# Causal dynamics of null horizons under linear perturbations

Peter K S Dunsby<EMAIL_ADDRESS>Cosmology and Gravity Group,
Department of Mathematics and Applied Mathematics, University of Cape Town,
Rondebosch 7701, Cape Town, South Africa  South African Astronomical
Observatory, Observatory 7925, Cape Town, South Africa  Centre for Space
Research, North-West University, Potchefstroom 2520, South Africa  Seoktae Koh
<EMAIL_ADDRESS>Department of Science Education, Jeju National
University, Jeju, 63243, South Korea  Abbas M Sherif<EMAIL_ADDRESS>Department of Science Education, Jeju National University, Jeju, 63243, South
Korea

###### Abstract

We study the causal dynamics of an embedded null horizon foliated by
marginally outer trapped surfaces (MOTS) for a locally rotationally symmetric
background spacetime subjected to linear perturbations. We introduce a simple
procedure which characterizes the transition of the causal character of the
null horizon. We apply our characterization scheme to non-dissipative
perturbations of the Schwarzschild and spatially homogeneous backgrounds. For
the latter, a linear equation of state was imposed. Assuming a harmonic
decomposition of the linearized field equations, we clarify the variables of a
formal solution to the linearized system that determine how the null horizon
evolves. For both classes of backgrounds, the shear and vorticity 2-vectors
are essential to the characterization, and their roles are made precise.
Finally, we discuss aspects of the relationship between the characterizing
conditions. Various properties related to the self-adjointness of the MOTS
stability operator are extensively discussed.

##  I Introduction

Background and motivation: The discovery of supermassive black holes as well
as the detection of gravitational waves emitted by inspiralling black holes
have ignited renewed interest in black hole physics. A key concept in
understanding the dynamics of black holes is the notion of a trapped surface,
those on which both ingoing and outgoing null rays are converging, introduced
in the proof of the 1965 singularity theorem by Roger Penrose [1].

Marginally outer trapped surfaces (MOTS) are the marginal case of trapped
surfaces, closed surfaces on which the outgoing null ray is neither converging
nor diverging. Slight iterations to the definition of MOTS can be found in the
literature from Wald’s marginally trapped surface [2] (note the “outer” here
has been dropped) which rather imposes that both null rays are non-diverging,
to Hayward’s marginal surface [3] which is in fact what we call a MOTS here,
but in Hayward’s case the choice is which of the null ray that is neither
converging nor diverging is a not fixed. (See the references [4, 5] for
excellent reviews and additional details. The notion of a MOTS will also be
rigorously defined in Section III).

MOTS foliate 3-dimensional hypersurfaces known as marginally outer trapped
tubes (MOTTs) [8], and where the causal character is fixed at all points of a
MOTT it will be referred to by the common name horizon, which under certain
conditions bound trapped regions containing trapped surfaces. Dynamical and
null horizons are well known cases of horizons (see for example [9]). The
timelike MOTTs are usually referred to simply as membranes as they, by
definition, cannot enclose trapped surfaces.

MOTS have also been fundamental to understanding the formation of black holes
due to gravitational collapse, as well as their dynamics and evolution. MOTS
admit a notion of stability [6, 7] (consequently leading to a notion of
stability of black holes). This is a way to determine whether one can deform a
MOTS $\mathcal{S}$ to another $\mathcal{S^{\prime}}$ along a unit direction
normal to $\mathcal{S}$ that points along the slice in which $\mathcal{S}$ is
embedded. This stability condition is given in terms of conditions on the
principal eigenvalue of a second order linear elliptic operator. It has been
shown that under physically reasonable conditions, strict stability, where the
principal eigenvalue is positive, a MOTS will evolve to foliate a DH [6] (the
same conclusion follows in the case that the operator has no zero eigenvalue).

Given a horizon embedded in a background spacetime which is subjected to a
perturbation, one expects the perturbation to in general affect the dynamics,
and hence causal character of the horizon. Perturbation of horizons have been
examined variously in different contexts using different approaches. For
example, in [10] linear perturbations of a null horizon by a gravitational
field was studied and it was established that such perturbations do not affect
the causal character of the horizon. This feature of the invariance of the
causal character, as have already been mentioned, will not be the case in
general, and this was neatly discussed in [11], where generic first and higher
order perturbations of non-expanding horizons were considered.

A more explicit example was considered by Pilkington et al. in [12]. Embedding
a stationary metric in the Weyl solution is a way of smoothly distorting
stationary blackholes (see for example [13]). The Weyl potentials tune the
behavior of horizons and are referred to as distortion potentials. The authors
demonstrated that while for small distortions the horizon is apparently ‘well
behaved’, for large distortions, even the MOTS structure of the horizon is not
assured, let alone the causal character.

Objectives: Our aim here is to introduce an approach which characterizes the
evolution of MOTS in a linearly perturbed locally rotationally symmetric (LRS)
background spacetime [14, 15]. More precisely we consider the following
problem: How does a null horizon foliated by MOTS, embedded in a LRS
background solution, evolve when the background is subjected to linear
perturbations?

We will employ the 1+1+2 covariant formalism [16, 17], which has been used in
several works to study MOTS and their evolution in background LRS spacetimes
[18, 19, 20, 21]. After formulating the characterization scheme, we will adapt
our approach to cases of some well known background solutions. Specifically,
we consider the null event horizon in the Schwarzschild background and null
horizons in hypersurface orthogonal and spatially homogeneous background
solutions. As will be seen, at least in these examples to which our scheme
will be applied, the shear and vorticity 2-vectors appear to consistently play
the pivotal roles in determining the causal behavior of the null horizons
under perturbations. (The roles of these 2-vectors will be made more precise
in the text).

We also aim to consider aspects of the self-adjointness of the stability
operator. As it will be seen, the 1-form that characterizes the self-
adjointness of the operator is defined in terms of the shear and vorticity
2-vectors as well, and so it is quite natural to consider this in relation to
the evolution of the horizons.

Structure: Section II of the paper gives a quick overview of the covariant
formalism that is used, and the procedure for linearizing a LRS background
solution, which provides the mapping between a background solution and a
perturbed one, is outlined. The process to harmonically decompose first order
scalars, 2-vectors and 2-tensor, in this formalism is briefly reviewed
following the standard literature. In Section III, we introduce an approach to
characterize MOTS evolution in the perturbed solution.

We start by briefly introducing the notion of a MOTS, which is well
proliferated in the literature but nonetheless necessary for the flow of the
paper. We then go on to introduce the characterization scheme where it becomes
transparent how the required gauge choice forces us to consider a null horizon
in the background. That is, our scheme does not work for a non-null horizon.
Section IV applies the characterization scheme to static and spatially
homogeneous LRS backgrounds. In both cases the evolution of the null horizon
is characterized, with the case of the latter an equations of state being
imposed and a partial characterization provided. In Section V, we discuss some
simple implications of the results of Section IV for the self-adjointness of
the MOTS stability. Several additional properties of the MOTS and aspects of
the stability operator are discussed. We conclude with discussion of results
in Section VI and discuss potential avenues for future research.

##  II Preliminaries

In this section we briefly introduce the 1+1+2 covariant formalism. We then
describe the general prescription for the perturbative scheme that have been
used extensively over the last few years, to carry out perturbations of
background LRS solutions.

###  II.1 Covariant decomposition

There is by now a well established trove of literature on the 1+1+2 covariant
formalism, with several references providing details of the steps in the
decomposition. A reader interested in more details is referred to the
following standard texts [16, 17] (Also see the references [22, 23, 24] for
some recent corrections to the full set of equations). As such, the
introduction here would be very brief, only necessary enough for the work
carried out in this paper.

In the well known and powerful 1+3 approach, a unit tangent direction - call
this $u^{a}$ \- along which timelike observers flow, is what threads the
spacetime. The field equations are decomposed along $u^{a}$ plus some
constraint equations. If there exists some unit spacelike vector $n^{a}$
obeying $u_{a}n^{a}=0$, the 3-space from the previous split can be further
decomposed along this direction as a $1+2$ product manifold. This, in addition
to the evolution equations along $u^{a}$, introduces a set of propagation
equations along $n^{a}$, as well as some constraint equations. The splitting
naturally introduces new derivative along $n^{a}$ and on the 2-space
(generally referred to as the sheet):

  * •

Along $u^{a}$:
$\dot{\psi}^{a\cdots b}_{\ \ \ \ c\cdots d}=u^{f}\nabla_{f}\psi^{a\cdots b}_{\
\ \ \ c\cdots d}$;

  * •

Along $n^{a}$:
$\hat{\psi}^{a\cdots b}_{\ \ \ \ c\cdots d}=n^{f}\nabla_{f}\psi^{a\cdots b}_{\
\ \ \ c\cdots d}$;

  * •

Along the sheet:
$\delta_{f}\psi^{a\cdots b}_{\ \ \ \ c\cdots d}=N^{\bar{c}}_{\ c}\cdots
N^{\bar{d}}_{\ d}N^{a}_{\ \bar{a}}\cdots N^{b}_{\ \bar{b}}N^{e}_{\
f}\nabla_{e}\psi^{\bar{a}\cdots\bar{b}}_{\ \ \ \ \bar{c}\cdots\bar{d}}$,

for any tensor $\psi^{a\cdots b}_{\ \ \ \ c\cdots d}$, where
$N^{ab}=g^{ab}+u^{a}u^{b}-n^{a}n^{b}$ projects vectors and tensors to the
sheet (it is the metric induced from the splitting, on the sheet).

A spacetime vector $\psi^{a}$ can accordingly be decomposed as $\psi^{a}=\Psi
u^{a}+\bar{\Psi}n^{a}+\Psi^{a}$ ($\Psi^{a}$ is the sheet component of
$\psi^{a}$), and a fully projected tensor ${}^{3}\psi_{ab}$ on the 3-space
decomposes as

${}^{3}\psi_{ab}=\Psi\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\Psi_{(a}n_{b)}+\Psi_{ab},$
(1)

with $\Psi=-^{3}\psi_{ab}N^{ab}$, $\Psi_{a}=N_{a}^{\ b}\ {}^{3}\psi_{bc}n^{c}$
and

$\displaystyle\Psi_{ab}=\left(N_{(a}^{\ c}N_{b)}^{\
d}-\frac{1}{2}N_{ab}N^{cd}\right)\ ^{3}\psi_{cd},$ (2)

is the symmetric, fully projected and trace-free part of ${}^{3}\psi_{ab}$.
The above can be seen as the 1+1+2 analogue of the 1+3 orthogonally projected
symmetric trace-free part of a 4-tensor:

$\displaystyle\psi^{\langle ab\rangle}=\left(h^{(a}_{\ c}h^{b)}_{\
d}-\frac{1}{3}h^{ab}h_{cd}\right)\psi^{cd},$ (3)

where $h_{ab}=g_{ab}+u_{a}u_{b}$ which introduces the derivative operator

$\displaystyle D_{f}\psi^{a\cdots b}_{\ \ \ \ c\cdots d}=h^{\bar{c}}_{\
c}\cdots h^{\bar{d}}_{\ d}h^{a}_{\ \bar{a}}\cdots h^{b}_{\ \bar{b}}h^{e}_{\
f}\nabla_{e}\psi^{\bar{a}\cdots\bar{b}}_{\ \ \ \ \bar{c}\cdots\bar{d}}.$

So, for example the shear, electric and magnetic Weyl 3-tensors decompose
respectively as

$\displaystyle\sigma_{ab}$
$\displaystyle=\sigma\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\Sigma_{(a}n_{b)}+\Sigma_{ab},$
(4) $\displaystyle E_{ab}$
$\displaystyle=\mathcal{E}\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\mathcal{E}_{(a}n_{b)}+\mathcal{E}_{ab},$
(5) $\displaystyle H_{ab}$
$\displaystyle=\mathcal{H}\left(n_{a}n_{b}-\frac{1}{2}N_{ab}\right)+\mathcal{H}_{(a}n_{b)}+\mathcal{H}_{ab}.$
(6)

Similarly, gradients of scalars decompose along these directions:

$\displaystyle\nabla_{a}\psi=-\dot{\psi}u_{a}+\hat{\psi}n_{a}+\delta_{a}\psi.$
(7)

The energy momentum tensor takes the covariant form

$\displaystyle T_{ab}=\rho u_{a}u_{b}+ph_{ab}+2q_{(a}u_{b)}+\pi_{ab}.$

Here, $\rho=T_{ab}u^{a}u^{b}$ is the (local) energy density, $3p=T_{ab}h^{ab}$
is the (isotropic) pressure, $q_{a}=T_{bc}h^{b}_{\ a}u^{c}$ is the heat flux
vector ($q_{a}=Qn_{a}+Q_{a}$), and $\pi_{ab}$, which can be decomposed using
(1) (with $\Pi=T_{ab}n^{a}n^{b}-p$), captures the degree of anisotropy.

