# Horizon area bound and MOTS stability in locally rotationally symmetric
solutions

Abbas M. Sherif<EMAIL_ADDRESS>Center for Geometry and Physics,
Institute for Basic Science (IBS), Pohang 37673, Korea  Peter K. S. Dunsby
<EMAIL_ADDRESS>Cosmology and Gravity Group, Department of Mathematics
and Applied Mathematics, University of Cape Town, Rondebosch 7701, South
Africa
South African Astronomical Observatory, Observatory 7925, Cape Town, South
Africa

###### Abstract

In this paper, we study the stability of marginally outer trapped surfaces
(MOTS), foliating horizons of the form $r=X(\tau)$, embedded in locally
rotationally symmetric class II perfect fluid spacetimes. An upper bound on
the area of stable MOTS is obtained. It is shown that any stable MOTS of the
types considered in these spacetimes must be strictly stably outermost, that
is, there are no MOTS “outside” of and homologous to $\mathcal{S}$. Aspects of
the topology of the MOTS, as well as the case when an extension is made to
imperfect fluids, are discussed. Some non-existence results are also obtained.
Finally, the “growth” of certain matter and curvature quantities on certain
unstable MOTS are provided under specified conditions.

###### Contents

  1. I Introduction
  2. II MOTS and MOTT in LRS solutions
  3. III Area bounds from stability and topological implications
    1. III.1 MOTS stability
    2. III.2 Area bounds
    3. III.3 Comments on the topology of the MOTS
  4. IV The growth of some covariant variables on horizons containing unstable MOTS
  5. V Conclusion

##  I Introduction

Stability of marginally outer trapped surfaces (MOTS) (see the references [1,
2, 3, 4, 5, 6, 7], or more recently, [8, 9, 10], where other interpretations
of stability have allowed direct comparisons to the geodesic Jacobi operator),
a Lorentzian analogue of the well studied notion of stability of minimal
surfaces in Riemannian geometry, has generated significant interest over the
years due to its utility is proving many useful results in general relativity.
A relatively recent interpretation of MOTS stability, [7], has proven very
successful in proving higher dimensional results for the topology of black
holes, which relies on showing that under certain assumptions on either the
initial data set containing the MOTS or the spacetime, the MOTS is of positive
Yamabe type, i.e., admits a metric of constant positive scalar curvature.

Area bounds for black holes have also been obtained under different scenarios.
Dain et al., [11, 12, 13], obtained lower bounds on the area in terms of the
charges and angular momenta. Andersson et al. [14], again, making use of the
stable properties of the MOTS $\mathcal{S}$, obtained that the area of the
black hole either has an upper bound which is a constant function of the
curvature, injectivity radius and the volume of the initial data set
containing $\mathcal{S}$ or, another MOTS $\mathcal{S}^{\prime}$ lie to the
outside of $\mathcal{S}$. The proof of this area inequality, in particular the
gluing approach used, shows that two stably outermost MOTS that are
sufficienctly close will have a MOTS surrounding them. This has been
interpreted as demonstrating a well known fact of black hole mergers: two
black holes that are sufficiently close will merge (see the reference [15]).

The main objective of this paper is to study the stability of MOTS in locally
rotationally symmetric (LRS) perfect fluids (though in certain cases the
results do extend to the case of anisotropic fluids with non-vanishing heat
flux) and examine its relationship to upper bounds on the horizon area, as
well as the implications for the topology of the MOTS. In doing this, we
consider additional restrictions on some geometric quantities.

Our interest is primarily in dynamical horisons (DH) and/or timelike membranes
(TLM), and as was demonstrated in [16], only the class II LRS solutions (those
with neither rotation nor spatial twist) admit DH or TLM of a certain
functional form. These will therefore be the focus of this work.

LRS II solutions are vorticity-free solutions, each point of which admits a
continuous isotropy group [18, 17]. The LRS class of solutions generalizes the
well studied spherically symmetric solutions of the Einstein’s field
equations. The line element of these solutions carries the specific form

$\displaystyle
ds^{2}=-C^{2}_{1}d\tau^{2}+C^{2}_{2}dr^{2}+C^{2}_{3}\left(dy^{2}+D^{2}dz^{2}\right),$

where $C_{1,2,3}$ are functions of $\tau$ and $r$, and $D$ is a function of
$y$ and a parameter $k$ which parametrizes the 2-surfaces of constant $r$ in
the $\tau=$ constant slices: the 2-surface geometry is spherical for $k=+1$,
hyperbolic for $k=-1$, and flat for $k=0$. (The $k=+1$ is the spherically
symmetric subclass.)

The approach to be employed in this work is a covariant (and hence gauge
invariant) approach, known as the 1+1+2 covariant formalism (see [19] and
references therein). This approach has recently been used to study horizons in
LRS spacetimes [20]. Like the powerful 1+3 formalism which allows for the
threading of the spacetime along the fluid flow lines with unit tangent vector
$u^{\mu}$, this approach is a specialization, where the 3-space orthogonal to
the fluid flow is further threaded along the vorticity, with this direction
specified by the unit vector field $e^{\mu}$. This allows for the projection
of tensor and vector quantities, as well as derivatives (a dot $\dot{\ }$
denotes derivative along $u^{\mu}$, a hat $\hat{\ }$, that along $e^{\mu}$,
and $\delta_{\mu}=N^{\nu}_{\mu}\nabla_{\nu}$ the derivative on the 2-space
which results from decomposing the 3-space, along these directions, with
$N^{\mu\nu}$ projecting vectors and tensors orthogonal to $u^{\mu}$ and
$e^{\mu}$, to the 2-space.) The derivative operator $\nabla_{\mu}$ denotes the
4-dimensional spacetime covariant derivative. The form of the field equations
$G_{\mu\nu}=T_{\mu\nu}$ is used instead of $G_{\mu\nu}=8\pi T_{\mu\nu}$.

In regards to horizons, the splitting, when considering the background (i.e.
exact) solutions, singles out surfaces in the foliation along $u^{\mu}$. The
particular surfaces we consider are picked out by the introduction of the
frame vector $e^{\mu}$. In our particular case in this work, the split here
singles out a “preferred” horizon. It must however be emphasized that, the
1+1+2 split works for perturbed LRS (non-exact) solutions, in which case we
cannot claim that the split picks up a preferred horizon, since the approach
is not frame-invariant. Different frame choices in the perturbed solution will
single out different horizons.

For LRS II solutions, one can characterize the spacetimes entirely by the set
of covariant scalars [19, 21, 22]

$\displaystyle\mathcal{D}:\equiv\\{{\rho,p,Q,\Pi,\mathcal{E},\mathcal{A},\Theta,\Sigma,\phi\\}},$

where $\rho\equiv T_{\mu\nu}u^{\mu}u^{\nu}$ is the energy density,
$p\equiv\left(1/3\right)h^{\mu\nu}T_{\mu\nu}$ is the pressure (isotropic),
$Q=-T_{\mu\nu}e^{\mu}u^{\nu}$ is the heat flux,
$\Pi=T_{\mu\nu}e^{\mu}e^{\nu}-p$ is the anisotropic stress,
$\mathcal{E}=E_{\mu\nu}e^{\mu}e^{\nu}$ encodes the electric part of the Weyl
tensor $E_{\mu\nu}$, $\mathcal{A}=\dot{u}_{\mu}e^{\mu}$ is the acceleration,
$\Theta\equiv D_{\mu}u^{\mu}$ is the expansion,
$\Sigma=e^{\mu}e^{\nu}D_{\langle\nu}u_{{\mu}\rangle}$ is the shear and
$\phi=\delta_{\mu}e^{\mu}$ denotes the expansion of the $2$-space (referred to
as the sheet expansion), with

$\displaystyle T_{\mu\nu}$ $\displaystyle=\rho
u_{\mu}u_{\nu}+2Qe_{(\mu}u_{\nu)}+\left(p+\Pi\right)e_{\mu}e_{\nu}+\left(p-\frac{1}{2}\Pi\right)N_{\mu\nu},$

being the stress energy tensor, and the derivative operator $D_{\mu}$ denoting
the covariant derivative on the hypersurface to which $u^{\mu}$ is
hypersurface orthogonal.

