# Gravitational waves in $f(Q)$ non-metric gravity without gauge fixing

Salvatore Capozziello<EMAIL_ADDRESS>_Dipartimento di Fisica "E.
Pancini", Università di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo,
Edificio G, Via Cinthia, I-80126, Napoli, Italy,_ _INFN Sezione di Napoli,
Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli,
Italy,_ _Scuola Superiore Meridionale, Largo S. Marcellino 10, I-80138,
Napoli, Italy,_ Maurizio Capriolo<EMAIL_ADDRESS>_Dipartimento di Fisica
Università di Salerno, via Giovanni Paolo II, 132, Fisciano, SA I-84084,
Italy._

(September 1, 2024)

###### Abstract

We investigate the polarization modes of gravitational waves in $f(Q)$ non-
metric gravity without gauge fixing. The main result of this study is that no
further scalar mode appears more than the two standard plus and cross
transverse polarizations of massless tensor gravitational radiation, typical
of General Relativity. This is because the first-order perturbation of
connection does not modify the linearized field equations in vacuum which
remain gauge invariant. Then, the world line equations of free point
particles, as well as the equations of their deviations, are obtained using
only the symmetric teleparallel connection. In $f(Q)$ gravity, test masses
follow timelike geodesics and not autoparallel curves. In the proper reference
frame, thanks to the geodesic deviation equation of the structure-less bodies
in free fall, we prove that, in any gauge, only the metric perturbations
$h_{\mu\nu}$, related to tensor modes, survive by exploiting the gauge
invariance. Besides, scalar modes disappear. This allows us to conclude that
only two degrees of freedom of linearized $f(Q)$ non-metric gravity propagate
as in General Relativity and in $f(T)$ teleparallel gravity. The situation is
different with respect to $f(R)$ gravity (with $f(R)\neq R$) where a further
scalar mode is found.

##  1 Introduction

General Relativity (GR) is the theory describing gravitational interaction
according to the geometric description formulated by Einstein and physically
based on the validity of the Equivalence Principle. Confirmations of this
theory range from Solar System dynamics, to the existence of gravitational
waves and black holes up to the dynamical formulation of the cosmological
problem. However, despite the undeniable successes of this theory, several
shortcomings emerge at infrared and ultraviolet scales. To overcome these
issues, extensions and modification of the Einstein picture have been
developed. See, e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9]. Some of these are based on
metric-affine geometry where the spacetime manifold is endowed with a metric
tensor and a general connection, which can be also not compatible with metric,
and features both torsion and curvature [10]. These two geometric objects,
metric and connection, are treated as independent, i.e., as fields whose
evolution describes gravitational interaction via equations of motion arising
from the given theory of gravity.

In principle, the number of degrees of freedom of metric-affine theories is of
$10$ components for the metric $g_{\mu\nu}$ and $64$ for the affine connection
$\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}$, necessary to encode the
gravitational field. If we limit ourselves to connections without torsion and
curvature, as in symmetric teleparallel gravity (STG) theories, this number is
drastically reduced to $10+4$, for details see Ref. [11]. A straightforward
extension of symmetric teleparallel equivalent General Relativity (STEGR),
described by a linear action of the non-metricity scalar $Q$, there is the so
called $f(Q)$ gravity, where $f$ is a generic analytic function. As we will
demonstrate in this study, only two degrees of freedom, responsible of two
tensor modes, propagate without fixing any gauge. In [12], the scalar mode in
$f(Q)$ gravity appears as a ghost, and can be eliminated by a constraint, and
therefore does not propagate. However, the number of degrees of freedom in STG
theories, and, in particular, the number of propagating modes, i.e. the
physical ones that cannot be eliminated, is matter of debate and it is still
an open question. In Ref. [13], authors claim that the degrees of freedom in
$f(Q)$ gravity are 8, while, in Ref. [14], it is argued that the degrees of
freedom should be equal to or less than 7.

In this paper, we investigate, in the framework of $f(Q)$ non-metric gravity,
the possible presence of further modes in gravitational radiation beyond the
two standard tensor ones of GR. We will perform the analysis in arbitrary
gauge, without fixing the coincident gauge as done in many studies [15, 16,
17, 18, 19, 20].

We will first derive the world-line equations of test bodies and then the
equation of deviations. After, we discuss the GW polarization modes by metric
perturbations $h_{\mu\nu}$ expressed in any gauge. Some astrophysical and
cosmological applications of $f(Q)$ gravity are reported in [21, 22, 23, 24,
25, 26, 27]. A different approach to GWs is reported in [28, 29]. In order to
study the production of GWs, it is necessary to derive the related
gravitational energy-momentum pseudo-tensor for. This object is derived in
[30] for non-local theories of gravity. Pseudo-tensors for higher-order
curvature-based gravity are discussed in [31, 32, 33], while, for teleparallel
gravity, see [34]. The propagation of GWs in non-local gravity is developed in
Refs. [35, 36, 37]. GWs in teleparallel and in metric gravity are discussed in
[38, 39, 40].

The present paper has the following structure. Sec. 2 is devoted to a summary
of the main geometric objects characterizing non-metric gravity. In
particular, we underline the difference between autoparallel and geodesic
curves. After deriving the field equations from a variational principle in
Sec. 3, we will linearize them in any gauge in Sec. 4. Specifically, we use
the linear perturbations of metric tensor and STG connection. Thanks to the
wave solutions, derived in Sec. 5 via the Fourier transformations, the
geodesic deviation equation can expressed in a local Lorentz frame (see Sec.
6). It can be solved in the metric perturbation $h_{\mu\nu}$ expressed in an
arbitrary gauge. See Sec. 7. Finally the polarizations and the nature of GW
modes is discussed in Sec. 7. Discussion and conclusions are reported in Sec.
8. The fall of free body and geodesic deviation equations are derived in the
two final appendices.

Adopted conventions are: $\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$ for the
Minkowski metric tensor;
$R^{\mu}_{\phantom{\mu}\nu\alpha\beta}=\partial_{\alpha}\Gamma^{\mu}_{\phantom{\mu}\nu\beta}\dots$
for the Riemann tensor;
$R_{\nu\beta}=R^{\lambda}_{\phantom{\lambda}\nu\lambda\beta}$ for the Ricci
tensor. Along the draft, $\Gamma$ stands for the teleparallel symmetric
connection as well as $\nabla$ stands for covariant derivative with respect to
the non-metric connection $\Gamma$.

##  2 A geometric summary

In an arbitrary metric-affine geometry, we can uniquely decompose the affine
connection $\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}$ in presence of
curvature, torsion and non-metricity, into three parts as [41, 42, 44, 43]

$\boxed{\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}=\genfrac{\\{}{\\}}{0.0pt}{1}{\alpha}{\mu\nu}+K^{\alpha}_{\phantom{\alpha}\mu\nu}+L^{\alpha}_{\phantom{\alpha}\mu\nu}}\
,$ (2.1)

where $\genfrac{\\{}{\\}}{0.0pt}{1}{\alpha}{\beta\gamma}$ are the _Christoffel
symbols_ defined as

$\boxed{\genfrac{\\{}{\\}}{0.0pt}{1}{\rho}{\mu\nu}\equiv\frac{1}{2}g^{\rho\alpha}\left(\partial_{\nu}g_{\mu\alpha}+\partial_{\mu}g_{\nu\alpha}-\partial_{\alpha}g_{\mu\nu}\right)}\
,$ (2.2)

$K^{\alpha}_{\phantom{\alpha}\mu\nu}$ is the contortion tensor defined as

$\boxed{K^{\alpha}_{\phantom{\alpha}\mu\nu}:=\frac{1}{2}g^{\alpha\lambda}\left(T_{\mu\lambda\nu}+T_{\nu\lambda\mu}+T_{\lambda\mu\nu}\right)}\
,$ (2.3)

through the torsion tensor $T^{\alpha}_{\phantom{\alpha}\mu\nu}$

$T^{\alpha}_{\phantom{\alpha}\mu\nu}=2\Gamma^{\alpha}_{\phantom{\alpha}[\mu\nu]}\
,$ (2.4)

and $L^{\alpha}_{\phantom{\alpha}\mu\nu}$ is the disformation tensor defined
as

$\boxed{\begin{aligned}
L^{\alpha}_{\phantom{\alpha}\mu\nu}&:=-\frac{1}{2}g^{\alpha\lambda}\left(Q_{\mu\lambda\nu}+Q_{\nu\lambda\mu}-Q_{\lambda\mu\nu}\right)\\\
&=\frac{1}{2}Q^{\alpha}_{\phantom{\alpha}\mu\nu}-Q_{(\mu\phantom{\alpha}\nu)}^{\phantom{(\mu}\alpha}\end{aligned}}\
,$ (2.5)

by means of $Q_{\alpha\mu\nu}$ which is the non-metricity tensor defined as

$\boxed{Q_{\alpha\mu\nu}=\nabla_{\alpha}g_{\mu\nu}=\partial_{\alpha}g_{\mu\nu}-\Gamma^{\beta}_{\phantom{\beta}\alpha\mu}g_{\beta\nu}-\Gamma^{\beta}_{\phantom{\beta}\alpha\nu}g_{\beta\mu}}\
.$ (2.6)