From the Einstein field equations with cosmological constant

$\displaystyle R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab}=T_{ab},$ (8)

where $R_{ab}$ and $R$ are respectively the spacetime Ricci tensor and scalar
curvature, one can write down the Ricci tensor of the spacetime in the
covariant form

$\displaystyle R_{ab}$
$\displaystyle=g_{1}u_{a}u_{b}+g_{2}n_{a}n_{b}+2\left(Qu_{(a}+\Pi_{(a}\right)n_{b)}$
$\displaystyle+\left(g_{2}-\frac{3}{2}\Pi\right)N_{ab}+2Q_{(a}u_{b)}+\Pi_{ab},$

where

$\displaystyle
g_{1}=\frac{1}{2}\left(\rho+3p-2\Lambda\right)\quad\mbox{and}\quad
g_{2}=\frac{1}{2}\left(\rho-p+2\Lambda+2\Pi\right).$

The covariant derivatives of the unit vector fields can be written down in
terms of the kinematic (geometrical) variables:

$\displaystyle\nabla_{a}u_{b}$
$\displaystyle=-u_{a}\left(\mathcal{A}e_{b}+\mathcal{A}_{b}\right)+\left(\frac{1}{3}\theta+\sigma\right)n_{a}n_{b}$
$\displaystyle+\frac{1}{2}\left(\frac{2}{3}\theta-\sigma\right)N_{ab}+\Omega\varepsilon_{ab}+\Sigma_{ab}$
$\displaystyle+2\left(n_{(a}\Sigma_{b)}+n_{[a}\varepsilon_{b]c}\Omega^{c}\right),$
(9a) $\displaystyle\nabla_{a}n_{b}$
$\displaystyle=-u_{a}\left(\mathcal{A}u_{b}+\alpha_{b}\right)+\left(\frac{1}{3}\theta+\sigma\right)n_{a}u_{b}$
$\displaystyle+\frac{1}{2}\phi
N_{ab}+\xi\varepsilon_{ab}+\zeta_{ab}+n_{a}a_{b}$
$\displaystyle+\left(\Sigma_{a}-\varepsilon_{ac}\Omega^{c}\right)u_{b}.$ (9b)

$\mathcal{A}=n_{a}\dot{u}^{a}$ is acceleration,
$\theta=h^{ab}\bar{D}_{a}u_{b}$ is expansion of $u^{a}$, $\phi$ is expansion
of $n^{a}$ (called sheet expansion), $\sigma=\sigma_{ab}n^{a}n^{b}$ is the
shear, $2\Omega=\varepsilon^{ab}\nabla_{a}u_{b}$ and
$2\xi=\varepsilon^{ab}\nabla_{a}n_{b}$ are the respective twists of $u^{a}$
and $n^{a}$ ($\Omega$ is usually referred to as vorticity or rotation, and
$\varepsilon_{ab}$ is the 2-dimensional alternating tensor); $\mathcal{A}^{a}$
is the sheet component of $\dot{u}_{a}$; $\alpha_{a}$ is the sheet component
of $\dot{n}_{a}$; $a_{a}=\hat{n}_{a}$ is the acceleration of $n^{a}$ and;
$\zeta_{ab}=\delta_{\\{a}n_{b\\}}$ is the shear of $n^{a}$.

The field equations then takes a covariant form as a set of evolution and
propagation equations of the covariant scalars, vectors and tensors, along
with some constraint equations. These are obtained from the field equations
themselves, and the Ricci and Bianchi identities for the unit fields $u^{a}$
and $n^{a}$. We do not need all of these equations here and will not enlist
them. The few equations that will be needed for the current work will be
introduced accordingly.

###  II.2 Mapping the background and ‘true’ spacetimes

Perturbative schemes to study LRS cosmologies have been recently implemented
using the 1+1+2 formalism. They have largely been applied to the LRS II class
of spacetimes, with a recent consideration being a generalization which
introduces dissipation of the fluid via the non-vanishing of the anisotropic
stress on the background (see for example these recent works [22, 23]).

In this section, we begin by first introducing LRS spacetimes in context of
the 1+1+2 formulation, and we briefly discuss the linearization procedure
(more details can be found in the references [16, 17]), up to where it is
relevant to this paper.

Locally rotationally symmetric (LRS) spacetimes admit a continuous isotropy
group at each point, as well as a preferred spatial direction (see the
references [14, 15] for more details).

The symmetry of these spacetimes allows one to write the metric in local
coordinates $(t,\mathcal{R},y,z)$ as (see [15] for example)

$\displaystyle ds^{2}$
$\displaystyle=-A_{1}^{2}dt^{2}+A_{2}^{2}dr^{2}+A_{3}^{2}dy^{2}$
$\displaystyle+\left(\left(DA_{3}\right)^{2}+\left(A_{2}h\right)^{2}-\left(A_{1}g\right)^{2}\right)dz^{2}$
$\displaystyle+2\left(A_{1}^{2}gdt-A_{2}^{2}hdr\right)dz,$ (10)

where $A_{i}=A_{i}\left(t,r\right)$, and $g=g(y)$, $h=h(y)$. $D=D(y,k)$ where
$k$ is a constant and specified $D$: For $k=-1$, $D=\sinh y$; for $k=0$, $D=y$
and; for $k=1$, $D=\sin y$.

In the limiting case that $g=0=h$, we recover the well studied LRS II class of
spacetimes, which generalizes spherically symmetric solutions to the Einstein
field equations. This class includes FLRW models, LTB spacetime model,
Schwardschild solution, Oppenheimer–Snyder dust model, etc.

The covariant set

$\displaystyle\mathcal{D}:\equiv\\{{\mathcal{A},\theta,\sigma,\phi,\xi,\Omega,\rho,p,Q,\Pi,\mathcal{E},\mathcal{H}\\}},$
(11)

specifies a LRS spacetime.

In general, the mapping between the background and perturbed spacetime
generally depends on the problem of physical interest. And for the specific
background under consideration, the linearization procedure may or may not be
particularly cumbersome. Essentially, the problem reduces to determining which
quantities vanish in the background and which do not. Here, we outline the
procedure for the perturbation. (A detailed discussion on this is found in
[17], formulated differently but unchanging underlying concept.)

  * •

We first define the background spacetime: start with a background LRS
spacetime $M$, and let the collection of scalars

$\displaystyle\mathcal{D}_{1}:\equiv\\{{\varphi^{i}\\}}_{i\in I},$ (12)

for some indexing set $I$, specify $M$, i.e.
$\mathcal{D}_{1}\subset\mathcal{D}$. (Note that all 2-vector and 2-tensor
quantities, those defined on the sheet, vanish in the background).

  * •

Define the set

$\displaystyle\mathcal{D}_{2}:\equiv\mathcal{D}\setminus\mathcal{D}_{1},$ (13)

of those covariant scalars in $\mathcal{D}$ that vanish in the background.

  * •

Further introduce the set of 2-vectors and 2-tensors:

$\displaystyle\mathcal{D}_{3}:\equiv\\{{\Psi_{a}^{j},\Psi_{ab}^{w}\\}}_{j\in
J,w\in W},$ (14)

with $J$ and $W$ being indexing sets, which also vanish in the background.

  * •

The full set of gauge invariant quantities is then

$\displaystyle\mathcal{D}_{2}\cup\mathcal{D}_{3},$ (15)

which follows from the Stewart-Walker Lemma [25]. One can then use the Ricci
and Bianchi identities for $u^{a}$ and the direction of anisotropy $n^{a}$ to
obtain the linearized equations for the set
$\bar{D}:\equiv\mathcal{D}_{1}\cup\mathcal{D}_{2}\cup\mathcal{D}_{3}$.

The linearized system is generally not gauge invariant. The evolution and/or
propagation equations of the background quantities may contain terms that do
not vanish in the background. This is resolved by introducing $\delta$
gradients of the background quantities whose evolution and propagation
equations replace those of the background scalars.

  * •

We note that the ‘type’ of perturbation being carried out will further
simplify the collection $\mathcal{D}_{2}\cup\mathcal{D}_{3}$. For example, one
may consider an irrotational or a shear-free perturbation, eliminating all
shear and vorticity quantities.

There is also a degree of freedom in choosing the frame as $n^{a}$ can be
freely chosen. This choice will depend on the particular physical
consideration at hand. One common choice when dealing with black hole
perturbation is to set the sheet component of the acceleration to zero [16]
which corresponds to a hovering observer. (At some point in this work we will
make this choice of frame to transparently demonstrate our characterization
scheme.) Other choices include the aligning of $n^{a}$ along the vorticity or
the electric Weyl scalar. That is, one makes the respective frame choices
$\Omega^{a}=0$ and $\mathcal{E}^{a}=0$.

  * •

In general the linearized equations are quite messy and seldom tractible,
analytically. This generally results from the presence of the angular
‘$\delta$’ derivative. One approach is to decompose first order variables into
harmonics, which converts the linearized equations into ordinary differential
equations which in some cases can be solved exactly.

In this work we do not seek to solve the linearized system. Rather, we suppose
that the harmonically decomposed linearized system can be solved exactly for
an appropriate basis. Then, we specify the roles that certain components of
the even (electric) vector harmonics (to be defined shortly) play in the
evolution of a null horizon embedded in a LRS background which is subject to a
perturbation.

###  II.3 Harmonic decomposition

We now present a quick overview of the harmonic decomposition procedure
adapted to the formalism. Here we again follow the general setup in the
references [16, 17].

One starts by introducing dimensionless harmonic functions that satisfy

$\displaystyle\delta^{2}\mathcal{Q}=-\frac{\bar{k}^{2}}{r^{2}}\mathcal{Q},\quad\hat{\mathcal{Q}}=\dot{\mathcal{Q}}=0,\quad\quad\quad\quad\mbox{($0\leq\bar{k}^{2}$)}$
(16)

on any LRS background, i.e. they are eigenfunctions of the sheet Laplacian.
The function $r$ is covariantly specified by the relations

$\displaystyle\dot{r}=\frac{r}{2}\left(\frac{2}{3}\theta-\sigma\right),\quad\hat{r}=\frac{r}{2}\phi,\quad\delta_{a}r=0.$
(17)

One may decompose any first order scalar $\Psi$ as

$\displaystyle\psi^{(1)}=\sum\psi_{S}^{(\bar{k})}\mathcal{Q}^{(\bar{k})},$
(18)

usually written simply as $\psi^{(1)}=\psi_{S}\mathcal{Q}$, with the sum
implicit over $\bar{k}$, and the $S$ subscript signalling a scalar
decomposition.