The LRS II class of spacetimes can be extended to the more general LRS class
by including rotation $\Omega=\omega^{a}e_{a}$ ($\omega_{a}$ is the vorticity
vector) and spatial twist (the twist of $e^{\mu}$)
$\xi=(1/2)\varepsilon^{\mu\nu}\delta_{\mu}e_{\nu}$, as well as
$\mathcal{H}=H_{\mu\nu}e^{\mu}e^{\nu}$ (encoding the magnetic part of the Weyl
tensor), in the set $\mathcal{D}$. (The use of the word vorticity should not
be confused with angular momentum)

While in general, for a 1+1+2 decomposed spacetimes, the 2-sheets are not
necessarily true surfaces, for LRS II solutions this is true as is seen from
the fact that the relation
$\delta_{\mu}\delta_{\nu}\psi=\delta_{\nu}\delta_{\mu}\psi$ holds when acting
on an arbitrary scalar $\psi$ (and trivially so for LRS scalars in the
covariant set $\mathcal{D}$). In adddition, $u^{\mu}$ and $e^{\mu}$ are
surface forming since their commutator has no sheet component [19, 21]:

$\displaystyle\hat{\dot{\psi}}-\dot{\hat{\psi}}=-\mathcal{A}\dot{\psi}+\left(\frac{1}{3}\Theta+\Sigma\right)\hat{\psi}.$

For more details, the interested reader is referred to [19].

The field equations for these spacetimes can be obtained using the Ricci
identities for the preferred unit directions and the contracted Ricci
identities (further details can be found in [19]). Throughout this work,
whenever needed, we will provide the relevant equations.

This paper has the following structure: In section II, we briefly introduce
the notion of MOTS in LRS spacetimes, formulated in the covariant approach
employed throughout this paper. A marginally outer trapped tube (MOTT) is also
introduced and the nature of their evolution discussed. In Section III, a
short overview of the notion of stability of MOTS is given. It is then applied
to perfect fluid LRS II spacetimes, and an upper bounds on the area are
provided as being necessary and/or sufficient for stability of the MOTS. The
topology of the MOTS are then briefly considered as it relates to bound on the
horizon area, with accompanying discussions. Section IV considers bounds on
growth of some of the spacetime variables in the neighborhood of certain
unstable MOTS. The results are discussed in Section V and ongoing work where
the results of this paper would prove useful are also mentioned.

##  II MOTS and MOTT in LRS solutions

Let $\mathcal{S}$ (assumed to be spacelike and closed) be an embedded
$2$-surface in a LRS spacetime, where $u^{\mu}$ and $e^{\mu}$ are both normal
to $\mathcal{S}$, with $u_{\mu}u^{\mu}=-1,e_{\mu}e^{\mu}=1$ and
$e_{\mu}u^{\mu}=0$. Then, the respective tangents to the outgoing and ingoing
null geodesics are given by (see [23])

$\displaystyle\begin{split}k^{\mu}&=\frac{1}{\sqrt{2}}\left(u^{\mu}+e^{\mu}\right),\\\
l^{\mu}&=\frac{1}{\sqrt{2}}\left(u^{\mu}-e^{\mu}\right).\end{split}$

(We note that, in general, there is a degree of freedom to scale the null
vectors as $k^{\mu}\rightarrow fk^{\mu}$ and $l^{\mu}\rightarrow
f^{-1}l^{\mu}$, for $f>0$. However, such detail will not be relevant to the
rest of this paper. This will be seen shortly as we discuss the causal
character of horizons. Furthermore, in the case of the stability operator that
will be introduced later, it was shown in [24] that the spectrum of the
eigenvalues remain invariant under such scaling.) The divergences of the
congruences generated by $k^{\mu}$ and $l^{\mu}$, also known as the null
expansion scalars with respect to the null directions, are calculated
respectively as [20]:

$\displaystyle\begin{split}\chi^{+}&=\chi^{+}\left(\tau,r\right)=\frac{1}{\sqrt{2}}\left(\frac{2}{3}\Theta-\Sigma+\phi\right),\\\
\chi^{-}&=\chi^{-}\left(\tau,r\right)=\frac{1}{\sqrt{2}}\left(\frac{2}{3}\Theta-\Sigma-\phi\right),\end{split}$

which are just the traces of the null second fundamental forms associated with
$k^{\mu}$ and $l^{\mu}$ respectively, where we have labelled as $\tau$ and $r$
the parameters of the integral curves along the vector fields $u^{\mu}$ and
$e^{\mu}$. $\mathcal{S}$ is then said to be marginally outer trapped (resp.
outer trapped) if $\chi^{+}=0$ (resp. $\chi^{+}<0$) [25, 26, 27]. (Henceforth,
we will simply write $\chi=\chi^{+}$.) In cases where both expansions vanish,
$\mathcal{S}$ is said to be minimal, a standard case in point being cross
sections of the Schwarzschild horizon in isotropic coordinates (notice that
for Schwarzschild, $\Theta=\Sigma=0$, so that
$\chi^{+}=\phi=0\implies\chi^{-}=-\phi=0$). A marginally outer trapped tube
(MOTT) is then a hypersurface foliated by MOTS. In the case that $\chi^{-}<0$,
the “outer” is dropped and the notations MTS and MTT are instead used (see for
example [28]). From the expressions for $\chi$ and $\chi^{-}$, it is clear
that if $\chi$ vanishes, then, one requires that $\phi>0$ in order for
$\chi^{-}<0$ on $\mathcal{S}$.

For a MOTT, one can make the choice of a vector field [28, 29]

$\displaystyle\mathcal{V}^{\mu}=k^{\mu}-\mathcal{C}l^{\mu},$

assuming that $\mathcal{V}^{\mu}$ is tangent to the MOTT and everywhere normal
to $\mathcal{S}$, for some smooth function $\mathcal{C}$ on the MOTT. Then,
there exists a unique vector field $\tilde{\mathcal{V}}^{\mu}$ which is normal
to the MOTT, and is given as

$\displaystyle\tilde{\mathcal{V}}^{\mu}=k^{\mu}+\mathcal{C}l^{\mu}.$

It is easily seen that the sign of $\mathcal{C}$ specifies the signature of
the induced metric on the MOTT.

While in general the sign of $\mathcal{C}$ may vary at different points of the
MOTT, we are interested in those cases where the sign of $\mathcal{C}$ remains
fixed on the MOTT. For a MOTT which is also an MTT (or alternatively a MOTT
with positive sheet expansion), the MOTT is called a dynamical horizon (DH) if
$\mathcal{C}>0$, a timelike membrane (TLM) if $\mathcal{C}<0$, and an isolated
horizon (IH) if $\mathcal{C}=0$ (in this case the sheet expansion may be
zero). Indeed it is necessarily true that, on DH, IH and TLM the sheet
expansion is non-negative, with the sheet expansion vanishing if and only if
the MTS is minimal.

In the rest of this paper we will be dealing with situations where
$\mathcal{C}$ is constant on each MOTS. In that case, the scalar $\mathcal{C}$
can be explicitly computed using the formula [28]

$\displaystyle\mathcal{C}=\frac{\mathcal{L}_{k}\chi}{\mathcal{L}_{l}\chi},$
(1)

which, for LRS II spacetimes is given as [20]

$\displaystyle\mathcal{C}=\frac{-\left(\rho+p+\Pi\right)+2Q}{\frac{1}{3}\left(\rho-3p\right)+2\mathcal{E}},$
(2)

where $\mathcal{L}_{k}$ (resp. $\mathcal{L}_{l}$) denotes the Lie derivative
along the vector field $k^{\mu}$ (resp. $l^{\mu}$). If the null energy
condition holds, then $\mathcal{L}_{k}\chi\leq 0$ so that the sign of the
metric signature is specified by the denominator of (2), a curvature condition
(see [20] for more details).

