha\beta}-P^{\alpha\beta}\Pi_{\alpha\beta}\left.\vphantom{\frac{1}{2}}\right]\
.$

Finally, substituting $\delta\tilde{f}_{\alpha}{}^{\beta\gamma\delta}$ from
(2.31) with $H^{\alpha\beta}=0$ into the action (2.51), performing an
integration by parts, and diagonalising the $P_{\alpha\beta}$ and
$B_{\alpha\beta}$ kinetic terms with the field redefinition
$\hat{B}_{\alpha\beta}={B}_{\alpha\beta}+2P_{\alpha\beta}$, we get the action

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\frac{1}{2}\left[\vphantom{\frac{1}{2}}\right.\mathring{R}(q)+\frac{1}{6}\partial_{\alpha}\hat{B}^{\alpha\beta}\partial_{\gamma}\hat{B}^{\gamma}_{\
\,\beta}-\frac{1}{2}\partial_{\alpha}P^{\alpha\beta}\partial_{\gamma}P^{\gamma}_{\
\,\beta}-\frac{1}{16}\partial_{\gamma}P_{\alpha\beta}\partial^{\gamma}P^{\alpha\beta}+\frac{1}{16}\partial_{\gamma}S_{\alpha\beta}\partial^{\gamma}S^{\alpha\beta}$
(2.52)
$\displaystyle-\frac{1}{48}\partial_{\alpha}S^{\beta}{}_{\beta}\partial^{\alpha}S^{\gamma}{}_{\gamma}-\frac{1}{12}\partial_{\alpha}S^{\alpha\beta}\partial_{\gamma}S^{\gamma}{}_{\beta}+\frac{1}{24}\partial_{\beta}S^{\beta\gamma}\partial_{\gamma}S^{\alpha}{}_{\alpha}-\frac{5}{4}\partial_{\alpha}P^{\alpha\beta}\partial_{\gamma}S^{\gamma}{}_{\beta}$
$\displaystyle+\frac{2}{3}\partial_{\alpha}\hat{B}^{\alpha\beta}\partial_{\gamma}S^{\gamma}{}_{\beta}+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+\hat{B}^{\alpha\beta}+2P^{\alpha\beta})\Sigma_{\alpha\beta}-S^{\alpha\beta}\Delta_{\alpha\beta}-P^{\alpha\beta}\Pi_{\alpha\beta}\left.\vphantom{\frac{1}{2}}\right].$

In contrast to the case when there was no a priori constraint on the
connection, dependence on any of the Ricci-type tensors now leads to new
degrees of freedom around Minkowski space.

Individually (i.e. ignoring cross terms) the kinetic terms of
$B_{\alpha\beta}$ (corresponding to $R_{[\alpha\beta]}$) are healthy, while
the kinetic terms of $P_{\alpha\beta}$ and $S_{\alpha\beta}$ (corresponding to
$\accentset{P}{R}_{\alpha\beta}$ and $\accentset{\Delta}{R}_{(\alpha\beta)}$,
respectively) lead to ghosts [56]. In the case $P^{\alpha\beta}=0$ ghost modes
remain after diagonalising the $B^{\alpha\beta}$ and $S^{\alpha\beta}$ kinetic
terms. In the case $P_{\alpha\beta}=S_{\alpha\beta}=0$, it is possible to
solve the connection exactly, and the extra degrees of freedom in
$B_{\alpha\beta}$ correspond to one vector field with a positive kinetic term
[29]. The theory can still be unstable if the field is tachyonic, which
depends on the form of $F$.

###  2.4 Zero non-metricity

Let us now consider the case when non-metricity is taken to vanish a priori.
This assumption is common in loop quantum gravity, and a theory that satisfies
it is called Einstein–Cartan theory. Again, projective invariance is broken,
and the constraint (2.27) does not apply. Unlike the zero torsion condition,
the assumption that non-metricity is zero depends on which metric we use: the
original metric $g_{\alpha\beta}$ or the new metric $q_{\alpha\beta}$.

Let us first consider the case $\overset{g}{Q}_{\gamma\alpha\beta}=0$. This is
the most natural option, as the theory is originally defined with the metric
$g_{\alpha\beta}$. The corresponding disformation is zero,
$J^{\alpha}{}_{\beta\gamma}=0$, so
$L^{\alpha}{}_{\beta}{}_{\gamma}=K^{\alpha}{}_{\beta}{}_{\gamma}$, where these
quantities are defined with the decomposition (2.1) in terms of
$g_{\alpha\beta}$. Writing the homothetic curvature tensor (2.1.2) in terms of
the Riemann tensor decomposition (2.7) and using the property
$K^{[\alpha}{}_{\beta}{}^{\gamma]}=0$ shows that
$\accentset{\sim}{R}_{\alpha\beta}=0$. We thus have $H^{\alpha\beta}=0$, just
as in the zero torsion case. Furthermore, (2.10) shows that
$\accentset{\Delta}{R}_{\alpha\beta}=0$, so $S^{\alpha\beta}=0$, and we can
also set $P^{\alpha\beta}=0$, as $\accentset{P}{R}_{\alpha\beta}$ is not
independent. So only $B^{\alpha\beta}$ remains.