The contortion tensor is antisymmetric in the first and third indices

$K_{\alpha\mu\nu}=-K_{\nu\mu\alpha}\ ,$ (2.7)

and its antisymmetric part is given by

$K^{\alpha}_{\phantom{\alpha}[\mu\nu]}=\frac{1}{2}T^{\alpha}_{\phantom{\alpha}\mu\nu}\
.$ (2.8)

The disformation tensor $L^{\alpha}_{\phantom{\alpha}\mu\nu}$ is symmetric in
the second and third indices

$L^{\alpha}_{\phantom{\alpha}[\mu\nu]}=0\ ,$ (2.9)

and the non-metricity tensor $Q_{\alpha\mu\nu}$ is symmetric in the last two
indices

$Q_{\alpha[\mu\nu]}=0\ ,$ (2.10)

and furthermore satisfies the following Bianchi identity

$\nabla_{[\alpha}Q_{\beta]\mu\nu}=0\ .$ (2.11)

Symmetric teleparallel gravity (STG) represents a class of non-metric theories
of gravity where both the curvature and the torsion of the affine connection
vanish, and therefore it is a formulation of gravity described in terms of
non-metricity only, i.e. ,

$\boxed{0=R^{\lambda}_{\phantom{\lambda}\mu\nu\sigma}}=\Gamma^{\lambda}_{\phantom{\lambda}\mu\sigma,\nu}-\Gamma^{\lambda}_{\phantom{\lambda}\mu\nu,\sigma}+\Gamma^{\beta}_{\phantom{\beta}\mu\sigma}\Gamma^{\lambda}_{\phantom{\lambda}\beta\nu}-\Gamma^{\beta}_{\phantom{\beta}\mu\nu}\Gamma^{\lambda}_{\phantom{\lambda}\beta\sigma}\
,$ (2.12)

and

$\boxed{T^{\alpha}_{\phantom{\alpha}\mu\nu}=0}\ .$ (2.13)

Therefore, in STG, the connection reads as

$\boxed{\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}=\genfrac{\\{}{\\}}{0.0pt}{1}{\alpha}{\mu\nu}+L^{\alpha}_{\phantom{\alpha}\mu\nu}}\
.$ (2.14)

The non-metricity scalar is

$\boxed{\begin{aligned}
Q\equiv&-\frac{1}{4}Q_{\alpha\beta\gamma}Q^{\alpha\beta\gamma}+\frac{1}{2}Q_{\alpha\beta\gamma}Q^{\gamma\beta\alpha}+\frac{1}{4}Q_{\alpha}Q^{\alpha}-\frac{1}{2}Q_{\alpha}\widetilde{Q}^{\alpha}\\\
=&\,g^{\mu\nu}\Bigl{(}L^{\alpha}_{\phantom{\alpha}\beta\mu}L^{\beta}_{\phantom{\beta}\nu\alpha}-L^{\alpha}_{\phantom{\alpha}\beta\alpha}L^{\beta}_{\phantom{\beta}\mu\nu}\Bigr{)}\end{aligned}}\
,$ (2.15)

and its two traces are

$Q_{\alpha}\equiv Q_{\alpha\phantom{\mu}\mu}^{\phantom{\alpha}\mu}\ ,$ (2.16)

and

$\widetilde{Q}^{\alpha}\equiv Q_{\mu}^{\phantom{\mu}\mu\alpha}\ .$ (2.17)

Accordingly, the contractions of the disformation tensor
$L^{\lambda}_{\phantom{\lambda}\alpha\beta}$ become

$L^{\lambda}_{\phantom{\lambda}\alpha\lambda}=-\frac{1}{2}Q_{\alpha}\quad\
,\quad
L^{\alpha\lambda}_{\phantom{\alpha\lambda}\lambda}=\frac{1}{2}Q^{\alpha}-\widetilde{Q}^{\alpha}\
.$ (2.18)

We can introduce a useful geometric object, the non-metricity conjugate
tensor, defined as

$\boxed{P^{\alpha}_{\phantom{\alpha}\mu\nu}=\frac{1}{2}\frac{\partial
Q}{\partial Q_{\alpha}^{\phantom{\alpha}\mu\nu}}}\ ,$ (2.19)

that, thanks to the symmetry of $Q_{\alpha\mu\nu}$ (2.10), is symmetric in the
last two indices

$P^{\alpha}_{\phantom{\alpha}\mu\nu}=P^{\alpha}_{\phantom{\alpha}(\mu\nu)}\ .$
(2.20)

The superpotential $P^{\alpha}_{\phantom{\alpha}\mu\nu}$ defined in Eq.
(2.19), thanks to the non-metricity scalar $Q$ given in (2.15), can be written
as

$\displaystyle P^{\alpha}_{\phantom{\alpha}\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Bigl{[}-Q^{\alpha}_{\phantom{\alpha}\mu\nu}+2Q_{(\mu\phantom{\alpha}\nu)}^{\phantom{(\mu}\alpha}+Q^{\alpha}g_{\mu\nu}-\widetilde{Q}^{\alpha}g_{\mu\nu}-\delta^{\alpha}_{(\mu}Q_{\nu)}\Bigr{]}$
(2.21) $\displaystyle=$
$\displaystyle-\frac{1}{2}L^{\alpha}_{\phantom{\alpha}\mu\nu}+\frac{1}{4}\bigl{(}Q^{\alpha}-\widetilde{Q}^{\alpha}\big{)}g_{\mu\nu}-\frac{1}{4}\delta^{\alpha}_{(\mu}Q_{\nu)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}L^{\alpha}_{\phantom{\alpha}\mu\nu}+\frac{1}{4}B^{\alpha}g_{\mu\nu}-\frac{1}{4}\delta^{\alpha}_{(\mu}Q_{\nu)}\
,$ (2.22)

where the vector $B^{\alpha}$ is defined as

$B^{\alpha}=Q^{\alpha}-\widetilde{Q}^{\alpha}\ .$ (2.23)

This allows us to rewrite the non-metricity scalar Q as

$Q=Q_{\alpha\mu\nu}P^{\alpha\mu\nu}\ .$ (2.24)

###  2.1 Autoparallel curves and geodesics

The autoparallel curves or affine geodesics are the straightest curves, that
is, a curve $x^{\mu}(s)$ along which the tangent vector $t=t^{\mu}{\bf
e}_{\mu}=\bigl{(}dx^{\mu}/ds\bigr{)}{\bf e}_{\mu}$ is parallel transported
with respect to the connection $\Gamma$, namely if

$\nabla_{t}t=0\longleftrightarrow
t^{\alpha}\nabla_{\alpha}t^{\mu}=0\longleftrightarrow\boxed{\frac{dt^{\alpha}}{ds}+\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma}t^{\beta}t^{\gamma}=0}\
.$ (2.25)

Then, the autoparallel curves only require the connection on the manifold. The
geodesic or metric geodesic are the "shortest" curves connecting two given
points, that is, let $s$ be the distance along the timelike curve
$x^{\mu}(s)$, the geodesic is the local extreme of the length functional $s$
obtained through the Euler-Lagrange equations, namely

$\delta
s=\delta\int_{s_{1}}^{s_{2}}\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}}ds=0\longleftrightarrow\boxed{\frac{dt^{\alpha}}{ds}+\genfrac{\\{}{\\}}{0.0pt}{1}{\alpha}{\beta\gamma}t^{\beta}t^{\gamma}=0}\
,$ (2.26)

where $\genfrac{\\{}{\\}}{0.0pt}{1}{\alpha}{\beta\gamma}$ are the Christoffel
symbols defined in Eq. (2.2). Thus Eq. (2.26) is the so-called geodesic
equation. Then the geodesic curves require only the metric tensor defined on
the manifold. In GR, the connection is metric-compatible and symmetric. From
Eq. (2.1), we have

$\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma}=\genfrac{\\{}{\\}}{0.0pt}{1}{\alpha}{\beta\gamma}\rightarrow\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma}=\hat{\Gamma}^{\alpha}_{\phantom{\alpha}\beta\gamma}\
,$ (2.27)

that is the connection $\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma}$ becomes
the Levi-Civita connection $\hat{\Gamma}^{\alpha}_{\phantom{\alpha}\mu\nu}$,
namely

$\boxed{\hat{\Gamma}^{\alpha}_{\phantom{\alpha}\mu\nu}:=\frac{1}{2}g^{\alpha\lambda}\left(\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu}\right)}\
.$ (2.28)

Therefore autoparallel curves and geodesics exactly coincide only in GR. For a
generic non-metric connection, this statement does not work. As we will see,
in STGs as $f(Q)$ gravity, autoparallel curves and geodesics do not coincide.
This means that falling bodies follow geodesics and not autoparallel curves.

##  3 Field and connection equations

In STG, spacetime is a Lorentz manifold equipped with a metric tensor
$g_{\mu\nu}$ and a non-metric, torsion-free and curvature-free connection, the
symmetric teleparallel connection $\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}$
as defined in (2.14). Metric and connection are governed by the respective
equations of motion. From now on, we will use only the non-metric, symmetric
and teleparallel connection without ever using the Levi-Civita one.

