A stellar model with diffusionin general relativity

# A stellar model with diffusion in general relativity

A. Alho
Center for Mathematical Analysis, Geometry and Dynamical Systems
Instituto Superior Técnico, Universidade de Lisboa
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
aalho@math.ist.utl.pt
S. Calogero
Department of Mathematical Sciences
Chalmers University of Technology, University of Gothenburg
Gothenburg, Sweden
calogero@chalmers.se
###### Abstract

We consider a spherically symmetric stellar model in general relativity whose interior consists of a pressureless fluid undergoing microscopic velocity diffusion in a cosmological scalar field. We show that the diffusion dynamics compel the interior to be spatially homogeneous, by which one can infer immediately that within our model, and in contrast to the diffusion-free case, no naked singularities can form in the gravitational collapse. We then study the problem of matching an exterior Bondi type metric to the surface of the star and find that the exterior can be chosen to be a modified Vaidya metric with variable cosmological constant. Finally, we study in detail the causal structure of an explicit, self-similar solution.

## 1 Introduction

The structure of singularities formed in the gravitational collapse of bounded matter distributions is a widely investigated problem in general relativity. The question of whether such singularities are naked, i.e., visible to far-away observers, or whether on the contrary they are safely hidden inside a black hole, has been the subject of innumerable works in the physical and mathematical literature. Nevertheless the problem remains poorly understood, even in the idealized case in which the collapsing body is spherically symmetric. A complete solution is only available for the gravitational collapse of a spherically symmetric dust cloud and can be found in the pioneering works by  Oppenheimer-Snyder [1] and Christodoulou [2]. (In [3, 4] Christodoulou analyses the question of existence and stability of naked singularities in the gravitational collapse of a massless scalar field. Notwithstanding the importance of these works, our focus is on matter models that describe material bodies, such as perfect fluids and kinetic particles [5].)

The Oppenheimer-Snyder model consists of a collapsing spatially homogeneous and isotropic dust interior, described by the contracting Friedmann-Lemaître solution, matched at a comoving boundary with a Schwarzschild exterior. Dropping the homogeneity assumption of the interior leads to the class of Lemaître-Tolman-Bondi solutions. These inhomogeneous stellar models were studied numerically by Eardley and Smarr [6], and analytically by Christodoulou [2]. It was shown that, in contrast to the case studied by Oppenheimer and Snyder, the spatially inhomogeneous collapse leads to the formation of naked singularities. See the prologue of [7] for an historical review on the gravitational collapse problem in general relativity.

In this paper we initiate the study of the gravitational collapse of matter subject to diffusion. We believe that the inclusion of diffusion dynamics in the gravitational collapse problem is meaningful both from a mathematical and physical point of view. From one hand it is well known that diffusion terms introduce a regularizing effect in the equations, which might prevent the formation of naked singularities in general relativity. On the other hand the physical relevance of diffusion phenomena is unquestionable and the applications in general relativity have been discussed in [8, 9, 10].

We begin our study with the simplest possible model, namely a spherically symmetric dust cloud undergoing diffusion in a cosmological scalar field. In this case the regularizing effect due to diffusion is overwhelming: The interior of the dust cloud is forced to be spatially homogeneous. By this fact one can easily infer that, in contrast to the diffusion-free scenario described above, naked singularities cannot form in the gravitational collapse of a spherically symmetric dust cloud in the presence of diffusion.

Another interesting property of our model is that, in contrast to the diffusion-free case [1, 11, 2], the exterior of the star cannot be static. The simplest spherically symmetric solution of the Einstein equations that can provide a suitable exterior region for our stellar model is given by a Vaidya type metric which includes a variable cosmological constant (a generalization of the radiating version of the Schwarzschild-(Anti-)de-Sitter family of solutions).

A detailed analysis of our model is given in the following sections. We conclude this introduction by outlining the diffusion theory of matter in general relativity. There exist two versions of this theory: a kinetic one [8], which is based on a Fokker-Planck equation for the particle density in phase-space, and a fluid one [9], which is the (formal) macroscopic limit of the kinetic theory. In the present paper we apply the fluid theory. We recall that the energy-momentum tensor and current density of a perfect fluid are

 Tμν=ρuμuν+p(gμν+uμuν),Jμ=nuμ, (1)

where is the rest-frame energy density, the pressure, the four-velocity and the particle density of the fluid. The diffusion behavior is imposed by postulating the equations

 ∇μTμν=σJν, (2a) ∇μ(nuμ)=0, (2b)

where is the diffusion constant, which measures the energy gained by the particles per unit of time due to the action of the diffusion forces. The second equation entails the conservation of the total number of fluid particles.

