# Moving Schwarzschild Black Hole and Modified Dispersion Relations

Cristian Barrera<EMAIL_ADDRESS>Justo López-
<EMAIL_ADDRESS>aDepartamento de Física, Universidad de
Santiago de Chile, Casilla 307, Santiago, Chile
bDepartamento de Física, Universidad Nacional de Colombia, 111321, Bogotá,
Colombia

###### Abstract

We study the thermodynamics of a moving Schwarzschild black hole, identifying
the temperature and entropy in a relativistic scenario. Furthermore, we set
arguments in a framework relating invariant geometrical quantities under
global spacetime transformations and the dispersion relation of the system. We
then extended these arguments in order to consider more general dispersion
relations, and identify criteria to rule them out.

##  I Introduction

The search for an unified picture of gravity and quantum principles has
deserved a lot of efforts and the approaches to this problem have taken
different roads and explored a wide class of scenarios. Theoretical proposals
like String Theory, Loop Quantum Gravity, or Causal Sets are examples of that,
and from there, one tries to find clues of some characteristic feature which
might survive at energy scales today attainable in observed phenomena, or some
tiny effects together with a sort of amplification mechanism which could offer
evidence. In this regard, the modification of the dispersion relation of
particles has received much attention in last years Mattingly ; Piran ;
Magueijo  as a possible signal of quantum gravity effects in the context of
the so called quantum gravity phenomenology, namely, the search of observable
relics of quantum gravity AmelinoLIVREV . Among the proposals sharing this
disctintive feature we can mention some limits of Loop Quantum Gravity Alfaro
, Double Special Relativity Amelino2000DSR ; Amelino2002DSR , String theory
String-MDR , etc. In this context, the dispersion relation that a macroscopic
body satisfies – taken as a body whose constituents satisfy a modified
dispersion relation – might be a source of difficulties when one adopt this
point of view, as noted already in early stages of these proposals
Soccerball2002 .

By other hand, black holes are probably one of the most suitable objects in
nature to test – at least theoretically – some features of quantum gravity.
For example, logarithmic corrections to the area-entropy law seems to be a
common characteristic shared by different proposals of quantum gravity (Kaul,
; Frolov, ; Carlip, ).

Under these considerations it seems natural to analyze if some imprints of
quantum gravity effects could be codified in the dispersion relation that a
black hole satisfies (Amelino, ). In this work we will try to gain some
insight in this matter, studying the kinematic consequences of a
generalization of symmetries in the movement of a massive object as a black
hole.

The layout of this paper is the following: in the present section we motivate
the problem and the questions that we would like to answer. In section II we
study the metric of a moving Schwarzschild black hole, or the Sexl-Aichelburg
metric, and calculate its temperature. Furthermore, in that section we will
generalize the geometrical thermodynamic arguments to this moving black hole,
identifying relevant parameters which will be useful later. In Section III we
extend the formalism carried out in the previous section to take into account
dispersion relations beyond the Lorentzian one. We will find certain
consistency relations that those modified relations have to fulfill. In
section IV we briefly discuss some interesting examples found in the
literature. Finally, we conclude in section V with final remarks, observations
and outlook.

##  II Moving Schwarzschild Black Hole

In this section we will analyze a Schwarzschild black hole moving with respect
to an observer, which will be carried out by means of a Lorentz boost to the
stationary solution. We calculate the geometrical temperature and analyze the
structure of its partition function.

###  II.1 Temperature of the moving black hole

Let us consider a black hole described by the Schwarzschild metric in
coordinates $(t^{\prime},r^{\prime},\theta^{\prime},\varphi^{\prime})$

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

where $M$ is the mass and we are using prime variables for the stationary
observer. In order to put the black hole in motion, we follow the approach of
Aichelburg and Sexl achsex . There, authors consider an observer moving
uniformly relative to the mass $M$ who sees a metric which is the previous one
deformed by a Lorentz transformation. That is, if $v$ is the velocity of the
black hole in the $z$ direction and $\gamma=(1-v^{2})^{-1/2}$ the relativistic
factor, we must take in (1) $t^{\prime}=\gamma(t-vz)$ and
$r^{\prime}=\sqrt{x^{2}+y^{2}+\gamma^{2}(z-vt)^{2}}$.

In order to study the thermodynamics of this object we will follow the path
integral approach of Euclidean path integral hawking  along with an
identification of spacetime points according to the system in movement. The
action one considers is the regularized gravitational action

$I=\frac{1}{16\pi}\int_{V}\sqrt{-g}R\,d^{4}x+\frac{1}{8\pi}\int_{\partial
V}[K]\sqrt{-h}\,d^{3}x,$ (2)

where $R$ is the scalar curvature, $g$ the metric determinant, $[K]$ the trace
of the extrinsic curvature regularized with respect to the flat spacetime and
$h$ the determinant of the induced metric on the boundary $\partial{V}$ of the
volume $V$. We will take $\partial{V}$ as the Lorentz transformed of the
stationary boundary for the Schwarzschild solution, i.e.
$r^{\prime}=r_{0}^{\prime}>2M$, rendering $r_{0}^{\prime}\to\infty$.

The value of the Euclidean action can be obtained straightforwardly by
noticing that both integrands in (2) are scalars under coordinate
transformations, and, as pointed out before, we are integrating over the
boosted Schwarzschild boundary. Then, the value of the action is the same as
computed in stationary coordinates with the Schwarzschild boundary, i.e.

$I=4\pi iM^{2}.$ (3)

By the other hand, in spite that the value of the Euclidean action does not
change, the temperature does. In fact, the partition function in the path-
integral approach must be integrated over periodic paths. Since the system is
moving in the $z$ direction with velocity $v$, the periodicity condition
implies the identification of points in spacetime as
$(t,x,y,z)\sim(t+i\beta,x,y,z+iv\beta)$. If we make the following coordinate
transformation in (1),

$\displaystyle\frac{y-i\xi}{y+i\xi}$ $\displaystyle=$ $\displaystyle
e^{t^{\prime}/2M},$ (4) $\displaystyle y^{2}+\xi^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{r^{\prime}}{2M}-1\right)e^{r^{\prime}/2M},$ (5)

the resulting Kruskal metric is free of conical singularities at
$r^{\prime}=2M$, for imaginary time $t^{\prime}=-i\tau^{\prime}$, if the
periodicity condition are consistent with the previous transformation, namely
$\Delta\tau^{\prime}=8\pi M$ and $\Delta r^{\prime}=0$. This implies that

$\Delta\tau^{\prime}=i\gamma(\Delta t-v\Delta z)=\gamma\beta(1-v^{2})=8\pi M,$
(6)

where $t=-i\tau$. Hence, from (6) we deduce that

$\beta=8\pi M\gamma,$ (7)

which is consistent for the case of Lorentz invariant systems Tolman .
Furthermore, $\Delta z^{\prime}=\gamma(\Delta z-v\Delta t)=0$, and then
$\Delta r^{\prime}=0$ as required.

Alternatively, one can use the Unruh Effect Hawkingevap ; Unruhevap  in order
to find the temperature of the system in movement as follows. Under the
transformations $\tilde{\tau}=t^{\prime}/(4M)$, $\rho^{2}=8M(r^{\prime}-2M)$
and $\tilde{x}=(2M\theta^{\prime},2M\phi^{\prime})$, the metric (1) can be
approximated in the vicinity of any point ($t_{0},x_{0},y_{0},z_{0}$) of the
event horizon $r^{\prime}=2M$ as the Rindler metric

$ds^{2}\approx-\rho^{2}d\tilde{\tau}^{2}+d\rho^{2}+d\tilde{x}^{2},$ (8)

which describes an accelerated frame, with acceleration $(4M)^{-1}$ respecting
the primed coordinates. If we focus on a moving point on the horizon (with
respect to the unprimed coordinates) we have
$\tilde{\tau}=\frac{\gamma}{4M}(t-vz)$. Then, in an interval of time $\Delta
t$, we have that $\Delta z=v\Delta t$, i.e.
$\Delta\tilde{\tau}=\frac{\gamma}{4M}(1-v^{2})\Delta t=\frac{\Delta
t}{4M\gamma}$. Hence, the surface gravity measured on the unprimed system is
$\kappa=(4M\gamma)^{-1}$, and then, by means of the Unruh Effect, the
temperature associated to the Hawking Radiation of the moving black hole is
$T=\beta^{-1}=\kappa(2\pi)^{-1}=(8\pi M\gamma)^{-1}$, as given by (7). Notice
that a global boost on the stationary system does not correspond to a local
boost in the proximity of the event horizon, but to a change in the surface
gravity.

###  II.2 Thermodynamics and Lorentz Symmetry

We now follow thermodynamical arguments to discuss the moving black hole
problem. The system is described by an observer ${\cal O}$ such that it has
conserved linear momentum ${\cal P}$ and Hamiltonian $H$ (for simplicity, we
consider the case where the relative velocity is aligned with one axis of the
reference system of the distant observer). In this scenario, we can write the
Massieu function as

$F(\beta,\eta)=-\ln Z=-\ln\left[\mbox{Tr}\left(e^{-\beta H+\eta
P}\right)\right],$ (9)

where $\beta$ and $\eta$ are chemical potentials (Lagrange multipliers)
associated to the conservation of charges. Notice that $F$ will coincide
numerically with the path integral formalism with imaginary time of quantum
mechanics. Its Legendre transformed function corresponds to the entropy
$S({{\cal U}},{{\cal P}})$, given by

$S({\cal U},{\cal P})=-F(\beta,\eta)+\beta\,{{\cal U}}-\eta\,{{\cal P}},$ (10)

with

${\cal U}=\left(\frac{\partial
F}{\partial\beta}\right)_{\eta},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\cal P}=-\left(\frac{\partial
F}{\partial\eta}\right)_{\beta}.$ (11)

We are interested in the connection that can be established between the
entropy of the system and the dispersion relation it satisfies. Given that
dispersion relations are associated to spacetime symmetries of the system, we
will start our analysis by considering the case of Lorentz invariance and we
will observe how, by imposing this, it is possible to connect the functional
dependence of $S$ and $F$ to certain invariants quantities built from the
intensive variables $\beta,\eta$. If the thermodynamical description done by
an observer ${\cal O}^{\prime}$ – at rest with the system – is related with
the description done by ${\cal O}$ through the Lorentz transformations on
quantities $H,P,\beta,\eta$, then $F$ must depend on a particular combination
of the variables $(\beta,\eta)$ which remains invariant. Indeed, let us
consider an unitary operator $U(\epsilon)$ which represents a Lorentz
transformation of relative rapidity $\epsilon$, then

$\displaystyle F(\beta,\eta)$ $\displaystyle=$
$\displaystyle-\ln\left[\mbox{Tr}\left(U^{\dagger}e^{-\beta H+\eta
P}U\right)\right]$ $\displaystyle=$
$\displaystyle-\ln\left[\mbox{Tr}\left(e^{-\beta H^{\prime}+\eta
P^{\prime}}\right)\right]$ $\displaystyle=$
$\displaystyle-\ln\left[\mbox{Tr}\left(e^{-\beta^{\prime}H+\eta^{\prime}P}\right)\right]\equiv
F(\beta^{\prime},\eta^{\prime}).$

where we have used fact that $H$ and $P$ transform linearly under the Lorentz
group, i.e.

