William Nelson
email: nelson@gravity.psu.edu

Lattice Refining LQC from an Isotropic Embedding of Anisotropic Cosmology

Keywords: numbers: 04.60.Kz, 04.60.Pp, 98.80.Qc

We demonstrate that it is possible to produce different isotropic embeddings of anisotropic Loop Quantum Cosmology, resulting in "lattice refinement" of the isotropic system. To introduce the general approach, we first use a simple model with only two anisotropic directions. We then employ the specific case of a Bianchi I model, to show how the method extends to three-dimensional systems. To concisely calculate the step-size of the resulting isotropic state, we define the "symmetric dual" of states and operators, for the two-and three-dimensional systems, respectively.

Introduction

Loop Quantum Gravity (LQG) quantises canonical General Relativity, in a background independent and non-perturbative manner. It does so by considering the threedimensional spatial slice to be a network of triads and then formulating the classical theory in terms of holonomies of the triad connections, around closed edges of the network, and fluxes of the triads, through the surfaces enclosed by these edges. Whilst the full, inhomogeneous theory has yet to be fully developed, significant progress has been made by applying the approach to symmetry reduced, mini-superspace models. In particular, by applying these principles of quantisation to Cosmology, one arrives at Loop Quantum Cosmology (LQC), which has proved itself especially successful [1].

Within LQC it has been possible, on the one hand, to explicitly show the avoidance of singularities that typically plague the classical versions of these cosmological theories [2,[START_REF] Bojowald | Chapter contributed to: "Beyond the Big Bang[END_REF] and, on the other hand, to develop powerful effective theories, that capture first order corrections to classical cosmology [START_REF] Bojowald | [END_REF]. In particular, it has been shown that, in order for the cosmological theory to be theoretically consistent [5,6], the physical volume (rather than the scale factor, or the triad component) must be the fundamental variable used. This conclusion has been backed up by various phenomenological [7,8] and theoretical [9] arguments. However, it is important to realise that since there is still lacking a complete derivation of LQC from the fundamental inhomogeneous and anisotropic theory, it is important to test LQC predictions for robustness (see e.g., Ref. [10]).

Various aspects of anisotropic cosmologies have been studied within LQC in the past [11,12,13], however the first full and consistent quantisation of a Bianchi I cosmology (the simplest of anisotropic cosmological models) was achieved in Ref. [14] ‡. Even more encouraging, the link back to the underlying full theory (LQG) has been strengthened, by considering the flux of the triads through surfaces consistent with the Bianchi I anisotropic case. With the quantisation of the Bianchi I model under control, it is possible to ask whether LQC features, obtained within the context of isotropic Friedmann-Lemaître-Roberston-Walker (FLRW) cosmology, are robust, at least with respect to this limited extension of the symmetries of the system.

In this paper, we will show that one can choose among different isotropic embeddings of the anisotropic Bianchi I model. We will then demonstrate that the choice of different embeddings has important consequences for the precise form of discretisation, produced by the loop quantisation procedure, in the isotropic sub-system. Note that, we use Bianchi I as an example of an anisotropic model. In other words, we do not imply that it is physically natural, nor that our results are constrained to hold only for this type of an anisotropic model.

In Section 2, the basics of LQC will be sketched, for both isotropic and anisotropic ‡ Performing a von Neumann stability analysis of the Hamiltonian constraint equation valid for anisotropic Bianchi I model, we have shown [15] that the the difference equation is unconditionally unstable. models. In Section 3, it will be demonstrated that an isotropic embedding of an anisotropic system can be realised in various ways, leading to consequences for the form of the induced isotropic system. This will be first demonstrated for a prototype two-dimensional system. In Section 4, we will define a "symmetric dual" of operators and states, to apply our formalism. In Section 5, the approach will be extended to the three-dimensional Bianchi I model and it will be shown that, with certain additional caveats, the same conclusions can be drawn. Finally, we will demonstrate that "lattice refinement" [8,16] can be motivated by alternative isotropic embeddings, while the difference from the standard quantisation of isotropic cosmology can be viewed simply as a different choice of basic observables.

Basics formalism of LQG/LQC

The full LQG theory is uniquely derived from the requirements that the theory is diffeomorphism invariant and satisfies the SU(2) gauge freedom. In both, the full theory and in LQC, the fundamental variables are the holonomies of the SU(2) connection, A i a , along a given edge, e, and the corresponding momentum, which turns out to be the flux of the triad through a two-surface S. The holonomies are given by

h e (A) = P exp e γµ (s)A i µ (γ(s)) τ i ds , (1)
where P infers path ordering on the exponential, γ µ is the tangent vector along the edge e, and τ i are the basis of the SU(2) Lie algebra. The corresponding momentum variable is

E(S, f ) = S abc E ci f i dx a dx b , (2)
where f i is an SU(2) valued test function and E a i is the densitised triad; i, j, k,

• • • are SU(2) indices, whilst a, b, c, • • • are coordinate indices.
Restricting to isotropic and homogeneous cosmologies allows us to consider only straight edges along integral curves of the basis vectors, X a i , that produce the network. Then the connection is given by a (dynamic) multiple of the basis one forms, ω i a , namely

A i a = c(t)ω i a , whilst the triad is E a i = 0 g p(t)X a i
, where 0 g is the determinant of the fiducial metric § and c stands for the connection component. This is simply a consequence of the symmetries imposed, since isotropy ensures that there can be no angular dependence of the connection or triads, whilst homogeneity ensures they are the same at every spatial point (but not necessarily on different time slices).

With these symmetries, the holonomies become simply

h i (A) = exp -iµ 0 σ i 2 c , = cos µ 0 c 2 + 2τ i sin µ 0 c 2 , (3)
§ The fiducial metric is a complication that arises only for an open universe and is used to define the volume to which spatial integrals are restricted to ensure they remain finite. Physical results should not depend on this volume [17].

where τ i , the basis of the SU(2) Lie algebra, are related to the Pauli matrices, σ i , by τ i = -iσ i /2 and µ 0 is the orientated length of the edge with respect to the fiducial metric. It is this µ 0 that is the ambiguity coming from the fact that we have imposed homogeneity on our system. In a full theory, the length of the edges would have some spectrum of values, in which case µ 0 would be the minimum such length, which has been shown to be non-zero [18]. Here we have treated µ 0 as a constant, however for consistent quantisation [6,9,7], it has been shown that it must vary as µ 0 ∼ µ -1/2 . In order to include this varying parameter, one must change variables from the triad component to the volume [5], however here we will briefly sketch the derivation for the constant µ 0 case, with more details given in Ref. [5].

By analogue with the full theory, the kinematic Hilbert space is extended via the Bohr compactification of the real line [18]. An orthonormal basis for this Hilbert space is {|µ }, where

c|µ = e i µc 2 . ( 4
)
Note that c is a re-definition of the connection component, so that the Poisson bracket of the connection component and the (re-defined) triad component is independent of the volume of the fiducial cell, namely {c, p} = κγ/3, with κ = 8πG and γ the Barbero-Immirzi parameter, fixed through a black hole entropy calculation and turns out to be γ ≈ 0.2375. The triad operator acts on these basis states as p|µ = -i κγ 3

∂ ∂c |µ = κγ 6 |µ||µ . (5)
Clearly, also the volume operator, V ≡ â3 = |p| 3/2 , has eigenstates in this basis, namely

V |µ = V µ |µ = κγ |µ| 6 3/2 |µ . (6)
To calculate the eigenvalues of the inverse volume operator, the classical expression [18],

p -1 = p L-1 1/(1-L) = 3 κγL {c, p L } 1/(1-L) , (7)
is used. This is a classical identity independent of L, but will be quantised to different operators for different L. Thus, L plays the rôle of a quantisation ambiguity.

It is now possible to quantise the classical Hamiltonian constraint, where the curvature is approximated via holonomies around a closed curve (which can not be shrunk to zero, due to the "area gap", given in terms of µ 0 ; it occurs by analogue with LQG), which gives Ĉgrav = 2i κ 2 γ 3 µ 3 0 tr i,j,k ijk ĥ(µ 0 ) i ĥ(µ 0 ) j ĥ(µ 0 )-1 i ĥ(µ 0 )-1 j ĥ(µ 0 )-1 k , V sgn (p) , (8) where, as in all quantum theories, there is an ambiguity in the factor ordering [9]; other quantum ambiguities have been fixed. Using Eq. (3), one can thus obtain a difference equation [17],

V µ+5µ 0 -V µ+3µ 0 + V µ+µ 0 -V µ-µ 0 Ψ µ+4µ 0 (φ) -4 V µ+µ 0 V µ-µ 0 Ψ µ (φ) + V µ-3µ 0 -V µ-5µ 0 + V µ+µ 0 -V µ-µ 0 Ψ µ-4µ 0 (φ) = - 4κ 2 γ 3 µ 3 0 3 H φ (µ)Ψ µ (φ) , (9)
of step 4µ 0 , for the wave-function coefficients Ψ (µ), defined by

|Ψ = µ Ψ (µ) |µ . ( 10
)
Note that, the matter Hamiltonian Ĥφ is assumed to act diagonally on the basis states with eigenvalue H φ (µ). By adapting the network to the symmetries of a Bianchi I model, one can similarly derive the quantum Hamiltonian for this anisotropic system [14]. In this case, we have three triad components (p 1 , p 2 , p 3 ), along the three directions of the anisotropic model. The gravitational sector of the kinematic Hilbert space consists of wave-functions Ψ(p 1 , p 2 , p 3 ), satisfying

Ψ(p 1 , p 2 , p 3 ) = Ψ(|p 1 |, |p 2 |, |p 3 |) . (11)
Neglecting the details of the derivation of the Hamiltonian constraint, which are of no relevance for our present study, we draw the attention of the reader to the fact that in both, the isotropic and the Bianchi I case, the holonomies act on a state as shift vectors.

Let us define three dimensionless variables, λ i , as

λ i = sgn(p i ) |p i | (4π|γ| √ ∆l 3 Pl ) 1/3 with i = 1, 2, 3 ; (12)
the quantum of area ∆l 2 Pl denotes the physical geometry, with ∆ = 4πγ √ 3. The Hamiltonian constraint for the wave-function Ψ(λ 1 , λ 2 , ν), where ν = 2λ 1 λ 2 λ 3 , is a difference equation. It reads [14]

∂ 2 φ Ψ(λ 1 , λ 2 , ν; φ) = πG 2 √ ν (ν + 2) √ ν + 4 Ψ + 4 (λ 1 , λ 2 , ν; φ) -(ν + 2) √ ν Ψ + 0 (λ 1 , λ 2 , ν; φ) -(ν -2) √ ν Ψ - 0 (λ 1 , λ 2 , ν; φ) + (ν -2) |ν -4| Ψ - 4 (λ 1 , λ 2 , ν; φ) , (13)
where Ψ ± 0,4 are defined as [14]

Ψ ± 0 (λ 1 , λ 2 , ν; φ) = Ψ ν ± 2 ν • λ 1 , ν ν ± 2 • λ 2 , ν; φ + Ψ ν ± 2 ν • λ 1 , λ 2 , ν; φ +Ψ ν ν ± 2 • λ 1 , ν ± 2 ν • λ 2 , ν; φ + Ψ ν ν ± 2 • λ 1 , λ 2 , ν; φ +Ψ λ 1 , ν ν ± 2 • λ 2 , ν; φ + Ψ λ 1 , ν ± 2 ν • λ 2 , ν; φ , (14)
and

Ψ ± 4 (λ 1 , λ 2 , ν; φ) = Ψ ν ± 4 ν ± 2 • λ 1 , ν ± 2 ν • λ 2 , ν ± 4; φ + Ψ ν ± 4 ν ± 2 • λ 1 , λ 2 , ν ± 4; φ +Ψ ν ± 2 ν • λ 1 , ν ± 4 ν ± 2 • λ 2 , ν ± 4; φ + Ψ ν ± 2 ν • λ 1 , λ 2 , ν ± 4; φ +Ψ λ 1 , ν ± 2 ν • λ 2 , ν ± 4; φ + Ψ λ 1 , ν ± 4 ν ± 2 • λ 2 , ν ± 4; φ , (15)
respectively. Note that, φ plays the rôle of internal time, exactly as in the isotropic case. Thus, as one can easily check from Eq. ( 13), as far as the ν dependence is concerned, the steps are uniform, since the argument of the wave-function involves ν -4, ν, ν + 4, exactly as in the isotropic case. However, the dependence on λ 1 , λ 2 is quite complicated, technically, to deal with. Writing the step-size in terms of the triad components, implies

Ψ iso (p) → Ψ iso p ± K |p| and Ψ (p 1 , p 2 , p 3 ) →          Ψ p 1 ± K | p 1 p 2 p 3 |, p 2 , p 3 Ψ p 1 , p 2 ± K | p 2 p 1 p 3 |, p 3 Ψ p 1 , p 2 , p 3 ± K | p 3 p 1 p 2 | , (16)
for some suitable constants K and K that are proportional to √ ∆l 3 Pl [14]. Note that, typically the K constants, appearing in the argument of the anisotropic wave-function, will be different.