The term horizon will be used to refer to MOTT (or MTT), and we will denote by
$\mathcal{H}$, whenever we refer to a DH. MOTS and MOTT will be consistently
used, and whenever there is ambiguity, clarity will be provided.

Of interest throughout the rest of the paper will be those TLM and DH of the
form $r=X(\tau)$ in the class of perfect fluid solutions with no anisotropy
and vanishing heat flux. It will be assumed throughout that, (at least) for
any point $\bar{p}$ on a MOTT, and a neighborhood $\mathcal{N}_{\bar{p}}$ of
$\bar{p}$, $\rho$ is non-negative on $\mathcal{N}_{\bar{p}}$. (It is possible
that this might hold true at all points of the ambient spacetime.) in the case
the $\rho=0$ applies, this will be made explicit. Otherwise, it will be
assumed that $\rho$ is positive.

##  III Area bounds from stability and topological implications

In this section we consider the relationship between stability of MOTS and the
area of the horizons they foliate. The focus of this section will be LRS II
perfect fluids, unless otherwise stated.

###  III.1 MOTS stability

The notion of stability for MOTS - analogous to the well understood notion of
stability of minimal surfaces - was introduced by Anderson et al. [4, 5].
Simply put, it addresses the following question: does the deformation of a
MOTS $\mathcal{S}$ along the unit normal direction $e^{\mu}$ leave
$\mathcal{S}$ marginally outer trapped or does $\mathcal{S}$ become untrapped?
In other words, denote by $\mathcal{S}_{t}$ the deformation of $\mathcal{S}$,
and let $\partial/\partial_{t}=\Psi e^{\mu}$ denote the tangent vector to the
curve which generates the variation, for some function $\Psi$. Is the
associated variation of the outward null expansion as

$\displaystyle\chi(\Psi):\equiv\left.\frac{\partial}{\partial
t}\right|_{t=0}\chi:\equiv\bar{\delta}_{\Psi e}\chi\geq 0,$

and positive somewhere on $\mathcal{S}$, or is $\chi(\Psi)<0$? A MOTS
$\mathcal{S}$ is said to be stable (or stably outermost) if $\chi(\Psi)\geq 0$
(with $\chi(\Psi)\not\equiv 0$), strictly stable (or strictly stably
outermost) if $\chi(\Psi)>0$ everywhere on $\mathcal{S}$, and unstable
otherwise. The above mentioned authors reduced this problem to an eigenvalue
problem by introducing a second order elliptic operator
$L_{\mathcal{S}}:C^{\infty}(\mathcal{S})\rightarrow C^{\infty}(\mathcal{S})$
acting on $\Psi$ as [4, 5]

$\displaystyle\chi(\Psi):\equiv
L_{\mathcal{S}}\Psi=-\Delta\Psi+2s^{\mu}\delta_{\mu}\Psi+\biggl{(}\frac{1}{2}R_{\mathcal{S}}-\left(\rho+J_{\mu}e^{\mu}\right)-\frac{1}{2}\chi_{\mu\nu}\chi^{\mu\nu}+\delta_{\mu}s^{\mu}-s_{\mu}s^{\mu}\biggr{)}\Psi,$
(3)

where $\Delta$ is the Laplacian on $\mathcal{S}$, $R_{\mathcal{S}}$ is the
scalar curvature of $\mathcal{S}$, $J_{\mu}=G_{\mu\nu}u^{\nu}$ is the local
momentum density along the horizon ($G_{\mu\nu}$ is the Einstein tensor),
$\chi_{\mu\nu}$ is the null second fundamental form associated with $k^{\mu}$,
and the one-form $s_{\mu}=-(1/2)l_{\nu}\delta_{\mu}k^{\nu}$ is the torsion of
the null normal field $k^{\mu}$ projected to $\mathcal{S}$.

In general, the operator $L_{\mathcal{S}}$ is not self-adjoint, a problem
which arises due to the presence of the second (linear) term in (3). However,
in certain specific cases - for example, if $\mathcal{S}$ lies in a time-
symmetric horizon (where $L_{\mathcal{S}}$ reduces to that of minimal
surfaces) or, in some cases where the one-form $s_{\mu}$ is a gradient -
$L_{\mathcal{S}}$ is self-adjoint (this is discussed in [5, 6] etc., and
references mentioned therein). This means that the eigenvalue spectrum is
generally complex. Nonetheless, there is always a principal eigenvalue
$\lambda$, which is real, and never bigger than the real part of any other
eigenvalue of $L_{\mathcal{S}}$ such that

$\displaystyle L_{\mathcal{S}}\Psi=\lambda\Psi,$ (4)

where $\Psi$ can be chosen to be strictly positive. Then, $\mathcal{S}$ being
stably outermost is equivalent to the condition that $\lambda\geq 0$, and
$\mathcal{S}$ being strictly stably outermost is equivalent to the condition
that $\lambda>0$. In either case one chooses the associated eigenfunction as
everywhere positive.

From our formulation, it is by now clear the fundamental role that the sheet
expansion $\phi$ plays in locating marginally trapped surfaces and the horizon
evolution. In particular, whether we can specialize from a MOTS to a MTS (and
hence from a MOTT to an MTT) depends on whether the unit normal $e^{\mu}$
diverges or converges on the MOTS. What are the implications that such
property of $\phi$ has for stability of the MOTS?

Indeed, strict positivity of $\lambda\implies\chi(\Psi)>0$ for positive
$\Psi$. Now, let us take the case of a MTS, considering the class of
spacetimes in this work, and denote by $\phi(\Psi)$ the associated variation
of the sheet expansion on $\mathcal{S}$. Suppose $\chi(\Psi)=0$ on
$\mathcal{S}$. Clearly, if $\phi(\Psi)\leq 0$, then $\chi^{-}(\Psi)\geq 0$.
Under which condition(s) will this affect the sign of $\chi^{-}$? Conversely,
suppose that the variation $\phi(\Psi)\geq 0$ (determining the sign of
$\phi(\Psi)$ can be written as an eigenvalue problem in a similar manner as
with the case of $\chi(\Psi)$). Does the associated eigenvalue - which we
denote by $\bar{\lambda}$ \- carry sufficient information about the stability
of $\mathcal{S}$? This could certainly be of interest. We will briefly return
to this in a short while, but a detailed analysis of the situation will be
deferred to a future work.

  * Remark 1.

If one were to consider a more general 2-surface $\underline{\mathcal{S}}$ in
an arbitrary spacetime admitting a 1+1+2 decomposition, the null vector field
$k^{\mu}$ (resp. $l^{\mu}$) acquires a sheet component - denote this by
$m^{\mu}$ (resp. - $m^{\mu}$) - orthogonal to both $u^{\mu}$ and $e^{\mu}$.
The induced metric on $\underline{\mathcal{S}}$ can be decomposed as

$\displaystyle\bar{N}_{\mu\nu}=N_{\mu\nu}+P_{\mu\nu},$

due to how the spacetime is decomposed, where $P_{\mu\nu}$ is some symmetric
2-tensor. The expansion scalars $\chi$ and $\chi^{-}$ will therefore acquire
the additional terms $w+P^{\mu\nu}\nabla_{\mu}k_{\nu}$ and
$-w+P^{\mu\nu}\nabla_{\mu}l_{\nu}$ ($w$ is some scalar), respectively. As per
the discussion in the previous paragraph, one would then consider whether the
quantity

$\displaystyle(\phi(\Psi)+w(\Psi))+Z(\Psi),$

carries sufficient information to determine stability, where we have denoted
by $Z(\Psi)$ the scalar $P^{\mu\nu}\nabla_{\mu}(e_{\nu}+m_{\nu})$ associated
to $\underline{\mathcal{S}}_{t}$.