$\displaystyle H^{\prime}=U^{\dagger}HU=H\cosh\epsilon-P\sinh\epsilon$ (12)
$\displaystyle P^{\prime}=U^{\dagger}PU=P\cosh\epsilon-H\sinh\epsilon,$ (13)

and then

$\displaystyle\beta^{\prime}=\beta\cosh\epsilon+\eta\sinh\epsilon$ (14)
$\displaystyle\eta^{\prime}=\eta\cosh\epsilon+\beta\sinh\epsilon.$ (15)

In the reference system ${\cal O}^{\prime}$ we have ${\cal P}^{\prime}=0$, and
then $\tanh\epsilon=v$ is the relative velocity between ${\cal O}$ and this
observer, i.e. $\cosh\epsilon=\gamma$, and $\sinh\epsilon=\gamma v$ and
therefore, in the present case, the function $F$ must depend on $(\beta,\eta)$
through the combination $\beta^{2}-\eta^{2}$. Consider now the particular
function $\mu(\beta,\eta)$ defined through

$\mu^{2}=\beta^{2}-\eta^{2},$ (16)

which invariant by construction. Notice that this quantity $\mu$ corresponds
to the temperature of the system at rest ($\eta=0$, $\beta=\beta_{0}$). For
the black hole under consideration, this quantity has a topological origin, in
virtue of the periodicity condition, and then it is appropriate to relate it
to the Schwarzschild radius $R_{s}$, concretely

$\mu=\beta_{0}=4\pi R_{s}.$ (17)

This is consistent with the fact that since the area of the event horizon $A$
is a marginally trapped surface, it will not change under a boost of the
Schwarzschild metric (1), and from (17) one finds the relation

$\mu^{2}=4\pi A.$ (18)

On the other hand, since the internal energy ${\cal U}$ and momentum ${\cal
P}$ are related to $F$ through (11) one obtains (for the invariant (16)) the
following relations

${\cal U}=\dot{F}(\mu)\frac{\beta}{\mu}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal
P}=-\dot{F}(\mu)\frac{\eta}{\mu}\,$ (19)

with $\dot{F}\equiv dF/d\mu$. Therefore, it is straightforward to obtain the
relativistic dispersion relation

${\cal U}^{2}-{\cal P}^{2}={\cal M}^{2},$ (20)

where we have denoted ${\cal M}=\dot{F}$, which is the inertial mass of the
black hole. Notice that in the previous argument, in addition to showing that
the internal energy and momentum of the black hole satisfy the relativistic
dispersion relation, we have found a nontrivial relation between the inertial
mass of the black hole and the Schwarzschild radius through the partition
function of the system. On the other hand, using equations (19) and (10) we
obtain

$S=-F(\mu)+\mu\dot{F},$ (21)

which expresses the entropy as the Legendre transform of $F$ with respect to
the invariant $\mu$. It is worth noting that entropy here appears as a
relativistic invariant, and its invariance under Lorentz transformations is
consistent with the approach to thermodynamics of moving objects by Tolman,
for example Tolman . This fact that appears trivial here will have interesting
consequences that will be discussed later. Moreover, expressing everything in
terms of ${\cal M}$ we obtain the relation $\mu=S^{\prime}({\cal M})$, where
$S^{\prime}$ is the derivative of the entropy with respect to ${\cal M}$. We
can read the velocity of the system as

$v\equiv\left(\frac{\partial U}{\partial P}\right)_{S}=\frac{\eta}{\beta}$
(22)

this is to say $\eta=\beta v$, and using (16) we obtain $\beta=\mu\gamma$. Let
us now identify the partition function of this system with the euclidean
action in the path integral formalism in the semiclassical approximation. The
value of the action (3) together with (17), and $R_{s}=2M$ imply that the
Massieu function describing the black hole is

$F_{\mbox{\tiny{BH}}}=\frac{\mu^{2}}{16\pi},$ (23)

and then ${\cal M}=\dot{F}=M$, namely, in the semiclassical approximation the
inertial mass coincides with the mass parameter in the Schwarzschild solution.
As a consequence, ${\cal M}^{2}=(R_{s}/2)^{2}=A/(16\pi)$, and then we obtain
the Smarr formula smarr ; Bardeen1973  for the moving black hole

${\cal U}^{2}=\frac{A}{16\pi}+{\cal P}^{2}.$ (24)

Finally, the entropy can be calculated with (21), and it corresponds to
$S=A/4$, and provided that $\mu=8\pi M$ the inverse of the temperature is then
$\beta=8\pi M\gamma$, as we proved by other arguments in the previous
subsection.

It is natural to expect that both the Bekenstein-Hawking entropy and the Smarr
formula will acquire corrections as one takes into account terms beyond the
semiclassical approximation. In fact, several approaches to quantum gravity
show corrections to the entropy-area law Medved ; Kaul ; Frolov ; Carlip ;
Chatterjee , which can be used to calculate $F(\mu)$ starting from (18) and
(21) as a differential equation. Given $S=S(A)$, the solution is

$F(\mu)=a\,\mu+\frac{\mu}{\sqrt{16\pi}}\int_{1}^{\mu^{2}/4\pi}\frac{{\cal
S}(A^{\prime})}{A^{\prime 3/2}}dA^{\prime},$ (25)

from which we also get

${\cal M}(\mu)=a+\frac{{\cal
S}(\mu)}{\mu}+\frac{1}{\sqrt{16\pi}}\int_{1}^{\mu^{2}/4\pi}\frac{{\cal
S}(A^{\prime})}{A^{\prime 3/2}}dA^{\prime}.$ (26)

In particular, certain quantum gravity models, such as Loop Quantum
GravityAshtekar ; Rovelli  and String theoryStrominger  coincide in that the
corrections to the area-entropy law have the form

$S=\frac{A}{4}+\rho\ln(A),$ (27)

where $\rho$ is a model dependent dimensionless quantity of order unity. Using
equations (25) and (26) for such case we obtain

$F(\mu)=\frac{\mu^{2}}{16\pi}-2\rho\left(1+\log\left(\frac{\mu}{4\pi}\right)\right)+a(\rho)\mu,$
(28)

and

${\cal M}=\ \frac{\mu}{8\pi}-\frac{2\rho}{\mu}+a(\rho).$ (29)

Here we have made explicit that the integration constant $a$ may depend on the
parameter $\rho$ and, concretely, it is direct to fix its value when $\rho=0$
in order to reproduce the semiclassical result. Considering the identification
of $\mu$ in terms of the Schwarzchild radius, equation (29) provides us two
solutions given by

$r_{\pm}=({\cal M}-a)\left(1\pm\sqrt{1+\frac{\rho}{\pi({\cal
M}-a)^{2}}}\right),$ (30)

where $r_{+}$ is continuously connected to the semiclassical Schwarzschild
radius, and $r_{-}$ to the singularity at the origin (given that $\rho<0$
independently of the model Kaul ). This can be observed by expanding both
solutions in $\rho/M$ powers

$\displaystyle r_{+}$ $\displaystyle=$ $\displaystyle 2({\cal
M}-a)+\frac{\rho}{2\pi({\cal M}-a)}+\mathcal{O}\left(\frac{\rho}{{\cal
M}^{3}}\right),$ (31) $\displaystyle r_{-}$ $\displaystyle=$
$\displaystyle-\frac{\rho}{2\pi({\cal
M}-a)}+\mathcal{O}\left(\frac{\rho}{{\cal M}^{3}}\right).$ (32)

These results resembles the ones obtained in Reuter1 , where authors start
from an effective metric whose Newton constant $G_{N}$ acquires quantum
corrections depending on $r$. In order to make contact with this idea, let us
assume that the inertial mass of the black hole has corrections with respect
to the semiclassical approximation, i.e.

${\cal M}=M+\rho\delta_{\cal M}.$ (33)

Writing equation (31) in terms of $M$ and collecting powers of $\rho/M$ we can
compare with the results obtained in Reuter1 ; Donoghue , identifying
$a=\rho\delta_{\cal M}$, and $\rho=-118/15$. It is worth noting that, even if
there is some discrepancy with the precise value obtained by other approaches,
both sign and order of magnitude of the model dependent parameter $\rho$ are
consistent.

##  III Generalization to Modified Dispersion Relations

Consider now the possibility that Lorentz symmetry is modified or even broken,
as the scenarios we mentioned in the first section, this is to say, we will
assume the existence of a deformed symmetry underlying the theory. We ask how
previous arguments will change in such case, expecting to be able to identify
the invariant $\mu$ in terms of $(\beta,\eta)$ with a more general functional
dependence.

A similar argument to equation (II.2) implies that the partition function will
be invariant under transformations of the form
$\beta^{\prime}=f_{\epsilon}(\beta,\eta)$, and
$\eta^{\prime}=g_{\epsilon}(\beta,\eta)$ where $\epsilon$ is any symmetry
parameter satisfying that $\beta=f_{0}(\beta,\eta)$ and
$\eta=g_{0}(\beta,\eta)$. Particularly, if we call $\mu=\beta$ when $\eta=0$,
and assume that the transformation connects the points in the ($\beta$,$\eta$)
plane in a qualitatively similar form as the Lorentz Transformations, then we
can always write down

$\displaystyle\beta=f_{\epsilon}(\mu,0)$ (34)
$\displaystyle\eta=g_{\epsilon}(\mu,0),$ (35)

such that $\mu=f_{0}(\mu,0)$ and $0=g_{0}(\mu,0)$. From here we should obtain
expressions such that

$\displaystyle\mu=\mu(\beta,\eta)$ (36)
$\displaystyle\epsilon=\epsilon(\beta,\eta),$ (37)

where $\mu(\beta,0)=\beta$ and $\epsilon(\beta,0)=0$. Finally, we can replace
the somewhat ambiguous parameter $\epsilon$ with the compatible parameter
$v\equiv\eta/\beta$.