The 2-vectors are also expanded in harmonics via the following defined basis

$\displaystyle\mathcal{Q}^{\bar{k}}_{a}$
$\displaystyle=r\delta_{a}\mathcal{Q}^{\bar{k}}\implies N^{b}_{\
a}\hat{\mathcal{Q}}_{b}=0=N^{b}_{\ a}\dot{\mathcal{Q}}_{b},$
$\displaystyle\delta^{2}\mathcal{Q}_{a}$
$\displaystyle=\frac{1}{r^{2}}\left(1-\bar{k}^{2}\right)\mathcal{Q}_{a};$
(19a) $\displaystyle\bar{\mathcal{Q}}^{\bar{k}}_{a}$
$\displaystyle=r\varepsilon_{ab}\delta^{b}\bar{\mathcal{Q}}^{\bar{k}}\implies
N^{b}_{\ a}\hat{\bar{\mathcal{Q}}}_{b}=0=N^{b}_{\
a}\dot{\bar{\mathcal{Q}}}_{b},$ $\displaystyle\delta^{2}\bar{\mathcal{Q}}_{a}$
$\displaystyle=\frac{1}{r^{2}}\left(1-\bar{k}^{2}\right)\bar{\mathcal{Q}}_{a},$
(19b)

respectively named even (electric) and odd (magnetic) vector harmonics. (For
all that are to follow, there should be no confusion that is to arise if we
drop the superscript $\bar{k}$ in relevant expressions.)

Notice that the harmonic basis vectors $\mathcal{Q}_{a}$ and
$\bar{\mathcal{Q}}_{a}$ can be obtained from one another, up to sign reversal,
by applying $\varepsilon_{ab}$:
$\bar{\mathcal{Q}}_{a}=\varepsilon_{ab}\mathcal{Q}^{b}\iff\mathcal{Q}_{a}=-\varepsilon_{ab}\bar{\mathcal{Q}}^{b}$.
For this reason, $\varepsilon_{ab}$ is understood as a parity operator, in the
sense of the harmonics introduced here. A quick check also shows that

$\displaystyle\delta_{a}\bar{\mathcal{Q}}^{a}$
$\displaystyle=0=\varepsilon_{ab}\delta^{a}\mathcal{Q}^{b},$
$\displaystyle\delta_{a}\mathcal{Q}^{a}$
$\displaystyle=-\frac{\bar{k}^{2}}{r}\mathcal{Q}=-\varepsilon_{ab}\delta^{a}\bar{\mathcal{Q}}^{b}.$

Since for each $\bar{k}$, $\mathcal{Q}_{a}\bar{\mathcal{Q}}^{a}=0$, first
order vectors ($2$-vectors) can always be decomposed as

$\displaystyle\Psi_{a}=\sum\Psi_{V}^{(\bar{k})}\mathcal{Q}_{a}+\bar{\Psi}_{V}^{(\bar{k})}\bar{\mathcal{Q}}_{a}=\Psi_{V}\mathcal{Q}_{a}+\bar{\Psi}_{V}\bar{\mathcal{Q}}_{a},$
(21)

again, where the sum over $\bar{k}$ is implicit, with the $V$ signalling a
vector decomposition. We note the following important point: for any first
order 2-vector $\Psi^{a}$, a quick check gives its sheet divergence as

$\displaystyle\delta_{a}\Psi^{a}=-\frac{\bar{k}^{2}}{r}\Psi_{V}Q.$ (22)

Thus, $\Psi^{a}$ is gradient if and only if $\Psi_{V}$ vanishes.

For a $2$-tensor $\Psi_{ab}$, the even and odd parity tensor harmonics are
defined respectively as

$\displaystyle\mathcal{Q}_{ab}$
$\displaystyle=r^{2}\delta_{\\{a}\delta_{b\\}}\mathcal{Q}\implies\hat{\mathcal{Q}}_{ab}=0=\dot{\mathcal{Q}}_{ab},$
(23a) $\displaystyle\bar{\mathcal{Q}}_{ab}$
$\displaystyle=r^{2}\varepsilon_{c\\{a}\delta^{c}\delta_{b\\}}\mathcal{Q}\implies\hat{\bar{\mathcal{Q}}}_{ab}=0=\dot{\bar{\mathcal{Q}}}_{ab}.$
(23b)

Again, we see that $\varepsilon_{ab}$ is a parity operator here:
$\bar{\mathcal{Q}}_{ab}=\varepsilon_{c\\{a}\mathcal{Q}_{b\\}}^{\ \
c}\iff\mathcal{Q}_{ab}=-\varepsilon_{c\\{a}\bar{\mathcal{Q}}_{b\\}}^{\ \ c}$.
Indeed, for each $\bar{k}$ $\mathcal{Q}_{ab}\bar{\mathcal{Q}}^{ab}=0$ so that
any $\Psi_{ab}$ can be decomposed as

$\displaystyle\Psi_{ab}=\Psi_{T}\mathcal{Q}_{ab}+\bar{\Psi}_{T}\bar{\mathcal{Q}}_{ab},$

with the $T$ signalling a tensor decomposition.

##  III Characterizing MOTS evolution

We introduce a characterization scheme for the evolution of MOTS orthogonal to
$u^{a}$ and $n^{a}$. We are interested in how MOTS in constant time slices in
linearized LRS background geometries evolve. We begin with a quick definition
of MOTS, after which we introduce the characterization. As will be seen, the
particular problem we are trying to address, or more precisely, how our
objective is formulated, is imposed by the required gauge invariance of the
evolution characterizing function.

###  III.1 MOTS

For the surfaces under consideration, the following must hold in order for
‘2-space’ to be a genuine surface rather than simply a collection of tangent
planes (see the reference [17] for discussion):

$\displaystyle\xi=\Omega=a^{a}$ $\displaystyle=0;$
$\displaystyle\Sigma^{a}+\varepsilon^{ab}\Omega_{b}-\alpha^{a}$
$\displaystyle=0.$ (24)

(The left hand side of the second line is known as the Greenberg vector.) In
particular, these conditions ensure that the Lie bracket of $u^{a}$ and
$n^{a}$ vanishes, i.e. $u^{a}$ and $n^{a}$ are surface forming, and that the
operator $\delta_{a}$ is a ‘true’ covariant derivative for the surface [17].

Consider a 2-surface $S$ in spacelike constant $t$ slices, with unit spacelike
normal $n^{a}$. Outgoing and ingoing null normals are respectively

$\displaystyle k^{a}=u^{a}+n^{a};\quad
l^{a}=\frac{1}{2}\left(u^{a}-n^{a}\right),$

normalized so that $l_{a}k^{a}=-1$. There is a degree of freedom to rescale
the null vectors as $k^{a}\rightarrow fk^{a},l^{a}\rightarrow f^{-1}l^{a}$,
for $f>0$. But as will be noted shortly, the MOTS character relevant here is
invariant under such scaling.

The induced metric on $S$ is

$\displaystyle\mathcal{F}_{ab}=g_{ab}+2k_{(a}l_{b)},$ (25)

and the ingoing and outgoing expansions are, respectively,

$\displaystyle\chi=\mathcal{F}^{ab}\nabla_{a}k_{b};\qquad\bar{\chi}=\mathcal{F}^{ab}\nabla_{a}l_{b}.$

(For our current purposes, $\mathcal{F}_{ab}$ here is just $N_{ab}$.) By MOTS,
we mean that on $S$, one has $\chi=0$. In the case that the sign of
$\bar{\chi}$ is constrained as $\bar{\chi}<0$ on $S$, MOTS is abbreviated to
MTS. (A MTS is generally considered to be associated to horizons that enclose
black holes in dynamical spacetimes.) One sees that, under the rescaling, the
expansion becomes $\chi_{f}=f\chi$ (similarly,
$\bar{\chi}_{f}=f^{-1}\bar{\chi}$). Thus, it is clear that any sign conditions
on $\chi$ and $\bar{\chi}$ is seen to be invariant under the freedom to scale
the null vectors.

###  III.2 MOTS evolution

Because of the MOTS condition $\chi=0$, we know that the gradient
$\nabla_{a}\chi$ is orthogonal to the MOTS. Hence the norm, which we notate by
$z$, specifies the causal character, pointwise:

$\displaystyle z>0\quad\mbox{(Timelike)};\quad z<0\quad\mbox{(Spacelike)};\
z=0\quad\mbox{(Null)}.$

Let us decompose the expansion $\chi$ into zeroth (background) and first order
scalars:

$\displaystyle\chi=\chi_{0}+\chi_{1},$

with the subscripts $\ast_{0}$ and $\ast_{1}$ denoting, respectively, zeroth
and first orders. (In general this will be the case as this is a linear
combination of the covariant scalars, some of which may or may not vanish in
the background.) We expand the norm $z$ as

$\displaystyle z$
$\displaystyle=\left(-\dot{\chi}_{0}^{2}+\hat{\chi}_{0}^{2}\right)+\left(-\dot{\chi}_{1}^{2}+\hat{\chi}_{1}^{2}\right)+2\left(-\dot{\chi}_{0}\dot{\chi}_{1}+\hat{\chi}_{0}\hat{\chi}_{1}\right)$
$\displaystyle+\left(\delta_{a}\chi_{0}\delta^{a}\chi_{0}+\delta_{a}\chi_{1}\delta^{a}\chi_{1}+2\delta_{a}\chi_{0}\delta^{a}\chi_{1}\right).$
(26)

As we neglect the products of first order quantities, the second and fourth
parenthesized terms of (III.2) vanish and we have

$\displaystyle
z=\left(-\dot{\chi}_{0}^{2}+\hat{\chi}_{0}^{2}\right)+2\left(-\dot{\chi}_{0}\dot{\chi}_{1}+\hat{\chi}_{0}\hat{\chi}_{1}\right).$
(27)

Indeed, it is clear that depending on the particular background solution one
begins with, the above expression will simplify as will later be observed.

Of course there will be some cases where even extending to the nonlinear
regime, i.e. in cases of nonlinear perturbations, the characterization can
proceed using the form of $z$ (27), for example when $z$ is optical, i.e. when
$\delta_{a}z\delta^{a}z$ vanishes on the MOTS. Let us consider some cases
where this is true for a particular frame choice or a class of backgrounds:
The commutation relations between the ‘$\delta_{a}$’, and the dot and hat
derivatives, acting on a scalar $\psi$, are given respectively as (we set
$\Omega=\xi=a_{a}=0$)

$\displaystyle\delta_{a}\dot{\psi}-N^{\
b}_{a}\left(\delta_{b}\psi\right)^{\cdot}$
$\displaystyle=-\mathcal{A}_{a}\dot{\psi}+\left(\Sigma_{a}-\varepsilon_{ab}\Omega^{b}+\alpha_{a}\right)\hat{\psi}$
$\displaystyle+\frac{1}{2}\left(\frac{2}{3}\theta-\sigma\right)\delta_{a}\psi,$
(28) $\displaystyle\delta_{a}\hat{\psi}-N^{\
b}_{a}\widehat{\left(\delta_{b}\psi\right)}$
$\displaystyle=-\left(\Sigma_{a}-\varepsilon_{ab}\Omega^{b}\right)\dot{\psi}+\frac{1}{2}\phi\delta_{a}\psi.$
(29)

Now notice that the function $r$, introduced in the harmonic decomposition
obeys

$\displaystyle\dot{r}+\hat{r}=\frac{r}{2}\chi.$

And upon using the equations (28) and (29) with $\psi=r$ we find that

$\displaystyle\delta_{a}\chi=\left(\mathcal{A}_{a}+\alpha_{a}\right)\phi.$
(30)

So, a frame choice $\mathcal{A}_{a}=\alpha_{a}=0$, $\delta_{a}\chi=0$. Also,
on any background with a vanishing sheet expansion, it is true that
$\delta_{a}\chi=0$ on the MOTS. Thus, analyzing MOTS in perturbed spatially
homogeneous or static LRS backgrounds, the form of the function $z$ suffices
even in nonlinear regimes, although the linearized equations is now more
formidable. However, we will keep the forthcoming applications modest and
restrict ourselves to linear perturbations.

As is easily seen from (27), the scalar $z$ will generally not be gauge
invariant: Indeed, the second parenthesized terms do vanish in the background
due to the presence of $\chi_{1}$ and derivatives thereof, but the first
parenthesized ones do not. Characterization of the evolution of the MOTS in
the perturbed spacetime obviously requires a gauge invariant $z$. In fact, $z$
is gauge invariant if and only if the combination of terms in the first
parenthesis vanishes. It indeed follows that, were one to start with a null
horizon $\bar{H}$, foliated by MOTS, in the background, $z$ is gauge invariant
with respect to $\bar{H}$ since the vanishing of the sum in the first
parenthesis characterizes the nullity of $\bar{H}$ in the first place. For
this reason, we can more precisely frame what is being done in this work:

Consider an exact LRS background spacetime $M$ and let $\bar{H}$ be a null
horizon in $M$, foliated by MOTS. Let us suppose that $M$ is subject to a
perturbation of linear order. How do we characterize the evolution of the
horizon $\bar{H}$ under the perturbation?

