eigenfrequencies

The system (198)–(205) found for the comoving frame and the one found for the
static frame, Eqs. (207)–(209), both describe the same setup. Nonetheless,
comparing the equations of each system, it is manifest that there is a trade-
off when considering one picture over the other: in the comoving frame, the
characterization of the matter fluid is simpler, but the dynamic description
of the perturbed star is more complex, whereas from the point of view of
static observers the description of the dynamics of the perturbed star is
simpler, but the characterization of the matter fluid is more complex. The
solutions of each system with appropriate boundary conditions are exactly the
same, however when analyzing the properties and computing the solutions, from
a technical point of view, it is easier to work in the static frame.

One key result that can be readily proven from Eqs. (207) and (208) is that if
the background spacetime is a solution of the Tolman-Oppenheimer-Volkoff
equations for energy density and pressure functions, $\mu_{0}$ and $p_{0}$,
that are real analytic in the interior and boundary of the star, we can find
real analytic solutions for the perturbations. Moreover, in those conditions
the coefficients $\Psi_{Q}$ verify a Sturm-Liouville eigenvalue problem with a
limit-point-non-oscillating (LPNO) endpoint and a regular endpoint, with
separated self-adjoint boundary conditions. This fact, in particular, allows
us to relate the properties of the eigenfrequencies $\upsilon$ with the
properties of the eigenvalues of the Sturm-Liouville problem. This section
focuses on the derivation of these results.

###  V.1 General solutions for the perturbations

To find explicit solutions for the system (207) and (208), we must break
covariance. In that regard, consider the parameter $r$ defined in Eq. (358).
Following Ref. [22], if the sheets are isometric to 2-spheres, $r^{-2}$ is, up
to a multiplicative constant, equal to the Gauss curvature of the sheets.
Therefore, the parameter $r$ represents the circumferential radius function
both in the background and the perturbed spacetime. Consequently, let $r=0$
represents the center $\mathfrak{C}$ of the star, and we will consider that
the background quantities
$\left\\{\mathcal{A}_{0},\phi_{0},\mathcal{E}_{0},\mu_{0},p_{0}\right\\}$ and
the coefficients $\Psi_{\mathsf{p}}$ and $\Psi_{Q}$ are functions of $r$.

Now, we will impose that the following regularity constraints hold:

  * •

the equilibrium fluid verifies the weak energy condition;

  * •

the background spacetime is a solution of the Tolman-Oppenheimer-Volkoff
equation for real analytic, non-trivial energy density and pressure functions
for the whole range within the equilibrium star;

  * •

the square of the speed of sound of the perturbed fluid, $f^{\prime}$, is
positive and real analytic in the interior and at the boundary of the star.

Requiring real analytical background solutions is a rather strong constraint;
nonetheless, to our knowledge, all classical exact solutions for compact
astrophysical objects, verify this hypothesis in some open neighborhood of the
center, at $r=0$. Therefore, the following results are appropriate for
treating perturbations of physically relevant setups. The radius of
convergence of the power series, of course, may or may not be greater than the
radius of the equilibrium star, in which case further treatment has to be
carried out, nonetheless, for the following results, it suffices that the
radius of convergence of the power series around the center is non-zero.

Imposing these conditions, Eqs. (188), (191) and (359) imply

$\displaystyle\phi_{0}$
$\displaystyle=\frac{2}{r}\sqrt{1-\frac{2M\left(r\right)}{r}}\,,$ (212)
$\displaystyle\mathcal{E}_{0}$
$\displaystyle=\frac{1}{3}\mu_{0}-\frac{2M\left(r\right)}{r^{3}}\,,$
$\displaystyle\mathcal{A}_{0}\phi_{0}$
$\displaystyle=p_{0}+\frac{2M\left(r\right)}{r^{3}}\,,$

where

$M\left(r\right):=\frac{1}{2}\int_{0}^{r}\mu_{0}x^{2}dx\,,$ (213)

is usually dubbed the mass function, and Eq. (59) yields

$\upsilon\left(r\right)=\lambda
e^{-\int\frac{2\mathcal{A}_{0}}{r\phi_{0}}dr}\,,$ (214)

relating the eigenfrequencies measured by an observer comoving with the fluid,
$\upsilon$, with the constant eigenfrequencies $\lambda$, measured by a free-
falling observer at spatial infinity. If the functions $\mu_{0}$ and $p_{0}$
verify the weak energy condition and are real analytic within the star, so are
the functions $\mathcal{A}_{0}$ and $\mathcal{E}_{0}$, and $\phi_{0}$ has a
simple pole at the center $\mathfrak{C}$, but is otherwise real analytic in
the interior and boundary of the star.

Hence, in the considered setup, Eqs. (207) and (208) form a system of ordinary
differential equations with real analytic coefficients in a neighborhood of
$\mathfrak{C}$, with a simple pole at $r=0$, and solutions can be found around
the singular point.

Using Eq. (358) to relate the hat derivatives with the derivatives with
respect to $r$ of first order quantities, the system of ODEs (207) and (208)
is given by

$\frac{r\phi_{0}}{2}\frac{d\Psi_{\mathsf{p}}}{dr}+\left[\frac{\mu_{0}+p_{0}}{\phi_{0}}+\left(2+\frac{1}{f^{\prime}\left(\mu_{0}\right)}\right)\mathcal{A}_{0}\right]\Psi_{\mathsf{p}}=\left[\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)+\frac{\mathcal{A}_{0}^{2}}{f^{\prime}\left(\mu_{0}\right)}+\frac{r\mathcal{A}_{0}\phi_{0}}{2\left(\mu_{0}+p_{0}\right)}\frac{d\mu_{0}}{dr}+\upsilon^{2}\right]\Psi_{Q}\,,$
(215)

$\frac{r\phi_{0}}{2}\frac{d\Psi_{Q}}{dr}+\left(\phi_{0}+2\mathcal{A}_{0}-\frac{\mathcal{A}_{0}}{f^{\prime}\left(\mu_{0}\right)}-\frac{r\phi_{0}}{2\left(\mu_{0}+p_{0}\right)}\frac{d\mu_{0}}{dr}-\frac{\mu_{0}+p_{0}}{\phi_{0}}\right)\Psi_{Q}=-\frac{1}{f^{\prime}\left(\mu_{0}\right)}\Psi_{\mathsf{p}}\,.$
(216)

To find solutions for this system, it is useful to write it in the matrix
form:

$\frac{d\mathds{W}}{dr}=\left(\frac{1}{r}\mathds{R}+\Theta\right)\mathds{W}\,,$
(217)

where

$\displaystyle\mathds{W}$
$\displaystyle=\left[\begin{array}[]{c}\Psi_{\mathsf{p}}\\\
\Psi_{Q}\end{array}\right]\,,$ $\displaystyle\mathds{R}$
$\displaystyle=\left[\begin{array}[]{lr}0&0\\\ 0&-2\end{array}\right]\,,$
(218)

and

$\Theta=-\frac{2}{r\phi_{0}}\left[\begin{array}[]{lr}\frac{\mu_{0}+p_{0}}{\phi_{0}}+2\mathcal{A}_{0}+\frac{\mathcal{A}_{0}}{f^{\prime}\left(\mu_{0}\right)}&\qquad-\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)-\frac{\mathcal{A}_{0}^{2}}{f^{\prime}\left(\mu_{0}\right)}-\frac{r\mathcal{A}_{0}\phi_{0}}{2\left(\mu_{0}+p_{0}\right)}\frac{d\mu_{0}}{dr}-\upsilon^{2}\\\
\frac{1}{f^{\prime}\left(\mu_{0}\right)}&2\mathcal{A}_{0}-\frac{\mathcal{A}_{0}}{f^{\prime}\left(\mu_{0}\right)}-\frac{r\phi_{0}}{2\left(\mu_{0}+p_{0}\right)}\frac{d\mu_{0}}{dr}-\frac{\mu_{0}+p_{0}}{\phi_{0}}\end{array}\right]\,.$
(219)

The regularity conditions that we have imposed on the thermodynamic variables
of the equilibrium configuration, $\mu_{0}$ and $p_{0}$, and on the equation
of state of the perturbed fluid in the comoving frame, $f$, guarantee that the
matrix $\Theta$ is analytic at $r=0$ and $r\phi_{0}$ does not vanish in the
interior or boundary of the star. These conditions imply that $r=0$ is a
regular singular point of the system.

To solve the system of ordinary differential equations, we will follow the
formalism in Ref. [34]. Since the $\Theta$ is assumed to be a real analytic
matrix at the center of the star, it can be expanded in a convergent power
series of the form

$\Theta\left(r\right)=\sum_{n=0}^{+\infty}\Theta_{n}r^{n}\,.$ (220)

Then, the solution matrix $\mathds{W}$ can be written in a power series
guaranteed to converge to the solution in a neighborhood of $r=0$. The radius
of convergence of the power series solution is equal, except for possibly at
$r=0$, to the radius of convergence of the power series of $\Theta$, which, of
course, depends on the equilibrium background spacetime considered.

Before proceeding, we remark that the general form of the solutions of the
boundary value problem (207)–(211) for a general static, spherically symmetric
spacetime can be rather complicated. However, imposing the previous regularity
conditions on $\mu_{0}$ and $p_{0}$, and assuming
$f^{\prime}\left(\mu_{0}\right)$ is not zero in a neighborhood of the center
of the star, it can be shown that various entries of the $\Theta_{0}$ and
$\Theta_{1}$ coefficient-matrices of the series (220) are zero, which greatly
simplifies the general family of solutions.

Taking into consideration the regularity of the background yields the power
series solution

$\left[\begin{array}[]{c}\Psi_{\mathsf{p}}\\\
\Psi_{Q}\end{array}\right]=\left[\begin{array}[]{cc}-\frac{1}{r}\left(\Theta_{0}\right)_{12}&\quad
1\\\
\frac{1}{r^{2}}&0\end{array}\right]\mathds{P}_{\mathds{W}}\left[\begin{array}[]{c}c_{1}\\\
c_{2}\end{array}\right]\,,$ (221)

where $c_{1}$ and $c_{2}$ are integration constants, which might depend on the
eigenfrequencies $\upsilon$, or equivalently on $\lambda$, defined in Eq.
(214). The notation $\left(\Theta_{n}\right)_{ij}$ is to be interpreted as the
$ij$-entry of the $n$th order coefficient of the power series of $\Theta$, and
$\mathds{P}_{\mathds{W}}$ is a real analytic matrix, such that

$\displaystyle\mathds{P}_{\mathds{W}}\left(r\right)$
$\displaystyle=\sum_{n=0}^{+\infty}\mathds{P}_{n}r^{n}$ (222)
$\displaystyle\mathds{P}_{0}$ $\displaystyle=\mathds{I}_{2}$
$\displaystyle\mathds{P}_{k}$
$\displaystyle=\frac{1}{k}\sum_{j=0}^{k-1}\mathds{A}_{k-1-j}\mathds{P}_{j}\,,\quad\text{for
}k\geq 1$

where $\mathds{I}_{2}$ represents the $2\times 2$ identity matrix, the matrix
$\mathds{A}$ is given by

$\mathds{A}=\left[\begin{array}[]{lr}\Theta_{22}-r\left(\Theta_{0}\right)_{12}\Theta_{21}&r^{2}\Theta_{21}\\\
\frac{\Theta_{12}-\left(\Theta_{0}\right)_{12}}{r^{2}}+\frac{\left(\Theta_{0}\right)_{12}\left(\Theta_{22}-\Theta_{11}\right)}{r}-\left(\Theta_{0}\right)_{12}^{2}\Theta_{21}&\quad\Theta_{11}+r\left(\Theta_{0}\right)_{12}\Theta_{21}\end{array}\right]\,,$
(223)

and $\mathds{A}_{n}$ represents the $n$th order coefficient of its power
series, that is,
$\mathds{A}\left(r\right)=\sum_{n=0}^{+\infty}\mathds{A}_{n}r^{n}$.

The general family of solutions of the system, for $0<r<a$, where
$a\in\mathbb{R}_{>0}$, is given by Eqs. (221) and (222). To select the
physically acceptable solutions and extend the domain to $r=0$, we will impose
the boundary conditions (i) and (ii) of Sec. IV.3. Considering
$\mathds{P}_{0}=\mathds{I}_{2}$, we can directly compute the lower order
coefficients of the power series expansion of $\mathds{W}$. Imposing the
boundary condition at the center sets the coefficient $c_{1}$ to be zero so
that the perturbations do not diverge at $r=0$ at all times. Then, we find

$\left[\begin{aligned} \Psi_{\mathsf{p}}\\\
\Psi_{Q}\end{aligned}\right]=\left[\begin{aligned}
c_{2}+\mathcal{O}\left(r^{2}\right)\\\
\mathcal{O}\left(r\right)\end{aligned}\right]\,.$ (224)

Provided the background spacetime, the equation of state of the perturbed
fluid, and the values of the eigenfrequencies, these results allow us to find
analytic solutions for the perturbations that verify the boundary conditions.
The eigenfrequencies themselves cannot be restricted directly from these
results. Nonetheless, in the following subsection, we will show that general
useful properties for the eigenfrequencies can be derived by relating
$\upsilon$ with the eigenvalues of a Sturm-Liouville problem.

###  V.2 Sturm-Liouville eigenvalue problem

We do not need to break covariance for the following result. Therefore, to
keep the discussion independent of a local coordinate system, we will consider
the hat derivatives without associating them with derivatives in a specific
coordinate system. Let $\ell$ be an affine parameter of the $e$ congruence,
and without loss of generality, we set that $\ell=0$ represents the center of
the star and $\ell=\ell_{\mathfrak{B}}$ the boundary of the equilibrium star.

Taking the hat derivative of Eq. (208) we find

$D_{e}\left[\exp{\left(\int_{\ell_{0}}^{\ell}F\left(x\right)dx\right)}\widehat{\Psi}_{Q}\right]+\exp{\left(\int_{\ell_{0}}^{\ell}F\left(x\right)dx\right)}G\left(\ell\right)\Psi_{Q}=-\frac{1}{f^{\prime}\left(\mu_{0}\right)}\exp{\left(\int_{\ell_{0}}^{\ell}F\left(x\right)dx\right)}\lambda^{2}\Psi_{Q}\,,$
(225)

where, for notational convenience, we have indicated the hat derivatives as
$D_{e}\equiv e^{\alpha}D_{\alpha}$, $\ell_{0}\in\mathbb{R}$, $\lambda$ is an
integration constant following from Eq. (59) and

$\displaystyle F\left(\ell\right)$
$\displaystyle=\phi_{0}+4\mathcal{A}_{0}-\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}+\frac{f^{\prime\prime}\left(\mu_{0}\right)\widehat{\mu}_{0}}{f^{\prime}\left(\mu_{0}\right)}\,,$
(226) $\displaystyle G\left(\ell\right)$
$\displaystyle=\frac{f^{\prime\prime}\left(\mu_{0}\right)\widehat{\mu}_{0}}{f^{\prime}\left(\mu_{0}\right)}\left(\phi_{0}+2\mathcal{A}_{0}-\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}-\frac{\mu_{0}+p_{0}}{\phi_{0}}\right)+\frac{1}{f^{\prime}\left(\mu_{0}\right)}\left(2\mathcal{A}_{0}\phi_{0}-p_{0}+\mathcal{A}_{0}^{2}\right)$
$\displaystyle+\mu_{0}+3p_{0}+\mathcal{A}_{0}\phi_{0}+2\mathcal{A}_{0}^{2}-\frac{1}{2}\phi_{0}^{2}-\frac{2\mathcal{A}_{0}\widehat{\mu}_{0}}{\mu_{0}+p_{0}}-\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}+\frac{\mu_{0}+p_{0}}{\phi_{0}}\right)$
$\displaystyle-\left[D_{e}\left(\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}\right)+D_{e}\left(\frac{\mu_{0}+p_{0}}{\phi_{0}}\right)\right]\,.$

Moreover, conditions (i) and (ii) of Sec. IV.3 imply the following set of
separated self-adjoint boundary conditions:

$\left\\{\begin{aligned} \left.\Psi_{Q}\right|_{\mathfrak{C}}&=0\\\
\left.\widehat{\Psi}_{Q}+\left(\phi_{0}+2\mathcal{A}_{0}-\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}-\frac{\mu_{0}+p_{0}}{\phi_{0}}\right)\Psi_{Q}\right|_{\mathfrak{B}}&=0\end{aligned}\right.$
(227)

Therefore, provided the background spacetime and $f^{\prime}$ are sufficiently
regular, $\Psi_{Q}$ verifies a formal Sturm-Liouville eigenvalue problem with
weight function

$w\left(\ell\right)=\frac{1}{f^{\prime}\left(\mu_{0}\right)}\exp{\left(\int_{\ell_{0}}^{\ell}F\left(x\right)dx\right)}\,,$
(228)

and eigenvalues $\lambda^{2}$. This last assumption, however, is not trivial,
and we have to specify the conditions under which the associated Sturm-
Liouville operator is self-adjoint.