In both, the isotropic and anisotropic cases, "consistency" is derived entirely within the model, i.e., the symmetries of the model are fixed from the outset and "consistency" is looked for, under these assumptions. This, of course, does not ensure that the model remains "consistent", if it derived from a less symmetric approximation. In particular, the isotropic case is consistently quantised for the step-size of the holonomies given in Eq. ( 16), only if one does not allow additional degree of freedom (and similarly for the Bianchi I model), however this does not ensure that an isotropic embedding of the Bianchi I model will have the same step-size as the isotropic case, given in Eq. ( 16). However, with both models fully quantised, it is possible to consider such an isotropic embedding of the anisotropic Bianchi I model and explicitly evaluate the type of step-size that is induced on the effective isotropic system.

This will be done in some detail below, but one can immediately see that setting p 1 = p 2 = p 3 = p, will produce the expected form of step-size, however this embedding procedure does not lead to a viable isotropic theory, instead one has to use a projection of the anisotropic degrees of freedom [14]. The reason for this is the following: for p 1 = p 2 = p 3 = p, we have λ 1 = λ 2 = λ 3 = (ν/2) 1/3 . Thus, the state which satisfies

p 1 = p 2 = p 3 = p is Ψ ν 2 1/3 , ν 2 1/3 , ν .
From Eq. ( 13), we see that applying the Hamiltonian to such a state involves states of the form Ψ λ1 , λ2 , ν , which do not satisfy λ1 = λ2 = λ3 = (ν/2) 1/3 . Thus, states for which p 1 = p 2 = p 3 = p, do not form a super-selection of states (i.e., the Hamiltonian does not preserve this property of states). The purpose of this paper is to show that it is, in fact, possible to find a superselection of states that are all (measured to be) isotropic. In particular, we will show that there are embeddings for which the expectation values of the variables p 1 , p 2 and p 3 are equal, and that are preserved under the action of the Hamiltonian. The Hamiltonian applied to such states does not produce the standard FLRW Hamiltonian of LQC, rather one finds a Hamiltonian constraint in which the discreteness length varies. Thus, one may view this as a form of lattice refinement on the isotropic sub-space. It is important to note however that the typical forms of lattice refining Hamiltonian used in the literature are not the same as those produced here; there is instead a simplified attempt to capture the missing underlying degrees of freedom (in this case the underlying anisotropic degrees of freedom).

The two-dimensional case

To demonstrate the approach, let us first consider an anisotropic wave-function that depends only on two directions, labelled 1, 2. The basic operators that are quantised are pi and ĥi , with i = 1, 2. Note that p i = a 2 i , while h i denote the holonomies. In general, the wave-function is anisotropic, namely

|Ψ (p 1 , p 2 ) = |Ψ (p 2 , p 1 ) . (17)
To be clear, the action of the operators pi is

p1 |Ψ (p 1 , p 2 ) = p 1 |Ψ (p 1 , p 2 ) p2 |Ψ (p 1 , p 2 ) = p 2 |Ψ (p 1 , p 2 ) ( 18
)
and

p1 |Ψ (p 2 , p 1 ) = p 2 |Ψ (p 2 , p 1 ) p2 |Ψ (p 2 , p 1 ) = p 1 |Ψ (p 2 , p 1 ) , (19)
and the action of the holonomies ĥi on a states |Ψ (p 1 , p 2 ) and |Ψ (p 2 , p 1 ) are,

ĥ1 |Ψ (p 1 , p 2 ) = |Ψ (p 1 + δ 1 (p 1 , p 2 ) , p 2 ) and ĥ2 |Ψ (p 1 , p 2 ) = |Ψ (p 1 , p 2 + δ 2 (p 1 , p 2 )) , (20)
where δ i is potentially a function of both p 1 and p 2 .

Standard isotropic embedding

Typically, one can go to the isotropic case by defining the operator Qsym , as

Qsym |Ψ (p 1 , p 2 ) ≡ |Ψ (p 1 , p 1 ) = |Ψ (p 2 , p 2 ) = |Ψ (p, p) . (22)
In other words, the operator Qsym sets p 1 = p 2 . It is easy to see that operating singly with either ĥ1 , or ĥ2 , on |Ψ (p 1 , p 2 ) , before operating with Qsym , implies δ i = 0, e.g.,

Qsym ĥ1 |Ψ (p 1 , p 2 ) = Qiso |Ψ (p 1 + δ 1 , p 2 ) = |Ψ (p 1 + δ 1 , p 1 + δ 1 ) = |Ψ (p 2 , p 2 ) ⇒ p 1 + δ 1 = p 2 , (23)
at least for singly defined wave-functions. Thus, the operation is not conserved under the action of a single holonomy, unless δ 1 = 0. This can be understood easily, since a general anisotropic state can be always made isotropic by setting p 1 = p 2 , but if we change either p 1 or p 2 , then the condition required to now make them equal will, of course, change.

If we consider only isotropic applications of the holonomies, e.g., ĥ1 ĥ2 , then the operation can be conserved under the action of Qsym , provided δ 1 = δ 2 . To be explicit,

Qsym ĥ1 ĥ2 |Ψ (p 1 , p 2 ) = Qsym |Ψ (p 1 + δ 1 , p 2 + δ 2 ) = |Ψ (p 1 + δ 1 , p 1 + δ 1 ) = |Ψ (p 2 + δ 2 , p 2 + δ 2 ) , ⇒ p 1 + δ 1 = p 2 + δ 2 , (24)
which preserves the condition p 1 = p 2 iff δ 1 = δ 2 , for singly defined wave-functions. So, we found that the value of p in either direction is the same and it changes in the same way, namely it is isotropic. Notice that, for consistency, this requires that we only consider isotropic applications of the holonomy operators (e.g., the Hamiltonian), which is physically acceptable since if our system is to end up isotropic, then it must evolve isotropically. However, in the Bianchi Hamiltonian (Eq. ( 13)) the holonomies act via multiplication, rather than addition, of the variables thus, Qsym ĥ1 ĥ2

|Ψ (p 1 , p 2 ) = Qsym |Ψ (δ 1 p 1 , δ 2 p 2 ) = |Ψ (δ 1 p 1 , δ 1 p 1 ) = |Ψ (δ 2 p 2 , δ 2 p 2 ) , ⇒ δ 1 p 1 = δ 2 p 2 , (25)
which again preserves the condition p 1 = p 2 iff δ 1 = δ 2 , (again for singly defined wavefunctions). The full Bianchi Hamiltonian (Eq. ( 13)) contains linear combinations of operations of the form of Eq. (25) that do not satisfy δ 1 = δ 2 . Thus, the condition p 1 = p 2 is not preserved. In order for the standard "improved dynamics" of the isotropic model to be recovered, one must project out the anisotropic degrees of freedom [14].

Alternative isotropic embedding

There is also another approach one could follow to forming an isotropic system from the fully (two-dimensional) anisotropic state |Ψ (p 1 , p 2 ) . Let us define the symmetrisation operator by

Qsym |Ψ (p 1 , p 2 ) = 1 2 |Ψ (p 1 , p 2 ) + |Ψ (p 2 , p 1 ) . (26)
Thus, whilst (for example) p1 Qsym |Ψ (p 1 , p 2 ) does not appear to be isotropic, the physically expectation value of p1 between two such states is. To be clear,

p1 Qsym |Ψ (p 1 , p 2 ) = 1 2 p 1 |Ψ (p 1 , p 2 ) + p 2 |Ψ (p 2 , p 1 ) , (27)
is not symmetric, however the inner product,

Ψ (p 1 , p 2 ) Qsym |p 1 | Qsym Ψ (p 1 , p 2 ) = 1 4 p 1 + p 1 Ψ (p 1 , p 2 ) |Ψ (p 2 , p 1 ) + p 2 Ψ (p 2 , p 1 ) |Ψ (p 1 , p 2 ) + p 2 = (p 1 + p 2 ) 4 [1 + Ψ (p 1 , p 2 ) |Ψ (p 2 , p 1 ) ] = Ψ (p 1 , p 2 ) Qsym |p 2 | Qsym Ψ (p 1 , p 2 ) , (28)
is clearly symmetric. Note that, we used X|Y = Y |X and the notation

Ψ (p 1 , p 2 ) Qsym | ≡ Qsym |Ψ (p 1 , p 2 ) † . (29)
Thus, the measured scale factor along either direction is the same.

We can now consider what the action of an holonomy is and, in particular, whether it is conserved under the symmetrisation operator. Consider, for example, the operation of ĥ1 :

Qsym ĥ1 |Ψ (p 1 , p 2 ) = 1 2 |Ψ (p 1 + δ 1 , p 2 ) + |Ψ (p 2 , p 1 + δ 1 ) . ( 30
)
We can again check that this state is symmetric, in that the expectation values of the two p i 's are the same, namely

Ψ (p 1 , p 2 ) ĥ1 Qsym |p 1 | Qsym ĥ1 Ψ (p 1 , p 2 ) = p 1 + p 2 + δ 1 4 1 + Ψ (p 1 + δ 1 , p 2 ) |Ψ (p 2 , p 1 + δ 1 ) , = Ψ (p 1 , p 2 ) ĥ1 Qsym |p 2 | Qsym ĥ1 Ψ (p 1 , p 2 ) . (31)
The above conclusion is however only valid, if the symmetrisation operator is always the last one to be taken on the state. This has some undesirable features, in particular,

Ô Qsym |Ψ = Qsym Ô|Ψ , (32)
namely that the symmetrisation operation does not commute with other operators, which is problematic, since the operation of a generic operator ( Ô) on a symmetric state, does not, on its own, lead to a symmetric state. In the next section, we will show how that this problem is avoided provided we are interested only in "symmetric operators" . In order to define such operators, it is useful to define a process we refer to as the "symmetric dual".

The "symmetric dual"

Here, we will define the "symmetric dual" of operators and states. Note that, the symmetrised version of an operator must be used even if p 1 = p 2 (we refer the reader to the previous section), since one has to ensure that only symmetric products of the holonomies are taken.