Now we emphasize again that we are not interested in solving linearized
system. It is assumed that an appropriate gauge (and possibly frame) choice
has been made, so that one has a mapping between the background and the
perturbed spacetimes. Independent of the gauge choice, Then, by simply
imposing the vanishing condition

$\displaystyle-\dot{\chi}_{0}^{2}+\hat{\chi}_{0}^{2}=0,$

the function $z$ will characterize the evolution of a null horizon from the
background. The gauge choice that maps the background and perturbed spacetimes
of course determines the simplification of $z$ as it is constructed from
elements of the linearized equations.

Now, to ensure a consistent sign of $z$ along a horizon foliated by the MOTS
under consideration, one requires $\delta_{a}z=0$. Explicitly, this is the
requirement that

$\displaystyle\dot{\chi}_{0}\delta_{a}\dot{\chi}-\hat{\chi}_{0}\delta_{a}\hat{\chi}=0.$
(31)

Again, once a background is specified, the expression (31) simplifies as will
be seen in Section IV.

  * Remark 1.

We point out that if one relaxes the $\delta_{a}z=0$ condition, one allows for
a range of possibilities, for example, horizons with different portions
exhibiting varying causal characters. This is beyond the scope of the current
work and will be deferred for a subsequent paper.

Let us now compute the forms of the outgoing and ingoing null expansion
scalars:

$\displaystyle\chi=\frac{2}{3}\theta-\sigma+\phi,\qquad\bar{\chi}=\frac{1}{2}\left(\frac{2}{3}\theta-\sigma-\phi\right).$
(32)

Thus, a MOTS is specified by the covariant triple $(\theta,\sigma,\phi)$, and
for any particular background, this splits into the background and first order
parts.

If our interest is in MOTS that are MTS, then the requirement that
$\bar{\chi}<0$ implies that necessarily $\phi>0$. For our purposes here,
however, we will not impose such restriction and will instead consider the
more general ‘MOTS’.

We note that the form of $\chi$, first equation of (32), is form invariant
with respect to the background and so the gauge problem is automatically
fixed. This is a consequence of the particular class of MOTS we are
considering for this work. For a different class of MOTS, this correspondence
may be broken and some additional condition may have to be imposed to fix the
gauge, in general. However, up to linear order for null horizons as is
considered here (or more generally nonexpanding horizons) $\chi$ vanishes on
the perturbed horizon as was established in [11].

  * Remark 2.

Before proceeding, we point out that caution is to be exercised when
considering the vanishing of the first parenthesized terms of (III.2). For
example if one is to consider a static background, then obviously,
$\dot{\chi}^{2}_{0}$ should vanish. However, $\hat{\chi}^{2}_{0}$ does not
vanish on a null horizon in the background. So, the terms should be taken
together. In fact, its easily seen that the first parenthesized terms of
(III.2) – we denote this as $F$ – factors as

$\displaystyle F=-\mathcal{L}_{k}\chi\mathcal{L}_{l}\chi.$ (33)

The vanishing condition in the background, assuming the null energy condition
is then $\mathcal{L}_{k}\chi=0$. So, one has to take care of the function $F$,
so that once a background is specified, $F$ vanishes on a null horizon in the
background.

The evolution and propagation equations for $\phi$ and $(2/3)\theta-\sigma$,
up to linear order, are [16]

$\displaystyle\dot{\phi}$
$\displaystyle=\left(\frac{2}{3}\theta-\sigma\right)\left(\mathcal{A}-\frac{1}{2}\phi\right)+Q+\delta_{a}\alpha^{a},$
(34a) $\displaystyle\hat{\phi}$
$\displaystyle=-\frac{1}{2}\phi^{2}+\left(\frac{2}{3}\theta-\sigma\right)\left(\frac{1}{3}\theta+\sigma\right)-\mathcal{E}-\frac{1}{2}\Pi$
$\displaystyle-\frac{2}{3}\left(\rho+\Lambda\right)+\delta_{a}a^{a},$ (34b)
$\displaystyle\frac{2}{3}\dot{\theta}-\dot{\sigma}$
$\displaystyle=\mathcal{A}\phi-\frac{1}{2}\left(\frac{2}{3}\theta-\sigma\right)^{2}+\mathcal{E}-\frac{1}{2}\Pi$
$\displaystyle-\frac{1}{3}\left(\rho+3p-2\Lambda\right)+\delta_{a}\mathcal{A}^{a},$
(34c) $\displaystyle\ \frac{2}{3}\hat{\theta}-\hat{\sigma}$
$\displaystyle=\frac{3}{2}\phi\sigma+Q+\delta_{a}\left(\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right).$
(34d)

Equation (34d) is obtained by projecting along $n^{a}$ the shear divergence
equation [26]

$\displaystyle
D^{b}\sigma_{ba}-\frac{2}{3}D_{a}\theta+\varepsilon_{abc}\left(D^{b}+2\dot{u}^{b}\right)\omega^{c}+q_{a}=0,$

where

$\displaystyle\omega^{a}=\Omega n^{a}+\Omega^{a},\quad q^{a}=Qn^{a}+Q^{a}.$

Equations (34a), (34b), and (34c) are obtained, respectively, by applying
$u^{a}N^{bc}$, $n^{a}N^{bc}$, and $-n^{a}u^{b}u^{c}$ to the Ricci identities
for $n^{a}$.

We can write down the evolution of the null expansion along the null rays as

$\displaystyle\mathcal{L}_{k}\chi$
$\displaystyle=\chi\left(\mathcal{A}-\frac{1}{2}\phi+\frac{3}{2}\sigma\right)-\left(\rho+p+\Pi\right)$
$\displaystyle+2Q+\delta_{a}\left(\alpha^{a}+\mathcal{A}^{a}+a^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right),$
(35) $\displaystyle\mathcal{L}_{l}\chi$
$\displaystyle=-\frac{1}{2}\chi\left(\chi-\frac{1}{2}\phi+\frac{3}{2}\sigma\right)+\frac{1}{3}\left(\rho-3p\right)+2\mathcal{E}$
$\displaystyle+\delta_{a}\left(\alpha^{a}+\mathcal{A}a-a^{a}-\Sigma^{a}-\varepsilon^{ab}\Omega_{b}\right).$
(36)

Note that $\chi$ vanishes on the horizon and hence the above are simplified
there. The scalar $F$ now expands, up to linear order on the horizon, to the
form

$\displaystyle F$
$\displaystyle=\left(\frac{1}{3}\left(\rho-3p\right)+2\mathcal{E}\right)$
$\displaystyle\times\left(\left(\rho+p+\Pi\right)-2Q-\delta_{a}Z^{a}\right)$
(37)

where we have defined

$\displaystyle
Z^{a}=\alpha^{a}+\mathcal{A}^{a}+a^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}.$
(38)

Depending on the particular class of background, and how one chooses the
$\\{u^{a}-n^{a}\\}$ frame by appropriately fixing the $n^{a}$ direction, the
scalar $F$ can be considerably simplified as will be seen in the next section.

#### The future outer trapping condition

A MOTS $\mathcal{S}$ on which the ingoing expansion $\bar{\chi}$ is strictly
negative and $\mathcal{L}_{l}\chi<0$ is known as a future outer trapped
surface (FOTS). A horizon foliated by FOTS is a future outer trapping horizon
(FOTH). These were introduced by Hayward in [3] and have since be extensively
studied [27, 28].) In the dynamical and isolated cases, respectively, assuming
the null energy condition, one has that $\mathcal{L}_{k}\chi<0$ and
$\mathcal{L}_{k}\chi=0$. In this case we have a dynamical FOTH and an isolated
FOTH. Crucially, the FOTH condition immediately implies the presence of a
black hole as it ensures the existence of trapped surfaces just to the inside
of the horizon. Thus, the conditions characterizing a FOTH for the dynamical
and isolated cases are $\bar{\chi}<0$, (31), plus $\mathcal{L}_{k}\chi<0$ and
$\mathcal{L}_{k}\chi=0$, respectively,

$\displaystyle z$ $\displaystyle<0,\dot{\chi}<0,\hat{\chi}<0,$ (39)
$\displaystyle z$ $\displaystyle=0,\dot{\chi}<0,\hat{\chi}>0.$ (40)

Henceforth, when specifying to a FOTH the condition $\bar{\chi}<0$ will be
assumed.

In the next section, we consider some backgrounds as particular examples, to
apply the horizon characterization scheme developed in this section to
horizons in the corresponding perturbed spacetime.

##  IV Applications

In this section we consider some particular background LRS solutions.
Specifically, we consider static and spatially homogeneous LRS backgrounds,
with restriction to the class II. We will also briefly comment on the case of
a background with both $t$ and $r$ dependence (see Remark 3).

###  IV.1 Static background

Let us take as an example a non-dissipative linear perturbation of the
Schwarzschild background with metric (we set the mass to unity here, which
does not, in any serious way, affect the calculations)

$\displaystyle
ds^{2}=-\left(1-\frac{2}{r}\right)dt^{2}+\left(1-\frac{2}{r}\right)^{-1}dr^{2}+r^{2}d\varsigma^{2},$
(41)

where $d\varsigma^{2}$ is the metric on the unit 2-sphere. We know the above
is time-symmetric, i.e. $\theta=0=\sigma$. Hence, we have the zeroth and first
orders decomposition of the null expansion $\chi$ as

$\displaystyle\chi_{0}=\phi\qquad\mbox{and}\qquad\chi_{1}=\frac{2}{3}\theta-\sigma.$

Therefore, $\dot{\chi}_{0}$ and $\dot{\chi}_{1}$ are first order quantities so
that $\dot{\chi}_{0}^{2}=\dot{\chi}_{0}\dot{\chi}_{1}=0$. The only non-
vanishing background covariant scalars are $\mathcal{A},\phi,\mathcal{E}$,
given in coordinate forms as

$\displaystyle\mathcal{A}=\frac{1}{r^{2}}\left(\sqrt{1-\frac{2}{r}}\right)^{-1};\quad\phi=\frac{2}{r}\sqrt{1-\frac{2}{r}};\quad\mathcal{E}=-\frac{2}{r^{3}}.$

In this background the MOTS is located at $\phi=0\iff r=2$. Now, the norm $z$
becomes

$\displaystyle z$ $\displaystyle=F+2\hat{\chi}_{0}\hat{\chi}_{1}$
$\displaystyle=-4\mathcal{E}\delta_{a}\left(Z^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right).$
(42)

Additionally one also requires, from (31), that (we have now set $a^{a}=0$ on
the horizon as it should be)

$\displaystyle
0=\hat{\chi}_{0}\delta_{a}\hat{\chi}=\delta_{a}\mathcal{E}+\delta_{a}\left[\delta_{b}\left(\Sigma^{b}+\varepsilon^{bc}\Omega_{c}\right)\right].$
(43)

Since $\mathcal{E}<0$, coupled with (43), the $r=2$ horizon in the
Schwarzschild background evolves into an isolated (resp. dynamical) horizon
provided that

$\displaystyle\delta_{a}\left(Z^{a}+\Sigma^{a}+\varepsilon^{ab}\Omega_{b}\right)=\mbox{({resp.}
$>$)}0.$ (44)

Separately, we examine the two cases of (44).

####  IV.1.1 Null case

We start with the null criterion, with the vanishing of (44), in harmonics,
yielding

$\displaystyle
0=-\frac{\bar{k}^{2}}{r^{2}}\left[\mathcal{A}_{V}+2\left(\Sigma_{V}-\bar{\Omega}_{V}\right)\right],$
(45)

which should hold for all $\bar{k}$.