Now, let us suppose that $F(\mu)$ is known. In such case, we can write

${\cal U}=\dot{F}\frac{\partial\mu}{\partial\beta},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal
P}=-\dot{F}\frac{\partial\mu}{\partial\eta},$

and in principle, variables $(\beta,\eta)$ can be obtained as function of
$({\cal U},{\cal P})$. We define the dispersion relation as

$m({\cal U},{\cal P})\equiv\mu(\beta({\cal U},{\cal P}),\eta({\cal U},{\cal
P})).$ (38)

The entropy, by other hand, turns out to be

$S({\cal U},{\cal
P})=-F(\mu)+\left(\beta\,\frac{\partial\mu}{\partial\beta}+\eta\,\frac{\partial\mu}{\partial\eta}\right)\dot{F}.$
(39)

Notice the following important point: if the expression in parenthesis in
equation (39) depends upon ($\beta$,$\eta$) in an arbitrary way, and not
through the invariant $\mu$, then the entropy would no longer be a movement
invariant, and that can carry us to some inconsistencies as, for example, the
number of accessible states being dependent upon the movement state of the
system. Then, we will take as an hypothesis the invariance of the entropy, and
consequently we must impose the term in parenthesis to be, in principle, an
arbitrary function of $\mu$. However, let us remember that $\mu=\beta$ at
rest, and in such case $S=-F+\beta{\cal U}=-F+\mu\dot{F}$. For compatibility
of the definitions of entropy in movement and at rest, it is only possible to
impose that

$\beta\,\frac{\partial\mu}{\partial\beta}+\eta\,\frac{\partial\mu}{\partial\eta}=\mu.$
(40)

This last equation is a particular case of the known Clairaut equation, whose
general solution is given by Euler’s homogeneous function theorem of degree 1.
Then, from equation (36) and the discussion in the paragraph following it, we
can deduce that

$\mu=\beta\mu(1,v)=\frac{\beta}{\Gamma(v)},$ (41)

where we have defined $\Gamma(v)=\mu^{-1}(1,v)$, which satisfies that
$\Gamma(0)=1$, and corresponds to a generalization of the relativistic
$\gamma$ factor. In such case, the entropy depends on the dispersion relation
and the following relations hold

$\displaystyle{\cal M}(\mu)$ $\displaystyle=$ $\displaystyle\frac{dF}{d\mu},$
(42) $\displaystyle\mu({\scriptstyle{\cal M}})$ $\displaystyle=$
$\displaystyle\frac{d{\cal S}}{d{\scriptstyle{\cal M}}}.$ (43)

Notice that this last result implies that the discussion at the end of the
previous section remains valid even for dispersion relations beyond the
relativistic one.

Now, we may ask the inverse problem, namely, knowing the entropy of the system
as a function of its internal energy and momentum, compatible with a certain
given dispersion relation

${\cal M}={\cal M}({\cal U},{\cal P}),$ (44)

that remains invariant at constant entropy, and where ${\cal M}$ reduces to
the rest energy, i.e. ${\cal U}={\cal M}({\cal U},0)$. Then, using the inverse
Legendre transform for $S$, and imposing that $F$ depends on an invariant
$\mu=S^{\prime}$, we have that

$F=-S+\left({\cal U}\frac{\partial{\cal M}}{\partial{\cal U}}+{\cal
P}\frac{\partial{\cal M}}{\partial{\cal P}}\right)S^{\prime}=-S+{\cal
M}S^{\prime}.$ (45)

This condition implies that

${\cal U}\frac{\partial{\cal M}}{\partial{\cal U}}+{\cal P}\frac{\partial{\cal
M}}{\partial{\cal P}}={\cal M},$ (46)

in other words, ${\cal M}$ has to be a homogeneous function of degree 1 with
respect to ${\cal U}$ and ${\cal P}$. This, again, imposes severe constraints
on the form that could have the dispersion relation of a thermodynamical
system.

Given a dispersion relation ${\cal M}({\cal U},{\cal P})$ satisfying (44) y
(46) we can write

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\mu\frac{\partial{\cal
M}}{\partial{\cal U}}$ (47) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle-\mu\frac{\partial{\cal M}}{\partial{\cal P}}$ (48)

and along with (41) we can identify

$\displaystyle v$ $\displaystyle=$ $\displaystyle-{\frac{\partial{\cal
M}}{\partial{\cal P}}}\Big{/}{\frac{\partial{\cal M}}{\partial{\cal
U}}}=\left(\frac{\partial{\cal U}}{\partial{\cal P}}\right)_{\cal M},$ (49)
$\displaystyle\Gamma(v)$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
M}}{\partial{\cal U}}.$ (50)

The RHS of these equations are homogeneous functions of degree 0 with respect
to ${\cal U}$ and ${\cal P}$, and then depend only on the combination ${\cal
P}/{\cal U}$. Consequently, from equation (49) we can find ${\cal P}/{\cal U}$
as function of $v$, and from (50) we can find explicitly the form of the
function $\Gamma(v)$. Finally, taking into account that $v=\eta/\beta$ and
$\mu=\beta/\Gamma$ we obtain the invariant $\mu$ as a function of
$(\beta,\eta)$.

##  IV Some Examples

In this section we will apply our previous discussion and review some
important cases: the relativistic dispersion relation and a relevant example
found in the phenomenological discussions beyond that frame.

The relativistic case has been already discussed in detail in the previous
sections, but we will analyze it briefly in terms of the arguments developed
in the previous one. Let us begin from the relativistic dispersion relation
${\cal U}^{2}=m^{2}+{\cal P}^{2}$. It is clear that the function ${\cal
M}=\sqrt{{\cal U}^{2}-{\cal P}^{2}}$ corresponds to equation (44) satisfying
(46). Following the steps at the end of the previous section, using equations
(49) and (50) we obtain that

$\displaystyle v$ $\displaystyle=$ $\displaystyle\frac{{\cal P}}{{\cal U}},$
(51) $\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-({\cal P}/{\cal U})^{2}}}=\gamma.$ (52)

Finally, making $v=\eta/\beta$ we find that

$\mu=\frac{\beta}{\Gamma}=\sqrt{\beta^{2}-\eta^{2}},$ (53)

as expected. Next, we discuss a more interesting case that is used as a probe
of dispersion relation beyond the Lorentz SymmetryMattingly ; Piran ; Ellis ;
Magueijo ; Garay , and that can be considered as a low energy limit of some
models Smolin ; Amelino-Smolin ; Alfaro ; Pullin . A typical way to write this
relation is

$E^{2}=m^{2}+p^{2}+\alpha l_{p}E^{3},$ (54)

where $\alpha$ is a dimensionless constant and $l_{p}$ a constant with length
dimensions typically of order of the inverse of Planck scale
$E_{p}=10^{19}\text{ GeV}$, such that for energies $E$ far below $E_{p}$ the
above relation is accurately approximated by the relativistic one.

As there is still no consensus on whether black holes should be considered
fundamental objects or composite systems, we are going to consider how our
arguments apply in each case. If one considers the black hole as a fundamental
object, then the dispersion relation (54) is applied directly, namely in our
notation

${\cal U}^{2}={\cal M}^{2}+{\cal P}^{2}+\alpha l_{p}{\cal U}^{3}.$ (55)

However, this dispersion relation clearly does not satisfy the condition
imposed by (46), and then our argument discards this dispersion relation
directly, as the homogeneity property is not fulfilled. This result is
consistent with the fact that if we consider the black hole as a fundamental
object, then clearly the LIV term in (54) can be large, as its mass can widely
exceed $M_{p}$, i.e. the Soccer Ball Problem would be present Soccerball2002 .

On the other hand, if we consider the black hole as a composite object, we can
find the function ${\cal M}$ of equation (38) compatible with the given
dispersion relation and satisfying the homogeneity property (46) using the
following implicit form

$(1-\delta){\cal M}^{2}={\cal U}^{2}-{\cal P}^{2}-\delta\frac{{\cal
U}^{3}}{{\cal M}},$ (56)

where $\delta$ is a dimensionless constant that cannot depend upon ${\cal M}$.
Notice that we have chosen the coefficients so that ${\cal U}={\cal M}({\cal
U},0)$, as required. If we consider this dispersion relation as a deviation
from the relativistic one, then $\delta\equiv\alpha l_{p}m<<1$. Calculating
the invariant $\mu$ in the limit $\delta\to 0$, which constitutes a small
departure from the relativistic scenario, we find at first order in $\delta$
that

$v=\frac{{\cal P}}{{\cal U}}\left(1+\frac{3\delta}{2}\frac{1}{\sqrt{1-({\cal
P}/{\cal U})^{2}}}+...\right),$ (57)

or conversely,

$\frac{{\cal P}}{{\cal U}}=v\left(1-\frac{3\delta}{2}\gamma+...\right).$ (58)

On the other hand we obtain

$\Gamma(v)=\gamma\left[1+\frac{\delta}{2}(1-3\gamma+2\gamma^{3})\right]+...$
(59)

and then the function $\mu(\beta,\eta)$ in this case is given by

$\mu=\sqrt{\beta^{2}-\eta^{2}}-\frac{\delta}{2}\left(\sqrt{\beta^{2}-\eta^{2}}-3\beta+2\frac{\beta^{3}}{\beta^{2}-\eta^{2}}\right).$
(60)

Notice that, in accordance with the previous discussion, we indeed have that
$\mu(\beta,0)=\beta$ also in this case. This invariant $\mu$ must be a direct
consequence of the symmetry that would replace the Lorentz Transformations,
and one should be able to identify the new transformations.