Considering the discussion in the previous subsection, in particular the
regularity conditions in subsec • ‣ V.1, the function $\phi_{0}$ has a simple
pole at $\mathfrak{C}$. This endpoint, at $\ell=0$, is limit-point since we
have found solutions that are not in
$L^{2}\left(\left]0,\delta\right[,w\right)$, for any
$\delta\in\mathbb{R}_{>0}$. However, in the conditions considered in the
previous subsection, we have explicitly shown that all solutions for
$\Psi_{Q}$ are real analytic in $\left]0,\varepsilon\right[$, for some
$\varepsilon\in\mathbb{R}_{>0}$, hence there are no non-trivial solutions that
have an infinite number of zeros in a right-neighborhood of $\ell=0$. We can
then conclude that the endpoint $\ell=0$ is LPNO. Moreover, the regularity
conditions guarantee that the coefficients of the differential equation verify

$w(\ell)>0,\quad\exp{\left(\int_{\ell_{0}}^{\ell}F\left(x\right)dx\right)}>0\,,$
(229)

almost everywhere and

$\exp{\left(-\int_{\ell_{0}}^{\ell}F\left(x\right)dx\right)},\;\exp{\left(\int_{\ell_{0}}^{\ell}F\left(x\right)dx\right)}G\left(\ell\right),\;w\left(\ell\right)\;\in
L_{\text{loc}}\left(\left]0,\ell_{\mathfrak{B}}\right[,\mathbb{R}\right)$
(230)

where
$L_{\text{loc}}\left(\left]0,\ell_{\mathfrak{B}}\right[,\mathbb{R}\right)$ is
the set of locally Lebesgue integrable real functions on
$\left]0,\ell_{\mathfrak{B}}\right[$. Given the previous considerations, we
can apply the results in Refs. [35, 36] to Eq. (225) together with the
boundary conditions (227), concluding that the eigenvalues verify several very
useful properties. In particular, the eigenvalues $\lambda^{2}$ are (i) real,
(ii) simple, (iii) countable, and (iv) have a minimum and are unbounded from
above, that is
$\lambda^{2}=\left\\{\lambda_{0},\lambda_{1},\lambda_{2},...\right\\}$, with
$\lambda_{0}<\lambda_{1}<\lambda_{2}<...\to+\infty$, and
$\lambda_{0}\in\mathbb{R}$. Indeed, in the following subsection, we show
explicitly that, in the considered setup, the eigenfrequencies are such that
$\upsilon^{2}$ are bounded from below and derive necessary constraints for the
minimum value.

###  V.3 Lower bound for the square of the eigenfrequencies

The result that $\Psi_{Q}$ verifies a Sturm-Liouville eigenvalue problem is of
great importance for the description of the behavior of the perturbed star.
The fact that the eigenvalues $\lambda^{2}$, hence $\upsilon^{2}$, are bounded
from below sets that either in the oscillating case or in the continuous
collapse or expanding scenarios, the eigenfrequencies of the excitable
eigenmodes are not arbitrarily small. In addition to those results, in the
considered setup, we will prove the following:

######  Proposition 1.

Let the boundary conditions (i) and (ii) of Sec. IV.3 and the regularity
conditions in Sec. • ‣ V.1 hold. If, in addition,
$\widehat{\mathcal{A}}_{0}\left(\ell_{\mathfrak{B}}\right)\geq 0$, then non-
trivial $\mathcal{C}^{1}$ solutions of the system (207) and (208) exist only
if

$\upsilon^{2}\geq\left.-\mathcal{A}_{0}\phi_{0}-\frac{1}{2}\mu_{0}\right|_{\ell_{\mathfrak{B}}}\,.$
(231)

The premises of Proposition 1 can be significantly relaxed, leading to the
following result:

######  Proposition 2.

Let the boundary conditions (i) and (ii) of Sec. IV.3 hold and consider the
following regularity conditions

  * •

the equilibrium fluid verifies the weak energy condition;

  * •

the background spacetime is a solution of the Tolman-Oppenheimer-Volkoff
equation characterized by non-trivial, $\mathcal{C}^{1}$ functions
$\mathcal{A}_{0}$, $\mu_{0}$ and $p_{0}$, and $\phi_{0}$ has a simple pole at
the center, but is otherwise of class $\mathcal{C}^{1}$ within the domain of
the solution;

  * •

$f^{\prime}\left(\mu_{0}\right)$ is positive in the interior of the perturbed
star.

Then, non-trivial $\mathcal{C}^{1}$ solutions of the system (207) and (208)
exist only if

$\max_{\ell\in\left]0,\ell_{\mathfrak{B}}\right[}\upsilon^{2}>-\max_{\ell\in\left]0,\ell_{\mathfrak{B}}\right[}\left[\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)+\frac{\mathcal{A}_{0}^{2}}{f^{\prime}\left(\mu_{0}\right)}+\frac{\mathcal{A}_{0}\widehat{\mu}_{0}}{\mu_{0}+p_{0}}\right]\,.$
(232)

######  Lemma 1.

If the premises of Proposition 2 hold, a non-trivial $\mathcal{C}^{1}$
solution $\Psi_{\mathsf{p}}$ has at least one root.

Notice that (231) is much simpler than (232). However, even under the
conditions where both inequalities can be used, it should be stressed that
without specifying the background spacetime and the equation of state of the
perturbed fluid, there is no way to infer which provides a tighter bound.
Indeed, the determination of the tighter bound depends critically of the
background spacetime. Nonetheless, finding a constraint on the minimum allowed
values of $\upsilon^{2}$ is of great importance when searching for solutions
to the system. Since the results in Ref. [36] also associate the number of
nodes of the eigenfunctions with the ordinal number of the eigenvalues in the
sequence $\left(\lambda_{n}\right)_{n\in\mathbb{N}}$, these constraints
provide a baseline to search for the eigenfrequencies algorithmically. Lemma 1
provides a further criterion useful to find the fundamental eigenfrequency.

As in the previous subsection, in the following proofs, we will consider
$\ell$ to be an affine parameter of the congruence associated with the
integral curves of the $e$ vector field, and without loss of generality, we
set that $\ell=0$ represents the center of the star and
$\ell=\ell_{\mathfrak{B}}$ the boundary of the equilibrium star.

###### Proof for Proposition 1.

The function $\phi_{0}$ has a simple pole at $\ell=0$. Therefore, imposing
regularity of $\Psi_{Q}$ and $\Psi_{\mathsf{p}}$ and their derivatives in the
interior and boundary of the star, Eq. (208) implies

$\Psi_{Q}\left(0\right)=0\,.$ (233)

In that case, we can disregard the solution with
$\Psi_{\mathsf{p}}\left(0\right)=0$ since this corresponds to the trivial
solution. Then, depending on the initial conditions, either
$\Psi_{\mathsf{p}}\left(0\right)>0$ or $\Psi_{\mathsf{p}}\left(0\right)<0$. As
expected, the reasoning for one case applies similarly to the other.
Therefore, we will explicitly treat only the case
$\Psi_{\mathsf{p}}\left(0\right)>0$. Then, assume
$\Psi_{\mathsf{p}}\left(0\right)>0$. Imposing
$f^{\prime}\left(\mu_{0}\right)>0$ and given that $\phi_{0}$ is positive, Eq.
(208) further implies

$\widehat{\Psi}_{Q}\left(0\right)\Psi_{\mathsf{p}}\left(0\right)<0\,.$ (234)

Therefore, for all $\ell\in\left]0,\epsilon\right[$, for some
$0<\epsilon<\ell_{\mathfrak{B}}$, we have
$\Psi_{\mathsf{p}}\left(\ell\right)\Psi_{Q}\left(\ell\right)<0$, that is, for
our choice of initial data

$\Psi_{Q}\left(\ell\right)<0\;,\text{for all
}\ell\in\left]0,\epsilon\right[\,.$ (235)

Now, the values of the eigenfrequencies are those that verify the boundary
conditions (i) and (ii), in particular Eq. (211). Imposing that the regularity
conditions in Sec. • ‣ V.1 hold, then $\mathcal{A}_{0}\geq 0$ within the
equilibrium star, being zero only at the center: $\ell=0$. Therefore, if
$\mathcal{C}^{1}$ solutions exist, for the fundamental mode, that is, for the
lowest value of $\upsilon^{2}$, choosing $\Psi_{\mathsf{p}}\left(0\right)>0$,
we must have

$\Psi_{\mathsf{p}}-\mathcal{A}_{0}\Psi_{Q}>0\;,\text{for
}\ell\in\left[0,\ell_{\mathfrak{B}}\right[\,,$ (236)

and

$\Psi_{\mathsf{p}}-\mathcal{A}_{0}\Psi_{Q}=0\;,\text{at
}\ell=\ell_{\mathfrak{B}}\,.$ (237)

These results then imply that at the boundary

$\left.D_{e}\left(\Psi_{\mathsf{p}}-\mathcal{A}_{0}\Psi_{Q}\right)\right|_{\ell_{\mathfrak{B}}}=\left.\widehat{\Psi}_{\mathsf{p}}-\mathcal{A}_{0}\widehat{\Psi}_{Q}-\widehat{\mathcal{A}}_{0}\Psi_{Q}\right|_{\ell_{\mathfrak{B}}}\leq
0\,,$ (238)

where, for notational convenience, we have indicated the hat derivative as
$D_{e}\equiv e^{\alpha}D_{\alpha}$. Since $\Psi_{Q}$ verifies a Sturm-
Liouville eigenvalue problem, for the fundamental eigenmode $\Psi_{Q}$ has no
zeros. Therefore, $\Psi_{Q}\left(\ell\right)<0$, for all
$\ell\in\left]0,\ell_{\mathfrak{B}}\right]$. Then, further imposing that
$\widehat{\mathcal{A}}_{0}\left(\ell_{\mathfrak{B}}\right)\geq 0$, the
previous inequality is verified if

$\left.\widehat{\Psi}_{\mathsf{p}}-\mathcal{A}_{0}\widehat{\Psi}_{Q}\right|_{\ell_{\mathfrak{B}}}\leq
0\,.$ (239)

At the boundary, $\ell=\ell_{\mathfrak{B}}$, Eqs. (207) and (208) reduce to

$\displaystyle\widehat{\Psi}_{\mathsf{p}}$
$\displaystyle=\left[\frac{\mu_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+\mathcal{A}_{0}\right)+\frac{\mathcal{A}_{0}\widehat{\mu}_{0}}{\mu_{0}}-2\mathcal{A}_{0}^{2}+\upsilon^{2}\right]\Psi_{Q}\,,$
(240) $\displaystyle\widehat{\Psi}_{Q}$
$\displaystyle=-\left(\phi_{0}+2\mathcal{A}_{0}-\frac{\widehat{\mu}_{0}}{\mu_{0}}-\frac{\mu_{0}}{\phi_{0}}\right)\Psi_{Q}\,.$

Substituting Eq. (240) in the inequality (239) we find

$\left.\left(\frac{1}{2}\mu_{0}+\phi_{0}\mathcal{A}_{0}+\upsilon^{2}\right)\Psi_{Q}\right|_{\ell_{\mathfrak{B}}}\leq
0\,.$ (241)

Remarking once again that for the fundamental eigenmode
$\Psi_{Q}\left(\ell\right)<0$, for all
$\ell\in\left]0,\ell_{\mathfrak{B}}\right]$, inequality (241) implies
inequality (231).

∎

###### Proof for Proposition 2 and Lemma 1.

For clarity, we will repeat some of the intermediate results found in the
proof of Proposition 1.

Consider the premises of Proposition 2. In those conditions,
$\mathcal{A}_{0}\geq 0$ within the equilibrium star, being zero only at the
center: $\ell=0$. Now, if $\mathcal{C}^{1}$ solutions of the boundary value
problem (207), (208) and (211) exist for all
$\ell\in\left[0,\ell_{\mathfrak{B}}\right]$, then

$\Psi_{\mathsf{p}}\left(\ell_{\mathfrak{B}}\right)\Psi_{Q}\left(\ell_{\mathfrak{B}}\right)\geq
0\,.$ (242)

On the other hand, since $\phi_{0}$ has a simple pole at $\ell=0$, from Eq.
(208), regularity of $\widehat{\Psi}_{Q}\left(0\right)$ implies

$\Psi_{Q}\left(0\right)=0\,.$ (243)

In particular, this allows us to disregard the solution with
$\Psi_{\mathsf{p}}\left(0\right)=0$, since this corresponds to the trivial
solution. Moreover, imposing $f^{\prime}\left(\mu_{0}\right)>0$, given that
$\phi_{0}$ is positive, Eq. (208) leads to the conclusion that

$\widehat{\Psi}_{Q}\left(0\right)\Psi_{\mathsf{p}}\left(0\right)<0\,.$ (244)

Therefore, for all $\ell\in\left]0,\epsilon\right[$, for some
$0<\epsilon<\ell_{\mathfrak{B}}$, we have

$\Psi_{\mathsf{p}}\left(\ell\right)\Psi_{Q}\left(\ell\right)<0\,.$ (245)

Considering Bolzano’s theorem, from Eqs. (242) and (245) we conclude that
there is a point $0<a\leq\ell_{\mathfrak{B}}$ where either
$\Psi_{\mathsf{p}}\left(\ell\right)$ or $\Psi_{Q}\left(\ell\right)$ vanishes.
Now, either $\Psi_{\mathsf{p}}\left(0\right)>0$ or
$\Psi_{\mathsf{p}}\left(0\right)<0$. As expected, the reasoning for one case
applies similarly to the other. Hence, we will explicitly treat only the case
$\Psi_{\mathsf{p}}\left(0\right)>0$. In that case, Eq. (245) implies that for
all $\ell\in\left]0,\epsilon\right[$, $\Psi_{Q}\left(\ell\right)<0$.
Continuing, assume $a\in\left]0,\ell_{\mathfrak{B}}\right[$ is such that

$\displaystyle\Psi_{Q}\left(a\right)$ $\displaystyle=0\,,$ (246)
$\displaystyle\widehat{\Psi}_{Q}\left(a\right)$ $\displaystyle\geq 0\,,$
$\displaystyle\Psi_{\mathsf{p}}\left(\ell\leq a\right)$ $\displaystyle>0\,.$

If several such values exist, we take $a$ to be the smallest one that verifies
Eq. (246). Notice that if $\Psi_{\mathsf{p}}\left(a\right)=0$, we would
recover the trivial solution, hence in light of Eq. (211), we can exclude the
case $a=\ell_{\mathfrak{B}}$. At $\ell=a$, Eq. (208) reads

$\widehat{\Psi}_{Q}=-\frac{1}{f^{\prime}\left(\mu_{0}\right)}\Psi_{\mathsf{p}}\,,$
(247)

however, further assuming $f^{\prime}\left(\mu_{0}\right)>0$, the above
equation contradicts the hypothesis. This result, in particular, proves Lemma
1, that is, either $\Psi_{Q}$ or $\Psi_{\mathsf{p}}$ must have a root in the
interior of the star, but independently of $\upsilon^{2}$, we have shown that
$\Psi_{Q}$ can only have a root at $\ell=a$ if $\Psi_{\mathsf{p}}$ has a root
at some $\ell<a$.

From the previous result, we conclude that for a valid solution of the
boundary value problem to exist, there must be either a point $\ell=b<a$ or,
if there is no value $a$ that verifies Eq. (246), a value
$b\in\text{$\left]0,\ell_{\mathfrak{B}}\right[$}$, such that

$\displaystyle\Psi_{\mathsf{p}}\left(b\right)$ $\displaystyle=0\,,$ (248)
$\displaystyle\widehat{\Psi}_{\mathsf{p}}\left(b\right)$ $\displaystyle\leq
0\,,$ $\displaystyle\Psi_{Q}\left(\ell\leq b\right)$ $\displaystyle<0\,.$

Once again, notice that if $\Psi_{Q}\left(b\right)=0$, we would recover the
trivial solution, hence in light of Eq. (211), we can exclude the case
$b=\ell_{\mathfrak{B}}$. At $\ell=b$, Eq. (207) then reads

$\widehat{\Psi}_{\mathsf{p}}=\left[\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)+\frac{\mathcal{A}_{0}^{2}}{f^{\prime}\left(\mu_{0}\right)}+\frac{\mathcal{A}_{0}\widehat{\mu}_{0}}{\mu_{0}+p_{0}}+\upsilon^{2}\right]\Psi_{Q}\,.$
(249)

However, for $\upsilon^{2}$ negative and sufficiently small we would have

$\left.\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)+\frac{\mathcal{A}_{0}^{2}}{f^{\prime}\left(\mu_{0}\right)}+\frac{\mathcal{A}_{0}\widehat{\mu}_{0}}{\mu_{0}+p_{0}}+\upsilon^{2}\right|_{\ell=b}<0\,,$
(250)

hence $\widehat{\Psi}_{\mathsf{p}}\left(b\right)\geq 0$, which would
contradict the hypothesis.

The case where $\Psi_{\mathsf{p}}\left(0\right)<0$ follows exactly the same
reasoning, confirming that in the considered setup, regular solutions of the
system with the considered boundary conditions exist only if the value of
$\upsilon^{2}$ is bounded from below.