The action in $f(Q)$ gravity is expressed in the framework of Palatini
formalism. Here $g_{\mu\nu}$ and $\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}$
are independent dynamical variables [45, 46]. It is

$S_{f(Q)}=\int_{\Omega}d^{4}x\,\Bigl{[}\frac{1}{2\kappa^{2}}\sqrt{-g}f\left(Q\right)+\lambda_{\alpha}^{\phantom{\alpha}\beta\mu\nu}R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}+\lambda_{\alpha}^{\phantom{\alpha}\mu\nu}T^{\alpha}_{\phantom{\alpha}\mu\nu}+\sqrt{-g}\mathcal{L}_{m}\bigl{(}g\bigr{)}\Bigr{]}\
,$ (3.1)

where $\kappa^{2}=8\pi G/c^{4}$ and
$\lambda_{\alpha}^{\phantom{\alpha}\beta\mu\nu}=\lambda_{\alpha}^{\phantom{\alpha}\beta[\mu\nu]}$,
$\lambda_{\alpha}^{\phantom{\alpha}\mu\nu}=\lambda_{\alpha}^{\phantom{\alpha}[\mu\nu]}$
are Lagrange multipliers. Varying the action $S_{f(Q)}$ with respect to the
metric tensor $g_{\mu\nu}$, we have [44, 45, 46, 52, 53, 47, 48, 49, 50, 51]

$\delta_{g}S_{f(Q)}=0\ .$

By requiring that the variation of metric tensor vanishes on the domain
boundary $\Omega$, we obtain the the field equations:

$\boxed{\frac{2}{\sqrt{-g}}\nabla_{\alpha}\left(\sqrt{-g}f_{Q}P^{\alpha}_{\phantom{\alpha}\mu\nu}\right)-\frac{1}{2}g_{\mu\nu}f+f_{Q}\Bigl{(}P_{\mu\alpha\beta}Q_{\nu}^{\phantom{\nu}\alpha\beta}-2Q^{\alpha\beta}_{\phantom{\alpha\beta}\mu}P_{\alpha\beta\nu}\Bigr{)}=\kappa^{2}T_{\mu\nu}}\
,$ (3.2)

where $f_{Q}=\partial f/\partial Q$ and $T_{\mu\nu}$ is the matter energy-
momentum tensor defined as

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\bigl{(}\sqrt{-g}\mathcal{L}_{m}(g)\bigr{)}}{\delta
g^{\mu\nu}}\ .$ (3.3)

Varying the action (3.1) with respect to the connection
$\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}$, the principle of last action gives
[54]

$\delta_{\Gamma}S_{f(Q)}=0\ .$

The connection field equations, with the variation of connection vanishing on
domain boundary, are

$\boxed{\nabla_{\mu}\nabla_{\nu}\bigl{(}\sqrt{-g}f_{Q}P^{\mu\nu}_{\phantom{\mu\nu}\alpha}\bigr{)}=0}\
.$ (3.4)

Constraints on the symmetric teleparallel theory are obtained by the vanishing
of variations with respect to the Lagrange multipliers, i.e.,

$\delta_{\lambda_{\alpha}^{\phantom{\alpha}\beta\mu\nu}}S_{f(Q)}=0\Rightarrow
R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}=0\ ,$ (3.5)

and

$\delta_{\lambda_{\alpha}^{\phantom{\alpha}\mu\nu}}S_{f(Q)}=0\Rightarrow
T^{\alpha}_{\phantom{\alpha}\mu\nu}=0\ .$ (3.6)

The two sets of above field equations give the $f(Q)$ gravity dynamics.

##  4 Linearized $f(Q)$ gravity in any gauge

In order to linearize the non-metric gravity described by action (3.1), let us
perturb around the Minkowski spacetime both the metric and connection up to
first order. These variables are considered independent. It is

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\quad\text{and}\quad\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}={\Gamma^{\alpha}}^{(0)}_{\mu\nu}+{\Gamma^{\alpha}}^{(1)}_{\mu\nu}\
,$ (4.1)

where $|h_{\mu\nu}|\ll 1$ and $|{\Gamma^{\alpha}}^{(1)}_{\mu\nu}|\ll 1$, with

${\Gamma^{\alpha}}^{(0)}_{\mu\nu}=0\ ,$ (4.2)

because the non-metric connection disappears to zero-order when gravity is
absent, having to reproduce the flat spacetime. For any gauge, to the first
order of metric and connection perturbations, we get the following linear
corrections

$Q_{\alpha\mu\nu}^{(1)}=\partial_{\alpha}h_{\mu\nu}-2\,\Gamma^{(1)}_{(\mu|\alpha|\nu)}\
,$ (4.3)
${Q^{\alpha}}^{(1)}=\partial^{\alpha}h-2\,{\Gamma^{\lambda\phantom{\lambda}\alpha}_{\phantom{\lambda}\lambda}}^{(1)}\
,$ (4.4)
${\widetilde{Q}}^{\alpha(1)}=\partial_{\beta}h^{\alpha\beta}-2\,{\Gamma^{(\beta\phantom{\beta}\alpha)}_{\phantom{\beta}\beta}}^{(1)}\
,$ (4.5)
${B^{\alpha}}^{(1)}={Q^{\alpha}}^{(1)}-{\widetilde{Q}}^{\alpha(1)}=\partial^{\alpha}h-\partial_{\beta}h^{\alpha\beta}+{\Gamma^{\alpha\phantom{\beta}\beta}_{\phantom{\alpha}\beta}}^{(1)}-{\Gamma^{\beta\phantom{\beta}\alpha}_{\phantom{\beta}\beta}}^{(1)}\
,$ (4.6)
${L^{\alpha}_{\phantom{\alpha}\mu\nu}}^{(1)}=\frac{1}{2}\partial^{\alpha}h_{\mu\nu}-\partial_{(\mu}h^{\alpha}_{\phantom{\alpha}\nu)}+{\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}}^{(1)}\
,$ (4.7)
${P^{\alpha}_{\phantom{\alpha}\mu\nu}}^{(1)}=-\frac{1}{4}\partial^{\alpha}h_{\mu\nu}+\frac{1}{2}\partial_{(\mu}h^{\alpha}_{\phantom{\alpha}\nu)}+\frac{1}{4}\bigl{(}\partial^{\alpha}h-\partial_{\beta}h^{\alpha\beta}\bigr{)}\eta_{\mu\nu}-\frac{1}{4}\delta^{\alpha}_{(\mu}\partial_{\nu)}h\\\
+\frac{1}{4}\bigl{(}{\Gamma^{\alpha\phantom{\beta}\beta}_{\phantom{\alpha}\beta}}^{(1)}-{\Gamma^{\beta\phantom{\beta}\alpha}_{\phantom{\beta}\beta}}^{(1)}\bigr{)}\eta_{\mu\nu}-\frac{1}{2}{\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}}^{(1)}+\frac{1}{2}\delta^{\alpha}_{(\mu}{\Gamma^{\lambda}_{\phantom{\lambda}\lambda|\nu)}}^{(1)}\
.$ (4.8)

Expanding $f(Q)$ as

$f(Q)=f(0)+f_{Q}(0)Q+O(Q^{2})\ ,$ (4.9)

we get

$f(Q)^{(0)}=f(0)=0\quad\text{and}\quad f(Q)^{(1)}=f_{Q}(0)Q^{(1)}=0\ ,$ (4.10)
$f_{Q}^{(0)}=f_{Q}(0)\quad\text{and}\quad f_{Q}^{(1)}=0\ ,$ (4.11)

assuming $f(0)=0$, and $f(Q)^{(1)}=0$ because, in $Q$, survive only second
order terms in $h_{\mu\nu}$ and ${\Gamma^{\alpha}}^{(1)}_{\mu\nu}$. Field Eqs.
(3.2) in vacuum, up to first order in $h$, become

$\partial_{\alpha}{P^{\alpha\mu}_{\phantom{\alpha\mu}\nu}}^{(1)}=0\ ,$ (4.12)

while connection equations of motion (3.4) at first order yield

$\partial_{\mu}\partial_{\alpha}{P^{\alpha\mu}_{\phantom{\alpha\mu}\nu}}^{(1)}=0\
,$ (4.13)

which does not add any other constraints either on the metric or on the
connection with respect to Eq. (4.12). Finally, linearized field Eqs. (4.12)
in vacuum, in any gauge, become

$\Box
h_{\mu\nu}-2\,\partial^{\alpha}\partial_{(\mu}h_{\alpha|\nu)}+\partial_{\mu}\partial_{\nu}h+2\,\partial_{\alpha}{\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}}^{(1)}-2\,\partial_{(\mu}{\Gamma^{\lambda}_{\phantom{\lambda}\lambda|\nu)}}^{(1)}=0\
.$ (4.14)

Taking into account the absence of torsion for the connection $\Gamma$, it
becomes symmetric, namely

