directly by the geometric procedure of moving S along ~ξ and then back to Σ are intuitively clear but typically weaker than those obtained by using elliptic theory results. There are some cases, however, where the reverse actually holds, and the geometric construction provides stronger results. We will present one of these cases in this subsection.
Corollary 3.4.3 gives restrictions on ξµlµ+|S for Killing vectors and homotheties
in spacetimes satisfying the NEC, provided ~ξ is future or past directed everywhere. However, when W vanishes identically, the result only gives useful information in the strictly stable case. The reason is that W ≡ 0 implies Lm~Q ≡ 0 and,
for marginally stable MOTS (i.e. when the principal eigenvalue of Lm~ vanishes),
the maximum principle is not strong enough to conclude that Q must have a sign. There is at least one case where marginally stable MOTS play an important role, namely after a jump in the outermost MOTS in a (3+1) foliation of the spacetime (see [1] for details). As we will see next, the geometric construction does give restrictions in this case even when W vanishes identically.
Theorem 3.4.8 Consider a spacetime (M, g(4)) possessing a Killing vector or a
homothety ~ξ and satisfying the NEC. Suppose M contains a compact spacelike
hypersurface ˜Σ with boundary consisting in the disjoint union of a weakly outer
trapped surface ∂−Σ and an outer untrapped surface ∂˜ +Σ (neither of which are˜
necessarily connected) and take ∂+Σ as a barrier with interior ˜˜ Σ. Without loss
of generality, assume that ˜Σ is defined locally by a level function T = 0 with T > 0 to the future of ˜Σ and let S be the outermost MOTS which is bounding
with respect to ∂+Σ. If ~˜ ξ(T ) ≤ 0 on some spacetime neighbourhood of S, then
ξµl+
µ ≤ 0 everywhere on S.
Remark 1. As usual, the theorem still holds if all the inequalities involving ~
ξ are reversed.
Remark 2. The simplest way to ensure that ~ξ(T ) ≤ 0 on some neighbour- hood of S is by imposing a condition merely on S, namely ξµnµ|S > 0, because
then ~ξ lies strictly below ˜Σ on S and this property is obviously preserved sufficiently near S (i.e. ~ξ points strictly below the level set of T on a sufficiently small spacetime neighbourhood of S). We prefer imposing directly the condition ~
76 3.4. Results provided Lm~Q has a sign on S
tangent to S.
Proof. First note that the hypersurface ˜Σ satisfies the assumptions of The- orem 2.2.30 which ensures that an outermost MOTS S which is bounding with respect to ∂+Σ does exist and, therefore, no weakly outer trapped surface can˜
penetrate in its exterior region. Then, the idea is precisely to use the geometric procedure described above to construct Sτ and use the fact that S is the outer-
most bounding MOTS to conclude that Sτ (τ > 0) cannot have points outside
S. Here we move S a small but finite amount τ , in contrast to the elliptic results before, which only involved infinitesimal displacements. We want to have infor- mation on the sign of the outer expansion of Sτ in order to make sure that a
weakly outer trapped surface forms. The first part of the displacement is along ~
ξ and gives S′
τ. Let us first see that all these surfaces are MOTS. For Killing
vectors, this follows at once from symmetry arguments. For homotheties we have the identity δ~ξθ+ = −12lα−L~ξl′+α(τ )− 2C θ+, (3.4.12)
which follows directly from (3.3.1) with ~η = ~l+ after using l+µLξ~l+µ′ (τ ) = 1
2aµν(~ξ )l µ
+lν+ = 0, see (3.3.8). Expression (3.4.12) holds for each one of the
surfaces {S′
τ}, independently of them being MOTS or not. Since this variation
vanishes on MOTS and the starting surface S has this property, it follows that each surface S′
τ (τ > 0) is also a MOTS. Moving back to ˜Σ along the null
hypersurface introduces, via the Raychaudhuri equation (2.2.13), a non-positive term NSW in the outer null expansion, provided the motion is to the future.
Hence, Sτ for small but finite τ > 0 is a weakly outer trapped surface provided
~
ξ moves to the past of ˜Σ. This is ensured if ~ξ(T )≤ 0 near S, because T cannot become positive for small enough τ . On the other hand, since a point p ∈ S moves initially along the vector field ν = ~ξ− NS~l+ = Q ~m + ~Yk, where Q = ξµlµ+
as usual, it follows that Q > 0 somewhere implies (for small enough τ ) that the bounding weakly outer trapped surface Sτ has a portion lying strictly to
the outside of S which, due to Theorem 2.2.30 by Andersson and Metzger, is a contradiction to S being the outermost bounding MOTS. Hence Q ≤ 0
everywhere and the theorem is proven.
It should be remarked that the assumption of ~ξ being a Killing vector or a homothety is important for this result. Trying to generalize it for instance to conformal Killings fails in general because then the right hand side of equation
(3.4.12) has an additional term 2~l+(φ), not proportional to θ+. This means that
moving a MOTS along a conformal Killing does not lead to another MOTS in general. The method can however, still give useful information if ~l+(φ) has the
appropriate sign, so that S′
τ is in fact weakly outer trapped. We omit the details.