Based on these results, we can find a lower bound for the value of
$\upsilon^{2}$. Since the quantity

$\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)+\frac{\mathcal{A}_{0}^{2}}{f^{\prime}\left(\mu_{0}\right)}+\frac{\mathcal{A}_{0}\widehat{\mu}_{0}}{\mu_{0}+p_{0}}+\upsilon^{2}\,,$
(251)

cannot be negative at all interior points of the perturbed star, non-trivial
$\mathcal{C}^{1}$ solutions of the boundary value problem necessarily verify
inequality (232). We remark, however, that inequality (232) is rather
conservative, so much so that even if $\upsilon^{2}$ verifies the inequality,
the quantity in (251) might still be negative at all points within the star. ∎

##  VI Conclusions

We developed a framework to describe general first-order perturbations of
static, locally rotationally symmetric of class II spacetimes within the
theory of general relativity. The new system of equations is completely
general and applicable to any equilibrium configuration. Moreover, by
construction, the framework is covariant and identification gauge invariant.
This last point is of pivotal importance when compared with perturbation
theory available in the literature. The classical metric-based approaches are
intrinsically dependent on a coordinate system and the gauge, which makes it
challenging to understand the dynamics of the perturbed spacetime
unambiguously. Indeed, the adopted covariant spacetime decomposition formalism
provides a manifest advantage over the standard theory: all quantities are
geometrically and physically motivated, which clearly shows the dependencies
and the source of the various physical properties of the perturbations. In
addition, the framework leads to a natural separation between even and odd
parity components of the perturbations, such that the systems of differential
equations for each parity are completely decoupled at the linear level.

As a first application, we considered the study of linear, isotropic, and
adiabatic perturbations. To do so, we have explicitly shown the importance of
choosing frames to describe the unperturbed and the perturbed spacetimes. To
our knowledge, this cannot be done in general in the metric-based linearized
theory available in the literature and makes the covariant gauge-invariant
approach all the more powerful. Given the symmetries of the problem, we
considered two meaningful frames: a frame associated with observers locally
comoving with the elements of volume of the fluid and a frame associated with
static observers with respect to an observer at spatial infinity. These frames
represent the classical Lagrangian and Eulerian pictures, respectively. The
adopted formalism makes it quite simple to change between frames and evaluate
the advantages of one frame over the other. In particular, we were able to
derive the relation between the equations of state of the comoving and static
frames in the perturbed spacetime, showing that, even at linear level, we have
to account for phase changes between the equilibrium and the perturbed fluids.
Moreover, we have shown that, since a perfect fluid in the comoving frame will
not be perceived as perfect in the static frame, and the net momentum flow
must be included in the equation of state to account for the correction to the
rate of change of the energy density and the pressure when compared to those
measured in the comoving frame.

Focusing on analyzing the general properties of linear, isotropic, and
adiabatic perturbations, we have rigorously shown that the problem can be cast
in the form of a singular Sturm-Liouville eigenvalue problem, inheriting the
standard properties for the eigenfunctions and eigenvalues. Moreover, under
rather general regularity conditions, we have found lower bounds for the
values of the eigenfrequencies. The application of the methods developed in
this article, the determination of the analytical solutions to the system of
equations, and the discussion regarding the stability of selected solutions of
the theory will be done elsewhere.

###### Acknowledgements.

PL acknowledges financial support provided under the European Union’s H2020
ERC Advanced Grant “Black holes: gravitational engines of discovery” grant
agreement no. Gravitas–101052587. Views and opinions expressed are, however,
those of the author only and do not necessarily reflect those of the European
Union or the European Research Council. Neither the European Union nor the
granting authority can be held responsible for them. PL acknowledges support
from the Villum Investigator program supported by the VILLUM Foundation (grant
no. VIL37766) and the DNRF Chair program (grant no. DNRF162) by the Danish
National Research Foundation. This project has received funding from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 101007855 and No 101131233. The work
of SC has been carried out in the framework of activities of the INFN Research
Project QGSKY.

##  Appendix A Field equations in the 1+1+2 formalism

Here, we present the Einstein field equations for the theory of General
Relativity, written in the 1+1+2 covariant spacetime decomposition formalism
introduced in Section II. Although the complete set of equations has already
been presented in literature, these expressions are known to contain several
typographical errors. To our knowledge, these errors do not affect published
works. However, the corrected equations are essential in studying general
linear perturbations of static LRS II spacetimes, hence the importance of
presenting them here.

###  A.1 Equations for the kinematical quantities associated with the timelike
congruence

We find the following evolution and propagation equations for the kinematical
quantities associated with the $u$ congruence.

Scalar equations:

$\displaystyle\widehat{\mathcal{A}}-\dot{\theta}$
$\displaystyle=\frac{1}{2}\left(\mu+3p\right)-\Lambda+\frac{1}{3}\theta^{2}+\frac{3}{2}\Sigma^{2}-2\Omega^{2}+2\Sigma_{\mu}\Sigma^{\mu}-\mathcal{A}\left(\mathcal{A}+\phi\right)$
(252)
$\displaystyle-2\Omega_{\mu}\Omega^{\mu}+\mathcal{A}_{\mu}\left(a^{\mu}-\mathcal{A}^{\mu}\right)+\Sigma_{\mu\nu}\Sigma^{\mu\nu}-\delta_{\mu}\mathcal{A}^{\mu}\,,$

$\displaystyle\dot{\Sigma}-\frac{2}{3}\widehat{\mathcal{A}}$
$\displaystyle=\frac{1}{2}\Pi-\mathcal{E}+\frac{1}{3}\left(2\mathcal{A}-\phi\right)\mathcal{A}-\left(\frac{2}{3}\theta+\frac{1}{2}\Sigma\right)\Sigma-\frac{2}{3}\Omega^{2}+\Sigma_{\mu}\left(2\alpha^{\mu}-\frac{1}{3}\Sigma^{\mu}\right)$
(253)
$\displaystyle-\frac{1}{3}\mathcal{A}_{\mu}\left(\mathcal{A}^{\mu}+2a^{\mu}\right)+\frac{1}{3}\Omega_{\mu}\Omega^{\mu}+\frac{1}{3}\Sigma_{\mu\nu}\Sigma^{\mu\nu}-\frac{1}{3}\delta_{\mu}\mathcal{A}^{\mu}\,,$

$\dot{\Omega}=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{A}_{\nu}+\mathcal{A}\xi+\left(\Sigma-\frac{2}{3}\theta\right)\Omega+\Omega_{\mu}\left(\Sigma^{\mu}+\alpha^{\mu}\right)\,,$
(254)

$\widehat{\Omega}=\Omega\left(\mathcal{A}-\phi\right)+\Omega_{\mu}\left(\mathcal{A}^{\mu}+a^{\mu}\right)-\delta_{\mu}\Omega^{\mu}\,,$
(255)
$\frac{2}{3}\widehat{\theta}-\widehat{\Sigma}=Q+\frac{3}{2}\Sigma\phi+2\Omega\xi+\delta_{\mu}\Sigma^{\mu}+\varepsilon_{\mu\nu}\delta^{\mu}\Omega^{\nu}-2\Sigma_{\mu}a^{\mu}+2\varepsilon_{\mu\nu}\mathcal{A}^{\mu}\Omega^{\nu}-\zeta_{\mu\nu}\Sigma^{\mu\nu}\,;$
(256)

Vector equations:

$\displaystyle
N_{\alpha}{}^{\mu}\dot{\Omega}_{\mu}+\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{\mathcal{A}}_{\mu}$
$\displaystyle=\frac{1}{2}\xi\mathcal{A}_{\alpha}-\left(\frac{2}{3}\theta+\frac{1}{2}\Sigma\right)\Omega_{\alpha}+\Omega\left(\Sigma_{\alpha}-\alpha_{\alpha}\right)$
(257)
$\displaystyle-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\left(\frac{1}{2}\phi\mathcal{A}_{\mu}+\mathcal{A}a_{\mu}-\delta_{\mu}\mathcal{A}\right)-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\zeta_{\mu\nu}\mathcal{A}^{\nu}+\Sigma_{\alpha\mu}\Omega^{\mu}\,,$
$\displaystyle
N_{\alpha}{}^{\mu}\dot{\Sigma}_{\mu}-\frac{1}{2}N_{\alpha}{}^{\mu}\widehat{\mathcal{A}}_{\mu}$
$\displaystyle=\frac{1}{2}\delta_{\alpha}\mathcal{A}+\left(\mathcal{A}-\frac{1}{4}\phi\right)\mathcal{A}_{\alpha}-\left(\frac{2}{3}\theta+\frac{1}{2}\Sigma\right)\Sigma_{\alpha}+\frac{1}{2}\mathcal{A}a_{\alpha}-\frac{3}{2}\Sigma\alpha_{\alpha}$
(258)
$\displaystyle-\Omega\Omega_{\alpha}-\frac{1}{2}\left(\zeta_{\alpha\mu}+\varepsilon_{\alpha\mu}\xi\right)\mathcal{A}^{\mu}+\Sigma_{\alpha\mu}\left(\alpha^{\mu}-\Sigma^{\mu}\right)-\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\,,$

$\displaystyle
N_{\alpha}{}^{\mu}\widehat{\Sigma}_{\mu}-\varepsilon_{\alpha}{}^{\mu}\widehat{\Omega}_{\mu}$
$\displaystyle=\frac{1}{2}\delta_{\alpha}\Sigma+\frac{2}{3}\delta_{\alpha}\theta-\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\Omega-\frac{3}{2}\phi\Sigma_{\alpha}+\varepsilon_{\alpha}{}^{\mu}\Sigma_{\mu}\xi-\Omega_{\alpha}\xi+\left(\frac{1}{2}\phi+2\mathcal{A}\right)\varepsilon_{\alpha}{}^{\mu}\Omega_{\mu}$
(259) $\displaystyle-\frac{3}{2}\Sigma
a_{\alpha}+\varepsilon_{\alpha}{}^{\mu}\Omega\left(a_{\mu}-2\mathcal{A}_{\mu}\right)-\delta_{\mu}\Sigma_{\alpha}{}^{\mu}-\zeta_{\alpha}{}^{\mu}\Sigma_{\mu}+\Sigma_{\alpha}{}^{\mu}a_{\mu}+\varepsilon_{\alpha\mu}\zeta^{\mu\nu}\Omega_{\nu}-Q_{\alpha}\,;$

Tensor equations:

$\displaystyle N^{\mu}{}_{\\{\alpha}N_{\beta\\}}{}^{\nu}\dot{\Sigma}_{\mu\nu}$
$\displaystyle=\delta_{\\{\alpha}\mathcal{A}_{\beta\\}}+\mathcal{A}_{\\{\alpha}\mathcal{A}_{\beta\\}}-\Sigma_{\\{\alpha}\Sigma_{\beta\\}}-2\Sigma_{\\{\alpha}\alpha_{\beta\\}}-\Omega_{\\{\alpha}\Omega_{\beta\\}}+\mathcal{A}\zeta_{\alpha\beta}$
(260)
$\displaystyle-\left(\frac{2}{3}\theta-\Sigma\right)\Sigma_{\alpha\beta}-\Sigma_{\\{\alpha|}{}^{\mu}\Sigma_{\mu|\beta\\}}-\mathcal{E}_{\alpha\beta}+\frac{1}{2}\Pi_{\alpha\beta}\,,$
$\displaystyle
N^{\mu}{}_{\\{\alpha}N_{\beta\\}}{}^{\nu}\widehat{\Sigma}_{\mu\nu}$
$\displaystyle=\delta_{\\{\alpha}\Sigma_{\beta\\}}+\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\Omega_{|\beta\\}}+\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{|\beta\\}}\Omega_{\mu}-\frac{1}{2}\Sigma_{\alpha\beta}\phi+\varepsilon^{\mu}{}_{\\{\alpha}\Sigma_{\beta\\}\mu}\xi+\frac{3}{2}\Sigma\zeta_{\alpha\beta}$
(261)
$\displaystyle-\varepsilon_{\mu\\{\alpha}\zeta_{\beta\\}}{}^{\mu}\Omega-2a_{\\{\alpha}\Sigma_{\beta\\}}+\varepsilon_{\\{\alpha|}{}^{\mu}\Omega_{\mu}\mathcal{A}_{|\beta\\}}+\varepsilon_{\\{\alpha|}{}^{\mu}\Omega_{|\beta\\}}\mathcal{A}_{\mu}-\Sigma_{\mu\\{\alpha}\zeta_{\beta\\}}{}^{\mu}-\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}{}^{\mu}\,;$

Constraint equations:

$\delta_{\mu}\Omega^{\mu}+\varepsilon^{\mu\nu}\delta_{\mu}\Sigma_{\nu}=\left(2\mathcal{A}-\phi\right)\Omega-3\Sigma\xi+\mathcal{H}+\varepsilon^{\mu\nu}\zeta_{\mu}{}^{\gamma}\Sigma_{\gamma\nu}\,,$
(262)
$\displaystyle\delta_{\alpha}\Sigma-\frac{2}{3}\delta_{\alpha}\theta+2\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\Omega+2\delta_{\mu}\Sigma_{\alpha}{}^{\mu}$
$\displaystyle=\phi\left(\varepsilon_{\alpha}{}^{\mu}\Omega_{\mu}-\Sigma_{\alpha}\right)-2\xi\left(\Omega_{\alpha}-3\varepsilon_{\alpha}{}^{\mu}\Sigma_{\mu}\right)-4\varepsilon_{\alpha}{}^{\mu}\Omega\mathcal{A}_{\mu}$
(263)
$\displaystyle+2\zeta_{\alpha\mu}\Sigma^{\mu}+2\varepsilon_{\alpha}{}^{\mu}\Omega^{\nu}\zeta_{\mu\nu}-2\varepsilon_{\alpha}{}^{\mu}\mathcal{H}_{\mu}-Q_{\alpha}\,.$

###  A.2 Equations for the kinematical quantities associated with the
spacelike congruence

We also find the following evolution and propagation equations for the
kinematical quantities associated with the $e$ congruence.

Scalar equations:

$\displaystyle\dot{\phi}$
$\displaystyle=Q+\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)\left(2\mathcal{A}-\phi\right)+2\Omega\xi+\delta_{\mu}\alpha^{\mu}-\alpha_{\mu}\left(a^{\mu}-\mathcal{A}^{\mu}\right)$
(264)
$\displaystyle-\left(\mathcal{A}^{\nu}+a^{\nu}\right)\left(\Sigma_{\nu}+\varepsilon_{\mu\nu}\Omega^{\mu}\right)-\Sigma^{\mu\nu}\zeta_{\mu\nu}\,,$

$\displaystyle\widehat{\phi}$
$\displaystyle=\left(\frac{1}{3}\theta+\Sigma\right)\left(\frac{2}{3}\theta-\Sigma\right)-\frac{1}{2}\phi^{2}+2\xi^{2}-\frac{2}{3}\left(\mu+\Lambda\right)-\frac{1}{2}\Pi-\mathcal{E}$
(265)
$\displaystyle+\delta_{\mu}a^{\mu}-a_{\mu}a^{\mu}-\Sigma_{\mu}\Sigma^{\mu}+\Omega_{\mu}\Omega^{\mu}+2\varepsilon_{\mu\nu}\alpha^{\mu}\Omega^{\nu}-\zeta_{\mu\nu}\zeta^{\mu\nu}\,,$
$\displaystyle\dot{\xi}$
$\displaystyle=\left(\frac{1}{2}\Sigma-\frac{1}{3}\theta\right)\xi+\frac{1}{2}\mathcal{H}+\Omega\left(\mathcal{A}-\frac{1}{2}\phi\right)+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\alpha_{\nu}$
(266)
$\displaystyle+\frac{1}{2}\left(\mathcal{A}_{\mu}+a_{\mu}\right)\left[\Omega^{\mu}+\varepsilon^{\mu\nu}\left(\Sigma_{\nu}+\alpha_{\nu}\right)\right]-\frac{1}{2}\varepsilon^{\mu\nu}\Sigma_{\mu\gamma}\zeta^{\gamma}{}_{\nu}\,,$

$\widehat{\xi}=\left(\frac{1}{3}\theta+\Sigma\right)\Omega-\phi\xi+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}a_{\nu}+\Omega^{\mu}\left(\alpha_{\mu}+\Sigma_{\mu}\right)-\frac{1}{2}\varepsilon^{\mu\nu}\zeta_{\mu\gamma}\zeta^{\gamma}{}_{\nu}\,;$
(267)

Vector equations:

$\displaystyle
N^{\mu}{}_{\alpha}\widehat{\alpha}_{\mu}-N^{\mu}{}_{\alpha}\dot{a}_{\mu}$
$\displaystyle=\varepsilon_{\mu\alpha}\mathcal{H}^{\mu}-\left(\mathcal{A}+\frac{1}{2}\phi\right)\alpha_{\alpha}+\varepsilon_{\alpha\mu}\alpha^{\mu}\xi+\left(\frac{1}{3}\theta+\Sigma\right)\left(\mathcal{A}_{\alpha}+a_{\alpha}\right)$
(268)
$\displaystyle+\left(\frac{1}{2}\phi-\mathcal{A}\right)\left(\Sigma_{\alpha}+\varepsilon_{\alpha\mu}\Omega^{\mu}\right)+\left(\Omega_{\alpha}-\varepsilon_{\alpha\mu}\Sigma^{\mu}\right)\xi$
$\displaystyle+\left(\Sigma^{\mu}+\varepsilon^{\mu\nu}\Omega_{\nu}-\alpha^{\mu}\right)\zeta_{\mu\alpha}+\frac{1}{2}Q_{\alpha}\,;$