$T^{\alpha}_{\phantom{\alpha}\mu\nu}\bigl{[}\Gamma\bigr{]}=0\Rightarrow\Gamma^{\alpha}_{\phantom{\alpha}[\mu\nu]}=0\
.$ (4.15)

This result has been already used in the previous equations. Since we are in
STG, the flatness of connection gives additional constraints, that is

$R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\bigl{[}\Gamma\bigr{]}=0\Rightarrow\Gamma^{\alpha}_{\phantom{\alpha}\beta\nu,\mu}-\Gamma^{\alpha}_{\phantom{\alpha}\beta\mu,\nu}+\Gamma^{\alpha}_{\phantom{\alpha}\lambda\mu}\Gamma^{\lambda}_{\phantom{\lambda}\beta\nu}-\Gamma^{\alpha}_{\phantom{\alpha}\lambda\nu}\Gamma^{\lambda}_{\phantom{\lambda}\beta\mu}=0\
.$ (4.16)

To the first order, constraints (4.16) become

${\Gamma^{\alpha}_{\phantom{\alpha}\beta\nu,\mu}}^{(1)}={\Gamma^{\alpha}_{\phantom{\alpha}\beta\mu,\nu}}^{(1)}\
,$ (4.17)

and by contracting the $\alpha$ and $\mu$ indices gives

$\partial_{\alpha}{\Gamma^{\alpha}_{\phantom{\alpha}\beta\nu}}^{(1)}=\partial_{\nu}{\Gamma^{\alpha}_{\phantom{\alpha}\beta\alpha}}^{(1)}\
,$ (4.18)

that, from the symmetry of connection, implies

$2\partial_{\alpha}{\Gamma^{\alpha}_{\phantom{\alpha}\beta\nu}}^{(1)}=2\partial_{(\beta}{\Gamma^{\alpha}_{\phantom{\alpha}\alpha|\nu)}}^{(1)}\
.$ (4.19)

Finally, from (4.19), Eqs. (4.14) are further simplified taking the following
form [28, 29]

$\boxed{\Box
h_{\mu\nu}-2\,\partial^{\alpha}\partial_{(\mu}h_{\alpha|\nu)}+\partial_{\mu}\partial_{\nu}h=0}\
,$ (4.20)

which are exactly the same differential equations obtained by linearizing the
$f(Q)$ field equations in the coincident gauge [55]. This means that, in any
gauge, to the first order in perturbations of metric and connection, the
linearized field equations in vacuum are independent of the connection
adopted, because the linearized connection terms in Eq. (4.14) vanish.
Furthermore, Eqs. (4.20) are gauge-invariant, because under gauge
transformation $x^{\prime\alpha}=x^{\alpha}+\epsilon^{\alpha}(x)$ with
infinitesimal vector $\epsilon^{\alpha}$, remain unchanged. Indeed, according
to the transformed perturbation

$h^{\prime}_{\mu\nu}(x^{\prime})=h_{\mu\nu}(x)-2\partial_{(\mu}\epsilon_{\nu)}(x)\
,$ (4.21)

Eqs. (4.20) transform as

$\Box
h^{\prime}_{\mu\nu}-2\,\partial^{\alpha}\partial_{(\mu}h^{\prime}_{\alpha|\nu)}+\partial_{\mu}\partial_{\nu}h^{\prime}=\Box
h_{\mu\nu}-2\,\partial^{\alpha}\partial_{(\mu}h_{\alpha|\nu)}+\partial_{\mu}\partial_{\nu}h\\\
+2\Big{[}\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\partial_{(\mu}\epsilon_{\nu)}+\partial^{\alpha}\partial_{\mu}\partial_{(\alpha}\epsilon_{\nu)}+\partial^{\alpha}\partial_{\nu}\partial_{(\alpha}\epsilon_{\mu)}-\eta^{\alpha\beta}\partial_{\mu}\partial_{\nu}\partial_{(\beta}\epsilon_{\alpha)}\Bigr{]}\
.$ (4.22)

The last line of Eqs. (4.22) cancels out and finally we obtain the gauge
invariance of linear field equations, namely

$\Box
h^{\prime}_{\mu\nu}-2\,\partial^{\alpha}\partial_{(\mu}h^{\prime}_{\alpha|\nu)}+\partial_{\mu}\partial_{\nu}h^{\prime}=\Box
h_{\mu\nu}-2\,\partial^{\alpha}\partial_{(\mu}h_{\alpha|\nu)}+\partial_{\mu}\partial_{\nu}h\
.$ (4.23)

##  5 Gravitational wave solutions

Let us now use the spatial Fourier transformation, in order to search for
wave-like solutions of Eqs. (4.20). We seek for solutions as

$h_{\mu\nu}(x)=\frac{1}{(2\pi)^{3/2}}\int\,d^{3}k\;\big{(}\widetilde{h}_{\mu\nu}\bigl{(}\bm{k}\bigr{)}e^{ik\cdot
x}+c.c.\bigr{)}\ ,$ (5.1)

with $\bm{k}$ the spatial wave vector and $c.c.$ the complex conjugate.
Afterward, the field Eqs. (4.20), in the momentum space, become

$\tilde{F}_{\mu\nu}=k^{2}\tilde{h}_{\mu\nu}-k_{\mu}k^{\alpha}\tilde{h}_{\alpha\nu}-k_{\nu}k^{\alpha}\tilde{h}_{\alpha\mu}+k_{\mu}k_{\nu}\tilde{h}=0\
,$ (5.2)

where c.c. terms have been omitted. Its trace reads as

$k^{2}\tilde{h}-k^{\alpha}k^{\beta}\tilde{h}_{\alpha\beta}=0\ .$ (5.3)

Let us now consider a wave propagating in the $+z$ direction with the wave
vector $k^{\mu}=\bigl{(}\omega,0,0,k_{z}\bigr{)}$, where
$k^{2}=\omega^{2}-k_{z}^{2}$. Thus the wave expansion (5.1) becomes

$h_{\mu\nu}(z,t)=\frac{1}{\sqrt{2\pi}}\int
dk_{z}\Bigl{(}\tilde{h}_{\mu\nu}(k_{z})e^{i(\omega t-k_{z}z)}+c.c.\Bigr{)}\ .$
(5.4)

The ten components of linear field Eqs. (5.2), in the $k$-space, take the
following form

$\displaystyle\tilde{F}_{00}$
$\displaystyle=\bigl{(}\omega^{2}+k_{z}^{2}\bigr{)}\tilde{h}_{00}+2\omega
k_{z}\tilde{h}_{03}-\omega^{2}\tilde{h}=0\ ,$ (5.5)
$\displaystyle\tilde{F}_{01}$ $\displaystyle=k_{z}^{2}\tilde{h}_{01}+\omega
k_{z}\tilde{h}_{13}=0\ ,$ $\displaystyle\tilde{F}_{02}$
$\displaystyle=k_{z}^{2}\tilde{h}_{02}+\omega k_{z}\tilde{h}_{23}=0\ ,$
$\displaystyle\tilde{F}_{03}$ $\displaystyle=\omega
k_{z}\bigl{(}\tilde{h}_{00}-\tilde{h}_{33}-\tilde{h}\bigr{)}=0\ ,$
$\displaystyle\tilde{F}_{11}$ $\displaystyle=k^{2}\tilde{h}_{11}=0\ ,$
$\displaystyle\tilde{F}_{12}$ $\displaystyle=k^{2}\tilde{h}_{12}=0\ ,$
$\displaystyle\tilde{F}_{13}$ $\displaystyle=\omega
k_{z}\tilde{h}_{01}+\omega^{2}\tilde{h}_{13}=0\ ,$
$\displaystyle\tilde{F}_{22}$ $\displaystyle=k^{2}\tilde{h}_{22}=0\ ,$
$\displaystyle\tilde{F}_{23}$ $\displaystyle=\omega
k_{z}\tilde{h}_{02}+\omega^{2}\tilde{h}_{23}=0\ ,$
$\displaystyle\tilde{F}_{33}$ $\displaystyle=2\omega
k_{z}\tilde{h}_{03}+\bigl{(}\omega^{2}+k_{z}^{2}\bigr{)}\tilde{h}_{33}+k_{z}^{2}\tilde{h}=0\,.$

The trace (5.3) becomes

$\omega^{2}\tilde{h}_{00}+2\omega
k_{z}\tilde{h}_{03}+k_{z}^{2}\tilde{h}_{33}-\bigl{(}\omega^{2}-k_{z}^{2}\bigr{)}\tilde{h}=0\
,$ (5.6)

where $\tilde{h}$ is the trace of the metric perturbation $h_{\mu\nu}$ in the
momentum space given by

$\tilde{h}=\tilde{h}_{00}-\tilde{h}_{11}-\tilde{h}_{22}-\tilde{h}_{33}\ .$
(5.7)

We first solve the set of Eqs. (5.5) and (5.6) for $k^{2}\neq 0$. It is
straightforward to derive the following solution with four independent
variables $\tilde{h}_{01}$, $\tilde{h}_{02}$, $\tilde{h}_{03}$ and
$\tilde{h}_{00}$, namely

$\displaystyle\tilde{h}_{11}$ $\displaystyle=\tilde{h}_{12}=\tilde{h}_{22}=0\
,$ (5.8) $\displaystyle\tilde{h}_{13}$
$\displaystyle=-\frac{k_{z}}{\omega}\tilde{h}_{01}\ ,$
$\displaystyle\tilde{h}_{23}$
$\displaystyle=-\frac{k_{z}}{\omega}\tilde{h}_{02}\ ,$
$\displaystyle\tilde{h}_{33}$
$\displaystyle=-2\frac{k_{z}}{\omega}\tilde{h}_{03}-\frac{k_{z}^{2}}{\omega^{2}}\tilde{h}_{00}\
.$

Therefore, for $k^{2}\neq 0$, we obtain four degrees of freedom, whose
physical nature will be studied below via the geodesic deviation.