For the rest of this work we shall assume that $\phi\geq 0$, with equality
holding only if $\mathcal{S}$ is minimal. That is, when “MOTS” is mentioned,
it is understood that the consideration allows for a negative sheet expansion.

Now, for a MOTS $\mathcal{S}$ embedded in a LRS II solution, $L_{\mathcal{S}}$
is self-adjoint. This follows from the fact that the projection of $s_{\mu}$
to $\mathcal{S}$ is zero:

$\displaystyle\begin{split}s_{\mu}&=-\frac{1}{2}N^{\mu\nu}\left(\mathcal{A}u_{\mu}+\left(\frac{1}{3}\Theta+\Sigma\right)e_{\mu}\right)&=0.\end{split}$

(This is consistent with the fact that this is true in spherically symmetric
LRS solutions.) In fact, the vanishing of $s_{\mu}$ will be true for
2-surfaces in a general LRS solutions with rotation and/or spatial twist. In
this case, the one-form $s_{\mu}$ will take the exact same form as above since
the rotation and twist terms scale the area form $\varepsilon_{\mu\nu}$, which
is zero when contracted with the unit vectors.

###  III.2 Area bounds

Let us now consider some results which relate stability of a MOTS to the size
of the area of the horizon in which the MOTS lie. Unless otherwise stated, all
considerations are with respect to MOTS (or MTS) in LRS II solutions where the
MOTS respect the LRS II symmetries. It will also be assumed that the isotropic
pressure $p$ is non-negative.

###### Proposition III.1

Let $\mathcal{S}$ be a MOTS embedded in a perfect fluid. Then, $\mathcal{S}$
is stably outermost if and only if the horizon area $\mathbb{A}$ satisfies

$\displaystyle\mathbb{A}\leq\frac{1}{2\rho}.$ (5)

  * Proof

The 4-covariant derivative in LRS spacetimes, pulled back to $\mathcal{S}$,
vanishes when acting on scalars
($\nabla_{\mu}\psi=-\dot{\psi}u_{\mu}+\hat{\psi}e_{\mu}$ $\forall$ smooth
functions $\psi$ in any LRS spacetime) so that the surface covariant
derivative and hence the Laplacian vanish when acting on scalars respecting
the LRS II symmetries. We note that “scalars” in LRS spacetimes, in context of
the formulation used, is understood to mean elements of the covariant set
$\mathcal{D}$. With respect to the current work, this means that our
considerations, for the remainder of this work, are those principal
eigenvalues with constant eigenfunctions $\Psi$.

Now, assume such positive $\Psi$ exists as in (4). Then, the principal
eigenvalue $\lambda$, which is real, simply reduces to

$\displaystyle\lambda=\frac{1}{2}R_{\mathcal{S}}-\left(\rho+J_{\mu}e^{\mu}\right)-\frac{1}{2}\chi_{\mu\nu}\chi^{\mu\nu},$
(6)

with $R_{\mathcal{S}}=2K$, where $K$ is the Gaussian curvature of
$\mathcal{S}$ given by (we note that $\mathcal{S}$ is a MOTS) (see [23])

$\displaystyle K=\frac{1}{3}\rho-\mathcal{E}.$

The local momentum density and the null second fundamental form associated to
$k^{\mu}$ are given by

$\displaystyle\begin{split}J_{\mu}&=-\left(\rho u_{\mu}+Qe_{\mu}\right),\\\
\chi_{\mu\nu}&=\frac{1}{2}\chi
N_{\mu\nu}\overset{\mathrm{\mathcal{S}}}{=}0,\end{split}$

so that for perfect fluids, $J_{\mu}e^{\mu}=0$. Explicitly, (6) takes the form

$\displaystyle\lambda=-\frac{2}{3}\rho-\mathcal{E}.$

(Note that the first term is non-positive by assumption.) Now, the Lie
derivative of $\chi$ along $l^{\mu}$ can be written as [28]

$\displaystyle\rho-p-\frac{1}{\mathbb{A}},$ (7)

up to a factor of $2\pi$ (which is $1/4$ in the natural units), which upon
comparing to the denominator of (2) (also up to a factor of $2\pi$) gives

$\displaystyle\mathbb{A}=\frac{3}{2\left(\rho-3\mathcal{E}\right)}.$ (8)

Therefore, if $\lambda\geq 0$, then, the inequality (5) follows from using (8)
to substitute for $\mathcal{E}$ in (5). It is also easy to see that if the
inequality (5) holds, then $\mathcal{E}\leq-(2/3)\rho\implies\lambda\geq 0$.

We note that in the non-vacuum case a necessary condition for stability of the
MOTS is that the Weyl scalar is strictly negative on the MOTS. In the vacuum
case, of course this is both necessary and sufficient for strict stability.

Indeed, if the spacetime in which the MOTS is embedded is conformally flat,
i.e. $\mathcal{E}=0$ (we could simply impose that this condition is horizon
compatible and does not need to hold on he entire spacetime), then the MOTS
are always unstable for positive $\rho$. A well known case of this
consideration is that of MOTS contained in TLM in the Oppenheimer-Snyder dust
collapse.

Another spacetime of wide ranging interests, which falls in the class of
spacetimes considered in this work, is the Lemaitre-Tolman-Bondi (LTB)
spacetimes. It was shown in [20] that DH in these spacetimes must satisfy
$\rho<-6\mathcal{E}$, and was further demonstrated that stability of MOTS
implies that the horizon is a DH. It is clear that DH in these spacetimes will
necessarily satisfy the following upper bound on the area:

$\displaystyle\mathbb{A}<\frac{1}{\rho}.$

We see that (5) is a refinement of the above inequality. This therefore
suggests that there may be DH in these spacetimes containing unstable MOTS. In
particular, MOTS that lie in a DH with area within the range

$\displaystyle\frac{1}{2\rho}<\mathbb{A}<\frac{1}{\rho},$

will be unstable.

Proposition III.1 suggests that the non blow-up of the horizon is necessary
for stability and it is ensured by $\mathcal{E}<0$. Here we see that the cut-
off on the area is rather a criteria and/or consequence of stability. Here we
have specified the upper bound on the horizon area as the inverse of the local
energy density.

Now, we had earlier pondered what implications, in the case of a MTS within
context of our formulation, would the sign of the variation of the sheet
expansion have on stability of the MTS. As an eigenvalue problem, in a general
1+1+2 spacetime, the variation of $\phi$ should give rise to principal
eigenvalue of the form

$\displaystyle\bar{\lambda}=\sqrt{2}\lambda-\mbox{extra term},$ (9)

where the “extra term” is a linear combination of any two of the three scalars
$\Theta,\Sigma,\phi$, and some scalars formed from sheet terms. Hence, it
appears that the sign of $\phi(\Psi)$ does carry information about stability
of the MOTS. Of course, for the present work, stability check is
straightforward because of the form $\lambda$ takes, and therefore, there is
no need for $\bar{\lambda}$. However, this could be useful in more general
settings.

Let us now prove the following:

###### Theorem III.2

Let $\mathcal{S}$ be a MOTS contained in a horizon embedded in a perfect
fluid. Then, the principal eigenvalue is non-zero.