By projecting (2a) along the direction of and onto the hypersurface orthogonal to , we obtain the following equations on the matter fields:

 ∇μ(ρuμ)+p∇μuμ=σn, (3a) (ρ+p)uμ∇μuν+uνuμ∇μp+∇νp=0. (3b)

The system (3) on the matter variables must be completed by assigning an equation of state between the pressure, the energy density and the particles number density. In this paper we assume that the fluid is pressureless (dust fluid). As the energy-momentum tensor of the fluid is not divergence-free, see (2a), we have to postulate the existence of an additional matter field in spacetime to re-establish the (local) conservation of energy. The role of this additional matter field is that of the solvent matter in which the diffusion of the fluid particles takes place. The simplest choice is to assume the existence of a vacuum energy scalar field with energy-momentum tensor , which leads to the following Einstein equations for the spacetime metric (in units ):

 Rμν−12gμνR+ϕgμν=Tμν. (4)

The evolution equation on the scalar field determined by (4), the Bianchi identities, and the diffusion equation (2a) is

 ∇νϕ=σJν. (5)

## 2 The stellar model

Throughout the rest of the paper we assume that spacetime is spherically symmetric. The particle number density and the four-velocity at each point are also spherically symmetric and satisfy (2b). Under these assumptions one can cover an open neighborhood of the center of symmetry by a coordinate system such that the metric and the four-velocity take the form

 g=−e2Φdt2+e2ΨdR2+r2dΩ2u=e−Φ∂tin U.

Here are functions of and is the standard metric on . The center of symmetry is defined by the timelike curve . This coordinate system is called comoving and it is defined up to a transformation of the time and radial coordinates.

### 2.1 The interior

We assume that the interior of the star defines a region of spacetime covered by comoving coordinates. In particular, we assume that there exist and such that and within . The timelike hypersurface given by

 Σ : R=Rb,

will be identified with the surface of the star. To identify the region as the interior, we assume that the matter fields are nowhere vanishing on . The freedom in the choice of the comoving coordinates will be used to impose

 Φ(t,Rb)=0, (6)

so that is the proper time of observers at rest with respect to the boundary of the star, and

 r(0,R)=R, (7)

so that the comoving radius coincides initially, i.e., at time , with the radius function of the group orbits. It will now be proved that, when the fluid is pressureless, the diffusion dynamics compel the interior of our model to be spatially homogeneous. We denote , , for any function , and similarly denotes the derivative of any function of one variable.

###### Theorem 2.1.

Let and let be a spherically symmetric solution of (3)-(5) on . Then , , are functions of only and there exist a positive function and a constant such that

 g=−dt2+a(t)2(dR21−kR2+R2dΩ2),u=∂t. (8)
###### Proof.

Equation (3b) for reads , which together with the boundary condition (6) implies . Hence , as claimed. Equation (2b) now reads

 ∂t(nr2eΨ)=0. (9)

Moreover (5) implies that and are functions of only, related by

 ϕ(t)=ϕ(0)−σ∫t0n(s)ds. (10)

The Einstein equations (4) are:

 −2r′′r+2r′rΨ′−(r′r)2+e2Ψ((˙rr)2+2˙rr˙Ψ)+e2Ψr2=(ρ+ϕ)e2Ψ, (11) e−2Ψ(r′r)2−1r2−(˙rr)2−2¨rr+ϕ=0, (12) −¨Ψ−˙Ψ2−¨rr−˙rr˙Ψ+e−2Ψ(r′′r−r′rΨ′)+ϕ=0, (13) r′˙Ψ−˙r′=0, (14)

where the only non-zero component of the energy momentum tensor is . The last Einstein equation gives

 e2Ψ=r′2f(R)2, (15)

where is an arbitrary positive function of the radial variable. Substituting into (9) we obtain

 ∂t(nr2r′)=0.

Integrating with the boundary condition (7) we obtain

 r(t,R)=R(n(0)n(t))1/3:=Ra(t), (16)

where we denoted . Substituting (15) and (16) into (12) we obtain the equation

 [2a(t)¨a(t)+˙a(t)2−ϕ(t)a(t)2]R2+1−f2(R)=0.