Then, for waves with $k^{2}=0$, the solution of Eqs. (5.5) and (5.6), where
$\omega=k_{z}$, depends on six independent variables $\tilde{h}_{12}$,
$\tilde{h}_{11}$, $\tilde{h}_{01}$, $\tilde{h}_{02}$, $\tilde{h}_{03}$ and
$\tilde{h}_{00}$, namely

$\displaystyle\tilde{h}_{22}$ $\displaystyle=-\tilde{h}_{11}\ ,$ (5.9)
$\displaystyle\tilde{h}_{13}$ $\displaystyle=-\tilde{h}_{01}\ ,$
$\displaystyle\tilde{h}_{23}$ $\displaystyle=-\tilde{h}_{02}\ ,$
$\displaystyle\tilde{h}_{33}$ $\displaystyle=-2\tilde{h}_{03}-\tilde{h}_{00}\
,$

and hence, six degrees of freedom which will be analyzed again through the
geodesic deviation. Even if $f(Q)$ gravity seems to have from four to six
degrees of freedom, we it is possible to show that only two propagate.

##  6 Geodesic deviation in $f(Q)$ gravity

Let $x^{\alpha}(\tau)$ be a timelike curve on a manifold whose timelike
tangent vector is $u^{\alpha}=dx^{\alpha}/d\tau$ with positive norm $\|u\|>0$
and $\tau$ the proper time. In $f(Q)$ non-metric gravity, the equation of
motion for a free test particle is

$\boxed{\frac{D^{\prime}u^{\lambda}}{d\tau}=L^{\lambda}_{\phantom{\lambda}\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}}\
,$ (6.1)

where

$\frac{D^{\prime}}{d\tau}=u^{\alpha}\nabla_{\alpha}\ ,$ (6.2)

which is a force equation for the presence of the extra-force term on the
right-hand side, see Appendix A.1 for details. Equivalently Eq. (6.1) can be
written as the geodesic equation

$\frac{du^{\lambda}}{d\tau}+\genfrac{\\{}{\\}}{0.0pt}{1}{\lambda}{\alpha\beta}u^{\alpha}u^{\beta}=0\
,$ (6.3)

that is, free point particles follow the timelike metric geodesic but not the
autoparallel curves with respect to the connection. As a consequence of the
zero Levi-Civita (LC) covariant divergence on the left-hand side of field Eqs
(3.2), on-shell, we have the LC covariant conservation of
$T^{\mu}_{\phantom{\mu}\nu}$ on-shell. The geodesic deviation equation in
$f(Q)$ gravity is

$\boxed{\frac{D^{\prime
2}\eta^{\mu}}{d\tau^{2}}-2u^{\alpha}L^{\mu}_{\phantom{\mu}\alpha\beta}\frac{D^{\prime}\eta^{\beta}}{d\tau}=\nabla_{\nu}L^{\mu}_{\phantom{\mu}\alpha\beta}\eta^{\nu}u^{\alpha}u^{\beta}}\
,$ (6.4)

where $\eta^{\mu}$ is the infinitesimal displacement vector connecting two
nearby world lines. It is the deviation or separation vector, see Appendix A.2
for details. Now we rewrite the geodesic deviation Eq. (6.4) in a local
Lorentz frame (LLF) that is in a coordinate system $\\{x^{\hat{\alpha}}\\}$
such as, in a point $\mathcal{P}_{0}$, the origin of the frame, the metric
tensor $g_{\mu\nu}$ reads [56, 57, 59, 58]

$ds^{2}=dx^{\hat{0}^{2}}-\delta_{\hat{i}\hat{j}}dx^{\hat{i}}dx^{\hat{j}}+O(|x^{\hat{j}}|^{2})dx^{\hat{\alpha}}dx^{\hat{\beta}}\
,$ (6.5)

likewise such that

$g_{\hat{\mu}\hat{\nu}}(\mathcal{P}_{0})=\eta_{\hat{\mu}\hat{\nu}}\quad\text{and}\quad\partial_{\hat{\alpha}}g_{\hat{\mu}\hat{\nu}}(\mathcal{P}_{0})=0\
,$ (6.6)

implying, in symmetric teleparallel theories,

$\Gamma^{\hat{\alpha}}_{\phantom{\alpha}\hat{\mu}\hat{\nu}}(\mathcal{P}_{0})=L^{\hat{\alpha}}_{\phantom{\alpha}\hat{\mu}\hat{\nu}}(\mathcal{P}_{0})=0\
,$ (6.7)

from the second relation of Eq. (4.1). In particular, let us choose a proper
reference frame of a freely falling test particle $A$, located at the origin
$\mathcal{P}_{0}$, and another neighboring freely falling test particle $B$.
The particle $B$ is initially at rest with respect to $A$ and can be described
by the same proper frame to lowest order. Then, let us consider the 4-vector
$\eta^{\hat{\alpha}}=(0,\vec{\chi)}$, where the spatial displacement
$\chi^{\hat{i}}$ represents the spatial coordinate of the particle $B$
relative to particle $A$, located at origin. The comoving 4-velocity of $A$ is
$u_{A}^{\hat{\alpha}}=(1,0,0,0)$, while to the first order in $h$, the
comoving 4-velocity of $B$ is $u_{B}^{\hat{\alpha}}=(1,0,0,0)$. To the first
order in the perturbation $h_{\mu\nu}$, we can identify the proper time with
the coordinate time, that is

$t=\tau+O(h)\ .$ (6.8)

In our freely falling frame, the following identity is fulfilled,

$\partial_{\hat{0}}\Gamma^{\hat{i}}_{\phantom{i}\hat{0}\hat{j}}(\mathcal{P}_{0})=\partial_{\hat{0}}L^{\hat{i}}_{\phantom{i}\hat{0}\hat{j}}(\mathcal{P}_{0})=0\
,$ (6.9)

and the expansion

$\frac{D^{\prime
2}\chi^{\hat{i}}}{d\tau^{2}}=\frac{d^{2}\chi^{\hat{i}}}{dt^{2}}+O(h^{2})\ ,$
(6.10)

has to hold. Thus, from Eqs. (6.4)–(6.10), the geodesic deviation equation
(6.4) in the proper reference frame reduces to

$\boxed{\ddot{\chi}^{\hat{i}}=2{\partial_{[\hat{j}}L^{\hat{i}}_{\phantom{\mu}\hat{0}|\hat{0}]}}^{(1)}{\chi^{\hat{j}}}}\
,$ (6.11)

where the dot stands for the derivative with respect to the coordinate time
$t$. In any gauge, from the linearized disformation tensor (4.7), we obtain

${\partial_{\nu}L^{\mu}_{\phantom{\mu}\alpha\beta}}^{(1)}=\frac{1}{2}\Bigl{(}\partial_{\nu}\partial^{\mu}h_{\alpha\beta}-2\partial_{\nu}\partial_{(\alpha}h^{\mu}_{\phantom{\mu}\beta)}+2\partial_{\nu}{\Gamma^{\mu}_{\phantom{\mu}\alpha\beta}}^{(1)}\Bigr{)}\
,$ (6.12)

and, from Eq. (4.17), we find

$2{\partial_{[\nu}L^{\mu}_{\phantom{\mu}\alpha|\beta]}}^{(1)}=\frac{1}{2}\Bigl{(}\partial_{\alpha}\partial_{\beta}h^{\mu}_{\phantom{\mu}\nu}+\partial_{\nu}\partial^{\mu}h_{\alpha\beta}-\partial^{\mu}\partial_{\beta}h_{\alpha\nu}-\partial_{\nu}\partial_{\alpha}h^{\mu}_{\phantom{\mu}\beta}\Bigr{)}\
,$ (6.13)

that is, the contribution of linearized connection disappears. For our
components in any gauge, we get

$2{\partial_{[j}L^{i}_{\phantom{\mu}0|0]}}^{(1)}=\frac{1}{2}\Bigl{(}\partial_{0}\partial_{0}h^{i}_{\phantom{\mu}j}+\partial_{j}\partial^{i}h_{00}-\partial^{i}\partial_{0}h_{0j}-\partial_{j}\partial_{0}h^{i}_{\phantom{\mu}0}\Bigr{)}\
.$ (6.14)

From the gauge invariance of Eq. (6.14), we have

$2{\partial_{[j}L^{i}_{\phantom{\mu}0|0]}}^{(1)}=2{\partial_{[\hat{j}}L^{\hat{i}}_{\phantom{\mu}\hat{0}|\hat{0}]}}^{(1)}\
,$ (6.15)

which allows us to put Eq. (6.11) into the following form

$\boxed{\ddot{\chi}^{\hat{i}}=2{\partial_{[j}L^{i}_{\phantom{\mu}0|0]}}^{(1)}{\chi^{\hat{j}}}}\
,$ (6.16)

which is the geodesic deviation equation of $fQ)$ gravity, in the proper
reference frame of freely falling particles where
${\partial_{[j}L^{i}_{\phantom{\mu}0|0]}}^{(1)}$ is expressed in any gauge, as
the metric perturbations $h_{\mu\nu}$ from Eq. (6.14). Eq. (6.16) can be
regarded as the relative acceleration between the two freely falling point
particles.

