# Static and LRS spacetimes of type II in $f(\mathcal{Q})$ gravity

Fabrizio Esposito  0000-0001-6883-152X<EMAIL_ADDRESS>DIME,
Università di Genova, Via all’Opera Pia 15, 16145 Genova, Italy  INFN Sezione
di Genova, Via Dodecaneso 33, 16146 Genova, Italy  Sante Carloni
0000-0003-2373-2653<EMAIL_ADDRESS>DIME, Università di Genova, Via
all’Opera Pia 15, 16145 Genova, Italy  INFN Sezione di Genova, Via Dodecaneso
33, 16146 Genova, Italy  Institute of Theoretical Physics, Faculty of
Mathematics and Physics, Charles University, Prague, V Holešovičkách 2, 180 00
Prague 8, Czech Republic  Stefano Vignolo  0000-0003-2926-0650
<EMAIL_ADDRESS>DIME, Università di Genova, Via all’Opera Pia 15,
16145 Genova, Italy  INFN Sezione di Genova, Via Dodecaneso 33, 16146 Genova,
Italy

(September 1, 2024)

###### Abstract

We investigate the $1+1+2$ covariant formalism in the presence of
nonmetricity. Focusing on static and Locally Rotationally Symmetric
spacetimes, we show how nonmetricity affects all the kinematic quantities
involved in the covariant $1+1+2$ decomposition. We apply the resulting
geometrical framework to study spherically symmetric solutions in the context
of $f(\mathcal{Q})$ gravity in vacuum. We obtain explicit solutions and
sufficient conditions for the existence of Schwarzschild-de Sitter type
solutions.

##  I introduction

General Relativity (GR) is a geometric theory of gravity that describes the
gravitational field in terms of the metric tensor $g_{ab}$ and relates the
matter distribution to the curvature of the spacetime, $R^{a}{}_{bcd}$. In
addition to GR, and since its very first formulation, other metric theories of
gravity were developed to extend the geometrization of fundamental forces and,
later, to address the limitations of GR on quantum, astrophysical, and
cosmological scales. These theories can contain additional fields linked to
the gravitational interaction or even have field equations containing higher-
than-second derivatives (of the metric tensor). Well-known examples are $f(R)$
gravity, a fourth-order theory [1, 2, 3], or Hordenski theories (see e.g.
[4]).

Further alternative theories are the ones based on torsion $T_{ab}{}^{c}$ and
nonmetricity $Q_{abc}$ tensors, which, together with the curvature, express
the properties of the affine connection defined on the spacetime. Among them,
undoubtedly, the most studied are the so-called teleparallel theories of
gravity, a.k.a Teleparallel Gravity (TG). TG represents a class of
gravitational gauge theory that assumes spacetime is flat and describes
gravitation through the torsion [5]. Recently, much attention has been devoted
to another class of teleparallel theories of gravity: Symmetric Teleparallel
Gravity (STG). STG is a class of theories in which one assumes a torsion-free
and curvature-free connection, but with nonmetricity that differs from zero
[6, 7, 8, 9]. Although teleparallel theories are geometric descriptions of
gravity different from GR, in some remarkable cases, they have field equations
equivalent to those of GR. These are called Teleparallel Equivalent of General
Relativity (TEGR) and Symmetric Teleparallel Equivalent of General Relativity
(STEGR). STGs have been extended to include additional invariants in their
action, generating what are now commonly known as $f(T)$-gravity and
$f(\mathcal{Q})$-gravity [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29].

This last class of theories, in particular, has been recently studied in great
detail in cosmological settings. The calculations in this context are
simplified by the choice of the so-called “coincident gauge”. In this specific
gauge, the full connection is assumed to be identically zero so that the
nonmetricity tensor can be thought to be expressed by a partial derivative of
the metric tensor [12]. Unfortunately, in the context of astrophysically
relevant spacetimes, like the spherically symmetric ones, the coincident gauge
has been shown to be incompatible with any non-linear form of the function
$f(\mathcal{Q})$. Because of this limitation, various other gauges have been
proposed. For example, the total connection is made to coincide with the Levi-
Civita one in the gravity-free case [30, 31, 32], or alternatively, via the
imposition of connection invariance under $SO(3)$ group transformations,
restrictions are found on the connection itself [33, 34, 35, 36].

In this paper, we aim to provide a new framework to study astrophysically
relevant metrics, including the static and spherically symmetric ones, in
$f(\mathcal{Q})$ gravity, using a formalism that precludes the assumptions on
the gauge and, therefore, treats the nonmetricity as a variable. More
specifically, our analysis will be performed on LRS spacetimes [37] using the
$1+1+2$ covariant formalism, a natural extension of $1+3$ formalism [38, 39,
40, 41, 42] that we have used in [16]. In this approach, in addition to
splitting spacetime in a preferred time direction and spatial hypersurfaces,
the space itself is split into a preferred spatial direction and surfaces
orthogonal to it. Therefore, it is well adapted to describe LRS spacetimes,
which have a local preferred spatial direction corresponding to the local axis
of symmetry. We will see, in particular, that the combination of the formalism
and the symmetries of LRS spacetimes will allow us to build a complete set of
scalar quantities able to describe the structure of static and spherically
symmetric spacetimes and to deduce the physical consequences of nonmetricity
in this framework.

The paper is organized as follows. In Sec. II we introduce the geometric
framework performing a detailed analysis of the $1+1+2$ formalism when both
curvature and nonmetricity tensor are present. Sec. III is dedicated to the
description of constraints induced by the assumptions of local rotational
symmetry and staticity. We also show how these constraints allow the
decomposition of the nonmetricity tensor in a set of scalars. Using these
results, in Sec. IV, we derive the 1+1+2 equations for $f(\mathcal{Q})$
gravity in LRS spacetimes. In Sec. V, we restrict to the static and spherical
case and consider some solutions. Finally, we discuss the results in Sec. VI.

Throughout the paper natural units ($c=8\pi G=1$) and metric signature
$(-,+,+,+)$ are used.

##  II Geometric framework

We consider a $4$-dimensional manifold endowed with a metric tensor $g_{ab}$
and an affine connection $\Gamma_{ab}{}^{c}$, which is assumed torsionless but
non-metric compatible. We denote by $\nabla$ the covariant derivative induced
by $\Gamma_{ab}{}^{c}$, and by

$Q_{cab}=\nabla_{c}g_{ab}.$ (1)

the corresponding nonmetricity tensor.

According to the above assumptions, the connection $\Gamma$ can be decomposed
as

$\Gamma_{ab}{}^{c}=\tilde{\Gamma}_{ab}{}^{c}+L_{ab}{}^{c},$ (2)

where

$\tilde{\Gamma}_{ab}{}^{c}=\frac{1}{2}g^{cd}\left(\partial_{a}g_{bd}+\partial_{b}g_{ad}-\partial_{d}g_{ab}\right),$
(3)

is the Levi-Civita connection induced by the metric tensor $g_{ab}$, and

$L_{ab}{}^{c}=\frac{1}{2}\left(Q^{c}{}_{ab}-Q_{a}{}^{c}{}_{b}-Q_{b}{}^{c}{}_{a}\right)$
(4)

is the disformation tensor. Throughout the paper, we will denote by a tilde
all quantities related to the Levi-Civita connection (3).

The Riemann tensor $R^{a}{}_{bcd}$ induced by the connection $\Gamma$ can be
expressed as

$\begin{split}R^{a}{}_{bcd}=&\partial_{c}\Gamma_{db}{}^{a}-\partial_{d}\Gamma_{cb}{}^{a}+\Gamma_{cp}{}^{a}\Gamma_{db}{}^{p}-\Gamma_{dp}{}^{a}\Gamma_{cb}{}^{p}=\\\
=&\tilde{R}^{a}{}_{bcd}+\tilde{\nabla}_{c}L_{db}{}^{a}-\tilde{\nabla}_{d}L_{cb}{}^{a}+L_{cp}{}^{a}L_{db}{}^{p}+\\\
&-L_{dp}{}^{a}L_{cb}{}^{p},\end{split}$ (5)

and it satisfies the following identities:

  * •

Antisymmetry in the last two indices,

$R^{a}{}_{bcd}=-R^{a}{}_{bdc};$ (6)

  * •

First Bianchi identity,

$R^{a}{}_{[bcd]}=0;$ (7)

  * •

Second Bianchi identity,

$\nabla_{[e|}R^{a}{}_{b|cd]}=0;$ (8)

Moreover, the Riemann tensor (5) has three independent traces:

$\begin{gathered}R_{bd}=R^{a}{}_{bad},\qquad\bar{R}_{ad}=R_{a}{}^{c}{}_{cd},\\\
\check{R}_{cd}=R^{a}{}_{acd}=-\partial_{[c}Q_{d]a}{}^{a}.\end{gathered}$ (9)

The contraction of Eq. (9) with the metric gives the Ricci scalar,

$R=g^{ab}R_{ab}=-g^{ab}\bar{R}_{ab}.$ (10)

From Eq. (5) it is easily seen that the Ricci scalar can be expressed as

$R=\tilde{R}+\tilde{\nabla}_{a}L_{b}{}^{ba}-\tilde{\nabla}_{b}L_{a}{}^{ba}+\mathcal{Q},$
(11)

where the quantity

$\begin{split}\mathcal{Q}&=L_{ap}{}^{a}L_{b}{}^{bp}-L_{bp}{}^{a}L_{a}{}^{bp}=\\\
=&\frac{1}{4}Q_{cab}Q^{cab}-\frac{1}{2}Q_{cab}Q^{abc}-\frac{1}{4}Q_{ab}{}^{b}Q^{a}{}_{b}{}^{b}+\frac{1}{2}Q_{ab}{}^{b}Q_{ba}{}^{b},\end{split}$
(12)

is the so-called nonmetricity scalar. The Riemann tensor can also be written
as

$R_{abcd}=C_{abcd}+\sum_{n=2}^{6}W^{(n)}_{abcd}+\sum_{n=2}^{5}Z^{(n)}_{abcd},$
(13)

where

$C_{abcd}=W^{(1)}_{abcd}+Z^{(1)}_{abcd}$ (14)

is the Weyl tensor. Further details are provided in Appendix A.

