# Gauge invariant perturbations of static spatially compact LRS II spacetimes

Paulo Luz<EMAIL_ADDRESS>Centro de Astrofísica e Gravitação -
CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade
de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal,  Departamento
de Matemática, ISCTE - Instituto Universitário de Lisboa, Portugal  Sante
Carloni<EMAIL_ADDRESS>Institute of Theoretical Physics, Faculty of
Mathematics and Physics, Charles University, Prague, V Holešovičkách 2, 180 00
Prague 8, Czech Republic,  DIME, Università di Genova, Via all’Opera Pia 15,
16145 Genova, Italy,  INFN Sezione di Genova, Via Dodecaneso 33, 16146 Genova,
Italy

###### Abstract

We present a framework to describe completely general first-order
perturbations of static, spatially compact, and locally rotationally symmetric
class II spacetimes within the theory of general relativity. The perturbation
variables are by construction covariant and identification gauge invariant and
encompass the geometry and the thermodynamics of the fluid sources. The new
equations are then applied to the study of isotropic, adiabatic perturbations.
We discuss how the choice of frame in which perturbations are described can
significantly simplify the mathematical analysis of the problem and show that
it is possible to change frames directly from the linear level equations. We
find explicitly that the case of isotropic, adiabatic perturbations can be
reduced to a singular Sturm-Liouville eigenvalue problem, and lower bounds for
the values of the eigenfrequencies can be derived. These results lay the
theoretical groundwork to analytically describe linear, isotropic, and
adiabatic perturbations of static, spherically symmetric spacetimes.

##  I Introduction

In the last few years, we have witnessed a true renaissance of relativistic
astrophysics. The detection of gravitational waves [1] and the first images of
the shadows of supermassive black holes [2] have brought a new wealth of data
to the research community. It has, therefore, become paramount to devise new
tools that allow for a clear interpretation of the latest data and deepen our
understanding of relativistic astrophysical systems.

Due to the fundamental nonlinear nature of the field equations of the theory
of general relativity (GR), finding exact solutions for these systems, even
assuming highly symmetric setups with simple source fields, is a formidable
task. Indeed, although several exact solutions have been found in vacuum or in
the presence of matter, many of the latter type of solutions are, in general,
not suitable to accurately model physically meaningful setups, especially in
the so-called strong field regime, where relativistic gravitational effects
play a pivotal role in the dynamics of the matter fields.

To circumvent the limitations in the applicability of idealized solutions,
various perturbative schemes were developed to linearize the field equations
and study perturbations of exact solutions. Indeed, the study of perturbations
of black holes has been an area of intense development in the past few
decades. Combining the advent of numerical relativity techniques with the
analytical results from linear perturbation theory for vacuum solutions
ultimately allowed for identifying patterns in the data and detecting
gravitational waves. On the other hand, in the case of compact stars, several
open problems remain in developing analytic tools to study perturbations of
this type of objects. Especially in the strong field regime, we have so far
relied almost exclusively on numerical methods to evolve the full nonlinear
equations of GR. These methods allow us to understand oscillations of compact
stellar objects in great generality, but carry the inevitable limitation of
purely numerical approaches. Hence, there is a need for analytic methods that
could complement the numerics in a synergic way.

The problem of linear perturbations of massive astrophysical objects is
markedly different from black hole perturbations. Black holes are described by
vacuum solutions of the Einstein field equations. In contrast, compact stellar
objects are described by solutions of GR with matter fluid sources
characterized by perfect fluids with complex equations of state, fluids with
non-trivial anisotropic pressure terms, or non-perfect multifluid models.
Understandably, the evolution of the perturbations of these solutions is
strongly tied to the properties of the matter fluid where, in general,
anisotropies and momentum flows may be generated, even at a linear level,
which then act as sources of shear and vorticity.

In 1964, by encoding the spacetime perturbations directly in the choice of
gauge, Chandrasekhar derived for the first time in Refs. [3, 4] an equation to
describe isotropic, adiabatic perturbations of static self-gravitating perfect
fluids with a barotropic equation of state. In the 1990s, in Refs. [5, 6],
structure equations were derived for nonradial perturbations of non-rotating
perfect fluids. In Refs. [7, 8, 9, 10], various extensions were proposed to
study perturbations of slowly rotating stars. However, understanding the
perturbative properties of relativistic stars using those frameworks has
remained challenging because of fundamental mathematical limitations and the
underlying methods used by those approaches. For instance, the original radial
pulsation equation by Chandrasekhar for isotropic, adiabatic perturbations
relies on the introduction of auxiliary trial functions, making it impossible
to assert the stability of the background solution unequivocally. Moreover, it
was later noticed that the choice of gauge in Ref. [5] leads to a higher-order
system of equations when compared to the choice of another gauge because of an
extraneous degree of freedom, adding unnecessary complication to the equations
[11]. Several works have tried to improve these results in the following
years, but all versions of those equations remain gauge-dependent.

In geometric gravity theories, the identification gauge problem in
perturbation theory arises because, even if the choice of perturbation
variables is physically reasonable, those might depend on the mapping between
the equilibrium and the perturbed manifold. Hence, their values and rates of
change are ambiguous. Therefore, to develop a rigorous perturbative framework
in geometric theories of gravity, the perturbation variables have to be
methodically chosen [12, 13].

In the case of vanishing or slow rotation, a Locally Rotationally Symmetric
(LRS) metric can be used to successfully describe the geometry of the
spacetime for the interior of compact stellar objects. This class of
spacetimes was first identified and classified according to their extra
symmetries by Ellis [14] and Ellis and Stewart [15] at the end of the 1960s
and is characterized by a local rotational symmetry at every event. Many
spacetimes of interest in astrophysics and cosmology belong to the LRS class.
Examples are the Bianchi cosmologies [16], the Lemaître-Tolman-Bondi spacetime
[17, 18, 19], the Oppenheimer-Snyder spacetime [20], and the vacuum
Schwarzschild spacetime.

Recently, a new formalism has been proposed that is especially suitable to
deal with LRS spacetimes: the 1+1+2 covariant approach [21, 22, 23]. This
formalism, analogous to the better-known covariant 1+3 formalism extensively
used in cosmology [24, 25, 26], is based on the procedure of covariant
spacetime threading. It can also be considered a semi-tetradic approach to the
description of spacetimes, which makes full use of the symmetries of the
spacetime. The covariant approaches have two important properties: (i) they
allow the maintenance of covariance at all stages of the calculations, and
(ii) they enable the description of spacetimes in terms of well-defined
physical quantities.

Analyzing LRS spacetimes with the 1+1+2 formalism can reveal many important
aspects of these spacetimes and their physical processes. For instance, in
Refs. [27, 28] it was found a covariant formulation of the Tolman-Oppenheimer-
Volkoff equation for perfect fluids and fluids with non-trivial anisotropic
pressure terms, allowing for the derivation of new solutions. In Ref. [29],
using this formalism, it was proved a generalization of Derrick’s theorem,
showing that the conclusion of the original theorem holds independently of the
geometric properties of the spacetime. In Ref. [30], the 1+1+2 formalism was
applied to the Einstein-Cartan theory to derive the general structure
equations for static, isotropic spacetimes, which were then used to find the
first known regular solutions suitable to model the interior of massive
astrophysical objects. However, the power of covariant formalisms is truly
revealed in the context of perturbation theory. Indeed, these approaches are
the cornerstone for constructing a covariant, gauge-invariant theory of
perturbations, which can be employed in many contexts. In particular, the
1+1+2 formalism has been used to describe tensor perturbations of
Schwarzschild black holes [21], to describe complex interaction between
gravitational and electromagnetic degrees of freedom [31], or to study
cosmological perturbations [32, 33].

The scope of this work is to construct a completely general covariant and
gauge-invariant perturbation theory of non-vacuum, static, spatially compact
LRS II spacetimes. Moreover, as a first application, we aim to describe the
dynamics of adiabatic isotropic perturbations. Using the covariant nature of
the equations, we will be able to describe the evolution of the perturbation
equations from the point of view of an observer locally comoving with the
fluid and one which is static with respect to an observer at spatial infinity.
For both cases, we propose a method to find a family of exact solutions in the
form of a power series for a wide variety of background solutions. We will
also prove that the perturbation equations with appropriate boundary
conditions constitute a singular Sturm-Liouville eigenvalue problem with a
limit-point-non-oscillating endpoint, which, to our knowledge, has not been
rigorously proven before. Using this property, we will be able to establish
lower bounds for the absolute value of the fundamental eigenfrequency in terms
of quantities of the background equilibrium solution.

The paper is organized as follows: Section II summarizes the 1+1+2 formalism
for a generic spacetime. In Section III, we derive the linearized covariant
gauge-invariant perturbation equations for a static, spatially compact LRSII
background, and we perform a harmonic decomposition, in particular, isolating
even and odd perturbations. In Section IV, we focus on adiabatic and isotropic
perturbations, writing the equations in a comoving frame and a static frame,
and for each frame, we properly define the boundary value problem. Here, we
show that the perturbation equations in the static frame form a Sturm-
Liouville eigenvalue problem and derive lower bounds for the values of the
fundamental eigenvalue. Last, we summarize and draw some conclusions in
Section VI. The paper also contains four appendices. In Appendix A, we present
the Einstein field equations in the language of the 1+1+2 formalism, and in
Appendices B and C we display the linearized field equations for the two sets
of gauge invariant quantities adopted in the body of the text. In Appendix D,
we present the definitions and properties of the eigenfunctions of the
covariant Laplace-Beltrami operator on 2-hypersurfaces. In Appendix E, we
discuss the effects on the equation of state of a generalized Lorentz boost
between two frames.

Throughout the article, we will work in the geometrized unit system where
$8\pi G=c=1$, and consider the metric signature $(-+++)$.

##  II The 1+1+2 decomposition

To construct a general set of gauge-invariant, covariant equations for the
perturbations of locally rotationally symmetric class II spacetimes, from
hereon LRS II [15], we will adopt the language of the 1+1+2 covariant
formalism [21, 22, 23]. In this section, we will then introduce the basic
quantities and conventions used throughout the article.

###  II.1 Projectors and the Levi-Civita volume form

Consider a Lorentzian manifold of dimension 4, $\left(\mathcal{M},g\right)$,
where $g$ represents the metric tensor, admitting in some open neighborhood
the existence of a congruence of timelike curves with tangent vector field
$u$. We will assume that the congruence to be affinely parameterized and
$u_{\alpha}u^{\alpha}=-1$. Without loss of generality, we can locally foliate
the manifold in 3-surfaces, $V$, orthogonal at each point to the curves of the
congruence, such that all tensor quantities are defined by their behavior
along the direction of $u$ and in $V$. This procedure is usually called 1+3
spacetime decomposition. Such decomposition of the spacetime manifold relies
on the existence of a pointwise projector to the cotangent space of $V$, which
can be naturally defined as

$h_{\alpha\beta}=g_{\alpha\beta}+u_{\alpha}u_{\beta}\,,$ (1)

where $g_{\alpha\beta}$ represents the components of the metric tensor in some
local coordinate system, with the following properties

$\displaystyle h_{\alpha\beta}$ $\displaystyle=h_{\beta\alpha}\,,$
$\displaystyle h_{\alpha\beta}h^{\beta\gamma}$
$\displaystyle=h_{\alpha}{}^{\gamma}\,,$ (2) $\displaystyle
h_{\alpha\beta}u^{\alpha}$ $\displaystyle=0\,,$ $\displaystyle
h_{\alpha}{}^{\alpha}$ $\displaystyle=3\,.$

The 1+1+2 decomposition builds from the 1+3 decomposition by defining a
congruence of spacelike curves with tangent vector field $e$ such that any
tensor quantity defined in the submanifold $V$ is defined by its behavior
along $e$ and the 2-surfaces $W$, orthogonal to both $u$ and $e$ at each
point. We shall refer to each surface $W$ as “sheet”. We will consider that
the spacelike congruence is affinely parameterized and
$e_{\alpha}e^{\alpha}=1$. We can then define a projector onto $W$ by

$N_{\alpha\beta}=h_{\alpha\beta}-e_{\alpha}e_{\beta}\,,$ (3)

verifying

$\displaystyle N_{\alpha\beta}$ $\displaystyle=N_{\beta\alpha}\,,$
$\displaystyle N_{\alpha\beta}N^{\beta\gamma}$
$\displaystyle=N_{\alpha}{}^{\gamma}\,,$ (4) $\displaystyle
N_{\alpha\beta}u^{\alpha}$ $\displaystyle=N_{\alpha\beta}e^{\alpha}=0\,,$
$\displaystyle N_{\alpha}{}^{\alpha}$ $\displaystyle=2\,.$

It is useful to introduce the following tensors derived from the covariant
Levi-Civita tensor $\varepsilon_{\alpha\beta\gamma\sigma}$,

$\displaystyle\varepsilon_{\alpha\beta\gamma}$
$\displaystyle=\varepsilon_{\alpha\beta\gamma\sigma}u^{\sigma}\,,$ (5)
$\displaystyle\varepsilon_{\alpha\beta}$
$\displaystyle=\varepsilon_{\alpha\beta\gamma}e^{\gamma}\,,$

with the following properties

$\displaystyle\varepsilon_{\alpha\beta\gamma}$
$\displaystyle=\varepsilon_{\left[\alpha\beta\gamma\right]}\,,$
$\displaystyle\varepsilon_{\alpha\beta}$
$\displaystyle=\varepsilon_{\left[\alpha\beta\right]}\,,$ (6)
$\displaystyle\varepsilon_{\alpha\beta\gamma}u^{\gamma}$ $\displaystyle=0\,,$
$\displaystyle\varepsilon_{\alpha\beta}u^{\alpha}$
$\displaystyle=\varepsilon_{\alpha\beta}e^{\alpha}=0\,,$
$\displaystyle\varepsilon_{\alpha\beta\gamma}\varepsilon^{\mu\nu\sigma}$
$\displaystyle=6h^{\mu}{}_{\left[\alpha\right.}h^{\nu}{}_{\beta}h_{\left.\gamma\right]}{}^{\sigma}$
$\displaystyle\varepsilon_{\alpha\beta}\varepsilon^{\mu\nu}$
$\displaystyle=N^{\mu}{}_{\alpha}N^{\nu}{}_{\beta}-N^{\mu}{}_{\beta}N^{\nu}{}_{\alpha}$
$\displaystyle\varepsilon_{\alpha\beta\gamma}\varepsilon^{\mu\nu\gamma}$
$\displaystyle=h_{\alpha}{}^{\mu}h_{\beta}{}^{\nu}-h_{\beta}{}^{\mu}h_{\alpha}{}^{\nu}\,,$
$\displaystyle\varepsilon_{\alpha}{}^{\gamma}\varepsilon_{\beta\gamma}$
$\displaystyle=N_{\alpha\beta}\,,$
$\displaystyle\varepsilon_{\alpha\mu\nu}\varepsilon^{\beta\mu\nu}$
$\displaystyle=2h_{\alpha}{}^{\beta}\,,$
$\displaystyle\varepsilon_{\alpha\beta\gamma}$
$\displaystyle=e_{\alpha}\varepsilon_{\beta\gamma}-e_{\beta}\varepsilon_{\alpha\gamma}+e_{\gamma}\varepsilon_{\alpha\beta}\,,$

where in the right-hand side of the relation for
$\varepsilon_{\alpha\beta\gamma}\varepsilon^{\mu\nu\sigma}$ the anti-
symmetrization is to be considered on all, and only, the lower indices. We
will adopt the convention to indicate the symmetric and anti-symmetric part of
a tensor using parentheses and brackets, such that for a 2-tensor $\chi$

$\displaystyle\chi_{\left(\alpha\beta\right)}$
$\displaystyle=\frac{1}{2}\left(\chi_{\alpha\beta}+\chi_{\beta\alpha}\right)\,,$
(7) $\displaystyle\chi_{\left[\alpha\beta\right]}$
$\displaystyle=\frac{1}{2}\left(\chi_{\alpha\beta}-\chi_{\beta\alpha}\right)\,.$

###  II.2 Covariant derivatives of $u$ and $e$

####  II.2.1 Decomposition of the covariant derivatives on the sheet $W$

Using the definitions of the projector operators onto the surfaces $V$ and
$W$, the covariant derivatives of the tangent vector fields $u$ and $e$ can
be, at each point, uniquely decomposed in their components along $u$, $e$ and
in $W$.111Although it is an abuse of language, we will refer to the tensor
fields $h$ and $N$ as projectors onto the space $V$ or $W$, being implicit
that we mean that the projection is onto the respective tangent and cotangent
spaces.