Tensor equations:

$\displaystyle N^{\mu}{}_{\\{\alpha}N_{\beta\\}}{}^{\nu}\dot{\zeta}_{\mu\nu}$
$\displaystyle=\left(\frac{1}{2}\Sigma-\frac{1}{3}\theta\right)\zeta_{\alpha\beta}+\varepsilon_{\mu\\{\alpha|}\zeta^{\mu}{}_{|\beta\\}}\Omega+\left(\mathcal{A}-\frac{1}{2}\phi\right)\Sigma_{\alpha\beta}-\Sigma_{\\{\alpha|\mu}\left(\zeta^{\mu}{}_{|\beta\\}}+\varepsilon^{\mu}{}_{|\beta\\}}\xi\right)$
(269)
$\displaystyle+\delta_{\\{\alpha}\alpha_{\beta\\}}+\left(\mathcal{A}_{\\{\alpha}-a_{\\{\alpha}\right)\alpha_{\beta\\}}-\left(\Sigma_{\\{\alpha}-\varepsilon_{\\{\alpha|\mu}\Omega^{\mu}\right)\left(\mathcal{A}_{\beta\\}}+a_{\beta\\}}\right)-\varepsilon^{\mu}{}_{\\{\alpha}\mathcal{H}_{\beta\\}\mu}\,,$
$\displaystyle
N^{\mu}{}_{\\{\alpha}N_{\beta\\}}{}^{\nu}\widehat{\zeta}_{\mu\nu}$
$\displaystyle=\delta_{\\{\alpha}a_{\beta\\}}-\phi\zeta_{\alpha\beta}-\zeta_{\\{\alpha|\mu}\zeta^{\mu}{}_{|\beta\\}}-a_{\\{\alpha}a_{\beta\\}}-\Sigma_{\\{\alpha}\Sigma_{\beta\\}}-\Omega_{\\{\alpha}\Omega_{\beta\\}}$
(270)
$\displaystyle+2\alpha_{\\{\alpha}\varepsilon_{\beta\\}\mu}\Omega^{\mu}+\left(\frac{1}{3}\theta+\Sigma\right)\Sigma_{\alpha\beta}-\mathcal{E}_{\alpha\beta}-\frac{1}{2}\Pi_{\alpha\beta}\,;$

Constraint equation:

$\displaystyle\frac{1}{2}\delta_{\alpha}\phi-\delta_{\mu}\zeta_{\alpha}{}^{\mu}-\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\xi=$
$\displaystyle-2\varepsilon_{\alpha\mu}a^{\mu}\xi-\Omega\left(\Omega_{\alpha}+\varepsilon_{\alpha\mu}\Sigma^{\mu}-2\varepsilon_{\alpha\mu}\alpha^{\mu}\right)+\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}\right)$
(271)
$\displaystyle-\mathcal{E}_{\alpha}-\frac{1}{2}\Pi_{\alpha}-\Sigma_{\alpha}{}^{\mu}\left(\Sigma_{\mu}-\varepsilon_{\mu\nu}\Omega^{\nu}\right)\,.$

###  A.3 Electric and magnetic parts of the Weyl tensor

Moreover, we have the following evolution and propagation equations the
electric and magnetic part of the Weyl tensor.

Scalar equations:

$\displaystyle\dot{\mathcal{E}}+\frac{1}{2}\dot{\Pi}+\frac{1}{3}\widehat{Q}$
$\displaystyle=3\mathcal{H}\xi+\mathcal{E}\left(\frac{3}{2}\Sigma-\theta\right)-\frac{1}{2}\Pi\left(\frac{1}{3}\theta+\frac{1}{2}\Sigma\right)+\frac{1}{3}Q\left(\frac{1}{2}\phi-2\mathcal{A}\right)-\frac{1}{2}\left(\mu+p\right)\Sigma$
(272)
$\displaystyle+\frac{1}{6}\delta_{\mu}Q^{\mu}+\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{H}_{\nu}+\frac{1}{3}Q_{\mu}\left(a^{\mu}+\mathcal{A}^{\mu}\right)+2\varepsilon_{\mu\nu}\mathcal{A}^{\mu}\mathcal{H}^{\nu}$
$\displaystyle+\mathcal{E}{}^{\mu}\left(\Sigma_{\mu}+2\alpha_{\mu}-\varepsilon_{\mu\nu}\Omega^{\nu}\right)-\frac{1}{2}\left(\frac{1}{3}\Sigma_{\mu}-2\alpha_{\mu}+\varepsilon_{\mu\nu}\Omega^{\nu}\right)\Pi^{\mu}$
$\displaystyle-\left(\mathcal{E}_{\mu\nu}-\frac{1}{6}\Pi_{\mu\nu}\right)\Sigma^{\mu\nu}+\varepsilon_{\mu\nu}\mathcal{H}_{\alpha}{}^{\mu}\zeta^{\nu\alpha}\,,$
$\displaystyle\widehat{\mathcal{E}}-\frac{1}{3}\widehat{\mu}+\frac{1}{2}\widehat{\Pi}=$
$\displaystyle-\delta_{\mu}\mathcal{E}^{\mu}-\frac{1}{2}\delta_{\mu}\Pi^{\mu}-\frac{3}{2}\phi\left(\mathcal{E}+\frac{1}{2}\Pi\right)+\left(\frac{1}{2}\Sigma-\frac{1}{3}\theta\right)Q+3\mathcal{H}\Omega$
(273)
$\displaystyle+2\left(\mathcal{E}_{\mu}+\frac{1}{2}\Pi_{\mu}\right)a^{\mu}+\frac{1}{2}\Sigma_{\mu}Q^{\mu}+3\mathcal{H}_{\mu}\Omega^{\mu}+\frac{3}{2}\varepsilon_{\mu\nu}Q^{\mu}\Omega^{\nu}$
$\displaystyle+\varepsilon^{\mu\nu}\Sigma_{\mu}\mathcal{H}_{\nu}+\varepsilon^{\mu\nu}\Sigma_{\mu\gamma}\mathcal{H}_{\nu}{}^{\gamma}+\left(\mathcal{E}^{\mu\nu}+\frac{1}{2}\Pi^{\mu\nu}\right)\zeta_{\mu\nu}\,,$

$\displaystyle\dot{\mathcal{H}}$
$\displaystyle=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\Pi_{\nu}-\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{E}_{\nu}+\Omega
Q-3\left(\mathcal{E}-\frac{1}{2}\Pi\right)\xi-\left(\theta-\frac{3}{2}\Sigma\right)\mathcal{H}$
(274)
$\displaystyle-2\varepsilon^{\mu\nu}\mathcal{A}_{\mu}\mathcal{E}_{\nu}+\mathcal{H}^{\mu}\left(2\alpha_{\mu}+\Sigma_{\mu}-\varepsilon_{\mu\nu}\Omega^{\nu}\right)-\mathcal{H}^{\mu\nu}\Sigma_{\mu\nu}$
$\displaystyle-\frac{1}{2}\left(\Omega_{\mu}+\varepsilon_{\mu\nu}\Sigma^{\nu}\right)Q^{\mu}+\varepsilon^{\mu\nu}\mathcal{E}_{\nu\gamma}\zeta_{\mu}{}^{\gamma}+\frac{1}{2}\varepsilon_{\mu}{}^{\nu}\Pi^{\gamma\mu}\zeta_{\nu\gamma}\,,$
$\displaystyle\widehat{\mathcal{H}}=$
$\displaystyle-\delta_{\mu}\mathcal{H}^{\mu}-\frac{1}{2}\varepsilon_{\mu\nu}\delta^{\mu}Q^{\nu}-\frac{3}{2}\mathcal{H}\phi-\left(3\mathcal{E}+\mu+p-\frac{1}{2}\Pi\right)\Omega-Q\xi$
(275)
$\displaystyle+2\mathcal{H}^{\mu}a_{\mu}-\left(3\mathcal{E}^{\mu}-\frac{1}{2}\Pi^{\mu}\right)\Omega_{\mu}+\varepsilon_{\mu\nu}\Sigma^{\nu}\left(\mathcal{E}^{\mu}+\frac{1}{2}\Pi^{\mu}\right)$
$\displaystyle-\varepsilon_{\nu\gamma}\Sigma^{\nu\mu}\left(\mathcal{E}_{\mu}{}^{\gamma}+\frac{1}{2}\Pi_{\mu}{}^{\gamma}\right)+\mathcal{H}^{\mu\nu}\zeta_{\mu\nu}\,;$

Vector equations:

$\displaystyle
N_{\alpha}{}^{\mu}\dot{\mathcal{E}}_{\mu}+\frac{1}{2}N_{\alpha}{}^{\mu}\dot{\Pi}_{\mu}$
$\displaystyle=\left(\frac{1}{2}\phi-\mathcal{A}\right)\left(\frac{1}{2}Q_{\alpha}+\varepsilon_{\alpha\mu}\mathcal{H}^{\mu}\right)-\frac{1}{2}Q\mathcal{A}_{\alpha}+\frac{3}{2}\mathcal{E}_{\alpha}\Sigma+3\mathcal{H}_{\alpha}\xi-\frac{3}{2}\left(\mathcal{E}+\frac{1}{2}\Pi\right)\alpha_{\alpha}$
(276)
$\displaystyle+\varepsilon_{\alpha\mu}\Omega\left(\mathcal{E}^{\mu}-\frac{1}{2}\Pi^{\mu}\right)-\left(\mathcal{E}_{\alpha}+\frac{1}{6}\Pi_{\alpha}\right)\theta+\frac{3}{2}\varepsilon_{\alpha\mu}\mathcal{A}^{\mu}\mathcal{H}+\frac{1}{4}\Sigma\Pi_{\alpha}$
$\displaystyle-\frac{1}{2}\left(\mu+p+\Pi\right)\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}\right)+\frac{1}{2}\left(\zeta_{\alpha\mu}+\varepsilon_{\alpha\mu}\xi\right)Q^{\mu}$
$\displaystyle+\mathcal{E}_{\alpha\mu}\left(2\Sigma^{\mu}+\alpha^{\mu}-2\varepsilon^{\mu\nu}\Omega_{\nu}\right)+\Sigma_{\alpha\mu}\left(\mathcal{E}^{\mu}-\frac{1}{2}\Pi^{\mu}\right)$
$\displaystyle-\varepsilon^{\mu\nu}\mathcal{H}_{\alpha\mu}\mathcal{A}_{\nu}-\varepsilon^{\mu\nu}\mathcal{H}_{\mu}\zeta_{\nu\alpha}+\frac{1}{2}\Pi_{\alpha\mu}\alpha^{\mu}$
$\displaystyle-\frac{1}{2}\delta_{\alpha}Q+\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\mathcal{H}+\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{H}_{\nu\alpha}\,,$

$\displaystyle
N^{\mu}{}_{\alpha}\widehat{\mathcal{E}}_{\mu}+\frac{1}{2}N^{\mu}{}_{\alpha}\widehat{\Pi}_{\mu}$
$\displaystyle=\frac{1}{2}Q\Sigma_{\alpha}+3\mathcal{H}_{\alpha}\Omega-\frac{3}{2}\mathcal{H}\Omega_{\alpha}-\left(\frac{1}{3}\theta+\frac{1}{4}\Sigma\right)Q_{\alpha}-\frac{3}{2}\left(\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\right)\phi$
(277)
$\displaystyle-\frac{3}{2}\left(\mathcal{E}+\frac{1}{2}\Pi\right)a_{\alpha}+\frac{3}{2}\varepsilon_{\alpha\mu}\left(\Sigma^{\mu}\mathcal{H}-\Sigma\mathcal{H}^{\mu}\right)+\varepsilon_{\alpha\mu}\left(\mathcal{E}^{\mu}+\frac{1}{2}\Pi^{\mu}\right)\xi$
$\displaystyle+\frac{3}{2}\varepsilon_{\alpha\mu}\left(Q^{\mu}\Omega-Q\Omega^{\mu}\right)+\frac{1}{2}\Sigma_{\alpha\mu}Q^{\mu}+\varepsilon_{\alpha\nu}\left(\mathcal{H}_{\mu}\Sigma^{\mu\nu}-\Sigma_{\mu}\mathcal{H}^{\mu\nu}\right)$
$\displaystyle+\left(\mathcal{E}_{\alpha\mu}+\frac{1}{2}\Pi_{\alpha\mu}\right)a^{\mu}-\left(\mathcal{E}^{\mu}+\frac{1}{2}\Pi^{\mu}\right)\zeta_{\mu\alpha}+3\mathcal{H}_{\alpha\mu}\Omega^{\mu}$
$\displaystyle+\frac{1}{2}\delta_{\alpha}\mathcal{E}+\frac{1}{3}\delta_{\alpha}\mu+\frac{1}{4}\delta_{\alpha}\Pi-\delta_{\mu}\mathcal{E}_{\alpha}{}^{\mu}-\frac{1}{2}\delta_{\mu}\Pi_{\alpha}{}^{\mu}\,,$

$\displaystyle\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{\mathcal{E}}_{\mu}-N_{\alpha}{}^{\mu}\dot{\mathcal{H}}_{\mu}-\frac{1}{4}\varepsilon_{\alpha}{}^{\mu}\widehat{\Pi}_{\mu}$
$\displaystyle=\frac{3}{4}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\mathcal{E}-\frac{3}{8}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\Pi+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{E}_{\nu\alpha}-\frac{1}{4}\varepsilon^{\mu\nu}\delta_{\mu}\Pi_{\nu\alpha}-\frac{3}{4}\mathcal{H}\left(\Sigma_{\alpha}-2\alpha_{\alpha}\right)$
(278)
$\displaystyle+\frac{3}{2}\varepsilon_{\alpha\mu}\mathcal{A}^{\mu}\mathcal{E}+\frac{3}{8}\varepsilon_{\alpha\mu}Q^{\mu}\Sigma-\frac{1}{4}\varepsilon_{\alpha\mu}\Sigma^{\mu}Q-\frac{3}{4}\varepsilon_{\alpha\mu}\Omega^{\mu}\mathcal{H}+\frac{5}{2}\xi\left(\mathcal{E}_{\alpha}-\frac{1}{2}\Pi_{\alpha}\right)$
$\displaystyle-\frac{3}{4}\left(Q_{\alpha}\Omega+Q\Omega_{\alpha}\right)-\frac{3}{4}\varepsilon_{\alpha\mu}a^{\mu}\left(\mathcal{E}-\frac{1}{2}\Pi\right)-\varepsilon_{\alpha\mu}\mathcal{E}^{\mu}\mathcal{A}+\mathcal{H}_{\alpha}\left(\theta-\frac{3}{4}\Sigma\right)$
$\displaystyle-\frac{1}{4}\varepsilon_{\alpha\mu}\left(\mathcal{E}^{\mu}-\frac{1}{2}\Pi^{\mu}\right)\phi+\frac{1}{2}\varepsilon_{\alpha\mu}\mathcal{H}^{\mu}\Omega+\frac{1}{2}\varepsilon_{\alpha\mu}a_{\nu}\left(\mathcal{E}^{\mu\nu}-\frac{1}{2}\Pi^{\mu\nu}\right)$
$\displaystyle-\frac{1}{2}\varepsilon^{\mu\nu}\zeta_{\nu\alpha}\left(\mathcal{E}_{\mu}-\frac{1}{2}\Pi_{\mu}\right)-\varepsilon_{\alpha\mu}\zeta^{\mu\nu}\left(\mathcal{E}_{\nu}-\frac{1}{2}\Pi_{\nu}\right)+\varepsilon^{\mu\nu}\mathcal{A}_{\mu}\mathcal{E}_{\nu\alpha}$
$\displaystyle-\frac{1}{2}\Sigma_{\alpha\mu}\left(3\mathcal{H}^{\mu}+\frac{1}{2}\varepsilon^{\mu\nu}Q_{\nu}\right)-\mathcal{H}_{\alpha\mu}\left(\frac{3}{2}\Sigma^{\mu}+\alpha^{\mu}-\frac{1}{2}\varepsilon^{\mu\nu}\Omega_{\nu}\right)\,,$