###  II.1 $1+1+2$ decomposition

Let us consider a congruence of timelike curves $x^{a}=x^{a}(\lambda)$,
expressed in terms of an affine parameter $\lambda$ and filling all spacetime.
The 4-vector field tangent to the congruence is

$u^{a}=\frac{dx^{a}}{d\lambda},\qquad u_{a}=g_{ab}u^{b},\qquad u^{a}u_{a}=-1.$
(15)

At any point, we can identify a $3$-dimensional subspace of the tangent bundle
orthogonal to $u^{a}$. The projector onto this spatial subspace is defined as

$h_{ab}=g_{ab}+u_{a}u_{b},$ (16)

and satisfies the properties

$h^{a}{}_{c}h^{c}{}_{b}=h^{a}{}_{b},\qquad h_{ab}u^{b}=0.$ (17)

The $1+1+2$ decomposition is implemented by introducing a congruence of
spacelike curves with tangent 4-vector field $e$ everywhere orthogonal to $u$:
$g_{ab}u^{a}e^{b}=0$. This allows us to split the spatial subspace into a
preferred direction parallel to $e$ and a $2$-dimensional subspace, called
“sheet”, orthogonal to $e$. The projection tensor onto the sheet is given by,

$N_{ab}=h_{ab}-e_{a}e_{b},$ (18)

with

$N^{a}{}_{c}N^{c}{}_{b}=N^{a}{}_{b},\qquad N_{ab}u^{b}=N_{ab}e^{b}=0.$ (19)

Norm conservation of vectors $u$ and $e$ along both the congruences, and with
respect to the covariant derivative induced by the full connection
$\Gamma_{ab}{}^{c}$, as well as the preservation of their orthogonality impose
restrictions on the nonmetricity tensor $Q_{cab}$:

$\begin{gathered}u^{c}\nabla_{c}\left(g_{ab}u^{a}u^{b}\right)=Q_{cab}u^{c}u^{a}u^{b}+2u_{b}u^{c}\nabla_{c}u^{b}=0\\\
\longrightarrow\quad
Q_{cab}u^{c}u^{a}u^{b}=2u^{b}u^{c}\nabla_{c}u_{b},\end{gathered}$ (20)
$\begin{gathered}e^{c}\nabla_{c}\left(g_{ab}e^{a}e^{b}\right)=Q_{cab}e^{c}e^{a}e^{b}+2e_{b}e^{c}\nabla_{c}e^{b}=0\\\
\longrightarrow\quad
Q_{cab}e^{c}e^{a}e^{b}=2e^{b}e^{c}\nabla_{c}e_{b},\end{gathered}$ (21)
$\begin{gathered}e^{c}\nabla_{c}\left(g_{ab}u^{a}u^{b}\right)=Q_{cab}e^{c}u^{a}u^{b}+2u_{b}e^{c}\nabla_{c}u^{b}=0\\\
\longrightarrow\quad
Q_{cab}e^{c}u^{a}u^{b}=2u^{b}e^{c}\nabla_{c}u_{b},\end{gathered}$ (22)
$\begin{gathered}u^{c}\nabla_{c}\left(g_{ab}e^{a}e^{b}\right)=Q_{cab}u^{c}e^{a}e^{b}+2e_{b}u^{c}\nabla_{c}e^{b}=0\\\
\longrightarrow\quad
Q_{cab}u^{c}e^{a}e^{b}=2e^{b}u^{c}\nabla_{c}e_{b},\end{gathered}$ (23)
$\begin{gathered}u^{c}\nabla_{c}\left(g_{ab}u^{a}e^{b}\right)=Q_{cab}u^{c}u^{a}e^{b}+g_{ab}u^{c}\nabla_{c}\left(u^{a}e^{b}\right)=0\\\
\longrightarrow\quad
Q_{cab}u^{c}u^{a}e^{b}=u^{b}u^{c}\nabla_{c}e_{b}+e^{b}u^{c}\nabla_{c}u_{b},\end{gathered}$
(24)
$\begin{gathered}e^{c}\nabla_{c}\left(g_{ab}u^{a}e^{b}\right)=Q_{cab}e^{c}u^{a}e^{b}+g_{ab}e^{c}\nabla_{c}\left(u^{a}e^{b}\right)=0\\\
\longrightarrow\quad
Q_{cab}e^{c}u^{a}e^{b}=u^{b}e^{c}\nabla_{c}e_{b}+e^{b}e^{c}\nabla_{c}u_{b}.\end{gathered}$
(25)

The normalization of $u^{a}$ ($g_{ab}u^{a}u^{b}=-1$) and $e^{a}$
($g_{ab}e^{a}e^{b}=1$) implies the following properties for the projector
tensors:

$h_{a}{}^{a}=3,\qquad\text{and}\qquad N_{a}{}^{a}=2.$ (26)

Making use of $h_{ab}$ and $N_{ab}$, we can define the projected symmetric
trace-free (PSTF) part of a tensor with respect to $u^{a}$ and $e^{a}$
respectively: given a second rank covariant tensor $\psi_{ab}$ we have,

$\psi_{\langle
ab\rangle}=\left[h_{(a}{}^{c}h_{b)}{}^{d}-\frac{1}{3}h_{ab}h^{cd}\right]\psi_{cd},$
(27)

and

$\psi_{\\{ab\\}}=\left[N_{(a}{}^{c}N_{b)}{}^{d}-\frac{1}{2}N_{ab}N^{cd}\right]\psi_{cd}.$
(28)

It is also useful to introduce the volume elements derived from the Levi-
Civita tensor $\epsilon_{abcd}$,

$\epsilon_{abc}=\epsilon_{abcd}u^{d}\qquad\qquad\epsilon_{ab}=\epsilon_{abc}e^{c},$
(29)

which are characterized by the following properties:

$\epsilon_{abc}=\epsilon_{[abc]},\qquad\qquad\epsilon_{ab}=\epsilon_{[ab]},$
(30)
$\epsilon_{abc}u^{c}=0,\quad\epsilon_{abc}\epsilon^{dec}=2h_{[a}{}^{d}h_{b]}{}^{e},\quad\epsilon_{abc}\epsilon^{abd}=2h_{c}{}^{d}$
(31)
$\begin{gathered}\epsilon_{ab}e^{b}=\epsilon_{ab}u^{b}=0,\qquad\qquad\epsilon_{a}{}^{c}\epsilon_{cb}=N_{ab},\\\
\epsilon_{abc}=e_{a}\epsilon_{bc}-e_{b}\epsilon_{ac}+e_{c}\epsilon_{ab}.\end{gathered}$
(32)

Given a generic tensor $\psi^{a\cdot\cdot\cdot}{}_{b\cdot\cdot\cdot}$, there
are several kinds of covariant derivative we can define in the $1+1+2$
formalism. In particular:

  * •

the time derivative: the covariant derivative along $u^{a}$,

$\dot{\psi}^{a\cdot\cdot\cdot}{}_{b\cdot\cdot\cdot}=u^{c}\nabla_{c}\psi^{a\cdot\cdot\cdot}{}_{b\cdot\cdot\cdot};$
(33)

  * •

the spatial derivative: the covariant derivative projected onto the
$3$-dimensional subspace orthogonal to $u^{a}$,

$D_{c}\psi^{a\cdot\cdot\cdot}{}_{b\cdot\cdot\cdot}=h_{c}{}^{d}h^{a}{}_{e}\cdot\cdot\cdot
h_{b}{}^{f}\cdot\cdot\cdot\nabla_{d}\psi^{e\cdot\cdot\cdot}{}_{f\cdot\cdot\cdot};$
(34)

  * •

the hat derivative: the covariant spatial derivative along $e^{a}$,

$\hat{\psi}^{a\cdot\cdot\cdot}{}_{b\cdot\cdot\cdot}=e^{c}D_{c}\psi^{a\cdot\cdot\cdot}{}_{b\cdot\cdot\cdot};$
(35)

  * •

the $\delta$-derivative: the covariant spatial derivative projected onto the
sheet orthogonal to $e^{a}$,

$\delta_{c}\psi^{a\cdot\cdot\cdot}{}_{b\cdot\cdot\cdot}=N_{c}{}^{d}N^{a}{}_{e}\cdot\cdot\cdot
N_{b}{}^{f}\cdot\cdot\cdot D_{d}\psi^{e\cdot\cdot\cdot}{}_{f\cdot\cdot\cdot}.$
(36)

We can now decompose the covariant derivative of the $4$-velocity,

$\begin{split}\nabla_{a}u_{b}=&D_{a}u_{b}-u_{a}\left(\mathcal{A}_{b}+e_{b}\mathcal{A}^{(u)}\right)+\frac{1}{2}u_{a}u_{b}Q_{cde}u^{c}u^{d}u^{e}+\\\
&-\frac{1}{2}u_{b}e_{a}Q_{cde}e^{c}u^{d}u^{e}-\frac{1}{2}u_{b}N_{a}{}^{c}Q_{cde}u^{d}u^{e}.\end{split}$
(37)

where

$\mathcal{A}^{(u)}=e^{a}u^{b}\nabla_{b}u_{a},\qquad\text{and}\qquad\mathcal{A}_{b}=N_{b}{}^{a}u^{c}\nabla_{c}u_{a}.$
(38)

Moreover, the spatial derivative of $u_{a}$ can be expressed as

$D_{a}u_{b}=\frac{1}{3}h_{ab}\theta+\sigma_{ab}+\omega_{ab},$ (39)

where

$\theta=h^{ab}D_{a}u_{b}$ (40)

represents the expansion of the time congruence,

$\begin{split}\sigma_{ab}=&\left[h_{(a}{}^{c}h_{b)}{}^{d}-\frac{1}{3}h_{ab}h^{cd}\right]D_{c}u_{d}=\\\
=&\Sigma_{ab}+2\Sigma_{(a}e_{b)}+\Sigma\left(e_{a}e_{b}-\frac{1}{2}N_{ab}\right)\end{split}$
(41)

represents the distortion, and it is called “shear” tensor, whereas

$\begin{gathered}\Sigma_{ab}=\sigma_{\\{ab\\}},\qquad\qquad\Sigma_{a}=N_{a}{}^{c}e^{b}\sigma_{cb},\\\
\Sigma=e^{a}e^{b}\sigma_{ab}=-N^{ab}\sigma_{ab},\end{gathered}$ (42)

are the tensor, vector, and scalar parts of the shear, respectively. Finally,
the quantity

$\omega_{ab}=D_{[a}u_{b]}=\epsilon_{ab}\Omega+\epsilon_{abc}\Omega^{c}$ (43)

is the “vorticity” tensor of the time congruence, with

$\Omega_{c}=\frac{1}{2}N_{cd}\epsilon^{abd}D_{a}u_{b},\qquad\Omega=\frac{1}{2}\epsilon^{ab}N_{a}{}^{c}N_{b}{}^{d}D_{c}u_{d}.$
(44)

In an analogous way, the covariant derivative of $e_{a}$ can be expressed as,

$\begin{split}\nabla_{a}e_{b}=&D_{a}e_{b}-u_{a}\alpha_{b}-\mathcal{A}^{(e)}u_{a}u_{b}+\left(\frac{1}{3}\theta+\Sigma\right)e_{a}u_{b}+\\\
&+\Sigma_{a}u_{b}-\epsilon_{ab}\Omega^{d}u_{b}-\frac{1}{2}u_{a}e_{b}Q_{cdf}u^{c}e^{d}e^{f}+\\\
&-N_{a}{}^{c}u_{b}Q_{cdf}u^{d}e^{f}-e_{a}u_{b}Q_{cdf}e^{c}u^{d}e^{f}.\end{split}$
(45)

The spatial derivative of $e_{a}$ is given by

$\begin{split}D_{a}e_{b}=&\frac{1}{2}N_{ab}\phi+\zeta_{ab}+\epsilon_{ab}\xi+e_{a}\mathrm{a}_{b}+\\\
&-N_{a}{}^{c}e_{b}L_{cd}^{f}e^{d}e_{f}+\frac{1}{2}e_{a}e_{b}Q_{cdf}e^{c}e^{d}e^{f}\end{split}$
(46)

where

$\phi=N^{ab}\delta_{a}e_{b}\qquad\zeta_{ab}=\delta_{\\{a}e_{b\\}}\qquad\xi=\frac{1}{2}\epsilon^{ab}\delta_{a}e_{b}$
(47)

are the analogous of expansion, shear, and vorticity for the spatial vector
$e_{a}$. The scalar

$\mathcal{A}^{(e)}=-u^{b}u^{a}\nabla_{a}e_{b}$ (48)

is the projection of the temporal acceleration of $e^{a}$ along $u^{a}$, the
vector

$\alpha_{b}=u^{c}N_{b}{}^{d}\nabla_{c}e_{d}$ (49)

is the projection of the temporal acceleration of $e^{a}$ onto the sheet, and

$\mathrm{a}_{b}=e^{c}N_{b}{}^{d}\nabla_{c}e_{d}$ (50)

is the projection of the spatial acceleration onto the sheet.