What if we instead take the non-metricity defined with the new metric to be
zero, $\overset{q}{Q}_{\gamma\alpha\beta}=0$? We can now decompose the
homothetic curvature tensor in terms of the new metric $q_{\alpha\beta}$ and
again find that $\accentset{\sim}{R}_{\alpha\beta}=0$, as discussed after
(2.11). So we still have $H^{\alpha\beta}=0$. In contrast, now
$\accentset{\Delta}{R}_{\alpha\beta}$ does not vanish in general. However, the
only difference to the previous case is that the trace $\hat{R}_{\alpha\beta}$
of the Riemann tensor in (2.1.2) is defined with $q^{\alpha\beta}$ instead of
$g^{\alpha\beta}$. As we work to first order, only the background metric
enters here, and if the background metrics are conformal to each other, we
have to first order $\accentset{\Delta}{R}_{\alpha\beta}=0$. In particular,
this is true for perturbations around Minkowski space. So we can again set
$S^{\alpha\beta}=P^{\alpha\beta}=0$, and only $B^{\alpha\beta}$ remains.

Varying the action with respect to the antisymmetric part of the connection
(the symmetric part is determined by the antisymmetric part when non-metricity
is zero) gives, according to (2.28),

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-q}\left[-\left(\nabla_{\gamma}+T_{\gamma}\right)\delta^{[\delta}{}_{\mu}+\frac{1}{2}T^{[\delta}{}_{\gamma\mu}\right]f_{\alpha}{}^{\beta]\gamma\mu}\delta\Gamma^{\alpha}_{\delta\beta}\
.$ (2.53)

Setting the variation to zero and expanding the equation of motion to first
order in perturbations around Minkowski space, we get

$\displaystyle T_{[\beta|\alpha|\gamma]}+\eta_{\alpha[\beta}T_{\gamma]}$
$\displaystyle=$ $\displaystyle
2\partial_{\delta}\delta\tilde{f}_{\alpha[\beta\gamma]}{}^{\delta}\ ,$ (2.54)

with the solution

$\displaystyle T_{\gamma\alpha\beta}$ $\displaystyle=$
$\displaystyle-6\partial_{\delta}\delta\tilde{f}_{[\alpha\beta\gamma]}{}^{\delta}+2\eta_{\gamma[\alpha}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{\beta]\delta\mu}-2\eta_{\gamma[\alpha}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{|\delta|\beta]\mu}\
.$ (2.55)

Inserting this back into the action and expanding to second order gives

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\frac{1}{2}\left[\vphantom{\frac{1}{2}}\right.\mathring{R}(q)-3\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\alpha[\beta\delta]}{}^{\mu}-\frac{5}{2}\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\beta\alpha\delta}{}^{\mu}+\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\beta\delta\alpha}{}^{\mu}$
(2.56)
$\displaystyle-2\partial^{\gamma}\delta\tilde{f}^{\alpha\beta}{}_{\alpha\gamma}\partial_{\mu}\delta\tilde{f}_{\beta\delta}{}^{\delta\mu}-2\partial_{\beta}\delta\tilde{f}_{\alpha}{}^{\alpha\beta\gamma}\partial_{\mu}\delta\tilde{f}_{\gamma\delta}{}^{\delta\mu}+\frac{3}{2}\partial_{\gamma}\delta\tilde{f}^{\alpha\beta\gamma\delta}\partial_{\mu}\delta\tilde{f}_{\delta\beta\alpha}{}^{\mu}$
$\displaystyle-4\partial_{\beta}\delta\tilde{f}_{\alpha}{}^{\alpha\beta\gamma}\partial^{\mu}\delta\tilde{f}^{\delta}{}_{[\gamma\delta]\mu}+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+B^{\alpha\beta})\Sigma_{\alpha\beta}\left.\vphantom{\frac{1}{2}}\right]\
.$

Substituting $\delta\tilde{f}_{\alpha}{}^{\beta\gamma\delta}$ from (2.31) with
$S^{\alpha\beta}$, $P^{\alpha\beta}$, and $H^{\alpha\beta}$ set to zero, the
action becomes

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\frac{1}{2}\left[\mathring{R}(q)+\frac{3}{4}\partial_{\gamma}B_{\alpha\beta}\partial^{\gamma}B^{\alpha\beta}+\frac{\sqrt{-g}}{\sqrt{-q}}F-(q^{\alpha\beta}+B^{\alpha\beta})\Sigma_{\alpha\beta}\right]\
.$ (2.57)

The kinetic term of $B_{\alpha\beta}$ corresponds to an unstable theory [56].
So in the metric-compatible case the theory is not healthy if it has any
dependence on $R_{[\alpha\beta]}$.