Define the symmetrised version of a state, |Ψ sym , as

|Ψ sym ≡ 1 A [|Ψ + |Ψ S ] , (33)
= 1 A [|Ψ (p 1 , p 2 ) + |Ψ (p 2 , p 1 ) ] , (34)
where by |Ψ S we denote the symmetric "dual" of a state |Ψ (p 1 , p 2 ) . Similarly, define the symmetrised version of an operator, Ôsym , as

Ôsym ≡ 1 B Ô + ÔS , (35)
where the operation of the "dual" operator is defined by requiring,

Ô|Ψ S = ÔS |Ψ S , (36)
and noting that |Ψ S S = |Ψ , by definition. We thus get

Ôsym |Ψ sym = 1 AB Ô|Ψ + Ô|Ψ S + ÔS |Ψ + ÔS |Ψ S , (37)
= 1 A | Ψ + | Ψ S , (38)
where

| Ψ = 1 B Ô|Ψ + ÔS |Ψ . (39)
Thus, we find that the operation of a symmetric operator on a symmetric state is (as expected) a symmetric state. This can be visualised on the two-dimensional (p 1 , p 2 ) plane. The case of p 1 = p 2 is the diagonal line, and if we restrict ourselves to states that lie on this line, then we must also restrict to operators that act diagonally, as one can see in Fig. 1. However, if we consider a symmetric superposition of states, then in order for the operation of a general operator Ô to give another symmetric state, one should require the operation to be symmetric. One easy way of doing this, is to define the dual operator, ÔS , and act with the symmetric combination Ôsym ∼ ( Ô + ÔS ). Note that, this procedure is by no means unique, however such symmetric operators maintain the symmetric subspace made up of states |Ψ sym , i.e., if we restrict ourselves to considering only such operators, then we have a super-selection of states |Ψ sym . In Fig. 1, we have taken the operator Ô to act on a state by shifting (p 1 , p 2 ). This is, of course, motivated by the fact that we want to consider holonomies, which we will assume act as shift operators, as in Eq. (20). Consider the restrictions placed on δ 1 (p 1 , p 2 ) and δ 2 (p 1 , p 2 ), by requiring that

ĥ1 ĥ2 = Ô = ÔS , (40)
i.e., that the action of a holonomy along each direction, is a symmetric operation in the sense defined above. As we will see, does not mean that the state is "shifted" diagonally. We have

ĥ1 ĥ2 |Ψ = ĥ1 ĥ2 |Ψ (p 1 , p 2 ) = |Ψ (p 1 + δ 1 (p 1 , p 2 ) , p 2 + δ 2 (p 1 , p 2 )) and ĥ1 ĥ2 |Ψ S = ĥ1 ĥ2 |Ψ (p 2 , p 1 ) = |Ψ (p 2 + δ 1 (p 2 , p 1 ) , p 1 + δ 2 (p 2 , p 1 )) .
Thus, to satisfy

Ô|Ψ S = ÔS |Ψ S = Ô|Ψ S , (41)
one should require that

|Ψ (p 2 + δ 2 (p 1 , p 2 ) , p 1 + δ 1 (p 1 , p 2 )) = |Ψ (p 2 + δ 1 (p 2 , p 1 ) , p 1 + δ 2 (p 2 , p 1 )) , (42)
which implies

δ 2 (p 1 , p 2 ) = δ 1 (p 2 , p 1 ) . (43)
For illustration, let us consider some simple examples. For δ 1 = δ 2 = const., the pair of holonomies acts diagonally, with a constant step. A more interesting example is

δ 1 = 1 √ p 1 = 1 √ p 2 = δ 2 ,
which is valid only when p 1 = p 2 , namely on the diagonal line. This is the analogue of the "new" quantisation (or "improved" quantisation) approach. Motivated by the three-dimensional anisotropic results of Ref. [14], one can take

δ 1 (p 1 , p 2 ) = p 1 p 2 , δ 2 (p 1 , p 2 ) = p 2 p 1 .
Clearly there are many other cases one may consider. The question then is, what would be the measured step-size between states separated by this symmetric operator. We have

sym Ψ (p 1 , p 2 ) | p1 |Ψ (p 1 , p 2 ) sym = sym Ψ (p 1 , p 2 ) | p2 |Ψ (p 1 , p 2 ) sym = p 1 + p 2 A 2 1 + Ψ (p 1 , p 2 ) |Ψ (p 2 , p 1 ) = p 1 + p 2 A 2 1 + Ψ|Ψ S = p 1 + p 2 , (44)
where in the last line, we used the fact that the symmetrised state should be normalised, and we have chosen

Ψ|Ψ S = Ψ (p 1 , p 2 ) |Ψ (p 2 , p 1 ) = A 2 -1 . (45)
Defining,

|ψ (p 1 , p 2 ) sym ≡ ĥ1 ĥ2 |Ψ (p 1 , p 2 ) sym , (46)
and taking the inner product, we find

sym ψ (p 1 , p 2 ) |p 1 |ψ (p 1 , p 2 ) sym = sym ψ (p 1 , p 2 ) |p 2 |ψ (p 1 , p 2 ) sym = p 1 + p 2 + δ 1 (p 1 , p 2 ) + δ 1 (p 2 , p 1 ) A 2 1 + Ψ ( p1 , p2 ) Ψ ( p2 , p1 ) , (47) where
( p1 , p2 ) = (p 1 + δ 1 (p 1 , p 2 ) , p 2 + δ 2 (p 1 , p 2 ))
and ( p2 , p1 ) = (p 2 + δ 1 (p 2 , p 1 ) , p 1 + δ 2 (p 2 , p 1 )) .

We can then evaluate what the measured step-size, ∆ sym , would be:

∆ sym ≡ sym ψ (p 1 , p 2 ) |p i |ψ (p 1 , p 2 ) sym -sym Ψ (p 1 , p 2 ) |p i |Ψ (p 1 , p 2 ) sym = (p 1 + p 2 ) (X -1) + (δ 1 (p 1 , p 2 ) + δ 2 (p 1 , p 2 )) X , ( 48
)
where X is defined as

X = 1 + Ψ (p 1 , p2 ) | Ψ (p 2 , p1 ) 1 + Ψ (p 1 , p 2 ) | Ψ (p 2 , p 1 ) . (49)
Thus, we have

X = 1 + χ| χ S 1 + Ψ| Ψ S , (50)
and we used | χ ≡ | Ψ ( p1 , p2 ) to emphasise the possibility of a difference from | Ψ (p 1 , p 2 ) . The normalisation of the symmetrised state gives the denominator (see, Eq. ( 45)), but we get X = 1 iff the norm of every state (or every state in the superselection, if one is possible) is the same. In particular, if we were to restrict ourselves to p 1 = p 2 ≡ p, this would imply that p1 = p2 and hence that

| χ = | χ S and | Ψ = | Ψ S .
This gives X = 1, which means ∆ = δ 1 (p 1 , p 2 ) + δ 1 (p 2 , p 1 ) = 2δ (p). However, for general ("symmetric dual") states, X can be greater, or less, than unity. Thus, the step-size would include a different dependence on p 1 and p 2 ; it can be seen as a "lattice refinement" analogue.

In Section 5, we will extend to three directions and explicitly use the Bianchi I model of Ref. [14] to evaluate the step-sizes for the two different symmetrisation schemes. Indeed, the above can be used directly for the case that two of the directions are set equal to each other. For example, if the restriction p 1 = p 2 = p 3 is made, then we are restricted to a sub-class of Bianchi I models, which are essentially two-dimensional. Such cylindrically symmetric models would be useful for investigating black holes, where the expectation value of p is different along the radial and angular directions.

Intuitively, what we have is that the use of superpositions to symmetrise a state, means that (for symmetric operators, such as pairs of holonomies with the correct type of step) we can find symmetric states which however have different step-sizes than in the standard case. Essentially, the standard case is just a special case of the above procedure, when the two states used in the superposition are the same.

To be clear, we have chosen to normalise the states, such that

Ψ (p 1 , p 2 ) |Ψ (p 2 , p 1 ) = A 2 -1 . (51)
However, the restriction δ 1 (p 1 , p 2 ) = δ 2 (p 2 , p 1 ), is not sufficient to simultaneously specify the normalisation of

Ψ [p 1 + δ 1 (p 1 , p 2 ) , p 2 + δ 2 (p 1 , p 2 )] |Ψ [p 2 + δ 1 (p 2 , p 1 ) , p 1 + δ 2 (p 2 , p 1 )] .
For

p 1 = p 2 , ∆ sym = δ 1 (p 1 , p 2 ) + δ 2 (p 1 , p 2 ) = 2δ (p) , (52)
where the last equality follows from the fact that δ 1 (p 1 , p 2 ) = δ 2 (p 2 , p 1 ) ≡ δ (p). For any other choice of (p 1 , p 2 ), such that p 1 = p 2 , this is not the case. Whilst we cannot rule out the possibility that the (symmetric) state evolves towards a state made up of a superposition of two even more anisotropic states, this seems unlikely.

If we take that, at least for large p 1 + p 2 , we have |Ψ (p 1 , p 2 ) → |Ψ (p, p) , then we expect X 1 (i.e., the inner product of the "symmetric dual" states tends to 1 from above; it can be understood by thinking of the states as vectors in a vector space). Thus,

∆ sym ≥ δ 1 (p 1 , p 2 ) + δ 2 (p 1 , p 2 ) , (53)
whilst tending to equality as p 1 and p 2 become large and tend to each other.

For the case of an isotropic system, made up of ever more anisotropic states, the converse is true, namely,

∆ sym ≤ δ 1 (p 1 , p 2 ) + δ 2 (p 1 , p 2 ) , (54)
whilst not necessarily tending to equality as p 1 and p 2 become large. Note that here we also require Eq. ( 43) to hold, which is what ensures that the measured step-size, ∆ sym , is indeed symmetric under interchange of p 1 , p 2 .

In conclusion, in general the step-size can be either larger or smaller (and has a different dependence on (p 1 , p 2 )) than in the p 1 = p 2 case. For the case of a state made up of a superposition of ever more isotropic states, the measured step-size tends to the standard step-size as p 1 , p 2 become large. This is precisely what is modelled by "lattice refinement" , albeit in an heuristic way, except that we might expect the "lattice refinement" to approach the special case of the "new" quantisation refinement rate, for large scales.

The three-dimensional case

In this section, we demonstrate a concrete example of the above procedure for the case of an anisotropic Bianchi I model, recently consistently derived in Ref. [14]. In this case the states are defined on the orthonormal basis |λ 1 , λ 2 , λ 3 , where λ i are essentially the scale factors along the three directions of the Bianchi model.

In the standard approach, isotropic states are taken to be those in which the dependence on λ 1 ,λ 2 and λ 3 is "integrated out", or more precisely, defining the volume of the state as ν = 2λ 1 λ 2 λ 3 to eliminate one of the directions (λ 3 ), the map

|Ψ (λ 1 , λ 2 , ν) → λ 1 ,λ 2 Ψ (λ 1 , λ 2 , ν) ≡ |Ψ (ν) , (55)
produces isotropic states. This projection produces exactly isotropic system with the equations given in terms of the volume (called the "ν quantisation"). Here we will show that it is also possible to have a super-selection of states that are isotropic and which evolve according to an alternative quantisation procedure which may be modelled by lattice refinement. We are going to work with the three scale factors λ i , to demonstrate that there is an ambiguity in exactly what the measured volume of such isotropic states

Here and throughout we use "lattice refinement" to refer to any quantisation in which the lattice defined with respect to µ is dynamically varying. In particular, then, "new" quantisation is simply a special case of such "lattice refinement". would be. Firstly, we want to find a symmetric superposition of states that is isotropic, in the sense defined in Section 3.2.

Consider the state

| Ψ (λ 1 , λ 2 , λ 3 ) ≡ 1 A |Ψ (λ 1 , λ 2 , λ 3 ) +|Ψ (λ 3 , λ 1 , λ 2 ) +|Ψ (λ 2 , λ 3 , λ 1 ) ,( 56
)
with the additional restriction

Ψ (λ 1 , λ 2 , λ 3 ) |Ψ (λ 3 , λ 1 , λ 2 ) = Ψ (λ 1 , λ 2 , λ 3 ) |Ψ (λ 2 , λ 3 , λ 1 ) = Ψ (λ 3 , λ 1 , λ 2 ) |Ψ (λ 2 , λ 3 , λ 1 ) ≡ f , (57)
on the anisotropic states being used. This additional requirement is not present in the two-dimensional case, since there is only one cross-correlator between two anisotropic states, which is trivially equal to itself. In the three-dimensional system however, discussed here, we have three cross-correlators, which need to be equal if the state is to be isotropic. This is a restriction that would be present in any dimension higher than 2.

The expectation values of the scale factors along each direction of such a state are λi =

λ 1 + λ 2 + λ 3 3 . (58)
Note that we have chosen A to normalise the state. It is worth mentioning that, by choosing a different state, the conditions required for the expectation values to match change. In particular, for the state

|χ (λ 1 , λ 2 , λ 3 ) ≡ 1 Ã Ψ (λ 3 , λ 1 , λ 2 ) |Ψ (λ 2 , λ 3 , λ 1 ) |Ψ (λ 1 , λ 2 , λ 3 ) + Ψ (λ 1 , λ 2 , λ 3 ) |Ψ (λ 3 , λ 1 , λ 2 ) |Ψ (λ 2 , λ 3 , λ 1 ) + Ψ (λ 1 , λ 2 , λ 3 ) |Ψ (λ 2 , λ 3 , λ 1 ) |Ψ (λ 3 , λ 1 , λ 2 ) , (59)
to be isotropic, we require

| Ψ (λ 1 , λ 2 , λ 3 ) |Ψ (λ 3 , λ 1 , λ 2 ) | 2 = | Ψ (λ 1 , λ 2 , λ 3 ) |Ψ (λ 2 , λ 3 , λ 1 ) | 2 = | Ψ (λ 3 , λ 1 , λ 2 ) |Ψ (λ 2 , λ 3 , λ 1 ) | 2 . ( 60
)
With such a state, one also finds that λi = (λ 1 + λ 2 + λ 3 ) /3, which explicitly demonstrates the non-uniqueness of the choice of isotropic embedding.