$\displaystyle
N^{\mu}{}_{\alpha}\widehat{\mathcal{H}}_{\mu}-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{Q}_{\mu}$
$\displaystyle=\frac{1}{2}\delta_{\alpha}\mathcal{H}-\delta_{\mu}\mathcal{H}_{\alpha}{}^{\mu}-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}Q-\frac{3}{2}\varepsilon_{\alpha\mu}\Sigma^{\mu}\left(\mathcal{E}+\frac{1}{2}\Pi\right)-\frac{3}{2}\mathcal{H}a_{\alpha}$
(279)
$\displaystyle+\frac{1}{2}\varepsilon_{\alpha\mu}Qa^{\mu}-\Omega\left(3\mathcal{E}_{\alpha}-\frac{1}{2}\Pi_{\alpha}\right)+\frac{3}{2}\varepsilon_{\alpha\mu}\Sigma\left(\mathcal{E}^{\mu}+\frac{1}{2}\Pi^{\mu}\right)$
$\displaystyle+\left(\varepsilon_{\alpha\mu}\mathcal{H}^{\mu}-\frac{1}{2}Q_{\alpha}\right)\xi-\Omega_{\alpha}\left(\mu+p+\frac{1}{4}\Pi-\frac{3}{2}\mathcal{E}\right)$
$\displaystyle-\frac{3}{2}\left(\mathcal{H}_{\alpha}-\frac{1}{6}\varepsilon_{\alpha\mu}Q^{\mu}\right)\phi-\Omega^{\mu}\left(3\mathcal{E}_{\alpha\mu}-\frac{1}{2}\Pi_{\alpha\mu}\right)$
$\displaystyle+\mathcal{H}_{\alpha\mu}a^{\mu}-\mathcal{H}^{\mu}\zeta_{\alpha\mu}+\varepsilon_{\alpha\nu}\Sigma_{\mu}\left(\mathcal{E}^{\mu\nu}+\frac{1}{2}\Pi^{\mu\nu}\right)$
$\displaystyle-\varepsilon_{\alpha\nu}\Sigma_{\mu}{}^{\nu}\left(\mathcal{E}^{\mu}+\frac{1}{2}\Pi^{\mu}\right)+\frac{1}{2}\varepsilon_{\alpha\mu}Q_{\nu}\zeta^{\mu\nu}\,;$

Tensor equations:

$\displaystyle
N^{\mu}{}_{\\{\alpha}N_{\beta\\}}{}^{\nu}\dot{\mathcal{E}}_{\mu\nu}+\frac{1}{2}N^{\mu}{}_{\\{\alpha}N_{\beta\\}}{}^{\nu}\dot{\Pi}_{\mu\nu}+N^{\mu}{}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\nu}\widehat{\mathcal{H}}_{\mu\nu}$
$\displaystyle=\varepsilon_{\\{\alpha|}{}^{\nu}\delta_{\nu}\mathcal{H}_{|\beta\\}}-\frac{1}{2}\delta_{\\{\alpha}Q_{\beta\\}}-\frac{1}{2}\left(\mu+p+3\mathcal{E}-\frac{1}{2}\Pi\right)\Sigma_{\alpha\beta}$
(280)
$\displaystyle-\frac{1}{2}Q\zeta_{\alpha\beta}-\frac{3}{2}\mathcal{H}\varepsilon^{\nu}{}_{\\{\alpha}\zeta_{\beta\\}\nu}-\left(\theta+\frac{3}{2}\Sigma\right)\mathcal{E}_{\alpha\beta}+\mathcal{H}_{\alpha\beta}\xi$
$\displaystyle+\varepsilon_{\mu\\{\alpha}\mathcal{E}_{\beta\\}}{}^{\mu}\Omega-\left(\frac{1}{6}\theta-\frac{1}{4}\Sigma\right)\Pi_{\alpha\beta}+\frac{1}{2}\varepsilon^{\mu}{}_{\\{\alpha}\Pi_{\beta\\}\mu}\Omega$
$\displaystyle+\left(\frac{1}{2}\phi+2\mathcal{A}\right)\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}{}^{\mu}+\left(3\mathcal{E}_{\\{\alpha}-\frac{1}{2}\Pi_{\\{\alpha}\right)\Sigma_{\beta\\}}$
$\displaystyle-\mathcal{A}_{\\{\alpha}Q_{\beta\\}}-\left(2\mathcal{E}_{\\{\alpha}+\Pi_{\\{\alpha}\right)\left(\alpha_{\beta\\}}-\frac{1}{2}\varepsilon_{\beta\\}}{}^{\mu}\Omega_{\mu}\right)$
$\displaystyle+\mathcal{H}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\mu}\left(2\mathcal{A}_{\mu}-a_{\mu}\right)+\mathcal{H}_{\mu}\varepsilon^{\mu}{}_{\\{\alpha}a_{\beta\\}}$
$\displaystyle-\mathcal{H}^{\mu}{}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\nu}\zeta_{\mu\nu}\,,$

$\displaystyle
N^{\mu}{}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\nu}\widehat{\mathcal{E}}_{\mu\nu}-\frac{1}{2}N^{\mu}{}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\nu}\widehat{\Pi}_{\mu\nu}-\dot{\mathcal{H}}_{\left\\{\alpha\beta\right\\}}$
$\displaystyle=\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\mathcal{E}_{|\beta\\}}-\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\Pi_{|\beta\\}}+\xi\left(\mathcal{E}_{\alpha\beta}-\frac{1}{2}\Pi_{\alpha\beta}\right)$
(281)
$\displaystyle+\frac{3}{2}\mathcal{H}\Sigma_{\alpha\beta}+\frac{1}{2}Q\varepsilon_{\mu\\{\alpha}\Sigma_{\beta\\}}{}^{\mu}+\left(\theta+\frac{3}{2}\Sigma\right)\mathcal{H}_{\alpha\beta}$
$\displaystyle-\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}{}^{\mu}\Omega-\frac{3}{2}Q_{\\{\alpha}\Omega_{\beta\\}}-\frac{1}{4}\phi\varepsilon_{\mu\\{\alpha}\Pi_{\beta\\}}{}^{\mu}$
$\displaystyle-\frac{1}{2}a^{\mu}\varepsilon_{\mu\\{\alpha}\Pi_{\beta\\}}-\left(3\mathcal{H}_{\\{\alpha}+\frac{1}{2}\varepsilon_{\mu\\{\alpha}Q^{\mu}\right)\Sigma_{\beta\\}}$
$\displaystyle+\Omega^{\mu}\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}-\frac{3}{2}\left(\mathcal{E}-\frac{1}{2}\Pi\right)\varepsilon_{\mu\\{\alpha}\zeta_{\beta\\}}{}^{\mu}$
$\displaystyle+\varepsilon_{\mu\\{\alpha}a_{\beta\\}}\left(\mathcal{E}^{\mu}-\frac{1}{2}\Pi^{\mu}\right)-3\mathcal{H}_{\mu\\{\alpha}\Sigma_{\beta\\}}{}^{\mu}$
$\displaystyle+2\mathcal{H}_{\\{\alpha}\alpha_{\beta\\}}+\left(\frac{1}{2}\phi+2\mathcal{A}\right)\varepsilon_{\mu\\{\alpha}\mathcal{E}_{\beta\\}}{}^{\mu}$
$\displaystyle+\zeta_{\mu\nu}\varepsilon^{\mu}{}_{\\{\alpha}\left(\mathcal{E}^{\nu}{}_{\beta\\}}-\frac{1}{2}\Pi^{\nu}{}_{\beta\\}}\right)$
$\displaystyle-\varepsilon_{\mu\\{\alpha}\mathcal{E}_{\beta\\}}\left(2\mathcal{A}^{\mu}-a^{\mu}\right)\,,$

where we have used the fact that for any two symmetric and traceless 2-tensors
on the sheet, $A$ and $B$, that is
$A_{\mu\nu}=N_{\\{\mu}{}^{\alpha}N_{\nu\\}}{}^{\beta}A_{\alpha\beta}$ and
$B_{\mu\nu}=N_{\\{\mu}{}^{\alpha}N_{\nu\\}}{}^{\beta}B_{\alpha\beta}$, the
contraction $A^{\delta}{}_{\\{a}B_{b\\}\delta}=0$.

###  A.4 Matter conservation laws

Lastly, the energy and momentum conservation equations are given by

$\dot{\mu}+\widehat{Q}=Q^{\mu}\left(a_{\mu}-2\mathcal{A}_{\mu}\right)-2\Pi^{\mu}\Sigma_{\mu}-\left(\phi+2\mathcal{A}\right)Q-\frac{3}{2}\Pi\Sigma-\theta\left(\mu+p\right)-\delta_{\mu}Q^{\mu}-\Pi^{\mu\nu}\Sigma_{\mu\nu}\,,$
(282) $\displaystyle\dot{Q}+\widehat{p}+\hat{\Pi}=$
$\displaystyle-\delta_{\mu}\Pi^{\mu}-\left(\frac{3}{2}\phi+\mathcal{A}\right)\Pi-\left(\frac{4}{3}\theta+\Sigma\right)Q-\left(\mu+p\right)\mathcal{A}$
(283)
$\displaystyle+\left(\alpha_{\mu}-\Sigma_{\mu}+\varepsilon_{\mu\nu}\Omega^{\nu}\right)Q^{\mu}+\Pi^{\mu}\left(2a_{\mu}-\mathcal{A}_{\mu}\right)+\Pi_{\mu\nu}\zeta^{\mu\nu}\,,$
$\displaystyle
N_{\alpha}{}^{\mu}\dot{Q}_{\mu}+N_{\alpha}{}^{\mu}\widehat{\Pi}_{\mu}=$
$\displaystyle-\delta_{\alpha}p+\frac{1}{2}\delta_{\alpha}\Pi-\delta_{\mu}\Pi_{\alpha}{}^{\mu}-Q\left(\alpha_{\alpha}+\Sigma_{\alpha}+\varepsilon_{\alpha\beta}\Omega^{\beta}\right)+\varepsilon_{\alpha\mu}Q^{\mu}\Omega$
(284) $\displaystyle-\frac{3}{2}\Pi
a_{\alpha}-\left(\frac{4}{3}\theta-\frac{1}{2}\Sigma\right)Q_{\alpha}-\left(\frac{3}{2}\phi+\mathcal{A}\right)\Pi_{\alpha}-\Pi^{\mu}\left(\zeta_{\alpha\mu}-\varepsilon_{\alpha\mu}\xi\right)$
$\displaystyle-\left(\mu+p-\frac{1}{2}\Pi\right)\mathcal{A}_{\alpha}-Q^{\mu}\Sigma_{\alpha\mu}+\Pi_{\alpha\mu}\left(a^{\mu}-\mathcal{A}^{\mu}\right)\,.$

##  Appendix B Linearized 1+1+2 equations for the angular gradients variables

In this appendix, we present a set of covariant, gauge invariant equations for
the linear order perturbations of a background spacetime assumed to be static
LRS II and permeated by a general matter fluid, such that the equilibrium
configuration is characterized by the scalars
$\left\\{\phi_{0},\mathcal{A}_{0},\mathcal{E}_{0},\mu_{0},p_{0},\Pi_{0},\Lambda\right\\}$.
Considering the variables in Eq. (55) and using the commutation relations
(47), we find the following set of linearized equations for the various
quantities that characterize the perturbed spacetime.

###  B.1 Linearized equations for the kinematical quantities associated with
the timelike congruence

  * •

Evolution and propagation equations for scalars and gradients of scalar
quantities:

$\displaystyle\left(\delta_{\alpha}\theta\right)^{\cdot}-\widehat{\mathbb{A}}_{\alpha}$
$\displaystyle=\left[\frac{1}{2}\left(\mu_{0}+3p_{0}\right)-\Lambda-\mathcal{A}_{0}\left(\mathcal{A}_{0}+\phi_{0}\right)\right]a_{\alpha}-\frac{1}{2}\left(\mathfrak{m}_{\alpha}+3\mathfrak{p}_{\alpha}\right)$
(285)
$\displaystyle+\mathcal{A}_{0}\mathbb{F}_{\alpha}+\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\mathbb{A}_{\alpha}+\delta_{\alpha}\left(\delta_{\mu}\mathcal{A}^{\mu}\right)\,,$
$\displaystyle\frac{2}{3}\left(\delta_{\alpha}\theta\right)^{\cdot}-\left(\delta_{\alpha}\Sigma\right)^{\cdot}$
$\displaystyle=\phi_{0}\mathbb{A}_{\alpha}+\mathcal{A}_{0}\left(\mathbb{F}_{\alpha}+\phi_{0}\mathcal{A}_{\alpha}\right)-\left(\frac{1}{3}\mu_{0}+p_{0}-\frac{2}{3}\Lambda+\frac{1}{2}\Pi_{0}-\mathcal{E}_{0}\right)\mathcal{A}_{\alpha}$
(286)
$\displaystyle-\frac{1}{3}\mathfrak{m}_{\alpha}-\mathfrak{p}_{\alpha}-\frac{1}{2}\mathbb{P}_{\alpha}+\mathbb{E}_{\alpha}+\delta_{\alpha}\left(\delta_{\mu}\mathcal{A}^{\mu}\right)\,,$
$\displaystyle\frac{3}{2}\widehat{\delta_{\alpha}\Sigma}-\widehat{\delta_{\alpha}\theta}$
$\displaystyle=\left(\mu_{0}+3p_{0}-2\Lambda+\frac{3}{2}\Pi_{0}-3\mathcal{E}_{0}-3\mathcal{A}_{0}\phi_{0}\right)\varepsilon_{\alpha\mu}\Omega^{\mu}$
(287)
$\displaystyle+\frac{1}{2}\phi_{0}\left(\delta_{\alpha}\theta-6\delta_{\alpha}\Sigma\right)-\frac{3}{2}\delta_{\alpha}Q-\frac{3}{2}\delta_{\alpha}\left(\delta_{\mu}\Sigma^{\mu}\right)-\frac{3}{2}\varepsilon_{\mu\nu}\delta_{\alpha}\left(\delta^{\mu}\Omega^{\nu}\right)\,,$

$\dot{\Omega}=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{A}_{\nu}+\mathcal{A}_{0}\xi\,,$
(288)

$\hat{\Omega}=\left(\mathcal{A}_{0}-\phi_{0}\right)\Omega-\delta_{\mu}\Omega^{\mu}\,;$
(289)

  * •

Evolution and propagation equations for vector quantities:

$\dot{\Omega}_{\alpha}+\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{\mathcal{A}}_{\mu}=-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\left(\frac{1}{2}\phi_{0}\mathcal{A}_{\mu}+\mathcal{A}_{0}a_{\mu}-\mathbb{A}_{\mu}\right)\,,$
(290)
$\displaystyle\dot{\Sigma}_{\alpha}-\frac{1}{2}\widehat{\mathcal{A}}_{\alpha}=$
$\displaystyle\left(\mathcal{A}_{0}-\frac{1}{4}\phi_{0}\right)\mathcal{A}_{\alpha}+\frac{1}{2}\mathcal{A}_{0}a_{\alpha}+\frac{1}{2}\mathbb{A}_{\alpha}-\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\,,$
(291)

$\displaystyle\widehat{\Sigma}_{\alpha}-\varepsilon_{\alpha}{}^{\mu}\widehat{\Omega}_{\mu}$
$\displaystyle=\frac{1}{2}\delta_{\alpha}\Sigma+\frac{2}{3}\delta_{\alpha}\theta-\frac{3}{2}\phi_{0}\Sigma_{\alpha}+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\varepsilon_{\alpha}{}^{\mu}\Omega_{\mu}$
(292)
$\displaystyle-\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\Omega-\delta_{\mu}\Sigma_{\alpha}{}^{\mu}-Q_{\alpha}\,;$

  * •

Evolution and propagation equations for tensor quantities:

$\dot{\Sigma}_{\alpha\beta}=\delta_{\\{\alpha}\mathcal{A}_{\beta\\}}+\mathcal{A}_{0}\zeta_{\alpha\beta}-\mathcal{E}_{\alpha\beta}+\frac{1}{2}\Pi_{\alpha\beta}\,,$
(293)
$\widehat{\Sigma}_{\alpha\beta}=\delta_{\\{\alpha}\Sigma_{\beta\\}}+\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\Omega_{|\beta\\}}+\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{|\beta\\}}\Omega_{\mu}-\frac{1}{2}\phi_{0}\Sigma_{\alpha\beta}-\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}{}^{\mu}\,;$
(294)

  * •

Constraint equations:

$\varepsilon^{\mu\nu}\delta_{\mu}\mathbb{A}_{\nu}=-2\widehat{\mathcal{A}}_{0}\xi\,,$
(295)
$\delta_{\mu}\Omega^{\mu}+\varepsilon^{\mu\nu}\delta_{\mu}\Sigma_{\nu}=\left(2\mathcal{A}_{0}-\phi_{0}\right)\Omega+\mathcal{H}\,,$
(296)
$\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\Omega+\delta_{\mu}\Sigma_{\alpha}{}^{\mu}=\frac{1}{2}\phi_{0}\left(\varepsilon_{\alpha}{}^{\mu}\Omega_{\mu}-\Sigma_{\alpha}\right)-\varepsilon_{\alpha}{}^{\mu}\mathcal{H}_{\mu}+\frac{1}{3}\delta_{\alpha}\theta-\frac{1}{2}\delta_{\alpha}\Sigma-\frac{1}{2}Q_{\alpha}\,.$
(297)

###  B.2 Linearized equations for the kinematical quantities associated with
the spacelike congruence

  * •

Evolution and propagation equations for scalars and gradients of scalar
quantities:

