General spherically symmetric elastic stars in Relativity

Abstract

The relativistic theory of elasticity is reviewed within the spherically symmetric context with a view towards the modeling of star interiors possessing elastic properties such as theones expected in neutron stars. Emphasis is placed on generality in the main sections of the paper, and the results are then applied to specific examples. Along the way, a few general results for spacetimes admitting isometries are deduced, and their consequences are fully exploited in the case of spherical symmetry relating them next to the the case in which the material content of the spacetime is some elastic material. This paper extends and generalizes the pioneering work by Magli and Kijowski [1], Magli [2] and [3], and complements, in a sense, that by Karlovini and Samuelsson in their interesting series of papers [4], [5] and [6]. Comment: 23 pages

Full text

arXiv:0911.1425v1 [gr-qc] 7 Nov 2009
General spherically symmetric elastic stars in
Relativity
I. Brito∗, J. Carot♮, E.G.L.R. Vaz∗
♮Departament de F´ısica, Universitat de les Illes Balears,
Cra Valldemossa pk 7.5, E-07122 Palma de Mallorca, Spain
∗Departamento de Matem´atica para a Ciˆencia e Tecnologia,
Universidade do Minho, Guimar˜aes, Portugal
Abstract
The relativistic theory of elasticity is reviewed within the spherically symmetric context
with a view towards the modeling of star interiors possessing elastic properties such as the
ones expected in neutron stars. Emphasis is placed on generality in the main sections of
the paper, and the results are then applied to specific examples. Along the way, a few
general results for spacetimes admitting isometries are deduced, and their consequences
are fully exploited in the case of spherical symmetry relating them next to the the case in
which the material content of the spacetime is some elastic material. This paper extends
and generalizes the pioneering work by Magli and Kijowski [1], Magli [2] and [3], and
complements, in a sense, that by Karlovini and Samuelsson in their interesting series of
papers [4], [5] and [6].
1 Introduction
The interest of a relativistic theory of elasticity is twofold; on the one hand there is
its purely theoretical interest, namely that of providing a relativistic extension of a
well-known (and very fruitful) classical theory; on the other hand and on theoretical
grounds, it is expected that neutron stars possess a solid crust with elastic properties
which may help explaining certain observational issues (see [4] for a thorough account
of these and other related features). Further, anisotropy in pressures is a phenomenon
occurring in many situations of equilibrium which are of interest in astrophysics and
whose corresponding dynamics has been thoroughly studied (see for instance [7] and
references therein); the assumed point of view however has been an heuristic one,
without providing mechanisms explaining how the anisotropy in pressures may arise
and using instead ad hoc assumptions. Magli and Kijowski [1] and Magli [3] have
shown that in spherical symmetry, the anisotropy in pressures arises quite naturally
as the relativistic extension of the classical (non-relativistic) non-isotropic stress in
elasticity theory. See also [8] for an excellent study of the static case in spherical
symmetry, where existence theorems for regular solutions near the center are proven
1

under rather mild, physically meaningful, hypotheses. Beig and Schmidt [9] have
shown that, in general, the field equations for elastic matter can be cast into a first-
order symmetric hyperbolic system and that as a consequence, local-in-time existence
and uniqueness theorems may be obtained under various circumstances.
The aim of this paper is to extend and generalize the work presented in [1] and [3]
as well as to set up a set of mathematical tools and equations that may facilitate
the obtention of exact solutions to Einstein´s Field Equations (EFE) describing the
interior of elastic materials and satisfying the Dominant Energy Condition (DEC).
The paper is organized as follows: in the next section we provide a brief account of
the theory of relativistic elasticity, much along the lines followed in [1], [3] and [4],
but we shall also include some comments on the relationship between the isometries
in the material space and in the spacetime. Most of the results in that section are
well known and could be found in the above references, but we are still including
them in order to set up the notation which will be followed in the remainder of the
paper as well as for making the present paper more self-contained. Section 3 contains
a digression on spherically symmetric spacetimes and the restrictions that such an
assumption imposes on the physics in these spacetimes, which we then apply it to the
case of elastic materials. In section 4 EFEs are obtained for the general case and some
particular cases are commented upon. In section 5 we analyze in detail the case of
shearfree solutions, paying special attention to the fulfilment of the dominant enrgy
condition as well as to the necessary and sufficient conditions that must be satisfied
for an equation of state to be admitted; finally we p resent a few selected examples;
these include the analysis of the elasticity difference tensor for the nonstatic case,
most along the lines followed in a previous paper by two of the authors [13].
2 Relativistic elasticity revisited
Let (M, g) be a spacetime, Mthen being a 4-dimensional Hausdorff, simply connected
manifold of class C2at least, and ga Lorentz metric of signature (−,+,+,+). The
material space Xis a 3-dimensional manifold endowed with a Riemannian metric γ,
the material metric; points in Xcan then be thought of as the particles of which
the material is made of. Coordinates in Mwill be denoted as xafor a= 0,1,2,3,
and coordinates in Xas yA,A= 1,2,3. The material metric γis not a dynamical
quantity of the theory, but it is frozen in the material, and it roughly describes
distances between neighboring particles in the relaxed state of the material.
The spacetime configuration of the material is said to be completely specified when-
ever a submersion ψ:M→Xis given; if one chooses coordinate charts in Mand X
as above, the coordinate representative of ψis given by three fields
yA=yA(xb), A = 1,2,3
and the physical laws describing the mechanical properties of the material can then
be expressed in terms of a hyperbolic second order system of PDE. The differential
map ψ∗:TpM→Tψ(p)Xis then represented in the above charts by the rank 3 matrix
yA
bp, yA
b=∂yA
∂xbA= 1,2,3, b = 0,1,2,3
2