For both these states, the measured scale factor is equal in each direction and is given by the average of the scale factors of the underlying, anisotropic states. However, the measured volume of such a state, found using the volume operator,

ν| Ψ (λ 1 , λ 2 , λ 3 ) = 2λ 1 λ 2 λ 3 | Ψ (λ 1 , λ 2 , λ 3 ) , (61)
is just ν = 2λ 1 λ 2 λ 3 , which is not the cube of the measured scale factor. Thus, whilst it is the eigenvalue of the anisotropically defined volume operator, it is not necessarily what we would measure as the volume. Essentially, this is because whilst the scale factors λ i are measured to be equal in each direction, they are not eigenvalues of the state, i.e., λi | Ψ (λ 1 , λ 2 , λ 3 ) = λ i | Ψ (λ 1 , λ 2 , λ 3 ) . However, both the average and the product (i.e., the volume) of the scale factors are eigenvalues. It is this ambiguity, that leads to the possibility of deviations from the standard isotropic case. In particular, for our choice of isotropic states, we have essentially defined the projection,

| Ψ (λ 1 , λ 2 , λ 3 ) → Ψ λ , ν ( λ ) , (62)
where the state depends only on the average, λ , of the three scale factors, and their product (i.e., the volume) can be viewed as a complicated function of this average. The Hamiltonian found in Ref. [14] is linear and can be written as a symmetric combination of three operators that act on each λ 1 , λ 2 and λ 3 equally. Thus the Hamiltonian operator satisfies our definition of a "symmetric operator", Eq. ( 35) and hence we can directly apply the above states to it. Alternatively one can directly evaluate the Hamiltonian acting on a symmetric state and show show that it contains only states that are themselves symmetric.

As it is explained in Ref. [14], the Hamiltonian acts on uniform steps along the ν direction, just as in the "new quantisation" of isotropic cosmologies. Just as in the standard case, one finds that the Hamiltonian, acting on symmetric states, produces a super-selection of points along the ν direction, that are equally spaced, however from Eq. ( 62) it is clear that this no longer implies a uniform spacing in either λ (the measured scale factor) or λ 3 (the measured volume). To deduce the step-size in the measured scale factor (which would give the step-size in the holonomies that should be used in the case of isotropic LQC), one would have to invert the function ν ( λ ). The use of "lattice refinement" in isotropic LQC is a heuristic first approximation to this, and whilst the power law like refinements typically used in the literature [8] are rather crude, they have the advantage of being analytically tractable.

There are many ways that such isotropic states can be produced from superpositions of anisotropic states. Certainly, those given in Eqs. ( 56) and (59) are not the most general ones, however they serve to demonstrate the possibility that whilst the underlying, anisotropic system requires specific, well defined holonomy step-sizes, the isotropic version contains an ambiguity in the precise value of the step-size that one can take. Here, we have focused only on the Bianchi I model, for which a full understanding of the quantisation exists, however one expects similar effects to occur when starting from other anisotropic and inhomogeneous models. In particular, the motivation for lattice refinement in LQC comes from the fact that one expects that the additional degrees of freedom present in the underlying, full theory, will not exactly produce the symmetry reduced models of cosmology. Here we have shown that the set of states that are measured to be isotropic form a super-selection of states within the Bianchi 1 model, which have a non-constant measured volume step. In this sense we have shown that the anisotropic degrees of freedom of the Bianchi 1 model, that are missing in the symmetric FLRW case, can produce a non-standard evolution of the FLRW volume, which is exactly what is modelled by lattice refinement. It is important to note however that the Hamiltonian governing the evolution of these isotropic states is not the standard FLRW Hamiltonian that is usually considered in lattice refinement models. Thus whilst it may be hoped that lattice refinement models capture some of the consequences of a varying discreteness size, the use of the standard Hamiltonian makes this a rather crude approach.

Finally, it is important to note that in Eq. ( 62), we chose to view the measured scale factor of the isotropic state as the primary variable, however one could have chosen to view the eigenvalue of the volume operator as fundamental, in which case the isotropic volume of the state would be ν and the isotropic Hamiltonian would indeed produce a super-selection with uniform steps volume. In this case however the scale factor of the state ( λ ) would be related to the volume through a complicated function (essentially the inverse of ν ( λ )). This would seem to be largely a matter of taste, however it is possible that future work may choose one view point over the other.

Conclusions

Historically, the quantisation of FLRW cosmologies within LQC, considered the triad element to be the fundamental variable that is to be quantised [18]. However, it was realised that this leads to an instability within the theory at large scales [11], which is particularly problematic for inflation [7]. It was later shown that these problems are cured [7] for the "new" quantisation approach in which the physical volume is considered as the fundamental object to be quantised ¶ [5].

In terms of the basic triad component p, the discretisation of the Hamiltonian occurs with step-size given by p A , where A = 0 for the original quantisation approach and A = -1/2 for the "new" quantisation. These two possibilities are the two limits in which, either only the labels of the edge spins, or the number of vertices dynamically change in the underlying lattice (respectively for the "old" and "new" quantisation approaches). Whilst the behaviour of the full Hamiltonian constraint on a general lattice has yet to be derived from LQG, it is expected that the constraint on A should be 0 < A < -1/2, since the Hamiltonian acts by a combination of both processes. A full derivation of this from an underlying LQG point of view, is not yet possible, and here we do not restrict the value of A and refer to the general case as "lattice refinement" + . There has been much work (and significant debate) on whether "lattice refinement" for a general A is theoretically consistent [5,6,9] and strictly working within the symmetry reduced FLRW models, it appears that only for A = -1/2 ("new" quantisation) is this the case [6]. ¶ Although the onset of inflation within LQC may remain problematic, even with the use of the "new" quantisation [19]; a question which has been recently addressed in Ref. [20]. + The term "lattice refinement" is also sometimes used to include only the cases with -1/2 < A < 0 case, which explicitly excluded the "new" quantisation approach. Here, as previously, we use the term "lattice refinement" in its most general sense, to refer all quantisations in which the step-size is given by p A , which includes both the "new" and "old" quantisations as special cases.

However, here we have shown that, given a (fully consistent) quantum anisotropic model, it is possible to find isotropic states for which the discrete step-size of the isotropically embedded Hamiltonian constraint is not (necessarily) that of "new" quantisation. This demonstrates the possibility of "lattice refinement" being due to the degrees of freedom that are absent in the isotropic model. It is of course possible to embed * the "new" quantisation approach within the anisotropic models and it may be that this is the more natural procedure, however without an a priori knowledge of which embedding process to follow, the conservative approach would be to consider both.

We have shown that the difference between the two procedures is essentially due to what one considers to be more fundamental, the volume of the underlying states (ν) or the measured volume of the symmetric state ( λ 3 ), which are not necessarily equal. Choosing ν leads to the "new" quantised Hamiltonian of isotropic cosmology, whilst choosing λ 3 results in some kind of different "lattice refinement". This "lattice refinement" is significantly more complicated that the single power law behaviour, p A , usually considered, however one may hope to capture some of its effects using the simple model.

It is important to note that here we have started from a Bianchi I model and looked for isotropic embeddings, however similar ideas should apply more general models. We are not suggesting that the Bianchi I model is the underlying geometry of our universe, however it is sufficient to demonstrate the ambiguities associated with finding an effective isotropic cosmology within a model that has more degrees of freedom. This can be the thought of as the source of the "lattice refinement" ambiguity of LQC, however more work is needed to find out if the simple power law model is a sufficiently accurate approximation of this ambiguity. More work is also necessary to see how this new formulation of "lattice refinement", in particular the new relationship between the effective scale factor, λ , and the volume, ν, effects the consistency of the isotropic theory.

Finally, one may be concerned with the fact that simply deciding which variables are the more fundamental one, can have important implications for the large scale physics of the theory, however this is not an unusual occurrence for quantum systems. In all quantum systems, we start from a classical action and follow specific procedures to produce the quantum version of the theory. Precisely which classically equivalent action to choose is a source of quantum ambiguities. From a theoretical point of view, one may hope to have the choice picked out from a more complete understanding of the underlying theory, whilst on a practical level, it is often more useful to parametrise our ignorance and fix these parameters from experiments and observations. In the latter case one simply assumes that the approach is an effective theory, approximating some, as yet unknown, full underlying theory, which is sufficiently accurate for the current data. In the case of LQC (and indeed classical cosmology) we are constantly working within the framework of an effective theory of universal dynamics, in which (it appears) the symmetry reduction is typically a valid approximation. In this context, "lattice refinement" is simply an effective parametrisation of the fact that cosmological symmetries are not exact symmetries of the physical universe. Whilst measuring any consequences of this effective parametrisation seems unlikely, it is possible that the largest scales of the Cosmic Microwave Background (which correspond to the earliest scales to leave the horizon) might retain a signature of this LQC regime, from which ambiguities such as "lattice refinement" may be fixed.

Figure 1 .

 1 Figure 1. Visualisation of the symmetrisation of states.

  and ĥ1 |Ψ (p 2 , p 1 ) = |Ψ (p 2 + δ 1 (p 1 , p 2 ) , p 1 ) and ĥ2 |Ψ (p 2 , p 1 ) = |Ψ (p 2 , p 1 + δ 2 (p 1 , p 2 )) ,

* Here the term "embedding" is used in a loose sense. The FRLW model is in fact a "projection" of a system with more degrees of freedom.

Acknowledgments

The work of M.S. is partially supported by the European Union through the Marie Curie Research and Training Network UniverseNet (MRTN-CT-2006-035863).


J L Flores

S Haesen

M Ortega

New Examples of Marginally Trapped Surfaces and Tubes in Warped Spacetimes

Keywords: 02.40-k 04.20-q Warped spacetimes, trapped surface, marginally trapped surface/tube, CMC surface, minimal surface, Hopf fibration, Clifford tori, twisted space

New examples of marginally trapped surfaces and tubes in warped spacetimes

Introduction

The notion of trapped surface was firstly introduced by Penrose [START_REF] Penrose | Gravitational collapse and space-time singularities[END_REF] in order to study singularities in General Relativity. These surfaces, and their various relatives, have been extensively studied in recent years (just to mention a few, see e.g. [START_REF] Beig | Vacuum Spacetimes with Future Trapped Surfaces[END_REF][START_REF] Bray | Proof of the Riemannian Penrose inequality using the positive mass theorem[END_REF][START_REF] Dafermos | The interior of charged black holes and the problem of uniqueness in general relativity[END_REF][START_REF] Lehner | Numerical Relativity: a review[END_REF]), since they are central not only for singularity theorems, but also to understand the evolution of black holes, the cosmic censorship hypothesis, the Penrose inequality... Trapped surfaces have the physical property that the two null congruences normal to the surface are both converging. From the mathematical point of view, the null converging condition means that the mean curvature vector, which measures the tension of the surface coming from the surrounding space, is a timelike vector everywhere on the surface. If, in addition, the mean curvature vector is future-or past-pointing all over the surface, the trapped surface is accordingly called future-or past-trapped.

In this paper, a closed surface is a compact surface without boundary, embedded in some other semi-Riemannian manifold. The existence of closed trapped surfaces has been investigated in several types of spacetimes. For example, the formation of these surfaces in several cosmological spacetimes have been studied in [START_REF] Ellis | Closed trapped surfaces in cosmology[END_REF][START_REF] Malec | Trapped surfaces in cosmological spacetimes[END_REF].

A spacelike surface in a 4-dimensional Lorentzian manifold is called marginally trapped1 if its mean curvature vector is null. When its mean curvature vector is zero all over the surface it is called extremal.