In this example $m$ is an arbitrary scale that can be fixed by convenience. If
we choose $m$ to be of the same order as the mass of the system ${\cal M}$
then it is clear that, in order to maintain the phenomenological consistency
for black holes with macroscopic masses, that is, in order to avoid the Soccer
Ball Problem, $\alpha<<1$, and then (54) implies that the LIV scale should be
far below the Planck scale $l_{p}$. Namely, the LIV term appearing in (54)
needs to be strongly suppressed by a scale far above $E_{p}$. This conclusion
is in agreement with the results in Relative-Soccer , where the Soccer Ball
Problem is investigated in the context of Relative Locality, and it is shown
that, for a composite system, the first correction to the relativistic
dispersion relation is suppressed not by $E_{p}$ but by $NE_{p}$, being $N$
the number constituents. Thus, the present analysis shows that the modified
dispersion relation 54 is consistent only if one indeed considers the black
hole as a composite system, and the LIV term is suppressed by a scale above
$E_{p}$, avoiding the Soccer Ball Problem. Moreover, one could think, for
instance, on the existence of additional adiabatically invariant quantities
defining the system, namely, entropy could depend not only on ${\cal U}$ and
${\cal P}$ but also on another extensive quantity Ghosh . In the framework of
black holes, these could be known quantities as angular momentum or charge, or
quantities characterizing the microscopic state of the system.

##  V Final Remarks and Outlook

In this work we have studied the thermodynamics of a moving Schwarzschild
black hole identifying the temperature and entropy in a relativistic scenario.
Furthermore, we have set arguments in a framework relating invariant
geometrical quantities under global spacetime transformations and the
dispersion relation of the system. Next, we have extended these arguments in
order to consider more general dispersion relations. The formalism just shown
may be interesting for analyzing LIV phenomenology in massive objects
presumably structureless as black holes. As we discussed at the end of section
IV, there are dispersion relations used in discussions on LIV phenomenology
that present some difficulties in virtue of our analysis. A possible solution
to such drawbacks could be to consider more general black holes or presume
certain internal structure yet unknown. We expect to exploit this last idea in
next works, firstly by considering rotating and charged black holes, and
afterwards by considering new degrees of freedom that may match our arguments
with phenomenological observations in a natural way.

###### Acknowledgements.

The authors would like to acknowledge the inspiring comments of Aurelio Grillo
and Fernando Mendez, and the support of FONDECYT Chile, through grant 1140243.

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Marginally outer trapped tubes in de Sitter spacetime

Marc Mars

Departamento de Física Fundamental, Universidad de Salamanca

Plaza de la Merced s/n, 37008 Salamanca, Spain

Carl Rossdeutscher, Walter Simon and Roland Steinbauer
Fakultät für Mathematik, Universität Wien

Oskar-Morgenstern Platz 1, 1090 Wien, Austria

###### Abstract

We prove two results which are relevant for constructing marginally outer
trapped tubes (MOTTs) in de Sitter spacetime. The first one (Theorem 1) holds
more generally, namely for spacetimes satisfying the null convergence
condition and containing a timelike conformal Killing vector with a ”temporal
function”. We show that all marginally outer trapped surfaces (MOTSs) in such
a spacetime are unstable. This prevents application of standard results on the
propagation of stable MOTSs to MOTTs.

On the other hand, it was shown recently [11] that, for every sufficiently
high genus, there exists a smooth, complete family of CMC surfaces embedded in
the round 3-sphere $\mathbb{S}^{3}$. This family connects a Lawson minimal
surface with a doubly covered geodesic 2-sphere. We show (Theorem 2) by a
simple scaling argument that this result translates to an existence proof for
complete MOTTs with CMC sections in de Sitter spacetime. Moreover, the area of
these sections increases strictly monotonically. We compare this result with
an area law obtained before for holographic screens.

##  1 Introduction

Our general setting is a smooth $n$-dimensional ($n\geq 3$), oriented and
time-oriented Hausdorff manifold $({\cal M},g)$ with a smooth metric of
signature $(-,+,...+)$. In Sect. 2 we require the Ricci tensor Ricg to satisfy
the null convergence condition Ric${}_{g}(\ell,\ell)\geq 0$ for all null
vectors $\ell$, while in Sect. 3 we restrict ourselves to de Sitter spacetime
(dS). We first recall well-known key definitions to also fix our notation.
Note that we use the ”physics” convention throughout, in particular in Def. 1
where most mathematics references (e.g. [11]) define the mean curvature by
$\widehat{H}=H/\mbox{dim}~{}{\cal U}$.

######  Definition 1.

Let ${\cal U}$ be a hypersurface (i.e. a submanifold of codimension one)
embedded in any semi-Riemannian manifold ${\cal N}$. The mean curvature $H$ of
${\cal U}$ is the trace of its second fundamental form.

######  Definition 2.

Let ${\cal F}$ be a spacelike surface of codimenion two embedded in ${\cal M}$
with future directed, orthogonal null vectors $\ell^{\pm}$, and denote the
null expansions of the corresponding families of emanating geodesics by
$\theta^{\pm}=tr_{\cal F}(\nabla\ell^{\pm})$.

  * •

A Marginally outer trapped surface (MOTS) or just Marginal surface ${\cal F}$
is defined by $\theta^{+}=0$. Here $\ell^{+}$ defines the ”outer” direction;
the latter is ill-defined for minimal surfaces for which $\theta^{-}=0$ as
well. (As to terminology cf. Remark 3 in Sect. 2).

  * •

A Marginally trapped surface (MTS) is a MOTS with either $\theta^{-}<0$ or
$\theta^{-}>0$ (we avoid the term ”past trapped” or ”antitrapped” for the
latter case).

  * •

A Marginally outer trapped tube (MOTT) ${\cal T}$ is a hypersurface foliated
by MOTS.

  * •

A Marginally trapped tube (MTT) or holographic screen is a hypersurface
foliated by MTSs.

MOTS, MTS or trapped surfaces which satisfy $\theta^{+}<0$ and $\theta^{-}<0$,
are key in singularity theorems which yield incomplete geodesics. However, in
a typical collapse scenario, there is not only the (outermost) “key MOTS” but
also a plethora of other, seemingly “spurious” MOTS (see e.g. [20, 6] and
references therein). One motivation of this work is to better understand such
spurious MOTS.

Apart from these collapse scenarios, MOTS are normally also abundant in non-
singular spacetimes which do not satisfy all requirements for the singularity
theorems. We mention two settings in which they are worth studying: Firstly,
under additional assumptions, null geodesics emanating from MOTS have
conjugate points while still being complete. This suggests a recent definition
[19] of the ”outward affine size” of a non-singular universe with a MOTS.
Secondly, for asymptotically dS spacetimes, there are results on the
”visibility” of future weakly trapped surfaces (defined by $\theta^{\pm}\leq
0)$) from infinity [12].

A key property of MOTS is their (in)stability, cf. Def. 3 and Remark 2 below.
Here we recall important properties of strictly stable MOTSs: They have
spherical topology, inherit all symmetries of the ambient space, and propagate
along achronal MOTTs in a given spacelike foliation of spacetime [3, 4].
Somewhat weaker results hold in the non-strictly stable case. Moreover,
stability also plays a role in a singularity theorem (cf. Thm. 7.1 and Remark
7.2 in [4]).

The present work has two somehwat independent aspects exposed in Sect. 2 and
3. In Sect. 2 we show absence of stable MOTS in a class of spacetimes
containing a timelike conformal Killing field with a ”temporal function”. The
precise result is as follows:

######  Theorem 1.

Let $({\cal M},g)$ be an $n$-dimensional ($n\geq 3$) spacetime satisfying the
null convergence condition. Assume $({\cal M},g)$ admits a future directed
timelike conformal Killing field $\xi$, i.e.

$\displaystyle\pounds_{\xi}g=2\Psi g$

where $\pounds_{\xi}$ is the Lie derivative and $\Psi$ is a temporal function,
i.e. its gradient is timelike and past directed. Then $({\cal M},g)$ admits no
stable MOTS.

In Sect. 3 we restrict ourselves to dS for which Ric${}_{g}=(n-1)\delta^{2}g$,
$\delta$ a positive constant. It can be globally written in the form

$\displaystyle g_{\mbox{\tiny
dS}}=\frac{1}{\delta^{2}\cos^{2}\sigma}\left(-d\sigma^{2}+g_{\mathbb{S}^{n-1}}\right)$
(1)

where $\sigma\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ is taken to
increase to the future, and $g_{\mathbb{S}^{n-1}}$ is the standard metric on
the round unit sphere $\mathbb{S}^{n-1}$.

Choosing $\xi=\frac{\partial}{\partial\sigma}$ in Thm. 1 we show in Corollary
1 that all MOTSs in dS are unstable. Hence the above-said results of [3, 4] on
topology, symmetry and evolution to achronal MOTTs do not apply. To analyze
MOTTs we now restrict ourselves to 3+1 dimensions and focus on umbilic and
constant curvature slicings of dS (with curvatures of all possible signs). It
is easy to show (Lemma 2 in Sect. 3.2.1) that, in this setting, MOTSs are
automatically MTSs and (as hypersurfaces within the slices) surfaces of
constant mean curvature (CMC). This allows, in particular, constructing a MTSs
from certain families of CMC surfaces in either flat $\mathbb{R}^{3}$, the
hyperbolic space $\mathbb{H}^{3}$ or the round $\mathbb{S}^{3}$. We consider
the former two settings (in Sects. 3.2.2 and 3.2.3) as a “warmup” and focus on
$\mathbb{S}^{3}$ (cf. Sect. 3.2.4).

In Sect. 3.2.4 we analyze, then, dS in terms of its ”standard” umbilic slicing
(1) by round three spheres ${\mathbb{S}}^{3}_{\sigma}$ of radius
$\rho=\delta^{-1}\cos^{-1}\sigma$. In this slicing, MOTS with $\theta^{\pm}=0$
correspond to CMC surfaces with mean curvature $H=\mp 2\delta\sin\sigma$, thus
in particular minimal for $\sigma=0$. The resulting MTSs are easily found
explicitly in the case of spherical and toroidal MTS-sections. In the former
case, the MTSs are null surfaces with sections of constant area, while in the
latter case, the MTSs are timelike and their area increases monotonically from
the Clifford torus at $\sigma=0$ to $\sigma\rightarrow\pm\frac{\pi}{2}$ (where
it diverges).

On the other hand, knowledge of CMC surfaces of genus ${\mathfrak{g}}$ larger
than one in $\mathbb{S}^{3}$ is rather rudimentary. However, for every
${\mathfrak{g}}$ sufficiently large (${\mathfrak{g}}\gg 1$), S. Charlton, L.
Heller, S. Heller and M. Traizet [11] recently proved existence of a complete
family of CMC surfaces $f_{\psi}^{\mathfrak{g}}$,
($\psi\in(-\frac{\pi}{4},\frac{\pi}{4}$)) embedded in the unit sphere
$\mathbb{S}^{3}$. This family connects the well-known Lawson minimal surface
$\xi_{1,{\mathfrak{g}}}$ (for $\psi=0$) with a doubly covered geodesic sphere
(for $\psi=\pm\frac{\pi}{4}$). Moreover, these authors show that the Willmore
energy of these surfaces, defined by

$\displaystyle W=A(\frac{H^{2}}{4}+1),$ (2)

where $A$ is the area, increases strictly monotonically from the Lawson
surfaces towards the limiting spheres.