##  3 Conclusions

We have investigated the stability of non-degenerate actions that depend
algebraically on Ricci-type tensors, i.e. the first traces (2.1.2) of the
Riemann tensor. We performed a Legendre transformation and introduced
auxiliary fields to make the action linear in the Riemann tensor. We then
solved perturbatively for the connection and inserted it back into the action
to shift the non-standard gravitational dynamics to the matter sector. We
considered three cases: the general case when no constraints are imposed on
the connection, the case with zero torsion, and the case with zero non-
metricity. Our results are summarised in table 1.

Tensor | General case |  | Zero torsion | Zero non-metricity
---|---|---|---|---
| Minkowski | FLRW | Minkowski | Minkowski
$R_{(\alpha\beta)}$ | no new dofs | - | no new dofs | no new dofs
$R_{[\alpha\beta]}$ | ghosts | - | healthy vector | ghosts
$\accentset{\Delta}{R}_{(\alpha\beta)}$ | no new dofs | ghosts | ghosts | 0
$\accentset{P}{R}_{\alpha\beta}$ | no new dofs | no new dofs | ghosts | not independent
$\accentset{\sim}{R}_{\alpha\beta}$ | no new dofs | no new dofs | not independent | 0
Table 1: Summary of which Ricci-type tensors lead to new degrees of freedom
(dofs), and which of those are ghosts. An entry of 0 means that the tensor is
zero, while “not independent” means that it depends linearly on the other
tensors. The new degrees of freedom due to
$\accentset{\Delta}{R}_{(\alpha\beta)}$ in the FLRW background in the general
case can be stable or unstable, depending on the background. In the FLRW case
we included $R_{(\alpha\beta)}$ only via its trace and did not include
$R_{[\alpha\beta]}$.

In the general case, dependence on $R_{[\alpha\beta]}$ or the projectively
invariant combination
$R_{[\alpha\beta]}-\accentset{\Delta}{R}_{[\alpha\beta]}$ leads to ghosts
around Minkowski space. Around Minkowski space none of the tensors
$\accentset{\Delta}{R}_{(\alpha\beta)}$, $\accentset{P}{R}_{\alpha\beta}$, and
$\accentset{\sim}{R}_{\alpha\beta}$ lead to new degrees of freedom. Around
FLRW space, $\accentset{P}{R}_{\alpha\beta}$ and
$\accentset{\sim}{R}_{\alpha\beta}$ also do not give new degrees of freedom,
but $\accentset{\Delta}{R}_{(\alpha\beta)}$ gives new degrees of freedom that
are unstable for some backgrounds. Note that
$\accentset{\Delta}{R}_{(\alpha\beta)}$ is projectively invariant. This
difference in the number of dynamical degrees of freedom for different
backgrounds may point to a strong coupling problem around Minkowski space,
casting doubt on the viability of a theory that depends on
$\accentset{\Delta}{R}_{(\alpha\beta)}$ [47, 48, 49, 50]. When we impose
constraints on the connection, the results are quite different.

In the case with zero torsion, $\accentset{\sim}{R}_{\alpha\beta}$ is not an
independent tensor. All other Ricci-type tensors apart from
$R_{(\alpha\beta)}$ now lead to new degrees of freedom. Due to couplings
between the fields, the kinetic sector is complicated. Taken in isolation,
$R_{[\alpha\beta]}$ leads to a healthy vector field, as is well known, and
true independent of the background [39, 40, 29]. In isolation, the other
Ricci-type tensors $\accentset{P}{R}_{\alpha\beta}$ and
$\accentset{\Delta}{R}_{(\alpha\beta)}$ lead to ghosts. If we drop
$\accentset{P}{R}_{\alpha\beta}$, the conclusions about $R_{[\alpha\beta]}$
and $\accentset{\Delta}{R}_{(\alpha\beta)}$ hold. We did not diagonalise the
kinetic sector when $\accentset{P}{R}_{\alpha\beta}$ is also present.

In the case with zero non-metricity we have
$\accentset{\sim}{R}_{\alpha\beta}=0$. If non-metricity is defined with
respect to the original metric $g_{\alpha\beta}$, we also have
$\accentset{\Delta}{R}_{\alpha\beta}=0$. If it is defined with respect to the
new metric $q_{\alpha\beta}$, this holds to linear order around Minkowski
space, but not in general. The only remaining Ricci-type tensor is
$R_{\alpha\beta}$. The symmetric part $R_{(\alpha\beta)}$ does not lead to new
degrees of freedom [35, 36, 37, 38], while $R_{[\alpha\beta]}$ gives ghosts
[28].

The only case with stable new degrees of freedom is when torsion is zero and
the action depends only on the Ricci tensor, which was known already. As our
focus is on whether a theory is healthy, we did not enumerate the number and
type of all degrees of freedom, only whether some of them are unstable. We
have in all cases required that the action is non-degenerate, i.e. that the
Legendre transformation is invertible. There can be degenerate actions that
depend on some of the tensors excluded above but nevertheless give a stable
theory.