##  7 Polarization of Gravitational Waves

Let us now consider the following expansion to the first order of spatial
separation vector $\chi^{\hat{i}}=(\chi_{x},\chi_{y},\chi_{z})$ in the LLF

$\chi^{\hat{i}}(t)=\chi^{(0)\hat{i}}+\delta x^{\hat{i}}(t)\ ,$ (7.1)

where $\chi^{(0)\hat{i}}=(x_{0},y_{0},z_{0})$ is the initial displacement,
i.e., the initial relative position of the particle $B$ with respect to the
particle $A$ and $\delta x^{\hat{i}}=(\delta x,\delta y,\delta y)$ the linear
perturbation of the spatial deviation vector. The relative variation of the
linear perturbation $\delta x^{\hat{i}}$ with respect to the displacement
$\chi^{(0)\hat{i}}$ is of the same order of $h$, namely

$\frac{|\delta x^{\hat{i}}|}{|\chi^{(0)\hat{i}}|}=O(h)\ .$ (7.2)

Inserting Eq. (6.14) into Eq. (6.11), the linear system of differential
equations, for a wave traveling along positive $z$-axis in a proper local
reference frame, to the first order according to Eq. (7.1) reads as

$\left\\{\begin{array}[]{lr}\ddot{\delta
x}=-\frac{1}{2}\bigl{(}h_{11,00}\bigr{)}x_{0}-\frac{1}{2}\bigl{(}h_{12,00}\bigr{)}y_{0}+\frac{1}{2}\bigl{(}h_{01,03}-h_{13,00}\bigr{)}z_{0}\\\
\ddot{\delta
y}=-\frac{1}{2}\bigl{(}h_{12,00}\bigr{)}x_{0}-\frac{1}{2}\bigl{(}h_{22,00}\bigr{)}y_{0}+\frac{1}{2}\bigl{(}h_{02,03}-h_{23,00}\bigr{)}z_{0}\\\
\ddot{\delta
z}=\frac{1}{2}\bigl{(}h_{01,03}-h_{13,00}\bigr{)}x_{0}+\frac{1}{2}\bigl{(}h_{02,03}-h_{23,00}\bigr{)}y_{0}+\frac{1}{2}\bigl{(}2h_{03,03}-h_{33,00}-h_{00,33}\bigr{)}z_{0}\end{array}\,\right.\
.$ (7.3)

We have taken into account that

${\partial_{j}L^{i}_{\phantom{i}00}}^{(1)}=-{\partial_{j}L_{i00}}^{(1)}\ ,$
(7.4)

where perturbations $h_{\mu\nu}$ are expressed in an arbitrary gauge. For a
plane wave at fixed $k_{z}$, with the angular frequency $\omega$, related by
$k^{2}=\omega^{2}-k_{z}^{2}$, as

$h_{\mu\nu}=\widetilde{h}_{\mu\nu}\bigl{(}k_{z}\bigr{)}e^{i(\omega
t-k_{z}x)}+c.c.\ ,$ (7.5)

system (7.3) yields

$\left\\{\begin{array}[]{ll}\ddot{\delta
x}(t)=\Bigl{[}\frac{1}{2}\omega^{2}\tilde{h}_{11}x_{0}+\frac{1}{2}\omega^{2}\tilde{h}_{12}y_{0}+\frac{1}{2}\bigl{(}\omega
k_{z}\tilde{h}_{01}+\omega^{2}\tilde{h}_{13}\bigr{)}z_{0}\Bigr{]}e^{i(\omega
t-k_{z}x)}\\\ \ddot{\delta
y}(t)=\Bigl{[}\frac{1}{2}\omega^{2}\tilde{h}_{12}x_{0}+\frac{1}{2}\omega^{2}\tilde{h}_{22}y_{0}+\frac{1}{2}\bigl{(}\omega
k_{z}\tilde{h}_{02}+\omega^{2}\tilde{h}_{23}\bigr{)}z_{0}\Bigr{]}e^{i(\omega
t-k_{z}x)}\\\ \ddot{\delta z}(t)=\Bigl{[}\frac{1}{2}\bigl{(}\omega
k_{z}\tilde{h}_{01}+\omega^{2}\tilde{h}_{13}\bigr{)}x_{0}+\frac{1}{2}\bigl{(}\omega
k_{z}\tilde{h}_{02}+\omega^{2}\tilde{h}_{23}\bigr{)}y_{0}\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{2}\bigl{(}2\omega
k_{z}\tilde{h}_{03}+\omega^{2}\tilde{h}_{33}+k_{z}^{2}\tilde{h}_{00}\bigr{)}z_{0}\Bigr{]}e^{i(\omega
t-k_{z}x)}\end{array}\right.+c.c.\ .$ (7.6)

In the case $k^{2}=M^{2}\neq 0$, we insert solution (5.8) in system (7.6),
assuming fixed $k_{z}$. It takes the form

$\left\\{\begin{array}[]{lr}\ddot{\delta x}=0\\\ \ddot{\delta y}=0\\\
\ddot{\delta z}=0\end{array}\ .\right.$ (7.7)

Also, imposing the initial conditions, such as the initial displacement
perturbation $\delta x^{\hat{i}}(0)=0$ and its initial velocity $\dot{\delta
x}^{\hat{i}}(0)=0$, after a double integration with respect to $t$ of system
(7.7), we obtain the solution

$\delta x(t)=0\ ,\quad\delta y(t)=0\ ,\quad\delta z(t)=0\ ,$ (7.8)

which is not a wave, i.e., there is no massive mode associated with $k^{2}\neq
0$. That is, when $k^{2}\neq 0$, none of the four degrees of freedom has a
physical meaning.

On the other hand, in the case $k^{2}=0$, implying $\omega=k_{z}$, by means of
solution (5.9), the geodesic deviation equation, to the first order in
perturbation for a fixed $k_{z}$ of a single plane wave (7.6), yields

$\left\\{\begin{array}[]{lr}\ddot{\delta
x}=\frac{1}{2}\omega^{2}\bigl{(}\tilde{h}^{(+)}x_{0}+\tilde{h}^{(\times)}y_{0}\bigr{)}e^{i\omega(t-z)}\\\
\ddot{\delta
y}=\frac{1}{2}\omega^{2}\bigl{(}\tilde{h}^{(\times)}x_{0}-\tilde{h}^{(+)}y_{0}\bigr{)}e^{i\omega(t-z)}\\\
\ddot{\delta z}=0\end{array}\right.+c.c.\ ,$ (7.9)

where $\tilde{h}_{11}=\tilde{h}^{(+)}$ and
$\tilde{h}_{12}=\tilde{h}^{(\times)}$. Hence, in any gauge, for $k^{2}=0$,
only two degrees of freedom of the initial six survive exactly as in the case
of GR. Then, after double integration with respect to $t$, the solution of
system (7.9) becomes

$\left\\{\begin{array}[]{lr}\delta
x(t)=-\frac{1}{2}\bigl{(}\tilde{h}^{(+)}x_{0}+\tilde{h}^{(\times)}y_{0}\bigr{)}e^{i\omega(t-z)}\\\
\delta
y(t)=-\frac{1}{2}\bigl{(}\tilde{h}^{(\times)}x_{0}-\tilde{h}^{(+)}y_{0}\bigr{)}e^{i\omega(t-z)}\\\
\delta z(t)=0\end{array}\right.+c.c.\ ,$ (7.10)

describing the response of a ring of freely falling masses hit by a
gravitational wave. When we have the pure tensor mode
$\tilde{h}^{(\times)}=0$, from Eq. (7.10), the solution is

$\left\\{\begin{array}[]{ll}\delta
x(t)=-\tilde{h}^{(+)}x_{0}\cos(\omega(t-z))\\\ \delta
y(t)=+\tilde{h}^{(+)}y_{0}\cos(\omega(t-z))\end{array}\right.\ ,$ (7.11)

that is, the effect of gravitational wave is to distort the circle of
particles into ellipses oscillating in a $+$ pattern. While in the pure tensor
mode $\tilde{h}^{(+)}=0$, the ring distorts into ellipses oscillating in a
$\times$ pattern rotated by $45$ degrees in a right-handed sense with respect
to it, because the solution of Eq. (7.10) is