By defining $Y_{a}=\delta_{a}\mathcal{E}$ and decomposing the equation (43)
into harmonics, we write it down as the pair of equations

$\displaystyle
Y_{V}=-\frac{\bar{k}^{2}}{r^{2}}\left(\Sigma_{V}-\bar{\Omega}_{V}\right);\quad\bar{Y}_{V}=0.$
(46)

Upon substituting the first equation of (46) into (45) gives

$\displaystyle 0=-\frac{\bar{k}^{2}}{r^{2}}\mathcal{A}_{V}+2Y_{V}.$ (47)

Evoking the vanishing of the Greenberg vector, (46) is
$r^{2}Y_{V}+\bar{k}^{2}\alpha_{V}=0$, so that (47) reduces to the condition

$\displaystyle\mathcal{A}_{V}+2\alpha_{V}=0.$ (48)

This is the required condition that preserves the null character of the null
horizon $r=2$ under linear perturbation.

Were one to fix the frame by setting $\mathcal{A}^{a}=0$ (so that $n^{a}$
points along $\dot{u}^{a}$), we will have that the condition (48) now becomes
$\alpha_{V}=0$. Otherwise, the horizon does not retain its null character.

On the other hand, by (48), the vanishing condition
$\Sigma_{V}-\bar{\Omega}_{V}=0$ would lead to the nullity condition fixing the
frame: $\mathcal{A}_{V}=0$.

FOTH: Let us consider the FOTH condition, $\dot{\chi}<0$ and $\hat{\chi}>0$.
If one fixes the frame by setting $\mathcal{A}^{a}=0$, the FOTH condition
trivially follows: Explicitly, the FOTH condition is the pair

$\displaystyle\mathcal{E}$ $\displaystyle<-\delta_{a}\alpha^{a},$ (49)
$\displaystyle\mathcal{E}$
$\displaystyle<\delta_{a}a^{a}+\delta_{a}\alpha^{a},$ (50)

where we have used the fact that the Greenberg vector must vanish. So, again,
how the black hole evolves under linear perturbation depends crucially on how
the frame is choosen, as one would expect.

Again, $a^{a}$ must vanish for our MOTS to be a genuine surface. As the
specification of the frame choice $\mathcal{A}^{a}=0$ gives $\alpha_{V}=0$,
the FOTH condition is simply the requirement that $\mathcal{E}<0$, which
always holds.

####  IV.1.2 Dynamical case

Let us now consider the case where the $r=2m$ null horizon possibly evolves to
acquire a different character. (Again for simplicity let us make the same
frame choice here by setting $\mathcal{A}^{a}=0$.) Then, with all
considerations of the vanishing Greenberg vector, as well as the necessity of
the vanishing of $a^{a}$, the characterizing norm $z$ takes the form

$\displaystyle
z=-12\frac{\bar{k}^{2}}{r}\mathcal{E}\left(\Sigma_{V}-\bar{\Omega}_{V}\right).$
(51)

We note that from the field equations the vectors $\Sigma_{a}$ and
$\Omega_{a}$ do not evolve independently [16, 17], and so the presence of one
ensures the non-vanishing of the other. Hence, we see that as long as there is
a nonvanishing shear, $z\neq 0$ for $\bar{k}^{2}\neq 0$, and if there is no
shear the horizon stays null as was seen in the previous case.

We do expect that under the perturbation and in the presence of a nonzero
shear the Schwarzschild event horizon becomes spacelike. This then imposes
that

$\displaystyle\Sigma_{V}-\bar{\Omega}_{V}<0,$ (52)

thereby imposing a constraint on the solution to the harmonically decomposed
linearized system, when restricted to the horizon. We summarize the above
discussion as follows:

Let $\tilde{M}$ be the spacetime obtained by linearly perturbing a
Schwarzschild background. Then the $r=2$ horizon is null in $\tilde{M}$ if and
only if the shear and vorticity 2-vectors obey
$\Sigma_{V}-\bar{\Omega}_{V}=0$, but otherwise spacelike in $\tilde{M}$ with
$\Sigma_{V}-\bar{\Omega}_{V}<0$.

Note: A nondissipative perturbation is assumed here. The above conclusion
drawn therefore may or may not be true in the presence of a nonzero $Q$ and
$\Pi$, depending on their relative magnitudes.

###  IV.2 Spatially homogeneous background

We consider a hypersurface orthogonal LRS II solution, which is spatially
homogeneous, with metric

$\displaystyle
ds^{2}=-dt^{2}+\bar{a}^{2}_{1}(t)dr^{2}+\bar{a}^{2}_{2}(t)d\bar{\varsigma}^{2}.$
(53)

(The 2-surface with metric $d\bar{\varsigma}^{2}$ is allowed different
geometries including a toroidal one.) In this case we have that
$\mathcal{A}=\phi=0$, with the non-vanishing background kinematic quantities
being the expansion and shear:

$\displaystyle\theta=\frac{\bar{a}_{1t}}{\bar{a}_{1}}+2\frac{\bar{a}_{2t}}{\bar{a}_{2}};\qquad\sigma=\frac{2}{3}\left(\frac{\bar{a}_{1t}}{\bar{a}_{1}}-\frac{\bar{a}_{2t}}{\bar{a}_{2}}\right),$
(54)

where the underscore ‘$t$’ indicates partial differentiation. Hence,

$\displaystyle\chi_{0}=\frac{2}{3}\theta-\sigma\qquad\mbox{and}\qquad\chi_{1}=\phi.$

Indeed, $\dot{\chi}_{1}$ and $\hat{\chi}_{0}$ are first order quantities so
that $\hat{\chi}_{0}^{2}=\hat{\chi}_{0}\hat{\chi}_{1}=0$. For simplicity, we
specify to a conformally flat perturbation, and set $\Lambda=0$. We further
restrict to the nondissipative case. This then gives the norm $z$ as (we
recall that this is always evaluated at $\chi=0$)

$\displaystyle z$ $\displaystyle=F-2\dot{\chi}_{0}\dot{\chi}_{1}$
$\displaystyle=\frac{1}{3}\left(\rho-3p\right)\left[\left(\rho+p\right)-\delta_{a}Z^{a}\right]-\frac{4}{3}\rho\delta_{a}\alpha^{a}.$
(55)

Also, the vanishing condition (31) becomes (again evaluating at $\chi=0$)

$\displaystyle 0=\dot{\chi}_{0}\delta_{a}\dot{\chi}$
$\displaystyle=-\frac{1}{3}\left(\rho+3p\right)\delta_{a}\left(\delta_{b}\mathcal{A}^{b}+\delta_{b}\alpha^{b}\right)$
$\displaystyle=\frac{\bar{k}^{2}}{3r^{2}}\left(\rho+3p\right)\left(\mathcal{A}_{V}+\alpha_{V}\right)\mathcal{Q}_{a}.$
(56)

So, either we fix the equation of state $\rho+3p=0$ or
$\mathcal{A}_{V}+\alpha_{V}=0$.

Now, for the former case this would require that $\rho$ (and consequently $p$)
vanishes in the background to ensure the gauge invariance of $z$, in which
case the horizon is always null.

Let us consider the latter case, and consider the particular frame choice
$\mathcal{A}^{a}=0$. This would impose that $\alpha_{V}=0$. (Note again that
this condition is compatible with $\Sigma_{V}-\bar{\Omega}_{V}=0$ by virtue of
the vanishing of the Greenberg vector.) Therefore, the characterizing function
$z$ reduces to

$\displaystyle z=\frac{1}{3}\left(\rho-3p\right)\left(\rho+p\right).$ (57)

Hence, assuming the weak energy condition, the character of a null horizon,
evolving under linear perturbation, behaves according to the sign of
$\rho-3p$: the horizon stays null, or becomes spacelike or timelike provided
$\rho=3p,\rho<3p$ or $\rho>3p$. So, for example, with a linear equation of
state (EoS) $p=w\rho$, the above conditions are the respective requirements
$w=0,w>0$ and $w<0$.

Of course, a choice different from $\mathcal{A}^{a}=0$ may be used to fix the
frame. In this case, we will have $z$ assuming the form

$\displaystyle
z=\frac{1}{3}\left[\left(\rho-3p\right)\left(\rho+p\right)+\frac{\bar{k}^{2}}{r}\left(5\rho-3p\right)\alpha_{V}\right].$
(58)

We see that even with a linear EoS of the form $p=w\rho$, the signature of $z$
above acquires some complexity an carries a crucial dependence on sign of the
component $\alpha_{V}$ (Note that $\alpha_{V}=\Sigma_{V}-\bar{\Omega}_{V}$).
We may however still comment on some possible characters of the perturbed
horizon (we are assuming a positive $\rho$). For example, it is not difficult
to see that for the range

$\displaystyle-1\leq w\leq\frac{1}{3},$ (59)

$\alpha_{V}>0$ precludes a spacelike or null character, allowing for only a
timelike character. This indeed differs from the timelike characterization of
the background as the extremes $w=-1$ and $w=1/3$ allow only for the null
character. (See for example the references [29, 19].) If we look outside the
bound (59) in the case $\alpha_{V}>0$, for $w<-1$, the horizon character will
depend on the magnitude of $\alpha_{V}$ at each $r$ in relation to each
$\bar{k}^{2}$. (In the background, this is quite straightforward as outside
the bound (59), the horizon is spacelike. Again, see the references [29, 19].)
This will also be the case for the bound

$\displaystyle\frac{1}{3}<w<\frac{5}{3},$ (60)

but for $w\geq 5/3$, only the timelike character is possible.

Now consider the case for $\alpha_{V}<0$. Then, a timelike or null character
is ruled out within the bound (60), allowing for only a spacelike character.
For the bounds

$\displaystyle-1<w<\frac{1}{3}\qquad\mbox{and}\qquad w>\frac{5}{3},$ (61)

the character again depends on the magnitude of $\alpha_{V}$ at each $r$ in
relation to each $\bar{k}^{2}$.

At the extreme $w=1/3$ the horizon is spacelike and timelike for

$\displaystyle\frac{1}{3}<w\leq\frac{5}{3},$ (62)

And for values $w<-1$, a timelike or null character is prohibited. (Notice
that at the critical point $w=5/3$ the horizon signature is insensitive to the
sign of $\alpha_{V}$.)

We summarize the above results as follows:

Let $\tilde{M}$ be the spacetime obtained by linearly perturbing a spatially
homogeneous background $M$ with EoS $p=w\rho$, and let $\mathcal{T}$ be a null
horizon foliated by MOTS in $M$. Then for $\Sigma_{V}-\bar{\Omega}_{V}=0$,
$\mathcal{T}$ is null, spacelike or timelike in $\tilde{M}$ provided $w=0,w>0$
or $w<0$, respectively. Otherwise, for $\Sigma_{V}-\bar{\Omega}_{V}\neq 0$,
see Table 1. (Note that $\alpha_{V}=\Sigma_{V}-\bar{\Omega}_{V}$ on the MOTS.)

Null horizon character under linear perturbation
---
$\alpha_{V}$ |  $w$ |  Character
|  $(-1\leq w\leq\frac{1}{3})\cup(w\geq\frac{5}{3})$ |  Timelike
$\alpha_{V}>0$ |  |
|  $(\frac{1}{3}<w<\frac{5}{3})\cup(w<-1)$ |  –
|  $(-1<w<\frac{1}{3})\cup(w>\frac{5}{3})$ |  –
$\alpha_{V}<0$ |  $\frac{1}{3}<w\leq\frac{5}{3}$ |  Timelike
|  $(w\leq-1)\cup(w=\frac{1}{3})$ |  Spacelike
Table 1: The behavior of a null horizon evolving under linear perturbation.
Assuming a linear equation of state $p=w\rho$, the above table shows the
character of the horizon for different ranges of the equation of state
parameter $w$, and the relationship with the sheet vector $\alpha^{a}$
decomposed harmonically. The ‘$-$’ indicates a non-trivial relationship which
is dependent on the magnitudes of $\alpha_{V}$ and $\rho$ as well as the value
of the nonnegative integer $\bar{k}^{2}$.