which is sometimes called relativistic deformation gradient. Since ψ∗has maximal
rank 3, its kernel is spanned at each point by a single timelike vector which we may
take as normalized to unity, the resulting vector field, say ~u =ua∂a, satisfies then
yA
bub= 0, uaua=−1, u0>0
the last condition stating that we choose it future oriented; ~u is called the velocity
field of the matter, and in the above picture in which the points in Xare material
points, it turns out that the spacetime manifold M(or, to be more precise, an open
submanifold of it) is then made up by the worldlines of the material particles, whose
tangent vector is precisely ~u.
The material space is said to be in a locally relaxed state at an event p∈Mif, at
p, it holds kab ≡(ψ∗γ)ab =hab where hab =gab +uaub. Otherwise, it is said to
be strained, and a measurement of the difference between kab and hab is the strain,
whose definition varies in the literature; thus, it can be defined simply as Sab =
−1
2(kab −hab) = −1
2(kab −uaub−gab). We shall follow instead the convention in [3]
and use
Kab ≡kab −uaub(1)
Notice that Ka
bub=ua, and therefore one of its eigenvalues is 1. Definitions using
the logarithm of the above tensor1also appear in the literature as that allows simple
interpretations of the associated algebraic invariants, see e.g. [1] and [8].
The strain tensor determines the elastic energy stored in an infinitesimal volume
element of the material space (or energy per particle), hence that energy will be a
scalar function of Kab. This function is called constitutive equation of the material,
and its specification amounts to the specification of the material. We shall represent
it as v=v(I1, I2, I3), where I1, I2, I3are any suitably chosen set of scalar invariants2
associated with and characterizing Kab completely. Following [3] we shall choose
I1=1
2(TrK−4)
I2=1
4TrK2−(TrK)2+ 3
I3=1
2(detK−1) ,
(2)
Notice that for Kab =gab (equivalently kab =hab ) the strain tensor Sab is zero, that is:
the induced metric on the rest frame of an observer moving with four-velocity ~u,h,
coincides with the material metric γ(its pull-back by ψ) describing the relaxed state
of the material; thus it makes sense to have zero elastic energy stored. It is immediate
to check from the above expressions that in this case one has I1=I2=I3= 0.
The energy density ρwill then be the particle number density ǫtimes the constitutive
equation, that is
ρ=ǫv(I1, I2, I3) = ǫ0√det K v(I1, I2, I3) (3)
1With one index raised, thus one has a linear operator which turns out to be positive and self-
adjoint, its logarithm being then well-defined
2Recall that one of the eigenvalues is 1, therefore, there exist three other scalars (in particular
they could be chosen as the remaining eigenvalues) characterizing Ka
bcompletely along with its
eigenvectors.
3

where ǫ0is the particle number density as measured in the material space, or rather,
with respect to the volume form associated with kab = (ψ∗γ)ab, and ǫis that with
respect to hab; see [10] for a proof of the above equation. In some references (e.g.
[3]), the names ρand ǫare exchanged and the density measured w.r.t. kab = (ψ∗γ)ab
(ǫ0in our notation) is then called “density of the relaxed material” (see the above
comments on the meaning of γ), whereas that measured w.r.t. hab is referred to as
the “density in the rest frame”.
We next turn our attention towards the energy-momentum tensor of an elastic ma-
terial. Before proceeding, it will be useful to recall that any symmetric, second order
covariant tensor field may be decomposed with respect to a timelike unit vector field
~v, vava=−1 as follows:
Tab =ρvavb+phab +Pab +vaqb+qavb(4)
where hab =gab +vavb,Pab =hm
ahn
b(Tmn −3phmn), qa=−(Tab vb+ρva), ρ=Tabvavb,
p=1
3habTab . From the definitions of these variables it readily follows
habvb= 0, Pab vb=gabPab = 0,and qava= 0.
In the case that Tab represents the energy-momentum tensor of some material distri-
bution, ρ, p, Pab, qaare respectively the energy density, isotropic pressure, anisotropic
pressure tensor and heat flow that a family of observers moving with four-velocity ~v
would measure at every point in the spacetime.
In the case of elastic matter, it can be seen using the standard variational principle
for the Lagrangian density Λ = √−gρ (see for instance [1] or [4] and the beginning
of section 5 for further details) that the energy-momentum takes the form, when
decomposed with respect to ~u, the velocity of the matter:
Tab =ρuaub+phab +Pab (5)
where all the definitions are the same as the ones given above substituting ~u for ~v,
i.e.: hab =gab +uaub,Pab =hm
ahn
b(Tmn −3phmn), ρ=Tabuaub,p=1
3habTab and they
satisfy habub= 0, Pab ub=gabPab = 0; thus in particular one gets qa= 0 and the
resulting tensor is of the diagonal Segre type {1,111}or any of its degeneracies, ~u
being its (unit) timelike eigenvector (see [11]).
This means that an orthonormal tetrad exists {ua, xa, ya, za}(with uaua=−1, xaxa=
yaya=zaza= +1 and the mixed products zero) with respect to which Tab may be
written as
Tab =ρuaub+p1xaxb+p2yayb+p3zazb, p =1
3(p1+p2+p3),
hab =xaxb+yayb+zazb,etc.(6)
It is interesting to mention that the Dominant Energy Condition (DEC), see for
instance [11], is fulfilled if and only if
ρ≥0,|pA| ≤ ρ, A = 1,2,3.(7)
4

3 On symmetries and their consequences on physics
Let (M, g) admit a Killing Vector (KV) ~
ξ, i.e.: in any coordinate chart xa,L~
ξgab = 0.
It is then immediate to show that L~
ξRab =L~
ξGab =L~
ξRa
bcd = 0, etc. and, from
EFEs it then follows that L~
ξTab = 0 where Tdenotes the energy-momentum tensor
describing the material in the spacetime.
It is easy to show that, if ~v is a non-degenerate unit (i.e.: vava=ǫwith ǫ=±1)
eigenvector of Tab with corresponding eigenvalue λ, then
L~
ξλ=L~
ξva= 0.(8)
We next include a short proof of this.
(i) Taking the Lie derivative of Tabvb, with respect to ~
ξand since we are
assuming that Tabvb=λvaone gets
L~
ξ(Tab vb) = L~
ξ(λ va) = λL~
ξva+vaL~
ξλ. (9)
On the other hand,
L~
ξ(Tab vb) = Tab L~
ξvb.(10)
Equating (9) and (10) and then contracting with va, yields
λ vb(L~
ξvb) = λ vaL~
ξva+L~
ξλ. (11)
Therefore L~
ξλ= 0,since as ~
ξis a KV vaL~
ξva=vaL~
ξva.
(ii) Substituting this result into (9) and using (10) one obtains Tab L~
ξvb=
λL~
ξva.Therefore, vaand the vector wa≡ L~
ξvaare eigenvectors of Tab
associated with the same eigenvalue. Since this is non-degenerate, ~v and
~w have to be proportional, i.e. L~
ξva=αva, for some real value α, however
this αmust be zero as ~v is unit and ~
ξis a KV, indeed
0 = L~
ξvava= 2vaL~
ξva= 2αvava,hence α= 0.
Under the hypothesis that ~
ξis a Killing vector, and on account of the above consid-
erations the following conditions hold in the case that Tab represents elastic matter
and is therefore of the form (5)
L~
ξgab = 0 ⇒ L~
ξρ= 0,L~
ξua= 0,L~
ξhab = 0,L~
ξPab = 0,L~
ξp= 0.
The first two are just the specialization of the above comments to the case ~v =~u and
λ=ρ. The vanishing of L~
ξhab follows then from the vanishing of the Lie derivative of
the metric and that of ua; notice that L~
ξhab = 0 as well. Next, since p=1
3habTab it also
follows that its Lie derivative with respect to the KV vanishes as those of Tab and hab
do, and the vanishing of Pab follows then immediately from the expression (5) and the
vanishing of the Lie derivatives of all the other terms. Therefore we have shown that
5