In addition to this, making use of the Weyl tensor (14), we introduce six new
tensors

$\begin{gathered}H_{cd}=\frac{1}{2}\epsilon_{c}{}^{ab}u^{e}C_{abde},\\\
\bar{H}_{ab}=\frac{1}{2}\epsilon_{a}{}^{cd}u^{e}C_{becd}=H_{ab}+\frac{1}{2}\epsilon_{a}{}^{cd}u^{e}Z^{(1)}{}_{becd},\\\
\check{H}_{ab}=\frac{1}{2}\epsilon_{a}{}^{cd}u^{e}C_{ebcd}=-H_{ab}+\frac{1}{2}\epsilon_{a}{}^{cd}u^{e}Z^{(1)}{}_{ebcd},\end{gathered}$
(51)
$\begin{gathered}E_{ac}=C_{abcd}u^{b}u^{d},\qquad\qquad\bar{E}_{bc}=C_{abcd}u^{a}u^{d},\\\
\check{E}_{cd}=C_{abcd}u^{a}u^{b},\end{gathered}$ (52)

which are the “magnetic” and “electric” parts of the Weyl tensor,
respectively. These tensors satisfy the following properties:

$\begin{gathered}H_{cd}=h_{c}{}^{a}h_{d}{}^{b}H_{ab},\quad H_{cd}=H_{dc},\\\
\quad H^{a}{}_{a}=\bar{H}^{a}{}_{a}=\check{H}^{a}{}_{a}=0\end{gathered}$ (53)
$\check{E}_{cd}=-\check{E}_{dc},\qquad
E^{a}{}_{a}=\bar{E}^{a}{}_{a}=\check{E}^{a}{}_{a}=0.$ (54)

The magnetic and electric parts can be decomposed as well:

$H_{ab}=\mathcal{H}_{ab}+\mathcal{H}_{a}e_{b}+e_{a}\mathcal{H}_{b}+\left(e_{a}e_{b}-\frac{1}{2}N_{ab}\right)\mathcal{H},$
(55)
$\begin{split}\bar{H}_{ab}=&\bar{\mathcal{H}}_{ab}+\bar{\mathcal{H}}_{a}e_{b}+e_{a}\mathbf{H}_{b}-\check{\bar{\mathcal{H}}}_{a}u_{b}+\epsilon_{ab}\mathbb{H}+\\\
&-e_{a}u_{b}\check{\bar{\mathcal{H}}}+\left(e_{a}e_{b}-\frac{1}{2}N_{ab}\right)\bar{\mathcal{H}},\end{split}$
(56)
$\begin{split}\check{H}_{ab}=&\check{\mathcal{H}}_{ab}+\check{\mathcal{H}}_{a}e_{b}+e_{a}\check{\mathbf{H}}_{b}-\mathfrak{H}_{a}u_{b}+\epsilon_{ab}\check{\mathbb{H}}+\\\
&-e_{a}u_{b}\mathfrak{H}+\left(e_{a}e_{b}-\frac{1}{2}N_{ab}\right)\check{\mathcal{H}},\end{split}$
(57)
$\begin{split}E_{ab}=&\mathcal{E}_{ab}+\mathcal{E}_{a}e_{b}+e_{a}\mathbf{E}_{b}-u_{a}\check{\mathcal{E}}_{b}+\epsilon_{ab}\mathbb{E}+\\\
&-u_{a}e_{b}\check{\mathcal{E}}+\left(e_{a}e_{b}-\frac{1}{2}N_{ab}\right)\mathcal{E},\end{split}$
(58)
$\begin{split}\bar{E}_{ab}=&\bar{\mathcal{E}}_{ab}+\bar{\mathcal{E}}_{a}e_{b}+e_{a}\bar{\mathbf{E}}_{b}-u_{a}\check{\bar{\mathcal{E}}}_{b}+\epsilon_{ab}\bar{\mathbb{E}}+\\\
&-u_{a}e_{b}\check{\bar{\mathcal{E}}}+\left(e_{a}e_{b}-\frac{1}{2}N_{ab}\right)\bar{\mathcal{E}},\end{split}$
(59)
$\check{E}_{ab}=\mathfrak{E}_{ab}+2\mathfrak{E}_{[a}e_{b]}-2\check{\mathfrak{E}}_{[a}u_{b]}.$
(60)

Eqs. (55)-(60) involve tensor, vector, and scalar quantities which are better
specified in Appendix B.

Finally, we remind the decomposition of the (symmetric) energy-momentum tensor
$\Psi_{ab}$ with respect to $u^{a}$, $e^{a}$ and $N_{ab}$:

$\begin{split}\Psi_{ab}=&\mu
u_{a}u_{b}+2qe_{(a}u_{b)}+\left(p-\frac{1}{2}\pi\right)N_{ab}+\\\
&+\left(p+\pi\right)e_{a}e_{b}+2\bar{q}_{(a}u_{b)}+2\pi_{(a}e_{b)}+\pi_{ab}\end{split}$
(61)

with

$\begin{gathered}\mu=u^{a}u^{b}\Psi_{ab},\quad q=-e^{a}u^{b}\Psi_{ab},\quad
p=\frac{1}{3}h^{ab}\Psi_{ab},\\\
\pi=\frac{1}{3}\left(2e^{a}e^{b}-N^{ab}\right)\Psi_{ab},\\\
q_{a}=-N_{a}{}^{b}u^{c}\Psi_{bc},\quad\quad\pi_{a}=N_{a}{}^{b}e^{c}\Psi_{bc},\\\
\pi_{ab}=\left(N_{a}{}^{c}N_{b}{}^{d}-\frac{1}{2}N_{ab}N^{cd}\right)\Psi_{cd}.\end{gathered}$
(62)

##  III Static Locally Rotationally Symmetric spacetime

A spacetime is said to be Locally Rotationally Symmetric in a neighborhood
$N(p)$ of a point $p$, _if at each point $p^{*}$ in $N(p)$ there exists a non-
discrete subgroup G of the Lorentz group in the tangent space $T_{p}$ which
leaves invariant $u^{a}$, the curvature tensor and their derivatives up to
third order_ [37]. If $G$ is one-dimensional, there is a single preferred
spatial direction that constitutes a local axis of symmetry. Therefore, the
LRS definition implies that all the spacelike projections of tensors in
spatial directions different from the axis of symmetry must be zero.

In the $1+1+2$ formalism, the natural choice for the local axis of symmetry is
the above-defined spatial vector $e^{a}$. The non-zero kinematic and
thermodynamic quantities are the scalars

$\begin{gathered}\\{\theta,\Sigma,\Omega,\phi,\xi,\mathcal{A}^{(u)},\mathcal{A}^{(e)},\mu,p,q,\pi\\\
\mathcal{H},\bar{\mathcal{H}},\bar{\mathbb{H}},\check{\bar{\mathcal{H}}},\mathcal{E},\mathbb{E},\check{\mathcal{E}},\bar{\mathcal{E}},\bar{\mathbb{E}},\check{\bar{\mathcal{E}}}\\}.\end{gathered}$
(63)

As we said above, in our framework the torsion is zero. On the other hand,
torsion is strictly related to vorticity (see [43] for details) and therefore
we assume the condition

$\Omega=\xi=0.$ (64)

Under this assumption, because of Frobenius’s theorem [44], both timelike and
spacelike congruences are hypersurface orthogonal, i.e., we are working within
an LRS spacetime of class II.111If only the spacelike congruence is
hypersurface orthogonal, the spacetime is called LRS of class I. Instead, when
only the timelike congruence is hypersurface orthogonal the corresponding
spacetime is LRS of class III.

###  III.1 Decomposition of nonmetricity tensor in an LRS spacetime of class
II

In a LRS spacetime, the decomposition of the nonmetricity tensor involves only
scalar quantities, according to222In LRS spacetime, both the vector and tensor
components are zero.

$\begin{split}Q_{abc}=&-Q_{0}u_{a}u_{b}u_{c}+Q_{1}e_{a}e_{b}e_{c}-Q_{2}u_{a}e_{b}e_{c}+\\\
&-2Q_{3}e_{a}u_{(b}e_{c)}+Q_{4}e_{a}u_{b}u_{c}+2Q_{5}u_{a}e_{(b}u_{c)}+\\\
&-\frac{1}{2}Q_{6}u_{a}N_{bc}-Q_{7}u_{(b}N_{c)a}+\frac{1}{2}Q_{8}e_{a}N_{bc}+\\\
&+Q_{9}e_{(b}N_{c)a}-Q_{10}\epsilon_{a(b}u_{c)}+Q_{11}\epsilon_{a(b}e_{c)},\end{split}$
(65)

where

$\begin{split}&Q_{0}=Q_{abc}u^{a}u^{b}u^{c},\qquad
Q_{1}=Q_{abc}e^{a}e^{b}e^{c},\\\ &Q_{2}=Q_{abc}u^{a}e^{b}e^{c},\qquad
Q_{3}=Q_{abc}e^{a}e^{b}u^{c},\\\ &Q_{4}=Q_{abc}e^{a}u^{b}u^{c},\qquad
Q_{5}=Q_{abc}u^{a}u^{b}e^{c},\\\ &Q_{6}=Q_{abc}u^{a}N^{bc},\qquad
Q_{7}=Q_{abc}N^{ab}u^{c},\\\ &Q_{8}=Q_{abc}e^{a}N^{bc},\qquad
Q_{9}=Q_{abc}N^{ab}e^{c},\\\ &Q_{10}=Q_{abc}\epsilon^{ab}u^{c},\qquad
Q_{11}=Q_{abc}\epsilon^{ab}e^{c}.\end{split}$ (66)

As a consequence, the nonmetricity scalar (12) is seen to assume the form

$\begin{split}\mathcal{Q}=&\frac{1}{2}Q_{0}\left(Q_{3}+Q_{7}\right)+\frac{1}{2}Q_{1}\left(-Q_{5}+Q_{9}\right)+\\\
&+\frac{1}{2}Q_{2}\left(Q_{3}+Q_{6}-Q_{7}\right)-\frac{1}{2}Q_{4}\left(Q_{5}-Q_{8}+Q_{9}\right)+\\\
&+\frac{1}{8}Q_{6}\left(-4Q_{3}+Q_{6}\right)-\frac{1}{8}Q_{8}\left(4Q_{5}+Q_{8}\right)+\\\
&-\frac{1}{2}Q_{10}{}^{2}+\frac{1}{2}Q_{11}{}^{2}.\end{split}$ (67)

In view of Eq. (65), Eqs. (37) and (45) are now written as

$\begin{split}\nabla_{a}u_{b}=&\frac{1}{3}h_{ab}\theta+\left(e_{a}e_{b}-\frac{1}{2}N_{ab}\right)\Sigma-
u_{a}e_{b}\mathcal{A}^{(u)}+\\\
&+\epsilon_{ab}\Omega+\frac{1}{2}u_{a}u_{b}Q_{0}-\frac{1}{2}e_{a}u_{b}Q_{4},\end{split}$
(68)

and

$\begin{split}\nabla_{a}e_{b}=&\frac{1}{2}N_{ab}\phi+\left(\frac{1}{3}\theta+\Sigma\right)e_{a}u_{b}-u_{a}u_{b}\mathcal{A}^{(e)}+\\\
&+\epsilon_{ab}\xi+\frac{1}{2}e_{a}e_{b}Q_{1}-\frac{1}{2}u_{a}e_{b}Q_{2}-e_{a}u_{b}Q_{3}.\end{split}$
(69)

For later use, it is useful to distinguish the contributions due to the Levi-
Civita connection from the ones due to nonmetricity in the following scalars,

$\begin{gathered}\theta=\tilde{\theta}-\frac{1}{2}Q_{2}+Q_{3}-\frac{1}{2}Q_{6}+Q_{7},\\\
\Sigma=\tilde{\Sigma}-\frac{1}{3}Q_{2}+\frac{2}{3}Q_{3}+\frac{1}{6}Q_{6}-\frac{1}{3}Q_{7},\\\
\phi=\tilde{\phi}-\frac{1}{2}Q_{8}+Q_{9},\qquad\mathcal{A}^{(u)}=\tilde{\mathcal{A}}+\frac{1}{2}Q_{4},\\\
\mathcal{A}^{(e)}=\tilde{\mathcal{A}}+\frac{1}{2}Q_{4}-Q_{5}.\end{gathered}$
(70)

###  III.2 Static spacetime

The spacetime is said “stationary” if it admits a timelike Killing vector
field $\mathbf{X}$ such that

$\mathcal{L}_{\mathbf{X}}g_{ab}=0,$ (71)

with $\mathcal{L}_{\mathbf{X}}$ denoting Lie derivative. In our geometrical
setting in which torsion is zero, we also require the further condition

$\mathcal{L}_{\mathbf{X}}Q_{abc}=0,$ (72)

i.e., both metric and nonmetricity tensors are invariant under the action of
the $1$-parameter group of local diffeomorphisms generated by $\mathbf{X}$.