Let it be emphasized that the linear choice of the EoS is for demonstrable
purposes for our particular application and there is no restriction here. Any
EoS is in principle allowable with (possible) caveat being the the analysis
acquiring a more complicated character.

We will now conclude this section with the following remark:

  * Remark 3.

In a general inhomogeneous case, $\chi_{1}=0$ and we see that the associated
norm is simplified:

$\displaystyle z=F,$

so that the characterization proceeds from (III.2). However, the consideration
of the $\delta_{a}z=0$ criterion is now significantly more complicated.
Relaxing this condition of course then gives a simple pointwise
characterization via (III.2). If one does not impose $\delta_{a}z=0$ a priori,
this allows for the possibility of varying character across the horizon. This
is clearly a more involving case and this is worth considering for a
subsequent work. However, as was just mentioned, while generally these cases
are expected to pose problems, we see that in a general inhomogeneous
background the condition provides a certain analytic simplicity, at least
point-wise.

##  V On the MOTS stabiliity operator

As have been seen from the previous section, the 2-vectors associated to the
shear and vorticity play a crucial role in how a null horizon in a LRS
background spacetime evolves under a linear perturbation. This particular
combination of the shear and vorticity evolves simultaneously according to the
field equations and have no independent evolution or propagation equations as
was mentioned in the previous section. This relationship has implications for
the self-adjointness of the MOTS stability operator as these same 2-vectors
also control the self-adjointness of the operator. We will begin by
introducing the MOTS stability operator after which we provide some commentary
on this relationship. We will then delve into several properties of the
operator in context of our current considerations as it relates to self-
adjointness. The evolution, propagation and sheet equations that will be
utilized here have been obtained in [16, 17] (with corrections in [22, 23] as
stated in Section III). So, especially the reference [16] is implicit wherever
these equations appear.

###  V.1 The MOTS stability operator

Given a MOTS $\mathcal{S}$ and a normal foliation by a collection
$\mathcal{S}_{v}$, with the subscript ‘$v$’ labelling the foliation, one may
define the tangent vector to curves generating the deformation as

$\displaystyle\partial_{v}=\bar{\psi}n^{a},$

for some smooth function $\bar{\psi}$ on $\mathcal{S}$. Then, stability of
$\mathcal{S}$ is captured via the variation of the expansion $\chi$ along
$\partial_{v}$ (several discussions of this can be found in the original
papers [6, 7], as well as, for example, the works [30, 31] and references
therein):

$\displaystyle\bar{\delta}_{\bar{\psi}n}\chi=L_{\mathcal{S}}\bar{\psi},$

where the operator $L_{\mathcal{S}}$ is

$\displaystyle L_{\mathcal{S}}=-\delta^{2}+2\tilde{s}^{a}\delta_{a}+\bar{F},$
(63)

with the definition

$\displaystyle\bar{F}=\frac{1}{2}\mathcal{R}_{\mathcal{S}}-\tilde{s}_{a}\tilde{s}^{a}+\delta_{a}s^{a}-G_{ab}k^{a}l^{b},$
(64)

(It is important to note that the scalar $\bar{\psi}$ completely characterizes
the variation of $\mathcal{S}$.) The one-form $\tilde{s}_{a}=-N^{c}_{\
a}l_{b}\nabla_{c}k^{b}$ is the projection of the torsion of $k^{a}$ onto
$\mathcal{S}$, $G_{ab}$ is the Einstein tensor and $\mathcal{R}_{\mathcal{S}}$
is the scalar curvature of $\mathcal{S}$.

$\mathcal{S}$ is said to be strictly or marginally stable if there exists a
nonnegative (and not identically zero) $\bar{\psi}$ obeying respectively
$L_{\mathcal{S}}\bar{\psi}>0$ or $L_{\mathcal{S}}\bar{\psi}=0$, and unstable
otherwise.

For a 1+1+2 decomposed spacetime, $\tilde{s}_{a}$ is explicitly

$\displaystyle\tilde{s}_{a}=\Sigma_{a}-\varepsilon_{ab}\Omega^{b},$ (65)

And in the harmonic basis,

$\displaystyle\tilde{s}_{a}=\underbrace{\left(\Sigma_{V}+\bar{\Omega}_{V}\right)}_{\text{$\tilde{s}_{V}$}}\mathcal{Q}_{a}+\underbrace{\left(\bar{\Sigma}_{V}-\Omega_{V}\right)}_{\text{$\bar{\tilde{s}}_{V}$}}\bar{\mathcal{Q}}_{a}.$
(66)

In general the operator $L_{\mathcal{S}}$ is not self-adjoint, which results
from the presence of the linear term $\tilde{s}^{a}\delta_{a}$ so that
$L_{\mathcal{S}}$ is allowed to have complex eigenvalues. The imaginary part
encodes information about rotation through the Komar angular momentum,
obtained by integrating over the MOTS the inner product of $\tilde{s}_{a}$ and
an axial Killing field.

Still, the principal eigenvalue, which we denote as $\bar{\lambda}$, is always
real [6]. The stability of a MOTS $\mathcal{S}$ then reduces to the sign of
$\bar{\lambda}$: $\mathcal{S}$ is strictly or marginally stable if
$\bar{\lambda}>0$ or $\bar{\lambda}=0$, and unstable otherwise.

Here, we are interested in the implications of the characterization conditions
obtained for the backgrounds in the previous sections, for the self-
adjointness of the stability operator. Crucially, the dependence on the sign
of $\alpha_{V}$ (or equivalently $\Sigma_{V}-\bar{\Omega}_{V}$).

Now, it is known that the self-adjointness of the MOTS stability operator is
guaranteed by the one-form $\tilde{s}_{a}$ being a gradient [31]. Particular,
if the 1-form $\tilde{s}_{a}$ is gradient, then the operator $L_{\mathcal{S}}$
is similar to a self-adjoint operator. (Of course, the case of vanishing
$\tilde{s}_{a}$ is trivial, and so we consider this possibility
$\tilde{s}_{a}\neq 0$.) This self-adjointness in the case that $\tilde{s}_{a}$
is gradient can be seen here in a rather trivial way. To see this, let us
consider the action of the operator $L_{\mathcal{S}}$ on a first order scalar
$\psi^{(1)}=\psi_{S}\mathcal{Q}$:

$\displaystyle
L_{\mathcal{S}}y^{(1)}=\left[\left(-\delta^{2}+\bar{F}\right)\mathcal{Q}+\frac{1}{r}\tilde{s}_{V}\right]\psi_{S}.$
(67)

(The normalization $\mathcal{Q}_{a}\mathcal{Q}^{a}=1$ has been assumed for
simplicity.) Since if the $\tilde{s}_{a}$ is gradient it has a vanishing
divergence, it is then clear that this requires the vanishing of
$\tilde{s}_{V}$, i.e.

$\displaystyle\Sigma_{V}+\bar{\Omega}_{V}=0.$ (68)

Thus, the eigenvalue problem for the MOTS stability operator simply reduces to
the following eigenvalue problem for $\mathcal{Q}$:

$\displaystyle-\delta^{2}\mathcal{Q}+F\mathcal{Q}=\lambda\mathcal{Q}.$ (69)

That is, the problem reduces to the eigenvalue problem for the self-adjoint
operator
$L^{\prime}_{\mathcal{S}}\mathcal{Q}=\left(-\delta^{2}+F\right)\mathcal{Q}=\lambda\mathcal{Q}$
so that

$\displaystyle\lambda_{\bar{k}^{2}}=\frac{\bar{k}^{2}}{r}+\bar{F},$ (70)

i.e. the eigenvalues are parametrized by $\bar{k}^{2}$. Thus, strict stability
is always guaranteed for

$\displaystyle\bar{F}=\frac{1}{2}\left(\mathcal{R}_{\mathcal{S}}-R\right)-\bar{\tilde{s}}_{V}^{2}-\left(p+Q\right)+2\Lambda>0.$
(71)

(recall that $R$ is the four dimensional scalar curvature.)

It will later be seen that $\bar{\tilde{s}}_{V}$ is necessarily zero on the
MOTS in the case that $\tilde{s}_{V}=0$, i.e. $\tilde{s}_{a}$ is a gradient.
So, in this case for $p+Q\leq 0$ and $\Lambda\geq 0$, whenever
$\mathcal{R}_{\mathcal{S}}>R$, the operator has no negative eigenvalue. The
principal eigenvalue and the MOTS is strictly stable. This is indeed the case,
say, for vacuum solutions undergoing a shear-free and a nondissipative
perturbation where one will have $\tilde{s}_{V}$ vanishing.

On the other hand, consider an irradiating ($Q=0$) spacetime with a positive
scalar curvature, a vanishing $\Lambda$ and a nonnegative pressure. Then, for
a sufficiently large $p$ it is possible that $\bar{F}<0$, so that the
principal eigenvalue is always negative and hence the MOTS will be unstable.

  * Remark 4.

Actually, in the linear regime any perturbation to the MOTS scalar curvature
$\mathcal{R}_{\mathcal{S}}$ is encoded in first order scalars. Particularly,
if the scalars $\mathcal{E},\Pi$, and $\Lambda$ (if $\Lambda$ is included as a
perturbation variable) are nonvanishing in the background, then
$\mathcal{R}_{\mathcal{S}}$ is not perturbed. However, nonlinear effects are
present under perturbation of the background as

$\displaystyle\mathcal{R}_{\mathcal{S}}=\
^{0}\mathcal{R}_{\mathcal{S}}+\mathcal{Z},$ (72)

where $\mathcal{Z}=\Sigma_{ab}\Sigma^{ab}-\zeta_{ab}\zeta^{ab}$ is just twice
the inner product of the projected shears of $k^{a}$ and $l^{a}$. So, it is
clear that once nonlinear effects are factored in any analysis, we do expect
the behavior of the eigenvalues to be modified.

Now, the commutation relations of the dot and hat derivatives for an arbitrary
2-vector $\Psi_{a}$ (imposing the vanishing of the Greenberg vector and taken
up to linear order)

$\displaystyle\hat{\dot{\Psi}}_{a}-\dot{\hat{\Psi}}_{a}=-\mathcal{A}\dot{\Psi}_{a}+\left(\frac{1}{3}\theta+\sigma\right)\hat{\Psi}_{a}+\mathcal{H}\varepsilon_{ab}\Psi^{b}.$
(73)

Harmonically decomposing the above relation and applying to the one-form
$\tilde{s}_{a}$ while setting the component $\tilde{s}_{V}=0$ says that

$\displaystyle\hat{\dot{\bar{\tilde{s}}}}_{V}-\dot{\hat{\bar{\tilde{s}}}}_{V}$
$\displaystyle=-\mathcal{A}\dot{\bar{\tilde{s}}}_{V}+\left(\frac{1}{3}\theta+\sigma\right)\hat{\bar{\tilde{s}}}_{V},$
(74) $\displaystyle\mathcal{H}\bar{\tilde{s}}_{V}$ $\displaystyle=0.$ (75)

The above is meant to suggest that for a magnetized ($\mathcal{H}\neq 0$)
linear perturbation $\tilde{M}$ of a LRS background spacetime $M$, if the
1-form $\tilde{s}_{a}$ is gradient in which case the MOTS stability operator
is self-adjoint (remember this is for the class of MOTS under consideration),
then $\tilde{s}_{a}$ necessarily vanishes on the MOTS. But as it will be seen
shortly, the quantities $\bar{\tilde{s}}_{V}$ and $\mathcal{H}$ are
proportional so that in the linear regime on any MOTS it is true that both
$\bar{\tilde{s}}_{V}$ and $\mathcal{H}$ vanish. More specifically,
$\mathcal{H}$ must be first order.

###  V.2 Comments on relations to the characterization of null horizons

Let us now specify our discussions to the backgrounds of the previous section.

####  V.2.1 Schwarzschild background

To begin with, consider the case of the Schwarzschild background. We will
establish the following statement (C1) and the corresponding implication (C2):

  1. (C1).

For a linear perturbation $\tilde{M}$ of a Schwarzschild background, the
1-form $\tilde{s}_{a}$ is gradient (in which case the MOTS stability operator
is self-adjoint) if and only if $\tilde{s}_{a}$ is identically zero;
$\implies$

  2. (C2).