For notational convenience, given a tensor quantity $\chi$, throughout the
article, we will use the compact notation

$\displaystyle\dot{\chi}$ $\displaystyle:=u^{\mu}\nabla_{\mu}\chi\,,$
$\displaystyle\widehat{\chi}$ $\displaystyle:=e^{\mu}\nabla_{\mu}\chi\,,$ (8)

to represent the derivative along the integral curves of the vector field $u$
and the derivative along the integral curves of the vector field $e$,
respectively.

The covariant derivatives of the tensor fields $u$ and $e$ on the sheet can be
uniquely decomposed as

$\delta_{\alpha}u_{\beta}\equiv
N_{\alpha}{}^{\mu}N_{\beta}{}^{\nu}\nabla_{\mu}u_{\nu}=\frac{1}{2}N_{\alpha\beta}\tilde{\theta}+\Sigma_{\alpha\beta}+\varepsilon_{\alpha\beta}\Omega\,,$
(9)

where

$\displaystyle\tilde{\theta}$ $\displaystyle=\delta_{\alpha}u^{\alpha}\,,$
(10) $\displaystyle\Sigma_{\alpha\beta}$
$\displaystyle=\delta_{\\{\alpha}u_{\beta\\}}\,,$ $\displaystyle\Omega$
$\displaystyle=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}u_{\nu}\,,$

and

$\delta_{\alpha}e_{\beta}=\frac{1}{2}N_{\alpha\beta}\phi+\zeta_{\alpha\beta}+\varepsilon_{\alpha\beta}\xi\,,$
(11)

with

$\displaystyle\phi$ $\displaystyle=\delta_{\mu}e^{\mu}\,,$ (12)
$\displaystyle\zeta_{\alpha\beta}$
$\displaystyle=\delta_{\\{\alpha}e_{\beta\\}}\,,$ $\displaystyle\xi$
$\displaystyle=\frac{1}{2}\varepsilon^{\mu\nu}\delta_{\mu}e_{\nu}\,,$

where the curly brackets represent the projected symmetric part without trace
of a tensor in $W$, that is, for a tensor $\chi_{\alpha\beta}$,

$\chi_{\left\\{\alpha\beta\right\\}}=\left[N^{\mu}{}_{(\alpha}N_{\beta)}{}^{\nu}-\frac{N_{\alpha\beta}}{2}N^{\mu\nu}\right]\chi_{\mu\nu}\,.$
(13)

Using the 2-form $\varepsilon_{\alpha\beta}$, a completely anti-symmetric
tensor defined on the sheet,
$\chi_{\alpha\beta}=\chi_{\left[\alpha\beta\right]}=N^{\mu}{}_{[\alpha}N_{\beta]}{}^{\nu}\chi_{\mu\nu}$,
can be written as

$\chi_{\left[\alpha\beta\right]}=\varepsilon_{\alpha\beta}\left(\frac{1}{2}\varepsilon^{\mu\nu}\chi_{\mu\nu}\right)\,.$
(14)

####  II.2.2 Decomposition of the covariant derivatives on $V$

The decomposition of the projected covariant derivatives of $u$ onto $V$ is
given by

$D_{\alpha}u_{\beta}=h_{\alpha}{}^{\mu}h_{\beta}{}^{\nu}\nabla_{\mu}u_{\nu}=\frac{1}{3}h_{\alpha\beta}\theta+\sigma_{\alpha\beta}+\omega_{\alpha\beta}\,,$
(15)

with

$\displaystyle\theta$ $\displaystyle=h^{\mu\nu}D_{\mu}u_{\nu}\,,$ (16)
$\displaystyle\sigma_{\alpha\beta}$
$\displaystyle=D_{\left\langle\alpha\right.}u_{\left.\beta\right\rangle}\,,$
(17) $\displaystyle\omega_{\alpha\beta}$
$\displaystyle=h^{\mu}{}_{[\alpha}h_{\beta]}{}^{\nu}D_{\mu}u_{\nu}\,,$ (18)

where we used the angular brackets to represent the projected symmetric part
without trace of a tensor on $V$, that is, for a tensor, $\chi_{\alpha\beta}$,

$\chi_{\left\langle\alpha\beta\right\rangle}=\left[h^{\mu}{}_{(\alpha}h_{\beta)}{}^{\nu}-\frac{h_{\alpha\beta}}{3}h^{\mu\nu}\right]\chi_{\mu\nu}\,.$
(19)

The scalar and tensor quantities in Eqs. (16)–(18) can themselves be further
decomposed in their contributions exclusively on $W$ and along $e$, such that

$\theta=\tilde{\theta}+\vartheta\,,$ (20)

where $\tilde{\theta}$ is defined in Eq. (10) and

$\vartheta=-u^{\mu}\left(e^{\nu}D_{\nu}e_{\mu}\right)=-u^{\mu}\,\widehat{e}_{\mu}\,;$
(21)
$\sigma_{\alpha\beta}=\Sigma_{\alpha\beta}+2\Sigma_{(\alpha}e_{\beta)}+\Sigma\left(e_{\alpha}e_{\beta}-\frac{1}{2}N_{\alpha\beta}\right)\,,$
(22)

with

$\displaystyle\Sigma_{\alpha\beta}$
$\displaystyle=\sigma_{\left\\{\alpha\beta\right\\}}\,,$ (23)
$\displaystyle\Sigma_{\alpha}$
$\displaystyle=N_{\alpha}{}^{\mu}e^{\nu}\sigma_{\mu\nu}\,,$
$\displaystyle\Sigma$
$\displaystyle=e^{\mu}e^{\nu}\sigma_{\mu\nu}=-N^{\mu\nu}\sigma_{\mu\nu}\,,$

and

$\omega_{\alpha\beta}=\varepsilon_{\alpha\beta\mu}\left(\Omega
e^{\mu}+\Omega^{\mu}\right)\,,$ (24)

where $\Omega$ is given in Eq. (10) and

$\Omega^{\alpha}=\frac{1}{2}N_{\gamma}{}^{\alpha}\varepsilon^{\mu\nu\gamma}D_{\mu}u_{\nu}\,.$
(25)

The quantities $\theta$, $\Sigma$, $\tilde{\theta}$ and $\vartheta$ are not
independent, in fact:

$\displaystyle\tilde{\theta}$ $\displaystyle=\frac{2}{3}\theta-\Sigma\,,$ (26)
$\displaystyle\vartheta$ $\displaystyle=\frac{1}{3}\theta+\Sigma\,;$ (27)

as such, when setting up the 1+1+2 formalism, only two of those quantities are
chosen. The convention followed here uses the variables $\theta$ and $\Sigma$.

For the projected covariant derivative of the vector field $e$ on $V$ we have

$D_{\alpha}e_{\beta}=h_{\alpha}{}^{\mu}h_{\beta}{}^{\nu}\nabla_{\mu}e_{\nu}=\delta_{\alpha}e_{\beta}+e_{\alpha}a_{\beta}\,,$
(28)

where $\delta_{\alpha}e_{\beta}$ is given by Eq. (11) and

$a_{\alpha}=e^{\mu}D_{\mu}e_{\alpha}=\widehat{e}_{\alpha}\,.$ (29)

####  II.2.3 Decomposition of the covariant derivatives on the full manifold

Finally, we can decompose the total covariant derivatives of $u^{\alpha}$ and
$e^{\alpha}$, such that

$\nabla_{\alpha}u_{\beta}=-u_{\alpha}\left(\mathcal{A}e_{\beta}+\mathcal{A}_{\beta}\right)+D_{\alpha}u_{\beta}\,,$
(30)

with

$\displaystyle\mathcal{A}$
$\displaystyle=-u_{\mu}u^{\nu}\nabla_{\nu}e^{\mu}=-u_{\mu}\dot{e}^{\mu}\,,$
(31) $\displaystyle\mathcal{A}_{\alpha}$
$\displaystyle=N_{\alpha\mu}\dot{u}^{\mu}\,,$

and

$\displaystyle\nabla_{\alpha}e_{\beta}$
$\displaystyle=D_{\alpha}e_{\beta}-u_{\alpha}\alpha_{\beta}-\mathcal{A}u_{\alpha}u_{\beta}+\left(\frac{1}{3}\theta+\Sigma\right)e_{\alpha}u_{\beta}+\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}\right)u_{\beta}\,,$
(32)

where

$\alpha_{\alpha}=h_{\alpha}{}^{\mu}\dot{e}_{\mu}\,.$ (33)

###  II.3 Covariant derivatives of the projectors and the Levi-Civita volume
form

Gathering the results in subsection II.2, the covariant derivatives of the
tangent vector fields $u$ and $e$ can be written as

$\displaystyle\delta_{\alpha}u_{\beta}=N_{\alpha}{}^{\mu}N_{\beta}{}^{\nu}\nabla_{\mu}u_{\nu}$
$\displaystyle=N_{\alpha\beta}\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)+\Sigma_{\alpha\beta}+\varepsilon_{\alpha\beta}\Omega\,,$
(34) $\displaystyle
D_{\alpha}u_{\beta}=h_{\alpha}{}^{\mu}h_{\beta}{}^{\nu}\nabla_{\mu}u_{\nu}$
$\displaystyle=\delta_{\alpha}u_{\beta}+\left(\frac{1}{3}\theta+\Sigma\right)e_{\alpha}e_{\beta}+2\Sigma_{(\alpha}e_{\beta)}-\varepsilon_{\alpha\mu}\Omega^{\mu}e_{\beta}+e_{\alpha}\varepsilon_{\beta\mu}\Omega^{\mu}\,,$
$\displaystyle\nabla_{\alpha}u_{\beta}$
$\displaystyle=D_{\alpha}u_{\beta}-u_{\alpha}\left(\mathcal{A}e_{\beta}+\mathcal{A}_{\beta}\right)\,,$

and

$\displaystyle\delta_{\alpha}e_{\beta}=N_{\alpha}{}^{\mu}N_{\beta}{}^{\nu}\nabla_{\mu}e_{\nu}=$
$\displaystyle\frac{1}{2}N_{\alpha\beta}\phi+\zeta_{\alpha\beta}+\varepsilon_{\alpha\beta}\xi\,,$
(35) $\displaystyle
D_{\alpha}e_{\beta}=h_{\alpha}{}^{\mu}h_{\beta}{}^{\nu}\nabla_{\mu}e_{\nu}=$
$\displaystyle\delta_{\alpha}e_{\beta}+e_{\alpha}a_{\beta}\,,$
$\displaystyle\nabla_{\alpha}e_{\beta}=$ $\displaystyle
D_{\alpha}e_{\beta}-u_{\alpha}\alpha_{\beta}-\mathcal{A}u_{\alpha}u_{\beta}+\left(\frac{1}{3}\theta+\Sigma\right)e_{\alpha}u_{\beta}+\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}\right)u_{\beta}\,.$

From these relations, we find the following expressions for the covariant
derivatives of the projector tensors:

$\displaystyle\delta_{\alpha}N_{\beta\gamma}$ $\displaystyle=0\,,$
$\displaystyle\nabla_{\rho}\varepsilon_{\alpha\beta\gamma\delta}$
$\displaystyle=0\,,$ (36) $\displaystyle e^{\mu}D_{\mu}N_{\alpha\beta}$
$\displaystyle=-2e_{(\alpha}a_{\beta)}\,,$ $\displaystyle
u^{\mu}\nabla_{\mu}\varepsilon_{\alpha\beta\gamma}$
$\displaystyle=\varepsilon_{\alpha\beta\gamma\mu}\left(\mathcal{A}e^{\mu}+\mathcal{A}^{\mu}\right)\,,$
$\displaystyle u^{\mu}\nabla_{\mu}N_{\alpha\beta}$
$\displaystyle=2u_{(\alpha}\mathcal{A}_{\beta)}-2e_{(\alpha}\alpha_{\beta)}\,,$
$\displaystyle D_{\alpha}\varepsilon_{\beta\gamma\delta}$ $\displaystyle=0\,,$
$\displaystyle D_{\alpha}h_{\beta\gamma}$ $\displaystyle=0\,,$ $\displaystyle
u^{\mu}\nabla_{\mu}h_{\alpha\beta}$
$\displaystyle=2u_{(\alpha|}\left(\mathcal{A}e_{|\beta)}+\mathcal{A}_{|\beta)}\right)\,.$

###  II.4 Weyl and stress-energy tensors

For the Levi-Civita connection, the Riemann tensor can be defined by the Ricci
identity valid for an arbitrary 1-form $\chi$:

$R_{\alpha\beta\delta}{}^{\rho}\chi{}_{\rho}=\left(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha}\right)\chi_{\delta}\,.$
(37)

In the case of a manifold of dimension 4, the components of the Riemann
curvature tensor, $R_{\alpha\beta\gamma\delta}$, can be written as the
following sum

$R_{\alpha\beta\gamma\delta}=C_{\alpha\beta\gamma\delta}+R_{\alpha\left[\gamma\right.}g_{\left.\delta\right]\beta}-R_{\beta\left[\gamma\right.}g_{\left.\delta\right]\alpha}-\frac{1}{3}R\,g_{\alpha\left[\gamma\right.}g_{\left.\delta\right]\beta}\,,$
(38)

where $C_{\alpha\beta\gamma\delta}$ represent the components of the Weyl
tensor, $R_{\alpha\beta}:=R_{\alpha\mu\beta}{}^{\mu}$ the components of the
Ricci tensor and $R$ the Ricci scalar. The Weyl tensor plays a pivotal role in
relativistic gravity, describing the tidal forces and the properties of
gravitational waves. Remarkably, the Weyl 4-tensor can be completely
characterized by two 2-tensors, defined as