The latter implies that there exists a constant such that

 1−f2(R)=kR2,2¨aa+(˙aa)2−ϕ=−ka2.

Finally (11) entails that is a function of only. ∎

From now on we assume that the interior fluid is pressureless. By rescaling the radial coordinate we may assume that , hence the spacetime metric in the interior of the star takes the Robertson-Walker form

 gint=−dt2+a(t)2(dR21−kR2+R2dΩ2),k=0,±1,R

where for , and the equations  (3)-(5) reduce to the following. The conservation of the particle number density (2b) gives , while satisfy

 (˙aa)2+ka2=13(ρ+ϕ), (18a) ¨aa=−16ρ+13ϕ, (18b) ˙ϕ=−3βa3, (18c) ˙ρ=−3ρ˙aa−˙ϕ, (18d)

where

 β=σn(0)a(0)3/3. (19)

The initial data for the system (18) are given on and consist of a quadruple , with , such that the constraint equation (18a) is satisfied at time , i.e.,

 (˙a0a0)2+ka20=13(ρ0+ϕ0).

In particular, the set of admissible initial data comprises a three-dimensional manifold, which we denote by . By a regular solution of (18) in the interval with initial data we mean a triple of functions , satisfying (18b)-(18d) and such that , for , and . Let be the maximal time of existence of a regular solution with a given set of initial data. We say that the regular solution is global if .

For , i.e., in the absence of diffusion, the interior reduces to a dust cloud in a spacetime with cosmological constant . In particular, for and , the exterior must be Schwarzschild, and we recover the well-known Oppenheimer-Snyder model [1]. The analysis of the latter model is greatly simplified by the fact that a closed formula solution is known for the Einstein equations, see eg. [12]. In contrast to this, the general solution to the system (18) is not known, and so the analysis of the stellar interior in the presence of diffusion is more complicated. This analysis has been carried out in [13], where a complete characterization of the qualitative behavior of solutions to the system (18) has been obtained using dynamical systems methods. The relevant properties for the present study are summarized in the following theorem:

###### Theorem 2.2.

The manifold of initial data can be written as the disjoint union of two three-dimensional submanifolds , such that

• For initial data in the submanifold , the corresponding regular solution of (18) is global and holds for all . We call these interior models “expanding”.

• For initial data in the submanifold , the corresponding regular solution of (18) exists only up to a finite time and as . Moreover there exists a constant such that

 a(t)∼c(t−ts)2/3,˙a(t)∼2c3(t−ts)−1/3,¨a(t)∼−2c3(t−ts)−5/3,as t→t−s.

We call these interior models “collapsing”.

The asymptotic behavior of the scalar factor claimed in (ii) corresponds to the property proved in [13] that collapsing (dust) solutions behave like the Friedmann-Lemaître diffusion-free solution in the limit toward the singularity. It implies that the spacelike singularity at is a curvature singularity, as the Kretschmann scalar satisfies

 K=O((t−ts)−4),as t→t−s.

In particular, spacetime is inextendible beyond the spacelike hypersurface and no outgoing light ray can emanate from the singularity. We obtain the following important corollary:

###### Corollary 2.1.

There exist no naked singularities in the spherical collapse of dust clouds undergoing diffusion in a cosmological scalar field.

We remark that the above result is independent of the spacetime exterior and applies to all collapsing models. In the absence of diffusion the formation of (locally and globally) naked singularities in the gravitational collapse of a dust cloud occurs for spatially inhomogeneous models, see [2]. In the present case the action of the diffusion forces prevents the formation of naked singularities by an explicit and compelling regularizing effect: it forces the dust interior to be spatially homogeneous.

Let us recall the definition of some important geometrical/physical quantities associated to the metric (17). Let

 χ(t,R)=1−kR2−˙a(t)2R2=(√1−kR2+˙a(t)R)(√1−kR2−˙a(t)R). (20)

The region in the interior where is the region of trapped surfaces, while defines the regular interior region. The hypersurface defines the apparent horizon. We define a local mass function through the identity

 χ=1−2mr−ϕ3r2,r(t,R)=a(t)R, (21)

which reduces to the standard definition of Misner-Sharp mass when  [11]. By (18a) we obtain the important identity

 m(t,R)=a(t)3R3ρ(t)6.