$\left\\{\begin{array}[]{ll}\delta
x(t)=-\tilde{h}^{(\times)}y_{0}\cos(\omega(t-z))\\\ \delta
y(t)=-\tilde{h}^{(\times)}x_{0}\cos(\omega(t-z))\end{array}\right.\ .$ (7.12)

In summary, we have obtained the well-known plus and cross, massless,
transverse, and spin 2 modes of GR. The metric perturbation induced by
gravitational wave can be put into the form

$h_{\mu\nu}(z,t)=\frac{1}{\sqrt{2\pi}}\int
d\omega\Bigl{[}\epsilon_{\mu\nu}^{(+)}\tilde{h}^{(+)}(\omega)+\epsilon_{\mu\nu}^{(\times)}\tilde{h}^{(\times)}(\omega)\Bigr{]}e^{i\omega(t-z)}+c.c.\
,$ (7.13)

where $\epsilon_{\mu\nu}^{(+)}$ and $\epsilon_{\mu\nu}^{(\times)}$ are the
polarization tensors defined as

$\epsilon^{(+)}_{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&0\end{pmatrix}\ ,$ (7.14)
$\epsilon^{(\times)}_{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0\\\
0&0&1&0\\\ 0&1&0&0\\\ 0&0&0&0\end{pmatrix}\ ,$ (7.15)

which, under spatial rotations around the $z$-axes, transform such that the
helicity of perturbation field $h_{\mu\nu}$ is equal two. Hence without fixing
any gauge in $f(Q)$ non-metric gravity, we obtain exactly the same GWs
predicted by GR in the TT gauge. Finally a gravitational wave propagating
along an arbitrary $\mathbf{k}$ direction in $f(Q)$ gravity, from Eq. (7.13),
becomes

$\boxed{h_{\mu\nu}(x)=\frac{1}{(2\pi)^{3/2}}\int
d^{3}k\Bigl{[}\epsilon_{\mu\nu}^{(+)}\tilde{h}^{(+)}(\mathbf{k})+\epsilon_{\mu\nu}^{(\times)}\tilde{h}^{(\times)}(\mathbf{k})\Bigr{]}e^{i(\omega
t-\mathbf{k}\cdot\mathbf{x})}+c.c.}\ .$ (7.16)

Finally, considering the total six degrees of freedom, only two with $k^{2}=0$
propagate, exactly $\tilde{h}^{(+)}$ and $\tilde{h}^{(\times)}$. The geodesic
deviation equation is valid as long as the modulus of the spatial deviation
vector $|\chi^{i}|$ is much smaller than typical scale over the gravitational
field changes. Thus for a gravitational wave of wavelength $\lambda$ and a GWs
detector of characteristic linear size $L$, such as resonant bar detector or
ground-based (LIGO-VIRGO, Einstein Telescope) and space-based (LISA)
interferometers, the following approximation

$L\ll\lambda\ ,$ (7.17)

must hold [57]. In conclusion, as in the case of $f(T)$ teleparallel gravity
[63], no further GW modes than those of GR emerge.

##  8 Discussion and Conclusions

In this paper, we considered the GW polarization modes in $f(Q)$ non-metric
gravity without fixing any gauge. Through the geodesic deviation equation, we
analyzed possible massive and massless scalar and tensor modes. The main
result is that, in any gauge, the gravitational radiation of $f(Q)$ gravity
exhibits only tensor modes without any scalar component. Specifically, in the
linearized theory, only two degrees of freedom propagate, that is $h^{(+)}$
and $h^{(\times)}$. They uniquely encode the gravitational field. Furthermore,
this radiation is transverse and massless, reproducing exactly the same plus
and cross polarizations of GR.

We first perturbed the metric tensor and the affine connection around the
Minkowski spacetime. We thus derived the field and connection equations of
$f(Q)$ in vacuum at the first order. These equations remain gauge invariant
because the linear perturbation of connection disappears. We therefore
obtained, at the first order, the same equations in vacuum obtained in the
coincidence gauge. This result is due to the fact that even if the gauge is
left free, the GW modes remain unchanged. Furthermore, we verified that in our
, symmetric, teleparallel, non-metric theory, the world lines of free point
particles follow timelike geodesics and not autoparallel curves as consequence
of LC covariant conservation of the energy-momentum tensor on-shell, namely
$\mathcal{D}_{\alpha}T^{\alpha}_{\phantom{\alpha}\beta}=0$. We hence obtained
the particle equations of motion using only the connection without imposing
the LC connection. This is because we have equipped our spacetime manifold
with a metric and the symmetric teleparallel connection. The equation for the
deviations between the nearby geodesics was obtained in the same way using
only our non-metric connection. Subsequently, we first solved the linearized
field equations in vacuum in the momentum space by seeking for wave solutions.
After assuming a LLF, the comoving one with free particles in free fall, we
solved the geodetic deviations equation thanks to the gauge invariance of the
derivatives of linearized disformation tensor
$L^{\alpha(1)}_{\phantom{\alpha}\mu\nu}$, which allows us to express the
metric perturbations $h_{\mu\nu}$ in any gauge. Finally, we showed that
massive gravitational radiation modes with $k^{2}\neq 0$ do not propagate
while massless radiation with $k^{2}=0$ propagate. That is, when a massless GW
hits a ring of particles, the first-order perturbations of their position from
the center are transverse tensors, exactly reproducing the two standard modes
of GR. To the first order in $h$ and $\Gamma^{(1)}$, no further scalar mode
oscillates either massive or massless, regardless of the gauge adopted.

Experimentally, if, for example, we take into account as GW source, a
coalescence of BH-BH binary system, emitting radiation with a frequency
$\nu\sim 100\,\text{Hz}$ ($\lambda\sim 3000\,\text{km}$) at cosmological
distances $R\sim 100\,\text{Mpc}$, the typical amplitude or strain is
$h\sim\,10^{-21}$. The arm length of the ground-based interferometers (LIGO-
VIRGO) is $L\sim 3\text{--}4\,\text{km}$ so the approximation (7.17) holds and
the strain falls within the sensitivity of GW Earth observatories.
Additionally, the LISA space interferometer [60] will increase the sensitivity
by a factor 10, $h\sim 10^{-22}$ allowing the detection of weaker GW signals
in low frequencies between $10^{-4}$ and $10^{-1}$ Hz, and the Einstein
Telescope will further improve the sensitivity allowing to detect signals with
even smaller strains $h\sim 10^{-23}$ in the frequency range $\nu=10^{0}\div
10^{3}\,\text{Hz}$ [61]. However, neither GR nor $f(T)$ gravity [62, 63, 64,
65] can be distinguished from the $f(Q)$ theory via GW measurements because
they exhibit the same tensor modes. On the other hand, $f(R)$ gravity could
exhibit detectable scalar radiation [3]. Onnthe other hand, if we add a
boundary term to $f(Q)$, it is possible to prove that the theory exhibits an
additional scalar mode like $f(R)$ [66]. The fact that the non-linear
extensions $f(R)$, $f(T)$, and $f(Q)$ are not equivalent while their linear
actions are equivalent, i.e., the geometric trinity of gravity, has
consequences on the physical degrees of freedom and could have important
consequences in cosmology in discriminating among concurring models [67, 68,
69, 70, 71, 72, 73, 74]. It is worth noticing that symmetric teleparallel
theories could lead to an overall amplification of the stochastic spectrum, as
recently pointed out in other modified theories of gravity [75].

##  Appendix A Appendices

###  A.1  The equations of motion for particles

The matter energy-momentum tensor $T^{\alpha}_{\phantom{\alpha}\beta}$ for a
perfect fluid, specifically for dust, in its rest frame is [41, 76, 17]

$T^{\alpha}_{\phantom{\alpha}\beta}=\rho u^{\alpha}u_{\beta}\ ,$ (A.1)

where $\rho$ is the proper mass-energy density of a particle of mass $m$
located at $\vec{x}_{0}$ i.e.,

$\rho(x)=\frac{m}{\sqrt{-g}}\delta^{3}\big{[}\vec{x}(\tau)-\vec{x}(0)\bigr{]}\
,$ (A.2)

with $\delta^{3}$ the spatial Dirac delta, $\tau$ the proper time and
$u^{\alpha}$ the comoving 4-velocity. Performing the covariant derivative of
Eq. (A.1), we have

$\nabla_{\alpha}T^{\alpha}_{\phantom{\alpha}\beta}=\rho\bigl{(}\nabla_{\alpha}u^{\alpha}\bigr{)}u_{\beta}+\rho\,u^{\alpha}\nabla_{\alpha}u_{\beta}\
.$ (A.3)

We can write the field Eqs (3.2) as

$F^{\alpha}_{\phantom{\alpha}\beta}=\kappa^{2}T^{\alpha}_{\phantom{\alpha}\beta}\
.$ (A.4)

where $F^{\alpha}_{\phantom{\alpha}\beta}$ is the left-hand side of Eqs.(3.2).
We have the vanishing LC covariant divergence on-shell, i.e., for metric
tensor $g_{\mu\nu}$ and connection $\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}$,
solutions of Eqs. (3.2) and (3.4), it is