If the congruence associated with $\mathbf{X}$ is hypersurface orthogonal, the
spacetime is said “static”. In the presence of nonmetricity, the Killing
equations assume the form

$\begin{split}0=\mathcal{L}_{\mathbf{X}}g_{ab}=&2\nabla_{(a}X_{b)}+2X^{c}L_{abc}=\\\
=&2\tilde{\nabla}_{(a}X_{b)}.\end{split}$ (73)

In the following analysis, the form involving the total covariant derivative
will be used.

We choose the Killing vector expressed as

$\mathbf{X}=C(x)u,$ (74)

where $C(x)$ is a generic smooth function of the coordinates. Contracting Eq.
(73) twice with $u^{a}$, we obtain

$\dot{C}=0,$ (75)

that is, the function $C$ is independent of time. On the other hand, the
contraction of Eq. (73) with $u^{a}$ and $e^{b}$ yields the relation,

$\hat{C}=\tilde{\mathcal{A}}C.$ (76)

Finally, from the contraction with $h_{c}{}^{a}h_{d}{}^{b}$ we find

$\begin{split}C&\left[N_{ab}\left(\frac{2}{3}\theta-\Sigma+\frac{1}{2}Q_{6}-Q_{7}\right)+\right.\\\
&\qquad\qquad\left.+e_{a}e_{b}\left(\frac{2}{3}\theta+2\Sigma+Q_{2}-2Q_{3}\right)\right]=0,\end{split}$
(77)

from which we derive

$\begin{gathered}\theta=-\frac{1}{2}Q_{2}+Q_{3}-\frac{1}{2}Q_{6}+Q_{7},\\\
\Sigma=-\frac{1}{3}Q_{2}+\frac{2}{3}Q_{3}+\frac{1}{6}Q_{6}-\frac{1}{3}Q_{7}.\end{gathered}$
(78)

The condition (72) and Eq. (78) ensure that the time derivative of the scalar
components of nonmetricity tensor is zero. Hence, because of the LRS
symmetries and the stationary conditions, the covariant derivatives of the
scalars (63) are zero except those along the radial direction $e^{a}$.

Given condition (64) we have that $\mathbf{X}$ is hypersurface orthogonal
since it is proportional to the $4$-velocity. Thus, our spacetime is static.

By a direct comparison of Eq. (70) with Eq. (78), we obtain the identities

$\tilde{\theta}=0,\qquad\text{and}\qquad\tilde{\Sigma}=0.$ (79)

This is not a surprising result, since $\tilde{\theta}$ and $\tilde{\Sigma}$
are respectively the expansion and shear we would have in GR where both
quantities are known to be null in a stationary LRS spacetime of class II.
Moreover, it is a direct consequence of the Killing equations (73) when using
the form with the Levi-Civita covariant derivative.

Equation (78) means that in the presence of a non-metric connection,
expansion, and shear can manifest exclusively due to the nonmetricity tensor.

##  IV $f(\mathcal{Q})$ gravity

$f(\mathcal{Q})$ gravity is a generalization of Symmetric Teleparallel Gravity
in which both curvature and torsion are zero. The action is given by

$\begin{split}A=&\int\hbox{d}^{4}x\left[-\frac{1}{2}\sqrt{-g}f(\mathcal{Q})+\lambda_{a}{}^{bcd}R^{a}{}_{bcd}+\right.\\\
&\left.+\lambda_{a}{}^{cd}T_{cd}{}^{a}+\sqrt{-g}\mathcal{L}_{m}\right],\end{split}$
(80)

where $g$ is the determinant of metric, $f(\mathcal{Q})$ is a generic function
of the nonmetricity scalar, $\mathcal{L}_{m}$ is the matter Lagrangian
density, $\lambda_{a}{}^{bij}$ and $\lambda_{a}{}^{ij}$ are Lagrange
multipliers introduced to impose the vanishing of curvature and torsion.
Variations with respect to Lagrange multipliers, metric, and connection yield
the field equations,

$R^{a}{}_{bcd}=0,\qquad T_{ab}{}^{c}=0,$ (81)
$\begin{split}&\frac{2}{\sqrt{-g}}\nabla_{c}\left(\sqrt{-g}f^{\prime}P^{c}{}_{ab}\right)+\frac{1}{2}g_{ab}f+\\\
&\qquad\qquad+f^{\prime}\left(P_{acd}Q_{b}{}^{cd}-2Q^{cd}{}_{a}P_{cdb}\right)=\Psi_{ij},\end{split}$
(82)

and

$\nabla_{d}\lambda_{c}{}^{bad}+\lambda_{c}{}^{ab}-\sqrt{-g}f^{\prime}P^{ab}{}_{c}=\Delta^{ab}{}_{c},$
(83)

with

$P^{c}{}_{ab}=-\frac{1}{4}Q^{c}{}_{ab}+\frac{1}{2}Q_{(ab)}{}^{c}+\frac{1}{4}q^{c}g_{ab}-\frac{1}{4}Q^{c}g_{ab}-\frac{1}{4}q_{(a}\delta^{c}_{b)},$
(84)

and

$\begin{gathered}\Psi_{ab}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\delta
g^{ab}},\\\
\Phi^{ab}{}_{c}=-\frac{1}{2}\frac{\delta\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\delta{\Gamma_{ab}{}^{c}}}.\end{gathered}$
(85)

In the above equations, the prime denotes the derivative with respect to the
argument of the function. Since we will consider perfect fluids, in our case
$\Phi^{ab}{}_{c}$ is identically zero, and the energy-momentum conservation
law is equal to the standard one,

$\tilde{\nabla}_{a}\Psi^{a}{}_{b}=0.$ (86)

The metric field equation Eq. (82) can be recast in a more suitable form for
the $1+1+2$ formalism,

$\begin{split}\tilde{R}_{ab}=&\frac{1}{f^{\prime}}\left(\Psi_{ab}-\frac{1}{2}g_{ab}\Psi\right)+\frac{1}{2}g_{ab}\left(\frac{f}{f^{\prime}}-\mathcal{Q}\right)+\\\
&-2\frac{f^{\prime\prime}}{f^{\prime}}\left(P^{c}{}_{ab}-\frac{1}{2}g_{ab}P^{cd}{}_{d}\right)\partial_{c}\mathcal{Q},\end{split}$
(87)

with $\Psi=g^{ab}\Psi_{ab}$. By replacing $f(\mathcal{Q})=\mathcal{Q}$ into
Eq. (87), we recover the field equations of GR,

$\tilde{R}_{ab}=\Psi_{ab}-\frac{1}{2}g_{ab}\Psi.$ (88)

###  IV.1 $1+1+2$ field equations in LRS of class II

To make explicit the field equations in the $1+1+2$ formalism, we need the
Ricci identity for the Levi-Civita Riemann tensor, written for both $u_{a}$
and $e_{a}$,

$\begin{split}\left[\tilde{\nabla}_{c}\tilde{\nabla}_{d}-\tilde{\nabla}_{d}\tilde{\nabla}_{c}\right]u_{b}=&-\tilde{R}^{a}{}_{bcd}u_{a},\\\
\left[\tilde{\nabla}_{c}\tilde{\nabla}_{d}-\tilde{\nabla}_{d}\tilde{\nabla}_{c}\right]e_{b}=&-\tilde{R}^{a}{}_{bcd}e_{a}.\end{split}$
(89)

From Eqs. (87) and (89), we obtain the following relations:

  * •

$\tilde{R}_{ad}u^{a}u^{d}=g^{bc}u^{d}\left[\tilde{\nabla}_{c}\tilde{\nabla}_{d}-\tilde{\nabla}_{d}\tilde{\nabla}_{c}\right]u_{b}$,

$\begin{split}&\hat{\tilde{\mathcal{A}}}+\tilde{\mathcal{A}}^{2}+\tilde{\mathcal{A}}\tilde{\phi}-\frac{1}{2}\frac{1}{f^{\prime}}\left(\mu+3p\right)+\frac{1}{2}\frac{f}{f^{\prime}}+\\\
&-\frac{1}{2}\mathcal{Q}-\frac{1}{2}\frac{f^{\prime\prime}}{f^{\prime}}\left(Q_{4}-2Q_{5}\right)\hat{\mathcal{Q}}=0;\end{split}$
(90)

  * •

$\tilde{R}_{ad}e^{a}e^{d}=g^{bc}e^{d}\left[\tilde{\nabla}_{c}\tilde{\nabla}_{d}-\tilde{\nabla}_{d}\tilde{\nabla}_{c}\right]e_{b}$,

$\begin{split}&\hat{\tilde{\phi}}+\frac{1}{2}\tilde{\phi}^{2}+\hat{\tilde{A}}+\tilde{A}^{2}+\frac{1}{2}\frac{1}{f^{\prime}}\left(\mu-p+2\pi\right)+\frac{1}{2}\frac{f}{f^{\prime}}+\\\
&-\frac{1}{2}\mathcal{Q}-\frac{1}{2}\frac{f^{\prime\prime}}{f^{\prime}}\left(Q_{4}-Q_{8}\right)\hat{\mathcal{Q}}=0;\end{split}$
(91)

  * •

$\tilde{R}_{ad}u^{a}e^{d}=g^{bc}e^{d}\left[\tilde{\nabla}_{c}\tilde{\nabla}_{d}-\tilde{\nabla}_{d}\tilde{\nabla}_{c}\right]u_{b}$,

$f^{\prime\prime}\left(Q_{0}+Q_{2}-Q_{6}\right)\hat{\mathcal{Q}}=q;$ (92)

  * •

$-\tilde{R}^{a}{}_{bcd}e_{a}u^{b}u^{c}e^{d}=u^{b}u^{c}e^{d}\left[\tilde{\nabla}_{c}\tilde{\nabla}_{d}-\tilde{\nabla}_{d}\tilde{\nabla}_{c}\right]e_{b}$,

$\begin{split}&\hat{\tilde{A}}+\tilde{A}^{2}-\tilde{\mathcal{E}}-\frac{1}{6}\frac{1}{f^{\prime}}\left(\mu+3p-3\pi\right)+\frac{1}{6}\frac{f}{f^{\prime}}-\frac{1}{6}\mathcal{Q}+\\\
&-\frac{1}{12}\frac{f^{\prime\prime}}{f^{\prime}}\left(4Q_{4}-4Q_{5}-Q_{8}-2Q_{9}\right)\hat{Q}=0;\end{split}$
(93)

  * •

$-\tilde{R}^{a}{}_{bcd}e_{a}u^{b}\epsilon^{cd}=\epsilon^{cd}u^{b}\left[\tilde{\nabla}_{c}\tilde{\nabla}_{d}-\tilde{\nabla}_{d}\tilde{\nabla}_{c}\right]e_{b}$,

$\tilde{\mathcal{H}}=0.$ (94)

In vacuum, for Eq. (92) to be satisfied, there are three possible solutions:
$f^{\prime\prime}=0$, and $\hat{\mathcal{Q}}=0$, from which only
Schwarzschild-de Sitter spacetime can be obtained [34], or
$Q_{0}+Q_{2}-Q_{6}=0$, which instead imposes a restriction on the connection.
Such a condition results in being automatically satisfied under the
assumptions we will use to solve the field equations.