Moreover by (18c)-(18d) we have

 ρ(t)a(t)3=ρ0a30+3βt,

hence the mass function is given by

 m(t,R)=m(0,R)+R32βt. (22)

Thus the local mass is conserved only in the absence of diffusion and it is otherwise linearly increasing in time. The latter behavior is intimately connected with the irreversibility of the diffusion process. In fact, using that the entropy of a dust fluid equals the energy per particle, i.e., (see [9]), and due to the conservation of the total number of particles, i.e., , we have

 S(t)=ρ(t)n(t)=ρ(t)a(t)3n(0)a(0)3=S(0)m(0,R)m(t,R),

hence by (22) and recalling the definition (19) of the parameter we obtain that the entropy is linearly increasing in time:

 S(t)=S(0)+σt.

### 2.2 The exterior

We write the metric on the exterior using Bondi coordinates:

where are functions of and . The time coordinate is the ingoing (advanced) null coordinate for and the outgoing (retarded) null coordinate for . We assume that the comoving boundary between the interior and the exterior region is expressed as in Bondi coordinates, i.e.,

 Σ:r=rΣ(w),

so that in (23). As the boundary is assumed to be timelike, we require that

 A(w,r(w))(B(w,r(w))−2ε˙rΣ(w))>0,˙rΣ=drΣdw, (24)

i.e., the first fundamental form of has the signature . Moreover, letting be the transformation of the time coordinate on the boundary, we require , which means that the time orientation of spacetime does not change across the boundary. We recall that two metrics and may be matched on if and only if they satisfy the junction conditions that they induce the same first and second fundamental form on , see [11] and references therein.

###### Theorem 2.3.

The metrics (17) and (23) satisfy the junction conditions on the comoving boundary if and only if

• There holds

 rΣ(w)=a(t(w))Rb; (25)
• The transformation of variable satisfies

 ˙t(w)=A(w,rΣ(w))(√1−kR2b−ε˙a(t(w))Rb); (26)
• There holds

 B(w,rΣ(w))=A(w,rΣ(w))(1−kR2b−˙a(t(w))2R2b); (27)
• There holds , where

 Q(w)= [(B−2ε˙rΣ(w))((B−ε˙rΣ(w))∂rA+ε∂wA) −A(2¨rΣ(w)+(B−3ε˙rΣ(w))∂rB−ε∂wB)]r=rΣ(w). (28)
###### Proof.

The first and second fundamental forms induced by the interior metric (17) on are given respectively by

 ¯gint=−dt2+a(t)2R2bdΩ2,¯Kint=Rb√1−kR2ba(t)dΩ2,

while the same quantities induced by the exterior metric (23) are

 ¯gext=−A(w,rΣ(w))[B(w,rΣ(w))−2ε˙rΣ(w)]dw2+rΣ(w)2dΩ2,
 ¯Kext =−Q(w)2[A(w,rΣ(w))(B(w,rΣ(w))−2ε˙rΣ(w))]1/2dw2 +rΣ(w)(B(w,rΣ(w))−ε˙rΣ(w))[A(w,rΣ(w))(B(w,rΣ(w))−2ε˙rΣ(w))]1/2dΩ2.

Hence the junction condition gives immediately (25) as well as

 ˙t(w)=[A(w,rΣ(w))(B(w,rΣ(w))−2ε˙rΣ(w))]1/2. (29)

The junction condition is equivalent to and

 rΣ(w)(B(w,rΣ(w))−ε˙rΣ(w))[A(w,rΣ(w))(B(w,rΣ(w))−2ε˙rΣ(w))]1/2=Rba(t(w))√1−kR2b.

Using (25) in the latter equation we obtain

 B(w,rΣ(w))[B(w,rΣ(w))−2ε˙rΣ(w)]+˙rΣ(w)2=(1−kR2b)A(w,rΣ(w))[B(w,rΣ(w))−2ε˙rΣ(w)].

In the term we use

and so doing we obtain (27). Replacing (25) and (27) into (29) gives (26). ∎

In principle, any exterior metric that satisfies the conditions in the theorem can be matched to the interior. However a particularly simple and natural choice can be made as follows. We assume that in the whole exterior, so that (27) gives

 B(w,rΣ(w))=1−kR2b−˙a(t(w))2R2b. (30)

Using (20) and (21) in (30) we obtain

 B(w,rΣ(w))=1−2m(t(w),Rb)rΣ(w)−ϕ(t(w))3rΣ(w)2. (31)

Moreover the junction condition (d) in Theorem 2.3 becomes

 [2¨rΣ(w)+(B−3ε˙rΣ(w))∂rB−ε∂wB]r=rΣ(w)=0.