The results demonstrate that in order to determine whether a theory can depend
on some Ricci-type tensor, it is necessary to consider them in combination, as
there can be cancellations. It remains open whether
$\accentset{\sim}{R}_{\alpha\beta}$ ever leads to new degrees of freedom, and
if so, whether they are healthy. From the fact that we do not get kinetic
terms for it around Minkowski space or FLRW space we cannot conclude that this
never happens. However, if there are new degrees of freedom around a different
background, this may point to a strong coupling problem. Wrong sign kinetic
terms from Ricci-type tensors can be cancelled by other terms, such as those
that depend explicitly on the torsion [28]. Our results show that projective
invariance is not a sufficient condition for a theory to be ghost-free. The
most general healthy Palatini theory remains to be determined.

## Acknowledgments

JA is supported by Academy of Finland projects 320123 and 345070.

##  Appendix A Solving the connection in the FLRW case

In this appendix we provide details on solving the connection in the FLRW
background studied in section 2.2.3.

Substituting the decompositions (2.42) and (2.43) into the equations of motion
for the connection at zeroth order gives four independent equations for the
coefficients $L_{i}$. Due to projective symmetry the coefficient $L_{2}$ is
left undetermined. One of the equations is simply $L_{5}=0$, corresponding to
the fact that there is no parity-violating source. From the rest we solve for
$L_{1}$, $L_{3}$, and $L_{4}$, obtaining

$\displaystyle
L_{1}=-\frac{S_{2}}{4}\left[2H+\frac{S_{2}^{\prime}}{a(S_{2}+2)}\right]\
,\quad
L_{3}=-\frac{S_{2}}{4}\left[2H+\frac{S_{2}^{\prime}}{a(S_{2}-2)}\right]$
$\displaystyle
L_{4}=-HS_{2}-\frac{(S_{2}^{2}+4)S_{2}^{\prime}}{2a(S_{2}^{2}-4)}\ .$ (A.1)

Substituting these back into (2.43), we get the non-zero components of the
background distortion:

$\displaystyle\bar{L}^{0}{}_{00}$ $\displaystyle=$
$\displaystyle-\frac{2S_{2}^{\prime}}{S_{2}^{2}-4}-aL_{2}\ ,\qquad\qquad\quad\
\bar{L}^{i}{}_{0j}=-aL_{2}\delta^{i}{}_{j}$ $\displaystyle\bar{L}^{i}{}_{j0}$
$\displaystyle=$
$\displaystyle\left[\frac{a^{\prime}}{2a}S_{2}+\frac{S_{2}S_{2}^{\prime}}{4(S_{2}-2)}\right]\delta^{i}{}_{j}\
,\quad\bar{L}^{0}{}_{ij}=-\left[\frac{a^{\prime}}{2a}S_{2}+\frac{S_{2}S_{2}^{\prime}}{4(S_{2}+2)}\right]\delta_{ij}\
.$ (A.2)

Substituting this solution into the background action gives (2.44).

After finding the background, we need to solve the connection to first order,
and substitute it back into the action to obtain the kinetic terms for the
perturbations. The equation of motion for the connection expanded to first
order (2.45) is in detail (indices are lowered with the background metric
$\bar{q}_{\alpha\beta}$ and raised with its inverse)

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Big{(}\bar{q}^{\gamma\nu}\delta^{\mu}{}_{\alpha}\delta^{\beta}{}_{\sigma}+\bar{q}^{\beta\mu}\delta^{\nu}{}_{\alpha}\delta^{\gamma}{}_{\sigma}-\bar{q}^{\mu\nu}\delta^{\gamma}{}_{\alpha}\delta^{\beta}{}_{\sigma}-\bar{q}^{\beta}{}^{\gamma}\delta^{\mu}{}_{\sigma}\delta^{\nu}{}_{\alpha}+\tfrac{1}{2}\bar{q}^{\beta}{}^{\gamma}\bar{S}_{\sigma}{}^{\mu}\delta^{\nu}{}_{\alpha}-\tfrac{1}{2}\bar{q}^{\gamma\nu}\bar{S}_{\alpha}{}^{\mu}\delta^{\beta}{}_{\sigma}$
(A.3)
$\displaystyle+\tfrac{1}{2}\bar{q}^{\mu\nu}\bar{S}_{\alpha}{}^{\gamma}\delta^{\beta}{}_{\sigma}-\tfrac{1}{2}\bar{q}^{\beta\mu}\bar{S}^{\gamma}{}_{\sigma}\delta^{\nu}{}_{\alpha}-\tfrac{1}{2}\bar{S}^{\mu\nu}\delta_{\alpha}{}^{\gamma}\delta^{\beta}{}_{\sigma}+\tfrac{1}{2}\bar{S}^{\beta}{}^{\mu}\delta^{\gamma}{}_{\sigma}\delta^{\nu}{}_{\alpha}-\tfrac{1}{2}\bar{S}^{\beta}{}^{\gamma}\delta^{\mu}{}_{\sigma}\delta^{\nu}{}_{\alpha}$
$\displaystyle+\tfrac{1}{2}\bar{S}^{\gamma}{}^{\nu}\delta^{\beta}{}_{\sigma}\delta^{\mu}{}_{\alpha}\Big{)}\delta
L^{\sigma}{}_{\mu\nu}+\rho_{\alpha}{}^{\beta\gamma}\ ,$