$\mathcal{D}_{\alpha}F^{\alpha}_{\phantom{\alpha}\beta}=0\ ,$ (A.5)

where $\mathcal{D}$ is the covariant derivative with respect to the
Christoffel symbols. This implies the LC covariant conservation of the energy-
momentum tensor on-shell, namely

$\mathcal{D}_{\alpha}T^{\alpha}_{\phantom{\alpha}\beta}=0\ .$ (A.6)

Then, from the identity

$\nabla_{\alpha}F^{\alpha}_{\phantom{\alpha}\beta}=\mathcal{D}_{\alpha}F^{\alpha}_{\phantom{\alpha}\beta}-\frac{1}{2}Q_{\lambda}F^{\lambda}_{\phantom{\lambda}\beta}+L^{\lambda}_{\phantom{\lambda}\alpha\beta}F^{\alpha}_{\phantom{\alpha}\lambda}\
,$ (A.7)

we derive the covariant divergence of stress-energy tensor

$\nabla_{\alpha}T^{\alpha}_{\phantom{\alpha}\beta}=-\frac{1}{2}Q_{\lambda}T^{\lambda}_{\phantom{\lambda}\beta}+L^{\lambda}_{\phantom{\lambda}\alpha\beta}T^{\alpha}_{\phantom{\alpha}\lambda}\
.$ (A.8)

From Eq. (A.3) and (A.8), we have

$\rho\bigl{(}\nabla_{\alpha}u^{\alpha}\bigr{)}u_{\beta}+\rho\,g_{\sigma\beta}u^{\alpha}\nabla_{\alpha}u^{\sigma}+\rho
Q_{\alpha\sigma\beta}u^{\alpha}u^{\sigma}=-\frac{1}{2}Q_{\lambda}T^{\lambda}_{\phantom{\lambda}\beta}+L^{\lambda}_{\phantom{\lambda}\alpha\beta}T^{\alpha}_{\phantom{\alpha}\lambda}\
.$ (A.9)

So, from the normalization condition of the four-velocity
$u^{\alpha}u_{\alpha}=1$, differentiating it with respect to the non-metric
connection, we get the following differential identity

$2u_{\alpha}\nabla_{\beta}u^{\alpha}+Q_{\beta\alpha\rho}u^{\alpha}u^{\rho}=0\
.$ (A.10)

Projecting Eqs. (A.9) on the 3-space, orthogonal to the four-velocity by the
projector operator $h^{\beta}_{\mu}=\delta^{\beta}_{\mu}-u^{\beta}u_{\mu}$ and
from the differential relation (A.10), we obtain

$g_{\sigma\mu}\frac{D^{\prime}u^{\sigma}}{d\tau}-\frac{1}{2}\bigl{(}Q_{\alpha\beta\sigma}+2L_{\sigma\beta\alpha}\bigr{)}u^{\alpha}u^{\beta}u^{\sigma}u_{\mu}+\bigl{(}L^{\sigma}_{\phantom{\sigma}\alpha\mu}+Q_{\alpha\mu}^{\phantom{\alpha\mu}\sigma}\bigr{)}u^{\alpha}u_{\sigma}=0\
.$ (A.11)

According to the following identities

$\bigl{(}Q_{\alpha\beta\sigma}+2L_{\sigma\beta\alpha}\bigr{)}u^{\alpha}u^{\beta}u^{\sigma}=0\
,$ (A.12)

and

$L^{\sigma}_{\phantom{\sigma}\alpha\mu}+Q_{\alpha\mu}^{\phantom{\alpha\mu}\sigma}=-L_{\mu\alpha}^{\phantom{\mu\alpha}\sigma}\
,$ (A.13)

Eqs. (A.11) finally read as

$\frac{D^{\prime}u^{\nu}}{d\tau}=L^{\nu}_{\phantom{\nu}\alpha\beta}u^{\alpha}u^{\beta}\
,$ (A.14)

that is, from Eq. (6.3), the equations of motion for metric geodesic are
governed by Christoffel symbols and therefore by the metric tensor
$g_{\mu\nu}$.

###  A.2  The geodesic deviation equation

Let us calculate the second covariant derivative of the deviation vector
connecting two nearby geodesics (A.14), i.e.,

$H^{\alpha}=\nabla_{u}\nabla_{u}\eta^{\alpha}\ ,$ (A.15)

where the covariant derivative along the tangent vector $u$ is

$\nabla_{u}=u^{\alpha}\nabla_{\alpha}\ .$ (A.16)

In symmetric teleparallel theories, the $\eta=\eta^{\alpha}{\bf e}_{\alpha}$
and $u=u^{\beta}{\bf e}_{\beta}$ four-vectors commute in a coordinate bases
$\\{{\bf e}_{\alpha}\\}$, since in a symmetric theory the torsion is zero,
that is their commutator vanishes

$[\eta,u]=0\ ,$ (A.17)

or equivalently

$\eta^{\alpha}\nabla_{\alpha}u^{\beta}=u^{\alpha}\nabla_{\alpha}\eta^{\beta}\
,$ (A.18)

or again

$\nabla_{\eta}u^{\beta}=\nabla_{u}\eta^{\beta}\ .$ (A.19)

Now from (A.15) and Eq. (A.17) and the commutativity of the covariant
derivatives $\nabla_{\rho}$ with respect to the connection that has neither
torsion nor curvature, it is

$\begin{split}H^{\alpha}&=\frac{{D^{\prime}}^{2}\eta^{\alpha}}{d\tau^{2}}=u^{\rho}\nabla_{\rho}\bigl{(}u^{\sigma}\nabla_{\sigma}\eta^{\alpha}\bigr{)}\\\
&=\bigl{(}u^{\rho}\nabla_{\rho}\eta^{\sigma}\bigr{)}\big{(}\nabla_{\sigma}u^{\alpha}\bigr{)}+u^{\rho}\eta^{\sigma}\nabla_{\sigma}\nabla_{\rho}u^{\alpha}\\\
&=\bigl{(}\eta^{\rho}\nabla_{\rho}u^{\sigma}\bigr{)}\big{(}\nabla_{\sigma}u^{\alpha}\bigr{)}+\eta^{\sigma}\nabla_{\sigma}\bigl{(}u^{\rho}\nabla_{\rho}u^{\alpha}\bigr{)}-\bigl{(}\eta^{\sigma}\nabla_{\sigma}u^{\rho}\bigr{)}\bigl{(}\nabla_{\rho}u^{\alpha}\bigr{)}\\\
&=\eta^{\sigma}\nabla_{\sigma}\bigl{(}u^{\rho}\nabla_{\rho}u^{\alpha}\bigr{)}\end{split}\
,$ (A.20)

where in the third line we used the Leibniz rule and Eq. (A.17) again. Then,
from the equation of motion (A.14), Eq. (A.20) becomes

$\begin{split}\frac{{D^{\prime}}^{2}\eta^{\nu}}{d\tau^{2}}&=\eta^{\sigma}\nabla_{\sigma}\bigl{(}L^{\nu}_{\phantom{\nu}\alpha\beta}u^{\alpha}u^{\beta}\bigr{)}\\\
&=\nabla_{\sigma}L^{\nu}_{\phantom{\nu}\alpha\beta}\eta^{\sigma}u^{\alpha}u^{\beta}+2L^{\nu}_{\phantom{\nu}\alpha\beta}\bigl{(}\eta^{\sigma}\nabla_{\sigma}u^{\alpha}\bigr{)}u^{\beta}\\\
&=\nabla_{\sigma}L^{\nu}_{\phantom{\nu}\alpha\beta}\eta^{\sigma}u^{\alpha}u^{\beta}+2L^{\nu}_{\phantom{\nu}\alpha\beta}\bigl{(}u^{\sigma}\nabla_{\sigma}\eta^{\alpha}\bigr{)}u^{\beta}\end{split}\
,$ (A.21)

where, in the last line, we always used Eq. (A.17). Finally Eq. (A.21) can be
put in the following form

$\frac{{D^{\prime}}^{2}\eta^{\nu}}{d\tau^{2}}=\nabla_{\sigma}L^{\nu}_{\phantom{\nu}\alpha\beta}\eta^{\sigma}u^{\alpha}u^{\beta}+2L^{\nu}_{\phantom{\nu}\alpha\beta}\frac{D^{\prime}\eta^{\alpha}}{d\tau}u^{\beta}\
,$ (A.22)

that is the geodesic deviation equation of $f(Q)$ gravity.

## Acknowledgements

Authors acknowledge the Istituto Nazionale di Fisica Nucleare (INFN) Sez. di
Napoli, Iniziative Specifiche QGSKY and MOONLIGHT2, and the Istituto Nazionale
di Alta Matematica (INdAM), gruppo GNFM, for the support. This paper is based
upon work from COST Action CA21136: Addressing observational tensions in
cosmology with systematics and fundamental physics (CosmoVerse) supported by
COST (European Cooperation in Science and Technology).

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