Using (18b) and (31) we may write the latter as

 [2r−6m(t(w),Rb)r2−2Br+∂rB]r=rΣ(w)=0. (32)

If we now require (32) to be valid for all , and not only on , and solve the resulting differential equation on subject to the boundary condition (31) at we obtain

 B(w,r)=1−2m(t(w),Rb)r−Λ(w)3r2,

where

 Λ(w)=ϕ(t(w)). (33)

Hence we obtain the following corollary.

###### Corollary 2.2.

The exterior metric can be chosen to be the Vaidya type metric with variable cosmological constant given by

 gext=−(1−2M(w)r−Λ(w)3r2)dw2+2εdwdr+r2dΩ2, (34)

where is given by (33) and

 M(w)=m(t(w),Rb) (35)

is the exterior mass function.

Conversely, if we now start with a metric of the form (34) and impose the matching conditions (31)-(32), we obtain the identities and , that is to say, the mass function and the cosmological scalar field must be continuous through the boundary in order for the metric (34) to be an admissible exterior.

The generalized Vaidya metric (34) solves the Einstein equation (4) with cosmological scalar field and the energy-momentum tensor

 Text=~ρdw2,~ρ=εr2(2dMdw+r33dΛdw),

where, by (18c), (22) and (26),

 dMdw=βR3b2(√1−kR2b−ε˙a(t(w))Rb), (36) dΛdw=−3βa(t(w))3(√1−kR2b−ε˙a(t(w))Rb). (37)

When , i.e., in the absence of diffusion, the functions and become constant and so . In this limit our model reduces to the Oppenheimer-Snyder model with cosmological constant [14]. The energy-momentum tensor describes a cloud of dust particles moving along the null directions orthogonal to . Observe that the dust particles in the exterior do not undergo diffusion in the cosmological scalar field. This outcome of the model is consistent with the well-known fact that particles moving along null directions are not subject to diffusion, see [15]. Requiring the weak energy condition to hold in the whole exterior region forces us to restrict to the outgoing Vaidya solution.

###### Proposition 2.1.

. Moreover the weak energy condition , for all , holds only for the outgoing Vaidya metric (34).

###### Proof.

By (36) and (37),

 (38)

by which the result follows immediately. ∎

We remark that the property implies that the surface of the star is not radiating, i.e., the star is thermally isolated from the exterior. In particular the null dust particles in the exterior are not emanated by the star, but rather they are created spontaneously by the decaying vacuum energy field in the exterior (see [9] for a similar behavior occurring in cosmology). Note also that for we have for . Hence if we consider a spatially homogeneous exterior, the interior can be chosen to be the ingoing Vaidya metric, leading to a void model with diffusion which generalises the well-known Einstein-Strauss model [16].

It is worth to point out the following important differences between our model and the analogous diffusion-free model studied in [17, 18]. Firstly, in the diffusion-free model the Vaidya solution can arise as the exterior of a fluid ball only if the pressure of the fluid is positive (dust is not allowed). Secondly the matching boundary in the diffusion-free model cannot be comoving. The latter property can be naturally regarded as being the cause of energy dissipation from the surface of the star as seen by comoving observers, see the discussion in [17, 18].

In view of Proposition 2.1 we restrict from now on to the generalized outgoing Vaidya metric (34), which we rewrite in standard notation as

 gext=−(1−2M(u)r−Λ(u)3r2)du2−2dudr+r2dΩ2. (39)

The apparent horizons of the metric (39) are the hypersurfaces in the exterior where

 B(u,r)=(1−2M(u)r−Λ(u)3r2)=0.

As , see (27), and owing to (20), the interior and exterior apparent horizons intersect on the boundary.