matter 4−velocity, pressure, density, anisotropic tensor all stay invariant along the
Killing vectors of the space-time, together with the projection tensor hab =gab +uaub.
It is interesting to notice that, for a general energy momentum tensor such as (4),
and if one assumes that L~
ξua= 0, it also follows that L~
ξqa= 0; but in this case that
assumption has to be made, as ~u is no longer an eigenvector of the energy-momentum
tensor.
Similar conclusions (although not the same) can be drawn when ~
ξis a proper ho-
mothetic vector field; i.e.: L~
ξgab = 2kgab, with k6= 0; in which case one also has
L~
ξTab = 0 but then, for instance, L~
ξua=−kuaand also L~
ξρ=kρ, etc.
In this paper we shall be concerned with the case of elastic materials in spheri-
cally symmetric spacetimes. We next explore the consequences that the existence of
symmetries has on the material content of a spacetime. Most of the developments
following are well known although disperse in the literature, we collect them here for
the sake of completeness.
As it is well known, for a spherically symmetric spacetime, coordinates xa=t, r, θ, φ
exist (and are non-unique) such that the line element can be written as
ds2=−a(r, t)dt2+b(r, t)dr2+r2dθ2+r2sin2θdφ2(12)
with aand bpositive and independent of θand φ. This metric possesses three
Killing vectors, namely ~
ξ1=−cos φ ∂θ+ cot θsin φ ∂φ,~
ξ2=∂φand ~
ξ3=
−sin φ ∂θ−cot θcos φ ∂φwhich generate the 3-dimensional Lie algebra so(3).
To start with, we show that any timelike vector field ~v that remains invariant along
the three Killing vectors is necessarily of the form
~v =vt(t, r)∂t+vr(t, r)∂r.
Using L~
ξ2va= 0 for a= 0,1,2,3 we conclude that all the components vaare inde-
pendent of φ. Then, the expression
L~
ξ1va=ξc
1va
,c −vcξa
1,c,
for a= 0,1 gives that v0and v1are also independent of θand for a= 2,3 yields
v2=v3= 0.
It should be noticed that it always exists a coordinate transformation taking t, r into
t′, r′such that ~v =vt′(t′, r′)∂t′and the metric (12) reads then
ds2=−a(r, t)dt2+b(r, t)dr2+Y2(r, t)dθ2+ sin2θdφ2(13)
where primes have been dropped for convenience. These new coordinates are the so
called comoving coordinates; one can see this either by direct computation, showing
that one such coordinate change is always possible, or else, by showing first that
any vector field such as ~v above is always hypersurface orthogonal, that is: ωab =
v[a;b]+ ˙v[avb]= 0, then it follows that va∝∂at′for some function t′, choosing this
function as the new time coordinate, one can readily show the above result.
Next, for any symmetric, second order tensor Pab, which is traceless, invariant under
the three KVs above, and orthogonal to a vector such as ~v above (i.e.: timelike
spherically symmetric), that is:
6

(i) Pabvb= 0
(ii) gabPab = 0
(iii) L~
ξAPab = 0,for A= 1,2,3
it follows that Pab is proportional to the shear tensor of ~v whenever the latter is
non-zero, namely:
Pab ∝σab, σab =v(a;b)+ ˙v(avb)−1
3θhab (14)
where θ=va
;ais the expansion, ˙va=va;bvbis the acceleration and round brackets
denote symmetrization as usual.
This can be proven easily by making use of the comoving coordinate system referred
to above, and imposing the various conditions (i - iii); also, a more general proof is
possible in the context of warped spacetimes (of which the spherically symmetric ones
are special instances), see [12].
In the comoving coordinate system above, one can see by direct computation that
the shear σab is
σab = diag 0,1
3
(btY−2Ytb)
√aY ,−1
6
Y(btY−2Ytb)
√aY ,−1
6
Y(btY−2Ytb)
√aY sin2θ(15)
and therefore this field is shearfree if and only if
b(r, t) = F2(r)Y2(r, t),
in which case it is always possible, by means of an obvious redefinition of the coordi-
nate r, bring the metric to the form
ds2=−a(r, t)dt2+Y2(r, t)dr2+dθ2+ sin2θdφ2.(16)
For the class of spacetimes we shall be interested in, namely elastic, spherically sym-
metric, all the above apply for ~u (the velocity of matter), as it is indeed the unit
timelike eigenvector of Tab given by (5), and Pab the anisotropic pressure tensor since,
as discussed previously, it is invariant under the KVs the spacetime possesses and is
also traceless and orthogonal to ~u, therefore we have that, whenever the shear of ~u is
non-zero
Pab = 2λσab, σab =u(a;b)+ ˙u(aub)−1
3θhab (17)
where θ=ua
;aand ˙ua=ua;bub, and λ=λ(t, r) is some function, therefore, for the
generic (non shearfree) case, it is always possible to treat, at least formally3, the
elastic material as a viscous fluid with zero heat flow. This interpretation would
indeed break down in the case in which ~u is shearfree.
3There is a further requirement for a fully physically meaningful interpretation as a viscous fluid,
namely that λ < 0 in which case the kinematical viscosity would be η=−λ > 0.
7