In order to gain some idea of the properties of marginally trapped surfaces in particular spacetimes, classification results were obtained for the case of having positive relative nullity in Lorentzian space forms [START_REF] Chen | Marginally trapped surfaces in Lorentzian space with positive relative nullity[END_REF] and in Robertson-Walker spaces [START_REF] Chen | Spatial and Lorentzian surfaces in Robertson-Walker spacetimes[END_REF]. In [START_REF] Haesen | Boost invariant marginally trapped surfaces in Minkowski 4-space[END_REF][START_REF] Haesen | Marginally trapped surfaces in Minkowski 4-space invariant under a rotation subgroup of the Lorentz group[END_REF][START_REF] Haesen | Screw Invariant Marginally Trapped Surfaces in Minkowski 4space[END_REF] marginally trapped surfaces invariant under symmetries of 4-dimensional Minkowski space were studied. A complete classification of spacelike surfaces in a 4-dimensional Lorentzian spacetime, containing the above cases, was recently given in [START_REF] Senovilla | Classification of spacelike surfaces in spacetime[END_REF].

Some results concerning the non-existence of closed marginally trapped surfaces can be also found in the literature. Among the classical ones, a result due to R. Penrose [START_REF] Penrose | Gravitational collapse and space-time singularities[END_REF] implies the non-existence of closed marginally trapped surfaces in the Minkowski spacetime when it bounds a compact domain. In [START_REF] Mars | Trapped surfaces and symmetries[END_REF] the non-existence of closed marginally trapped surfaces is shown for strictly stationary spacetimes. Finally, in [START_REF] Carrasco | On marginally outer trapped surfaces in stationary and static spacetimes[END_REF] the authors have shown the nonexistence of marginally trapped surfaces bounding a domain and entering a region of a static spacetime where the Killing vector field is timelike, and with the additional assumptions of dominant energy condition and an outer untrapped barrier.

The main aim of this paper is to provide new examples of marginally trapped surfaces and tubes in warped spacetimes. [START_REF] Andersson | Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes[END_REF]. In Section 2, we establish some existence/non-existence results on trapped and marginally trapped surfaces in FLRW spacetimes (Corollaries 2.2, 2.4) by using a simple, but fundamental, relation between these surfaces and constant mean curvature surfaces in 3manifolds (Theorem 2.1). In particular, we show the existence of closed marginally trapped surfaces with any genus in closed FLRW spacetimes.

2. In Section 3, we develop a method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on an extension of the classical Hopf map to a submersion between closed FLRW spacetimes of dimension 4 and 3. We illustrate it with a simple example in Subsection 3.4. In order to show the utility of this constructive method, in Subsection 3.5 we apply it to provide marginally trapped surfaces crossing expanding and collapsing regions of a closed FLRW spacetime.

3. Section 4 is devoted to studying marginally trapped tubes. They are defined as smooth hypersurfaces foliated by marginally trapped surfaces. Then, we give some existence/nonexistence results for these objects in closed FLRW spacetimes (Corollaries 4.2, 4.3), and provide examples of them with any type of causal behavior, Subsection 4.1. Finally, in Section 5 we extend the approach introduced in this paper to twisted spaces (Theorem 5.1).

Marginally trapped surfaces in t-slices of warped spacetimes

In general, given an immersion x : Σn → Σ m of a n-dimensional semi-Riemannian manifold into another m-dimensional semi-Riemannian manifold, the second fundamental form and the mean curvature vector will be denoted by h x and H x = trace(h x )/n, respectively. Let f : I ⊂ R → (0, ∞) be a smooth function (t ∈ I), (M 3 , g 3 ) a 3-dimensional Riemannian manifold and S a surface. Denote by M 4 1 = I × f M 3 the Lorentzian warped product manifold given by the product manifold I ×M 3 endowed with metric g 4 = -dt 2 +f 2 g 3 . Let ϕ : S → M 3 be an immersion of S in M 3 , ψ : M 3 → I × f M 3 the embedding of M 3 in I × f M 3 and φ : S → I × f M 3 the corresponding immersion of S in the warped product, both in the t slice (t-slice for short). According to a well-known result (see [9, p. 79]), the following relation holds between the corresponding second fundamental forms:

h φ (X, Y ) = h ϕ (X, Y ) + h ψ (X, Y ), whereby X, Y ∈ X(S).
(2.1)

The expression of h ψ is also known (see e.g. [23, p. 344]):

h ψ (X, Y ) = -g 4 (X, Y ) grad g 4 f f = g 4 (X, Y ) f f ∂ t ,
where we have used grad

g 4 f = -f ∂ t .
Hence, we obtain

h φ (X, Y ) = h ϕ (X, Y ) + g 4 (X, Y ) f f ∂ t .
Taking one half of the trace of the above expression, using an orthonormal frame {∂ t , {E i } 3 i=1 } w.r.t. the metric g 4 , i.e. E i = e i f whereby {e i } 3 i=1 is the corresponding orthonormal frame w.r.t. the metric g 3 on M 3 (and S), one obtains

H φ = H ϕ f 2 + f f ∂ t , (2.2)
where H φ and H ϕ stand for the mean curvature vectors associated with h φ and h ϕ , respectively.

Recall that a surface S is called of constant mean curvature, CMC for short, if the length of its mean curvature vector is a constant function.

Theorem 2.1 A surface S contained in a t 0 -slice of M 4 1 = I × f M 3 is trapped (respectively, marginally trapped) iff it is a CMC surface in M 3 with H ϕ < |f (t 0 )| (respectively, H ϕ = |f (t 0 )|).
Proof. We compute the length of the mean curvature H φ , by making use of (2.2):

f (t 0 ) 2 ḡ( H φ , H φ ) = f (t 0 ) 2 H ϕ f (t 0 ) 2 + f (t 0 ) f (t 0 ) ∂ t 2 = H ϕ 2 -f (t 0 ) 2 .
This readily gives the results.

From this result one can deduce some simple consequences for FLRW spacetimes, i.e. warped spacetimes with fiber M 3 = R 3 , S 3 or H 3 . When the fiber is M 3 = S 3 , we will say that our FLRW is closed.

First, recall that the so-called Clifford tori C u in S 3 are given by

C u := (z 1 , z 2 ) ∈ C 2 : |z 1 | = cos(u), |z 2 | = sin(u) , u ∈ (0, π/2).
These are closed surfaces in S 3 with constant mean curvature satisfying

H u = |2 cot(2u)|. (2.3)
Of course, other CMC tori can be obtained by applying to them isometries of S 3 . In addition, by making surgery on a finite number of Clifford tori, Butscher-Packard [START_REF] Butscher | Doubling Constant Mean Curvature Tori in S 3[END_REF] obtained closed surfaces in S 3 that are also CMC with arbitrary genus. Moreover, as far as we know, these are the only known surfaces in S 3 which are closed, CMC, non-minimal and with arbitrary genus.

Corollary 2.2 (Existence result

). There exist closed trapped and closed marginally trapped surfaces with arbitrary genus in closed FLRW spacetimes.

Proof. From Theorem 2.1, any surface in M 3 with constant mean curvature H ϕ = |f (t 0 )| can be seen as a marginally trapped surface in the t 0 -slice of I × f M 3 . Notice also that there exist closed CMC surfaces in M 3 = S 3 with arbitrary genus (standard spheres; Clifford tori; Butscher-Packard surfaces [START_REF] Butscher | Doubling Constant Mean Curvature Tori in S 3[END_REF]). Therefore, there exist closed marginally trapped surfaces in I × f S 3 with arbitrary genus. From Theorem 2.1, any minimal surface S in M 3 can be seen as a trapped surface in any t 0 -slice of I × f M 3 with f (t 0 ) = 0. Notice also that there exist closed minimal surfaces in M 3 = S 3 with arbitrary genus [START_REF] Lawson | Complete Minimal Surfaces in S 3[END_REF]. Therefore, there exist closed trapped surfaces in I × f S 3 with arbitrary genus whenever f ≡ cte.

Remark 2.3 Standard spheres and embedded CMC tori can be chosen with any constant value of its mean curvature function, and so, there is no restriction for the warping function f in Corollary 2.2. However, the mean curvature function of a Butscher-Packard's surface has to be sufficiently small, due to the gluing process. Thus, in this case Corollary 2.2 only applies to warping functions of sufficiently small derivative. This observation must be also taken into account in Corollary 4.2.

Corollary 2.4 (Non-existence result). Let M 4 1 = I × f M 3 be a FLRW spacetime with fiber M 3 = H 3 . There are no closed marginally trapped surfaces contained in any t 0 -slice such that |f (t 0 )| ≤ 1.

Proof. According to a result by do Carmo and Lawson [START_REF] Do Carmo | On Alexandrov-Bernstein theorems in hyperbolic space[END_REF], if S is a closed CMC surface in H 3 , it must be a geodesic sphere with mean curvature satisfying H ϕ > 1. Therefore, the proof directly follows from Theorem 2.1.

Remark 2.5 Formula (2.2) implies that H φ cannot be future-directed at t = t 0 if f (t 0 ) ≤ 0, and so, the following result [START_REF] Senovilla | Novel results on trapped surfaces[END_REF] is reobtained: there are no future trapped (resp. marginally trapped) surfaces in any slice of collapsing (i.e. f (t) ≤ 0 for all t) warped spacetimes. Analogously, H φ cannot be past-directed at t = t 0 if f (t 0 ) ≥ 0, hence: there are no past trapped (resp. marginally trapped) surfaces in any slice of expanding (i.e. f (t) ≥ 0 for all t) warped spacetimes.

Remark 2.6

The stability result in [START_REF] Carrasco | On marginally outer trapped surfaces in stationary and static spacetimes[END_REF] (which can be applied to more general surfaces than the ones contained in a t-slice, assumed some additional conditions) suggests that the marginally trapped surfaces found in this section should be unstable.

Marginally Trapped surfaces in closed FLRW spacetimes:

A constructive method

In the present section we are going to construct marginally trapped surfaces, non-necessarily contained in a t-slice of warped spacetimes, by using the classical Hopf map. The price to pay is that we will need to restrict our ambient space to closed FLRW spacetimes. Very roughly, the idea is as follows. We can see closed FLRW spacetimes I × f S 3 as a semi-Riemannian submersion over I × f S 2 (1/2) such that the lift of any curve in I × f S 2 (1/2) gives rise to a surface in I × f S 3 whose geometric properties depend on the base curve. Thus, by choosing appropriate curves in the base, we can obtain embedded surfaces in I × f S 3 with mean curvature vector as desired, i.e. spacelike, timelike or lightlike.

To develop our approach, first we need to recall some notions about semi-Riemannian submersions and the Hopf map.

Semi-Riemannian submersions

Let π : (M, g M ) → (B, g B ) be a surjective map between semi-Riemannian manifolds. Assume that π has maximal rank. The fibers are π

-1 (b), with b ∈ B. A tangent vector to M is called vertical (resp. horizontal) if it is tangent (resp. orthogonal) to the fibers. The vertical part of π at a point m ∈ M is ker(dπ) m ⊂ T m M. If for each point m ∈ M, π * satisfies g M (u, v) = g B (π * u, π * v), (3.1)
for any horizontal tangent vectors u, v at m ∈ M, then π is called a semi-Riemannian submersion. This lemma summarizes the basic properties of semi-Riemannian submersions, [START_REF] O'neill | The fundamental equations of a submersion[END_REF].

Lemma 3.1 Let π : (M, g M ) → (B, g B ) be a semi-Riemannian submersion.

• Given X ∈ X(B), there exists a horizontal lift X ∈ X(M) of X such that X is horizontal and π * X = X

• Given a curve γ : I → B, t 0 ∈ I and a point m ∈ π -1 (γ(t 0 )), there exists a unique horizontal lift γ :

I → M of γ, i.e. it satisfies γ(t 0 ) = m, π • γ = γ and γ is horizontal.
In particular, γ is unitary if, and only if, so is γ.

• If ∇ M and ∇ B are the Levi-Civita connections of M and B, resp., then for any

X, Y, Z ∈ X(B), g M (∇ M X Ỹ , Z) = g B (∇ B X Y, Z).

The Hopf map and closed FLRW spacetimes

Let C be the field of complex numbers, with i = √ -1 the complex unit, |z| the modulus of z ∈ C and z its complex conjugate. Firstly, the round 3-sphere in C 2 can be seen as

S 3 = {(z, w) ∈ C 2 : |z| 2 + |w| 2 = 1}
, with standard metric g 3 . Also, we can see the round 2-sphere of radius 1/2 as S 2 (1/2) = {(z, x) ∈ C × R : |z| 2 + x 2 = 1/4}, with standard metric g 2 . We recall the classical Hopf map

π : S 3 → S 2 (1/2), π(z, w) = zw, 1 2 |z| 2 - 1 2 |w| 2 ,
where ω is the complex conjugate of ω. It is well-known that π is a Riemannian submersion with totally geodesic fibers. In fact, this Riemannian submersion π is the quotient map of the following isometry group action:

S 1 × S 3 → S 3
, (e iθ , (z, w)) → (e iθ z, e iθ w).