###  IV.2 Weyl tensor: magnetic and electric parts

Keeping in mind that the Weyl tensor is zero because of Eq. (81), we can make
explicit the contributions due to Levi-Civita connection and nonmetricity in
the magnetic and electric parts of the Weyl tensor in order to derive
constraints on nonmetricity. Equations
$\check{\mathcal{E}}=\check{\bar{\mathcal{E}}}=0$ are automatically satisfied,
so they do not provide any constraint. On the other hand, from the scalar part
of (55), we have

$\begin{split}\mathcal{H}=0=&\tilde{\mathcal{H}}-\frac{1}{8}\tilde{\phi}Q_{10}+\frac{1}{4}\tilde{\mathcal{A}}Q_{10}+\frac{1}{4}\hat{Q}_{10}+\\\
&+\frac{1}{16}\left(Q_{8}-2Q_{9}\right)Q_{10}-\frac{1}{8}\left(Q_{6}-2Q_{7}\right)Q_{11},\end{split}$
(95)

from which, considering Eq. (94), we derive

$Q_{10}=Q_{11}=0.$ (96)

The condition (96) ensures
$\bar{\mathcal{H}}=\mathbb{H}=\check{\mathcal{H}}=\check{\mathcal{H}}=\check{\mathbb{H}}=\mathfrak{H}=\mathbb{E}=\bar{\mathbb{E}}=0$
as well. Now, we just have to write the scalars $\mathcal{E}$ and
$\bar{\mathcal{E}}$, which are derived by Eqs. (194) and (198):

$\begin{split}\mathcal{E}=0=&\tilde{\mathcal{E}}-\frac{1}{24}\tilde{\mathcal{A}}\left(6Q_{1}+2Q_{4}+10Q_{5}-2Q_{8}-Q_{9}\right)+\frac{1}{48}\tilde{\phi}\left(6Q_{1}-4Q_{4}+10Q_{5}-5Q_{8}-Q_{9}\right)+\\\
&-\frac{1}{48}Q_{0}\left(6Q_{2}-2Q_{3}-4Q_{6}+Q_{7}\right)+\frac{1}{48}Q_{1}\left(-6Q_{4}+10Q_{5}-3Q_{8}+5Q_{9}\right)+\\\
&-\frac{1}{48}Q_{2}\left(-2Q_{3}-2Q_{6}-5Q_{7}+6Q_{2}\right)+\frac{1}{48}Q_{3}Q_{6}+\frac{1}{48}Q_{4}\left(10Q_{5}+2Q_{8}+Q_{9}-6Q_{4}\right)+\\\
&-\frac{5}{48}Q_{5}Q_{8}+\frac{1}{32}Q_{6}Q_{7}+\frac{Q_{6}^{2}}{96}-\frac{3}{32}Q_{8}Q_{9}+\frac{5Q_{8}^{2}}{96}+\frac{1}{6}\hat{Q}_{4}-\frac{5}{12}\hat{Q}_{5}+\frac{1}{12}\hat{Q}_{8}-\frac{5}{24}\hat{Q}_{9},\end{split}$
(97)

and

$\begin{split}\bar{\mathcal{E}}=0=&-\tilde{\mathcal{E}}-\frac{1}{24}\tilde{\mathcal{A}}\left(6Q_{1}+10Q_{4}+2Q_{5}+2Q_{8}-5Q_{9}\right)+\frac{1}{48}\tilde{\phi}\left(6Q_{1}+4Q_{4}+2Q_{5}-Q_{8}-5Q_{9}\right)+\\\
&-\frac{1}{48}Q_{0}\left(6Q_{2}-10Q_{3}-4Q_{6}+5Q_{7}\right)+\frac{1}{48}Q_{1}\left(-6Q_{4}+2Q_{5}-3Q_{8}+Q_{9}\right)+\\\
&-\frac{1}{48}Q_{2}\left(-10Q_{3}+2Q_{2}Q_{6}+Q_{7}+6Q_{2}\right)+\frac{5}{48}Q_{3}Q_{6}+\frac{1}{48}Q_{4}\left(5Q_{5}-2Q_{8}+5Q_{9}-6Q_{4}\right)+\\\
&-\frac{1}{48}Q_{5}Q_{8}-\frac{3}{32}Q_{6}Q_{7}+\frac{5Q_{6}^{2}}{96}+\frac{1}{32}Q_{8}Q_{9}+\frac{Q_{8}^{2}}{96}-\frac{1}{12}\hat{Q}_{4}-\frac{1}{12}\hat{Q}_{5}-\frac{1}{12}\hat{Q}_{8}-\frac{1}{24}\hat{Q}_{9}.\end{split}$
(98)

###  IV.3 Riemann tensor contractions

Besides the Weyl tensor, we can deduce constraints from the Riemann tensor
itself by considering all its possible contractions with $u^{a}$, $e^{a}$,
$N^{ab}$, $\epsilon^{ab}$ and imposing Eq. (81). Making use of the Ricci
identities for the full connection,

$\begin{split}\left[\nabla_{c}\nabla_{d}-\nabla_{d}\nabla_{c}\right]u_{b}=&-R^{a}{}_{bcd}u_{a},\\\
\left[\nabla_{c}\nabla_{d}-\nabla_{d}\nabla_{c}\right]e_{b}=&-R^{a}{}_{bcd}e_{a},\end{split}$
(99)

we deduce then the following relations:

  * •

$R^{a}{}_{bcd}u_{a}u_{b}u_{c}e^{d}=0$,

$\hat{Q}_{0}+3\Sigma\left(\mathcal{A}^{(e)}-\mathcal{A}^{(u)}\right)-\frac{1}{3}\theta\left(Q_{4}+2Q_{5}\right)-\Sigma\left(Q_{4}-Q_{5}\right)+\mathcal{A}^{(u)}\left(Q_{0}+2Q_{3}\right)-\frac{1}{2}Q_{0}Q_{4}-\frac{1}{2}Q_{2}Q_{4}+Q_{3}Q_{4}=0;$
(100)

  * •

$R^{a}{}_{bcd}e_{a}e_{b}u_{c}e^{d}=0$,

$\begin{split}\hat{Q}_{2}+2\left(\Sigma+\frac{1}{3}\theta\right)\left(\mathcal{A}^{(e)}-\mathcal{A}^{(u)}-\frac{1}{2}Q_{1}\right)+\mathcal{A}^{(u)}\left(Q_{2}+2Q_{3}\right)-\frac{1}{2}Q_{1}Q_{2}+Q_{1}Q_{3}-\frac{1}{2}Q_{2}Q_{4}=0;\end{split}$
(101)

  * •

$R^{a}{}_{bcd}N_{a}{}^{b}u^{c}e^{d}=0$,

$\begin{split}&\hat{Q}_{6}+\tilde{\mathcal{A}}Q_{6}=0;\end{split}$ (102)

  * •

$R^{a}{}_{bcd}u_{a}e^{b}u^{c}e^{d}=0$,

$\hat{\mathcal{A}}^{(u)}+\mathcal{A}^{(u)}{}^{2}+\frac{1}{2}\mathcal{A}^{(u)}Q_{1}-\left(\Sigma+\frac{1}{3}\theta\right)^{2}-\frac{1}{2}\left(\Sigma+\frac{1}{3}\theta\right)\left(Q_{0}+2Q_{2}-2Q_{3}\right)=0;$
(103)

  * •

$R^{a}{}_{bcd}e_{a}u^{b}u^{c}e^{d}=0$,

$\begin{split}&\hat{\mathcal{A}}^{(e)}+\mathcal{A}^{(e)}\mathcal{A}^{(u)}-\frac{1}{2}\mathcal{A}^{(e)}\left(Q_{1}+2Q_{4}\right)-\left(\Sigma+\frac{1}{3}\theta\right)^{2}+\frac{1}{2}\left(\Sigma+\frac{1}{3}\theta\right)\left(Q_{0}+4Q_{3}\right)+\\\
&+\frac{1}{2}\phi\left(\mathcal{A}^{(e)}-\mathcal{A}^{(u)}+Q_{5}\right)-Q_{3}^{2}-\frac{1}{2}Q_{0}Q_{3}=0;\end{split}$
(104)

  * •

$R^{a}{}_{bcd}N_{a}{}^{c}u^{b}u^{d}=0$,

$\begin{split}\left(\Sigma-\frac{2}{3}\theta\right)^{2}+\left(\Sigma-\frac{2}{3}\theta\right)\left(Q_{0}+2Q_{7}\right)-2\phi\mathcal{A}^{(u)}+2\phi
Q_{5}+2\mathcal{A}^{(e)}Q_{9}+Q_{0}Q_{7}+Q_{7}^{2}=0;\end{split}$ (105)

  * •

$R^{a}{}_{bcd}N_{a}{}^{c}e^{b}e^{d}=0$,

$\begin{split}\hat{\phi}-\hat{Q}_{9}+\frac{\phi^{2}}{2}+\frac{1}{2}Q_{9}^{2}+\Sigma^{2}-\frac{2}{9}\theta^{2}-\frac{1}{3}\theta\Sigma+\frac{1}{2}\phi\left(Q_{1}-2Q_{9}\right)+\frac{1}{3}\theta
Q_{7}+\Sigma Q_{7}-\frac{1}{2}Q_{1}Q_{9}=0;\end{split}$ (106)

  * •

$R^{a}{}_{bcd}N_{a}{}^{c}u^{b}e^{d}=0$,

$\begin{split}\hat{\Sigma}-\frac{2}{3}\hat{\theta}+\hat{Q}_{7}+\frac{3}{2}\Sigma\phi-\frac{1}{2}\Sigma\left(Q_{4}+3Q_{9}\right)-\left(\phi-
Q_{9}\right)\left(Q_{3}-\frac{1}{2}Q_{7}\right)+\frac{1}{3}\theta
Q_{4}-\frac{1}{2}Q_{4}Q_{7}=0;\end{split}$ (107)

  * •

$R^{a}{}_{bcd}N_{a}{}^{c}e^{b}u^{d}=0$,

$\begin{split}\frac{1}{2}\Sigma\phi-\frac{1}{3}\theta\phi-\left(\Sigma-\frac{2}{3}\theta\right)\left(Q_{5}+\frac{1}{2}Q_{9}+\mathcal{A}^{(e)}\right)-\frac{1}{2}\left(\phi-
Q_{9}\right)\left(Q_{2}-Q_{7}\right)-\mathcal{A}^{(u)}Q_{7}=0;\end{split}$
(108)

  * •

$R^{a}{}_{bcd}N^{bc}u_{a}u^{d}=0$,

$\begin{split}\left(\Sigma-\frac{2}{3}\theta\right)^{2}-2\phi\mathcal{A}^{(u)}-\left(\Sigma-\frac{2}{3}\theta\right)\left(Q_{0}+Q_{6}-Q_{7}\right)=0;\end{split}$
(109)

  * •

$R^{a}{}_{bcd}N^{bc}e_{a}e^{d}=0$,

$\begin{split}\hat{\phi}+\frac{\phi^{2}}{2}+\Sigma^{2}-\frac{2}{9}\theta^{2}-\frac{1}{3}\theta\Sigma-\Sigma
Q_{3}+\frac{2}{3}\theta
Q_{3}-\frac{1}{2}\phi\left(Q_{1}-Q_{8}+Q_{9}\right)=0;\end{split}$ (110)

  * •

$R^{a}{}_{bcd}N^{bc}u_{a}e^{d}=0$,

$\begin{split}\hat{\Sigma}-\frac{2}{3}\hat{\theta}+\frac{3}{2}\Sigma\phi+\frac{1}{2}\left(\Sigma-\frac{2}{3}\theta\right)\left(Q_{4}+Q_{8}-Q_{9}\right)=0;\end{split}$
(111)

  * •

$R^{a}{}_{bcd}N^{bc}e_{a}u^{d}=0$,

$\begin{split}\mathcal{A}^{(e)}\left(4\theta-6\Sigma\right)+\phi\left(-2\theta+3Q_{2}-3Q_{6}+3Q_{7}+3\Sigma\right)=0;\end{split}$
(112)

All the remaining contractions, which are not explicitly expressed above, are
identically null.