4 Elasticity in spherical symmetry
Let us now consider in more detail the problem of elasticity in a spherically symmetric
spacetime (M, ¯g) with associated material space (X, ¯γ).
The results given in this section generalize those in [3] in the sense that here we
consider a non flat material metric ¯γ, while, when referring to quantities and results
in [3], we shall use non-barred quantities (hence the bars on the spacetime metric and
the material metric in our notation).
Recalling the notation and results in section 2, we shall demand that the submersion
ψ:M−→ Xpreserves the KVs, that is: ψ∗(~
ξA) = ~ηAare also KVs on X.
This implies that the metric ¯γis also spherically symmetric and therefore coordinates
yA= (y, ˜
θ, ˜
φ) exist with y=y(t, r), ˜
θ=θand ˜
φ=φ, and are such that ~ηA=~
ξAare
KVs of the metric ¯γ. Thus, the line elements of ¯gand ¯γmay be written as:
d¯s2=−¯a(t, r)dt2+¯
b(t, r)dr2+r2dθ2+r2sin2θdφ2(18)
d¯
Σ2=f2(y)(dy2+y2dθ2+y2sin2θdφ2),(19)
Notice that this last expression is completely general, as any 3-dimensional spherically
symmetric metric is necessarily conformally flat, as it is immediate to show.
The results in [3] correspond to f(y) = 1, and the relation between ¯γand the flat
material metric γused in [3] is given by
¯γAB =f2(y)γAB.(20)
Next, attention should be payed to the canonical definition of the energy-momentum
tensor used by [3]:
Ta
b=1
√−g∂Λ
∂yA
a
yA
b−δa
bΛ.(21)
As shown in [14], this canonical definition of the energy-momentum tensor coincides
with the symmetric definition of the energy-momentum tensor, used by other authors,
up to a sign, which is a particular case of the general Belinfante-Rosenfeld theorem
[15], [16].
Denoting by ¯
kthe pull-back by ψof the material metric ¯γ, that is: ¯
k=ψ∗(¯γ), one
has:
¯
ka
b= ¯gac¯
kcb = ¯gac¯γCB yC
cyB
b=f2(y)¯gacγCB yC
cyB
b=f2(y)¯gackcb
=f2(y)¯gac[ ˙y2δ0
cδ0
b+ ˙yy′(δ0
cδ1
b+δ0
bδ1
c) + y′2δ1
bδ1
c+y2δ2
bδ2
c+y2sin2θδ3
bδ3
c],
or
¯
ka
b=



−f2(y)( ˙y2/¯a)−f2(y)( ˙yy′/¯a) 0 0
f2(y)( ˙yy′/¯
b)f2(y)(y′2/¯
b) 0 0
0 0 f2(y)y2/r20
0 0 0 f2(y)y2/r2




,(22)
8

where a dot indicates a derivative with respect to tand a prime a derivative with
respect to r.
The velocity field of the matter, defined by the conditions ¯uayA
a= 0, ¯gab ¯ua¯ub=−1
and ¯u0>0, can be expressed as
¯ua=¯
Γ
√¯a1,−˙y
y′,0,0,(23)
where
¯
Γ≡ 1−¯
b
¯a˙y
y′2!−1
2
.(24)
Therefore the projection tensor is
¯
ha
b=δa
b+ ¯ua¯ub=



1−¯
Γ2−¯
Γ2(¯
b˙y/¯ay′) 0 0
¯
Γ2( ˙y/y′) 1 + ¯
Γ2(¯
b/¯a)( ˙y/y′)20 0
0 0 1 0
0 0 0 1




.(25)
We will use an orthonormal tetrad and write the metric as ¯gab =−¯ua¯ub+ ¯xa¯xb+
¯ya¯yb+ ¯za¯zb,such that:
¯ua=¯
Γ
√¯a,−˙y
y′
¯
Γ
√¯a,0,0¯ua=−√¯a¯
Γ,−¯
b
√¯a
˙y
y′
¯
Γ,0,0
¯xa= −√¯
b
¯a
˙y
y′
¯
Γ,¯
Γ
√¯
b,0,0!¯xa=˙y
y′p¯
b¯
Γ,p¯
b¯
Γ,0,0
¯ya=0,0,1
r,0¯ya= (0,0, r, 0)
¯za=0,0,0,1
rsin θ¯za= (0,0,0, r sin θ),
where ¯
Γ is the auxiliary quantity given in (24). Here, ¯uais the matter velocity and
¯xa, ¯yaand ¯zaare spacelike eigenvectors of the pulled-back material metric ¯
ka
b. From
our developments in section 2, it is immediate to see that the pressure tensor has the
same eigenvectors as ¯
kab and can be written, for the space-time under consideration
as ¯
Pab = ¯p1¯xa¯xb+ ¯p2( ¯ya¯yb+ ¯za¯zb). Therefore, (6) yields
¯
Tab = ¯ρ¯ua¯ub+ ¯p1¯xa¯xb+ ¯p2(¯ya¯yb+ ¯za¯zb),(26)
where ¯ρis the energy density, ¯p1, the radial pressure and ¯p2, the tangential pressure.
The results in [3] can be easily recovered by setting f(y) = 1 above.
Now, much clarity is gained by making use of the comoving coordinates adapted to
~u, the timelike eigenvector of the energy-momentum tensor, which were introduced
in the above section. The form of the metric is given by
ds2=−¯a(r, t)dt2+¯
b(r, t)dr2+¯
Y2(r, t)dθ2+ sin2θdφ2,(27)
~u, being then
ua=1
√¯a,0,0,0, ua=−√¯a, 0,0,0,(28)
9

hence, we have for the material space (M, ¯γ) that coordinates yA= (y, ˜
θ, ˜
φ) exist
with y=y(r), ˜
θ=θand ˜
φ=φ, as follows from the condition yA
aua= 0 and the
requirement that ψ∗(~
ξA) = ~ηAare KVs of the metric ¯γ.
Further, and since the line element of the material space is
d¯σ2=f2(y)dy2+y2dθ2+ sin2θdφ2,
with y=y(r), no generality is lost by setting y=r, as this amounts to a redefinition
of the rcoordinate in spacetime, and leaves unchanged the form of the metric (27)
as well as that of the velocity field of the matter (28). We shall do that in the sequel.
Thus, the pulled-back material metric ¯
k(22) is
¯
ka
b= ¯gac¯
kcb = ¯gac¯γCB yC
cyB
b=f2(y)¯gacγCB yC
cyB
b=f2(y)¯gackcb
=f2(y)¯gac[y′2δ1
bδ1
c+y2δ2
bδ2
c+y2sin2θδ3
bδ3
c],
where a prime indicates a derivative with respect to r, which upon setting y=ras
discussed above it simplifies further to:
¯
ka
b=