After quoting and slightly extending the results of [11] as Theorem 3 in Sect.
3 we describe its straightforward application to dS. In fact we only need to
rescale the geometry from the $\mathbb{S}^{3}$ of unit radius to the spherical
sections of dS whose radius blows up with the cosmological time $\sigma$ as
sketched above. Remarkably, this scaling entails that the Willmore energy of
the CMC surfaces found in [11] translates just to the area of the MTS-sections
in the dS setting and yields the corresponding monotonicity statement. Thus
our adaptation of the results of [11] reads as follows.

######  Theorem 2.

Our setting is de Sitter spacetime in coordinates (1) for $n=4$. For every
${\mathfrak{g}}\in\mathbb{N}$, ${\mathfrak{g}}\gg 1$ , there exist a smooth
function
$\sigma:(-\frac{\pi}{4},\frac{\pi}{4})\ni\psi\mapsto\sigma(\psi)\in(-\sigma_{m},\sigma_{m})$
and a smooth family of conformal CMC embeddings ${\cal
F}_{\psi}^{\mathfrak{g}}:{\cal
R}^{\mathfrak{g}}\rightarrow\mathbb{S}^{3}_{\sigma}$ of a Riemann surface
${\cal R}^{\mathfrak{g}}$ of genus ${\mathfrak{g}}$ into the round 3-sphere
$\mathbb{S}^{3}_{\sigma}$ of radius $\rho_{\sigma}=\delta^{-1}\cos^{-1}\sigma$
such that

  1. 1.

${\cal F}_{0}^{\mathfrak{g}}$ is the Lawson surface $\xi_{1,{\mathfrak{g}}}$
of genus ${\mathfrak{g}}$.

  2. 2.

For $\psi\rightarrow\pm\frac{\pi}{4}$, ${\cal F}_{\psi}^{\mathfrak{g}}$
smoothly converges to a doubly covered geodesic 2-sphere with
$2{\mathfrak{g}}+2$ branch points; the family ${\cal F}_{\psi}^{\mathfrak{g}}$
cannot be extended in $\psi$ in the space of immersions.

  3. 3.

${\cal F}_{\psi}^{\mathfrak{g}}={\cal F}_{-\psi}^{\mathfrak{g}}$ up to
reparametrization and isometries of $\mathbb{S}^{3}_{\sigma}$, and accordingly
the constant mean curvatures satisfy ${\cal H}_{\psi}^{\mathfrak{g}}=-{\cal
H}_{-\psi}^{\mathfrak{g}}$.

  4. 4.

${\cal H}_{\psi}^{\mathfrak{g}}$ decreases strictly monotonically from zero at
$\psi=0$ to a minimum ${\cal
H}_{\psi_{m}}^{\mathfrak{g}}=-2\delta\sin\sigma^{\mathfrak{g}}_{m}$ at some
$\psi=\psi_{m}$, from which it increases strictly monotonically to zero for
$\psi\rightarrow\frac{\pi}{4}$. The monotonicity behaviour of ${\cal
H}_{\psi}^{\mathfrak{g}}$ for $\psi\in(-\frac{\pi}{4},0)$ is then determined
by property 3.

  5. 5.

The area of ${\cal F}_{\psi}^{\mathfrak{g}}$ increases strictly monotonically
from $\psi=0$ towards both $\psi=\pm\frac{\pi}{4}$ where it takes the values
$A_{\pm\pi/4}=8\pi\delta^{-2}$.

This theorem has the following obvious Corollary:

######  Corollary 1.

For every ${\mathfrak{g}}\gg 1$, the smooth family of conformal embeddings
constucted in Thm. 2 determines a smooth MTT whose MTS sections have
properties 1-5.

In the final Sect. 4 we note that the monotonicity of the area of the MTS-
sections of MTTs (holographic screens) is the key ingredient in their
thermodynamic interpretation along the lines of the ”second law”. We briefly
discuss the area law established in [8].

##  2 A class of spacetimes with no stable MOTSs

We start with definitions, setting out from Def. 2. We normalize the null
vectors to satisfy $\langle\ell^{+},\ell^{-}\rangle=-2$. Capital letters
($A,B,..$) denote indices on objects on ${\cal F}$. The induced metric on
${\cal F}$ is denoted by $j$ or $j_{AB}$, while its Levi-Civita derivative by
$D$ or $D_{A}$ and $\Delta_{j}:=D_{A}D^{A}$ is the Laplacian and $R_{j}$ the
scalar curvature. The null second fundamental form of ${\cal F}$ with respect
to $\ell^{+}$ is denoted by $k^{+}$ ($k_{AB}^{+}$) and the torsion one form
$s~{}(s_{A})$ is defined by

$\displaystyle
s(X)=-\frac{1}{2}\langle\ell^{-},\nabla_{X}\ell^{+}\rangle\quad\forall~{}\mbox{vector
fields}~{}X~{}\mbox{on}~{}{\cal F}.$

######  Remark 1.

The motivation and inspiration for the following definition and the results on
(in)stability of MOTS (cf. Sects. 3-5 of [3]) come from corresponding material
on minimal surfaces of codimension 1 in manifolds with a Riemannian metric. We
recall, however, that in the minimal surface case stability can be defined via
the second variation of the area functional. This is no longer the case for
MOTS, which entails substantial differences (cf. Remark 2 below).

The key definitions for this section read as follows.

######  Definition 3 (Stability).

We denote the Einstein tensor by
$\mathrm{Ein}_{g}=\mathrm{Ric}_{g}-\frac{1}{2}{\mathrm{R}}_{g}g$.

  * •

The stability operator $L$ of a MOTS ${\cal F}$ is defined for smooth
functions $w$ by

$\displaystyle
L(w)=-\Delta_{j}w+2s^{A}D_{A}w+\frac{1}{2}\Big{(}\mathrm{R}_{j}-\mathrm{Ein}_{g}(\ell^{+},\ell^{-})-2s^{A}s_{A}+2D_{A}s^{A}\Big{)}w$
(3)

  * •

$L$ admits a principal eigenvalue $\lambda$, which is the eigenvalue with
smallest real part. This eigenvalue is necessarily real and its eigenspace is
one-dimensional and spanned by an everywhere positive function $\phi$, called
principal eigenfunction.

  * •

The MOTS ${\cal F}$ is called strictly stable (resp. stable, marginally stable
or unstable) provided $\lambda>0$ ($\lambda\geq 0$, $\lambda=0$ or
$\lambda<0$).

######  Remark 2.

The stability operator is relevant for calculating the variation
$\delta_{\xi}\theta^{+}$ of the expansion $\theta^{+}$ with respect to an
arbitrary vector field $\xi$ normal to ${\cal F}$. In Lemma 3.1 of [3] it was
shown that

$\displaystyle\delta_{\xi}\theta^{+}=L(u)-2BZ$ (4)

where the functions $u,B$ and $Z$ are defined on ${\cal F}$ by
$u:=\langle\xi,\ell^{+}\rangle$, $B:=-\langle\xi,\ell^{-}\rangle$ and
$Z:=k^{+}_{AB}k_{+}^{AB}+\mathrm{Ein}_{g}(\ell^{+},\ell^{+})$.

The “stability operator $L_{v}$ in the direction of a normal vector $v$”
introduced in Def.3.1 of [3] is related to (4) as follows:

$\displaystyle\delta_{\xi}\theta^{+}=L_{v}(u)~{}~{}\mbox{for}~{}\xi=u\,v$ (5)

In particular, $L(u)$ as defined in (3) agrees with $L_{v}(u)$ for
$v=-\frac{1}{2}\ell^{-}$.

The stability operator of a stable MOTS satisfies a maximum principle, cf.
Lemma 4.2 of [3]:

######  Lemma 1.

Let ${\cal F}$ be a MOTS and $L$ the corresponding stability operator. Let
$\lambda$ be the principal eigenvalue and $\phi>0$ the principal
eigenfunction. Let $\psi$ be a smooth solution of $L\psi=f$ with $f\geq 0$.
then

  * (i)

If $\lambda=0$, then $f=0$ and $\psi=C\phi$ for some constant $C$.

  * (ii)

If $\lambda>0$ and $f$ is not identically zero then $\psi>0$.

  * (iii)

If $\lambda>0$ and $L\psi=0$, then $\psi=0$.

From this lemma we can prove the following general result on the non-existence
of stable MOTS:

######  Proposition 1.

Let ${\cal F}$ be a MOTS in an n-dimensional ($n\geq 3$) spacetime $({\cal
M},g)$ satisfying the null convergence condition. Assume that there exists a
future causal vector field $\xi$ along ${\cal F}$ which is not everywhere
proportional to $\ell^{+}$ and such that $\delta_{\xi}\theta^{+}\geq 0$. Then
${\cal F}$ cannot be strictly stable. Moreover, if $\delta_{\xi}\theta^{+}$ is
positive somewhere then ${\cal F}$ cannot be marginally stable either.

###### Proof.

The basis $\\{\ell^{+},\ell^{-}\\}$ is future directed, so $\xi$ being future
causal implies $u\leq 0$ and $B\geq 0$. Moreover $u$ is not identically zero,
because $\xi$ is not everywhere proportional to $\ell^{+}$. Together with Equ.
(4) and the hypotheses of the proposition this implies $L(u)\geq 0$. If ${\cal
F}$ is strictly stable (i.e. $\lambda>0)$, items (ii) and (iii) of Lemma 1
imply that $u$ is either strictly positive or identically zero, which is a
contradition. This proves the first claim of the proposition. Finally, if
${\cal F}$ is marginally stable then item (i) of the Lemma yields
$\delta_{\xi}\theta^{+}+BZ=0$, which cannot happen if $\delta_{\xi}\theta^{+}$
is positive somewhere. ∎

We are now ready to prove Thm. 1 stated in the Introduction. We use greek
indices on spacetime objects and sum over repeatedly occurring ones.