  * Proof

Suppose the principal eigenvalue is zero. Using the propagation equations [19]

$\displaystyle\begin{split}\frac{2}{3}\hat{\Theta}-\hat{\Sigma}&=\frac{3}{2}\phi\Sigma,\\\
\hat{\phi}&=-\frac{1}{2}\phi^{2}+\left(\frac{1}{3}\Theta+\Sigma\right)\left(\frac{2}{3}\Theta-\Sigma\right)-\frac{2}{3}\rho-\mathcal{E},\end{split}$

it is easily checked that $\lambda=\hat{\chi}$. Therefore if $\lambda=0$, we
have that $\hat{\chi}=0$. Then, we have that $\mathcal{C}=1$, and the horizon
is a DH. From the definition of $\mathcal{V}^{\mu}$, we have that $u^{\mu}$ is
everywhere orthogonal to the horizon, i.e. a constant time slice. Now, of
course all evolution equations on such a hypersurface are constraints. The
constraint from the evolution of the local energy density in this case is
(again, see [19] for the full set of equations)

$\displaystyle 0=\Theta\left(\rho+p\right),$

so that $\Theta$ must vanish of the DH since $\rho+p\neq 0$. We therefore
have, from the evolution equations for $(2/3)\Theta-\Sigma$ and $\phi$, the
respective constraints

$\displaystyle\begin{split}0&=-\left(\rho+p\right)+\phi\left(\mathcal{A}-\frac{1}{2}\phi\right)+\lambda,\\\
0&=\phi\left(\mathcal{A}-\frac{1}{2}\phi\right),\end{split}$

from which we have (by setting $\lambda=0$) $\rho+p=0$, again clearly not
possible on a DH. Hence, $\lambda$ cannot be zero.

Actually, it can be easily checked that the above result extends to MOTS in
any LRS II solution.

###### Corollary III.3

A MOTS $\mathcal{S}$ contained in a horizon embedded in a perfect fluid, is
either unstable or strictly stably outermost. In particular, $\mathcal{S}$ is
stably outermost if an only if the horizon area satisfies

$\displaystyle\mathbb{A}<\frac{1}{2\rho}.$ (10)

###  III.3 Comments on the topology of the MOTS

Here we provide some comments related to the topology of stable MOTS in
perfect fluids and the accompanying implications. Additional analysis is
carried out in the case that one extends to imperfect fluids.

In LRS II solutions, the 2-surfaces under consideration are allowed to have
spherical, flat (including toroidal) as well as hyperbolic topologies. Let us
integrate (6) over the area of $\mathcal{S}$, and use the Gauss-Bonnet theorem
(noting that $\mathcal{S}$ is closed) so that we have

$\displaystyle
2\pi\underline{\chi}(\mathcal{S})=\lambda[\mathcal{S}]+\int_{\mathcal{S}}\rho,$
(11)

with $\underline{\chi}(\mathcal{S})$ denoting the Euler characteristics of
$\mathcal{S}$. Hence, for a stably outermost MOTS, $\lambda[\mathcal{S}]>0$
and so $\mathcal{S}$ can only have a spherical topology. (Note that the second
term on the RHS of (11) ie non-negative by assumption ($\rho\geq 0\implies$)
DEC holds.) This topological result is a particular case of the following more
general result due to Galloway [7], extending to higher dimensions:

######  Theorem III.4 (Galloway)

Let $(V^{n},h,\mathcal{K})$ be an initial data set satisfying the dominant
energy condition (DEC), $\rho\geq J_{\mu}J^{\mu}$. If $\tilde{\Sigma}^{n-1}$
is an outermost MOTS in $(V^{n},h,\mathcal{K})$, then $\tilde{\Sigma}^{n-1}$
admits a metric of positive scalar curvature.

(Of course in case $n=3$, $\tilde{\Sigma}$ is a topological 2-sphere.)
However, we would like to know if the causal character of the horizon
containing the MOTS is restricted by the stability criteria and/or the
topology. Of course we can immediately rule out stable pressureless TLM. We
can however completely rule out the possible existence of a stable MOTS in a
TLM embedded in a perfect fluid: for a TLM in LRS II, the area satisfies

$\displaystyle\mathbb{A}>\frac{1}{\rho-p}.$

Comparing the above to (10) we see that the MOTS are stable provided that
$\rho-p>2\rho\implies-\left(\rho+p\right)>0$, which is not possible given that
here the NEC is assumed (also implies the weak energy condition strictly
$\rho+p>0$). We therefore state this as the following non-existence result

###### Proposition III.5

A TLM embedded in a perfect fluid cannot contain a stable MOTS.

Also, notice how (11) allows us to rule out the other topologies if we assume
stability: $\underline{\chi}(\mathcal{S})=$ (resp. $<0$) would require that
$\lambda=$ (resp. $<-\rho$).

Now, suppose that we wish to extend our considerations to imperfect fluids
with $\Pi=0$, so that $J\not\equiv 0$. If $Q<0$, there is no problem here
since $\rho-Q$ is always positive. (Of course this means that we can
immediately rule out stable non-spherical 2-surface geometries.) We will show
the following.

###### Theorem III.6

Any stable MOTS $\mathcal{S}$ in an imperfect LRS II fluid with vanishing
stress anisotropy is a topological sphere.

  * Proof

If $Q\neq 0$, then, (11) becomes

$\displaystyle
2\pi\underline{\chi}(\mathcal{S})=\lambda[\mathcal{S}]+\int_{\mathcal{S}}(\rho-Q).$
(12)

It is then quite easy to show that $Q\leq\rho$, so that if $\mathcal{S}$ is
stable, then, $\underline{\chi}(\mathcal{S})>0$: it can be checked that the
denominator of (5) acquires the additional term $-2Q$. It is not difficult to
see that for S to be stable, it is necessary that $Q\geq\mathcal{E}$. From (8)
we have the desired area bound:

$\displaystyle\mathbb{A}=\frac{3}{2\left(\rho-3Q\right)},$

from which it follows that $Q\leq\rho$.

Notice that in the above result, we do not impose the DEC. However, the DEC in
this case reduces to

$\displaystyle\rho\geq Q^{2}-\rho^{2},$ (13)

which is always satisfied for $Q\leq\rho$. Furthermore, there is no sign
fixing on the horizon and hence the above result would appear to hold true for
arbitrary signature. However, just as before, we can completely rule out the
existence of stable MOTS in a TLM embedded in an imperfect LRS fluid, where
the argument follows similarly as the perfect fluid case:

###### Proposition III.7

A TLM embedded in an imperfect fluid with vanishing anisotropy cannot contain
a stable MOTS.

We point out that equality holds for $Q\leq\rho$ if an only if the horizon in
which the MOTS lie is isolated, with the particular equation of state
$\rho=p$. For pressure-less imperfect fluids with no anisotropy, we see that
for stable MOTS, away from isolation (more specifically, in a DH), $Q$ is
bounded from both sides:

$\displaystyle\mathcal{E}<Q<\rho.$

(For a DH we assume the infalling of radiation so that $Q>0$. Using (2))

$\displaystyle\mathcal{E}<Q<-6\mathcal{E}.$ (14)

Observe that $\mathcal{E}$ cannot be zero, and it is clear that we must have
$\mathcal{E}<0$. It therefore follows that on any MOTS contained in a DH in an
LRS II spacetime with no anisotropy, and on which the pressure $p$ vanishes
and $Q>0$, $\mathcal{E}<0$. This is of course expected for otherwise, the
horizon cannot be a DH, i.e. the denominator of $\mathcal{C}$ is non-negative.