where the source term $\rho_{\alpha}{}^{\beta\gamma}$ is

$\displaystyle\rho_{\alpha}{}^{\beta\gamma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\bar{L}{}^{\beta}{}^{\delta}{}_{\delta}\bar{S}{}^{\gamma}{}^{\mu}\delta^{\nu}{}_{\alpha}-\bar{L}{}^{\beta}{}^{\delta}{}^{\gamma}\bar{S}{}_{\delta}{}^{\mu}\delta^{\nu}{}_{\alpha}+2\bar{L}{}^{\delta}{}_{\delta}{}_{\alpha}\bar{q}^{\beta\mu}\bar{q}^{\gamma\nu}-\bar{L}{}^{\delta}{}^{\lambda}{}_{\alpha}\bar{S}{}_{\delta}{}_{\lambda}\bar{q}^{\beta\mu}\bar{q}^{\gamma\nu}-\bar{L}{}^{\beta}{}^{\delta}{}^{\mu}\bar{S}{}_{\alpha}{}^{\gamma}\delta^{\nu}{}_{\delta}$
(A.4)
$\displaystyle-\bar{L}{}^{\delta}{}^{\beta}{}_{\alpha}\bar{S}{}^{\gamma}{}^{\mu}\delta^{\nu}{}_{\delta}-2\bar{L}{}^{\gamma}{}^{\delta}{}_{\alpha}\bar{q}^{\beta\mu}\delta^{\nu}{}_{\delta}+\bar{L}{}^{\delta}{}^{\mu}{}_{\alpha}\bar{S}{}^{\gamma}{}_{\delta}\bar{q}^{\beta\nu}-2\bar{L}{}^{\beta}{}_{\alpha}{}^{\delta}\bar{q}^{\gamma\mu}\delta^{\nu}{}_{\delta}+\bar{L}{}^{\beta}{}^{\delta}{}^{\mu}\bar{S}{}_{\alpha}{}_{\delta}\bar{q}^{\gamma\nu}$
$\displaystyle+2\bar{L}{}^{\beta}{}^{\delta}{}^{\mu}\delta{}_{\alpha}{}^{\gamma}\delta^{\nu}{}_{\delta}+\bar{L}{}^{\delta}{}^{\lambda}{}_{\alpha}\bar{S}{}_{\lambda}{}^{\nu}\bar{q}{}^{\beta}{}^{\gamma}\delta^{\mu}{}_{\delta}+\bar{q}^{\beta\mu}\bar{q}^{\nu\gamma}\bar{\nabla}{}_{\delta}\bar{S}{}_{\alpha}{}^{\delta}-\delta^{\mu}{}_{\alpha}\bar{q}{}^{\beta}{}^{\gamma}\bar{\nabla}{}_{\lambda}\bar{S}{}^{\nu}{}^{\lambda}$
$\displaystyle+\delta^{\mu}{}_{\alpha}\bar{\nabla}{}^{\beta}\bar{S}{}^{\gamma\nu}-\bar{q}^{\beta\mu}\bar{q}^{\delta\nu}\bar{\nabla}{}_{\delta}\bar{S}{}_{\alpha}{}^{\gamma}\Big{)}\delta
q_{\mu\nu}+\frac{1}{4}\Big{(}\bar{S}^{\mu\nu}\bar{q}{}^{\beta}{}^{\gamma}\delta^{\delta}{}_{\alpha}-\bar{S}{}^{\beta}{}^{\nu}\bar{q}^{\gamma\delta}\delta^{\mu}{}_{\alpha}+\bar{S}{}_{\alpha}{}^{\nu}\bar{q}^{\mu\gamma}\bar{q}^{\beta\delta}$
$\displaystyle+\bar{S}^{\mu\nu}\bar{q}^{\beta\delta}\delta{}_{\alpha}{}^{\gamma}-\bar{S}{}_{\alpha}{}^{\nu}\bar{q}^{\beta\mu}\bar{q}^{\delta\gamma}-\bar{S}{}^{\beta}{}^{\delta}\bar{q}^{\mu\nu}\delta{}_{\alpha}{}^{\gamma}-\bar{S}{}_{\alpha}{}^{\delta}\bar{q}{}^{\beta}{}^{\gamma}\bar{q}^{\mu\nu}+2\bar{S}{}^{\gamma}{}^{\delta}\bar{q}^{\beta\nu}\delta^{\mu}{}_{\alpha}$