$\displaystyle E_{\alpha\beta}$
$\displaystyle=C_{\alpha\mu\beta\nu}u^{\mu}u^{\nu}\,,$ (39) $\displaystyle
H_{\alpha\beta}$
$\displaystyle=\frac{1}{2}\varepsilon_{\alpha}{}^{\mu\nu}C_{\mu\nu\beta\delta}u^{\delta}\,,$
(40)

respectively referred as the “electric” and “magnetic” part of the Weyl
tensor, both symmetric and traceless tensors, such that

$C_{\alpha\beta\gamma\delta}=-\varepsilon_{\alpha\beta\mu}\varepsilon_{\gamma\delta\nu}E^{\nu\mu}-2u_{\alpha}E_{\beta\left[\gamma\right.}u_{\left.\delta\right]}+2u_{\beta}E_{\alpha\left[\gamma\right.}u_{\left.\delta\right]}-2\varepsilon_{\alpha\beta\mu}H^{\mu}{}_{\left[\gamma\right.}u_{\left.\delta\right]}-2\varepsilon_{\mu\gamma\delta}H^{\mu}{}_{\left[\alpha\right.}u_{\left.\beta\right]}\,.$
(41)

In the 1+1+2 spacetime decomposition formalism, the components of the Weyl
tensor are given by

$\displaystyle E_{\alpha\beta}$
$\displaystyle=\mathcal{E}\left(e_{\alpha}e_{\beta}-\frac{1}{2}N_{\alpha\beta}\right)+\mathcal{E}_{\alpha}e_{\beta}+e_{\alpha}\mathcal{E}_{\beta}+\mathcal{E}_{\alpha\beta}\,,$
(42) $\displaystyle H_{\alpha\beta}$
$\displaystyle=\mathcal{H}\left(e_{\alpha}e_{\beta}-\frac{1}{2}N_{\alpha\beta}\right)+\mathcal{H}_{\alpha}e_{\beta}+e_{\alpha}\mathcal{H}_{\beta}+\mathcal{H}_{\alpha\beta}\,,$

where

$\displaystyle\mathcal{E}$
$\displaystyle=E_{\mu\nu}e^{\mu}e^{\nu}=-N^{\mu\nu}E_{\mu\nu}\,,$
$\displaystyle\mathcal{H}$
$\displaystyle=e^{\mu}e^{\nu}H_{\mu\nu}=-N^{\mu\nu}H_{\mu\nu}\,,$ (43)
$\displaystyle\mathcal{E}_{\alpha}$
$\displaystyle=N_{\alpha}{}^{\mu}e^{\nu}E_{\mu\nu}=e^{\mu}N_{\alpha}{}^{\nu}E_{\mu\nu}\,,$
$\displaystyle\mathcal{H}_{\alpha}$
$\displaystyle=N_{\alpha}{}^{\mu}e^{\nu}H_{\mu\nu}=e^{\mu}N_{\alpha}{}^{\nu}H_{\mu\nu}\,,$
$\displaystyle\mathcal{E}_{\alpha\beta}$
$\displaystyle=E_{\left\\{\alpha\beta\right\\}}\,,$
$\displaystyle\mathcal{H}_{\alpha\beta}$
$\displaystyle=H_{\left\\{\alpha\beta\right\\}}\,.$

We shall also decompose the metric stress-energy tensor, with components
$T_{\alpha\beta}$ in a local coordinate system, in terms of its pointwise
projections onto $u$, $e$, and $W$:

$T_{\alpha\beta}=\mu\,u_{\alpha}u_{\beta}+\left(p+\Pi\right)e_{\alpha}e_{\beta}+\left(p-\frac{1}{2}\Pi\right)N_{\alpha\beta}+2Qe_{\left(\alpha\right.}u_{\left.\beta\right)}+2Q_{\left(\alpha\right.}u_{\left.\beta\right)}+2\Pi_{\left(\alpha\right.}e_{\left.\beta\right)}+\Pi_{\alpha\beta}\,,$
(44)

with

$\displaystyle\mu$ $\displaystyle=u^{\mu}u^{\nu}T_{\mu\nu}\,,$ $\displaystyle
Q_{\alpha}$
$\displaystyle=-N_{\alpha}{}^{\mu}u^{\nu}T_{\mu\nu}=-u^{\mu}N_{\alpha}{}^{\nu}T_{\mu\nu}\,,$
(45) $\displaystyle p$
$\displaystyle=\frac{1}{3}\left(e^{\mu}e^{\nu}+N^{\mu\nu}\right)T_{\mu\nu}\,,$
$\displaystyle\Pi_{\alpha}$
$\displaystyle=N_{\alpha}{}^{\mu}e^{\nu}T_{\mu\nu}=e^{\mu}N_{\alpha}{}^{\nu}T_{\mu\nu}\,,$
$\displaystyle\Pi$
$\displaystyle=\frac{1}{3}T_{\alpha\beta}\left(2e^{\alpha}e^{\beta}-N^{\alpha\beta}\right)\,,$
$\displaystyle\Pi_{\alpha\beta}$
$\displaystyle=T_{\left\\{\alpha\beta\right\\}}\,.$ $\displaystyle Q$
$\displaystyle=-e^{\mu}u^{\nu}T_{\mu\nu}=-u^{\mu}e^{\nu}T_{\mu\nu}\,,$

Moreover, we have the following equations

$\displaystyle p_{r}$ $\displaystyle=e^{\mu}e^{\mu}T_{\mu\nu}=p+\Pi\,,$ (46)
$\displaystyle p_{\perp}$
$\displaystyle=\frac{1}{2}N^{\mu\nu}T_{\mu\nu}=p-\frac{1}{2}\Pi\,,$

relating, respectively, the 1+1+2 variables $p$ and $\Pi$ with the “radial”
and “tangential” components of the pressure.

###  II.5 Commutation relations

Given a scalar field $\chi$ in $\left(\mathcal{M},g\right)$, using the 1+1+2
spacetime decomposition, we have the following useful commutation relations
for its covariant derivatives:

$\displaystyle\widehat{\dot{\chi}}-\dot{\hat{\chi}}$
$\displaystyle=\left(\frac{1}{3}\theta+\Sigma\right)\widehat{\chi}-\mathcal{A}\dot{\chi}+\left(\Sigma_{\mu}+\varepsilon_{\mu\nu}\Omega^{\nu}-\alpha_{\mu}\right)\delta^{\mu}\chi\,,$
(47)
$\displaystyle\delta_{\alpha}\dot{\chi}-N_{\alpha}{}^{\mu}\left(\delta_{\mu}\chi\right)^{\cdot}$
$\displaystyle=\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)\delta_{\alpha}\chi+\left(\Sigma_{\alpha\mu}+\varepsilon_{\alpha\mu}\Omega\right)\delta^{\mu}\chi+\left(\Sigma_{\alpha}-\varepsilon_{\alpha\mu}\Omega^{\mu}+\alpha_{\alpha}\right)\widehat{\chi}-\mathcal{A}_{\alpha}\dot{\chi}\,,$
$\displaystyle\delta_{\alpha}\widehat{\chi}-N_{\alpha}{}^{\mu}\left(\widehat{\delta_{\mu}\chi}\right)$
$\displaystyle=\frac{1}{2}\phi\delta_{\alpha}\chi+\left(\zeta_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\xi\right)\delta_{\mu}\chi-2\varepsilon_{\alpha\mu}\Omega^{\mu}\dot{\chi}+\widehat{\chi}a_{\alpha}\,,$
$\displaystyle\delta_{\alpha}\delta_{\beta}\chi-\delta_{\beta}\delta_{\alpha}\chi$
$\displaystyle=2\varepsilon_{\alpha\beta}\Omega\dot{\chi}-2\varepsilon_{\alpha\beta}\xi\widehat{\chi}\,.$

In the case of the covariant derivatives of a 1-form field $\chi$ defined on
the sheet, that is $\chi_{\alpha}=N_{\alpha}{}^{\mu}\chi_{\mu}$, the
commutation relations take the form:

$\displaystyle
N_{\alpha}{}^{\mu}\widehat{\dot{\chi}_{\mu}}-N_{\alpha}{}^{\mu}\left(\widehat{\chi}{}_{\mu}\right)^{\cdot}$
$\displaystyle=\left(\frac{1}{3}\theta+\Sigma\right)N_{\alpha}{}^{\mu}\widehat{\chi}_{\mu}-\mathcal{A}N_{\alpha}{}^{\mu}\dot{\chi}{}_{\mu}+\chi_{\mu}\left(\Sigma^{\mu}+\varepsilon^{\mu\nu}\Omega_{\nu}\right)\mathcal{A}_{\alpha}$
(48)
$\displaystyle+\left(\Sigma^{\mu}+\varepsilon^{\mu\nu}\Omega_{\nu}-\alpha^{\mu}\right)\left(\delta_{\mu}\chi_{\alpha}\right)+\varepsilon_{\alpha}{}^{\mu}\chi_{\mu}\mathcal{H}\,,$
$\displaystyle\delta_{\alpha}\dot{\chi}_{\beta}-N_{\alpha}{}^{\mu}N_{\beta}{}^{\nu}\left(\delta_{\mu}\chi{}_{\nu}\right)^{\cdot}$
$\displaystyle=\left(\Sigma_{\alpha}+\alpha_{\alpha}-\varepsilon_{\alpha\lambda}\Omega^{\lambda}\right)N_{\beta}{}^{\mu}\widehat{\chi}_{\mu}-\mathcal{A}_{\alpha}N_{\beta}{}^{\mu}\dot{\chi}{}_{\mu}-\frac{1}{2}N_{\alpha\beta}Q^{\mu}\chi_{\mu}+\mathcal{H}_{\alpha}\varepsilon_{\beta}{}^{\mu}\chi_{\mu}$
(49)
$\displaystyle+\left[N_{\alpha}{}^{\mu}\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)+\Sigma_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\Omega\right]\left(\delta_{\mu}\chi_{\beta}\right)-\chi_{\mu}\left(\zeta_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\xi\right)\alpha_{\beta}$
$\displaystyle+\chi_{\alpha}\left[\mathcal{A}_{\beta}\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)+\frac{1}{2}Q_{\beta}-\frac{1}{2}\phi\alpha_{\beta}\right]+\chi_{\mu}\left(\Sigma_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\Omega\right)\mathcal{A}_{\beta}\,,$
$\displaystyle\delta_{\alpha}\widehat{\chi}_{\beta}-N_{\alpha}{}^{\mu}N_{\beta}{}^{\nu}\left(\widehat{\delta_{\mu}\chi_{\nu}}\right)$
$\displaystyle=a_{\alpha}N_{\beta}{}^{\mu}\widehat{\chi}_{\mu}-2\varepsilon_{\alpha\mu}\Omega^{\mu}N_{\beta}{}^{\nu}\dot{\chi}_{\nu}+\left(\frac{1}{2}N_{\alpha}{}^{\mu}\phi+\zeta_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\xi\right)\delta_{\mu}\chi_{\beta}$
(50)
$\displaystyle+\chi_{\alpha}\left[\left(\Sigma_{\beta}+\varepsilon_{\beta\mu}\Omega^{\mu}\right)\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)-\mathcal{E}_{\beta}-\frac{1}{2}\Pi_{\beta}-\frac{1}{2}\phi
a_{\beta}\right]$
$\displaystyle+\chi_{\mu}\left[\left(\Sigma_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\Omega\right)\left(\Sigma_{\beta}+\varepsilon_{\beta\lambda}\Omega^{\lambda}\right)-\left(\zeta_{\alpha}{}^{\mu}+\varepsilon_{\alpha}{}^{\mu}\xi\right)a_{\beta}\right]$
$\displaystyle+N_{\alpha\beta}\chi_{\mu}\left[\mathcal{E}^{\mu}+\frac{1}{2}\Pi^{\mu}-\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)\left(\Sigma^{\mu}+\varepsilon^{\mu\nu}\Omega_{\nu}\right)\right]$
$\displaystyle-\chi_{\mu}\left(\Sigma_{\alpha\beta}+\varepsilon_{\alpha\beta}\Omega\right)\left(\Sigma^{\mu}+\varepsilon^{\mu\nu}\Omega_{\nu}\right)\,,$
$\displaystyle\delta_{\alpha}\delta_{\beta}\chi_{\gamma}-\delta_{\beta}\delta_{\alpha}\chi_{\gamma}$
$\displaystyle=2\varepsilon_{\alpha\beta}N_{\gamma}{}^{\mu}\left(\dot{\chi}_{\mu}\Omega-\widehat{\chi}_{\mu}\xi\right)+2\chi_{\left[\alpha\right.}N_{\left.\beta\right]\gamma}\left[\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)^{2}-\frac{1}{4}\phi^{2}+\mathcal{E}+\frac{1}{2}\Pi-\frac{1}{3}\left(\mu+\Lambda\right)\right]$
(51)
$\displaystyle+2\left[\left(\Sigma_{\left[\alpha\right|}{}^{\mu}+\varepsilon_{\left[\alpha\right|}{}^{\mu}\Omega\right)\left(\Sigma_{\left|\beta\right]\gamma}+\varepsilon_{\left|\beta\right]\gamma}\Omega\right)-\left(\zeta_{\left[\alpha\right|}{}^{\mu}+\varepsilon_{\left[\alpha\right|}{}^{\mu}\xi\right)\left(\zeta_{\left|\beta\right]\gamma}+\varepsilon_{\left|\beta\right]\gamma}\xi\right)\right]\chi_{\mu}$
$\displaystyle+2\left[\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)\left(\Sigma_{\left[\alpha\right|}{}^{\mu}+\varepsilon_{\left[\alpha\right|}{}^{\mu}\Omega\right)-\frac{1}{2}\phi\left(\zeta_{\left[\alpha\right|}{}^{\mu}+\varepsilon_{\left[\alpha\right|}{}^{\mu}\xi\right)-\frac{1}{2}\Pi_{\left[\alpha\right|}{}^{\mu}\right]N_{\left|\beta\right]\gamma}\chi_{\mu}$
$\displaystyle+2\chi_{\left[\alpha\right|}\left[\left(\frac{1}{3}\theta-\frac{1}{2}\Sigma\right)\left(\Sigma_{\left|\beta\right]\gamma}+\varepsilon_{\left|\beta\right]\gamma}\Omega\right)-\frac{1}{2}\phi\left(\zeta_{\left|\beta\right]\gamma}+\varepsilon_{\left|\beta\right]\gamma}\xi\right)-\frac{1}{2}\Pi_{\left|\beta\right]\gamma}\right]\,.$

###  II.6 The set of 1+1+2 equations

The 1+1+2 variables introduced in the previous subsections completely describe
the geometry of the spacetime manifold and the properties of the matter fields
that permeate it. Using the Ricci and Bianchi identities and the Einstein
field equations for the theory of General Relativity,

$R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R+g_{\alpha\beta}\Lambda=T_{\alpha\beta}\,,$
(52)

where $\Lambda$ represents the cosmological constant, we can find a set of
evolution, propagation, and constraint equations for the 1+1+2 variables. In
Appendix A we give the general set of these equations. Using the freedom of
choice of frame to describe the setup and providing an equation of state for
the matter fields, those equations form a closed system, describing the
geometry of the manifold and the dynamics of the permeating matter fields.