###  IV.4 The final system of equations

Before proceeding to write the final system of equations to be solved, some
preliminary considerations are in order.

First of all, we are looking for a vacuum solution, so $\mu=p=q=\pi=0$.
Moreover, from the static nature of spacetime, we have already deduced the
relations $\tilde{\theta}=\tilde{\Sigma}=0$. In addition to this, we decide to
impose the conditions $\theta=\Sigma=0$ too. Both the total rate of expansion
$\theta$ and shear $\Sigma$ are set to zero to simplify the mathematics and
because we want a distortion-free spacetime as well. Hence, from Eq. (78) we
get the identities

$Q_{2}=2Q_{3},\quad\text{and}\quad Q_{6}=2Q_{7}.$ (113)

Furthermore, we require that the curves of the timelike congruence are
autoparallel, i.e., $u^{c}\nabla_{c}u^{a}=0$, which implies

$Q_{0}=0,\quad\mathcal{A}^{(e)}=0,\quad\text{and}\quad\mathcal{A}^{(u)}=Q_{5}.$
(114)

Equations (113), (114), together with Eqs. (100)-(112), give rise to the
identities

$Q_{2}=Q_{3}=Q_{6}=Q_{7}=0,$ (115)

which are in agreement with $Q_{0}+Q_{2}-Q_{6}=0$ derived from the off
diagonal equation (92) in vacuum.

By collecting all the obtained results, we end up with the final system of
field equations:

$\hat{\tilde{\mathcal{A}}}+\tilde{\mathcal{A}}^{2}+\tilde{\mathcal{A}}\tilde{\phi}+\frac{1}{2}\frac{f}{f^{\prime}}-\frac{1}{2}\mathcal{Q}-\frac{1}{2}\frac{f^{\prime\prime}}{f^{\prime}}\left(Q_{4}-2Q_{5}\right)\hat{\mathcal{Q}}=0,$
(116)

$\begin{split}\tilde{\mathcal{E}}+\tilde{\mathcal{A}}\tilde{\phi}&+\frac{1}{3}\frac{f}{f^{\prime}}-\frac{1}{3}\mathcal{Q}-\frac{1}{6}\frac{f^{\prime\prime}}{f^{\prime}}Q_{4}\hat{\mathcal{Q}}+\frac{2}{3}\frac{f^{\prime\prime}}{f^{\prime}}Q_{5}\hat{\mathcal{Q}}+\\\
&\qquad\qquad-\frac{1}{12}\frac{f^{\prime\prime}}{f^{\prime}}Q_{8}\hat{\mathcal{Q}}-\frac{1}{6}\frac{f^{\prime\prime}}{f^{\prime}}Q_{9}\hat{\mathcal{Q}}=0,\end{split}$
(117)

$\hat{\tilde{\phi}}+\frac{1}{2}\tilde{\phi}^{2}-\tilde{\mathcal{A}}\tilde{\phi}-\frac{f^{\prime\prime}}{f^{\prime}}Q_{5}\hat{\mathcal{Q}}+\frac{1}{2}\frac{f^{\prime\prime}}{f^{\prime}}Q_{8}\hat{\mathcal{Q}}=0,$
(118)

$\hat{Q}_{5}+Q_{5}{}^{2}+\frac{1}{2}Q_{1}Q_{5}=0,$ (119)

$\hat{\phi}-\hat{Q}_{9}+\frac{\phi^{2}}{2}+\frac{Q_{9}^{2}}{2}-\phi\left(Q_{9}-\frac{Q_{1}}{2}\right)-\frac{1}{2}Q_{1}Q_{9}=0,$
(120)

$Q_{5}\phi=0,$ (121)

$\hat{\phi}+\frac{1}{2}\phi^{2}-\frac{1}{2}\left(Q_{1}-Q_{8}+Q_{9}\right)\phi=0,$
(122)
$\begin{split}0=&\tilde{\mathcal{E}}-\frac{1}{24}\tilde{\mathcal{A}}\left(6Q_{1}+2Q_{4}+10Q_{5}-2Q_{8}-Q_{9}\right)+\\\
&+\frac{1}{48}\tilde{\phi}\left(6Q_{1}-4Q_{4}+10Q_{5}-5Q_{8}-Q_{9}\right)+\\\
&-\frac{1}{8}Q_{1}Q_{4}+\frac{5}{24}Q_{1}Q_{5}-\frac{1}{16}Q_{1}Q_{8}+\frac{5}{48}Q_{1}Q_{9}+\\\
&+\frac{5}{24}Q_{4}Q_{5}+\frac{1}{24}Q_{4}Q_{8}+\frac{1}{48}Q_{4}Q_{9}-\frac{Q_{4}^{2}}{8}+\\\
&-\frac{5}{48}Q_{5}Q_{8}-\frac{3}{32}Q_{8}Q_{9}+\frac{5Q_{8}^{2}}{96}+\\\
&+\frac{1}{6}\hat{Q}_{4}-\frac{5}{12}\hat{Q}_{5}+\frac{1}{12}\hat{Q}_{8}-\frac{5}{24}\hat{Q}_{9},\end{split}$
(123)
$\begin{split}\mathcal{Q}=&\frac{1}{2}Q_{1}\left(-Q_{5}+Q_{9}\right)-\frac{1}{2}Q_{4}\left(Q_{5}-Q_{8}+Q_{9}\right)+\\\
&-\frac{1}{8}Q_{8}\left(4Q_{5}+Q_{8}\right).\end{split}$ (124)

Eq. (98) is not included in the final system of equations since one can prove
that the combination (97)+(98) can always be obtained by suitable combinations
of Eqs. (119)-(122).

##  V Spherically symmetric solutions

In this section, we specialize the system of covariant equations (116)-(124)
to a spherically symmetric scenario. For this purpose, we introduce the
following expression of a static spherically symmetric metric,

$\text{d}s^{2}=-\mathbb{A}(r)\text{d}t^{2}+\mathbb{B}(r)\text{d}r^{2}+r^{2}\left(\text{d}\Theta^{2}+\sin^{2}{\Theta}\text{d}\varphi^{2}\right),$
(125)

where $\mathbb{A}(r)$ and $\mathbb{B}(r)$ are generic positive functions of
the radial coordinate $r$. In this coordinate system, the $4$-velocity $u^{a}$
and the spatial $4$-vector $e^{a}$ are expressed as

$u^{a}\equiv\bigg{\\{}\frac{1}{\sqrt{\mathbb{A}}},\>0,\>0,\>0\bigg{\\}},\quad\text{and}\quad
e^{a}\equiv\bigg{\\{}0,\>\frac{1}{\sqrt{\mathbb{B}}},\>0,\>0\bigg{\\}}.$ (126)

Using Eq. (126) and the definitions of $\tilde{\phi}$ and
$\tilde{\mathcal{A}}$, we obtain the relations,

$\frac{\mathbb{A}^{\prime}(r)}{\mathbb{A}(r)}=\frac{4\tilde{\mathcal{A}}(r)}{r\tilde{\phi}(r)},$
(127) $\mathbb{B}(r)=\frac{4}{r^{2}}\frac{1}{\tilde{\phi}(r)^{2}},$ (128)
$e^{a}\nabla_{a}\psi=e^{a}\tilde{\nabla}_{a}\psi=e^{a}\partial_{a}\psi=\frac{1}{2}\tilde{\phi}\>r\>\partial_{r}\psi,$
(129)

where $\psi$ is an arbitrary scalar function. From Eq. (121) two conditions
arise:

$\phi=0\qquad\text{or}\qquad Q_{5}=0.$ (130)

To test the viability of both solutions (130), we consider the function
$f(\mathcal{Q})=\mathcal{Q}$ and verify that, in this case, the equations
yield the Schwarzschild spacetime as a solution. The condition $\phi=0$ leads
to an unacceptable result for $f(\mathcal{Q})=\mathcal{Q}$: all kinetic
scalars are null. For this reason, we discard this branch of solutions, and we
will narrow down our investigation to the branch $Q_{5}=0$. In such a
circumstance, we easily obtain the following relations for $\phi$ and
$\mathcal{A}$ when $f(\mathcal{Q})=\mathcal{Q}$,

$\tilde{\phi}=\frac{2}{r}\sqrt{1-\frac{r_{s}}{r}},\qquad\tilde{\mathcal{A}}=-\frac{1}{2}Q_{4}=\frac{r_{s}}{2r\sqrt{r^{2}-rr_{s}}},$
(131)

which substituted into Eqs. (127) and (128) give the Schwarzschild metric as
solution,

$\mathbb{A}=\mathbb{B}^{-1}=1-\frac{r_{s}}{r},$ (132)

being $r_{s}$ the Schwarzschild radius, as expected.

###  V.1 Schwarzschild-de Sitter solutions

In order for Eqs. (127) and (128) admit solutions of Schwarzschild-de Sitter
type, from the well-known results of GR we must have that

$\hat{\tilde{\phi}}+\frac{1}{2}\tilde{\phi}^{2}-\tilde{\mathcal{A}}\tilde{\phi}=0.$
(133)

Consequently, Eq. (118) and the condition $Q_{5}=0$ imply

$f^{\prime\prime}Q_{8}\hat{\mathcal{Q}}=0.$ (134)

As we already know, Eq. (134) tells us that to have a Schwarzschild-de Sitter
metric as a solution either the function $f(\mathcal{Q})$ must be linear or
$\mathcal{Q}$ must be constant. But there is also a third possibility given by
the requirement $Q_{8}=0$.

The condition $Q_{8}=0$ perfectly matches the gauge choice made in other works
(e.g. [30, 31]) in which static spherically symmetric spacetimes are studied
and the full connection is assumed to coincide with the Levi-Civita one of a
Minkowski spacetime in spherical coordinates.

Now, we can solve the final system of equations with the constraint $Q_{8}=0$.
From Eqs. (118), (120) and (122) we get the relation

$Q_{1}=Q_{4},$ (135)

which substituted into Eq. (124) gives $\mathcal{Q}=0$. The vanishing of
nonmetricity scalar implies

$\begin{gathered}f(\mathcal{Q})|_{\mathcal{Q}=0}=f_{0},\qquad
f^{\prime}(\mathcal{Q})|_{\mathcal{Q}=0}=f^{\prime}_{0},\\\
f^{\prime\prime}(\mathcal{Q})|_{\mathcal{Q}=0}=f^{\prime\prime}_{0},\end{gathered}$
(136)

i.e., the function $f$ and its derivatives are constant on-shell. The
remaining equations admit the solution

$\begin{gathered}\tilde{\phi}=\frac{2}{r}\sqrt{1-\frac{r_{s}}{r}-\frac{1}{6}\frac{f_{0}}{f^{\prime}_{0}}r^{2}},\\\
\tilde{\mathcal{A}}=-\frac{1}{2}Q_{4}=\frac{3f^{\prime}_{0}r_{s}-f_{0}r^{3}}{6r^{2}\sqrt{{f^{\prime}_{0}}^{2}\left(1-\frac{r_{s}}{r}-\frac{1}{6}\frac{f_{0}}{f^{\prime}_{0}}r^{2}\right)}},\end{gathered}$
(137)
$Q_{9}=\frac{6f^{\prime}_{0}r_{s}+f_{0}r^{3}}{3r^{2}\sqrt{{f^{\prime}_{0}}^{2}\left(1-\frac{r_{s}}{r}-\frac{1}{6}\frac{f_{0}}{f^{\prime}_{0}}r^{2}\right)}},\quad\tilde{\mathcal{E}}=-\frac{r_{s}}{r^{3}},$
(138)
$\mathbb{A}=\mathbb{B}^{-1}=1-\frac{r_{s}}{r}-\frac{1}{6}\frac{f_{0}r^{2}}{f^{\prime}_{0}}.$
(139)

which represents the Schwarzschild-de Sitter spacetime.