$\displaystyle+\bar{S}{}^{\beta}{}^{\delta}\bar{q}^{\nu\gamma}\delta^{\mu}{}_{\alpha}+\bar{S}{}_{\alpha}{}^{\delta}\bar{q}^{\mu\beta}\bar{q}^{\nu\gamma}-2\bar{S}{}^{\nu}{}^{\delta}\bar{q}{}^{\beta}{}^{\gamma}\delta^{\mu}{}_{\alpha}-2\bar{S}{}^{\nu}{}^{\delta}\bar{q}^{\beta\mu}\delta{}_{\alpha}{}^{\gamma}+\bar{S}{}^{\beta}{}^{\nu}\bar{q}^{\mu\gamma}\delta^{\delta}{}_{\alpha}\Big{)}\bar{\nabla}{}_{\delta}\delta
q_{\mu\nu}$
$\displaystyle+\frac{1}{2}\Big{(}\bar{L}{}^{\beta}{}^{\delta}{}_{\delta}\bar{q}_{\alpha\mu}\delta^{\gamma}{}_{\nu}-\bar{L}{}^{\beta}{}_{\nu}{}^{\gamma}\bar{q}_{\alpha\mu}-\bar{L}{}^{\delta}{}_{\delta}{}_{\alpha}\delta^{\beta}{}_{\mu}\delta^{\gamma}{}_{\nu}+\bar{L}{}^{\gamma}{}_{\nu}{}_{\alpha}\delta^{\beta}{}_{\mu}+\bar{L}{}^{\beta}{}_{\alpha}{}_{\nu}\delta^{\gamma}{}_{\mu}-\bar{L}{}_{\nu}{}^{\beta}{}_{\alpha}\delta^{\gamma}{}_{\mu}$
$\displaystyle+\bar{L}{}_{\nu}{}_{\mu}{}_{\alpha}\bar{q}{}^{\beta}{}^{\gamma}-\bar{L}{}^{\beta}{}_{\nu}{}_{\mu}\delta{}_{\alpha}{}^{\gamma}\Big{)}\delta
S^{\mu\nu}+\frac{1}{2}\Big{(}\delta^{\delta}{}_{\alpha}\delta^{\beta}{}_{\mu}\delta^{\gamma}{}_{\nu}+\bar{q}_{\alpha\mu}\bar{q}^{\delta\beta}\delta^{\nu}{}_{\gamma}-\bar{q}_{\alpha\mu}\bar{q}{}^{\beta}{}^{\gamma}\delta^{\delta}{}_{\nu}$
$\displaystyle-\delta^{\gamma}{}_{\alpha}\delta^{\beta}{}_{\mu}\delta^{\delta}{}_{\nu}\Big{)}\bar{\nabla}_{\delta}\delta
S^{\mu\nu}+\frac{1}{2}\Big{(}\bar{L}{}^{\beta}{}_{\nu}{}^{\gamma}\bar{q}_{\alpha\mu}-\bar{L}{}^{\beta}{}^{\delta}{}_{\delta}\bar{q}_{\alpha\mu}\delta^{\gamma}{}_{\nu}-\bar{L}{}^{\delta}{}_{\delta}{}_{\alpha}\delta^{\beta}{}_{\mu}\delta^{\gamma}{}_{\nu}+\bar{L}{}^{\gamma}{}_{\nu}{}_{\alpha}\delta^{\beta}{}_{\mu}$
$\displaystyle-\bar{L}{}^{\beta}{}_{\alpha}{}_{\nu}\delta^{\gamma}{}_{\mu}-\bar{L}{}_{\nu}{}^{\beta}{}_{\alpha}\delta^{\gamma}{}_{\mu}-\bar{L}{}_{\mu}{}_{\nu}{}_{\alpha}\bar{q}{}^{\beta}{}^{\gamma}+\bar{L}{}^{\beta}{}_{\mu}{}_{\nu}\delta{}_{\alpha}{}^{\gamma}\Big{)}\delta
P^{\mu\nu}+\frac{1}{2}\Big{(}\delta^{\delta}{}_{\alpha}\delta^{\beta}{}_{\mu}\delta^{\gamma}{}_{\nu}$
$\displaystyle-\bar{q}_{\alpha\mu}\bar{q}^{\delta\beta}\delta^{\gamma}{}_{\nu}+\bar{q}_{\alpha\mu}\bar{q}{}^{\beta}{}^{\gamma}\delta^{\delta}{}_{\nu}-\delta{}_{\alpha}{}^{\gamma}\delta^{\beta}{}_{\mu}\delta^{\delta}{}_{\nu}\Big{)}\bar{\nabla}_{\delta}\delta
P^{\mu\nu}-2\delta{}_{\alpha}{}^{\beta}\bar{\nabla}{}_{\delta}H{}^{\gamma}{}^{\delta}\
.$