##  III Linearized equations for the perturbed spacetime

###  III.1 Background spacetime

Having properly introduced the 1+1+2 covariant formalism, we are now in a
position to derive a set of covariant and gauge invariant equations to
describe linear order perturbations of a spacetime assumed to be static, LRS
II, and permeated by a general matter fluid. As is customary in relativistic
perturbation theory, throughout the article, we will refer to the unperturbed
spacetime as the “background spacetime”.

For a proper choice of frame, in an LRS II spacetime, all covariantly defined
vector and tensor quantities of the 1+1+2 decomposition can be made to vanish
identically. If, in addition, the spacetime is static, we can align the $u$
vector field with the timelike, hypersurface orthogonal Killing vector field,
such that all dot-derivatives of the covariantly defined quantities are zero
and the scalars $\left\\{\theta,\Sigma,\Omega,\xi,\mathcal{H},Q\right\\}$ also
vanish. Hence, a static LRS II background spacetime can be completely
characterized by the quantities
$\left\\{\phi_{0},\mathcal{A}_{0},\mathcal{E}_{0},\mu_{0},p_{0},\Pi_{0},\Lambda\right\\}$,
which satisfy the following equations

$\displaystyle\mathcal{\widehat{A}}_{0}$
$\displaystyle=\frac{1}{2}\left(\mu_{0}+3p_{0}\right)-\Lambda-\mathcal{A}_{0}\left(\mathcal{A}_{0}+\phi_{0}\right)\,,$
(53) $\displaystyle\widehat{\phi}_{0}$
$\displaystyle=-\frac{1}{2}\phi_{0}^{2}-\frac{2}{3}\left(\mu_{0}+\Lambda\right)-\frac{1}{2}\Pi_{0}-\mathcal{E}_{0}\,,$
$\displaystyle\widehat{p}_{0}+\widehat{\Pi}_{0}$
$\displaystyle=-\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{0}-\left(\mu_{0}+p_{0}\right)\mathcal{A}_{0}\,,$
$\displaystyle\widehat{\mathcal{E}}_{0}-\frac{1}{3}\widehat{\mu}_{0}+\frac{1}{2}\widehat{\Pi}_{0}$
$\displaystyle=-\frac{3}{2}\phi_{0}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\,,$

and the constraint

$\displaystyle\mu_{0}+3p_{0}-2\Lambda-3\mathcal{A}_{0}\phi_{0}$
$\displaystyle=-\frac{3}{2}\Pi_{0}+3\mathcal{E}_{0}\,,$ (54)

where the subscript “$0$” from hereon will be used to refer to the quantities
that characterize the equilibrium configuration. The system is closed by
providing an equation of state that relates the pressure components, $p_{0}$
and $\Pi_{0}$, with the energy density, $\mu_{0}$, or some functional
dependencies for these matter variables are imposed.

###  III.2 Gauge invariant variables

Following the Stewart-Walker lemma [12], to write a set of equations for the
linear perturbations that are identification gauge invariant, we will consider
perturbation variables that either vanish in the background spacetime or are
scalars with the same constant value in the background and the perturbed
spacetimes (e.g., the cosmological constant). Since the background spacetime
is assumed to be static LRS II, provided the choice of frame adopted in the
previous subsection, all vector and tensor quantities can be used as
identification gauge invariant perturbation quantities. Moreover, for such
spacetimes, the scalars
$\left\\{\theta,\Sigma,\Omega,\xi,\mathcal{H},Q\right\\}$ are identically
zero, hence are identification gauge invariant quantities. Then, we only have
to find a set of variables that vanish in the background spacetime, and that
can be used to characterize the linear perturbations of the remaining scalars,
i.e. $\left\\{\mathcal{A},\phi,\mathcal{E},\mu,p,\Pi\right\\}$. Given the
symmetries of the background spacetime, two natural sets of variables can be
adopted. The first one is given by

$\displaystyle\mathbb{A}_{\alpha}$
$\displaystyle:=\delta_{\alpha}\mathcal{A}\,,$
$\displaystyle\mathbb{F}_{\alpha}$ $\displaystyle:=\delta_{\alpha}\phi\,,$
$\displaystyle\mathbb{E}_{\alpha}$
$\displaystyle:=\delta_{\alpha}\mathcal{E}\,,$ (55)
$\displaystyle\mathfrak{m}_{\alpha}$ $\displaystyle:=\delta_{\alpha}\mu\,,$
$\displaystyle\mathfrak{p}_{\alpha}$ $\displaystyle:=\delta_{\alpha}p\,,$
$\displaystyle\mathbb{P}_{a}$ $\displaystyle:=\delta_{\alpha}\Pi\,,$

which represent the gradients on the sheets of the various scalar quantities
that describe the background spacetime. The second one instead contains

$\displaystyle\mathsf{A}$ $\displaystyle:=\dot{\mathcal{A}}\,,$
$\displaystyle\mathsf{F}$ $\displaystyle:=\dot{\phi}\,,$
$\displaystyle\mathsf{E}$ $\displaystyle:=\dot{\mathcal{E}}\,,$ (56)
$\displaystyle\mathsf{m}$ $\displaystyle:=\dot{\mu}\,,$
$\displaystyle\mathsf{p}$ $\displaystyle:=\dot{p}\,,$
$\displaystyle\mathcal{P}$ $\displaystyle:=\dot{\Pi}\,,$

which represent the dot-derivatives of the various scalar quantities that
describe the spacetime. In a static LRS II background, both the
$\delta$-gradients in Eq. (55), which are vectors, and the quantities in Eq.
(56), which are related to the proper-time variation, vanish. Therefore, the
quantities of the sets above can be used as gauge invariant variables that
characterize the linear perturbations of a static LRS II spacetime.

In principle, we can use either set among Eq. (55) and (56). Moreover, as
shown in Eq. (355), the above perturbation variables are not independent.
Then, in general, we can also construct a set of variables containing elements
of both Eq. (55) and (56), provided that they form a closed system. Note,
however, that dependence does not imply equivalence. For example, the
$\delta$-gradients variables in Eq. (55) preserve the information regarding
the degrees of freedom on the sheet, however, if we only perturb the
background spacetime in directions parallel to the $u$ and $e$ vector fields,
these variables by themselves are not suitable to fully describe the perturbed
spacetime, as they will remain identically zero. Conversely, the variables in
Eq. (56) can be used to fully characterize linear perturbations of
$\left\\{\mathcal{A},\phi,\mathcal{E},\mu,p,\Pi\right\\}$, only losing
information in the case of constant-in-time perturbations. Thus, this second
set of variables can appear more convenient to analyze the dynamics of the
perturbations. The drawback is that some of the linearized 1+1+2 equations for
the variables (56) are second order in time (cf. Appendix C), introducing
extra complexity to the problem. Then, in what follows, we will opt for using
the $\delta$-gradients in Eq. (55) only to describe vector and tensor
perturbations, since those will be characterized by first-order equations.
Conversely, we will use the variables in Eq. (56) to describe the scalar
modes.

###  III.3 The linearization procedure

The procedure to find the set of gauge invariant structure equations for
linear perturbations of static, LRS II spacetimes is relatively
straightforward, although rather laborious. After choosing the gauge-invariant
quantities from those in Eqs. (55) or (56), we deduce the equations for these
quantities by applying the projected derivative operators “dot" and
$\delta_{\alpha}$ to the 1+1+2 equations in Appendix A. Then, using the
commutation relations in subsection II.5, we express all gauge-dependent terms
as a combination of gauge-independent quantities. Successively, the terms
containing the products of two or more first-order quantities are discarded as
higher order. We present in Appendices B and C the general set of gauge
independent field equations, valid at linear level, for the perturbations for
both the $\delta$-gradients and the dot-derivatives variables, respectively.
As explained above, we will opt to use a mix of these equations to describe
general linear perturbations. However, the set of equations in each appendix
mentioned above is completely general and can be used by themselves.

###  III.4 Harmonic decomposition

The sets of the linearized equations in Appendices B and C form two systems of
partial differential equations. The appearance of $\delta$-derivatives makes
the system particularly complicated to integrate, such that finding solutions,
even in relatively simple setups, an intractable problem. Following Refs. [21,
22, 23], to transform this system into a system of ordinary differential
equations (ODEs) valid at linear level, we can use the fact that the
background spacetime is static and LRS II.

Using the eigenfunctions of the projected covariant Laplace-Beltrami operator,
$\delta^{2}\equiv\delta_{\mu}\delta^{\mu}$, locally defined on the sheets of
the LRSII background spacetime, we can make a harmonic decomposition and write
the various quantities that characterize the perturbed spacetime as a linear
combination of these eigenfunctions. In Appendix D, we list the definitions
and various useful properties of the scalar, vector, and tensor harmonics of
the $\delta^{2}$ operator. Then, given some scalar, 1-tensor, or symmetric,
traceless 2-tensor quantity defined on a sheet of the perturbed spacetime, say
$\chi$, $\chi_{\alpha}$ or $\chi_{\alpha\beta}$, at linear level we can
formally write these as the following infinite sum

$\displaystyle\chi$
$\displaystyle=\Psi_{\chi}^{\left(k,S\right)}\mathcal{Q}^{\left(k\right)}\,,$
$\displaystyle\chi_{\alpha}$
$\displaystyle=\Psi_{\chi}^{\left(k,V\right)}\mathcal{Q}_{\alpha}^{\left(k\right)}+\overline{\Psi}_{\chi}^{\left(k,V\right)}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\,,$
(57) $\displaystyle\chi_{\alpha\beta}$
$\displaystyle=\Psi_{\chi}^{\left(k,T\right)}\mathcal{Q}_{\alpha\beta}^{\left(k\right)}+\overline{\Psi}_{\chi}^{\left(k,T\right)}\bar{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}\,,$

where summation or integration in the Laplace-Beltrami eigenvalue $k$ (the so-
called “modes” ) is assumed, depending on the geometry of the sheets. We will
use the compact notation $\Psi_{\chi}^{\left(k,S\right)}$ to refer to the
harmonic coefficients associated with the eigenfunctions
$\mathcal{Q}^{\left(k\right)}$ of the scalar quantity $\chi$,
$\Psi_{\chi}^{\left(k,V\right)}$ and
$\overline{\Psi}_{\chi}^{\left(k,V\right)}$ to refer to the coefficients
associated with the eigenfunctions $\mathcal{Q}_{\alpha}^{\left(k\right)}$ or
$\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}$ of the 1-tensor quantity
$\chi_{\alpha}$, and $\Psi_{\chi}^{\left(k,T\right)}$ and
$\overline{\Psi}_{\chi}^{\left(k,T\right)}$ to refer to the coefficients
associated with the eigenfunctions
$\mathcal{Q}_{\alpha\beta}^{\left(k\right)}$ or
$\bar{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}$ of the 2-tensor quantity
$\chi_{\alpha\beta}$.

Additionally, since the background spacetime is assumed to be static, we can
Fourier transform the harmonic coefficients of Eq. (III.4), explicitly
factorizing their time dependence, writing them as a linear combination of the
eigenfunctions of the Laplace operator in $\mathbb{R}$ (or some subset of it)
for the appropriate boundary conditions. That is, we can write all
coefficients for linear perturbations: $\Psi_{\chi}^{\left(k,I\right)}$, where
$I=\left\\{S,V,T\right\\}$, as linear combinations of the eigenfunctions
$e^{i\upsilon\tau}$, where $\tau$ represents the proper time of an observer
with 4-velocity $u$ and $\upsilon$ represent the eigenvalues of the Laplace
operator in that manifold. The scalar $\upsilon$ takes discrete or continuous
values depending on the chosen boundary conditions. To make this idea precise,
consider the time-harmonic functions $T^{\left(\upsilon\right)}$ with the
following properties

$\displaystyle\dot{T}^{\left(\upsilon\right)}$ $\displaystyle=i\upsilon
T^{\left(\upsilon\right)}\,,$ (58)
$\displaystyle\widehat{T}^{\left(\upsilon\right)}=\delta_{\alpha}T^{\left(\upsilon\right)}$
$\displaystyle=0\,,$ $\displaystyle\dot{\upsilon}=\delta_{\alpha}\upsilon$
$\displaystyle=0\,,$

where $i$ represents the imaginary unit. These properties imply that
$\widehat{\dot{T}}^{\left(\upsilon\right)}=-\mathcal{A}_{0}\dot{T}^{\left(\upsilon\right)}$,
from which we find

$\widehat{\upsilon}=-\mathcal{A}_{0}\upsilon\,.$ (59)

Knowing the function $\mathcal{A}_{0}=\mathcal{A}_{0}\left(x^{\alpha}\right)$
in the background, where $\left\\{x^{\alpha}\right\\}$ represent some local
coordinate system, Eq. (59) allow us to relate the eigenfrequencies $\upsilon$
defined with respect to $\tau$ and the eigenfrequencies defined with respect
to the $x^{0}$ time coordinate.

Gathering these results, formally, we can expand any scalar, 1-tensor or a
symmetric, traceless 2-tensor characterizing first-order quantities as

$\displaystyle\chi$
$\displaystyle=\Psi_{\chi}^{\left(\upsilon,k,S\right)}\mathcal{Q}^{\left(k\right)}T^{\left(\upsilon\right)}\,,$
(60) $\displaystyle\chi_{\alpha}$
$\displaystyle=\left(\Psi_{\chi}^{\left(\upsilon,k,V\right)}\mathcal{Q}_{\alpha}^{\left(k\right)}+\overline{\Psi}_{\chi}^{\left(\upsilon,k,V\right)}\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\right)T^{\left(\upsilon\right)}\,,$
$\displaystyle\chi_{\alpha\beta}$
$\displaystyle=\left(\Psi_{\chi}^{\left(\upsilon,k,T\right)}\mathcal{Q}_{\alpha\beta}^{\left(k\right)}+\overline{\Psi}_{\chi}^{\left(\upsilon,k,T\right)}\bar{\mathcal{Q}}_{\alpha\beta}^{\left(k\right)}\right)T^{\left(\upsilon\right)}\,,$

where either discrete sums or multiple integrals in $\upsilon$ and $k$ are
assumed, once again, depending on the boundary conditions of the problem.

Using the harmonic decomposition just described, we can turn the sets of
partial differential equations in Appendices B and C, in a set of ODEs. We
remark, though, that since the different harmonics are not defined for all
values of $k$, we have to distinguish between the modes where the vector and
tensor harmonics are not identically zero and those where these harmonics are
zero. Then, we have the following set of gauge invariant equations for the
linear perturbations that characterize the perturbed spacetime.