0 0 0 0
0f2(r)(1/¯
b) 0 0
0 0 f2(r)r2/Y 20
0 0 0 f2(r)r2/Y 2




.(29)
The operator ¯
Ka
b= ¯gac¯
kcb −¯ua¯ub, introduced in section 2 and used to measure the
state of strain of the material has one eigenvalue equal to 1 (corresponding to the
eigenvector ~u), while the other eigenvalues are
¯s=f2(y)y2
¯
Y2=f2(r)r2
¯
Y2
¯η=f2(y)y′2
¯
b=f2(r)
¯
b,
(30)
and ¯shas algebraic multiplicity two.
The three invariants I1, I2, I3of ¯
Kintroduced in (2) have the following expressions
¯
I1=1
2Tr ¯
K−4=1
2(¯η+ 2¯s−3)
¯
I2=1
4hTr ¯
K2−Tr ¯
K2i+ 3 = −1
2¯s2+ 2¯η¯s+ ¯η+ 2¯s−3
¯
I3=1
2det ¯
K−1=1
2¯η¯s2−1
(31)
In [3], the energy-momentum tensor was calculated from these invariants for a flat
material metric. A similar calculation shows that, for the non-flat material metrics
under consideration, the same expression holds so that
¯
Ta
b= ¯ρ δa
b−∂¯ρ
∂¯
I3
det ¯
K¯
ha
b+Tr ¯
K∂¯ρ
∂¯
I2−∂¯ρ
∂¯
I1¯
ka
b−∂¯ρ
∂¯
I2
¯
ka
c¯
kc
b.(32)
10

Therefore, the nonzero components are
¯
T0
0= ¯ρ,
¯
T1
1= ¯ρ−y′2
¯
bX,
¯
T2
2=¯
T3
3= ¯ρ−y2
¯
Y2X+∂¯ρ
∂¯
I2−f2(y)y2
¯
Y2
∂¯ρ
∂¯
I3f4(y)y2
¯
Y2−f4(y)y′2
¯
b,(33)
where
X=f2(y)∂¯ρ
∂¯
I1−∂¯ρ
∂¯
I21 + 2 f2(y)y2
¯
Y2+∂¯ρ
∂¯
I3
f4(y)y4
¯
Y4.(34)
The rest frame energy per unit volume4, ¯ρ, is defined by
¯ρ= ¯ǫ¯v=ǫ0¯s√¯η¯v(¯s, ¯η),(35)
where, as discussed in section 2, ¯v= ¯v(¯
I1,¯
I2,¯
I3) = ¯v(¯s, ¯η) represents the constitutive
equation, ǫ0, the density of the relaxed material (density w.r.t the pulled-back material
metric ¯
k) and
¯ǫ=ǫ0pdet ¯
K=ǫ0¯s√¯η, (36)
the density calculated in the rest frame (that is, w.r.t. h).
Then, using (31), one can prove the following relations:
∂¯ρ
∂¯η=1
2f2X,(37)
∂¯ρ
∂¯s=1
f2X+f2∂¯ρ
∂¯
I2−f4y2
¯
Y2
∂¯ρ
∂¯
I3y2
¯
Y2−y′2
¯
b.(38)
Alternatively, one can express the components of the energy-momentum tensor in
terms of the eigenvalues ¯sand ¯ηby substituting the last results in (33):
¯
T0
0= ¯ǫ¯v,
¯
T1
1=−¯ǫ2 ¯η∂¯v
∂¯η,
¯
T2
2=−¯ǫ¯s∂¯v
∂¯s.
(39)
The Einstein field equations ¯
Ga
b= 8π¯
Ta
bcan be written as follows:
¯
G0
0= 8π¯
T0
0:
−
˙
¯
Y
¯
Y2¯a−
˙
¯
Y
¯
Y
˙
¯
b
¯a¯
b+2¯
Y′′
¯
Y¯
b+¯
Y′2
¯
Y2¯
b−¯
Y′
¯
Y
¯
b′
¯
b2−1
¯
Y2= ¯ǫ¯v8π(40)
¯
G1
0= 8π¯
T1
0:
2˙
¯
Y′−¯a′
¯a
˙
¯
Y−
˙
¯
b
¯
b¯
Y′= 0,(41)
4In [3] the quantities ¯ρand ¯ǫare ǫand ρ, respectively.
11

¯
G1
1= 8π¯
T1
1:
−
˙
¯
Y2
¯
Y2¯a+
˙
¯
Y
¯
Y
˙
¯a
¯a2+¯
Y′
¯
Y
¯a′
¯a¯
b+¯
Y′2
¯
Y2¯
b−2¨
¯
Y
¯
Y¯a−1
¯
Y2=−¯ǫ2 ¯η∂¯v
∂¯η8π, (42)
¯
G2
2= 8π¯
T2
2:
1
2
˙
¯
Y˙
¯a
¯
Y¯a2−1
2
˙
¯
Y˙
¯
b
¯
Y¯a¯
b−1
4
¯a′2
¯a2¯
b+1
2
¯
Y′¯a′
¯
Y¯a¯
b−1
4
¯a′¯
b′
¯a¯
b2+¯
Y′′
¯
Y¯
b−1
2
¯
Y′¯
b′
¯
Y¯
b2+
1
2
¯a′′
¯a¯
b−1
2
¨
¯
b
¯a¯
b+1
4
˙
¯a˙
¯
b
¯a2¯
b+1
4
˙
¯
b2
¯a¯
b2−
¨
¯
Y
¯
Y¯a=
−¯ǫ¯s∂¯v
∂¯s8π.
(43)
It is interesting to express the contracted Bianchi identities for ¯
Ta
bin terms of ¯vand
its derivatives w.r.t the quantities ¯ηand ¯s. Thus, from ¯
Ta
b;a= 0 one has:
¯
Ta
b,a +∂n(ln √−¯g)¯
Tn
b−¯
Γn
ba ¯
Ta
n= 0 (44)
and specifying this equation to b= 0,1 one gets respectively (for non-stationary
solutions):
∂t(¯ǫ¯v) +
˙
¯
b
¯
b1
2¯ǫ¯v+ ¯ǫ¯η∂¯v
∂¯η+
˙
¯
Y
¯
Y(2ǫ¯v+ 2¯ǫ¯s∂¯v
∂¯s) = 0,(45)
−2¯ǫ¯η∂¯v
∂¯η,r −1
2¯ǫ¯v¯a′
¯a−¯ǫ¯η∂¯v
∂¯η¯a′
¯a+ 4 ¯
Y′
¯
Y+ 2¯ǫ¯s∂¯v
∂¯s
¯
Y′
¯
Y= 0.(46)
The remaining equations for b= 2,3 which can be obtained from (44) are identically
satisfied.
Equation (45) for non-stationary solutions, implies readily
¯ǫ=1
√¯
b¯
Y2ǫ0(r),(47)
which can then be substituted into (46) to get a slightly simplified equation.
From this point onwards, we shall drop the bars, as no confusion may arise with the
results in [3].
5 Shearfree solutions. Examples
In this section we shall consider in detail the case of spacetimes with a material content
that may be represented by some elastic material such that the velocity of the matter
is shearfree, in which case coordinates exist such that the metric can be written in
the form (16). For this case, the interpretation as a viscous fluid with kinematical
viscosity is not possible, and therefore the anisotropy in the pressures must be a
12