Indeed, if the sign of $Q$ is known, then, (14) provides us with a way to
quickly check the (non)-existence of stable MOTS in a pressure-less scenario.
For example, if for some constant $\bar{c}\leq-6$ we have that
$Q=\bar{c}\mathcal{E}$ on a DH $\mathcal{H}$, then, $\mathcal{H}$ cannot
contain a stable MOTS. For the case of vanishing acceleration, we can use the
field equations and the commutator of the dot and hat derivatives to show that
this condition is precisely

$\displaystyle\hat{\mathcal{E}}\geq-\phi\left(\frac{9}{2}\mathcal{E}+\frac{2}{3}\Theta^{2}\right),$

where the number $c$ is just (note that $\phi\neq 0$ on a DH, and
$\mathcal{E}\neq 0$ as discussed above)

$\displaystyle\bar{c}=\frac{1}{\phi\mathcal{E}}\left(2\hat{\mathcal{E}}+\phi\left(3\mathcal{E}+\frac{4}{3}\Theta^{2}\right)\right).$

We also point out that the ruling out of stable surfaces of other topologies
in the perfect fluid case also holds here with $Q\neq 0$. In fact, we can make
the following stronger claim: Horizons in LRS II solutions with no anisotropy,
of the type considered in this work, cannot be foliated by non-spherical MOTS
(even unstable ones). To see this, we observe that for both the perfect and
imperfect case with no anisotropy (without assuming any sign restriction on
$\lambda$), the area $\mathbb{A}$ blows up if
$\underline{\chi}(\mathcal{S})=0$ (an infinitely large black hole which we can
rule out) and $\mathbb{A}<0$ (which is clearly impossible) if
$\underline{\chi}(\mathcal{S})<0$. However, if we throw in anisotropy, then
such foliation is in principle possible for $\Pi>0$. This is due to the
introduction of the $(1/2)\Pi$ term in $\lambda$, a contribution that comes
from the Gaussian curvature of the MOTS.

Let us end this section with the following remark:

  * Remark 2.

While we have neither explicitly constructed MOTS nor seen explicit examples
of them in LRS solutions with at least one of $\xi$ or $\Omega$ non-vanishing,
to our understanding we are not able to rule out their existence. Therefore,
one could attempt to generalize these results to LRS solutions with rotation
and spatial twist if they do admit MOTS. However, there are two points to
note:

    1. 1.

Firstly, rotation and spatial twist are obstructions to nonminimal MOTS, so
that, for the MOTS we consider, if either one of the rotation or spatial twist
is zero, the MOTS can only foliate minimal horizons as was earlier alluded to.
The problem is, since the main focus of this is relating horizon area bound
and stability, the minimal cases are not very much of interest here.
Furthermore, for the eigenvalue, the only extra contribution from non LRS II
quantities can only possibly be from the scalar curvature of the MOTS. In
particular, this will be a factor of the square of the rotation scalar
$\Omega$, and so the stability of the minimal MOTS is determined by the
difference between $\mathcal{E}$ and the factor of the square of the rotation;

    2. 2.

Secondly, with respect to the topological considerations for this subsection,
all the results applicable to the null case will apply if we include rotation
and spatial twist, as the contributing term is encoded in
$\underline{\chi}(\mathcal{S})$.

##  IV The growth of some covariant variables on horizons containing unstable
MOTS

Let us now look at some supplementary results where the magnitude of the
function $\mathcal{C}$ provides sign constraints on the propagation of the
electric Weyl scalar and energy density. As a consequence, the following
result will be demonstrated.

###### Proposition IV.1

Let $\mathcal{H}$ be a DH in a perfect fluid with $\mathcal{C}>1$ and
$\Theta\geq 0$, and let $\mathcal{S}$ be a MOTS in $\mathcal{H}$. Suppose the
isotropic pressure is sufficiently small so that first order derivatives are
negligible. Then, for small $t>0$, the variation of the outgoing null
expansion of $\mathcal{S}=\mathcal{S}_{0}$ is less than that of
$\mathcal{S}_{t}$.

To prove the above result, we will go through a series of intermediate results
- and accompanying discussions - which transparently demonstrates the
interplay between the curvature quantities appearing in $\lambda$ and their
propagation along the direction along which the MOTS is varied, i.e.,
$e^{\mu}$. (Notice that, if the pressure is sufficiently small relative to the
local energy density, then, on a DH we must have $\mathcal{E}<0$. Therefore,
whenever the smallness of the pressure is assumed, it will be understood that
$\mathcal{E}$ is strictly negative.)

Before proceeding, we make a few observations. We observe that the following
set of statements is true for a MOTS contained in a DH in any LRS II solution:

$\displaystyle\begin{split}0<\mathcal{C}<1&\implies\lambda>0\quad\mbox{(strictly
stable)};\\\
\mathcal{C}>1&\implies\lambda<0\quad\mbox{(unstable)}.\end{split}$

Indeed, we know that $\lambda\neq 0$ on a MOTS contained in $\mathcal{H}$
($\lambda=0\implies\mathcal{C}=1$), and hence, while $\lambda$ increases along
$e^{\mu}$, the sign stays fixed. This also provides a picture of the evolution
of an unstable MOTS: if a MOTS is unstable, then, one can tell its future
evolution - assuming all reasonable energy conditions are satisfied - by the
sign of $\mathcal{L}_{l}\chi$, i.e., whether it evolves to foliate a TLM or a
DH.

Now, Proposition III.5 confirms a widely accepted view that only stable MOTS
with globally spacelike world tubes (i.e., they foliate a DH) can be
identified as black hole boundaries. The conditions on the magnitude, as it
relates to stability of the MOTS we have considered in this work, have the
following implication: On a MOTS, if the quantity $\mathcal{C}$ takes a value
in the interval $0<\mathcal{C}<1$, then, that MOTS will evolve into a MOTT
that is a suitable black hole boundary.

Let us now begin by taking the Lie derivative of (8) along the tangent vector
field $\mathcal{V}^{\mu}$ to obtain

$\displaystyle\frac{1}{3}\mathcal{L}_{\mathcal{V}}\rho-\mathcal{L}_{\mathcal{V}}\mathcal{E}=-\frac{1}{2\mathbb{A}^{2}}\mathcal{L}_{\mathcal{V}}\mathbb{A}.$

As is known, if the horizon is a TLM, the area is decreasing and hence
$\mathcal{L}_{\mathcal{V}}\mathbb{A}<0$. And if it is a DH, the area is non-
decreasing and $\mathcal{L}_{\mathcal{V}}\mathbb{A}\geq 0$. We therefore have
that

$\displaystyle\frac{1}{3}\mathcal{L}_{\mathcal{V}}\rho-\mathcal{L}_{\mathcal{V}}\mathcal{E}>(resp.\leq)\
0,$

for a TLM (resp. DH). Explicitly, the above expression becomes

$\displaystyle\left(1-\mathcal{C}\right)\left(\frac{1}{3}\dot{\rho}-\dot{\mathcal{E}}\right)+\left(1+\mathcal{C}\right)\left(\frac{1}{3}\hat{\rho}-\hat{\mathcal{E}}\right)>(resp.\leq)\
0,$

which, on the horizon, simplifies to

$\displaystyle\phi\left(\frac{1}{2}\left(1-\mathcal{C}\right)\left(\rho+p\right)+3\mathcal{C}\mathcal{E}\right)>(resp.\leq)\
0,$

where we have used the evolution and propagation equations [19]

$\displaystyle\dot{\mathcal{E}}-\frac{1}{3}\dot{\rho}$
$\displaystyle=-\left(\frac{2}{3}\Theta-\Sigma\right)\left(\frac{1}{2}\left(\rho+p\right)-\frac{3}{2}\mathcal{E}\right),$
(15a) $\displaystyle\hat{\mathcal{E}}-\frac{1}{3}\hat{\rho}$
$\displaystyle=-\frac{3}{2}\phi\mathcal{E}.$ (15b)

Away from minimality, we know that $\phi$ is strictly positive, and hence we
have the following

$\displaystyle\left(1-\mathcal{C}\right)\left(\rho+p\right)>(resp.\leq)\
-6\mathcal{C}\mathcal{E}$ (16)

Now, take the case of a DH $\mathcal{H}$ and suppose that the Weyl scalar is
strictly negative. Then, clearly for $\mathcal{E}<0$, (16) is always true for
$\mathcal{C}>1$ (note that $\rho+p>0$ is always true since $\mathcal{H}$ is
non-null and, $\rho\geq
0\implies\mbox{DEC}\implies\mbox{NEC}\implies\rho+p>0$). However, for
$0<\mathcal{C}<1$, (16) is a constraint.