###  III.5 Even sector

####  III.5.1 $k$-Modes where
$\left\\{\mathcal{Q}_{\alpha\beta}^{\left(k\right)},\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}\right\\}\neq
0$

##### Equations for the kinematical quantities associated with the timelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of gradients
of scalar quantities:

$\displaystyle\frac{1}{r}i\upsilon\Psi_{\mathbb{\theta}}^{\left(\upsilon,k,S\right)}-\widehat{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\mathcal{\widehat{A}}_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\left(\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+3\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}\right)-\frac{k^{2}}{r^{2}}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}$
(61)
$\displaystyle+\mathcal{A}_{0}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\frac{2}{3r}i\upsilon\Psi_{\mathbb{\theta}}^{\left(\upsilon,k,S\right)}-\frac{1}{r}i\upsilon\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}$
$\displaystyle=\phi_{0}\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\frac{k^{2}}{r^{2}}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}$
(62)
$\displaystyle-\frac{1}{3}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}\,,$
$\frac{3}{2}\widehat{\Psi}_{\Sigma}^{\left(\upsilon,k,S\right)}-\widehat{\Psi}_{\mathbb{\theta}}^{\left(\upsilon,k,S\right)}=\frac{3k^{2}}{2r}\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\Psi_{Q}^{\left(\upsilon,k,S\right)}-\frac{9}{4}\phi_{0}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}\,;$
(63)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$2i\upsilon\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}+\widehat{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}=\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\phi_{0}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}-\mathcal{A}_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}\,,$
(64)
$i\upsilon\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\,,$
(65)

$\displaystyle\widehat{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}+\widehat{\overline{\Psi}}_{\Omega}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{1}{2r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}+\frac{2}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}-\frac{3}{2}\phi_{0}\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}-\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}$
(66)
$\displaystyle-\frac{2-k^{2}}{2r}\Psi_{\Sigma}^{\left(\upsilon,k,T\right)}-\Psi_{Q}^{\left(\upsilon,k,V\right)}\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of tensor
quantities:

$i\upsilon\Psi_{\Sigma}^{\left(\upsilon,k,T\right)}=\frac{1}{r}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\Psi_{\zeta}^{\left(\upsilon,k,T\right)}-\Psi_{\mathcal{E}}^{\left(\upsilon,k,T\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,T\right)}\,,$
(67)
$\widehat{\Psi}_{\Sigma}^{\left(\upsilon,k,T\right)}=\frac{1}{r}\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{2}\phi_{0}\Psi_{\Sigma}^{\left(\upsilon,k,T\right)}+\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}\,;$
(68)

  * •

Constraint equations for the harmonic coefficients:

$\phi_{0}\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)+\frac{1}{r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}-\frac{2}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}+\Psi_{Q}^{\left(\upsilon,k,V\right)}-2\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{2-k^{2}}{r}\Psi_{\Sigma}^{\left(\upsilon,k,T\right)}=0\,.$
(69)

##### Equations for the kinematical quantities associated with the spacelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of gradients
of scalar quantities:

$\displaystyle i\upsilon\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\widehat{\phi}_{0}\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\alpha}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)+\frac{1}{r}\Psi_{Q}^{\left(\upsilon,k,S\right)}$
(70)
$\displaystyle+\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\left(\frac{2}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}-\frac{1}{r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}\right)-\frac{k^{2}}{r^{2}}\Psi_{\alpha}^{\left(\upsilon,k,V\right)}\,,$
$\widehat{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\left(\widehat{\phi}_{0}+\frac{k^{2}}{r^{2}}\right)\Psi_{a}^{\left(\upsilon,k,V\right)}-\frac{3}{2}\phi_{0}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\frac{2}{3}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}\,;$
(71)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle\widehat{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}-i\upsilon\Psi_{a}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\left(\mathcal{A}_{0}+\frac{1}{2}\phi_{0}\right)\Psi_{\alpha}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{Q}^{\left(\upsilon,k,V\right)}$
(72)
$\displaystyle+\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of tensor
quantities:

$i\upsilon\Psi_{\zeta}^{\left(\upsilon,k,T\right)}=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Psi_{\Sigma}^{\left(\upsilon,k,T\right)}+\frac{1}{r}\Psi_{\alpha}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}\,,$
(73)
$\widehat{\Psi}_{\zeta}^{\left(\upsilon,k,T\right)}=\frac{1}{r}\Psi_{a}^{\left(\upsilon,k,V\right)}-\phi_{0}\Psi_{\zeta}^{\left(\upsilon,k,T\right)}-\Psi_{\mathcal{E}}^{\left(\upsilon,k,T\right)}-\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,T\right)}\,;$
(74)

  * •

Constraint equations for the harmonic coefficients:

$\frac{2-k^{2}}{r}\Psi_{\zeta}^{\left(\upsilon,k,T\right)}-\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-2\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-\Psi_{\Pi}^{\left(\upsilon,k,V\right)}=0\,.$
(75)

##### Equations for the Weyl tensor components and the matter variables

  * •

Evolution and propagation equations for the harmonic coefficients of gradients
of scalar quantities:

$\displaystyle\frac{1}{r}\widehat{\Psi}_{Q}^{\left(\upsilon,k,S\right)}+i\upsilon\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\widehat{\mu}_{0}\left(\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}+\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\alpha}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{r}\left(\mu_{0}+p_{0}\right)\Psi_{\theta}^{\left(\upsilon,k,S\right)}$
(76)
$\displaystyle-\frac{3}{2r}\Pi_{0}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}-\frac{1}{r}\left(\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{Q}^{\left(\upsilon,k,S\right)}+\frac{k^{2}}{r^{2}}\Psi_{Q}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\frac{1}{3}i\upsilon\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}i\upsilon\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-i\upsilon\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}-3\mathcal{E}_{0}\right)\left(\frac{1}{2r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}-\frac{1}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}\right)$
(77)
$\displaystyle-\frac{3}{2}\phi_{0}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}+\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\alpha}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle+\frac{k^{2}}{r^{2}}\left(\frac{1}{2}\Psi_{Q}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{2r}\phi_{0}\Psi_{Q}^{\left(\upsilon,k,S\right)}\,,$

$\displaystyle\frac{1}{r}i\upsilon\Psi_{Q}^{\left(\upsilon,k,S\right)}+\widehat{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\widehat{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\left(\mu_{0}+p_{0}\right)\left(\mathcal{A}_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}-\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)-\mathcal{A}_{0}\left(\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)$
(78)
$\displaystyle-\frac{1}{2}\phi_{0}\left(\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+4\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)+\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}$
$\displaystyle-\Pi_{0}\left(\frac{3}{2}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)+\frac{k^{2}}{r^{2}}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\,,$

$\displaystyle\widehat{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{3}\widehat{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-2\phi_{0}\left(\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{12}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}\right)$
(79)
$\displaystyle-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\phi_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle+\frac{k^{2}}{r^{2}}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle
i\upsilon\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle=\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\frac{1}{2}\Psi_{Q}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{\alpha}^{\left(\upsilon,k,V\right)}$
(80)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+\Pi_{0}\right)\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle-\frac{1}{2r}\Psi_{Q}^{\left(\upsilon,k,S\right)}+\frac{2-k^{2}}{2r}\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}\,,$
$\displaystyle\widehat{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{3}{2}\phi_{0}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{a}^{\left(\upsilon,k,V\right)}$
(81)
$\displaystyle+\frac{1}{2}\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{3}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\frac{1}{4}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}$
$\displaystyle-\frac{2-k^{2}}{2r}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,T\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,T\right)}\right)\,,$
$\displaystyle\frac{1}{2}\widehat{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-i\upsilon\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{4}\widehat{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{3}{2}\mathcal{E}_{0}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\frac{3}{4}\left(\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{4}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{a}^{\left(\upsilon,k,V\right)}$
(82)
$\displaystyle-\frac{1}{4}\phi_{0}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)-\mathcal{A}_{0}\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}$
$\displaystyle+\frac{2-k^{2}}{4r}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,T\right)}-\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,T\right)}\right)\,,$
$\displaystyle\widehat{\overline{\Psi}}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\widehat{\Psi}_{Q}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{2-k^{2}}{2r}\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}-\frac{1}{2r}\Psi_{Q}^{\left(\upsilon,k,S\right)}-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}$
(83)
$\displaystyle-\left(\mu_{0}+p_{0}+\frac{1}{4}\Pi_{0}-\frac{3}{2}\mathcal{E}_{0}\right)\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}-\frac{3}{2}\phi_{0}\left(\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{6}\Psi_{Q}^{\left(\upsilon,k,V\right)}\right)\,,$
$\displaystyle
i\upsilon\Psi_{Q}^{\left(\upsilon,k,V\right)}+\widehat{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\frac{3}{2}\Pi_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}-\frac{2-k^{2}}{2r}\Psi_{\Pi}^{\left(\upsilon,k,T\right)}$
(84)
$\displaystyle-\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Psi_{\Pi}^{\left(\upsilon,k,V\right)}-\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of tensor
quantities:

$\displaystyle
i\upsilon\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,T\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,T\right)}\right)+\widehat{\overline{\Psi}}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}=$
$\displaystyle-\frac{1}{r}\left(\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{Q}^{\left(\upsilon,k,V\right)}\right)-\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}$
(85)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+3\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\Sigma}^{\left(\upsilon,k,T\right)}\,,$
$\displaystyle\frac{1}{2}\widehat{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}-\widehat{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}-i\upsilon\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}$
$\displaystyle=\frac{1}{r}\left(\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}-\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{4}\phi_{0}\Psi_{\Pi}^{\left(\upsilon,k,T\right)}$
(86)
$\displaystyle-\frac{3}{2}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\zeta}^{\left(\upsilon,k,T\right)}+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{\mathcal{E}}^{\left(\upsilon,k,T\right)}\,.$

####  III.5.2 $k$-Modes where
$\left\\{\mathcal{Q}_{\alpha}^{\left(k\right)},\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\right\\}\neq
0\wedge\left\\{\mathcal{Q}_{\alpha\beta}^{\left(k\right)},\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}\right\\}=0$:
$k^{2}=2$

For the $k^{2}=2$ modes, the tensor harmonics are not defined, but in general,
the vector and scalar harmonics do not vanish. For those modes, we find the
following equations for the harmonic coefficients.

##### Equations for the kinematical quantities associated with the timelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of gradients
of scalar quantities:

$\displaystyle\frac{1}{r}i\upsilon\Psi_{\mathbb{\theta}}^{\left(\upsilon,k,S\right)}-\widehat{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\mathcal{\widehat{A}}_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}-\frac{2}{r^{2}}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\left(\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+3\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}\right)$
(87)
$\displaystyle+\mathcal{A}_{0}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\frac{2}{3r}i\upsilon\Psi_{\mathbb{\theta}}^{\left(\upsilon,k,S\right)}-\frac{1}{r}i\upsilon\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}$
$\displaystyle=\phi_{0}\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\frac{2}{r^{2}}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}$
(88)
$\displaystyle-\frac{1}{3}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}\,,$
$\frac{3}{2}\widehat{\Psi}_{\Sigma}^{\left(\upsilon,k,S\right)}-\widehat{\Psi}_{\mathbb{\theta}}^{\left(\upsilon,k,S\right)}=\frac{3}{r}\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\Psi_{Q}^{\left(\upsilon,k,S\right)}-\frac{9}{4}\phi_{0}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}\,,$
(89)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$2i\upsilon\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}+\widehat{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}=\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\phi_{0}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}-\mathcal{A}_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}\,,$
(90)
$i\upsilon\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\,,$
(91)

$\widehat{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}+\widehat{\overline{\Psi}}_{\Omega}^{\left(\upsilon,k,V\right)}=\frac{1}{2r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}+\frac{2}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}-\frac{3}{2}\phi_{0}\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}-\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}-\Psi_{Q}^{\left(\upsilon,k,V\right)}\,;$
(92)

  * •

Constraint equations for the harmonic coefficients:

$\phi_{0}\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)+\frac{1}{r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}-\frac{2}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}+\Psi_{Q}^{\left(\upsilon,k,V\right)}-2\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}=0\,.$
(93)

##### Equations for the kinematical quantities associated with the spacelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of gradients
of scalar quantities:

$\displaystyle i\upsilon\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\widehat{\phi}_{0}\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\alpha}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)+\frac{1}{r}\Psi_{Q}^{\left(\upsilon,k,S\right)}$
(94)
$\displaystyle+\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\left(\frac{2}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}-\frac{1}{r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}\right)-\frac{2}{r^{2}}\Psi_{\alpha}^{\left(\upsilon,k,V\right)}\,,$
$\widehat{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\left(\widehat{\phi}_{0}+\frac{2}{r^{2}}\right)\Psi_{a}^{\left(\upsilon,k,V\right)}-\frac{3}{2}\phi_{0}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\frac{2}{3}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}\,;$
(95)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle\widehat{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}-i\upsilon\Psi_{a}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\left(\mathcal{A}_{0}+\frac{1}{2}\phi_{0}\right)\Psi_{\alpha}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{Q}^{\left(\upsilon,k,V\right)}$
(96)
$\displaystyle+\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)\,;$

  * •

Constraint equations for the harmonic coefficients:

$\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+2\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\Psi_{\Pi}^{\left(\upsilon,k,V\right)}=0\,.$
(97)

##### Equations for the Weyl tensor components and the matter variables

  * •

Evolution and propagation equations for the harmonic coefficients of gradients
of scalar quantities:

$\displaystyle\frac{1}{r}\widehat{\Psi}_{Q}^{\left(\upsilon,k,S\right)}+i\upsilon\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\widehat{\mu}_{0}\left(\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}+\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\alpha}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{r}\left(\mu_{0}+p_{0}\right)\Psi_{\theta}^{\left(\upsilon,k,S\right)}$
(98)
$\displaystyle-\frac{3}{2r}\Pi_{0}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}-\frac{1}{r}\left(\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{Q}^{\left(\upsilon,k,S\right)}+\frac{2}{r^{2}}\Psi_{Q}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\frac{1}{3}i\upsilon\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}i\upsilon\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-i\upsilon\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}-3\mathcal{E}_{0}\right)\left(\frac{1}{2r}\Psi_{\Sigma}^{\left(\upsilon,k,S\right)}-\frac{1}{3r}\Psi_{\theta}^{\left(\upsilon,k,S\right)}\right)$
(99)
$\displaystyle-\frac{3}{2}\phi_{0}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}+\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\alpha}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle+\frac{2}{r^{2}}\left(\frac{1}{2}\Psi_{Q}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{2r}\phi_{0}\Psi_{Q}^{\left(\upsilon,k,S\right)}\,,$

$\displaystyle\frac{1}{r}i\upsilon\Psi_{Q}^{\left(\upsilon,k,S\right)}+\widehat{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\widehat{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\left(\mu_{0}+p_{0}\right)\left(\mathcal{A}_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}-\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)-\mathcal{A}_{0}\left(\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)$
(100)
$\displaystyle-\frac{1}{2}\phi_{0}\left(\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+4\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)+\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}$
$\displaystyle-\Pi_{0}\left(\frac{3}{2}\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+\Psi_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)+\frac{2}{r^{2}}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\,,$