###  V.2 $\mathcal{Q}$-Gravastars

Among the exotic objects that have been proposed as alternatives for black
holes, there are the so-called gravitational vacuum condensate stars or
Gravastars [45, 46]. A gravastar is essentially a compact object made of a
dark energy condensate. As the nonmetricity terms in equations (87) can be
considered as an effective fluid which can have negative pressure, one might
ask if $f(\mathcal{Q})$ gravity can admit gravastar solutions without invoking
explicitly the presence of a cosmological constant. We dub these solutions
“Q-gravastars” and give their simplest realization in the following.

First, let us show that the $\mathcal{Q}$ dependent terms in (87) can generate
an effective cosmological constant. Rewriting these equations in the form

$\tilde{R}_{ab}-\frac{1}{2}g_{ab}\tilde{R}=\frac{1}{f^{\prime}}T_{ab}-\frac{1}{2}g_{ab}\left(\frac{f}{f^{\prime}}-\mathcal{Q}\right)-2\frac{f^{\prime\prime}}{f^{\prime}}P^{c}{}_{ab}\partial_{c}\mathcal{Q},$
(140)

we have that, in the case $\mathcal{Q}=\mathcal{Q}_{*}=const.$, the effective
energy-momentum tensor,

$T^{eff}_{ab}=-\frac{1}{2}g_{ab}\left(\frac{f(\mathcal{Q}_{*})}{f^{\prime}(\mathcal{Q}_{*})}-\mathcal{Q}_{*}\right)=-g_{ab}\Lambda_{*},$
(141)

can be thought of as a fluid characterized by a negative pressure,

$\frac{1}{3}T^{eff}_{ab}h^{ab}=p_{eff}=-\rho_{eff}=T^{eff}_{ab}u^{a}u^{b},$
(142)

if

$\Lambda_{*}=\frac{1}{2}\left(\frac{f(\mathcal{Q}_{*})}{f^{\prime}(\mathcal{Q}_{*})}-\mathcal{Q}_{*}\right)>0,$
(143)

which corresponds to a function $f$, which grows slower than linear. It is now
clear that, with our assumptions, considering a Schwarzschild radius equals
zero, the solution of the field equations will be just the de Sitter solution,

$\mathbb{A}=\mathbb{B}^{-1}=1-\frac{1}{3}\Lambda_{*}r^{2}.$ (144)

A critical aspect of the gravastar models is the compatibility with a vacuum
(Schwarzschild) exterior. Such compatibility is obtained by the well-known
Israel-Darmois junction conditions [47] (see also [48] and [49]), which we
briefly summarize for our specific case in the 1+1+2 language (see [50] for
some additional details on the strategy and [51] for a more general treatment
of 1+1+2 junction conditions).

Let $\mathcal{S}$ be a $3$-dimensional hypersurface with a spacelike normal
separating the spacetime in two regions denoted by $\mathcal{M}^{+}$ and
$\mathcal{M}^{-}$, in which Schwarzschild and de Sitter solutions hold
respectively. Then, in the 1+1+2 framework, we can choose the vector $e^{a}$
as equal to the normal vector, and the induced metric on the boundary will be
given by

$q_{ab}=N_{ab}-u_{a}u_{b}=g_{ab}-e_{a}e_{b}.$ (145)

Assuming smooth $u^{a}$, $e^{a}$ and $N_{ab}$ and indicating with
$\tilde{K}_{ab}$ the Levi-Civita extrinsic curvature of $\mathcal{S}$, the
standard Israel-Darmois junction conditions can be formulated as

$\displaystyle\left[q_{ab}\right]$ $\displaystyle=0,$ (146)
$\displaystyle\left[\tilde{K}_{ab}\right]$ $\displaystyle=0,$ (147)

where

$\begin{split}\tilde{K}_{ab}=&\left(N_{a}{}^{c}-u_{a}u^{c}\right)\left(N_{b}{}^{d}-u_{b}u^{d}\right)\tilde{\nabla}_{c}e_{d}=\\\
=&\frac{1}{2}N_{ab}\tilde{\phi}-u_{a}u_{b}\tilde{\mathcal{A}},\end{split}$
(148)

and the brackets indicate the jump of a generic geometric quantity $W$ across
the hypersurface,

$\left[W\right]=W^{+}-W^{-}.$ (149)

Following [52], we recognize, however, that the above conditions are
insufficient to guarantee a non-singular geometry. The additional conditions
that are required in the case of $f(\mathcal{Q})$ gravity are

$\displaystyle\left[L_{ab}{}^{c}\right]$ $\displaystyle=0,$ (150)
$\displaystyle\left[\mathcal{Q}\right]$ $\displaystyle=0,$ (151)

which are derived directly from the field equations. All the above equations
must be evaluated in the same coordinate system, which must be well defined in
$\mathcal{M}^{+}$ and $\mathcal{M}^{-}$.

If the conditions on the extrinsic curvature and the disformation tensor are
not satisfied, then the geometry can be regular only if we assume the presence
of a thin shell separating the two regions $\mathcal{M}^{+}$ and
$\mathcal{M}^{-}$. Such a thin shell can be proven to possess an energy-
momentum tensor of the form

$\begin{split}\Psi^{S}_{ab}=&\left(N_{ab}-u_{a}u_{b}\right)\left[\tilde{K}\right]-\left[\tilde{K}_{ab}\right]=\\\
=&-\left[\tilde{\phi}\right]u_{a}u_{b}+\left(\frac{1}{2}\left[\tilde{\phi}\right]+\left[\tilde{\mathcal{A}}\right]\right)N_{ab}.\end{split}$
(152)

Let us now apply these relations to our specific case. First, note that (151)
implies that the value of $\mathcal{Q}$ must be the same on the boundary.
Since we have seen that a Schwarzschild solution is characterized by relation
(134), we can conclude that the junction can be satisfied
$\mathcal{Q}_{*}=0$333This is not the only possibility, though. We will
explore these additional possibilities in a forthcoming work.. With this
assumption, we have

$\Lambda_{*}=\frac{1}{2}\frac{f_{0}}{f^{\prime}_{0}},$ (153)

with $f_{0}=f(0)$ and $f^{\prime}_{0}=f^{\prime}(0)$. Therefore, the effective
cosmological constant depends on the value of the function $f$ and its
derivative in zero. At this point, it is necessary to observe that two
functions $f(\mathcal{Q})$ and $g(\mathcal{Q})$, which differ by a constant
term, can be considered the “same” theory of gravity. This is no different
than considering GR as the same theory of gravity in the presence or absence
of a cosmological constant. Consequently, it is meaningful to perform the
junction between a solution of $f(\mathcal{Q})$ and $g(\mathcal{Q})$.
Alternatively, one can write

$f(\mathcal{Q})=2\Lambda_{*}f^{\prime}_{0}+g(\mathcal{Q}),$ (154)

where $g(0)=0$, and therefore consider the constant term as a cosmological
constant, albeit of non-metric origin, present in a theory with Lagrangian
proportional to $g(\mathcal{Q})$.

With this idea in mind, we can apply the junction conditions. We assume that
in the region $\mathcal{M}^{-}$ we have $\Lambda_{*}\neq 0$ and that
$\Lambda_{*}=0$ in $\mathcal{M}^{+}$. Therefore, we have the de Sitter
solution in $\mathcal{M}^{-}$,

$\begin{split}\text{d}s^{2}=&g_{ab}dX^{a}dX^{b}=-\left(1-\frac{1}{3}\Lambda_{*}R^{2}\right)\text{d}T^{2}\\\
&+\left(1-\frac{1}{3}\Lambda_{*}R^{2}\right)^{-1}\text{d}R^{2}\\\
&+R^{2}\left(\text{d}\Theta^{2}+\sin^{2}{\Theta}\text{d}\varphi^{2}\right),\end{split}$
(155)

and the Schwarzschild one in $\mathcal{M}^{+}$,

$\begin{split}\text{d}s^{2}=&g_{ab}dx^{a}dx^{b}=\\\
=&-\left(1-\frac{r_{s}}{r}\right)\text{d}t^{2}+\left(1-\frac{r_{s}}{r}\right)^{-1}\text{d}r^{2}\\\
&+r^{2}\left(\text{d}\Theta^{2}+\sin^{2}{\Theta}\text{d}\varphi^{2}\right).\end{split}$
(156)

Notice that in writing these metrics we have already aligned the angular
coordinates. This is possible because both metrics are spherically symmetric.
Assuming $e_{a}$ as a vector normal to a sphere of radius $r$ in Schwarzschild
spacetime, we have

$e_{a}^{+}dx^{a}=\left(1-\frac{r_{s}}{r}\right)^{1/2}dr,$ (157)

in $\mathcal{M}^{+}$ and, upon changing the radial coordinate $r$ into $R$, we
have an expression of the normal in the interior spacetime

$e_{a}^{-}dX^{a}=\left(1-\frac{1}{3}\Lambda_{*}R^{2}\right)^{1/2}dR.$ (158)

Then the induced metrics are

$\begin{split}q^{-}_{ab}dX^{a}dX^{b}=&(g^{-}_{ab}-e^{-}_{a}e^{-}_{b})dX^{a}dX^{b}=\\\
=&-\left(1-\frac{1}{3}\Lambda_{*}r_{*}^{2}\right)\text{d}T^{2}\\\
&+r_{*}^{2}\left(\text{d}\Theta^{2}+\sin^{2}{\Theta}\text{d}\varphi^{2}\right),\end{split}$
(159)
$\begin{split}q^{+}_{ab}dX^{a}dX^{b}=&(g^{+}_{ab}-e^{+}_{a}e^{+}_{b})dx^{a}dx^{b}\\\
=&-\left(1-\frac{r_{s}}{r_{*}}\right)\text{d}t^{2}\\\
&+r_{*}^{2}\left(\text{d}\Theta^{2}+\sin^{2}{\Theta}\text{d}\varphi^{2}\right),\end{split}$
(160)

and (146) gives the transformation between the two coordinate systems at the
boundary $r_{*}$

$\begin{split}t=&\sqrt{\frac{1-\frac{1}{3}\Lambda_{*}r_{*}^{2}}{1-\frac{r_{s}}{r_{*}}}}(T-T_{0}),\\\
r=&R,\end{split}$ (161)

where $T_{0}$ is an integration constant.

Using these results we can verify that in $r_{*}$ Eq. (147) is not satisfied.
In fact, we have

$\begin{split}\left[\tilde{\phi}\right]&=\frac{2\sqrt{r_{*}-r_{s}}}{r_{*}^{3/2}}-\frac{2\sqrt{1-\frac{1}{3}\Lambda_{*}r_{*}^{2}}}{r_{*}},\\\
\left[\tilde{\mathcal{A}}\right]&=\frac{r_{s}}{2r_{*}^{3/2}\sqrt{r_{*}-r_{s}}}+\frac{\Lambda_{*}r_{*}}{\sqrt{9-3\Lambda_{*}r_{*}^{2}}},\end{split}$
(162)

which cannot be put simultaneously to zero. Thus, a regular geometry can only
be recovered, assuming the presence of a thin shell of matter separating the
two regions. The stress-energy tensor of this form of matter can be deduced by
the formula (152) and has

$\begin{split}\mu^{S}=&\Psi^{S}_{ab}u^{a}u^{b}=\frac{2\sqrt{1-\frac{1}{3}\Lambda_{*}r_{*}^{2}}}{r_{*}}-\frac{2\sqrt{r_{*}-r_{s}}}{r_{*}^{3/2}},\\\
p^{S}=&\frac{1}{2}\Psi^{S}_{ab}N^{ab}=\frac{2\Lambda_{*}r_{*}^{2}-3}{r_{*}\sqrt{9-3\Lambda_{*}r_{*}^{2}}}+\frac{2r_{*}-r_{s}}{2r_{*}^{3/2}\sqrt{r_{*}-r_{s}}}.\end{split}$
(163)

The above quantities are positive for

$0<\Lambda_{*}<\frac{3}{r_{s}^{2}},\qquad
r_{s}<\sqrt[3]{\frac{3r_{s}}{\Lambda_{*}}}<r_{*}<\sqrt[3]{\frac{3}{\Lambda_{*}}}.$
(164)

As $r_{*}$ lies beyond the Schwarzschild radius, we can conclude that the
above version of the Q-gravastar, despite its simplicity, is a consistent
model for black hole alternatives in the context of $f(\mathcal{Q})$ gravity.