$\displaystyle\dot{\mathbb{F}}_{\alpha}$
$\displaystyle=\left[\frac{1}{2}\phi_{0}^{2}+\frac{2}{3}\left(\mu_{0}+\Lambda\right)+\frac{1}{2}\Pi_{0}+\mathcal{E}_{0}\right]\left(\Sigma_{\alpha}+\alpha_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}\right)$
(298)
$\displaystyle+\delta_{\alpha}Q+\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\left(\frac{2}{3}\delta_{\alpha}\theta-\delta_{\alpha}\Sigma\right)+\delta_{\alpha}\left(\delta_{\mu}\alpha^{\mu}\right)\,,$
$\displaystyle\widehat{\mathbb{F}}_{\alpha}$
$\displaystyle=\left[\frac{1}{2}\phi_{0}^{2}+\frac{2}{3}\left(\mu_{0}+\Lambda\right)+\frac{1}{2}\Pi_{0}+\mathcal{E}_{0}\right]a_{\alpha}+\delta_{\alpha}\left(\delta_{\mu}a^{\mu}\right)$
(299)
$\displaystyle-\frac{2}{3}\mathfrak{m}_{\alpha}-\frac{1}{2}\mathbb{P}_{\alpha}-\mathbb{E}_{\alpha}-\frac{3}{2}\phi_{0}\mathbb{F}_{\alpha}\,,$
$\dot{\xi}=\frac{1}{2}\mathcal{H}+\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Omega+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\alpha_{\nu}\,,$
(300)

$\widehat{\xi}=-\phi_{0}\xi+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}a_{\nu}\,;$
(301)

  * •

Evolution and propagation equations for vector quantities:

$\widehat{\alpha}_{\alpha}-\dot{a}_{\alpha}=\varepsilon_{\mu\alpha}\mathcal{H}^{\mu}-\left(\mathcal{A}_{0}+\frac{1}{2}\phi_{0}\right)\alpha_{\alpha}+\frac{1}{2}Q_{\alpha}+\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\Sigma_{\alpha}+\varepsilon_{\alpha\mu}\Omega^{\mu}\right)\,;$
(302)

  * •

Evolution and propagation equations for tensor quantities:

$\dot{\zeta}_{\alpha\beta}=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Sigma_{\alpha\beta}+\delta_{\\{\alpha}\alpha_{\beta\\}}-\varepsilon^{\mu}{}_{\\{\alpha}\mathcal{H}_{\beta\\}\mu}\,,$
(303)
$\widehat{\zeta}_{\alpha\beta}=\delta_{\\{\alpha}a_{\beta\\}}-\phi_{0}\zeta_{\alpha\beta}-\mathcal{E}_{\alpha\beta}-\frac{1}{2}\Pi_{\alpha\beta}\,;$
(304)

  * •

Constraint equations:

$\varepsilon^{\mu\nu}\delta_{\mu}\mathbb{F}_{\nu}=\left[\phi_{0}^{2}+\frac{4}{3}\left(\mu_{0}+\Lambda\right)+\Pi_{0}+2\mathcal{E}_{0}\right]\xi\,,$
(305)
$\delta_{\mu}\zeta_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\xi=\frac{1}{2}\mathbb{F}_{\alpha}+\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\,.$
(306)

###  B.3 Linearized equations for the Weyl tensor components and the matter
variables

  * •

Evolution and propagation equations for scalars and gradients of scalar
quantities:

$\displaystyle\widehat{\delta_{\alpha}Q}+\dot{\mathfrak{m}}_{\alpha}$
$\displaystyle=\widehat{\mu}_{0}\left(\varepsilon_{\alpha\mu}\Omega^{\mu}-\Sigma_{\alpha}-\alpha_{\alpha}\right)-\frac{3}{2}\Pi_{0}\delta_{\alpha}\Sigma-\delta_{\alpha}\left(\delta_{\mu}Q^{\mu}\right)$
(307)
$\displaystyle-\left(\mu_{0}+p_{0}\right)\delta_{\alpha}\theta-\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\delta_{\alpha}Q\,,$

$\displaystyle\frac{1}{3}\dot{\mathfrak{m}}_{\alpha}-\frac{1}{2}\dot{\mathbb{P}}_{\alpha}-\dot{\mathbb{E}}_{\alpha}$
$\displaystyle=\frac{3}{2}\phi_{0}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\varepsilon_{\alpha\mu}\Omega^{\mu}-\Sigma_{\alpha}-\alpha_{\alpha}\right)-\frac{1}{2}\phi_{0}\delta_{\alpha}Q$
(308)
$\displaystyle+\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}-3\mathcal{E}_{0}\right)\left(\frac{1}{2}\delta_{\alpha}\Sigma-\frac{1}{3}\delta_{\alpha}\theta\right)$
$\displaystyle-\frac{1}{2}\delta_{\alpha}\left(\delta_{\mu}Q^{\mu}\right)-\varepsilon^{\mu\nu}\delta_{\alpha}\left(\delta_{\mu}\mathcal{H}_{\nu}\right)\,,$

$\displaystyle\left(\delta_{\alpha}Q\right)^{\cdot}+\widehat{\mathbb{P}}_{\alpha}+\widehat{\mathfrak{p}}_{\alpha}$
$\displaystyle=\left(\mu_{0}+p_{0}\right)\left(\mathcal{A}_{0}a_{\alpha}-\mathbb{A}_{\alpha}\right)-\mathcal{A}_{0}\left(\mathfrak{m}_{\alpha}+\mathfrak{p}_{\alpha}+\mathbb{P}_{\alpha}\right)$
(309)
$\displaystyle-\frac{1}{2}\phi_{0}\left(\mathfrak{p}_{\alpha}+4\mathbb{P}_{\alpha}\right)+\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{0}a_{\alpha}$
$\displaystyle-\Pi_{0}\left(\frac{3}{2}\mathbb{F}_{\alpha}+\mathbb{A}_{\alpha}\right)-\delta_{\alpha}\left(\delta_{\mu}\Pi^{\mu}\right)\,,$

$\displaystyle\frac{3}{2}\widehat{\mathbb{P}}_{\alpha}+3\widehat{\mathbb{E}}_{\alpha}-\widehat{\mathfrak{m}}_{\alpha}=$
$\displaystyle-6\phi_{0}\left(\mathbb{E}_{\alpha}+\frac{1}{2}\mathbb{P}_{\alpha}-\frac{1}{12}\mathfrak{m}_{\alpha}\right)-\frac{9}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\mathbb{F}_{\alpha}-\phi_{0}a_{\alpha}\right)$
(310)
$\displaystyle-3\delta_{\alpha}\left(\delta_{\mu}\mathcal{E}^{\mu}\right)-\frac{3}{2}\delta_{\alpha}\left(\delta_{\mu}\Pi^{\mu}\right)\,,$

$\dot{\mathcal{H}}=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\Pi_{\nu}-\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{E}_{\nu}-3\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\xi\,,$
(311)
$\widehat{\mathcal{H}}=-\delta_{\mu}\mathcal{H}^{\mu}-\frac{1}{2}\varepsilon_{\mu\nu}\delta^{\mu}Q^{\nu}-\frac{3}{2}\phi_{0}\mathcal{H}-\left(3\mathcal{E}_{0}+\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\Omega\,;$
(312)

  * •

Evolution and propagation equations for vector quantities:

$\displaystyle\dot{\mathcal{E}}_{\alpha}+\frac{1}{2}\dot{\Pi}_{\alpha}$
$\displaystyle=\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\frac{1}{2}Q_{\alpha}+\varepsilon_{\alpha\mu}\mathcal{H}^{\mu}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\alpha_{\alpha}$
(313)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+\Pi_{0}\right)\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}\right)-\frac{1}{2}\delta_{\alpha}Q$
$\displaystyle+\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\mathcal{H}+\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{H}_{\nu\alpha}\,,$

$\displaystyle\widehat{\mathcal{E}}_{\alpha}+\frac{1}{2}\widehat{\Pi}_{\alpha}$
$\displaystyle=-\frac{3}{2}\phi_{0}\left(\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)a_{\alpha}-\frac{1}{2}\delta_{\mu}\Pi_{\alpha}{}^{\mu}$
(314)
$\displaystyle+\frac{1}{2}\mathbb{E}_{\alpha}+\frac{1}{3}\mathfrak{m}_{\alpha}+\frac{1}{4}\mathbb{P}_{\alpha}-\delta_{\mu}\mathcal{E}_{\alpha}{}^{\mu}\,,$

$\displaystyle\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{\mathcal{E}}_{\mu}-\dot{\mathcal{H}}_{\alpha}-\frac{1}{4}\varepsilon_{\alpha}{}^{\mu}\widehat{\Pi}_{\mu}$
$\displaystyle=\frac{3}{4}\varepsilon_{\alpha}{}^{\mu}\mathbb{E}_{\mu}-\frac{3}{8}\varepsilon_{\alpha}{}^{\mu}\mathbb{P}_{\mu}+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{E}_{\nu\alpha}-\frac{1}{4}\varepsilon^{\mu\nu}\delta_{\mu}\Pi_{\nu\alpha}$
(315)
$\displaystyle+\frac{3}{2}\mathcal{E}_{0}\varepsilon_{\alpha\mu}\mathcal{A}^{\mu}-\frac{3}{4}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\varepsilon_{\alpha\mu}a^{\mu}-\mathcal{A}_{0}\varepsilon_{\alpha\mu}\mathcal{E}^{\mu}$
$\displaystyle-\frac{1}{4}\phi_{0}\varepsilon_{\alpha\mu}\left(\mathcal{E}^{\mu}-\frac{1}{2}\Pi^{\mu}\right)\,,$

$\displaystyle\widehat{\mathcal{H}}_{\alpha}-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{Q}_{\mu}$
$\displaystyle=\frac{1}{2}\delta_{\alpha}\mathcal{H}-\delta_{\mu}\mathcal{H}_{\alpha}{}^{\mu}-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}Q-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\varepsilon_{\alpha\mu}\Sigma^{\mu}$
(316)
$\displaystyle-\left(\mu_{0}+p_{0}+\frac{1}{4}\Pi_{0}-\frac{3}{2}\mathcal{E}_{0}\right)\Omega_{\alpha}-\frac{3}{2}\phi_{0}\left(\mathcal{H}_{\alpha}-\frac{1}{6}\varepsilon_{\alpha\mu}Q^{\mu}\right)\,,$
$\displaystyle\dot{Q}_{\alpha}+\widehat{\Pi}_{\alpha}$
$\displaystyle=\frac{1}{2}\mathbb{P}_{\alpha}-\mathfrak{p}_{\alpha}-\frac{3}{2}\Pi_{0}a_{\alpha}-\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{\alpha}$
(317)
$\displaystyle-\delta_{\mu}\Pi_{\alpha}{}^{\mu}-\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\mathcal{A}_{\alpha}\,;$

  * •

Evolution and propagation equations for tensor quantities:

$\displaystyle\dot{\mathcal{E}}_{\alpha\beta}+\frac{1}{2}\dot{\Pi}_{\alpha\beta}+N^{\mu}{}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\nu}\widehat{\mathcal{H}}_{\mu\nu}$
$\displaystyle=\varepsilon_{\\{\alpha|}{}^{\nu}\delta_{\nu}\mathcal{H}_{|\beta\\}}-\frac{1}{2}\delta_{\\{\alpha}Q_{\beta\\}}+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}{}^{\mu}$
(318)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+3\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\Sigma_{\alpha\beta}\,,$

$\displaystyle
N^{\mu}{}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\nu}\widehat{\mathcal{E}}_{\mu\nu}-\frac{1}{2}N^{\mu}{}_{\\{\alpha}\varepsilon_{\beta\\}}{}^{\nu}\widehat{\Pi}_{\mu\nu}-\dot{\mathcal{H}}_{\alpha\beta}$
$\displaystyle=\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\mathcal{E}_{|\beta\\}}-\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\Pi_{|\beta\\}}-\frac{3}{2}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\varepsilon_{\mu\\{\alpha}\zeta_{\beta\\}}{}^{\mu}$
(319)
$\displaystyle+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\varepsilon_{\mu\\{\alpha}\mathcal{E}_{\beta\\}}{}^{\mu}-\frac{1}{4}\phi_{0}\varepsilon_{\mu\\{\alpha}\Pi_{\beta\\}}{}^{\mu}\,;$

  * •

Constraint equations:

$\frac{1}{3}\varepsilon^{\mu\nu}\delta_{\mu}\mathfrak{m}_{\nu}-\varepsilon^{\mu\nu}\delta_{\mu}\mathbb{E}_{\nu}-\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\mathbb{P}_{\nu}=-3\phi_{0}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\xi\,,$
(320)

$\varepsilon^{\mu\nu}\delta_{\mu}\mathfrak{p}_{\nu}=-2\widehat{p}_{0}\xi\,,$
(321)

##  Appendix C Linearized 1+1+2 equations for the dot-derivatives variables

In this appendix, we present a set of covariant, gauge invariant equations for
the linear order perturbations of a background spacetime assumed to be static
LRS II and permeated by a general matter fluid, such that the equilibrium
configuration is characterized by the scalars
$\left\\{\phi_{0},\mathcal{A}_{0},\mathcal{E}_{0},\mu_{0},p_{0},\Pi_{0},\Lambda\right\\}$.
Considering the variables in (56) and using the commutation relations (47), we
find the following set of linearized equations for the various quantities that
characterize the perturbed spacetime. Here, we also show that the variables in
Eq. (55) and (56) are not fully independent.

###  C.1 Linearized equations for the kinematical quantities associated with
the timelike congruence

  * •

Evolution and propagation equations for scalars and dot-derivatives of scalar
quantities:

$\widehat{\mathsf{A}}-\ddot{\theta}=\frac{1}{2}\left(\mathsf{m}+3\mathsf{p}\right)+\widehat{\mathcal{A}}_{0}\left(\frac{1}{3}\theta+\Sigma\right)-\left(3\mathcal{A}_{0}+\phi_{0}\right)\mathsf{A}-\mathcal{A}_{0}\mathsf{F}-\left(\delta_{\mu}\mathcal{A}^{\mu}\right)^{\cdot}\,,$
(322)

$\ddot{\Sigma}-\frac{2}{3}\ddot{\theta}=\frac{1}{3}\left(\mathsf{m}+3\mathsf{p}\right)+\frac{1}{2}\mathcal{P}-\mathsf{E}-\mathcal{A}_{0}\mathsf{F}-\phi_{0}\mathsf{A}-\left(\delta_{\mu}\mathcal{A}^{\mu}\right)^{.}\,,$
(323)

$\dot{\Omega}=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{A}_{\nu}+\mathcal{A}_{0}\xi\,,$
(324)

$\widehat{\Omega}=\left(\mathcal{A}_{0}-\phi_{0}\right)\Omega-\delta_{\mu}\Omega^{\mu}\,,$
(325)
$\frac{2}{3}\widehat{\theta}-\widehat{\Sigma}=Q+\frac{3}{2}\phi_{0}\Sigma+\delta_{\mu}\Sigma^{\mu}+\varepsilon_{\mu\nu}\delta^{\mu}\Omega^{\nu}\,;$
(326)

  * •

Evolution and propagation equations for vector quantities:

$\dot{\Omega}_{\alpha}+\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{\mathcal{A}}_{\mu}=-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\left(\frac{1}{2}\phi_{0}\mathcal{A}_{\mu}+\mathcal{A}_{0}a_{\mu}-\mathbb{A}_{\mu}\right)\,,$
(327)
$\dot{\Sigma}_{\alpha}-\frac{1}{2}N_{\alpha}{}^{\mu}\widehat{\mathcal{A}}_{\alpha}=\frac{1}{2}\mathbb{A}_{\alpha}+\left(\mathcal{A}_{0}-\frac{1}{4}\phi_{0}\right)\mathcal{A}_{\alpha}+\frac{1}{2}\mathcal{A}_{0}a_{\alpha}-\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\,,$
(328)

$\displaystyle\hat{\Sigma}_{\alpha}-\varepsilon_{\alpha}{}^{\mu}\widehat{\Omega}_{\mu}$
$\displaystyle=\frac{1}{2}\delta_{\alpha}\Sigma+\frac{2}{3}\delta_{\alpha}\theta-\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\Omega+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\varepsilon_{\alpha}{}^{\mu}\Omega_{\mu}$
(329)
$\displaystyle-\frac{3}{2}\phi_{0}\Sigma_{\alpha}-\delta_{\mu}\Sigma_{\alpha}{}^{\mu}-Q_{\alpha}\,;$

  * •

Evolution and propagation equations for tensor quantities:

$\dot{\Sigma}_{\alpha\beta}=\delta_{\\{\alpha}\mathcal{A}_{\beta\\}}+\mathcal{A}_{0}\zeta_{\alpha\beta}-\mathcal{E}_{\alpha\beta}+\frac{1}{2}\Pi_{\alpha\beta}\,,$
(330)
$\widehat{\Sigma}_{\alpha\beta}=\delta_{\\{\alpha}\Sigma_{\beta\\}}+\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\Omega_{|\beta\\}}+\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{|\beta\\}}\Omega_{\mu}-\frac{1}{2}\phi_{0}\Sigma_{\alpha\beta}-\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}{}^{\mu}\,;$
(331)