## 3 Self-similar solutions

In this section we study in detail the causal structure of a particular solution to our model. Recall that the interior is assumed to be pressureless. Assuming in addition that it is self-similar, we obtain the following solution of (4):

 a(t)=δkt, (40a) ϕ(t)=3β2δka(t)−2, (40b) ρ(t)=3βδka(t)−2, (40c) where δk is the real (positive) solution of the polynomial equation δ3+kδ−3β2=0. (40d)

The Penrose diagrams of this solution are given in Figure 1. It has been shown in [13] that, for an open set of initial data, solutions of both types and have an intermediate behavior that is close to the special solution (40).

Note that we shifted the origin of time so that the singularity appears at . This is a curvature singularity as the Ricci scalar curvature satisfies

 Rint=9βδka(t)−2→+∞as a(t)→0+.

Toward the future the solution is forever expanding () without acceleration (). There is only one apparent horizon for this solution, located at

 RAH=1√δ2k+k. (41)

Let the boundary of the star. We distinguish three cases:

• ; the interior has an apparent horizon in this case.

• ; the boundary of the star coincides with the apparent horizon.

• ; the interior has no apparent horizon.

Now, let the matching surface as seen by exterior observers. We assume that the exterior metric is given by (39). From (25), (26) we obtain

 u=Ckt,rΣ(u)=xΣu,xΣ=RbδkCk,Ck=1√1−kR2b+δkRb>0,

and (39) becomes

 gext=−(1−λ1ur−λ2r2u2)du2−2dudr+r2dΩ2,r>xΣu, (42)

where

 λ1=βR3bCk,λ2=βC2k2δ3k.

The mass and the cosmological scalar field in the exterior are given by

 M(u)=λ1u/2,Λ(u)=3λ2/u2.

Note also that the metric (42) has a curvature singularity at , for its Kretschmann scalar is given by

 K=12λ21u2r6+24λ22u4.

The apparent horizons of the metric (42) are the hypersurfaces where , where

 B(u,r)=h(ru),h(x)=1−λ1x−λ2x2. (43)
###### Proposition 3.1.

The following holds:

• In case (i) there is no apparent horizon in the exterior region and for all .

• In case (ii) the apparent horizon of the metric (42) coincides with the apparent horizon of the interior, as well as with the matching surface:

 rAH(u)=rΣ(u)≡RAH=Rb.

Moreover for all .

• In case (iii) there exists such that the metric (34) has an apparent horizon at . Moreover for and for .

###### Proof.

For the proof it suffices to notice that the function in (43) attains its maximum at and . ∎

The model under discussion is self-similar, with the lines , , on the -plane being tangent to the homothetic vector field in the exterior. We call such curves homothetic curves.

###### Theorem 3.1.

There exists such that the homothetic curve is spacelike for , null for and timelike for . In case (iii) there holds . Moreover the homothetic curve is the first ingoing radial null geodesics that escapes to null-infinity.

###### Proof.

The metric induced on the hypersurface in the exterior is given by

 −f(x)du2+r2dΩ2,f(x)=h(x)+2x,

where is the function defined in (43). The hypersurface is spacelike if , timelike if and null if . A straightforward calculation shows that

 f(xΣ)=1C2k>0.

As when , there exists one (and only one) such that for , and for . The first part of the result follows. Moreover and so when the exterior apparent horizon exists we have . As to the last statement, the equation for the ingoing radial null geodesics is

 drdu=−12B(u,r(u)).

Consider a solution such that , for some . Let . As

 ˙x(u)=−12uf(x(u)),

then is decreasing for . It follows that , for all . Hence , for all and therefore

 x(u)≤x∗−C2loguu0.

We conclude that for large enough the ingoing light ray must hit the boundary in cases (i)-(ii) and the apparent horizon in case (iii). In the latter case the ingoing light ray will enter the regular region and therefore hit the surface of the star in finite time. In conclusion, all ingoing light rays escaping to infinity other than the null homothetic curve must satisfy , for all , which concludes the proof of the theorem. ∎

The result of the previous theorem leads to the Penrose diagram shown in Figure 2.

## Acknowledgements

A. A. is supported by the projects EXCL/MAT-GEO/0222/2012, PTDC/MAT-ANA/1275/2014, by CAMGSD, Instituto Superior Técnico, by FCT/Portugal through UID/MAT/04459/2013, and by the FCT Grant No. SFRH/BPD/85194/2012.

The authors would like to thank J. Natário for numerous discussions and the anonymous referee for valuable comments that helped to improve the paper. Furthermore, A. A. thanks the Department of Mathematics at Chalmers University, Sweden, for the very kind hospitality.

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