An immediate consequence of the finite construction of Sτ is the extension of
point (ii) of Corollary 3.4.4 to locally outermost MOTS.
Theorem 3.4.9 Let (M, g(4)) be a spacetime satisfying NEC and admitting a
causal Killing vector or homothety ~ξ which is future (past) directed on a locally
outermost MOTS S⊂ Σ. Then ~ξ ∝ ~l+ everywhere on S.
Proof. As before, let Σ be defined locally by a level function T = 0 with T > 0 to the future of Σ. Assume that ~ξ is past directed (the future directed case is similar). Then, the assumption ~ξ(T ) ≤ 0 on some spacetime neighbourhood of S of Theorem 3.4.8 is automatically satisfied. Then we can use the finite construction therein to find a weakly outer trapped surface which, due to the fact that ~ξ is causal (and past directed), does not penetrate in the interior part of the two-sided neighbourhood of S. In fact, this new trapped surface will have points strictly outside S if on some point of S ~ξ6∝ ~l+ which proves the result.
Finally, Theorem 3.4.9 together with Theorem 2.2.30 lead to the following result.
Theorem 3.4.10 Consider a spacelike hypersurface (Σ, g, K) possibly with
boundary in a spacetime satisfying the NEC and possessing a Killing vector or a homothety ~ξ with squared norm ξµξµ = −λ. Assume that Σ possesses a barrier
Sb with interior Ωb which is outer untrapped with respect to the direction pointing
outside of Ωb.
Consider any surface S which is bounding with respect to Sb. Let us denote by
Ω the exterior of S in Ωb. If S is weakly outer trapped and Ω⊂ {λ > 0}, then λ
cannot be strictly positive on any point p∈ S.
Remark. When weakly outer trapped surface is replaced by the stronger condition of being a weakly trapped surface with non-vanishing mean curvature, then this theorem can be proven by a simple argument based on the first variation of area [80]. In that case, the assumption of S being bounding becomes unnecessary. It would be interesting to know if Theorem 3.4.10 holds for arbitrary
78 3.4. Results provided Lm~Q has a sign on S 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 PSfrag S Sb {λ > 0} Σ Ω
Figure 3.4: Theorem 3.4.10 excludes the possibility pictured in this figure, where S (in blue) is a weakly outer trapped surface which is bounding with respect to the outer trapped barrier Sb. The grey (both light and dark) regions represent
the region where λ > 0. The dark grey region represents the interior of Sb, while
the striped area corresponds to Ω, which is the exterior of S in Ωb.
Proof. We argue by contradiction. Suppose a weakly outer trapped surface S satisfying the assumptions of the theorem and with λ > 0 at some point. Theorem 2.2.30 implies that an outermost MOTS ∂topT+ which is bounding
with respect to Sb exists in the closure of the exterior Ω of S in Ωb. In particular,
∂topT+ is a locally outermost MOTS. The hypothesis Ω ⊂ {λ > 0} implies
that the vector ~ξ is causal everywhere on ∂topT+, either future or past directed.
Moreover, the fact that λ > 0 on some point of S implies that the Killing vector is timelike in some non-empty set of ∂topT+, which contradicts Theorem 3.4.9.
The following result is a particularization of Theorem 3.4.10 to the case when the hypersurface Σ possesses an asymptotically flat end.
Theorem 3.4.11 Let (Σ, g, K) be a spacelike hypersurface in a spacetime satis-
fying the NEC and possessing a Killing vector or homothety ~ξ. Suppose that Σ
possesses an asymptotically flat end Σ∞ 0 .
Consider any bounding surface S (see Definition 2.3.6). Let us denote by Ω the exterior of S in Σ. If S is weakly outer trapped and Ω ⊂ {λ > 0}, then λ
cannot be strictly positive on any point p∈ S.
Proof. The result is a direct consequence of Theorem 3.4.10. Two immediate corollaries follow.
Corollary 3.4.12 Consider a spacelike hypersurface (Σ, g, K) in a spacetime sat-
isfying the NEC and possessing a Killing vector or a homothety ~ξ. Assume that Σ has a selected asymptotically flat end Σ∞
0 and λ > 0 everywhere on Σ. Then
Corollary 3.4.13 Let (Σ, g, K) be a spacelike hypersurface of the Minkowski
spacetime. Then there exists no bounding weakly outer trapped surface in Σ.
The second Corollary is obviously a particular case of the first one because the vector ∂t in Minkowskian coordinates is strictly stationary everywhere, in
particular on Σ. The non-existence result of a bounding weakly outer trapped surface in a Cauchy surface of Minkowski spacetime is however, well-known as this spacetime is obviously regular predictable (see [63] for definition) and then the proof of Proposition 9.2.8 in [63] gives the result.
So far, all the results we have obtained require that the quantity LmQ does
not change sign on the MOTS S. In the next section we will relax this condition.