consequence of the elastic properties of the material. The study of solutions with
non-vanishing shear tensor and their possible interpretations as viscous fluids, will be
carried out elsewhere as this would render the present paper too lengthy.
We will study separately the cases of static and non-static solutions, presenting ex-
amples of each instance which are regular at the origin, posses an equation of state
and satisfy the dominant energy condition (at least in some open submanifold of the
spacetime).
Consider the metric (16) which we rewrite here for convenience:
ds2=−a(r, t)dt2+Y2(r, t)dr2+dθ2+ sin2θdφ2(48)
From the field equations it follows that Gtr= 0 which in turn implies that
a=L(t)˙
Y
Y(49)
whenever ˙
Y6= 0, L(t) being a function of time.
If ˙
Y= 0 then Gtr= 0 is identically satisfied, and from (30) follows that sand η, and
therefore v(η, s) are functions of ralone, then (45) implies that ǫ=ǫ(r); further from
the field equation Gr
r=−8πǫ2η∂v
∂η it follows that Gr
rcan only depend on ras well,
which in turn implies that a(t, r) = a0(t)a1(r), the solution being then static, as a
trivial redefinition of the coordinate tcoordinate shows.
It is interesting now to see that in the shear-free case, if one sets either ηor sequal to
1, so that matter is strained in tangential directions (but not in the radial direction
η= 1), or it is strained only in the radial direction (s= 1), from the definition of these
quantities it follows that Y=Y(r), and according to the statements in the above
paragraph, it follows that the solution must be static, and therefore the results in [8]
apply. Thus, we have proven that: if the velocity field of the matter is shear-free and
the matter is stressed either in the radial direction only or in the tangential directions
only, the spacetime is necessarily static.
5.1 Static shearfree solutions
In the static, shear-free case (metric (48) with no dependence on t), the field equations
yield
ǫv8π= 2 Y′′
Y3−Y′2
Y4−1
Y2,(50)
−2ǫη ∂v
∂η 8π=a′
a
Y′
Y3+Y′2
Y4−1
Y2,(51)
−ǫs∂v
∂s 8π=Y′′
Y3+1
2
a′′
aY 2−1
4
a′2
a2Y2−Y′2
Y4,(52)
solving (50) for ǫand substituting it in (51) and (52) one gets two equations which
depend only on rand elementary considerations show that for given a(r) and Y(r),
13

functions y(r), f(y) and vcan be found so that the two equations are satisfied. It
remains to be seen, though, that the DEC are satisfied and therefore the solution is
physically acceptable.
The following simple example shows that solutions with these characteristics do indeed
exist.
Example 1
Consider the line element
ds2=−Y−2(r)dt2+Y2(r)dr2+dθ2+ sin2θdφ2.(53)
A direct calculation yields
8πǫv = 2 Y′′
Y3−Y′2
Y4−1
Y2,−16πǫη ∂v
∂η =−Y′2
Y4−1
Y2,−8πǫs ∂v
∂s =Y′2
Y4.(54)
The dominant energy condition (7) implies:
8π ρ = 2Y′′
Y3−Y′2
Y4−1
Y2≥0,(55)
8π(ρ−p1) = 2Y′′
Y3≥0,(56)
8π(ρ+p1) = 2 Y′′
Y3−Y′2
Y4−1
Y2≥0 (57)
8π(ρ−p2) = 2Y′′
Y3−2Y′2
Y4−1
Y2≥0,(58)
8π(ρ+p2) = 2Y′′
Y3−1
Y2≥0,(59)
where we put ρ=ǫv,p1=−2ǫη ∂v
∂η and p2=−ǫs ∂v
∂s .
Now, it is immediate to see that the above conditions are all satisfied if and only if
(57) is, which in turn can be written as
1
Y2Y′′
Y−Y′2
Y2−1≥0⇔(ln Y)′′ −1≥0,(60)
which is equivalent to
Y= exp r2/2f2(r) such that (ln f)′′ ≥0.(61)
Take, for instance,
Y= exp(5/2r2),(62)
14

one then has
ρ=ǫv =1
8πe−5r2(25r2+ 9),
p1=−2ǫη ∂v
∂η =−1
8πe−5r2(25r2+ 1), p2=−ǫs∂v
∂s =1
8π25r2e−5r2(63)
which is obviously well behaved: satisfies the dominant energy condition and is non-
singular at the origin. Notice that the radial pressure is negative (compressed mate-
rial) and the tangential pressures are zero at the centre, as one would expect.
The field equations in this case read:
ǫv =1
8πe−5r2(25r2+ 9) (64)
−2ǫη ∂v
∂η =−1
8πe−5r2(25r2+ 1) (65)
−ǫs∂v
∂s =1
8πe−5r225r2(66)
and one has that
η=f2(r)e−5r2, s =r2f2(r)e−5r2.(67)
Now, dividing (65) and (66) through by (64), and setting E≡ln η, Σ≡ln s, one gets
∂ln v
∂E =1
2
25r2+ 1
25r2+ 9,∂ln v
∂Σ=−25r2
25r2+ 9.(68)
From the expressions for ηand sone has that
E= 2 ln f(r)−5r2,Σ = E+ 2 ln r, (69)
hence one can express ras a function of E, and Σ as a function of Eas well, thus
∂ln v
∂E =∂Σ
∂E
∂ln v
∂Σ
from where it follows that
∂r
∂E =−3
4r−1
100r,or else E=−2
3ln(75r2+ 1),
that is
r=r1
75 e−3
2E−1.(70)
Plugging the expression of Ein terms of rinto (69) one gets that
f(r) = e5
2r2
(75r2+ 1)1
3
,(71)
15