(3.

2)

The fibers of π are the orbits of the action, i.e. given a point p = (z, w) ∈ S 3 , the orbit is {e iθ • p = (e iθ z, e iθ w) : e iθ ∈ S 1 }, which is a big circle (geodesic) of S 3 . We also remark that the vertical part of π at p = (z, w) ∈ S 3 is spanned by ip = (iz, iw). In other words, ker(dπ) p = Span{ip}. Given f :

I ⊂ R → (0, ∞) a smooth function (t ∈ I), consider the closed FLRW spacetime M 4 1 = I × f S 3 .
We also consider the toy model M 1 onto I is a semi-Riemannian submersion, whose horizontal part is spanned by ∂ t . Then, a vertical vector is orthogonal to ∂ t . Given a vector field X ∈ X(S 3 ), there exists a vertical lift X tangent to M 4 1 such that X ⊥ ∂ t . Given Z a tangent vector to M 4 1 , nor(Z) is the orthogonal projection onto the horizontal part, whereas tan(Z) is the orthogonal projection onto the vertical part. Formally, there is a similar situation for M 3 1 , so that we can use the same notation. Thus, we obtain [START_REF] O'neill | Semi-Riemannian Geometry with Applications to Relativity[END_REF]: Lemma 3.2 Let X, Y be tangent vector fields to S 3 (resp. S 2 (1/2)) and X, Ỹ be vertical lifts to

M 4 1 (resp. M 3 1 ): 1. nor(∇ X Ỹ ) = -g 4 ( X, Ỹ ) f grad g 4 (f ) (resp. nor(∇ X Ỹ ) = -g 3 ( X, Ỹ ) f grad g 3 (f )) 2. tan(∇ X Ỹ ) is the vertical lift of D X Y (resp. tan(∇ X Ỹ ) is the vertical lift of ∇ 2 X Y ).

Constructing the surface

From now on, we will make use of Lemma 3.1, sometimes without indicating it explicitly. We define the projection π :

M 4 1 → M 3 1 as π(t, p) := (t, π(p)).
Lemma 3.3 The map π is a semi-Riemannian submersion with vertical part at (t, p) spanned by (0, ip).

Proof. We denote by ∂ t both, the vector field tangent to M

π * ∂ t | (t,p) = d ds | s=0 π(t + s, p) = d ds | s=0 (t + s, π(p)) = ∂ t | (t,π(p)) .
Therefore,

g 3 (π * ∂ t , π * ∂ t ) = -1 = g 4 (∂ t , ∂ t ).
For any (t, p) ∈ M 4 1 , consider the curve α(s) = (t, cos(s)p + sin(s)ip). Taking into account that ker(dπ) p = Span(ip), we deduce:

π * (0, ip) = d ds | s=0 π(α(s)) = (0, π * (ip)) = (0, 0).
Finally, take (0, X) ∈ T (t,p) M 4 1 which is orthogonal to ip. Then, π * (0, X) = (0, π * X). Taking into account that π is a semi-Riemannian submersion, we deduce

g 3 (π * (0, X), π * (0, X)) = g 3 ((0, π * X), (0, π * X)) = f 2 g 2 (π * X, π * X) = f 2 g 3 (X, X) = g 4 ((0, X), (0, X)).
We recall that the Hopf map π is the quotient map of the isometry group action (3.2). We call Γ θ : S 3 → S 3 , Γ θ (z, w) = (e iθ z, e iθ w), which is an isometry of S 3 . We extend it to M 4 1 as follows. For each e iθ ∈ S 1 , we define the map

Γ θ : M 4 1 → M 4 1 , Γ θ (t, p) = (t, Γ θ (p)). Given (t, p) ∈ M 4 1 , consider T t I ≡ R and T p S 3 ⊂ C 2 .
Thus, it is possible to let Γ θ act on tangent vectors under these natural identifications. Lemma 3.4 For each e iθ ∈ S 1 , the map Γ θ is an isometry of M 4 1 with (Γ θ ) * = Γ θ under previous identifications.

Proof. Firstly, we are going to show that (Γ θ ) * = Γ θ . Observe that:

(Γ θ ) * (∂ t | (t,p) ) = d ds | s=0 Γ θ (t + s, p) = d ds | s=0 (t + s, Γ θ (p)) = ∂ t | (t,e iθ p) = Γ θ (∂ t | (t,p) ).
On the other hand, given a curve γ in S 3 such that γ(0) = p, γ(0) = X, we have

(Γ θ ) * (0, X) = d ds | s=0 Γ θ (0, γ(s)) = d ds | s=0 (0, e iθ γ(s)) = (0, e iθ X) = Γ θ (0, X).
Finally, in order to prove that Γ θ is isometry, notice that g 4 ((Γ θ ) * (0, X), (Γ θ ) * (0, X)) = g 4 ((0, e iθ X), (0, e iθ X)) = f 2 g 3 (e iθ X, e iθ X) = f 2 g 3 (X, X) = g 4 ((0, X), (0, X)).

Since S 2 (1/2) is an orientable manifold, M 3
1 is also orientable. We choose the orientation on M

∇ T T = 2 κN, ∇ T N = κT + 3 τ B, ∇ T B = -2 τ N, (3.3)
where 2 = g 3 (N, N ), 3 = g 3 (B, B), 2 = -3 = ±1, and {T, N, B} is a positive basis along α. Consider α(s) = (t(s), α 2 (s)), where t :

J → I, α 2 : J → S 2 (1/2). By Lemma 3.1, let β : J ⊂ R → M 4
1 be a horizontal lift of α. Since β is orthogonal to the vertical part of π, we have π

• β = α, β = (t, β 2 ), π • β 2 = α 2 , β2 ⊥ iβ 2 .
Now, we are able to construct a spacelike surface in M 4 1 with the help of Γ θ and β. Define:

φ : S = J × S 1 → M 4 1 , φ(s, θ) = Γ θ (β(s)) = (t(s), e iθ β 2 (s)). (3.4)
It is clear that the derivatives of φ are φ s = ( ṫ, e iθ β2 ), φ θ = (0, ie iθ β 2 ).

By using Lemma 3.3, the coefficients of the first fundamental form of φ * g 4 are the following:

E = g 4 (φ s , φ s ) = g 4 (( ṫ, e iθ β2 ), ( ṫ, e iθ β2 )) = -ṫ2 + f 2 g 3 (e iθ β2 , e iθ β2 ) = -ṫ2 + f 2 g 3 ( β2 , β2 ) = g 4 ( β, β) = 1, F = g 4 (φ s , φ θ ) = g 4 (( ṫ, e iθ β2 ), (0, ie iθ β 2 )) = f 2 g 3 (e iθ β2 , ie iθ β 2 ) = f 2 g 3 ( β2 , iβ 2 ) = 0, G = g 4 (φ θ , φ θ ) = g 4 ((0, ie iθ β 2 ), (0, ie iθ β 2 )) = f 2 g 3 (ie iθ β 2 , ie iθ β 2 ) = f 2 g 3 (β 2 , β 2 ) = f 2 .
Therefore,

{U 1 = φ s , U 2 = (1/f )φ θ } is a globally defined orthonormal tangent frame to S in M 4
1 . We also need to construct an orthonormal normal frame. To do so, we use the isometries Γ θ and the vectors N , B along α. By Lemma 3.1, let Ñ and B be horizontal lifts of N and B, resp., along β. Define

η N , η B : S → T M 4 1 , η N = (Γ θ ) * Ñ , η B = (Γ θ ) * B.
Lemma 3.5 The set {η N , η B } is a globally defined, orthonormal, normal frame to S.

Proof. We note that

φ s = (Γ θ ) * β, φ θ = (Γ θ ) * (0, iβ 2 ).
Bearing in mind these two expressions, (3.1) and Lemma 3.4, we deduce

g 4 (η N , φ s ) = g 4 ((Γ θ ) * Ñ , (Γ θ ) * β) = g 4 ( Ñ , β) = g 3 (N, T ) = 0, g 4 (η N , φ θ ) = g 4 ((Γ θ ) * Ñ , (Γ θ ) * (0, iβ 2 )) = g 4 ( Ñ , (0, iβ 2 )) = 0,
where the last equality holds because Ñ is horizontal and (0, iβ 2 ) is vertical. On the other hand,

g 4 (η B , φ s ) = g 4 ((Γ θ ) * B, (Γ θ ) * β) = g 4 ( B, β) = g 3 (B, T ) = 0 g 4 (η B , φ θ ) = g 4 ((Γ θ ) * B, (Γ θ ) * (0, iβ 2 )) = g 4 ( B, i β) = 0 g 4 (η N , η N ) = g 4 ((Γ θ ) * Ñ , (Γ θ ) * Ñ ) = g 4 ( Ñ , Ñ ) = g 3 (N, N ) = 2 .
Similarly, we deduce g 4 (η N , η B ) = 0 and g 4 (η B , η B ) = 3 .

Let h φ be the second fundamental form of the inmersion φ : S → M

H φ (s, e iθ ) = 2 2 κ(s) + f (t(s)) f (t(s)) g 3 (∂ t | α(s) , N (s)) • η N (s) + 3 2 f (t(s)) f (t(s)) g 3 (∂ t | α(s) , B(s)) • η B (s).
Proof. Since we already know U 1 , U 2 , η N and η B , we have

2 H φ = Σ 2 i=1 { 2 g 4 (h φ (U i , U i ), η N )η N + 3 g 4 (h φ (U i , U i ), η B )η B }.
We are going to compute all these four products. By Lemma 3.1 and (3.3)

g 4 (h φ (U 1 , U 1 ), η N ) = g 4 (∇ U 1 U 1 , η N ) = g 4 (∇ φs φ s , η N ) = g 4 (∇ (Γ θ ) * β (Γ θ ) * β, (Γ θ ) * Ñ ) = g 4 (∇ β β, Ñ ) = g 3 (∇ α α, N ) = g 3 ( 2 κN, N ) = κ.
Similarly, we have

g 4 (h φ (U 1 , U 1 ), η B ) = g 4 (∇ U 1 U 1 , η B ) = g 4 (∇ β β, B) = g 3 (∇ α α, B) = g 3 ( 2 κN, B) = 0.
For U 2 , we make use of Lemma 3.2. We consider the surface in S 3 given by ξ : J × S 1 → S 3 , ξ(s, e iθ ) = e iθ β 2 (s).

Since ξ θ = ie iθ β 2 (s), it is clear that φ θ = (0, ξ θ ). Also, note that

U 2 = 1 f φ θ = 1 f (0, ξ θ ). Since h φ (U 2 , U 2 ) = 1 f 2 h φ (φ θ , φ θ ), we compute tan(∇ φ θ φ θ ) = tan(∇ (0,ξ θ ) (0, ξ θ )) = (0, D ξ θ ξ θ ). (3.5)
Thus, we have to compute D ξ θ ξ θ . To do so, we recall that the position vector χ : S 3 → C 2 is a unit normal vector field with second fundamental form h χ (X, Y ) = -g 3 (X, Y )χ for any X, Y tangent to S 3 . Let D be the Levi-Civita connection of C 2 . By the Gauss formula, and by the fact that ξ θ is unit,

D ξ θ ξ θ = Dξ θ ξ θ -h χ (ξ θ , ξ θ ) = Dξ θ ξ θ + χ • ξ.
Now, we consider the curve in J × S 1 given by α(u) = (s, e i(θ+u) ). Since α(0) = (s, e iθ ) and α (0) = ∂ θ | (s,e iθ ) , we obtain

Dξ θ ξ θ = d du u=0 ξ θ (α(u)) = d du u=0 ξ θ (s, e i(θ+u) ) = d du u=0 i e i(θ+u) β 2 (s) = -e i(θ) β 2 (s).
Finally, we see

D ξ θ ξ θ = -e iθ β 2 (s) + (χ • ξ)(s, e iθ ) = 0.
By (3.5), we see that tan(∇ φ θ φ θ ) = 0. On the other hand, by Lemma 3.2

nor(∇ φ θ φ θ ) = nor(∇ (0,Z) (0, Z)) = - g 4 ((0, Z), (0, Z)) f grad g 4 (f ) = f g 3 (Z, Z)f ∂ t = f f ∂ t .
Therefore, we obtain ∇ φ θ φ θ = f f ∂ t . As a consequence, we get

h φ (U 2 , U 2 ) = 2 f 2 g 4 (∇ φ θ φ θ , η N )η N + 3 f 2 g 4 (∇ φ θ φ θ , η B )η B = 2 f f g 3 (∂ t , N )η N + 3 f f g 3 (∂ t , B)η B .
Proposition 3.7 Given (s, e iθ ) ∈ S, the mean curvature vector of the immersion φ is spacelike (resp. lightlike/zero, timelike) if, and only if,

2 κ + f f g 3 (∂ t , N ) 2 + 3 f f g 3 (∂ t , B) 2 > 0 (resp. = 0, < 0). (3.6)
Remark 3.8 Let us assume that the curve α satisfies that for some s 0 ∈ J, the point t 0 = t(s 0 ) is such that f (t 0 ) = 0. According to (3.6), the mean curvature vector of φ is lightlike/zero (respectively, timelike or spacelike) iff 2 κ 2 (s 0 ) = 0 (resp. < 0 or > 0). Moreover, in case the equality holds, the curvature at s 0 must vanish, i.e. κ(s 0 ) = 0, and so our procedure is no longer valid at this point (recall that we assumed a Frenet basis with the assumption κ > 0). As we will see later, sometimes we can overcome this difficulty by using a continuity argument.