As an aside, a point of consideration, where the subtlety of the above
discussion is crucial, is the case of a slowly evolving horizon [29, 30],
usually defined in terms of the smallness of a so-called slowly evolving
parameter $\epsilon$. It has been suggested that [20], at least for the class
of locally rotaionally symmetric solutions, slowly evolving horizons can
simply be interpreted as those for which $|\mathcal{C}|$ is sufficiently small
(in fact, this is easily seen to lead to an equivalent interpretation of
slowly evolving horizons: those for which the magnitude of the principal
eigenvalue of the stability operator is sufficiently small). In this case, one
sees from (16) that the WEC must be violated in this limit, which is clearly
not possible for the cases considered here. This would therefore suggest the
following: A DH $\mathcal{H}$, embedded in a perfect fluid and containing a
stable MOTS (the “stable MOTS” requirement can be relaxed to the requirement
that the Weyl scalar is strictly negative) cannot slowly evolve for a
“sufficiently large” $\rho$ on $\mathcal{H}$.

The necessity of the qualification that the energy density be large enough is
easily seen from (16) as the inequality would hold if $\rho$ is small enough.
Take as an example the LTB solution, which is pressureless with its metric in
local coordinates given as

$\displaystyle ds^{2}=-d\bar{\tau}^{2}+\frac{\mathcal{R}^{\prime
2}}{1-\frac{2\mathcal{M}}{\bar{r}}}d\bar{r}^{2}+\mathcal{R}^{2}dS^{2},$ (17)

with the prime denoting differentiation with respect to the radial coordinate
$\bar{r}$, where $\mathcal{M}=\mathcal{M}(\bar{r})$ corresponds to the Misner-
Sharp mass, $\mathcal{R}=\mathcal{R}(\bar{\tau},\bar{r})$, and $dS^{2}$ is the
spatial 2-surfaces metric. The coordinate expression of the energy density is
given as

$\displaystyle\rho=\frac{\left(\bar{r}^{3}\mathcal{M}\right)^{\prime}}{\mathcal{R}^{2}\mathcal{R}^{\prime}}.$
(18)

It is clear that, for a fixed $\bar{r}$ if the change in the mass
$\mathcal{M}$ is sufficiently small, so is $\rho$, which implies that
$\mathcal{C}$ is also very small. Of course, for sufficiently small
$\mathcal{C}$, $\rho$ is sufficiently small, which implies
$\mathcal{M}^{\prime}$ is very small.

MOTS in these horizons are stable and the horizon always slowly evolves.
Detailed considerations of slowly evolving horizons in the LTB solution can be
found in [30], where for the cases considered on a DH, the slowly evolving
parameter takes the form

$\displaystyle\epsilon^{2}=8\frac{\mathcal{M}^{\prime}}{\mathcal{R}^{\prime}-\mathcal{M}^{\prime}},$

from which it is clear that $\mathcal{M}^{\prime}$ being sufficiently small
implies $\epsilon$ is sufficiently small.

Note that if the stability requirement for the MOTS is relaxed, then in
general the sign of $\mathcal{E}$ is not constrained. However, if
$0<\mathcal{C}<1$, then $\mathcal{E}$ has to be strictly negative (not
considering the slowly evolving case).

The next set of results examines the “growth” of the Weyl curvature and local
energy variables. Let us begin with the following result.

###### Proposition IV.2

Suppose $\mathcal{H}$ is DH embedded in a perfect fluid, with
$\mathcal{C}>1,\mathcal{A}\geq 0$ and $\Theta\geq 0$. Then, at least one of
the following is true:

  1. 1.

The isotropic pressure decreases along $u^{\mu}$; or

  2. 2.

The local energy density decreases along $e^{\mu}$.

  * Proof

Taking the Lie derivative of (1) along $\mathcal{V}^{\mu}$, and noting that
$\mathcal{C}$ is constant, we obtain

$\displaystyle\left(\mathcal{L}_{\mathcal{V}}\mathcal{L}_{l}\chi\right)\left(\mathcal{L}_{k}\chi\right)=\left(\mathcal{L}_{\mathcal{V}}\mathcal{L}_{k}\chi\right)\left(\mathcal{L}_{l}\chi\right).$

Since the DEC is always implicit as it is assumed that $\rho\geq 0$, the NEC
always holds so that $\mathcal{L}_{\mathcal{V}}\mathcal{L}_{k}\chi<0$. And
since $\mathcal{L}_{k}\chi<0$ and $\mathcal{L}_{l}\chi<0$, we have that

$\displaystyle\mathcal{L}_{\mathcal{V}}\mathcal{L}_{l}\chi\leq 0.$ (19)

Using $\mathcal{L}_{l}\chi=\rho-p-1/\mathbb{A}$, the above inequality can be
explicitly written as (where we again use the fact that
$\mathcal{L}_{\mathcal{V}}\mathbb{A}\geq 0$)

$\displaystyle\left[\left(\mathcal{C}-1\right)\Theta+\left(1+\mathcal{C}\right)\mathcal{A}\right]\left(\rho+p\right)+\left(1+\mathcal{C}\right)\hat{\rho}+\left(\mathcal{C}-1\right)\dot{p}\leq
0.$ (20)

Indeed, since $\mathcal{A}\geq 0$ and $\Theta\geq 0$, for $\mathcal{C}>1$ it
follows that $\hat{\rho}$ and $\dot{p}$ cannot simultaneously be positive.

The following corollary is a consequence of Proposition IV.2.

###### Corollary IV.3

Let $\mathcal{H}$ be a DH with $\mathcal{C}>1$, $\mathcal{A}\geq 0$ and
$\Theta\geq 0$, embedded in a perfect fluid. If $\mathcal{L}_{l}\chi$ is
monotone along $e^{\mu}$ and the Weyl scalar is strictly negative on
$\mathcal{H}$, then, the local energy density must decrease along $e^{\mu}$.

  * Proof

We prove this corollary by showing that, under the assumptions, $\dot{p}$ is
strictly positive. We expand (19) as

$\displaystyle\left(1-\mathcal{C}\right)\left(\dot{\chi}-\hat{\chi}\right)^{\cdot}+\left(1+\mathcal{C}\right)\widehat{\left(\dot{\chi}-\hat{\chi}\right)}\leq
0,$

from which, for $\mathcal{C}>1$, we have

$\displaystyle
0<\frac{\mathcal{C}+1}{\mathcal{C}-1}\leq\frac{\left(\dot{\chi}-\hat{\chi}\right)^{\cdot}}{\widehat{\left(\dot{\chi}-\hat{\chi}\right)}}$
(21)

Clearly, the numerator and denominator of (21) must have the same sign. And
since $\mathcal{L}_{l}\chi$ is monotone along $e^{\mu}$, the denominator is
negative $\implies$ numerator must also be negative, and therefore we have
that

$\displaystyle\dot{p}>-3\phi\mathcal{E}+\left(\Theta+\phi\right)\left(\rho+p\right),$

where we have used the evolution equation (15a). Since $\mathcal{E}$ is
strictly negative, the right hand side of the above expression is positive,
from which it follows that $\dot{p}>0$. Hence, by Proposition IV.2,
$\hat{\rho}$ must be negative.

###### Proposition IV.4

Let $\mathcal{H}$ be a DH embedded in a perfect fluid with $\mathcal{C}>1$ and
$\Theta\geq 0$, and suppose the Weyl scalar is strictly negative. If the
pressure is sufficiently small so that first order derivatives are negligible,
then the Weyl distortion increases along $e^{\mu}$.

Proposition IV.4 follows as a corollary to the following proposition.