$\displaystyle\widehat{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{3}\widehat{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-2\phi_{0}\left(\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{12}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}\right)$
(101)
$\displaystyle-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\Psi_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\phi_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle+\frac{2}{r^{2}}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle
i\upsilon\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle=\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\frac{1}{2}\Psi_{Q}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{\alpha}^{\left(\upsilon,k,V\right)}$
(102)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+\Pi_{0}\right)\left(\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{2r}\Psi_{Q}^{\left(\upsilon,k,S\right)}\,,$
$\displaystyle\widehat{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{3}{2}\phi_{0}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{a}^{\left(\upsilon,k,V\right)}$
(103)
$\displaystyle+\frac{1}{2}\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{3}\Psi_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\frac{1}{4}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\frac{1}{2}\widehat{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-i\upsilon\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{4}\widehat{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{3}{2}\mathcal{E}_{0}\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\frac{3}{4}\left(\Psi_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{4}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{a}^{\left(\upsilon,k,V\right)}$
(104)
$\displaystyle-\frac{1}{4}\phi_{0}\left(\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\right)-\mathcal{A}_{0}\Psi_{\mathcal{E}}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\widehat{\overline{\Psi}}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\widehat{\Psi}_{Q}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{1}{2r}\Psi_{Q}^{\left(\upsilon,k,S\right)}-\left(\mu_{0}+p_{0}+\frac{1}{4}\Pi_{0}-\frac{3}{2}\mathcal{E}_{0}\right)\overline{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}$
(105)
$\displaystyle-\frac{3}{2}\phi_{0}\left(\overline{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{6}\Psi_{Q}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{\Sigma}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle
i\upsilon\Psi_{Q}^{\left(\upsilon,k,V\right)}+\widehat{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{1}{2}\Psi_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\Psi_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\mathcal{A}}^{\left(\upsilon,k,V\right)}$
(106)
$\displaystyle-\frac{3}{2}\Pi_{0}\Psi_{a}^{\left(\upsilon,k,V\right)}-\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Psi_{\Pi}^{\left(\upsilon,k,V\right)}\,.$

####  III.5.3 $k$-Modes where $\mathcal{Q}^{\left(k\right)}\neq
0\wedge\left\\{\mathcal{Q}_{\alpha}^{\left(k\right)},\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\right\\}=0\wedge\left\\{\mathcal{Q}_{\alpha\beta}^{\left(k\right)},\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}\right\\}=0$:
$k=0$

As mentioned above, for the $k$-modes where the vector and tensor harmonics
are not defined, we cannot use the equations in Appendix B, since those only
relate to vector and tensor quantities, that is, those equations are trivially
verified and do not contain information regarding linear perturbations that
are only along the $u$ and the $e$ directions. Then, to describe the dynamics
of the perturbations that are not along the directions on the sheets, we will
use, instead, the relations in Appendix C. The equations for the harmonic
coefficients for the $k=0$ mode are

$\displaystyle\widehat{\Psi}_{\mathsf{A}}^{\left(\upsilon,0,S\right)}+\upsilon^{2}\Psi_{\theta}^{\left(\upsilon,0,S\right)}$
$\displaystyle=\frac{1}{2}\left(\Psi_{\mathsf{m}}^{\left(\upsilon,0,S\right)}+3\Psi_{\mathsf{p}}^{\left(\upsilon,0,S\right)}\right)+\widehat{\mathcal{A}}_{0}\left(\frac{1}{3}\Psi_{\theta}^{\left(\upsilon,0,S\right)}+\Psi_{\Sigma}^{\left(\upsilon,0,S\right)}\right)$
(107)
$\displaystyle-\left(3\mathcal{A}_{0}+\phi_{0}\right)\Psi_{\mathsf{A}}^{\left(\upsilon,0,S\right)}-\mathcal{A}_{0}\Psi_{\mathsf{F}}^{\left(\upsilon,0,S\right)}\,,$
$\displaystyle\upsilon^{2}\left(\frac{2}{3}\Psi_{\theta}^{\left(\upsilon,0,S\right)}-\Psi_{\Sigma}^{\left(\upsilon,0,S\right)}\right)$
$\displaystyle=\frac{1}{3}\left(\Psi_{\mathsf{m}}^{\left(\upsilon,0,S\right)}+3\Psi_{\mathsf{p}}^{\left(\upsilon,0,S\right)}\right)+\frac{1}{2}\Psi_{\mathcal{P}}^{\left(\upsilon,0,S\right)}-\Psi_{\mathsf{E}}^{\left(\upsilon,0,S\right)}$
(108)
$\displaystyle-\mathcal{A}_{0}\Psi_{\mathsf{F}}^{\left(\upsilon,0,S\right)}-\phi_{0}\Psi_{\mathsf{A}}^{\left(\upsilon,0,S\right)}\,,$

$\frac{2}{3}\widehat{\Psi}_{\theta}^{\left(\upsilon,0,S\right)}-\widehat{\Psi}_{\Sigma}^{\left(\upsilon,0,S\right)}=\Psi_{Q}^{\left(\upsilon,0,S\right)}+\frac{3}{2}\phi_{0}\Psi_{\Sigma}^{\left(\upsilon,0,S\right)}\,,$
(109)

$\displaystyle-\upsilon^{2}\Psi_{Q}^{\left(\upsilon,0,S\right)}+\widehat{\Psi}_{\mathsf{p}}^{\left(\upsilon,0,S\right)}+\widehat{\Psi}_{\mathcal{P}}^{\left(\upsilon,0,S\right)}$
$\displaystyle=\left(\widehat{p}_{0}+\widehat{\Pi}_{0}\right)\left(\frac{1}{3}\Psi_{\theta}^{\left(\upsilon,0,S\right)}+\Psi_{\Sigma}^{\left(\upsilon,0,S\right)}\right)-\left(\mu_{0}+p_{0}\right)\Psi_{\mathsf{A}}^{\left(\upsilon,0,S\right)}$
(110)
$\displaystyle-\Pi_{0}\left(\frac{3}{2}\Psi_{\mathsf{F}}^{\left(\upsilon,0,S\right)}+\Psi_{\mathsf{A}}^{\left(\upsilon,0,S\right)}\right)-\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{\mathcal{P}}^{\left(\upsilon,0,S\right)}$
$\displaystyle-\mathcal{A}_{0}\left(\Psi_{\mathsf{m}}^{\left(\upsilon,0,S\right)}+2\Psi_{\mathsf{p}}^{\left(\upsilon,0,S\right)}\right)\,,$

$\Psi_{\mathsf{m}}^{\left(\upsilon,0,S\right)}+\widehat{\Psi}_{Q}^{\left(\upsilon,0,S\right)}=-\left(\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{Q}^{\left(\upsilon,0,S\right)}-\frac{3}{2}\Pi_{0}\Psi_{\Sigma}^{\left(\upsilon,0,S\right)}-\left(\mu_{0}+p_{0}\right)\Psi_{\theta}^{\left(\upsilon,0,S\right)}\,;$
(111)

and the constraints

$\Psi_{\mathsf{F}}^{\left(\upsilon,0,S\right)}=\Psi_{Q}^{\left(\upsilon,0,S\right)}+\left(2\mathcal{A}_{0}-\phi_{0}\right)\left(\frac{1}{3}\Psi_{\theta}^{\left(\upsilon,0,S\right)}-\frac{1}{2}\Psi_{\Sigma}^{\left(\upsilon,0,S\right)}\right)\,,$
(112)

$\Psi_{\mathsf{E}}^{\left(\upsilon,0,S\right)}+\frac{1}{2}\Psi_{\mathcal{P}}^{\left(\upsilon,0,S\right)}-\frac{1}{3}\Psi_{\mathsf{m}}^{\left(\upsilon,0,S\right)}=\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}-3\mathcal{E}_{0}\right)\left(\frac{1}{3}\Psi_{\theta}^{\left(\upsilon,0,S\right)}-\frac{1}{2}\Psi_{\Sigma}^{\left(\upsilon,0,S\right)}\right)+\frac{1}{2}\phi_{0}\Psi_{Q}^{\left(\upsilon,0,S\right)}\,.$
(113)

###  III.6 Odd sector

####  III.6.1 $k$-Modes where
$\left\\{\mathcal{Q}_{\alpha\beta}^{\left(k\right)},\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}\right\\}\neq
0$

##### Equations for the kinematical quantities associated with the timelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of scalars
and gradients of scalar quantities:

$\displaystyle\widehat{\overline{\Psi}}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\mathcal{\widehat{A}}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}$
(114)
$\displaystyle-\mathcal{A}_{0}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\left(\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+3\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}\right)\,,$
$i\upsilon\Psi_{\Omega}^{\left(\upsilon,k,S\right)}=\frac{k^{2}}{2r}\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(115)
$\widehat{\Psi}_{\Omega}^{\left(\upsilon,k,S\right)}=\left(\mathcal{A}_{0}-\phi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}+\frac{k^{2}}{r}\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\,;$
(116)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$i\upsilon\Psi_{\Omega}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\widehat{\overline{\Psi}}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}=\frac{1}{2}\left(\frac{1}{2}\phi_{0}\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)\,,$
(117)

$i\upsilon\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\,,$
(118)

$\displaystyle\widehat{\overline{\Psi}}_{\Sigma}^{\left(\upsilon,k,V\right)}-\widehat{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{1}{r}\Psi_{\Omega}^{\left(\upsilon,k,S\right)}-\frac{3}{2}\phi_{0}\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}$
(119)
$\displaystyle+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,V\right)}+\frac{2-k^{2}}{2r}\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,T\right)}\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of tensor
quantities:

$i\upsilon\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,T\right)}=-\frac{1}{r}\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\overline{\Psi}_{\zeta}^{\left(\upsilon,k,T\right)}-\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}\,,$
(120)
$\widehat{\overline{\Psi}}_{\Sigma}^{\left(\upsilon,k,T\right)}=-\frac{1}{r}\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)-\frac{1}{2}\phi_{0}\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,T\right)}-\Psi_{\mathcal{H}}^{\left(\upsilon,k,T\right)}\,;$
(121)

  * •

Constraint equations for the harmonic coefficients:

$\frac{k^{2}}{2r}\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}+\hat{\mathcal{A}}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}=0\,,$
(122)
$-\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}+\left(\phi_{0}-2\mathcal{A}_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}+\frac{k^{2}}{r}\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)=0\,,$
(123)

$\phi_{0}\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)+\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}+2\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{2}{r}\Psi_{\Omega}^{\left(\upsilon,k,S\right)}-\frac{2-k^{2}}{r}\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,T\right)}=0\,,$
(124)
$\phi_{0}\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\frac{1}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}=0\,.$
(125)

##### Equations for the kinematical quantities associated with the spacelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of scalars
and gradients of scalar quantities:

$i\upsilon\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\widehat{\phi}_{0}\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)\,,$
(126)
$\widehat{\overline{\Psi}}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\widehat{\phi}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\frac{2}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{3}{2}\phi_{0}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}\,,$
(127)
$i\upsilon\Psi_{\xi}^{\left(\upsilon,k,S\right)}=\frac{1}{2}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}+\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}+\frac{k^{2}}{2r}\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}\,,$
(128)
$\widehat{\Psi}_{\xi}^{\left(\upsilon,k,S\right)}=-\phi_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}+\frac{k^{2}}{2r}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}\,;$
(129)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle\widehat{\overline{\Psi}}_{\alpha}^{\left(\upsilon,k,V\right)}-i\upsilon\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\left(\mathcal{A}_{0}+\frac{1}{2}\phi_{0}\right)\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}$
(130)
$\displaystyle+\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of tensor
quantities:

$i\upsilon\overline{\Psi}_{\zeta}^{\left(\upsilon,k,T\right)}=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,T\right)}-\frac{1}{r}\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}-\Psi_{\mathcal{H}}^{\left(\upsilon,k,T\right)}\,,$
(131)

$\widehat{\overline{\Psi}}_{\zeta}^{\left(\upsilon,k,T\right)}=-\frac{1}{r}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\phi_{0}\overline{\Psi}_{\zeta}^{\left(\upsilon,k,T\right)}-\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}-\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}\,;$
(132)

  * •

Constraint equations for the harmonic coefficients:

$\frac{k^{2}}{2r}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\widehat{\phi}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(133)
$\frac{2}{r}\Psi_{\xi}^{\left(\upsilon,k,S\right)}-\frac{2-k^{2}}{r}\overline{\Psi}_{\zeta}^{\left(\upsilon,k,T\right)}-\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-2\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}=0\,.$
(134)

##### Equations for the Weyl tensor components and the matter variables

  * •

Evolution and propagation equations for the harmonic coefficients of scalars
and gradients of scalar quantities:

$\displaystyle
i\upsilon\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\hat{\mu}_{0}\left(\Psi_{\Omega}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}\right)\,,$
(135)

$i\upsilon\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}i\upsilon\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}=\left(\widehat{\mathcal{E}}_{0}+\frac{1}{2}\widehat{\Pi}_{0}\right)\left(\Psi_{\Omega}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}\right)\,,$
(136)

$\displaystyle\widehat{\overline{\Psi}}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\widehat{\overline{\Psi}}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\left(\mu_{0}+p_{0}\right)\left(\mathcal{A}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)-\mathcal{A}_{0}\left(\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)$
(137)
$\displaystyle-\frac{1}{2}\phi_{0}\left(\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+4\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)+\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}$
$\displaystyle-\Pi_{0}\left(\frac{3}{2}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)\,,$

$\displaystyle\widehat{\overline{\Psi}}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\overline{\Psi}}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{3}\widehat{\overline{\Psi}}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-2\phi_{0}\left(\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{12}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}\right)$
(138)
$\displaystyle-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\phi_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}\right)\,,$

$i\upsilon\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}=\frac{k^{2}}{r}\left(\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}\right)+3\left(\frac{1}{2}\Pi_{0}-\mathcal{E}_{0}\right)\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(139)
$\widehat{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,S\right)}=\frac{k^{2}}{r}\left(\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\phi_{0}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}-\left(3\mathcal{E}_{0}+\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}\,;$
(140)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle
i\upsilon\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle=\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\frac{1}{2}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}+\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}$
(141)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+\Pi_{0}\right)\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle+\frac{1}{2r}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}+\frac{2-k^{2}}{2r}\Psi_{\mathcal{H}}^{\left(\upsilon,k,T\right)}\,,$
$\displaystyle\widehat{\overline{\Psi}}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\overline{\Psi}}_{\Pi}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{3}{2}\phi_{0}\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}$
(142)
$\displaystyle+\frac{1}{2}\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\frac{1}{4}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\frac{2-k^{2}}{2r}\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}\right)\,,$
$\displaystyle\frac{1}{4}\widehat{\overline{\Psi}}_{\Pi}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\widehat{\overline{\Psi}}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-i\upsilon\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{3}{4}\left(\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)+\frac{2-k^{2}}{4r}\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}-\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}\right)$
(143)
$\displaystyle-\frac{3}{2}\mathcal{E}_{0}\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\frac{1}{4}\phi_{0}\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle+\frac{3}{4}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\widehat{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\overline{\Psi}}_{Q}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{1}{2r}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}-\frac{2-k^{2}}{2r}\Psi_{\mathcal{H}}^{\left(\upsilon,k,T\right)}+\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}$
(144)
$\displaystyle-\frac{3}{2}\phi_{0}\left(\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{1}{6}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}\right)-\left(\mu_{0}+p_{0}+\frac{1}{4}\Pi_{0}-\frac{3}{2}\mathcal{E}_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\,,$