  * •

Constraint equations:

$\delta_{\mu}\Omega^{\mu}+\varepsilon^{\mu\nu}\delta_{\mu}\Sigma_{\nu}=\left(2\mathcal{A}_{0}-\phi_{0}\right)\Omega+\mathcal{H}\,,$
(332)
$\delta_{\alpha}\Sigma-\frac{2}{3}\delta_{\alpha}\theta+2\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\Omega+2\delta_{\mu}\Sigma_{\alpha}{}^{\mu}=\phi_{0}\left(\varepsilon_{\alpha}{}^{\mu}\Omega_{\mu}-\Sigma_{\alpha}\right)-2\varepsilon_{\alpha}{}^{\mu}\mathcal{H}_{\mu}-Q_{\alpha}\,.$
(333)

###  C.2 Linearized equations for the kinematical quantities associated with
the spacelike congruence

  * •

Evolution and propagation equations for scalars and dot-derivatives of scalar
quantities:

$\mathsf{F}=Q+\left(2\mathcal{A}_{0}-\phi_{0}\right)\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)+\delta_{\mu}\alpha^{\mu}\,,$
(334)

$\widehat{\mathsf{F}}=\hat{\phi}_{0}\left(\frac{1}{3}\theta+\Sigma\right)-\left(\mathcal{A}_{0}+\phi_{0}\right)\mathsf{F}-\frac{2}{3}\mathsf{m}-\frac{1}{2}\mathcal{P}-\mathsf{E}+\left(\delta_{\mu}a^{\mu}\right)^{.}\,,$
(335)
$\dot{\xi}=\frac{1}{2}\mathcal{H}+\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Omega+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\alpha_{\nu}\,,$
(336)

$\widehat{\xi}=-\phi_{0}\xi+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}a_{\nu}\,;$
(337)

  * •

Evolution and propagation equations for vector quantities:

$\widehat{\alpha}_{\alpha}-\dot{a}_{\alpha}=\varepsilon_{\mu\alpha}\mathcal{H}^{\mu}-\left(\mathcal{A}_{0}+\frac{1}{2}\phi_{0}\right)\alpha_{\alpha}+\frac{1}{2}Q_{\alpha}+\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\Sigma_{\alpha}+\varepsilon_{\alpha\mu}\Omega^{\mu}\right)\,;$
(338)

  * •

Evolution and propagation equations for tensor quantities:

$\dot{\zeta}_{\alpha\beta}=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Sigma_{\alpha\beta}+\delta_{\\{\alpha}\alpha_{\beta\\}}-\varepsilon^{\mu}{}_{\\{\alpha}\mathcal{H}_{\beta\\}\mu}\,,$
(339)
$\widehat{\zeta}_{\alpha\beta}=\delta_{\\{\alpha}a_{\beta\\}}-\phi_{0}\zeta_{\alpha\beta}-\mathcal{E}_{\alpha\beta}-\frac{1}{2}\Pi_{\alpha\beta}\,;$
(340)

  * •

Constraint equation:

$\delta_{\mu}\zeta_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\xi-\frac{1}{2}\mathbb{F}_{\alpha}=\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\,.$
(341)

###  C.3 Linearized equations for the Weyl tensor components and the matter
variables

  * •

Evolution and propagation equations for scalars and dot-derivatives of scalar
quantities:

$\displaystyle\mathsf{E}+\frac{1}{2}\mathcal{P}+\frac{1}{3}\widehat{Q}$
$\displaystyle=\mathcal{E}_{0}\left(\frac{3}{2}\Sigma-\theta\right)-\frac{1}{2}\Pi_{0}\left(\frac{1}{3}\theta+\frac{1}{2}\Sigma\right)+\frac{1}{3}\left(\frac{1}{2}\phi_{0}-2\mathcal{A}_{0}\right)Q$
(342)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}\right)\Sigma+\frac{1}{6}\delta_{\mu}Q^{\mu}+\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{H}_{\nu}\,,$
$\displaystyle\widehat{\mathsf{E}}-\frac{1}{3}\widehat{\mathsf{m}}+\frac{1}{2}\widehat{\mathcal{P}}$
$\displaystyle=\left(\widehat{\mathcal{E}}_{0}-\frac{1}{3}\hat{\mu}_{0}+\frac{1}{2}\widehat{\Pi}_{0}\right)\left(\frac{1}{3}\theta+\Sigma\right)-\left(\delta_{\mu}\mathcal{E}^{\mu}\right)^{.}-\frac{1}{2}\left(\delta_{\mu}\Pi^{\mu}\right)^{.}$
(343)
$\displaystyle+\frac{1}{3}\mathcal{A}_{0}\mathsf{m}-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\mathsf{F}-\left(\mathcal{A}_{0}+\frac{3}{2}\phi_{0}\right)\left(\mathsf{E}+\frac{1}{2}\mathcal{P}\right)\,,$

$\dot{\mathcal{H}}=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\Pi_{\nu}-\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{E}_{\nu}-3\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\xi\,,$
(344)
$\widehat{\mathcal{H}}=-\delta_{\mu}\mathcal{H}^{\mu}-\frac{1}{2}\varepsilon_{\mu\nu}\delta^{\mu}Q^{\nu}-\frac{3}{2}\phi_{0}\mathcal{H}-\left(3\mathcal{E}_{0}+\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\Omega\,,$
(345)

$\mathsf{m}+\widehat{Q}=-\left(\phi_{0}+2\mathcal{A}_{0}\right)Q-\frac{3}{2}\Pi_{0}\Sigma-\left(\mu_{0}+p_{0}\right)\theta-\delta_{\mu}Q^{\mu}\,,$
(346) $\displaystyle\ddot{Q}+\widehat{\mathsf{p}}+\widehat{\mathcal{P}}$
$\displaystyle=\left(\widehat{p}_{0}+\widehat{\Pi}_{0}\right)\left(\frac{1}{3}\theta+\Sigma\right)-\Pi_{0}\left(\frac{3}{2}\mathsf{F}+\mathsf{A}\right)-\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\mathcal{P}$
(347)
$\displaystyle-\mathcal{A}_{0}\left(\mathsf{m}+2\mathsf{p}\right)-\left(\mu_{0}+p_{0}\right)\mathsf{A}-\left(\delta_{\mu}\Pi^{\mu}\right)^{.}\,;$

  * •

Evolution and propagation equations for vector quantities:

$\displaystyle\dot{\mathcal{E}}_{\alpha}+\frac{1}{2}\dot{\Pi}_{\alpha}$
$\displaystyle=\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\frac{1}{2}Q_{\alpha}+\varepsilon_{\alpha\mu}\mathcal{H}^{\mu}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\alpha_{\alpha}$
(348)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+\Pi_{0}\right)\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}\right)$
$\displaystyle-\frac{1}{2}\delta_{\alpha}Q+\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}\mathcal{H}+\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{H}_{\nu\alpha}\,,$

$\displaystyle\widehat{\mathcal{E}}_{\alpha}+\frac{1}{2}\widehat{\Pi}_{\alpha}$
$\displaystyle=-\frac{3}{2}\phi_{0}\left(\mathcal{E}_{\alpha}+\frac{1}{2}\Pi_{\alpha}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)a_{\alpha}$
(349)
$\displaystyle+\frac{1}{2}\mathbb{E}_{\alpha}+\frac{1}{3}\mathfrak{m}_{\alpha}+\frac{1}{4}\mathbb{P}_{\alpha}-\delta_{\mu}\mathcal{E}_{\alpha}{}^{\mu}-\frac{1}{2}\delta_{\mu}\Pi_{\alpha}{}^{\mu}\,,$

$\displaystyle\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{\mathcal{E}}_{\mu}-\dot{\mathcal{H}}_{\alpha}-\frac{1}{4}\varepsilon_{\alpha}{}^{\mu}\widehat{\Pi}_{\mu}$
$\displaystyle=\frac{3}{4}\varepsilon_{\alpha}{}^{\mu}\mathbb{E}_{\mu}-\frac{3}{8}\varepsilon_{\alpha}{}^{\mu}\mathbb{P}_{\mu}+\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{E}_{\nu\alpha}-\frac{1}{4}\varepsilon^{\mu\nu}\delta_{\mu}\Pi_{\nu\alpha}$
(350)
$\displaystyle+\frac{3}{2}\varepsilon_{\alpha\mu}\mathcal{E}_{0}\mathcal{A}^{\mu}-\frac{3}{4}\varepsilon_{\alpha\mu}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)a^{\mu}-\varepsilon_{\alpha\mu}\mathcal{E}^{\mu}\mathcal{A}$
$\displaystyle-\frac{1}{4}\varepsilon_{\alpha\mu}\phi_{0}\left(\mathcal{E}^{\mu}-\frac{1}{2}\Pi^{\mu}\right)\,,$

$\displaystyle\widehat{\mathcal{H}}_{\alpha}-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\widehat{Q}_{\mu}$
$\displaystyle=\frac{1}{2}\delta_{\alpha}\mathcal{H}-\delta_{\mu}\mathcal{H}_{\alpha}{}^{\mu}-\frac{1}{2}\varepsilon_{\alpha}{}^{\mu}\delta_{\mu}Q-\frac{3}{2}\varepsilon_{\alpha\mu}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Sigma^{\mu}$
(351)
$\displaystyle-\left(\mu_{0}+p_{0}+\frac{1}{4}\Pi_{0}-\frac{3}{2}\mathcal{E}_{0}\right)\Omega_{\alpha}-\frac{3}{2}\phi_{0}\left(\mathcal{H}_{\alpha}-\frac{1}{6}\varepsilon_{\alpha\mu}Q^{\mu}\right)\,,$

$\dot{Q}_{\alpha}+\widehat{\Pi}_{\alpha}=-\mathfrak{p}_{\alpha}+\frac{1}{2}\mathbb{P}_{\alpha}-\delta_{\mu}\Pi_{\alpha}{}^{\mu}-\frac{3}{2}\Pi_{0}a_{\alpha}-\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{\alpha}-\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\mathcal{A}_{\alpha}\,;$
(352)

  * •

Evolution and propagation equations for tensor quantities:

$\displaystyle\dot{\mathcal{E}}_{\alpha\beta}+\frac{1}{2}\dot{\Pi}_{\alpha\beta}+\varepsilon_{\\{\alpha|}{}^{\nu}\widehat{\mathcal{H}}_{|\beta\\}\nu}$
$\displaystyle=\varepsilon_{\\{\alpha|}{}^{\nu}\delta_{\nu}\mathcal{H}_{|\beta\\}}-\frac{1}{2}\left(\mu_{0}+p_{0}+3\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\Sigma_{\alpha\beta}$
(353)
$\displaystyle+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\varepsilon_{\mu\\{\alpha}\mathcal{H}_{\beta\\}}{}^{\mu}-\frac{1}{2}\delta_{\\{\alpha}Q_{\beta\\}}\,,$

$\displaystyle\varepsilon_{\\{\alpha|}{}^{\nu}\widehat{\mathcal{E}}_{|\beta\\}\nu}-\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\nu}\widehat{\Pi}_{|\beta\\}\nu}-\dot{\mathcal{H}}_{\left\\{\alpha\beta\right\\}}$
$\displaystyle=\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\mathcal{E}_{|\beta\\}}-\frac{1}{2}\varepsilon_{\\{\alpha|}{}^{\mu}\delta_{\mu}\Pi_{|\beta\\}}-\frac{1}{4}\phi_{0}\varepsilon_{\mu\\{\alpha}\Pi_{\beta\\}}{}^{\mu}$
(354)
$\displaystyle-\frac{3}{2}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\varepsilon_{\mu\\{\alpha}\zeta_{\beta\\}}{}^{\mu}+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\varepsilon_{\mu\\{\alpha}\mathcal{E}_{\beta\\}}{}^{\mu}\,.$

###  C.4 Extra equations

The variables
$\left\\{\mathfrak{m},\mathfrak{p},\mathbb{P},\mathbb{F},\mathbb{A},\mathbb{E}\right\\}$
and
$\left\\{\mathsf{m},\mathsf{p},\mathcal{P},\mathsf{F},\mathsf{A},\mathsf{E}\right\\}$
are not fully independent. Using the commutation relations (47) we find

$\displaystyle\delta_{\alpha}\mathsf{m}-\dot{\mathfrak{m}}_{\alpha}$
$\displaystyle=\widehat{\mu}_{0}\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}+\alpha_{\alpha}\right)\,,$
(355) $\displaystyle\delta_{\alpha}\mathsf{p}-\dot{\mathfrak{p}}_{\alpha}$
$\displaystyle=\widehat{p}_{0}\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}+\alpha_{\alpha}\right)\,,$
$\displaystyle\delta_{\alpha}\mathcal{P}-\dot{\mathbb{P}}_{\alpha}$
$\displaystyle=\widehat{\Pi}_{0}\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}+\alpha_{\alpha}\right)\,,$
$\displaystyle\delta_{\alpha}\mathsf{F}-\dot{\mathbb{F}}_{\alpha}$
$\displaystyle=\widehat{\phi}_{0}\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}+\alpha_{\alpha}\right)\,,$
$\displaystyle\delta_{\alpha}\mathsf{A}-\dot{\mathbb{A}}_{\alpha}$
$\displaystyle=\widehat{\mathcal{A}}_{0}\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}+\alpha_{\alpha}\right)\,,$
$\displaystyle\delta_{\alpha}\mathsf{E}-\dot{\mathbb{E}}_{\alpha}$
$\displaystyle=\widehat{\mathcal{E}}_{0}\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}+\alpha_{\alpha}\right)\,.$

Moreover, we also have

$\displaystyle\varepsilon^{\alpha\beta}\delta_{\alpha}\mathfrak{m}_{\beta}$
$\displaystyle=-2\widehat{\mu}_{0}\xi\,,$ (356)
$\displaystyle\varepsilon^{\alpha\beta}\delta_{\alpha}\mathfrak{p}_{\beta}$
$\displaystyle=-2\widehat{p}_{0}\xi\,,$
$\displaystyle\varepsilon^{\alpha\beta}\delta_{\alpha}\mathbb{P}_{\beta}$
$\displaystyle=-2\widehat{\Pi}_{0}\xi\,,$
$\displaystyle\varepsilon^{\alpha\beta}\delta_{\alpha}\mathbb{F}_{\beta}$
$\displaystyle=-2\widehat{\phi}_{0}\xi\,,$
$\displaystyle\varepsilon^{\alpha\beta}\delta_{\alpha}\mathbb{A}_{\beta}$
$\displaystyle=-2\widehat{\mathcal{A}}_{0}\xi\,,$
$\displaystyle\varepsilon^{\alpha\beta}\delta_{\alpha}\mathbb{E}_{\beta}$
$\displaystyle=-2\widehat{\mathcal{E}}_{0}\xi\,.$

##  Appendix D Scalar, vector and tensor harmonics

We list some of the properties of the scalar, vector, and tensor
eigenfunctions of the covariantly defined Laplace-Beltrami operator on
2-hypersurfaces. For concreteness, we consider a locally rotationally
symmetric of class II spacetime, such that it admits the existence of spatial
sections that can be described by 2-hypersurfaces.