###### Proof of Thm. 1.

In Corollary 1 of [10] the following general identity is proved for variations
of $\theta^{+}$ of a MOTS along an arbitrary vector field $\xi$ in terms of
the deformation tensor $a(\xi):=\pounds_{\xi}g$ of $\xi$

$\displaystyle\delta_{\xi}\theta^{+}=-\frac{1}{4}\theta^{-}a(\xi)_{\alpha\beta}\ell^{\alpha}_{+}\ell^{\beta}_{+}-a(\xi)_{\alpha\beta}e_{A}^{\alpha}e_{B}^{\beta}k_{+}^{AB}+j^{AB}e_{A}^{\alpha}e_{B}^{\beta}\ell^{\nu}_{+}\left(\frac{1}{2}\nabla_{\nu}a(\xi)_{\alpha\beta}-\nabla_{\alpha}a(\xi)_{\beta\nu}\right)$
(6)

where $\\{e^{\alpha}_{A}\\}$ is any basis of tangent vectors to ${\cal F}$.
Assume now that $\xi$ is a conformal Killing vector, so that $a(\xi)=2\Psi g$.
Inserting into (6) yields

$\displaystyle\delta_{\xi}\theta^{+}=j^{AB}e_{A}^{\alpha}e_{B}^{\beta}\ell^{\nu}_{+}\left(\nabla_{\nu}\Psi
g_{\alpha\beta}-2\nabla_{\alpha}\Psi
g_{\beta\nu}\right)=(n-2)\ell^{\nu}_{+}\nabla_{\nu}\Psi.$

Since $\ell^{+}$ is future directed and $\Psi$ is a temporal function, we have
$\delta_{\xi}\theta^{+}>0$ and the result is a consequence of Proposition 1. ∎

######  Remark 3.

Recall that MOTS were defined in Def. 2 by the vanishing of (at least) one of
the null expansions. As the “outer” amendment apparently plays no role,
neither in that Definition nor in the discussion above, the simpler notion
“marginal surface” (introduced by Hayward [17]) would suffice. To indicate
some benefit of the “MOTS” terminology, we recall that in Def.3.1 of [3],
stability of the MOTS with respect to any direction $v$ was defined by
requiring that the operator $L_{v}(w)$ (Equ. (5)) has non-negative lowest
eigenvalue. This is equivalent to non-negativity of either side of (5)
provided $\ell^{+}$ points in the same direction as $\xi=u\,v$, viz.
$u=\langle\xi,\ell^{+}\rangle\geq 0$. Calling these directions “outer” (or
“inner”) is just more efficient terminology than some “… points in the same
(or opposite) direction…” amendment. The next section provides corresponding
“unstable examples”.

##  3 MOTSs and MOTTs in de Sitter spacetime

###  3.1 Instability of MOTS

The following easy consequence of Thm. 1 was already sketched in the
Introduction.

######  Corollary 2.

All MOTS in de Sitter spacetime are unstable.

###### Proof.

In terms of the coordinates (1), $\xi=\frac{\partial}{\partial\sigma}$ is a
future directed, timelike conformal Killing vector field satisfying

$\displaystyle\pounds_{\xi}g_{\mbox{\tiny dS}}=2\tan\sigma\,g_{\mbox{\tiny
dS}}.$

Since $\tan$ is an increasing function and $\sigma$ is a temporal function, so
is $\Psi=\tan\sigma$. Hence Thm. 1 implies the claim. ∎

######  Remark 4.

There is a certain analogy between de Sitter spacetime on which we focus in
this section and round spheres, in the sense that both are maximally symmetric
spaces of constant positive curvature. Recalling also the analogy between the
stability definitions for minimal surfaces and MOTS (cf. Remark 1), Corollary
2 above can be seen as a counterpart to a result of [21] (namely case $p=n-1$
of Thm 5.1.1) that all minimal surfaces of codimension 1 on the round
${\mathbb{S}}^{n}$ are unstable. However, there is no obvious analogy between
the proofs, and we do not make any attempt of establishing one here.

###  3.2 MOTS in umbilic slicings

From now on we restrict ourselves to dS in 3+1 dimensions. Our aim is to
locate MOTTs in the three umbilic and constant curvature slicings of de
Sitter, namely flat and hyperboloidal and spherical. Our focus will be on the
latter case.

####  3.2.1 Preliminaries

We consider a spacelike hypersurface $({\cal N},h,K)$ with metric $h$, scalar
curvature ${}^{3}R$, covariant derivative ${}^{3}\nabla$, extrinsic curvature
$K$ (w.r.t. a future-pointing unit normal $N$) and mean curvature $tr_{h}K$
(i.e. again with ”physics convention”, cf. Def. 1). The constraints take the
form

$\displaystyle R_{h}-|K|_{h}^{2}+(tr_{h}K)^{2}$ $\displaystyle=$
$\displaystyle 6\delta^{2}$ (7) $\displaystyle
div_{h}\left(K-(tr_{h}K)h\right)$ $\displaystyle=$ $\displaystyle 0.$ (8)

######  Lemma 2.

Let $({\cal N},h,K)$ be an umbilic slice in de Sitter, i.e. the extrinsic
curvature satisfies $K=\beta h$ for some function $\beta$. Then $\beta$ is
constant on ${\cal N}$. Furthermore, any MOTS ${\cal F}\subset{\cal N}$ with
$\theta^{+}=0$ is a MTS and CMC as well, with mean curvature
$H=div_{h}X=-2\beta$ where $X$ is a unit outward normal vector to ${\cal F}$
in ${\cal N}$, i.e. $\langle X,\ell^{+}\rangle>0$.

###### Proof.

The constancy of $\beta$ follows from the constraint (8). For the second
statement, we use the normalisation $\langle\ell^{+},\ell^{-}\rangle=-2$ and
the decomposition $\ell^{\pm}=N\pm X$. This induces the following
decomposition of $\theta^{\pm}$

$\theta^{\pm}=\pm H+tr_{j}K=\pm H+2\beta$ (9)

which implies the assertion, with $\theta^{-}=4\beta$ on the MOTS
$\theta^{+}=0$. ∎

######  Remark 5.

In the following subsections we consider as examples umbilic slicings $({\cal
N},h,K)$ of constant curvature (i.e. $h$ is flat, spherical or hyperboloidal)
in dS. In this special situation, a recent result of [7] implies instability
of all MOTS ${\cal F}$ contained in such slices. The argument goes as follows.
The spaces ${\cal N}$ in question are homogeneous in the sense that their
isometry groups act transitively. In particular, for a given MOTS ${\cal F}$
any pair of points $p\in{\cal F}$ and $q\not\in{\cal F}$ can be joined by a
group element $\rho$, i.e. $q=\rho(p)$. Hence the Killing vector $\xi$ of
${\cal N}$ tangent to this orbit cannot be everywhere tangent to ${\cal F}$.
Now by Thm. 1.6 of [7] or by Thm. 8.1 of [3], ${\cal F}$ is either unstable or
marginally stable, with $\xi$ nowhere tangent to ${\cal F}$ in the latter
case. But this case is ruled out by Thm. 5.1 of [7] since by Lemma 2 above,
$H=-2\beta=$ const. on an umbilic slice.

While the results of [7] also cover other situations than the aforementioned
one (as discussed in that paper), there is not much further overlap with the
general settings considered in Proposition 1 and Corollary 2. Of course, the
latter results apply to umbilic slices.

These slicings are conveniently defined in terms of the dS hyperboloid

$\displaystyle-
x_{0}^{2}+x_{\alpha}x_{\alpha}=\delta^{-2}\qquad\alpha,\beta\in\\{1,2,3,4\\}.$
(10)

embedded in 5d Minkowski space

$\displaystyle ds^{2}=-dx_{0}^{2}+dx_{\alpha}dx_{\alpha}$ (11)

Above and henceforth, repeated indices are summed over. A useful reference for
the slicings considered below is Sect. 4 of [16].

####  3.2.2 Flat slicing

In terms of the coordinate transformation

$\displaystyle x_{0}$ $\displaystyle=$ $\displaystyle\delta^{-1}\sinh\delta
t+\frac{\delta}{2}r^{2}e^{\delta t}$ (12) $\displaystyle x_{1}$
$\displaystyle=$ $\displaystyle\delta^{-1}\cosh\delta
t-\frac{\delta}{2}r^{2}e^{\delta t}$ (13) $\displaystyle x_{i}$
$\displaystyle=$ $\displaystyle e^{\delta t}y_{i}\qquad(i\in\\{2,3,4\\})\qquad
r^{2}=y_{i}y_{i}$ (14)

and upon introducing polar coordinates on $\mathbb{S}^{2}$, the induced metric
on the hyperboloid (10) reads

$ds^{2}=-dt^{2}+e^{2\delta t}[dr^{2}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\phi^{2})]\quad r\in[0,\infty),~{}\vartheta\in[0,\pi],~{}\phi\in[0,2\pi]$
(15)

Clearly, the $t=$const. [$t\in(-\infty,\infty)$] slicing is flat and umbilic
(with $\beta=\delta$) but covers only half of dS as described by (1).

In order to determine the MOTS in these slices, we recall that closed CMC
surfaces embedded in flat 3-space must necessarily be round spheres [2].

A sphere of radius $r=R$ in a slice $t=T$ obviously has the induced metric

$ds_{R}^{2}=e^{2\delta T}R^{2}(d\vartheta^{2}+\sin^{2}\vartheta d\phi^{2}).$
(16)

We now choose the counter-intuitive label “outgoing” for the normal vector $X$
pointing towards decreasing $r$, viz.:

$X=-e^{-\delta T}\frac{\partial}{\partial r}.$ (17)

We find for the mean curvature w.r.t. $X$

$H=div_{h}X=-\frac{2}{R}e^{-\delta T}.$ (18)

It is only with this choice that (9) can be solved to determine the location
of a marginally outer trapped tube with MOTS sections $\theta^{+}=0$, namely:

$e^{-\delta T}=\delta R$ (19)

in consistency with Def. 2. We note that $H=-2\delta=const.$ along the MOTT.

####  3.2.3 Hyperboloidal slicing

We perform the coordinate transformation

$\displaystyle x_{0}$ $\displaystyle=$ $\displaystyle\delta^{-1}\sinh\delta
t\cosh r$ (20) $\displaystyle x_{1}$ $\displaystyle=$
$\displaystyle\delta^{-1}\cosh\delta t$ (21) $\displaystyle x_{i}$
$\displaystyle=$ $\displaystyle\delta^{-1}z_{i}\sinh\delta t\sinh
r\qquad(i\in\\{2,3,4\\})\qquad z_{i}z_{i}=1$ (22)

which yields for the induced metric on the hyperboloid (10)

$\displaystyle ds^{2}=-dt^{2}+\delta^{-2}\sinh^{2}(\delta
t)[dr^{2}+\sinh^{2}r(d\vartheta^{2}+\sin^{2}\vartheta d\phi^{2})]$ (23)

where $t\in(-\infty,\infty)$, $r\in[0,\infty)$ and
$\vartheta\in[0,\pi],~{}\phi\in[0,2\pi)$. This slicing is umbilic with
$\beta=\delta\coth\delta T$ for $t=T=$const.; again it covers only part of dS
given by (1), cf. Sect. 4.4.2 of [16].