For a linear perturbation $\tilde{M}$ of a Schwarzschild background, if the
one-form $\tilde{s}_{a}\neq 0$ and the vorticity 2-vector has a nonzero
contribution from the odd sector, the $r=2$ null horizon necessarily evolves
to a spacelike character.

Note: As was mentioned in the previous section, away from the null case the
horizon can only evolve to the spacelike case with the even parity component
of $\Sigma_{a}+\varepsilon_{ab}\Omega^{a}$ being negative. (A timelike
character would imply that timelike curves do enter the trapped region, which
is of course ruled out.) Thus, the second statement above, (C2), follows from
the fact that $\tilde{s}_{V},\bar{\Omega}_{V}\neq 0$ ensures that
$\Sigma_{V}-\bar{\Omega}_{V}\neq 0$.

Let us apply the tensor $\varepsilon^{ab}n^{c}$ to the Ricci identities for
the unit field $n^{a}$ to get

$\displaystyle\delta_{a}\left(\Omega^{a}+\varepsilon^{ab}\Sigma_{b}\right)=\left(2\mathcal{A}-\phi\right)\Omega+\mathcal{H},$
(76)

and upon noting $\Omega$ should be zero on the MOTS, the above reduces to

$\displaystyle-\frac{\bar{k}^{2}}{r}\bar{\tilde{s}}_{V}\mathcal{Q}=\mathcal{H},$
(77)

so that $\mathcal{H}$ is a first order quantity that vanishes on the
background, i.e. $\mathcal{H}$ is decomposed into the harmonic $\mathcal{Q}$
basis.

Then, by (75) it follows that $\mathcal{H}=\bar{\tilde{s}}_{V}=0$. It
therefore follows that $\tilde{s}_{V}=0\iff\tilde{s}_{a}=0$, i.e.
$\tilde{s}_{a}$ is gradient if and only if $\tilde{s}_{a}$ vanishes
identically.

Finally, the linearized evolution and propagation equations of $\mathcal{H}$
are respectively given by (these are respectively the $u^{a}$ and $n^{a}$
components of the magnetic Weyl divergence equation [26])

$\displaystyle\dot{\mathcal{H}}$
$\displaystyle=-3\mathcal{E}\xi-\varepsilon_{ab}\delta^{a}\mathcal{E}^{b},$
(78) $\displaystyle\hat{\mathcal{H}}$
$\displaystyle=-3\mathcal{E}\Omega-3\phi\mathcal{H}-\delta_{a}\mathcal{H}^{a},$
(79)

which translates to, by the vanishing of $\mathcal{H}$ as well as noting that
$\Omega=\xi=0$ on the MOTS,

$\displaystyle\mathcal{H}_{V}=0;\quad\bar{\mathcal{E}}_{V}=0.$ (80)

This is to say that, necessarily, on the MOTS one requires that
$\mathcal{H}^{a}$ is solenoidal (divergence-free) and that $\mathcal{E}^{a}$
has no odd parity contribution on the MOTS.

While there appears to be freedom in the quantities $\mathcal{E}_{V}$ and
$\bar{\mathcal{H}}_{V}$, they are actually constrained by a coupling of the
sheet expansion $\phi$ and the 1-form $\tilde{s}_{a}$ (see the discussions of
the next subsection). For example, for a vanishing $\tilde{s}_{a}$ in the case
of a nondissipative perturbation, it follows that one requires (again, see the
next subsection for more details on this)

$\displaystyle\mathcal{E}_{V}+\bar{\mathcal{H}}_{V}=0.$

Of course, we do not rule out self-adjointness of the operator in the case of
a spacelike character. This is in principle possible depending on the nature
of the perturbation. This case will however impose that $\Sigma_{V}<0$.

On the other hand the null character does not necessarily imply the self-
adjointness of the operator. The self-adjointness due to vanishing of
$\tilde{s}_{a}$, in this case, would imply (and is implied by)
$\Sigma_{V}=\bar{\Omega}_{V}=0$.

####  V.2.2 Hypersurface orthogonal and spatially homogeneous background

We now consider the spatially homogeneous case of the previous section, and
stick with a nondissipative perturbation. In this case the equations
corresponding to (78), (79) and (76) are respectively

$\displaystyle\dot{\mathcal{H}}$
$\displaystyle=-\frac{3}{2}\left(\frac{2}{3}\theta-\sigma\right)\mathcal{H}-3\mathcal{E}\xi-\varepsilon_{ab}\delta^{a}\mathcal{E}^{b},$
(81) $\displaystyle\hat{\mathcal{H}}$
$\displaystyle=-3\left(\mathcal{E}+\rho+p\right)\Omega-3\phi\mathcal{H}-\delta_{a}\mathcal{H}^{a},$
(82) $\displaystyle\mathcal{H}$
$\displaystyle=\delta_{a}\left(\Omega^{a}+\varepsilon^{ab}\Sigma_{b}\right)+3\xi\sigma.$
(83)

The vanishing of $\Omega,\xi=0$ and $\mathcal{H}$ then implies the result
$\mathcal{H}_{V},\bar{\mathcal{E}}_{V},\bar{\tilde{s}}_{V}=0$ follows as in
the Schwarzschild case. For this reason, the statement (C1) of the previous
subsection also holds here, although for obvious reasons the statement (C2)
does not hold.

  * Remark 5.

We do expect these three vanishing properties to be generic to the MOTS we are
considering in the sense that they are independent of the particular
background one works with. It will be quite interesting to see how much these
vanishing conditions further constrains the MOTS in full generality. This is
however beyond the scope of the current work.

In the self-adjoint case with a vanishing $\tilde{s}_{a}$, obviously either
one of the components $\Sigma_{V}$ or $\bar{\Omega}_{V}$, paired with the
parameter ranges suffice for the characterization of the horizon.

On the other hand, if one is to impose the condition that
$|\Sigma_{V}|\neq|\bar{\Omega}_{V}|$, this of course ensures that
$\tilde{s}_{a}$ is nonvanishing, in which case one may then check which
conditions this further impose on the horizon character. However, this does
not in by itself rule out self-adjointness of the operator. In principle, if
one can expand a nonvanishing $\tilde{s}_{V}$ in the harmonic $\mathcal{Q}$
basis, then the operator can still be brought to a self-adjoint form. In the
forthcoming subsection we will see a particular class of perturbations where
this may be possible.

###  V.3 More on the MOTS and stability operator

We obtain some additional results required to be satisfied on the MOTS, given
properties derived from the one-form $\tilde{s}_{a}$. As we shall see, this
will provide some interesting insights into properties of the 1-form
$\tilde{s}_{a}$.

The magnetic Weyl tensor is written down as a constraint to the field equation
[26]:

$\displaystyle H_{ab}=\varepsilon_{cd\langle a}D^{c}\sigma_{b\rangle}^{\
d}-2\dot{u}_{\langle a}\omega_{b\rangle}-D_{\langle a}\omega_{b\rangle}.$ (84)

One may interpret the above as the magnetic Weyl tensor measuring the degree
of distortion of vorticity.

Taking the $n^{a}$ component of (84) produces the constraint equation, up to
linear order,

$\displaystyle-\delta_{a}\left(\frac{2}{3}\theta-\sigma\right)$
$\displaystyle=-2\delta^{b}\Sigma_{ab}-\phi\tilde{s}_{a}$
$\displaystyle-2\varepsilon_{ab}\left[\mathcal{H}^{b}+\left(\delta^{b}+2\mathcal{A}^{b}\right)\Omega\right]$
$\displaystyle+\xi\left(\Omega_{a}-3\varepsilon_{ab}\Sigma^{b}\right).$ (85)

By setting $\Omega=\xi=0$ we have

$\displaystyle-\delta_{a}\left(\frac{2}{3}\theta-\sigma\right)=-2\left(\delta^{b}\Sigma_{ab}+\varepsilon_{ab}\mathcal{H}^{b}\right)-\phi\tilde{s}_{a}.$
(86)

Also, fully projecting the Ricci identities for $n^{a}$ taken up to linear
order and imposing the MOTS condition and the vanishing of $\xi$ and $\Omega$
as required we obtain

$\displaystyle-\delta_{a}\phi=2\left(\mathcal{E}_{a}-\delta^{b}\zeta_{ab}\right)-\Pi_{a}-\phi\tilde{s}_{a},$
(87)

and upon combining with (86) and (87) we have

$\displaystyle-\cancel{\delta}_{a}\chi=2\left[\mathcal{E}_{a}-\delta^{b}\left(\Sigma_{ab}+\zeta_{ab}\right)-\varepsilon_{ab}\mathcal{H}^{b}\right]-\phi\tilde{s}_{a},$
(88)

where we have written $\cancel{\delta}_{a}=\delta_{a}-\tilde{s}_{a}$. One may
then use the Codazzi equations to establish the following

$\displaystyle-
Q_{a}+Z_{ab}k^{b}=2\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{H}^{b}\right)-\phi\tilde{s}_{a},$
(89)

where $Z_{ab}=N^{\ c}_{a}C_{cbde}k^{d}l^{e}$, with $C_{abcd}$ being the
spacetime Weyl tensor.

We will now utilize a curvature property of the MOTS to establish that the
second term on the left hand side of (89) is of nonlinear order on the MOTS.
With this we will have established a relationship between the Weyl and heat
flux 2-vectors and the rotation 1-form.

The curvature 2-form on the MOTS, i.e. the curvature of $\tilde{s}_{a}$ –
which we denote by $\tilde{\Omega}_{ab}$ – contracted along the null direction
$k^{a}$ can be shown to obey the relation

$\displaystyle k^{b}\tilde{\Omega}_{ba}=-N^{\ c}_{a}Z_{cb}k^{b},$

so that

$\displaystyle
k^{b}\tilde{\Omega}_{ba}=-\left[2\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{H}^{b}\right)-\phi\tilde{s}_{a}+Q_{a}\right].$
(90)

Explicitly, the left hand side of (90) can be reduced to

$\displaystyle k^{b}\tilde{\Omega}_{ba}$
$\displaystyle=-\frac{1}{2}\Sigma_{b}\delta_{a}k^{b}$
$\displaystyle=-\frac{1}{2}\left(\Sigma_{ab}+\zeta_{ab}\right)\Sigma^{b},$

which we ignore as it is of second order. Thus,

$\displaystyle\phi\tilde{s}_{a}=Q_{a}+2\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{H}^{b}\right).$
(91)

Furthermore, imposing the MOTS condition on (29) we have

$\displaystyle\cancel{\delta}_{a}\phi=0,$

so that by (87) it follows

$\displaystyle\Pi_{a}=2\left(\mathcal{E}_{a}-\delta^{b}\zeta_{ab}\right).$
(92)

Clearly, away from the case of a vanishing $\phi$, the following statement is
concluded:

The MOTS stability operator in a conformally flat and nondissipative (this
imposes that $\zeta_{ab}$ is divergence-free) linearly perturbed inhomogeneous
LRS background is self-adjoint.

So for example, the above will be true for a nondissipative perturbation of a
Lemaitre-Tolman-Bondi type solutions.

Notice that if the tensor $\delta_{a}n_{b}$ is pure trace, dissipation on the
MOTS is generated entirely by contributions from the Weyl tensor.

On the other hand, without restricting to the nondissipative case, using the
vanishing of $\bar{\mathcal{E}}_{V},\mathcal{H}_{V}$ and $\bar{\tilde{s}}_{V}$
we have the following pair of equations from (91).

$\displaystyle\bar{Q}_{V}$ $\displaystyle=0,$ (93)
$\displaystyle\phi\tilde{s}_{V}$
$\displaystyle=Q_{V}+2\left(\mathcal{E}_{V}+\bar{\mathcal{H}}_{V}\right).$
(94)

In any case, we can draw some immediate conclusions from both (91) and (92)
about the generation of rotation on a horizon:

  * •

In a conformally flat linearly perturbed inhomogeneous LRS background,
rotation of a horizon foliated by MOTS can be entirely sourced by the heat
flux 2-vector along the MOTS. This then implies that self-adjointness of the
MOTS stability operator in this particular case may be obtained by switching
off any heat flux contribution along the MOTS.