Due to projective invariance the equation of motion leaves one vector
undetermined. We choose this to be the torsion vector
$T_{\alpha}=L^{\gamma}{}_{\alpha\gamma}-L^{\gamma}{}_{\gamma\alpha}$. Note
that $\rho_{\alpha}{}^{\alpha\beta}=0$, so
$\rho{}_{i}{}^{i}{}^{0}{}=-\rho_{0}{}^{00}$ and
$\rho{}_{j}{}^{j}{}^{i}=-\rho{}_{0}{}^{0}{}^{i}$. The equations of motion
(A.3) then read component by component

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-\tfrac{1}{2}(S_{2}-2)\delta
L{}^{i}{}_{i}{}_{0}+\tfrac{1}{2}(S_{2}+2)\delta
L{}_{0}{}^{i}{}_{i}-\rho_{0}{}^{0}{}_{0}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}(S_{2}-2)\delta
L{}^{i}{}_{00}-\tfrac{1}{2}(S_{2}+2)\delta
L{}_{0}{}_{0}{}^{i}-\rho{}_{0}{}_{0}{}^{i}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}a^{2}(S_{2}+2)\delta
L{}^{i}{}^{k}{}_{k}-\tfrac{1}{2}(S_{2}+2)\delta
L{}_{0}{}^{i}{}_{0}-\rho{}_{0}{}^{i}{}_{0}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\delta L{}^{i}{}_{j}{}_{0}-\delta
L{}^{k}{}_{k}{}_{0}\delta{}^{i}{}_{j}-\tfrac{1}{2}(S_{2}+2)\delta
L^{0}{}_{00}\delta{}^{i}{}_{j}+\tfrac{1}{2}(S_{2}+2)\delta
L{}^{i}{}_{0}{}_{j}+\rho{}_{0}{}^{i}{}_{j}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}(S_{2}-2)\delta
L{}^{k}{}_{k}{}_{i}+\tfrac{1}{2}(S_{2}-2)\delta
L{}^{0}{}_{i}{}_{0}-\rho{}_{i}{}^{0}{}_{0}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\delta L{}^{0}{}_{i}{}_{j}-\delta
L{}^{0}{}^{k}{}_{k}\bar{q}_{ij}-\tfrac{1}{2}(S_{2}-2)\delta
L{}_{j}{}^{0}{}_{i}+\tfrac{1}{2}(S_{2}-2)\delta
L^{00}{}_{0}\bar{q}_{ij}+\rho{}_{i}{}^{0}{}{}_{j}$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\tfrac{1}{2}(S_{2}-2)\delta
L{}^{j}{}_{i}{}_{0}-\tfrac{1}{2}(S_{2}+2)\delta
L{}_{0}{}^{j}{}_{i}-\rho{}_{i}{}^{j}{}_{0}{}$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\delta L{}^{j}{}_{i}{}_{k}+\delta
L{}_{k}{}^{j}{}_{i}-\delta
L{}^{l}{}_{l}{}_{i}\delta{}^{j}{}_{k}-\tfrac{1}{2}(S_{2}+2)\delta
L{}^{0}{}_{0}{}_{i}\delta{}^{j}{}_{k}-\tfrac{1}{2}(S_{2}-2)\delta
L{}^{j}{}_{00}\delta{}_{i}{}_{k}-\delta
L{}^{j}{}^{l}{}_{l}\bar{q}{}_{i}{}_{k}+\rho{}_{i}{}^{j}{}_{k}\ .$