whence expressions for η,sand ǫ=ǫ0s√ηcan be easily derived.
Next, from (70) and the first equation in (68), one can easily find an expression for
the equation of state, namely:
v=F(Σ) 


e3
2E
e−3
2E+ 2612 


1
39
,(72)
where F(Σ) must satisfy
∂ln F
∂Σ=−25r2
25r2+ 9,
where rin the right hand side of the equation has to be expressed in terms of Σ.
From the second equation in (69) it follows that rmust be the only real solution of
r3−75e3
2Σr2−e3
2Σ= 0,
which has a rather complicated form. In any case, one gets
F(Σ(r)) = 25r2+ 9−12
39 75r2+ 1−1
39 ,
and thus we have proven that a solution exists, which is regular at the origin r= 0,
satisfies the dominant energy condition and possesses an equation of state which can
be given in a closed form.
5.2 Non-static shearfree solutions
Assume now that ˙
Y6= 0, so that a(t, r) takes the form (49), substituting this into
(48) and redefining the coordinate tso as to absorb the arbitrary function L(t) one
has
ds2=−˙
Y
Ydt2+Y2(r, t)dr2+dθ2+ sin2θdφ2(73)
From Gtr= 0 it follows now that Y(r, t) = A(t)B(r), which substituted above, yields,
after a trivial redefinition of the coordinate t:
ds2=−dt2+A2(t)B2(r)dr2+dθ2+ sin2θdφ2(74)
A direct computation of the EFEs for the above metric give gives
8πǫv =−B′2−2B′′B+ 3 ˙
A2B4+B2
A2B4,(75)
−16πǫη ∂v
∂η =−2AB4¨
A−B′2+˙
A2B4+B2
A2B4,(76)
−8πǫs ∂v
∂s =−2AB4¨
A−B′′B+˙
A2B4+B′2
A2B4.(77)
16

where the energy density, radial and tangential pressures are
ρ=ǫv, p1=−2ǫη ∂v
∂η , p2=−ǫs ∂v
∂s .
On the other hand, the dominant energy condition, ρ≥0, ρ ±p1≥0 and ρ±p2≥0
implies
−(B′2−2B′′B+ 3 ˙
A2B4+B2)≥0,(78)
−B′2+B′′B−˙
A2B4+AB4¨
A≥0,(79)
−(AB4¨
A−BB′′ + 2 ˙
A2B4+B2)≥0,(80)
BB′′ −2˙
A2B4−B2+ 2AB4¨
A≥0,(81)
−(2B′2−3B′′B+ 4 ˙
A2B4+B2+ 2AB4¨
A)≥0.(82)
We next address the question of the existence of an equation of state v=v(η, s)
where in this case, ηand sare given by (see (30))
η=f2(r)
A2(t)B2(r), s =r2f2(r)
A2(t)B2(r).(83)
Dividing (76) and (77) by (75), and defining as before E= ln η, Σ = ln s, we get
∂ln v
∂E =−1
2
2AB4¨
A−B′2+˙
A2B4+B2
B′2−2B′′B+ 3 ˙
A2B4+B2≡VE,(84)
∂ln v
∂Σ=−2AB4¨
A−B′′B+˙
A2B4+B′2
B′2−2B′′B+ 3 ˙
A2B4+B2≡VS.(85)
In order for an equation of state v=v(η, s) (or equivalently v=v(E, Σ)) to exist, it
must be that
∂2ln v
∂Σ∂E =∂2ln v
∂E∂Σ⇔∂VE
∂Σ=∂VΣ
∂E .(86)
Notice that
∂E=∂t
∂E ∂t+∂r
∂E ∂r, ∂Σ=∂t
∂Σ∂t+∂r
∂Σ∂r.(87)
Now, from the expression (83) and the corresponding one for Eand Σ, it follows that
r=e1
2(Σ−E), A(t) = f
Be−1
2E; (88)
17

and differentiating them with respect to Eand Σ and applying the chain rule, we get
∂r
∂E =−1
2r, ∂r
∂Σ=1
2r, (89)
∂t
∂E =−1
2˙
Arf
B′+f
BAB
f,∂t
∂Σ=1
2˙
Arf
B′AB
f.(90)
Substituting the above expressions into (87), equation (86) reads, after some manip-
ulations
A
˙
Arf′
f−B′
B∂t(VE+VS) + r ∂r(VE+VS) + A
˙
A∂tVS= 0.(91)
Solving for f′
fwe get, after some algebra,
3rB2f′
f=−Kh(S+ 3B4˙
A2)Attt −6B4˙
A¨
A2+ 3B4A−1˙
A3i+MA−1˙
A¨
A
h(S+ 3B4˙
A2)Attt −6B4˙
A¨
A2+ 3B4A−1˙
A3i+SA−1˙
A¨
A
,(92)
where, K, S and Mare functions of ralone given by:
K= 2B−3rB′, S =B′2−2BB′′ +B2,
M=−6rB2Brrr + 2B(9rB′+B)B′′ + (2BB′−9rB′2−3rB2)B′−4B3.
Since the left hand side of (92) depends only on rthis implies that the time derivative
of the right hand side must vanish. A careful but otherwise trivial analysis, reveals
that there are only three possibilities, assuming the metric is n on-static, namely
1. M+KS = 0, in which case B(r) is determined by the resulting ordinary third
order differential equation. A(t) is in principle arbitrary, and f(r) is fixed by
(92) once the solution for B(r) to the equation M+KS = 0 is give n. We have
not been able to find an integral for B(r) in closed form, but in this case one
has for f(r):
f′
f=B′
B2−2
3rB .
2. ¨
A= 0, which in turn implies A=twithout loss of generality, since the two
constants of integration may be absorbed by suitable redefinitions of tand B.
In this case, B(r) is free, constrained only by the requirements imposed by t he
DEC, and once it is chosen, f(r) is determined through (92), which implies as
in the previous case
f′
f=B′
B2−2
3rB .
18