Here we restrict ourselves to calculating round CMC spheres; as to more
general shapes cf. Remark 6 below. Then the discussion becomes quite similar
to the flat case of the previous subsection. The induced metric on slices
$t=T=$const., $r=R=$const. reads

$\displaystyle ds_{R}^{2}=\delta^{-2}\sinh^{2}(\delta
T)\sinh^{2}R(d\vartheta^{2}+\sin^{2}\vartheta d\phi^{2})$

We now take as outgoing unit normal vector to the $R=const$ slices

$\displaystyle X=-\delta\sinh^{-1}(\delta T)\frac{\partial}{\partial r}$

which yields for the mean curvatures of the MOTSs,

$H=div_{h}X=-2\delta\sinh^{-1}(\delta T)\coth R.$ (24)

It then follows from (9) that the MOTT with spherical sections are determined
by the equation

$\cosh\delta T=\coth R.$ (25)

For the mean curvatures of the MOTS sections of these MOTTs we obtain

$\displaystyle H=-2\delta\cosh R\leq-2\delta$ (26)

######  Remark 6.

By the maximum principle, the mean curvature of a closed CMC surface embedded
in the unit hyperboloid $\mathbb{H}^{3}$ must satisfy either $H\geq 0$ or
$H\leq-2$ (cf. e.g. [13]), in consistency with (26). We also recall the
“Lawson correspondence” [18, 13], which is a locally bijective relation
between CMC surfaces in 3-dim space forms. In particular, CMC surfaces in the
unit hyperboloid $\mathbb{H}^{3}$ with $H\leq-2$ have corresponding CMC or
minimal surfaces in flat space or in the unit sphere $\mathbb{S}^{3}$. We
restrict the detailed discussion to the latter case.

####  3.2.4 Complete spherical slicing

Introducing the coordinates

$\displaystyle x_{0}$ $\displaystyle=$ $\displaystyle\delta^{-1}\sinh\delta t$
(27) $\displaystyle x_{\alpha}$ $\displaystyle=$
$\displaystyle\delta^{-1}z_{\alpha}\cosh\delta
t\qquad(\alpha\in\\{1,2,3,4\\})\qquad z_{\alpha}z_{\alpha}=1$ (28)

we obtain

$\displaystyle ds^{2}=-dt^{2}+\delta^{-2}\cosh^{2}(\delta
t)[d\tau^{2}+\sin^{2}\tau(d\vartheta^{2}+\sin^{2}\vartheta
d\phi^{2})]\qquad\tau\in[0,\pi]$ (29)

This is equivalent to (1) where the time coordinates are related by
$\tan\frac{\sigma}{2}=\tanh\frac{\delta t}{2}$. We proceed the discussion in
terms of (1).

The extrinsic and mean curvatures of the surfaces $\sigma=const$ read

$K=(\delta\sin\sigma)\,h\qquad tr_{h}K=3\delta\sin\sigma.$ (30)

Therefore $\beta=\delta\sin\sigma$ in the notation of Lemma 2. In contrast to
the previous slicings Sects. 3.2.2 and 3.2.3, we can now solve (9) for either
sign which yields

$\displaystyle H=\mp 2\delta\sin\sigma.$ (31)

3.2.4.1 Spherical MOTS

In this and the following subsection 3.2.4.2 our presentation largely follows
[5, Sect. 5] and [9, Ch. 5].

In polar coordinates (1) takes the form

$ds^{2}=\delta^{-2}\cos^{-2}\sigma\left[d\tau^{2}+\sin^{2}\tau\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right)\right];~{}~{}\tau,\vartheta\in[0,\pi],\varphi\in[0,2\pi)$
(32)

For round 2-spheres $\tau=const.$ we obtain from (9):

$\theta^{\pm}=2\delta[\pm\cos\sigma\cot\tau+\sin\sigma].$ (33)

The MOTSs given by $\tau_{0}^{\pm}=\pm\sigma_{0}+\frac{\pi}{2}$ are spheres
with radius $\delta^{-1}$.

Fig. 1 shows, via a conformal rescaling (”Einstein cylinder”, conformal factor
$\delta^{-2}\cos^{-2}\sigma$) the MOTTs ${\cal T}^{\pm}$ determined by the two
families of MOTSs. ${\cal T}^{\pm}$ are null surfaces. The $\varphi$ and
$\vartheta$ directions are suppressed, hence spheres reduce to two points; the
dots at $\sigma=0$ correspond to the equators on $\mathbb{S}^{3}$ while the
conical convergence towards $\sigma=\pm\frac{\pi}{2}$ is due to the conformal
factor. The drawing is sketchy (i.e. not a faithful result of calculation).

Figure 1: The MOTTs ${\cal T}^{-}$ (left) and ${\cal T}^{+}$ (right) with
spherical CMC sections

3.2.4.2 Toroidal MOTS

We recall the toroidal foliation of $\mathbb{S}^{3}$ in terms of the following
coordinates.

$ds^{2}=\delta^{-2}\cos^{-2}\sigma(d\tau^{2}+\sin^{2}\tau\,d\gamma^{2}+\cos^{2}\tau\,d\xi^{2});\quad\tau\in[0,\frac{\pi}{2}],\quad\gamma,\xi\in[0,2\pi)$
(34)

On a torus ${\cal F}$ given by $\tau=const.$ we find from (9):

$\theta^{\pm}=2\delta[\pm\cos\sigma\cot 2\tau+\sin\sigma]$ (35)

which formally differs just by a factor of $2$ in the $\cot$-term from the
spherical case (33). This entails that the MOTSs determined by
$\theta^{\pm}=0$ and given via $\tau=\tau^{\pm}$ on some slice
$\sigma=\sigma_{0}$ are now tori located at
$\tau_{0}^{\pm}=\pm\frac{\sigma_{0}}{2}+\frac{\pi}{4}$. As above the MOTS
exist for all $\sigma\in(-\frac{\pi}{2},\frac{\pi}{2})$ but now they form
timelike MOTTs. The area of its toroidal sections

$\frac{A}{4\pi}=|\frac{\sin\tau\cos\tau}{\delta^{2}\cos^{2}\sigma}|=|\frac{\sin
2\tau}{2\delta^{2}\cos^{2}\sigma}|=|\frac{\sin(\pm\sigma+\pi/2)}{2\delta^{2}\cos^{2}\sigma}|=\frac{1}{2\delta^{2}\cos\sigma}$
(36)

diverges when $\sigma\rightarrow\pm\frac{\pi}{2}$ and approaches the minimum
$\frac{1}{2\delta^{2}}$ at $\sigma=0$.

Fig. 2 shows again the Einstein cylinder with two dimensions suppressed; Each
MOTS ${\cal T}^{\pm}$ reduces to 4 points; the Clifford torus is indicated by
4 dots at $\sigma=0$. The MOTSs are timelike, which is vizualized by the null
vectors $\l^{\pm}$. As before this figure is also sketchy.

Figure 2: The MOTTs ${\cal T}^{-}$ (left) and ${\cal T}^{+}$ (right) with
toroidal CMC sections

3.2.4.3 MOTS of higher genus

We now turn to MOTSs and MOTTs of higher genus ${\mathfrak{g}}$. A natural
strategy is to set out from a minimal surface at the time-symmetric slice
$\sigma=0$, as such surfaces also have played special roles in the cases of
genus 0 and 1. We would like to show existence of their evolution to MOTTs,
and determine the causal character of the latter as well as the evolution of
the area of their MOTS sections. Unfortunately, for genus ${\mathfrak{g}}>1$
explicit expressions for the embedding functions and other geometric
quantitites seem not to be available. Moreover, instability of the MOTS
prevents us from applying the general results of [3, 4].

Following [11] we adopt here the following strategy. It is known that any CMC
surface in $\mathbb{R}^{3}$ or $\mathbb{S}^{3}$ gives a solution to a certain
Lax pair; vice versa one can construct CMC surfaces as solutions to certain
Lax pairs using the Sym-Bobenko formula. Making use of this, the DPW method
[14] is a way of constructing CMC surfaces from holomorphic potentials on
Riemann surfaces by solving a certain Cauchy problem. To ensure that the
resulting CMC surface is well-defined one has to solve in addition the so
called monodromy problem. In this way the authors of [11] construct families
of CMC embeddings on $\mathbb{S}^{3}$ for every sufficiently high genus
${\mathfrak{g}}$; the latter restriction comes from the use of an implicit
function theorem argument near $t=0$, where
$t=\frac{1}{2{\mathfrak{g}}+2}{\cal K}^{-1/2}$ and the quantity ${\cal K}$
arises in the monodronomy construction, cf. Prop. 9 of [11].

As Thm. 3 below we just recall a combination of Thm. 1 and a part of Thm. 2 of
[11]; item 4. below also contains a slight extension of the aforementioned
results. Moreover, we change the parametrisation from $\varphi$ to
$\psi=\varphi-\frac{\pi}{4}$ which simplifies the presentation.

######  Theorem 3 (Thms. 1 and 2 of [11]).

Our setting is the 3-dim. unit sphere $\mathbb{S}^{3}$. For every
${\mathfrak{g}}\in\mathbb{N}$ sufficiently large, there exists a smooth family
of conformal CMC embeddings $f_{\psi}^{\mathfrak{g}}:{\cal
R}^{\mathfrak{g}}\rightarrow\mathbb{S}^{3}$ from a Riemann surface with genus
$g$ and parameter $\psi\in(-\frac{\pi}{4},\frac{\pi}{4})$ satisfying

  1. 1.

$f_{0}^{\mathfrak{g}}$ is the Lawson surface $\xi_{1,{\mathfrak{g}}}$ of genus
${\mathfrak{g}}$.

  2. 2.