  * •

In a nondissipative linearly perturbed inhomogeneous LRS background, rotation
of a horizon foliated by MOTS can be sourced by the 2-vector components of the
Weyl curvature tensor. This is to say that in this case the contributing
sources are tidal forces acting along the MOTS.

  * Remark 6.

We remark here that the form of (94) suggests that it is possible, in
principle, to have a nonvanishing $\tilde{s}_{V}$ – i.e. a nonvanishing
$\tilde{s}_{a}$ – even in the case of a conformally flat perturbation with a
vanishing heat flux and vanishing $\phi$. Thus, in the first bulleted point
above we were quite cautious with semantics when we used “may be” when
switching off the heat flux contribution along the MOTS to obtain self-
adjointness of the stability operator. Only when $\phi$ is nonzero in the
perturbed spacetime that this is true, at least in our current context.

Finally, let us describe a simple case where $\tilde{s}_{V}$ is non-vanishing
but the stability operator may take a self-adjoint form. More specifically, we
may be able to express $\tilde{s}_{V}$ in terms of the $\mathcal{Q}$ basis. We
shall consider a static background, i.e. the shear and expansion scalars
vanish on the background.

We note that for any 2-tensor $\Psi_{ab}$ one has as the divergence

$\displaystyle\delta^{b}\Psi_{ab}=\frac{1}{r}\left(\bar{k}^{2}-2\right)\left(-\bar{\Psi}_{T}\mathcal{Q}_{a}+\Psi_{T}\bar{\mathcal{Q}}_{a}\right).$
(95)

Thus, one can harmonically decompose (86) to obtain the pair of constraint
equations

$\displaystyle 0$
$\displaystyle=\frac{1}{r}\left(\bar{k}^{2}-2\right)\Sigma_{T}+\mathcal{H}_{V},$
(96) $\displaystyle 0$
$\displaystyle=\frac{1}{r}\left(\bar{k}^{2}-2\right)\bar{\Sigma}_{T}+\bar{\mathcal{H}}_{V}$
$\displaystyle+\frac{1}{2r}\left(\frac{2}{3}\theta_{S}-\sigma_{S}\right)-\phi\tilde{s}_{V}.$
(97)

Indeed, from the first equation (96) it follows that the tensorial part
$\Sigma_{T}$, for $\bar{k}^{2}\neq 2$, vanishes by virtue of the vanishing of
the quantity $\mathcal{H}_{V}$. Furthermore, by imposing the MOTS condition
$\chi=0$, the second equation (97) can be recast as

$\displaystyle
0=\frac{1}{r}\left(\bar{k}^{2}-2\right)\bar{\Sigma}_{T}+\bar{\mathcal{H}}_{V}+\phi\left(\frac{1}{2r\mathcal{Q}}-\tilde{s}_{V}\right).$
(98)

Therefore, as long as the MOTS is nonminimal in the perturbed spacetime
(recall that by nonminimal it is meant that $\phi$ and $(2/3)\theta-\sigma$
cannot vanish simultaneously on the MOTS), for a perturbation with no
contributions from the odd vector harmonics part of $\mathcal{H}_{a}$ and the
shear 2-tensor $\Sigma_{ab}$, the 1-form component $\tilde{s}_{V}$ can
effectively be cast as

$\displaystyle\tilde{s}_{V}=\left(\frac{1}{2r\mathcal{Q}^{2}}\right)\mathcal{Q}.$
(99)

(Of course $\tilde{s}_{V}$ is nonzero and so $\tilde{s}_{a}$ is not a
gradient.) This then allows us to write down the eigenvalues of the operator
as

$\displaystyle\lambda_{\bar{k}^{2}}=\frac{\bar{k}^{2}}{r}+\frac{1}{2r^{2}\mathcal{Q}^{2}}+\bar{F}.$
(100)

Consequently the positivity of the modification term means that $\bar{F}>0$
again ensures nonnegativity of the eigenvalues.

##  VI Summary and outlook

Summary – In this work we have considered the behavior of a null horizon
foliated by MOTS in a LRS background spacetime subjected to linear
perturbations. The problem is fixed by the gauge invariant requirement of the
characterizing function. More precisely, we set out to characterize the
evolution of a MOTS in a linearized spacetime admitting a 1+1+2 decomposition,
which lies in the slice orthogonal to the $u^{a}$. The gauge invariance of the
chacterizing function introduced is possible if and only if a certain function
vanishes in the background. This vanishing condition is equivalent to the
condition for a horizon in the background to be null, thus phrasing the
problem in terms of the evolutionary behavior of a null horizon in a LRS
background subjected to linear perturbations. In other words – and emphasis is
placed on this – this characterization fails in the case of a non-null horizon
in the background. While this work has been dedicated to the case of linear
perturbations, it was demonstrated that the characterizing function is form
invariant even when extending to non-linear regimes, for a certain frame
choice or provided that the perturbed spacetime has a vanishing sheet
expansion.

We focused the application of our characterization to non-dissipative
perturbations. As was described in the introduction, perturbing a seed metric
generally affects the dynamics of an embedded horizon, and that this is the
case of the null event horizon in the Schwarzschild spacetime. The presence of
shear ensures a spacelike transition of the null horizon. We have assumed a
harmonic decomposition of first order scalars $\Psi$ and 2-vectors $\Psi_{a}$,
where the notations $\Psi_{V}$ and $\bar{\Psi}_{V}$ are used respectively for
the components of $\Psi_{a}$ along the even and odd vector harmonics. It is
established here that this spacelike transition requires that a particular
relationship between the shear and the vorticity 2-vectors holds,
specifically, $\Sigma_{V}-\bar{\Omega}_{V}<0$. In order for the causal
preservation of the horizon character, $\Sigma_{V}-\bar{\Omega}_{V}=0$. The
result was obtained for the particular consideration of a non-dissipative
perturbation where the scalars, tensors and vector quantities associated to
the heat flux and anisotropic pressure are vanish.

We also considered the case of a spatially homogeneous background solution and
wrote down the form of the horizon characterizing function. In the
Schwarzschild case, any transition from a null character is necessarily
spacelike as the horizon encloses a black hole (note that we are assuming the
MOTS condition is preserved). However, for the spatially homogeneous case null
horizons do not enclose black holes, and therefore the causal character could
transition to a timelike one under the perturbation. Imposing a linear
equation of state, for a particular frame choice, i.e., fixing the spatial
unit vector to align with the acceleration of the temporal unit vector (which
coincides with the vanishing of the difference between the 2-vector
components, i.e., $\Sigma_{V}-\bar{\Omega}_{V}=0$), the sign of the equation
of state parameter $w$ alone characterizes the evolution: null if $w=0$,
spacelike if $w>0$, and timelike if $w<0$. Without this frame choice, the
evolution of the horizon is a bit more involved. We nonetheless present a
partial characterization of the horizon dynamics in yhje presence of
perturbations. In particular, where the difference between these vector
components is strictly negative or strictly positive, we find the ranges of
the equation of state parameter where the induced metric on the horizon has
absolute sign, i.e., the causal character is timelike or spacelike. However,
there are ranges of the parameter for which the causal character is determined
by a complex relationship between the 2-vectors, the energy density and the
eigenvalues of MOTS Laplacian, which we could not analytically determine.

Finally, we considered the relationship between the shear and vorticity
2-vectors which play a crucial role in the characterization of the horizon, to
the MOTS stability operator. These 2-vectors also specify the 1-form
$\tilde{s}_{a}$ which controls the self-adjointness of the operator, and
points to an obvious connection to the characterization we have introduced. We
established that the component $\bar{\tilde{s}}_{V}$ vanishes for our
particular consideration in this work, and that
$\tilde{s}_{V}=\Sigma_{V}+\bar{\Omega}_{V}$. This implied that for our
consideration the 1-form $\tilde{s}_{a}$ is a gradient (so that the MOTS
stability operator is self-adjoint) if and only if $\tilde{s}_{a}$ vanishes
identically. As a consequence, in the case of perturbation of a Schwarzschild
background, it follows that when the stability operator is not self-adjoint
and $\bar{\Omega}_{V}\neq 0$, the event horizon necessarily transitions to a
spacelike character. In the case of a spatially homogeneous background, in the
self-adjoint case with a vanishing $\tilde{s}_{a}$, it is clear that only one
of the components $\Sigma_{V}$ or $\bar{\Omega}_{V}$ is needed for the
characterization.

Several additional results were also obtained, restricting the form of
2-tensors and 2-vectors on the MOTS. For example, it was established that (and
this appears independent of the choice of background) the components of the
even and odd vector harmonics, respectively, of the magnetic and electric Weyl
2-vectors necessarily vanish on the MOTS. In fact, it was demonstrated that
the bi-implication $\bar{\tilde{s}}_{V}=0\iff\mathcal{H}=0$ follows, which is
due to the fact that $\mathcal{H}$ is necessarily a first order quantity.
Furthermore, the component of the odd tensor harmonics of the shear 2-tensor
also vanishes on the MOTS.

As the 1-form $\tilde{s}_{a}$ is tied to rotation generation on the horizon,
we provide a clear picture, at least in the linear regime, of the precise
source of rotation on the horizon: for a non-dissipative linear perturbation
of a LRS background spacetime, rotation is entirely sourced by contributions
from the Weyl 2-vectors on the MOTS. If one includes dissipation, then
rotation gains a contribution from the heat flux 2-vector. Actually, there
could be dissipation from the anisotropic 2-vector and no contribution from
the heat flux 2-vector, and it remains true that horizon rotation will be
entirely sourced by Weyl contributions. It then follows that for a conformally
flat and non-dissipative perturbation of an inhomogeneous LRS background
spacetime, the stability operator is necessarily self-adjoint.

We also presented a class of perturbations for which the 1-form
$\tilde{s}_{a}$ is non-vanishing but one may nonetheless expand the component
$\bar{\tilde{s}}_{V}$ in the harmonic basis so that the MOTS stability
operator can be brought to a self-adjoint form.

Outlook – There are several interesting directions that could be explored. In
this work, we have restricted ourselves to first order perturbations. What
happens when non-linearities are incorporated into the analysis of the horizon
evolution? Naturally the problem becomes a bit (or a lot) more complicated as
seen from the characterizing function $z$ in (27). For example, there will be
additional contributions from the non-linearities to the scalar curvature of
the MOTS, so that large distortions are introduced on the MOTS. These
distortions are not present at first order as was discussed earlier.
Modification to the eigenvalue in the self-adjoint case will present in the
form of the inclusion of dissipation and the cosmological constant $\Lambda$
(if $\Lambda$ is considered a perturbation ‘variable’). Any non-linearity
introduced is an extra contribution to the principal eigenvalue through the
scalar curvature of the MOTS.

Another example is the following. At every stage of our analysis we have
imposed the MOTS condition $\chi=0$, meaning that the built-in assumption here
is that a background horizon, under perturbations of its ambient background
spacetime, evolves to a horizon foliated by MOTS. However, it is quite
possible that under the perturbation the horizon is no longer sliced into
MOTS. This is the case of the Weyl-distorted Schwarzschild geometry discussed
in the introduction, where by the tuning up the distortion parameter the $r=2$
horizon is no longer a MOTS [12]. This degeneracy may be a feature of non-
linearities under the perturbation and so might not happen in the linear
perturbative case considered in this work. However, as mentioned in the second
point above, extending our analysis to non-linear perturbations, it would
suffice as a more general consideration, not to impose the MOTS condition and
analyze the non-linear equations.

##  VII Acknowledgement

AS and SK research is supported by the Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of
education (grant numbers) (NRF-2022R1I1A1A01053784) and
(NRF-2021R1A2C1005748). PKSD acknowledges support through the First Rand Bank,
South Africa. AS would also like to thank the IBS Center for Theoretical
Physics of the Universe, Daejeon, South Korea, for its hospitality, where a
part of this work was carried out.

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