Finally, we point out that given a curve α 2 : J ⊂ R → S 2 (1/2), the lift of α 2 to S 3 via the Hopf projection π is classically called a Hopf tube. If, in addition, J = R, α 2 is periodic and α 2 has no self-intersection points (in other words, the image of α 2 is homeomorphic to a circle), the Hopf tube is an (embedded) Clifford torus in S 3 . This clearly extends to our curves α and immersions φ; that is, if the image of α : R → M

A simple example

Let γ : J ⊂ R → S 2 (1/2) be a Frenet unit curve. Let ∇ 2 be the Levi-Civita connection of S 2 (1/2). The Frenet apparatus of γ is {dγ/ds, n} with geodesic curvature c. In other words,

∇ 2 dγ/ds dγ/ds = cn, ∇ 2 dγ/ds n = -c • dγ/ds,
and {dγ/ds, n} is a positive basis for the usual orientation on S 2 (1/2). Given t 0 ∈ I, let us define α 2 : J ⊂ R → S 2 (1/2), α 2 (s) = γ(s/f (t 0 )), and the curve α : J → M 3 1 , α(s) = (t 0 , α 2 (s)). Simple computations give α = (0, α2 ) = 0, 1 f dγ ds . We call v = 1 f dγ ds . Note that (0, v) is unit for ḡ3 , and therefore, α is a spacelike unit curve in M 3 1 . Then, we easily obtain

∇ 2 v v = c f 2 n, ∇ 2 v n = -cv.
We put T = α = (0, v). From Lemma 3.2, we have

∇ T T = ∇ (0,v) (0, v) = tan ∇ (0,v) (0, v) + nor ∇ (0,v) (0, v) = 0, ∇ 2 v v -ḡ3 ((0,v),(0,v)) f grad g 3 f = f f , c f 2 n . The square g 3 -norm of ∇ T T is ḡ3 (∇ T T, ∇ T T ) = -(f ) 2 f 2 + f 2 g 2 c f 2 n, c f 2 n = c 2 -(f ) 2 f 2
. In order to obtain a Frenet curve, we must assume c 2 -(f ) 2 = 0 everywhere. We define

∆ := c 2 -(f ) 2 , δ := sign(∆) = ±1, κ := √ δ∆ f . (3.7)
According to previous computations and notation, we have

∇ T T = δ √ δ∆ f f δ √ δ∆ , δcn f √ δ∆ .
Therefore, we can choose

N = f δ √ δ∆ , δcn f √ δ∆ , B = c √ δ∆ , f n f √ δ∆ .
In fact, it is easy to check that {T, N, B} is an orthormal basis along α, with 2 = δ = -3 .

Let us check that it is also positive. We recall that we should compare it with {∂ t , γ , n}, bearing in mind (3.7); that is to say, we compute

det(T, N, B) = 0 f δ √ δ∆ c √ δ∆ 1/f 0 0 0 δc f √ δ∆ f f √ δ∆ = -1 f 2 ∆ f c c f = 1 f 2 > 0.
With this curve α, we construct an immersion φ as in (3.4). By (3.6), we can study the causal character of the mean curvature vector H φ of φ. Thus, we have

2 κ + f f g 3 ∂ t , N 2 + 3 f f g 3 ∂ t , B 2 = δ √ δ∆ f - δ(f ) 2 f √ δ∆ 2 -δ (f ) 2 f 2 c √ δ∆ 2 = = δ (c 2 -2(f ) 2 ) 2 -(f ) 2 c 2 δf 2 ∆ .
This means that H φ is spacelike (resp. lightlike/zero, timelike) if, and only if,

S := δ (c 2 -2f (t 0 ) 2 ) 2 -f (t 0 ) 2 c 2 > 0 (resp. = 0, < 0).
For instance, if the chosen level t 0 gives rise to a critical slide f (t 0 ) = 0, the mean curvature H φ is always spacelike.

Proposition 3.9 There exist infinitely many embedded tori φ : S 1 × S 1 → M 4 1 which are trapped, marginally trapped or untrapped (whenever f is not constant everywhere).

Proof. We resort to previous example. Pick a point t 0 such that, say, f (t 0 ) > 0, and consider a curve γ with constant geodesic curvature c (i.e., the curve γ is a small circle of S 2 (1/2)). If c = 2f (t 0 ) holds, simple computations show δ = 1 and S = 0. This means that the embedding φ is a marginally trapped surface. On the other hand, if c > 2f (t 0 ), we have S > 0, which is the spacelike case. Finally, if we take f (t 0 ) < c < 2f (t 0 ), then the mean curvature vector of φ is timelike. Remark 3.10 Previous proposition can be directly obtained from Theorem 2.1, just by considering suitable CMC tori in M 3 = S 3 .

Marginally trapped surfaces crossing expanding and collapsing regions

Consider the toy model M

3 1 = I × f S 2 (1/2) associated to a closed FLRW spacetime M 4 1 = I × f S 3 . Let α : J ⊂ R → M 3 1 , s → (t(s), α 2 (s)) be a curve with ṫ = h(t); in particular, ẗ(s) = h (t) ṫ(s) = h (t)h(t).
If we impose the vector field α = T = h(t)∂ t + α2 to be unitary, i.e.

g 3 ( α, α) = -h(t) 2 + f (t) 2 g 2 ( α2 , α2 ) = 1, we deduce, g 2 ( α2 , α2 ) = 1 + h(t) 2 f (t) 2 .
Consider the unitary reparametrization α2 of α 2 , i.e.

α2 = ρ α2 , ρ = 1 + h(t) 2 f (t) , g 2 ( α2 , α2 ) = 1 ⇒ g( α2 , α2 ) = f (t) 2
Then, T can be rewritten as

T = h(t)∂ t + ρ α2 . Moreover ∇ α2 α2 = ∇ 2 α2 α2 - g 3 ( α2 , α2 ) f (t) grad g 3 (f ) = ∇ 2 α2 α2 + f (t)f (t)∂ t ∇ α2 ∂ t = ∇ ∂t α2 = f (t) f (t) α2 , ∇ ∂t ∂ t = 0.
Therefore, we deduce:

DT (s) ds = D ds ṫ(s)∂ t + ρ α2 = ẗ(s)∂ t + ṫ(s) D∂ t ds + ρ α2 + ρ D α2 ds = ẗ(s)∂ t + ṫ(s) 2 ∇ ∂t ∂ t + ṫ(s)∇ α2 ∂ t + ρ α2 + ρ ṫ(s)∇ ∂t α2 + ∇ α2 α2 = h (t)h(t)∂ t + h(t)ρ∇ α2 ∂ t + ρ α2 + h(t)ρ∇ ∂t α2 + ρ 2 ∇ α2 α2 = h (t)h(t)∂ t + 2h(t)ρ f (t) f (t) α2 + ρ α2 + ρ 2 ∇ α2 α2 = h (t)h(t)∂ t + 2h(t)ρ f (t) f (t) + ρ α2 + ρ 2 ∇ 2 α2 α2 + f (t)f (t)∂ t = h (t)h(t) + ρ 2 f (t)f (t) ∂ t + 2h(t)ρ f (t) f (t) + ρ α2 + ρ 2 ∇ 2 α2 α2 ,
where the partner function of α2 becomes

2h(t)ρ f (t) f (t) + ρ = 2h(t)ρ f (t) f (t) -h(t) (1 + h(t) 2 )f (t) -f (t)h(t)h (t) f (t) 2 √ 1 + h(t) 2 = h(t)h (t) + (1 + h(t) 2 ) f (t) f (t) h(t) f (t) 1 + h(t) 2 .
Next, let { α2 , N } be some orthonormal basis of T S 2 (1/2) along α2 . Assume that α2 is a circle (non-necessarily maximum) in S 2 (1/2). Then,

∇ 2 α2 α2 = kN .
In conclusion, we deduce

∇ T T = hh + (1 + h 2 ) f f ∂ t + h √ 1 + h 2 f α2 + (1 + h 2 ) f 2 kN .
In particular, if we recall that ∇ T T = 2 κN , we have

g 3 (∇ T T, ∇ T T ) = hh + (1 + h 2 ) f f 2 -1 1+h 2 + (1 + h 2 ) 2 k 2 f 2 = 2 κ 2 .
Therefore, from the last equality in this expression:

κ 2 = 2 (1+h 2 ) 2 k 2 f 2 -1 1+h 2 hh + (1 + h 2 ) f f 2 = 2 (1 + h 2 ) (1+h 2 )k 2 f 2 -(ln √ 1 + h 2 ) + (ln f ) 2 . (3.8)
Next, we are going to compute the vector field B. To this aim, we write it as:

B = A∂ t + B α2 + CN By imposing g 3 (T, B) = -hA + B √ 1 + h 2 f = 0 we deduce B = h √ 1 + h 2 f A. (3.9)
On the other hand, by imposing g 3 (N, B) = 0 we have g 3 (∇ T T, B) = 0, and so,

hh + (1 + h 2 ) f f -A + hf √ 1 + h 2 B + (1 + h 2 )kC = 0. (3.10)
Taking into account (3.9) in (3.10) we deduce:

0 = - hh 1 + h 2 + f f A + (1 + h 2 )kC.
Therefore, if k = 0 we have

C = 1 (1 + h 2 ) 2 k hh + (1 + h 2 ) f f A. (3.11) Next, we impose g 3 (B, B) = -A 2 + f 2 (B 2 + C 2 ) = 3 .
Taking into account (3.8), (3.9) and (3.11) in this expression, one deduces:

A 2 = 3 -1 1+h 2 + f 2 (1+h 2 ) 4 k 2 hh +(1+h 2 ) f f 2 = - 3 k 2 (1+h 2 ) 2 f 2 (1+h 2 )k 2 f 2 -((ln √ 1+h 2 ) +(ln f ) ) 2 = k 2 (1+h 2 ) 3 f 2 κ 2 .
Next, we impose that the surface S generated by the curve α is marginally trapped (3.6):

κ + f f g 3 (∂ t , N ) 2 = f f g 3 (∂ t , B) 2 κ + f f 2 κ g 3 (∂ t , ∇ T T ) 2 = (f /f ) 2 A 2 2 κ 2 -(ln f ) hh + (1 + h 2 )(ln f ) 2 = ((ln f ) ) 2 κ 2 A 2 .
If we develop both members of the equation, we deduce:

2((ln f ) ) 2 + 3(ln 1 + h 2 ) + √ 1 + h 2 k f (ln f ) + ((ln 1 + h 2 ) ) 2 - (1 + h 2 )k 2 f 2 = 0,
which is a second order equation for (ln f ) . The discriminant of this equation is:

D = (ln 1 + h 2 ) + 3 √ 1 + h 2 k f 2 .
Therefore, the solutions are

(ln f ) ± = -3(ln √ 1 + h 2 ) - √ 1+h 2 k f ± (ln √ 1 + h 2 ) + 3 √ 1+h 2 k f 4 , that is, f + = -1 2 (ln √ 1 + h 2 ) f + + 2 √ 1 + h 2 k f -= -(ln √ 1 + h 2 ) f -- √ 1 + h 2 k.
The solution (ln f ) -implies κ = 0 (recall (3.8)). So, take the solution (ln f ) + with = +1. Take also t(s) = sin(s), and thus, h(t) = ± √ 1 -t 2 . In this case, the warping function f must satisfy the differential equation

f (t) = t 2(2 -t 2 ) f (t) + √ 2 -t 2 2 k, f (0) = f 0 > 0. (3.12)
So, according to previous construction, we will obtain a marginally trapped surface crossing expanding and collapsing regions along a closed FLRW spacetime with, say, I = (-1.3, 1.3), if, in addition, the following two properties hold for f 0 big enough:

(i) f (t) > 0 for all t ∈ [-1.3, 1.3], and

(ii) f (t) changes its sign along (-1, 1).