For $\psi\rightarrow\pm\frac{\pi}{4}$ the embedding $f_{\psi}^{\mathfrak{g}}$
smoothly converges to a doubly covered geodesic 2-sphere with
$2{\mathfrak{g}}+2$ branch points, i.e. the family $f_{\psi}^{\mathfrak{g}}$
cannot be extended in the parameter $\psi$ in the space of immersions.

  3. 3.

$f_{\psi}^{\mathfrak{g}}=f_{-\psi}^{\mathfrak{g}}$ up to reparametrization and
(orientation reversing) isometries of $\mathbb{S}^{3}$, and accordingly the
constant mean curvatures satisfy
$H_{\psi}^{\mathfrak{g}}=-H_{-\psi}^{\mathfrak{g}}$.

  4. 4.

The mean curvature $H_{\psi}^{\mathfrak{g}}$ of $f_{\psi}^{\mathfrak{g}}$
decreases strictly monotonically from zero at $\psi=0$ to some minimal value
$H_{\psi_{m}}^{\mathfrak{g}}$ for $\psi=\psi_{m}$ from which it increases
strictly monotonically to zero for $\psi\rightarrow\frac{\pi}{4}$. The
monotonicity behaviour for $\psi\in(0,\frac{\pi}{4})$ then follows from
property 3.

  5. 5.

The Willmore energy Equ. (2) of $f_{\psi}^{\mathfrak{g}}$ increases strictly
monotonically from $\psi=0$ towards both $\psi\rightarrow\pm\frac{\pi}{4}$
where $W_{\pm\pi/4}=8\pi$.

###### Proof.

Items 1 - 3 and 5 follow as in the proofs of Thms. 1 and 2 of [11]; to show
item 4 we note that from Proposition 34 of [11], the mean curvature
$H^{\mathfrak{g}}_{\psi}=H(t({\mathfrak{g}}),\psi)$ with
$t({\mathfrak{g}})=\frac{1}{2{\mathfrak{g}}+2}{\cal K}^{-1/2}(t,\psi)$ is
smooth and its Taylor expansion in $t$ takes the form

$\displaystyle H(t,\psi)=4t\cos
2\psi\ln\left[\tan(\psi+\frac{\pi}{4})\right]+O(t^{2}).$ (37)

A calculation shows that at the values $\psi=\psi_{0}$ and $\psi=-\psi_{0}$,
$H_{0}(\psi)=(\partial H/\partial t)(0,\psi)$ takes on its unique non-
degenerate minimum and maximum, respectively, i.e.

$\displaystyle\frac{dH_{0}}{d\psi}(\pm\psi_{0})=0\qquad\frac{d^{2}H_{0}}{d\psi^{2}}(\pm\psi_{0})\neq
0,$ (38)

where $\pm\psi_{0}$ are given implicitly by

$\displaystyle(\sin
2\psi_{0})\ln\left[\tan\left(\psi_{0}+\frac{\pi}{4}\right)\right]=1.$ (39)

Moreover, there hold the monotonicity properties

$\displaystyle\frac{dH_{0}}{d\psi}$ $\displaystyle>$ $\displaystyle
0\qquad\forall~{}\psi\in\left[\left.-\frac{\pi}{4},-\psi_{0}\right)\right.~{}\mbox{and}~{}\psi\in\left.\left(\psi_{0},\frac{\pi}{4}\right.\right]$
(40) $\displaystyle\frac{dH_{0}}{d\psi}$ $\displaystyle<$ $\displaystyle
0\qquad\forall~{}\psi\in(-\psi_{0},\psi_{0})$ (41)

We now observe that there exists a $\psi_{m}(\psi,t)=\psi_{0}+O(t)$ such that,
for sufficiently small $t$, properties (38), (40) and (41) hold for
$H(t,\psi)$ as well, with $\psi_{0}$ replaced by $\psi_{m}$ everywhere. Away
from the zeros of $H$ which are at $\psi=0,\pm\frac{\pi}{4}$ (cf. Prop. 18 of
[11]) this follows from smoothness and from the non-degeneracy (38), while in
a neighbourhood of the zeros of $H$, the assertion holds by virtue of Prop. 34
of [11]. ∎

######  Remark 7.

The convergence properties of the CMC surfaces for
$\psi\rightarrow\pm\frac{\pi}{4}$ (point 2 of Theorem 3) are explained in
Prop. 26 of [11].

We are now ready to prove Thm. 2 stated in the Introduction.

###### Proof of Thm. 2.

We define the conformal rescaling ${\cal
I}_{\sigma}:\mathbb{S}^{3}\rightarrow\mathbb{S}^{3}_{\sigma}$ from the unit
sphere to the sphere of radius $\rho=\delta^{-1}\cos^{-1}\sigma$. This clearly
induces a scaling for all embedded surfaces; we note that the mean curvature
of such surfaces changes from $H$ to ${\cal H}_{\sigma}=\delta H\cos\sigma$.
We now define the embedding ${\cal F}_{\psi}^{\mathfrak{g}}:{\cal
R}^{\mathfrak{g}}\rightarrow\mathbb{S}^{3}_{\sigma}$ by ${\cal
F}_{\psi}^{\mathfrak{g}}={\cal I}_{\sigma(\psi)}\circ f_{\psi}^{\mathfrak{g}}$
where $\sigma(\psi)$ is constructed as follows. We recall from (31) that any
CMC surface with mean curvature ${\cal H}_{\sigma}$ on $\sigma=$ const.
corresponds to a MTS if ${\cal H}_{\sigma}=\mp 2\delta\sin\sigma$. In
particular, if the mean curvatures $H_{\psi}^{\mathfrak{g}}$ of the CMC
surfaces $f_{\psi}^{\mathfrak{g}}$ constructed in Thm. 3 satisfy

$\displaystyle H_{\psi}^{\mathfrak{g}}=\frac{{\cal
H}_{\sigma}}{\delta\cos\sigma}=\mp 2\tan\sigma$ (42)

they build up MTTs, which serves to define $\sigma({\psi})$.

In order to prove the final statement 5. on monotonicity of the area, we show
that the Willmore functional ${\cal W}_{\sigma}$ (2) built from the rescaled
quantities is just the area of the embedded MOTS. To see this we note that at
time $\sigma$, the rescaled area is ${\cal
A}_{\sigma}=A\delta^{-2}\cos^{-2}\sigma$ while the rescaled mean curvature is
${\cal H}_{\sigma}=\delta H\cos\sigma$. Inserting in (2) we obtain
$W_{\sigma}=\delta^{2}A_{\sigma}$ which finishes the proof. ∎

######  Remark 8.

It is stated below Thm. 1 in [11] that ”the moduli space of genus
${\mathfrak{g}}$ CMC surfaces is 1-dimensional at the Lawson surface
$\xi_{1,\mathfrak{g}}$”. This is not to be understood in the sense that the
MTTs constructed in Thm. 2 above are unique, as also indicated in point 3. of
that Theorem. In fact isometries of $\mathbb{S}^{3}$ which are not tangent to
${\cal F}_{\psi}^{\mathfrak{g}}$ (and which exist, cf. Remark 5) will “move
around” ${\cal F}_{\psi}^{\mathfrak{g}}$ at any parameter value $\psi$.

######  Remark 9.

We recall that there are no non-trivial isometries of ${\cal
F}_{\psi}^{\mathfrak{g}}$ in the case ${\mathfrak{g}}\geq 2$ in which the
Euler number $\chi=2(1-\mathfrak{g})$ is negative. Assuming the contrary, the
Poincaré-Hopf theorem implies that $\chi$ is also the sum of all indices of
the (isolated) zeros of the corresponding Killing field. But if an isometry in
2 dim. has fixed points, it is necessarily a rotation in a neighbourhood.
Hence all Killing indices are $+1$ which implies that $\chi$ is non-negative.
This restricts the topology of ${\cal F}_{\psi}^{\mathfrak{g}}$ to the sphere
or the torus if it has isometries.

######  Remark 10.

We note that the conformal rescaling ${\cal
I}_{\sigma}:\mathbb{S}^{3}\rightarrow\mathbb{S}^{3}_{\sigma}$ involved in
passing from Thm. 3 to Thm. 2 is precisely the inverse of the one used to
construct the Penrose diagrams Figs. 1 and 2 of dS. Hence a Penrose-type
diagram arises just by directly “stacking” the family of CMC surfaces
$f_{\psi}^{\mathfrak{g}}$ constructed in Thm. 3. We recall, however, that the
embedding parameter $\psi$ is not a monotonic function of the cosmological
time $\sigma$ \- rather, the constructed MOTT “turns around” at
$\psi=\pm\psi_{m}$.

##  4 Discussion

Note that monotonicity of area holds along MTTs with MTS sections of
spherical, toroidal and high-genus topology in the complete spherical slicing
of dS as described in Sect. 3.2.4. Except for spherical MTS whose area stays
constant, the area increase is even strictly monotonic upon moving away from
the time-symmetric surface. In view of the extensively discussed
correspondence between area and entropy, our MTTs are thus candidates for
satisfying the “second law” of thermodynamics in an appropriate setting.

Such settings arise in attempts of understanding the ”information paradox” of
black holes, and in attempts of making sense of the idea of AdS/CFT
correspondence (cf. e.g. [15] and references therein). In the latter context,
MTTs run under the name ”holographic screens”. In fact an area theorem has
been obtained in [8] (cf. Thm. IV.3) and [1] for so-called ”regular
holographic screens” whose definition consists of four somewhat subtle
conditions (cf. Def. II.8 of [8]). We have not been able to extract the
required information from our Thm. 2 to check these requirements; in
particular the causal character of the constructed MTT is unclear except that
they must be spacelike near the turning points $\pm\psi_{m}$ where the mean
curvatures achieve their extrema $\pm{\cal H}^{\mathfrak{g}}_{\psi_{m}}$ (cf.
item 4 of Theorem 2). In order to obtain monotonicity of area of the MTS (item
5 of Thm. 2) we are thus bound to the argument involving their rescaled
Willmore energy (item 5 of Thm. 3).

Acknowledgement. M.M. acknowledges financial support under Project
PID2021-122938NB-I00 (Spanish Ministerio de Ciencia e Innovación and FEDER “A
way of making Europe”).

This research was also funded in part by the Austrian Science Fund (FWF)
[Grant-DOI 10.55776/P35078] (W.S. & C.R.) and [Grant DOI 10.55776/P33594]
(R.S.). For open access purposes, the authors have applied a CC BY public
copyright license to any author accepted manuscript version arising from this
submission.

Conflict of Interest. On behalf of all authors, the corresponding author
states that there is no conflict of interest.

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