$\displaystyle
i\upsilon\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}+\widehat{\overline{\Psi}}_{\Pi}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\frac{3}{2}\Pi_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}+\frac{2-k^{2}}{2r}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}$
(145)
$\displaystyle-\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}-\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}\,;$

  * •

Evolution and propagation equations for the harmonic coefficients of tensor
quantities:

$\displaystyle
i\upsilon\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}+\frac{1}{2}i\upsilon\overline{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}-\widehat{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,T\right)}=$
$\displaystyle-\frac{1}{r}\left(\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}\right)+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{\mathcal{H}}^{\left(\upsilon,k,T\right)}$
(146)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+3\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,T\right)}\,,$

$\displaystyle\widehat{\overline{\Psi}}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}-\frac{1}{2}\widehat{\overline{\Psi}}_{\Pi}^{\left(\upsilon,k,T\right)}-i\upsilon\Psi_{\mathcal{H}}^{\left(\upsilon,k,T\right)}$
$\displaystyle=\frac{1}{r}\left(\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}\right)+\frac{1}{4}\phi_{0}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,T\right)}$
(147)
$\displaystyle+\frac{3}{2}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\zeta}^{\left(\upsilon,k,T\right)}-\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,T\right)}\,;$

  * •

Constraint equations for the harmonic coefficients:

$\displaystyle\frac{k^{2}}{r}\left(\frac{1}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle=-3\phi_{0}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(148)

$\frac{k^{2}}{2r}\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+\widehat{p}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}=0\,.$
(149)

####  III.6.2 $k$-Modes where
$\left\\{\mathcal{Q}_{\alpha}^{\left(k\right)},\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\right\\}\neq
0\wedge\left\\{\mathcal{Q}_{\alpha\beta}^{\left(k\right)},\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}\right\\}=0$:
$k^{2}=2$

As was discussed in the even sector, for the $k^{2}=2$ modes, the tensor
harmonics are not defined, but the vector and scalar harmonics do not vanish
necessarily. For those modes, we find the following equations for the harmonic
coefficients.

##### Equations for the kinematical quantities associated with the timelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of scalars
and gradients of scalar quantities:

$\widehat{\overline{\Psi}}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}=-\mathcal{\widehat{A}}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\mathcal{A}_{0}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\left(\frac{3}{2}\phi_{0}+2\mathcal{A}_{0}\right)\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\left(\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+3\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}\right)\,,$
(150)

$i\upsilon\Psi_{\Omega}^{\left(\upsilon,k,S\right)}=\frac{1}{r}\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(151)
$\widehat{\Psi}_{\Omega}^{\left(\upsilon,k,S\right)}=\left(\mathcal{A}_{0}-\phi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}+\frac{2}{r}\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\,;$
(152)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$2i\upsilon\Psi_{\Omega}^{\left(\upsilon,k,V\right)}-\widehat{\overline{\Psi}}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}=\frac{1}{2}\phi_{0}\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\,,$
(153)

$i\upsilon\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)=\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\,,$
(154)

$\widehat{\overline{\Psi}}_{\Sigma}^{\left(\upsilon,k,V\right)}-\widehat{\Psi}_{\Omega}^{\left(\upsilon,k,V\right)}=-\frac{1}{r}\Psi_{\Omega}^{\left(\upsilon,k,S\right)}-\frac{3}{2}\phi_{0}\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}+\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\,;$
(155)

  * •

Constraint equations for the harmonic coefficients:

$\frac{1}{r}\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}+\widehat{\mathcal{A}}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}=0\,,$
(156)
$-\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}+\left(\phi_{0}-2\mathcal{A}_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}+\frac{2}{r}\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)=0\,,$
(157)

$\phi_{0}\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)+\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}+2\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{2}{r}\Psi_{\Omega}^{\left(\upsilon,k,S\right)}=0\,,$
(158)
$\phi_{0}\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}+\mathcal{A}_{0}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\frac{1}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}=0\,.$
(159)

##### Equations for the kinematical quantities associated with the spacelike
congruence

  * •

Evolution and propagation equations for the harmonic coefficients of scalars
and gradients of scalar quantities:

$i\upsilon\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\widehat{\phi}_{0}\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)\,,$
(160)
$\widehat{\overline{\Psi}}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\widehat{\phi}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\frac{2}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{3}{2}\phi_{0}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}\,,$
(161)
$i\upsilon\Psi_{\xi}^{\left(\upsilon,k,S\right)}=\frac{1}{2}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}+\left(\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}+\frac{1}{r}\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}\,,$
(162)
$\widehat{\Psi}_{\xi}^{\left(\upsilon,k,S\right)}=-\phi_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}+\frac{1}{r}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}\,;$
(163)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle\widehat{\overline{\Psi}}_{\alpha}^{\left(\upsilon,k,V\right)}-i\upsilon\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\left(\mathcal{A}_{0}+\frac{1}{2}\phi_{0}\right)\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}$
(164)
$\displaystyle+\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}+\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)\,;$

  * •

Constraint equations for the harmonic coefficients:

$\frac{1}{r}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}=-\widehat{\phi}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(165)

$\frac{2}{r}\Psi_{\xi}^{\left(\upsilon,k,S\right)}-\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-2\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}=0\,.$
(166)

##### Equations for the Weyl tensor components and the matter variables

  * •

Evolution and propagation equations for the harmonic coefficients of scalars
and gradients of scalar quantities:

$\displaystyle
i\upsilon\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\widehat{\mu}_{0}\left(\Psi_{\Omega}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}\right)\,,$
(167)

$i\upsilon\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}i\upsilon\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}=\left(\widehat{\mathcal{E}}_{0}+\frac{1}{2}\widehat{\Pi}_{0}\right)\left(\Psi_{\Omega}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}\right)\,,$
(168)

$\displaystyle\widehat{\overline{\Psi}}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}+\widehat{\overline{\Psi}}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\left(\mu_{0}+p_{0}\right)\left(\mathcal{A}_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)-\mathcal{A}_{0}\left(\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)$
(169)
$\displaystyle-\frac{1}{2}\phi_{0}\left(\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+4\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)+\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\Pi_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}$
$\displaystyle-\Pi_{0}\left(\frac{3}{2}\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}+\overline{\Psi}_{\mathbb{A}}^{\left(\upsilon,k,V\right)}\right)\,,$

$\displaystyle\widehat{\overline{\Psi}}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\overline{\Psi}}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{3}\widehat{\overline{\Psi}}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-2\phi_{0}\left(\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\frac{1}{12}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}\right)$
(170)
$\displaystyle-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\left(\overline{\Psi}_{\mathbb{F}}^{\left(\upsilon,k,V\right)}-\phi_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}\right)\,,$

$i\upsilon\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}=\frac{2}{r}\left(\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}\right)+3\left(\frac{1}{2}\Pi_{0}-\mathcal{E}_{0}\right)\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(171)
$\widehat{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,S\right)}=\frac{2}{r}\left(\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\phi_{0}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}-\left(3\mathcal{E}_{0}+\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,S\right)}\,;$
(172)

  * •

Evolution and propagation equations for the harmonic coefficients of vector
quantities:

$\displaystyle
i\upsilon\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle=\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\frac{1}{2}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}+\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\alpha}^{\left(\upsilon,k,V\right)}$
(173)
$\displaystyle-\frac{1}{2}\left(\mu_{0}+p_{0}+\Pi_{0}\right)\left(\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\right)+\frac{1}{2r}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}\,,$
$\displaystyle\widehat{\overline{\Psi}}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\overline{\Psi}}_{\Pi}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{3}{2}\phi_{0}\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\right)-\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}$
(174)
$\displaystyle+\frac{1}{2}\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}+\frac{1}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}+\frac{1}{4}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\frac{1}{4}\widehat{\overline{\Psi}}_{\Pi}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\widehat{\overline{\Psi}}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-i\upsilon\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}=$
$\displaystyle-\frac{3}{4}\left(\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)+\frac{3}{4}\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}$
(175)
$\displaystyle-\frac{3}{2}\mathcal{E}_{0}\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}+\frac{1}{4}\phi_{0}\left(\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle+\mathcal{A}_{0}\overline{\Psi}_{\mathcal{E}}^{\left(\upsilon,k,V\right)}\,,$
$\displaystyle\widehat{\Psi}_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{1}{2}\widehat{\overline{\Psi}}_{Q}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{1}{2r}\Psi_{\mathcal{H}}^{\left(\upsilon,k,S\right)}-\frac{3}{2}\phi_{0}\left(\Psi_{\mathcal{H}}^{\left(\upsilon,k,V\right)}+\frac{1}{6}\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}\right)$
(176)
$\displaystyle+\frac{3}{2}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\Sigma}^{\left(\upsilon,k,V\right)}-\left(\mu_{0}+p_{0}+\frac{1}{4}\Pi_{0}-\frac{3}{2}\mathcal{E}_{0}\right)\Psi_{\Omega}^{\left(\upsilon,k,V\right)}\,,$

$\displaystyle
i\upsilon\overline{\Psi}_{Q}^{\left(\upsilon,k,V\right)}+\widehat{\overline{\Psi}}_{\Pi}^{\left(\upsilon,k,V\right)}$
$\displaystyle=\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}-\left(\frac{3}{2}\phi_{0}+\mathcal{A}_{0}\right)\overline{\Psi}_{\Pi}^{\left(\upsilon,k,V\right)}$
(177)
$\displaystyle-\frac{3}{2}\Pi_{0}\overline{\Psi}_{a}^{\left(\upsilon,k,V\right)}-\left(\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\overline{\Psi}_{\mathcal{A}}^{\left(\upsilon,k,V\right)}\,;$

  * •

Constraint equations for the harmonic coefficients:

$\displaystyle\frac{2}{r}\left(\frac{1}{3}\overline{\Psi}_{\mathfrak{m}}^{\left(\upsilon,k,V\right)}-\overline{\Psi}_{\mathbb{E}}^{\left(\upsilon,k,V\right)}-\frac{1}{2}\overline{\Psi}_{\mathbb{P}}^{\left(\upsilon,k,V\right)}\right)$
$\displaystyle=-3\phi_{0}\left(\mathcal{E}_{0}+\frac{1}{2}\Pi_{0}\right)\Psi_{\xi}^{\left(\upsilon,k,S\right)}\,,$
(178)

$\frac{1}{r}\overline{\Psi}_{\mathfrak{p}}^{\left(\upsilon,k,V\right)}+\widehat{p}_{0}\Psi_{\xi}^{\left(\upsilon,k,S\right)}=0\,.$
(179)

####  III.6.3 $k$-Modes where $\mathcal{Q}^{\left(k\right)}\neq
0\wedge\left\\{\mathcal{Q}_{\alpha}^{\left(k\right)},\bar{\mathcal{Q}}_{\alpha}^{\left(k\right)}\right\\}=0\wedge\left\\{\mathcal{Q}_{\alpha\beta}^{\left(k\right)},\mathcal{\bar{Q}}_{\alpha\beta}^{\left(k\right)}\right\\}=0$:
$k=0$

Once again, as was discussed in the even sector, for $k$-modes where the
vector and tensor harmonics are not defined, to describe the dynamics of the
perturbations that are not along the directions on the sheets, we will use the
relations in Appendix C. The equations for the harmonic coefficients for the
$k=0$ mode are:

$i\upsilon\Psi_{\Omega}^{\left(\upsilon,0,S\right)}=\mathcal{A}_{0}\Psi_{\xi}^{\left(\upsilon,0,S\right)}\,,$
(180)
$\widehat{\Psi}_{\Omega}^{\left(\upsilon,0,S\right)}=\left(\mathcal{A}_{0}-\phi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,0,S\right)}\,,$
(181) $i\upsilon\Psi_{\xi}^{\left(\upsilon,0,S\right)}=0\,,$ (182)
$\widehat{\Psi}_{\xi}^{\left(\upsilon,0,S\right)}=-\phi_{0}\Psi_{\xi}^{\left(\upsilon,0,S\right)}\,,$
(183)
$i\upsilon\Psi_{\mathcal{H}}^{\left(\upsilon,0,S\right)}=-3\left(\mathcal{E}_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\xi}^{\left(\upsilon,0,S\right)}\,,$
(184)
$\widehat{\Psi}_{\mathcal{H}}^{\left(\upsilon,0,S\right)}=-\frac{3}{2}\phi_{0}\Psi_{\mathcal{H}}^{\left(\upsilon,0,S\right)}-\left(3\mathcal{E}_{0}+\mu_{0}+p_{0}-\frac{1}{2}\Pi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,0,S\right)}\,;$
(185)

and the constraint

$\left(2\mathcal{A}_{0}-\phi_{0}\right)\Psi_{\Omega}^{\left(\upsilon,0,S\right)}+\Psi_{\mathcal{H}}^{\left(\upsilon,0,S\right)}=0\,.$
(186)

##  IV Adiabatic isotropic perturbations

As a first application of the system of covariant, gauge invariant equations
presented in the previous section, in the remainder of the article we will
consider linear isotropic perturbations of a static, spherically symmetric
spacetime permeated by a perfect fluid, which we will call, for brevity and
its immediate physical application, “star”. Due to the scope of this article,
here we will analyze the general properties of the system of differential
equations and relegate finding and discussing the solutions for specific
background spacetimes to another article.

###  IV.1 The equilibrium spacetime

In the language of the 1+1+2 decomposition, static, spherically symmetric
spacetimes with a perfect fluid source can be completely characterized by the
six scalars
$\left\\{\mathcal{A}_{0},\phi_{0},\mathcal{E}_{0},\mu_{0},p_{0},\Lambda\right\\}$,
such that all their “dot derivatives” vanish. For simplicity, we will set the
cosmological constant to zero: $\Lambda=0$. Then, the covariant defined
scalars verify (cf. Eqs. (53) and (54)):

$\displaystyle\widehat{p}_{0}$
$\displaystyle=-\left(\mu_{0}+p_{0}\right)\mathcal{A}_{0}\,,$ (187)
$\displaystyle\widehat{\phi}_{0}$
$\displaystyle=-\frac{1}{2}\phi_{0}^{2}-\frac{2}{3}\mu_{0}-\mathcal{E}_{0}\,,$
(188) $\displaystyle\widehat{\mathcal{A}}_{0}$
$\displaystyle=\frac{3}{2}\mathcal{E}_{0}+\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\mathcal{A}_{0}\,,$
(189)
$\displaystyle\widehat{\mathcal{E}}_{0}+\frac{3}{2}\phi_{0}\mathcal{E}_{0}$
$\displaystyle=\frac{1}{3}\widehat{\mu}_{0}\,,$ (190)

and the constraint

$\displaystyle\frac{1}{3}\left(\mu_{0}+3p_{0}\right)-\mathcal{A}_{0}\phi_{0}$
$\displaystyle=\mathcal{E}_{0}\,.$ (191)

We will consider the setup where two solutions of the Einstein field equations
are smoothly matched at a common timelike hypersurface. The interior of the
star is described by a static, spatially compact solution with a perfect fluid
source, while the exterior spacetime is described by an asymptotically flat
branch of the vacuum Schwarzschild solution with no event horizons.