###  D.1 Scalar harmonics

Let $\mathcal{Q}^{\left(k\right)}$ represent the scalar eigenfunctions of the
covariantly defined Laplace-Beltrami operator
$\delta^{2}\equiv\delta^{\mu}\delta_{\mu}$, where the $\delta$ operator was
defined in Eq. (9), such that

$\displaystyle\delta^{2}\mathcal{Q}^{\left(k\right)}$
$\displaystyle=-\frac{k^{2}}{r^{2}}\mathcal{Q}^{\left(k\right)}\,,$ (357)
$\displaystyle\widehat{\mathcal{Q}}^{\left(k\right)}=\dot{\mathcal{Q}}^{\left(k\right)}$
$\displaystyle=0\,,$

where we have assumed that the harmonic index verifies $k^{2}\geq 0$ and the
function $r$ is covariantly defined as

$\displaystyle\frac{\hat{r}}{r}$ $\displaystyle=\frac{1}{2}\phi\,,$ (358)
$\displaystyle\frac{\dot{r}}{r}$
$\displaystyle=\frac{1}{3}\theta-\frac{1}{2}\Sigma\,,$
$\displaystyle\delta_{\alpha}r$ $\displaystyle=0\,,$

or, using the 1+1+2 equations in Appendix A in the particular case of an LRS
II spacetime, we have:

$\frac{1}{r^{2}}=\frac{\phi^{2}}{4}-\mathcal{E}+\frac{1}{3}\left(\mu+\Lambda\right)-\frac{\Pi}{2}-\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)^{2}\,.$
(359)

###  D.2 Vector harmonics

From the scalar harmonics $\mathcal{Q}^{\left(k\right)}$, we may define a
complete set of 1-tensors that can be used as a basis for sufficiently smooth
1-tensor fields defined on the 2-hypersurfaces. The vector harmonics are
defined as

$\displaystyle\mathcal{Q}_{\alpha}^{\left(k\right)}$
$\displaystyle=r\delta_{\alpha}\mathcal{Q}^{\left(k\right)}\,,$ (360)
$\displaystyle\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}$
$\displaystyle=r\varepsilon_{\alpha\mu}\delta^{\mu}\mathcal{Q}^{\left(k\right)}\,,$

where $\mathcal{Q}_{\alpha}^{\left(k\right)}$ are referred as the “even”
vector harmonics and $\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}$ as the
“odd” vector harmonics. These have the following properties

$\displaystyle u^{\mu}\nabla_{\mu}{\mathcal{Q}}_{\alpha}^{\left(k\right)}$
$\displaystyle=0\,,$ (361) $\displaystyle
e^{\mu}D_{\mu}{\mathcal{Q}}_{\alpha}^{\left(k\right)}$ $\displaystyle=0\,,$
$\displaystyle\delta^{\mu}\mathcal{Q}_{\mu}^{\left(k\right)}$
$\displaystyle=-\frac{k^{2}}{r}\mathcal{Q}^{\left(k\right)}\,,$
$\displaystyle\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{Q}_{\nu}^{\left(k\right)}$
$\displaystyle=0\,,$
$\displaystyle\delta^{2}\mathcal{Q}_{\alpha}^{\left(k\right)}$
$\displaystyle=\frac{1-k^{2}}{r^{2}}\mathcal{Q}_{\alpha}^{\left(k\right)}\,;$

and

$\displaystyle u^{\mu}\nabla_{\mu}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}$
$\displaystyle=0\,,$ (362) $\displaystyle
e^{\mu}D_{\mu}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}$
$\displaystyle=0\,,$
$\displaystyle\delta^{\mu}\bar{\mathcal{Q}}_{\mu}^{\left(k\right)}$
$\displaystyle=0\,,$
$\displaystyle\varepsilon^{\mu\nu}\delta_{\mu}\bar{\mathcal{Q}}_{\nu}^{\left(k\right)}$
$\displaystyle=\frac{k^{2}}{r}\mathcal{Q}^{\left(k\right)}\,,$
$\displaystyle\delta^{2}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}$
$\displaystyle=\frac{1-k^{2}}{r^{2}}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\,.$

Moreover, we have

$\displaystyle\bar{Q}_{\alpha}^{\left(k\right)}$
$\displaystyle=\varepsilon_{\alpha}{}^{\mu}Q_{\mu}^{\left(k\right)}\,,$ (363)
$\displaystyle Q_{\alpha}^{\left(k\right)}$
$\displaystyle=-\varepsilon_{\alpha}{}^{\mu}\bar{Q}_{\mu}^{\left(k\right)}\,,$

which can be used to readily verify
$N^{\mu\nu}\mathcal{Q}_{\mu}^{\left(k\right)}\bar{\mathcal{Q}}_{\nu}^{\left(k\right)}=0$,
valid for each value of $k$.

###  D.3 Tensor harmonics

The scalar and vector harmonics introduced in the previous subsection can
themselves be used to define a complete set of symmetric, traceless 2-tensors
that form a basis for sufficiently smooth, symmetric, traceless 2-tensor
fields defined on the 2-hypersurfaces. The tensor harmonics are defined as

$\displaystyle\mathcal{Q}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=r^{2}\delta_{\\{\alpha}\delta_{\beta\\}}\mathcal{Q}^{\left(k\right)}\,,$
(364) $\displaystyle\bar{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=r^{2}\varepsilon_{\mu\\{\alpha|}\delta^{\mu}\delta_{|\beta\\}}\mathcal{Q}^{\left(k\right)}\,,$

where, similarly to the vector harmonics above,
$\mathcal{Q}_{\alpha\beta}^{\left(k\right)}$ are referred as the “even” tensor
harmonics and $\bar{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}$ are referred
to as the “odd” tensor harmonics. These have the following properties

$\displaystyle
u^{\mu}\nabla_{\mu}{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=0\,,$ (365) $\displaystyle
e^{\mu}D_{\mu}{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=0\,,$
$\displaystyle\delta^{\beta}\mathcal{Q}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=\frac{2-k^{2}}{2r}\mathcal{Q}_{\alpha}^{\left(k\right)}\,,$
$\displaystyle\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{Q}_{\nu\alpha}^{\left(k\right)}$
$\displaystyle=\frac{2-k^{2}}{2r}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\,;$

and

$\displaystyle
u^{\mu}\nabla_{\mu}\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=0\,,$ (366) $\displaystyle
e^{\mu}D_{\mu}\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=0\,,$
$\displaystyle\delta^{\mu}\mathcal{\bar{Q}}_{\alpha\mu}^{\left(k\right)}$
$\displaystyle=-\frac{2-k^{2}}{2r}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\,,$
$\displaystyle\varepsilon^{\mu\nu}\delta_{\mu}\mathcal{\bar{Q}}_{\nu\alpha}^{\left(k\right)}$
$\displaystyle=\frac{2-k^{2}}{2r}\mathcal{Q}_{\alpha}^{\left(k\right)}\,.$

Moreover, we have

$\displaystyle\mathcal{Q}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=-\varepsilon^{\mu}{}_{\\{\alpha}\bar{\mathcal{Q}}_{\beta\\}\mu}^{\left(k\right)}\,,$
(367) $\displaystyle\bar{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}$
$\displaystyle=\varepsilon^{\mu}{}_{\\{\alpha}\mathcal{Q}_{\beta\\}\mu}^{\left(k\right)}\,,$

which can be used to readily prove that
$N^{\alpha\mu}N^{\beta\nu}\mathcal{Q}_{\alpha\beta}^{\left(k\right)}\bar{\mathcal{Q}}_{\mu\nu}^{\left(k\right)}=0$,
valid for each value of $k$.

##  Appendix E Change in the equation of state under isotropic frame
transformations

In this Appendix, we make use of generalized Lorentz boosts in the 1+1+2
formalism to derive the transformation law for the equation of state when
changing between the comoving frame and the static frame.

###  E.1 Isotropic frame transformations and projectors

Let the dyad $\left(u,e\right)$ be formed, respectively, by a timelike and a
spacelike vector field in a spacetime, such that $u^{\alpha}u_{\alpha}=-1$ and
$e^{\alpha}e_{\alpha}=1$, and let

$\displaystyle\bar{u}^{\alpha}$
$\displaystyle=\left(1+a\right)u^{\alpha}+be^{\alpha}+m^{\alpha}\,,$ (368)
$\displaystyle\bar{e}^{\alpha}$
$\displaystyle=cu^{\alpha}+\left(1+d\right)e^{\alpha}+k^{\alpha}\,,$

represent the components in some local coordinate system of a dyad
$\left(\bar{u},\bar{e}\right)$, with $\left\\{a,b,c,d\right\\}$ being scalar
coefficients and

$\displaystyle u^{\alpha}m_{\alpha}$ $\displaystyle=0\,,$ $\displaystyle
u^{\alpha}k_{\alpha}$ $\displaystyle=0\,,$ (369) $\displaystyle
e^{\alpha}m_{\alpha}$ $\displaystyle=0\,,$ $\displaystyle
e^{\alpha}k_{\alpha}$ $\displaystyle=0\,.$

In the expressions above, we can assume, without loss of generality, that the
vector fields $m$ and $k$ are associated with rotational displacements. Then,
for our purposes, it suffices to consider that the frame transformation is
isotropic, such that $m^{\alpha}=0$ and $k^{\alpha}=0$.

Imposing that the new dyad verifies

$\displaystyle\bar{u}^{\alpha}\bar{u}_{\alpha}$ $\displaystyle=-1\,,$ (370)
$\displaystyle\bar{e}^{\alpha}\bar{e}_{\alpha}$ $\displaystyle=+1\,,$
$\displaystyle\bar{u}^{\alpha}\bar{e}_{\alpha}$ $\displaystyle=0\,,$

we find that the transformation coefficients verify

$\displaystyle\left(1+d\right)^{2}$ $\displaystyle=\left(1+a\right)^{2}$ (371)
$\displaystyle\left(1+a\right)^{2}$ $\displaystyle=1+c^{2}$ $\displaystyle
b^{2}$ $\displaystyle=c^{2}\,.$

Assuming, $b,c>0$, hence $a,d>0$, that is, the transformation does not swap
the direction between the corresponding vector fields, i.e., imposing that
$u^{\alpha}\bar{u}_{\alpha}\leq 0$ and $e^{\alpha}\bar{e}_{\alpha}\geq 0$, we
see that the relation between $a$ and $c$ can be made obvious by introducing a
single hyperbolic angle $\beta$ such that

$\displaystyle 1+a$ $\displaystyle=\cosh\beta\,,$ (372) $\displaystyle c$
$\displaystyle=\sinh\beta\,.$

Then, we have the following relation between the two dyads

$\displaystyle\bar{u}^{\alpha}$
$\displaystyle=u^{\alpha}\cosh\beta+e^{\alpha}\sinh\beta\,,$ (373)
$\displaystyle\bar{e}^{\alpha}$
$\displaystyle=u^{\alpha}\sinh\beta+e^{\alpha}\cosh\beta\,.$

Given relations (373), the projectors $h$ and $N$ transform accordingly as

$\displaystyle\bar{h}_{\alpha\beta}$
$\displaystyle=h_{\alpha\beta}+\left(u_{\alpha}u_{\beta}+e_{\alpha}e_{\beta}\right)\sinh^{2}\beta+\frac{1}{2}\left(u_{\alpha}e_{\beta}+e_{\alpha}u_{\beta}\right)\sinh\left(2\beta\right)\,,$
(374) $\displaystyle\bar{N}_{\alpha\beta}$ $\displaystyle=N_{\alpha\beta}\,,$
(375)

which verify $\bar{h}_{\alpha\beta}\bar{u}^{\beta}=0$,
$\bar{N}_{\alpha\beta}\bar{u}^{\beta}=0$ and
$\bar{N}_{\alpha\beta}\bar{e}^{\beta}=0$, as expected.

###  E.2 Fluid variables transformations

Consider now a general symmetric stress-energy tensor decomposed accordingly
with Eq. (44). Under the transformation (373) we have

$\displaystyle\bar{\mu}$
$\displaystyle=\mu-Q\sinh\left(2\beta\right)+\left(\mu+p+\Pi\right)\sinh^{2}\beta\,,$
(376) $\displaystyle\bar{p}$
$\displaystyle=p-\frac{1}{3}Q\sinh\left(2\beta\right)+\frac{1}{3}\left(\mu+p+\Pi\right)\sinh^{2}\beta\,,$
$\displaystyle\bar{Q}$
$\displaystyle=Q\cosh\left(2\beta\right)-\frac{1}{2}\left(\mu+p+\Pi\right)\sinh\left(2\beta\right)\,,$
$\displaystyle\bar{\Pi}$
$\displaystyle=\Pi\left(1+\frac{2}{3}\sinh^{2}\beta\right)-\frac{2}{3}Q\sinh\left(2\beta\right)+\frac{2}{3}\left(\mu+p\right)\sinh^{2}\beta\,,$

where we have used the intuitive notation of keeping an overline to indicate
variables measured in the frame associated with the dyad
$\left(\bar{u},\bar{e}\right)$. Moreover, we have the following relations
between the scalar kinematical quantities

$\displaystyle\bar{\theta}$
$\displaystyle=\theta\cosh\beta+\left(\phi+\mathcal{A}\right)\sinh\beta+\nabla_{u}\cosh\beta+\nabla_{e}\sinh\beta\,,$
(377) $\displaystyle\bar{\Sigma}$
$\displaystyle=\Sigma\cosh\beta-\frac{1}{3}\left(\phi-2\mathcal{A}\right)\sinh\beta+\frac{2}{3}\nabla_{u}\cosh\beta+\frac{2}{3}\nabla_{e}\sinh\beta\,,$

where $\nabla_{u}\equiv u^{\alpha}\nabla_{\alpha}$ and $\nabla_{e}\equiv
e^{\alpha}\nabla_{\alpha}$.

###  E.3 Relation between equations of state of the comoving and static
observers

The comoving and static observer frames considered in Sec. IV are related by
an isotropic change of frame. Then, we can use the results in the previous
subsections of this appendix to find the equation of state experienced by a
static observer, knowing the equation of state measured by the comoving
observer.

Assuming the dyad $\left(u,e\right)$ to be associated to the comoving frame
and the dyad $\left(\bar{u},\bar{e}\right)$ to be associated with the static
frame, we can particularize Eq. (376) to the case when $Q=0$ and $\Pi=0$, such
that

$\displaystyle\bar{\mu}$
$\displaystyle=\mu+\left(\mu+p\right)\sinh^{2}\beta\,,$ (378)
$\displaystyle\bar{p}$
$\displaystyle=p+\frac{1}{3}\left(\mu+p\right)\sinh^{2}\beta\,,$
$\displaystyle\bar{Q}$
$\displaystyle=-\frac{1}{2}\left(\mu+p\right)\sinh\left(2\beta\right)\,,$
$\displaystyle\bar{\Pi}$
$\displaystyle=\frac{2}{3}\left(\mu+p\right)\sinh^{2}\beta\,,$

relating the nontrivial matter variables of the two frames. Moreover, since in
the static frame — the one associated with the barred variables — we have
$\frac{3}{2}\bar{\Sigma}=\bar{\theta}$, then

$\tanh\beta=\frac{1}{\phi}\left(\Sigma-\frac{2}{3}\theta\right)\,,$ (379)

which upon substitution in Eq. (378) yields

$\bar{Q}=-\frac{\mu+p}{\phi}\left(\Sigma-\frac{2}{3}\theta\right)\cosh^{2}\beta\,.$
(380)

Assuming the background spacetime to be static and LRS II, Eq. (379)
explicitly shows that $\beta$ is a first-order quantity with respect to the
background, i.e., $\beta$ vanishes in the background. This is expected since
comoving observers are also static observers for a static LRS II spacetime.
Moreover, these results confirm the expectation that the anisotropic pressure
measured by the static observer, $\bar{\Pi}$, is zero at linear level for
adiabatic, radial perturbations.

We can now find the linear order correction with respect to the background of
the equation of state in the static frame. Taking the derivative along
$\bar{u}$ of $\bar{p}$ and $\bar{\mu}$, and using Eqs. (373) and (378) we
find, in general,

$\displaystyle\nabla_{\bar{u}}\bar{\mu}$
$\displaystyle=\dot{\mu}\cosh\beta+\widehat{\mu}\sinh\beta+\left[\left(\nabla_{u}\sinh^{2}\beta\right)\left(\mu+p\right)+\sinh^{2}\beta\left(\dot{\mu}+\dot{p}\right)\right]\cosh\beta$
(381)
$\displaystyle+\left[\left(\nabla_{e}\sinh^{2}\beta\right)\left(\mu+p\right)+\sinh^{2}\beta\left(\widehat{\mu}+\widehat{p}\right)\right]\sinh\beta\,,$
$\displaystyle\nabla_{\bar{u}}\bar{p}$
$\displaystyle=\dot{p}\cosh\beta+\widehat{p}\sinh\beta+\frac{1}{3}\left[\left(\nabla_{u}\sinh^{2}\beta\right)\left(\mu+p\right)+\sinh^{2}\beta\left(\dot{\mu}+\dot{p}\right)\right]\cosh\beta$
$\displaystyle+\frac{1}{3}\left[\left(\nabla_{e}\sinh^{2}\beta\right)\left(\mu+p\right)+\sinh^{2}\beta\left(\widehat{\mu}+\widehat{p}\right)\right]\sinh\beta\,,$

where $\dot{\mu}=u^{\alpha}\nabla_{\alpha}\mu$ and
$\widehat{\mu}=e^{\alpha}\nabla_{\alpha}\mu$, and similarly for $p$. Using the
fact that $\beta$ is a first-order quantity with respect to the background, in
first-order perturbation theory Eq. (381) simplifies to

$\displaystyle\nabla_{\bar{u}}\bar{\mu}$
$\displaystyle=\dot{\mu}+\widehat{\mu}_{0}\beta\,,$ (382)
$\displaystyle\nabla_{\bar{u}}\bar{p}$
$\displaystyle=\dot{p}+\widehat{p}_{0}\beta\,.$

Also, from Eq. (378), at first order, we have

$\beta=-\frac{\bar{Q}}{\mu_{0}+p_{0}}\,.$ (383)

Gathering the intermediate results and using the equations that characterize
the background solution (cf. subsection IV.1) we find the transformed equation
of state for the static frame

$\nabla_{\bar{u}}\bar{\mu}=\frac{1}{f^{\prime}\left(\mu_{0}\right)}\nabla_{\bar{u}}\bar{p}-\frac{1}{\mu_{0}+p_{0}}\left(\widehat{\mu}_{0}-\frac{\widehat{p}_{0}}{f^{\prime}\left(\mu_{0}\right)}\right)Q\,,$
(384)

where $\dot{p}=f^{\prime}\left(\mu_{0}\right)\dot{\mu}$, that is,
$f^{\prime}\left(\mu_{0}\right)$ represents the square of the adiabatic speed
of sound measured in the comoving frame.

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