3. In this case, both A(t) and B(r) are determined as the solutions of the following
two third order differential equations:
−2BB′′ +B′2+B2−kB4= 0, k = constant,
AAttt k
3+˙
A¨
A−2A˙
A¨
A2+˙
A3−q˙
A¨
A= 0, k, q = constant.
Since A(t) and B(r) are fixed, so is f(r), and (92) implies in this case
3rB2f′
f=M+ (k−3q)KB4
3qB4.
As in the first case above, we have not been able to find integrals for A(t) or
B(r) in closed form.
Example 2
Let us next investigate in some detail the second case above, that is A=t. Substi-
tuting this into the EFEs we get
ρ=1
8π t22B′′
B3−B′2
B4−1
B2−3, p1=1
8π t2B′2
B4−1
B2−1, p2=1
8π t2B′′
B3−B′2
B4−1.(93)
On the other hand, the DEC (78)-(82) are all satisfied if and only if the following
three inequalities hold:
3BB′′ −2B′2−B2−4B4≥0, BB′′ −B′2−B4≥0, BB′′ −B2−2B4≥0 (94)
which, upon setting B≡ebare equivalent to
3b′′ +b′2−1−4e2b≥0, b′′ −e2b≥0, b′′ +b′2−1−2e2b≥0.(95)
It is easy to see that these conditions can be satisfied, at least for certain fanges of
the radial coordinate r∈[0, R), for suitably chosen functions b(r), such as
b(r) = 3
2ln 2
3+ sinh2r−r0
c−ln c⇔B(r) = √3
9cln −1 + 3 cosh2r−r0
c3
2
(96)
The form of f(r) can be given explicitly up to a quadrature.
While we do not claim that it has any particular significance, it provides a relatively
simple instance of solution with the desired properties.
Example 3
Another simple example with similar characteristics is provided by the following choice
of B(r):
b(r) = 3
2ln 2
3+r2⇔B(r) = √3
92 + 3r23
2,(97)
19

in which case f(r) can be integrated out yielding:
f(r) = exp 


−15 −9r2+3
2(2 + 3r2)√4 + 6r2tanh−12
√4+6r2
√6 + 9 r2(2 + 3 r2)


Similar remarks to the ones in the previous case regarding its physical significance,
apply also here.
6 The elasticity difference tensor for non-static solutions
Here we obtain the elasticity difference tensor, defined in [4], for non-static spherically
symmetric spacetimes and analyze this tensor following the procedure developed in
[13], where the static, spherically spacetime case was presented as an example.
This third order tensor, symmetric on the two covariant indices, is completely flow-line
orthogonal and is related with the (pulled back) material metric according to
Sa
bc =1
2k−am(Dbkmc +Dckmb −Dmkbc).(98)
Here k−am is such that k−amkmb =ha
band Drepresents the spatially projected con-
nection obtained from the spacetime connection ∇associated with gby
Datb...
c... =hd
ahb
e...hf
c...∇dte...
f... ,(99)
with the property Dahbc = 0.
The non zero components of Sa
bc for non-static, spherically symmetric spacetimes,
using the space-time metric (27) and the pulled-back material metric (29) can be
written as:
Sr
rr =f′
f−¯
b′
2¯
b
Sθ
θr =f′
f+1
r−¯
Y′
¯
Y
Sφ
φr =f′
f+1
r−¯
Y′
¯
Y
Sr
θθ =−f′r2
f−1
r+¯
Y′¯
Y
¯
b
Sr
φφ =−r2f′sin2θ
f−rsin2θ
1+¯
Y′¯
Ysin2θ
¯
b.
In this case, the pulled-back material metric kab is
kab =n2
1xaxb+n2
2(yayb+zazb).(100)
Here, x, y, z are eigenvectors of ka
bwith eigenvalues n2
1and n2
2=n2
3which depend on
tand raccording to
n2
1=f21
bn2
2=n2
3=f2r2
Y2.(101)
20

The elasticity difference tensor can be decomposed along the directions determined
by the eigenvectors of ka
bas follows
Sa
bc =Mbc
1
xa+Mbc
2
ya+Mbc
3
za,(102)
where M
i,i= 1,2,3 are second order, symmetric tensors (see [13]). It should be
noticed that the eigenvectors
Here we determine the eigenvectors and eigenvalues of these tensors, complementing
the results obtained in [13] for the static case, the result being summarized in Tables
1,2 and 3:
Table 1 - Eigenvectors and eigenvalues for M
1
Eigenvectors Eigenvalues
x µ1=f′
√¯
bf −¯
b′
2¯
b√¯
b
y µ2=¯
Y′
√¯
b¯
Y−r2f′√¯
b
¯
Y2f−r√¯
b
¯
Y2
z µ3=µ2
Table 2 - Eigenvectors and eigenvalues for M
2
Eigenvectors Eigenvalues
x+y µ4=f′
√¯
bf +1
√¯
br −¯
Y′
¯
Y√¯
b
x−y µ5=−µ4
z µ6= 0
Table 3 - Eigenvectors and eigenvalues for M
3
Eigenvectors Eigenvalues
x+z µ7=µ4
x−z µ8=−µ4
y µ9= 0
Therefore, the canonical forms for the three tensors M
iare:
Mbc
1
=µ1xbxc+µ2(ybyc+zbzc)
Mbc
2= 2µ4(xbyc+ybxc)
Mbc
3
= 2µ4(xbzc+zbxc).
(103)
Although the eigenvalues are different from the ones obtained in the static case, the
eigenvectors of the above tensors are the same for the static and non-static case.
7 Conclusions
In this paper we have considered spherically symmetric spacetimes with elastic mate-
rial content . We started considering the symmetries these sapcetimes posses in order
21

to exploit their consequences on physics for elastic spacetime configurations. By doing
this, we have generalized previous work done by [1] and [3] for non-static spherically
symmetric configurations, where only flat material metrics were considered. In fact,
we have shown that all material metrics compatible with a given spacetime are con-
formally related and, moreover, are conformally flat. Next we have used comoving
coordinates to relate the EFEs with quantities characterizing elasticity properties
(constitutive equation, material and energy density, eigenvalues of the pulled back
material metric) as well as the conformal factor referred to above. The case in which
the velocity of the matter is shearfree has been considered in detail, giving the neces-
sary and sufficient condition for a constitutive equation be admitted; further, we have
provided three examples of static and non-static shearfree solutions. For non-static
spherically symmetric space-times, the elasticity difference tensor has been studied,
thus extending some previous work for the static case.
Acknowledgements
One of the authors acknowledges financial support from the Spanish Ministry of
Education (MEC) through grant No.: HP2006-0074. Partial financial support from
(MEC) through grant FPA-2007-60220 and from the “Govern de les Illes Balears”
is also acknowledged. Further, this author wishes to express his gratitude for the
hospitality at the Universidade do Minho, where most of this work was done.
The other authors acknowledge financial support from (CRUP) through grant No.:
E-89/07 and from FCT and CMAT. They also express their thanks and gratitude for
the hospitability at the Universitat de les Iles Balears.
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