For (i), first we are going to prove that f (t) > 0 on [0, 1.3]. By contradiction, we assume there exists t 1 ∈ [0, 1.3] such that f (t 1 ) ≤ 0. Since f (0) = k > 0, f has to start to decrease at a certain point, and therefore there exists t 0 ∈ (0, t 1 ) such that f (t 0 ) = 0, f (t 0 ) > 0. This is in contradiction to the fact that

f (t 0 ) = t 0 2(2 -t 2 0 ) f (t 0 ) + 2 -t 2 0 2 k > 0.
Next, we focus on [-1.3, 0]. Let g(t) be the solution of the problem:

g (t) = √ 2 -t 2 2 k, g(0) = f 0 .
It is straightforward to check that

g(t) = 1 4 kt 2 -t 2 + k 2 arcsin(t/ √ 2) + f 0 on [-1.3, 0].
In particular, we notice that g(t) > 0 on [-1.3, 0] whenever f 0 is big enough. We are going to show that, under these conditions, g(t) ≤ f (t) on [-1.3, 0]. To this aim, define

Γ = {t ∈ [-1.3, 0] : g(t) ≤ f (t)}. Since g(0) = f 0 = f (0), it is 0 ∈ Γ = ∅. Moreover, Γ = (g-f ) -1 ((-∞, 0]) is closed in [-1.3, 0].
In order to show that Γ = [-1.3, 0], it suffices to prove that Γ is open, or, equivalently, if [t, 0] ⊂ Γ then [t -, 0] ⊂ Γ for some > 0. So, assume that [t, 0] ⊂ Γ. Then, f (t) ≥ g(t), and thus,

f (t) -g (t) = t 2(2 -t 2 ) f (t) < 0, for all t ∈ [t, 0]. Therefore, 0 t f (t)dt < 0 t g (t)dt, thus, f (t) > g(t).
By continuity, there exists > 0 such that f (t) ≥ g(t) for all t ∈ [t -, 0], and so, [t -, 0] ⊂ Γ. Summarizing, we have proved that property (i) above holds whenever f 0 is big enough. For property (ii), observe that f (0

) > 0. Moreover, f (-1) ≥ g(-1) ∞ if f 0 ∞. Therefore, f (-1) = - f (-1) 2 + k 2 < 0 if f 0 is big enough.
Hence, property (ii) also holds whenever f 0 is big enough.

In conclusion, we have proved the existence of a closed FLRW spacetime admitting a marginally trapped surface crossing expanding and collapsing regions. Remark 3.11 Two important subtleties have been omitted in previous development:

1. Notice that our approach breaks down at the points where function h becomes zero, i.e. for t(s) = ±1; and so, we cannot ensure, a priori, that our surface is marginally trapped at the corresponding points. This difficulty is overcome just by noting that the continuity of the length of the mean curvature vector, joined to the fact that this length is zero at the rest of the points, ensures that it is also zero here.

2. A similar argument shows that our surface is marginally trapped at the points where f vanishes (which must exist by property (ii)). From (3.12) it is straightforward to check that these points are isolated in I, and so, again a continuity argument on the length of the mean curvature vector ensures that the surface is also marginally trapped there.

Marginally trapped tubes in FLRW spacetimes

In the present paper we propose the following definition of marginally trapped tube (compare with the definition of MOTT in [START_REF] Andersson | Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes[END_REF]):

Definition 4.1 A smooth manifold G which admits a foliation {S λ :∈ λ ∈ Λ}, is a marginally trapped tube in a spacetime M if there is a smooth immersion of codimension 1, Φ : G → M, such that:

(A) Each Φ(S λ ) (λ ∈ Λ) is a marginally trapped surface in M, and
Next, we pay attention to the causal character of the embedding φ. In this sense, we are going to compute the first fundamental form induced by ḡ4 on our surface. To this aim, first we compute the derivatives φ t = 1, -e iθ sin(h(t))h (t), e iν cos(h(t))h (t) , φ θ = 0, ie iθ cos(h(t)), 0 , φ ν = 0, 0, ie iν sin(h(t)) .

A straightforward computation shows

φ * ḡ4 ≡   (f (t)h (t)) 2 -1 0 0 0 (f (t) cos(h(t))) 2 0 0 0 (f (t) sin(h(t))) 2   .
Since 0 < h(t) < π/2 for any t ∈ I, the derivatives φ θ and φ ν are always spacelike. Thus, everything depends on the derivative φ t . By recalling the expressions of φ t and h (t), we obtain

z(t) := ḡ4 (φ t , φ t ) = -1 + f (t)f (t) 4 + f (t) 2 2 . ( 4.1)
This expression can take any value, positive, negative or zero, depending only on the warping function f . We show a list of particular cases:

1. We choose non-negative real constants a and b such that a 2 = 4 + b 2 . In particular, a > b, which makes the function f : I = R → (0, ∞), f (t) = a cosh(t) + b sinh(t) welldefined. A simple computation shows z(t) = 0. Therefore, φ t is everywhere lightlike, and so is the corresponding marginally trapped tube.

2. We choose real constants c 1 , c 2 > 0. Then, the function f : R → (0, ∞), f (t) = 4+c 2 1 4c 2 t 2 + c 1 t + c 2 is well-defined. A simple computation shows z(t) = -3/4. This implies that φ t is everywhere timelike, and so is the corresponding marginally trapped tube.

3. We define the function f : (-1, 1) → (0, ∞), f (t) = If we show that z(t) ≥ 1, then we obtain that φ t is always spacelike, and thus, the marginally trapped tube is also spacelike. By simple computations, we have that for any t ∈ (-1, 1), z(t) + 1 ≥ 2 ⇐⇒ 0 ≥ t 2 (t 6 -4t 4 + 6t 2 -3). Standard computations give that the only real roots of the equality are t = 0, ±1. This readily proves z(t) ≥ 1 for any t ∈ (-1, 1).

4. We define the function f : R → (0, ∞), f (t) = 3 + cos(2t). A straightforward computation gives

z(t) = -1 + f (t)f (t) 4 + f (t) 2 2 = -1 + 4 cos 2 (2t)(3 + cos(2t)) 2 (3 -cos(4t)) 2 .
It is easy to check z(0) = 15 and z(π/4) = -1. Therefore, the marginally trapped tube changes its causal character with time.

5 Marginally trapped surfaces in t-slices of twisted spaces

Assume that the warping function f also depends on the fiber M 3 , i.e. f : I × M 3 → (0, ∞). Denote again by M 4 1 = I × f M 3 the Lorentzian twisted manifold given by the product manifold I × M 3 endowed with metric g 4 = -dt 2 + f 2 g 3 . As in Section 2, let ϕ : S → M 3 be an immersion of S in M 3 , ψ : M 3 → I × f M 3 the embedding of M 3 in I × f M 3 and φ : S → I × f M 3 the corresponding immersion of S in the twisted product, both in a tslice. Again from [9, p. 79], we have the following relation between the corresponding second fundamental forms: h φ (X, Y ) = h ϕ (X, Y ) + h ψ (X, Y ), whereby X, Y ∈ X(S).

From [26, Prop. 2], the second fundamental form h ψ is

h ψ (X, Y ) = -g 4 (X, Y ) grad g 4 f f = g 4 (X, Y ) ∂ t f f ∂ t - 1 f 3 grad g 3 f ,
where we have used that

grad g 4 f   = ij g ij 4 ∂f ∂x i ∂ ∂x j   = -∂ t f ∂ t + 1 f 2 grad g 3 f.
Hence,

h φ (X, Y ) = h ϕ (X, Y ) + g 4 (X, Y ) ∂ t f f ∂ t - 1 f 3 grad g 3 f .
Taking one half of the trace of the above expression, using an orthonormal frame {∂ t , {E i } 3 i=1 } w.r.t. the metric g 4 , i.e. E i = e i f whereby {e i } 3 i=1 is the corresponding orthonormal frame w.r.t. the metric g 3 on M 3 (and S), one obtains

H φ = 1 f 2 H ϕ - 1 f grad g 3 f + ∂ t f f ∂ t , (5.1)
where H φ and H ϕ stand for the mean curvature associated to h φ and h ϕ , respectively.

Theorem 5.1 A surface S contained in a t 0 -slice of twisted M 4 1 = I × f M 3 is marginally trapped iff its mean curvature vector satisfies:

H ϕ (•) - 1 f (t 0 , •) grad g 3 f (t 0 , •) = |∂ t f (t 0 , •)|.
On the other hand, S is trapped iff

H ϕ (•) - 1 f (t 0 , •) grad g 3 f (t 0 , •) < |∂ t f (t 0 , •)|.

3 1 =

 1 I × f S 2 (1/2), i.e. the 3-dimensional Lorentzian manifold formed by the product manifold I × S 2 (1/2) endowed with metric g 3 = -dt 2 + f 2 g 2 . Let ∇, D, ∇ and ∇ 2 be the Levi-Civita connections of M 4 1 , S 3 , M 3 1 and S 2 (1/2), resp. Note that the natural projection map of M

4

 4

3 1 1 . 3 1

 313 in such a way that for any local positive tangent frame {X, Y } on S 2 (1/2), the set {∂ t , X, Y } is a local positive frame on M 3 Let α : J ⊂ R → M be a unit spacelike Frenet curve with frenet apparatus {T = α, N, B} and κ > 0, τ . This means that the Frenet equations are

4 1 , and H φ = 1 2 4 1 . Then: Lemma 3 . 6

 41436 trace g 4 h φ the corresponding mean curvature vector of S in M The mean curvature of φ is given by

3 1

 3 is homeomorphic to a circle, the associated lift φ is a torus without boundary embedded in M 4 1 .

This definition may appear slightly modified in the bibliography.

Acknowledgments

The authors are partially supported by the Spanish MEC-FEDER Grant MTM2007-60731 and by the Grant P09-FQM-4496 (Consejera de Innovacin J. Andaluca with FEDER funds). Also, the second author is partially supported by the Research Foundation -Flanders project G.07.0432.

The authors thank Professors Marc Mars and J.M.M. Senovilla for their useful comments and suggestions. JLF also thanks the Department of Mathematics, Katholieke Universiteit Leuven, where part of this work started, for its kind hospitality during his stay there.

(B) Φ(S λ ) ∩ Φ(S µ ) = ∅ for any µ, λ ∈ Λ, µ = λ.

The second condition is required in order to avoid self-intersections in the direction of propagation of the tube.

With this definition in mind, we are going to apply our approach to obtain some information about marginally trapped tubes in FLRW spacetimes. We begin with the following direct consequences of the corresponding existence/non-existence results for marginally trapped surfaces in Section 2 (Cor. 2.2, 2.4):

Examples of marginally trapped tubes with different causality

Given a closed FLRW spacetime M 4 1 = I × f S 3 , first we are going to construct a marginally trapped tube foliated by Clifford tori (see Section 2) and defined for any time.

We define the smooth function

whose derivative is

Next, we define the embedding

φ(t, e iθ , e iν ) = t, e iθ cos(h(t)), e iν sin(h(t)) .

We notice that for each t ∈ I, the surface φ(t, -, -) :

, is a Clifford torus embedded in the t-slice. By comparing with the expression of the Clifford torus, we see that the length of the mean curvature of the torus (see (2.3)) at t is H u , with u = h(t). A straightforward computation shows

By Theorem 2.1, for each t, the torus is a marginally trapped surface.