##  VI Discussion and conclusions

We have developed the $1+1+2$ covariant formalism for static LRS spacetimes of
class II in the presence of nonmetricity. The resulting geometrical setting
has been applied for studying the features of vacuum solutions arising in
$f(\mathcal{Q})$ gravity, including the Birkhoff theorem and the presence of
Gravastar solutions.

We found that also in the case of $f(\mathcal{Q})$ gravity, the $1+1+2$
covariant formalism is very efficient in describing LRS spacetime and allows
us to understand in a very detailed way the effect of nonmetricity precluding
from a specific choice of gauge. Special attention has been devoted to a
complete decomposition of the Weyl tensor, which has revealed itself to be
fundamental to properly analyzing spacetime dynamics.

Our analysis shows that nonmetricity gives rise to significant kinematical and
dynamical differences with the purely metric case. For example, as it was
already pointed out in [28], it is evident that a given congruence of
worldlines, which is associated with a class of observers in the covariant
formalism, can be associated with two different types of acceleration,
provided by Eqs. (38) and (48) related to the geometric properties of the
congruence itself, like, e.g., autoparallelism.

As a first application of the general formalism, we have analyzed the case of
static and spherically symmetric spacetimes. In particular, using some
simplifying assumptions on the nonmetricity tensor connected with the
requirement of autoparallelism for the timelike congruence, we have deduced a
self-consistent system of algebraic/differential equations for the
investigation of vacuum solutions. These equations allow the identification of
the conditions under which the theory produces the same Einstein-like field
equations as GR. In addition, we have derived sufficient conditions that
ensure the existence of a Schwarzschild-de Sitter solution in vacuum,
extending the ones presented so far in the literature.

A second study case concerns the presence of gravastars in the context of
$f(\mathcal{Q})$. We have shown that the simplest possible gravastar solution
can be achieved by setting the nonmetricity scalar $\mathcal{Q}$ to zero. In
this hypothesis, the nonmetricity acts as an effective cosmological constant,
and the solution requires the presence of a shell that can be composed of
standard matter. However, the equations show that this toy model is only one
possible realization of this scenario. Other realizations, based, for example,
on the case of non-constant $\mathcal{Q}$, might lead to gravastars that do
not require external shells. In this respect, these solutions would be akin to
Brill–Hartle Geons (see, e.g., [53, 54]). Future studies will investigate the
existence and, above all, the dynamics of such solutions in $f(\mathcal{Q})$
gravity.

##  Appendix A Weyl tensor

In a metric theory, the Weyl tensor is defined by the Ricci decomposition as:

$\begin{split}\tilde{C}_{abcd}=&\tilde{R}_{abcd}+\\\
&+\frac{1}{2}\left(g_{ad}\tilde{R}_{bc}-g_{ac}\tilde{R}_{bd}+g_{bc}\tilde{R}_{ad}-g_{bd}\tilde{R}_{ac}\right)+\\\
&+\frac{1}{6}\tilde{R}\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right).\end{split}$
(165)

However, in the presence of nonmetricity, the Riemann tensor loses its
antisymmetry in the first two indices, and therefore, we must also consider
the homothetic curvature $\check{R}_{ab}$ [55, 56]:

$R_{abcd}=W_{abcd}+Z_{abcd}$ (166)

with

$R_{[ab]cd}=W_{abcd}=\sum_{n=1}^{6}W^{(n)}_{abcd}$ (167)
$W^{(1)}_{abcd}=W_{abcd}-\sum_{n=2}^{6}W^{(n)}_{abcd}$ (168)
$W^{(2)}_{abcd}=\frac{1}{2}\left(W_{abcd}-W_{cdab}\right)-\left(R_{[mn]}-\bar{R}_{[mn]}\right)\delta^{m}_{\left[a\right.}g_{\left.b\right]\left[c\right.}\delta^{n}_{\left.d\right]}$
(169) $W^{(3)}_{abcd}=W_{[abcd]}=R_{[abcd]}$ (170)
$W^{(4)}_{abcd}=-\left(R_{(mn)}-\bar{R}_{(mn)}\right)\delta^{m}_{\left[a\right.}g_{\left.b\right]\left[c\right.}\delta^{n}_{\left.d\right]}-\frac{1}{2}g_{c[a}g_{b]d}R$
(171)
$W^{(5)}_{abcd}=\left(R_{[mn]}-\bar{R}_{[mn]}\right)\delta^{m}_{\left[a\right.}g_{\left.b\right]\left[c\right.}\delta^{n}_{\left.d\right]}$
(172) $W^{(6)}_{abcd}=\frac{1}{6}g_{c[a}g_{b]d}R$ (173)
$R_{(ab)cd}=Z_{abcd}=\sum_{n=1}^{5}Z^{(n)}_{abcd}$ (174)
$Z^{(1)}_{abcd}=Z_{abcd}-\sum_{n=2}^{5}Z^{(n)}_{abcd}$ (175)
$\begin{split}Z^{(2)}_{abcd}=&\frac{1}{2}\left(Z_{abcd}-Z_{c(ab)d}+Z_{d(ab)c}\right)-\frac{1}{4}\left(R_{[mn]}+\right.\\\
&\left.+\bar{R}_{[mn]}-\check{R}_{mn}\right)\left(2\delta^{m}_{(a}g_{b)[c}\delta^{n}_{d]}-g_{ab}\delta^{m}_{[c}\delta^{n}_{d]}\right)\end{split}$
(176)
$\begin{split}Z^{(3)}_{abcd}=&\frac{1}{6}\left(R_{[mn]}+\bar{R}_{[mn]}+\right.\\\
&\left.-\frac{1}{2}\check{R}_{mn}\right)\left(4\delta^{m}_{(a}g_{b)[c}\delta^{n}_{d]}-g_{ab}\delta^{m}_{[c}\delta^{n}_{d]}\right)\end{split}$
(177)
$Z^{(4)}_{abcd}=\frac{1}{2}\left(R_{(mn)}+\bar{R}_{(mn)}\right)\delta^{m}_{(a}g_{b)[c}\delta^{n}_{d]}$
(178) $Z^{(5)}_{abcd}=\frac{1}{4}g_{ab}\check{R}_{cd}$ (179)

We call “general Weyl tensor” the following expression,

$C_{abcd}=W^{(1)}_{abcd}+Z^{(1)}_{abcd}.$ (180)

##  Appendix B Tensor, vector and scalar elements of magnetic and electric
parts of Weyl tensors

We introduce the several quantities involved in Eqs. (55)-(60):

$\mathcal{H}_{ab}=\left(N_{a}{}^{c}N_{b}{}^{d}-\frac{1}{2}N_{ab}N^{cd}\right)H_{cd},$
(181)
$\mathcal{H}_{a}=N_{a}{}^{c}e^{d}H_{cd},\qquad\mathcal{H}=e^{a}e^{b}H_{ab}=-N^{ab}H_{ab}$
(182)
$\bar{\mathcal{H}}_{ab}=\left(N_{(a}{}^{c}N_{b)}{}^{d}-\frac{1}{2}N_{ab}N^{cd}\right)\bar{H}_{cd},$
(183)
$\bar{\mathcal{H}}_{a}=N_{a}{}^{c}e^{d}\bar{H}_{cd},\qquad\mathbf{H}_{b}=e^{c}N_{b}{}^{d}\bar{H}_{cd},$
(184)
$\check{\bar{\mathcal{H}}}_{a}=N_{a}{}^{c}u^{d}\bar{H}_{cd},\qquad\bar{\mathbb{H}}=\frac{1}{2}\epsilon^{ab}\bar{H}_{ab},$
(185)
$\check{\bar{\mathcal{H}}}=e^{a}u^{b}\bar{H}_{ab},\qquad\bar{\mathcal{H}}=e^{a}e^{b}\bar{H}_{ab}=-N^{ab}\bar{H}_{ab}$
(186)
$\check{\mathcal{H}}_{ab}=\left(N_{(a}{}^{c}N_{b)}{}^{d}-\frac{1}{2}N_{ab}N^{cd}\right)\check{H}_{cd},$
(187)
$\check{\mathcal{H}}_{a}=N_{a}{}^{c}e^{d}\check{H}_{cd},\qquad\check{\mathbf{H}}_{b}=e^{c}N_{b}{}^{d}\check{H}_{cd},$
(188)
$\mathfrak{H}_{a}=N_{a}{}^{c}u^{d}\check{H}_{cd},\qquad\check{\mathbb{H}}=\frac{1}{2}\epsilon^{ab}\check{H}_{ab},$
(189)
$\mathfrak{H}=e^{a}u^{b}\check{H}_{ab},\qquad\check{\mathcal{H}}=e^{a}e^{b}\check{H}_{ab}=-N^{ab}\check{H}_{ab}$
(190)
$\mathcal{E}_{ab}=\left(N_{(a}{}^{c}N_{b)}{}^{d}-\frac{1}{2}N_{ab}N^{cd}\right)E_{cd},$
(191)
$\mathcal{E}_{a}=N_{a}{}^{c}e^{d}E_{cd},\quad\mathbf{E}_{b}=e^{c}N_{b}{}^{d}E_{cd},\quad\check{\mathcal{E}}_{b}=u^{c}N_{b}{}^{d}E_{cd}$
(192)
$\mathbb{E}=\frac{1}{2}\epsilon^{ab}E_{ab},\qquad\check{\mathcal{E}}=u^{a}e^{b}E_{ab},$
(193) $\mathcal{E}=e^{a}e^{b}E_{ab}=-N^{ab}E_{ab}$ (194)
$\bar{\mathcal{E}}_{ab}=\left(N_{(a}{}^{c}N_{b)}{}^{d}-\frac{1}{2}N_{ab}N^{cd}\right)E_{cd},$
(195)
$\bar{\mathcal{E}}_{a}=N_{a}{}^{c}e^{d}E_{cd},\quad\bar{\mathbf{E}}_{b}=e^{c}N_{b}{}^{d}E_{cd},\quad\check{\bar{\mathcal{E}}}_{b}=u^{c}N_{b}{}^{d}E_{cd}$
(196)
$\bar{\mathbb{E}}=\frac{1}{2}\epsilon^{ab}\bar{E}_{ab},\qquad\check{\bar{\mathcal{E}}}=u^{a}e^{b}\bar{E}_{ab}=\check{\mathcal{E}},$
(197) $\bar{\mathcal{E}}=e^{a}e^{b}\bar{E}_{ab}=-N^{ab}\bar{E}_{ab}$ (198)
$\mathfrak{E}_{ab}=N_{a}{}^{c}N_{b}{}^{d},\qquad\mathfrak{E}_{a}=N_{a}{}^{c}e^{d}\check{E}_{cd},$
(199) $\check{\mathfrak{E}}_{a}=N_{a}{}^{c}u^{d}\check{E}_{cd}$ (200)

## Acknowledgment

This work has been carried out in the framework of activities of the INFN
Research Project QGSKY.

## References

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