###### Proposition IV.5

Let $\mathcal{H}$ be a DH embedded in a perfect fluid with $\mathcal{C}>1$ and
$\Theta\geq 0$, and suppose the Weyl scalar is strictly negative. If the
isotropic pressure is sufficiently small so that first order derivatives are
negligible, then $\mathcal{L}_{l}\chi$ is increasing along $e^{\mu}$.

  * Proof

From Proposition IV.2, if the isotropic pressure is small enough so that its
derivative along $u^{\mu}$ and $e^{\mu}$ are negligible, then $\hat{\rho}$ is
negative for $\mathcal{C}>1$. It is also seen from the propagation for the
isotropic pressure [19]

$\displaystyle\hat{p}=-\mathcal{A}\left(\rho+p\right),$

that $\mathcal{A}$ is neglible as well so that $\mathcal{H}$ is taken as non-
accelerating. Assume to the contrary that $\mathcal{L}_{l}\chi$ is decreasing
along $e^{\mu}$. By Corollary IV.3, $\mathcal{L}_{l}\chi$ is also decreasing
along $u^{\mu}$ and so we have that

$\displaystyle-3\phi\mathcal{E}+\left(\Theta+\phi\right)\left(\rho+p\right)<0.$

For the above inequality to hold, we must have that $\mathcal{E}$ is positive,
which of course contradicts the assumption that $\mathcal{E}<0$, and therefore
$\mathcal{L}_{l}\chi$ must instead be increasing along $u^{\mu}$. And since
the signs of the derivatives of $\mathcal{L}_{l}\chi$ along $u^{\mu}$ and
$e^{\mu}$ must be the same from (21), $\mathcal{L}_{l}\chi$ must be increasing
along $e^{\mu}$.

Proof of Proposition IV.4: To prove Proposition IV.4, we note that the
statement that $\mathcal{L}_{l}\chi$ is increasing along $e^{\mu}$ can be
written explicitly as

$\displaystyle\hat{\rho}>3\phi\mathcal{E}.$ (22)

Now, we can rewrite (15b) as

$\displaystyle\hat{\mathcal{E}}=-\frac{1}{6}\hat{\rho}+\frac{1}{2}\left(\hat{\rho}-3\phi\mathcal{E}\right),$

so that the above expression is strictly positive, where we use (22) and the
fact that $\hat{\rho}<0$.

Indeed, it then follows that

$\displaystyle-\frac{2}{3}\hat{\rho}-\hat{\mathcal{E}}>0,$

which verifies the claim of Proposition IV.1.

##  V Conclusion

In this work we have carried out a careful analysis of stability of cross
sections of certain black hole horizons and their relationship to the horizon
area and the topology. We employed the 1+1+2 covariant formulation and limited
our focus to the case of LRS II spacetimes with a perfect fluid geometry, and
where the topology is concerned, we extended to the imperfect case with non-
vanishing heat flux.

Firstly, it was shown that the MOTS stability operator is self-adjoint, since
the linear term vanishes as it is acted on by the vanishing rotation one-form
on the MOTS. (The vanishing of this one-form is true for more general LRS
spacetimes which include rotation and/or spatial twist. Hence, if these
solutions admit MOTS, the operator is self adjoint.) In spherical symmetry,
the stability operator simplifies significantly and the constant function
$Psi=const$ is the eigenfunction of the principal eigenvalue. An upper area
bound on the horizon area containing stable MOTS was obtained, with a strict
inequality demonstrated, after it was shown that the eigenvalue of the
operator is non-zero. Stable MOTS in timelike membranes were also ruled out,
even with the introduction of heat flux. These findings confirms the rarity of
stable MOTS as well as DH that one may associate with black holes.

We briefly discussed the topological restrictions on the MOTS and in the case
where one extends to an imperfect fluid with non-vanishing heat flux,
stability imposes a relationship between the flux and the local energy
density, which ensure spherical topology of the MOTS. This relationship, in
the case of positive heat flux leads to a non-existence result, for which the
precise condition in terms of the spacetime variables is provided. In
particular, it is shown that for a non-accelerating imperfect LRS II solution
with vanishing anisotropy, if on a DH one has that

$\displaystyle\hat{\mathcal{E}}\geq-\phi\left(\frac{9}{2}\mathcal{E}+\frac{2}{3}\Theta^{2}\right),$

then, the DH cannot contain a stable MOTS.

Finally, since the eigenvalue is a linear combination of the local energy
density and the electric Weyl scalar and can be written as the directional
derivative of the outgoing null expansion scalar along the unit normal to the
MOTS, we considered how these quantities behave along this direction. There
are values of the function $\mathcal{C}$ (even for the case of DH) - used to
determine the horizon signature - which imply that the MOTS are necessarily
unstable. Under certain mild assumptions, it is shown that for an unstable
MOTS contained in a DH the local energy decreases along the unit normal to the
MOTS while the Weyl distortion increases. It then follows that, for small
$t>0$, the variation of the outgoing null expansion of
$\mathcal{S}=\mathcal{S}_{0}$ is less than that of $\mathcal{S}_{t}$.

The stability analysis here could be extended to more general MOTS embedded in
general 1+1+2 spacetimes. We briefly mentioned this in Remark 1 and propose a
variation of the stability that could be performed. However, as a start, one
could study the stability using the formulation as developed by Anderson et
al.. In this case one would certainly encounter much more difficulty, not only
due to the presence of more terms for the operator, but also the fact that
$s_{\mu}$ will take a much more complicated form. In addition, the Laplacian
will not vanish and will take a very messy form. As a first step towards such
a generalisation one can impose on the spacetime the conditions for which
$s_{\mu}=0$ to bring the operator into a self-adjoint form. In fact, one can
even replicate the form of the operator in this work by further assuming that
scalars can be expanded up to second order on the MOTS so that second surface
derivatives are negligible.

There is another reason for interest in this paper, which is linked to an
ongoing work by the same authors. The aim is to try to look at MOTS in
perturbed LRS spacetimes and the behaviour of the horizons they foliate.
Indeed, the one-form $s_{\mu}$, as well as the Laplacian, is in general non-
zero. It is of interest to see whether there are constraints on the
contributing terms to the Euler characteristic $\underline{\chi}(\mathcal{S})$
of MOTS in the perturbed spacetime and how these constraints influence the
form of the operator. Additionally, it will be a good exercise to study and
obtain conditions (and implications) under which the form of the stability
operator remains invariant under the perturbation of the underlying spacetime.
In fact, this problem is intricately linked to the problem of finding interior
MOTS that have recently been investigated (see [9, 10, 31, 32]). In [31, 32],
the authors studied MOTS in the interior of a Schwarzschild black hole by
adapting to the Painlevè-Gullstrand “horizon penetrating coordinates”. In
these coordinates, the constant ‘time” slices are flat but non-static. (The
flatness property allows one to study surfaces using analogous approaches in
the standard Euclidean case and the non-staticity means that one also captures
non-minimal MOTS). The transformed metric takes a form with a twist term. This
therefore would suggest that, in order to study MOTS not lying in the $r=2m$
hypersurface, from context of the 1+1+2 approach, one may consider the
transformed metric metric as a perturbation of the Schwarzschild spacetime in
standard coordinates. In this case, it is then possible to conduct a detailed
analytic study of these interior MOTS, as perturbation of Schwarzschild
spacetime in our approach has been well developed [21]. Furthermore,
successful implementation of this task would allow us to generalise to a
general LRS spacetime, which could potentially capture interior MOTS in a
range of spacetimes for which details are only known of those standard MOTS on
the boundary, i.e. LTB, FLRW, etc.

## Acknowledgement

The authors would like to thank the anonymous referees for valuable comments
and suggestions, which have significantly improved the quality and clarity of
the manuscript. AS acknowledges that this work was supported by the Institute
for Basic Science (IBS-R003-D1). PKSD is supported by the First Rand Bank,
South Africa.

## References

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