It is straightforward to solve the components of the connection, with the
result

$\displaystyle\delta L^{0}{}_{00}$ $\displaystyle=$
$\displaystyle\frac{1}{12(S_{2}+2)}[(S_{2}+6)\rho{}_{0}{}^{i}{}_{i}+(S_{2}+2)\rho{}_{i}{}_{0}{}{}^{i}-4\rho_{0}{}^{0}{}_{0}]+\frac{1}{3}T_{0}$
(A.6) $\displaystyle\delta L{}^{i}{}_{00}$ $\displaystyle=$
$\displaystyle-\frac{1}{2(S_{2}-2)^{2}(S_{2}+2)}\Big{[}a^{2}(S_{2}^{2}-4)\rho{}^{i}{}^{k}{}_{k}-a^{2}(S_{2}^{2}-4)\rho{}_{j}{}^{i}{}^{j}-2(S_{2}+2)\rho{}^{i}{}_{00}$
(A.7)
$\displaystyle-2(S_{2}-2)\rho{}_{0}{}^{i}{}_{0}-2(S_{2}^{2}-4)\rho{}_{0}{}_{0}{}^{i}\Big{]}$
$\displaystyle\delta L{}^{0}{}_{i}{}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{6(S_{2}^{2}-4)(S_{2}+2)}\Big{[}(S_{2}^{2}-4)(S_{2}+3)\rho{}_{i}{}^{k}{}_{k}+(S_{2}+1)(S_{2}^{2}-4)\rho{}_{k}{}_{i}{}^{k}$
(A.8)
$\displaystyle+2(S_{2}+2)(S_{2}+3)\rho{}_{i0}{}^{0}-2(S_{2}-2)(3S_{2}+5)\rho{}_{0}{}_{i}{}^{0}+2(S_{2}^{2}-4)(2S_{2}+3)\rho{}_{0}{}^{0}{}_{i}\Big{]}$
$\displaystyle+\frac{1}{3}T_{i}$ $\displaystyle\delta L{}^{0}{}_{0}{}_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{2(S_{2}^{2}-4)(S_{2}+2)}\Big{[}(S_{2}^{2}-4)\rho{}_{i}{}^{k}{}_{k}-(S_{2}^{2}-4)\rho{}_{k}{}_{i}{}^{k}+2(S_{2}+2)\rho{}_{i0}{}^{0}$
(A.9)
$\displaystyle+2(S_{2}^{2}-4)\rho{}_{0}{}^{0}{}_{i}-2(S_{2}-2)\rho{}_{0}{}_{i}{}^{0}\Big{]}$
$\displaystyle\delta L{}^{i}{}_{j}{}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4(S_{2}-2)}\Big{[}(S_{2}+2)\delta{}^{i}{}_{j}\rho{}_{k}{}_{0}{}{}^{k}-2(S_{2}+2)\rho{}^{i}{}_{0}{}{}_{j}-2(S_{2}-2)\rho{}_{0}{}_{j}{}^{i}+(S_{2}-2)\delta{}^{i}{}_{j}\rho{}_{0}{}^{k}{}_{k}$
(A.10) $\displaystyle+4\rho{}_{j}{}^{i}{}_{0}{}\Big{]}$ $\displaystyle\delta
L{}^{j}{}_{0}{}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{12(S_{2}^{2}-4)}\Big{[}12(S_{2}+2)\rho{}_{i}{}_{0}{}{}^{j}-24\rho{}^{j}{}_{i}{}_{0}{}+(S_{2}^{2}-4)\delta^{j}{}_{i}\rho{}_{k}{}_{0}{}{}^{k}-12(S_{2}-2)\rho{}_{0}{}^{j}{}_{i}$
(A.11)
$\displaystyle+(S_{2}-2)(S_{2}+6)\delta^{j}{}_{i}\rho{}_{0}{}^{k}{}_{k}+4(S_{2}+4)\delta^{j}{}_{i}\rho_{0}{}^{00}\Big{]}+\frac{1}{3}\delta^{j}{}_{i}T_{0}$
$\displaystyle\delta L{}^{0}{}_{i}{}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{4(S_{2}+2)}\Big{[}(S_{2}+2)\bar{q}{}_{i}{}_{j}\rho{}_{k}{}^{0}{}{}^{k}-2(S_{2}+2)\rho{}_{i}{}^{0}{}{}_{j}-2(S_{2}-2)\rho{}^{0}{}_{j}{}_{i}-4\rho{}_{j}{}_{i}{}^{0}$
(A.12)
$\displaystyle+(S_{2}-2)\bar{q}{}_{i}{}_{j}\rho{}^{0}{}_{k}{}^{k}\Big{]}$
$\displaystyle\delta L{}^{i}{}_{j}{}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{12(S_{2}^{2}-4)(S_{2}+2)}\Big{[}3(S_{2}+2)\bar{q}{}_{j}{}_{k}\Big{\\{}(S_{2}^{2}-4)(-\rho{}^{i}{}^{l}{}_{l}+\rho{}_{l}{}^{i}{}^{l})-2(S_{2}+2)\rho{}^{i}{}_{00}$
(A.13)
$\displaystyle-2(S_{2}-2)\rho{}_{0}{}^{i}{}_{0}\Big{\\}}-3(S_{2}+2)\Big{\\{}2(S_{2}^{2}-4)(-\rho{}^{i}{}_{k}{}_{j}+\rho{}_{j}{}^{i}{}_{k}+\rho{}_{k}{}_{j}{}^{i})$
$\displaystyle+\delta^{i}{}_{j}\big{[}(S_{2}^{2}-4)(\rho{}_{l}{}_{k}{}^{l}-\rho{}_{k}{}^{l}{}_{l})+2(S_{2}+2)\rho{}_{k}{}^{0}{}_{0}+2(S_{2}-2)\rho{}_{0}{}_{k}{}^{0}\big{]}\Big{\\}}$
$\displaystyle+2\delta^{i}{}_{k}\Big{\\{}(S_{2}^{2}-4)(S_{2}+3)\rho{}_{j}{}^{l}{}_{l}+(S_{2}^{2}-4)(S_{2}+1)\rho{}_{l}{}_{j}{}^{l}-2(S_{2}+2)(2S_{2}+3)\rho{}_{j}{}^{0}{}_{0}$
$\displaystyle+2(S_{2}-2)\rho{}_{0}{}_{j}{}^{0}+S_{2}(S_{2}^{2}-4)\rho{}_{0}{}^{0}{}_{j}\Big{\\}}\Big{]}+\frac{1}{3}\delta^{i}{}_{k}T_{j}\
.$

Because of projective invariance we can choose the projective mode freely. We
choose it so that the torsion vector vanishes, $T_{\alpha}=0$. For the
background decomposition (2.43) this translates into $L_{2}=L_{3}$.
Substituting (A) and (A.6)–(A.13), with the source (A.4), into the action
(2.39) gives the kinetic terms.

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