###  IV.2 Choice of frame in the perturbed spacetime

The sheets of the background spacetime have spherical symmetry. Hence, by
choosing the $e$ tangent vector field of the background spacetime to be
aligned with the gradient of the circumferential radius, $r$, we can
particularize the harmonics $\mathcal{Q}^{\left(k\right)}$ to be the spherical
harmonics: $Y_{lm}$, where the eigenvalues $k$ verify
$k^{2}=l\left(l+1\right)$, with $l\geq 0$, and $-l\leq m\leq l$. Moreover,
this choice of frame implies that,

$\mathcal{Q}_{\alpha}^{\left(0\right)}=0,\quad\bar{\mathcal{Q}}_{\alpha}^{\left(0\right)}=0,\quad\mathcal{Q}_{\alpha\beta}^{\left(0\right)}=\mathcal{Q}_{\alpha\beta}^{\left(1\right)}=0,\quad\bar{\mathcal{Q}}_{\alpha\beta}^{\left(0\right)}=\bar{\mathcal{Q}}_{\alpha\beta}^{\left(1\right)}=0.$
(192)

The perturbed spacetime is assumed to maintain spherical symmetry, therefore,
it is useful also to consider the spacelike vector field $e$ to not have
angular components and dependencies in the perturbed spacetime. Notice that,
at this point, we are free to choose any smooth mapping between the background
and the perturbed spacetimes given that covariantly defined tensors, vectors,
and the quantities in Eqs. (55) and (56) are identification gauge invariant.
Considering the equations in subsections III.5 and III.6, since the
perturbations are spherically symmetric, all coefficients, in both even and
odd sectors, with $l\geq 1$ vanish identically. Then, only the coefficients
for the $l=0$ mode, the monopole, may not be trivial.

In addition to the imposition that the perturbation maintains the spherical
symmetry of the spacetime, for simplicity, we will further assume that the
perturbation does not generate anisotropic pressure and is adiabatic, that is,
in the comoving matter frame, the perturbation will not give rise to heat
flows within the fluid. Moreover, we will also impose that the vorticity of
the timelike congruence is identically zero in the perturbed spacetime. These
extra conditions set the coefficients

$\Psi_{\mathcal{P}}^{\left(\upsilon,0,S\right)},\
\Psi_{\Omega}^{\left(\upsilon,0,S\right)},\
\Psi_{\xi}^{\left(\upsilon,0,S\right)},\
\Psi_{\mathcal{H}}^{\left(\upsilon,0,S\right)}=0\,,$ (193)

and in the comoving frame

$\Psi_{Q}^{\left(\upsilon,0,S\right)}=0\,.$ (194)

Now, the choice of frame is not completely determined by the choice of frame
in the background spacetime and the choice that the $e$ vector field does not
have angular components and dependencies in the perturbed spacetime. Since the
background spacetime is assumed to be static, comoving observers with the
fluid in equilibrium are also static. Therefore, to describe the perturbed
fluid, we can choose either to consider a congruence that describes the world-
lines of observers locally comoving with the elements of the fluid or a
congruence that describes the world-lines of static, $\dot{r}=0$, observers.
Both choices are valid, and both have advantages. For the former choice, the
matter field is described by the source fluid in its rest frame, i.e., a
perfect fluid model. Hence, it can be characterized simply by its energy
density and pressure. Moreover, all quantities that characterize the observer
directly characterize the elements of the fluid, allowing for a clear
interpretation of the quantities and how their dependencies give rise to the
various physical effects. In the latter case, we have the constraint
$\frac{2}{3}\theta=\Sigma$ between the expansion and shear kinematical
variables, Eq. (358). However, for static observers, the fluid will no longer
be perfect: such observers will measure momentum flows, and we have to
consider a heat flow term in the perturbed stress-energy tensor: $Q\neq 0$
(cf. Appendix E). Nonetheless, the description of the problem in this frame is
greatly simplified, allowing us to find some general properties of the
solutions easily. Below, we show the system of equations for both frames.

In order to lighten the notation, here and in the following sections, we will
indicate the perturbation-coefficients as

${\Psi}_{\chi}^{\left(\upsilon,0,S\right)}={\Psi}_{\chi}\,.$ (195)

where $\chi$ is a generic perturbation variable. This will not compromise
clarity as only the monopole perturbations mode is nontrivial for the
considered setup.

###  IV.3 Comoving observers

In the comoving frame, the perturbed fluid can be modeled by a perfect fluid.
Hence, it can be fully characterized by its energy density and pressure. To
close the system, we have to provide a matter model for the perturbed fluid.
As a simplifying assumption, we will consider that the perturbed matter fluid
still verifies a barotropic equation of state such that

$p=f\left(\mu\right)\,,$ (196)

where $f$ is assumed to be non-vanishing and of class $\mathcal{C}^{1}$ in
some neighborhood containing $\mu_{0}$. Then, at linear order

$\mathsf{p}\approx f^{\prime}\left(\mu_{0}\right)\mathsf{m}\,,$ (197)

where prime represents derivative with respect to the function’s parameter, so
that $f^{\prime}\left(\mu_{0}\right)$ represents the square of the adiabatic
speed of sound, to be assumed non-vanishing in the interior of the perturbed
star. Notice that the function $f$ does not have to be equal to the equation
of state of the background configuration, in particular, the equilibrium fluid
is not even required to verify a barotropic equation of state.

Imposing the previous conditions in the equations of subsections III.5.3 and
III.6.3, we are left with the following system for the non-trivial
coefficients of the perturbation variables

$\widehat{\Psi}_{\mathsf{p}}+2\mathcal{A}_{0}\left(1+\frac{1}{3f^{\prime}\left(\mu_{0}\right)}\right)\Psi_{\mathsf{p}}=-\left(\mu_{0}+p_{0}\right)\left(\Psi_{\mathsf{A}}+\mathcal{A}_{0}\Psi_{\Sigma}\right)\,,$
(198)
$\widehat{\Psi}_{\mathsf{A}}+\left(3\mathcal{A}_{0}-\frac{1}{2}\phi_{0}\right)\Psi_{\mathsf{A}}=\frac{\mathcal{E}_{0}}{\left(\mu_{0}+p_{0}\right)f^{\prime}\left(\mu_{0}\right)}\Psi_{\mathsf{p}}-\frac{3}{2}\left(\upsilon^{2}+\mathcal{A}_{0}^{2}+\frac{1}{3}\mu_{0}-2\mathcal{E}_{0}\right)\Psi_{\Sigma}\,,$
(199)
$\displaystyle\widehat{\Psi}_{\Sigma}+\left(\frac{3}{2}\phi_{0}-\frac{2\mathcal{A}_{0}}{3f^{\prime}\left(\mu_{0}\right)}\right)\Psi_{\Sigma}$
$\displaystyle=\frac{2}{3\left(\mu_{0}+p_{0}\right)f^{\prime}\left(\mu_{0}\right)}\left[\frac{3f^{\prime\prime}\left(\mu_{0}\right)\widehat{\mu}_{0}+2\mathcal{A}_{0}}{3f^{\prime}\left(\mu_{0}\right)}+\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}+\mathcal{A}_{0}\right]\Psi_{\mathsf{p}}$
(200)
$\displaystyle+\frac{2}{3f^{\prime}\left(\mu_{0}\right)}\Psi_{\mathsf{A}}\,,$

and the constraints

$\left(\upsilon^{2}+\mathcal{A}_{0}\phi_{0}+\mathcal{A}_{0}^{2}-p_{0}\right)\left(\frac{2}{3}\Psi_{\theta}-\Psi_{\Sigma}\right)=\Psi_{\mathsf{p}}-\phi_{0}\Psi_{\mathsf{A}}\,,$
(201)

$\Psi_{\mathsf{E}}=\mathcal{E}_{0}\left(\frac{3}{2}\Psi_{\Sigma}+\frac{\Psi_{\mathsf{p}}}{f^{\prime}\left(\mu_{0}\right)\left(\mu_{0}+p_{0}\right)}\right)-\frac{1}{2}\left(\mu_{0}+p_{0}\right)\Psi_{\Sigma}\,,$
(202)

$\Psi_{\mathsf{F}}=\left(\frac{1}{2}\phi_{0}-\mathcal{A}_{0}\right)\left(\frac{2\Psi_{\mathsf{p}}}{3f^{\prime}\left(\mu_{0}\right)\left(\mu_{0}+p_{0}\right)}+\Psi_{\Sigma}\right)\,,$
(203) $\Psi_{\mathsf{m}}=-\left(\mu_{0}+p_{0}\right)\Psi_{\theta}\,,$ (204)
$\Psi_{\mathsf{p}}=f^{\prime}\left(\mu_{0}\right)\Psi_{\mathsf{m}}\,,$ (205)

where Eq. (205) follows from Eq. (197). It can be readily shown that Eqs.
(198)–(201) are consistent with the relation (59).

To select the physically acceptable solutions and formalize the boundary value
problem, we impose the following boundary conditions:

  1. (i)

the energy density and the pressure perturbations at the center of the star
must be finite in a neighborhood of the initial instant;

  2. (ii)

the spacetime interior to the perturbed star can be smoothly matched to an
exterior vacuum Schwarzschild spacetime at a timelike hypersurface
$\mathfrak{B}$, which will represent the “surface of the star”.

From the point of view of the comoving observer, the boundary condition (ii)
implies that the pressure of the perturbed fluid is identically zero at all
times at the surface of the star and thus, $\mathsf{p}$, hence
$\Psi_{\mathsf{p}}$, are also identically zero at the boundary $\mathfrak{B}$.

###  IV.4 Static observers

An advantage of the adopted covariant formalism is the ability to readily
change frames without deriving a new set of equations for the perturbations.
Hence, an alternative way to describe isotropic adiabatic perturbations is to
consider the frame of static observers.

As was discussed previously, static observers will not describe the perturbed
fluid as a perfect fluid since, in general, these will measure a local net
momentum flow. Therefore, the relation between the energy density, pressure
and heat flow terms of the perturbed stress-energy tensor is more complex than
that of the comoving frame above. It is shown in Appendix E that a barotropic
equation of state in the comoving frame leads to the following relation
between the matter variables measured in a static frame

$\mathsf{m}=\frac{1}{f^{\prime}\left(\mu_{0}\right)}\mathsf{p}-\frac{1}{\mu_{0}+p_{0}}\left(\widehat{\mu}_{0}-\frac{\widehat{p}_{0}}{f^{\prime}\left(\mu_{0}\right)}\right)Q\,,$
(206)

where $f^{\prime}\left(\mu_{0}\right)$ represents the square of the speed of
sound measured in the comoving frame. We highlight that although we have kept
the nomenclature of the previous subsection, $\mathsf{m}$ and $\mathsf{p}$
refer here, respectively, to the dot derivatives of the energy density and
pressure, and $Q$ to the scalar heat flow density, all three measured by a
radially static observer. In this form, it is immediate to see that if the
equilibrium and the perturbed fluids verify the same equation of state, the
second term in the right-hand side of Eq. (206) vanishes. Therefore, the extra
term accounts for the phase change between the two fluids. This is a very
interesting result, showing that in the static frame the correction to the
equation of state has to be taken into account already at linear level.

Imposing condition $\Psi_{\Sigma}=\frac{2}{3}\Psi_{\theta}$, Eq. (358), and
using Eq. (206) in the equations of subsections III.5.3 and III.6.3, we find
the following system for the non-trivial coefficients of the perturbation
variables in the static frame

$\widehat{\Psi}_{\mathsf{p}}+\left[\frac{\mu_{0}+p_{0}}{\phi_{0}}+\left(2+\frac{1}{f^{\prime}\left(\mu_{0}\right)}\right)\mathcal{A}_{0}\right]\Psi_{\mathsf{p}}=\left[\frac{\mu_{0}+p_{0}}{\phi_{0}}\left(\frac{1}{2}\phi_{0}+2\mathcal{A}_{0}\right)+\frac{\mathcal{A}_{0}^{2}}{f^{\prime}\left(\mu_{0}\right)}+\frac{\mathcal{A}_{0}\widehat{\mu}_{0}}{\mu_{0}+p_{0}}+\upsilon^{2}\right]\Psi_{Q}\,,$
(207)

$\widehat{\Psi}_{Q}+\left(\phi_{0}+2\mathcal{A}_{0}-\frac{\mathcal{A}_{0}}{f^{\prime}\left(\mu_{0}\right)}-\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}-\frac{\mu_{0}+p_{0}}{\phi_{0}}\right)\Psi_{Q}=-\frac{1}{f^{\prime}\left(\mu_{0}\right)}\Psi_{\mathsf{p}}\,;$
(208)

and the constraints

$\displaystyle\Psi_{\mathsf{m}}$
$\displaystyle=\frac{1}{f^{\prime}\left(\mu_{0}\right)}\Psi_{\mathsf{p}}-\left(\frac{\mathcal{A}_{0}}{f^{\prime}\left(\mu_{0}\right)}+\frac{\widehat{\mu}_{0}}{\mu_{0}+p_{0}}\right)\Psi_{Q}\,,$
(209) $\displaystyle\Psi_{\mathsf{A}}$
$\displaystyle=\frac{1}{\phi_{0}}\left[\Psi_{\mathsf{p}}-\left(\frac{1}{2}\phi_{0}+\mathcal{A}_{0}\right)\Psi_{Q}\right]\,,$
$\displaystyle\Psi_{\mathsf{E}}$
$\displaystyle=\frac{1}{2}\phi_{0}\Psi_{Q}+\frac{1}{3f^{\prime}\left(\mu_{0}\right)}\Psi_{\mathsf{p}}\,,$
$\displaystyle\Psi_{\theta}$ $\displaystyle=-\frac{1}{\phi_{0}}\Psi_{Q}\,,$
$\displaystyle\Psi_{\Sigma}$ $\displaystyle=\frac{2}{3}\Psi_{\theta}\,,$
$\displaystyle\Psi_{\mathsf{F}}$ $\displaystyle=\Psi_{Q}\,.$

To select the physically relevant solutions, we impose the boundary conditions
(i) and (ii). In particular, for the static observer, condition (ii) implies
that at the surface of the star, at all times, we must have (cf, Eq. (382))

$\left.\mathsf{p}-\mathcal{A}_{0}Q\right|_{\mathfrak{B}}=0\,,$ (210)

hence

$\left.\Psi_{\mathsf{p}}-\mathcal{A}_{0}\Psi_{Q}\right|_{\mathfrak{B}}=0\,.$
(211)

##  V General solutions, Sturm-Liouville problem and properties of