Analysing the elasticity difference tensor of general relativity

Abstract

The elasticity difference tensor, used in [1] to describe elasticity properties of a continuous medium filling a space-time, is here analysed from the point of view of the space-time connection. Principal directions associated with this tensor are compared with eigendirections of the material metric. Examples concerning spherically symmetric and axially symmetric space-times are then presented.

Full text

arXiv:0706.2319v1 [gr-qc] 15 Jun 2007
ANALYSING THE ELASTICITY DIFFERENCE TENSOR OF
GENERAL RELATIVITY
E.G.L.R.VAZ∗AND IRENE BRITO♮
Abstract. The elasticity difference tensor, used in [1] to describe elasticity
properties of a continuous medium filling a space-time, is here analysed from
the point of view of the space-time connection. Principal directions associated
with this tensor are compared with eigendirections of the material metric.
Examples concerning spherically symmetric and axially symmetric space-times
are then presented.
1. Introduction
In recent years there has been a growing interest in the theory of general relativis-
tic elasticity. Based on the classical Newtonian elasticity theory going back to the
17th century and Hooke’s law, some authors began to adapt the theory of elasticity
to the relativity due to the necessity to study many astrophysical problems as the
interaction between the gravitational field and an elastic solid body in the descrip-
tion of stellar matter, as well as to understand the interaction of gravitational waves
and gravitational radiation and to study deformations of neutron star crusts. One
of the first elastic phenomenon considered in the relativistic context was Weber’s
observation of the elastic response of an aluminium cylinder to gravitational radi-
ation and the detection of gravitational waves [2], [3] and [4]. Neutron stars have
attracted attention since it has been argued [5] that the crusts of neutron stars are
in elastic states and since it has been established the existence of a solid crust and
speculated the possibility of solid cores in neutron stars, [6], [7], [8].
There were many attempts to formulate a relativistic version of elasticity theory.
Thereby laws of non relativistic continuum mechanics had to be reformulated in a
relativistic way. The study of elastic media in special relativity was firstly carried
out by Noether [9] in 1910 and by Born [10], Herglotz [11] and Nordstr¨om [12] in
1911. The discussion of elasticity theory in general relativity started with Synge
[13], De Witt [14], Rayner [15], Bennoun [16], [17], Hernandez [18] and Maugin
[19] 1. In 1973 Carter and Quintana [20] developed a relativistic formulation of
the concept of a perfectly elastic solid and constructed a quasi-Hookean perfect
elasticity theory suitable for applications to high-pressure neutron star matter. Re-
cently, Karlovini and Samuelsson [1] gave an important contribution to this topic,
extending the results of Carter and Quintana (see also [21], [22]). Other relevant
formulations of elasticity in the framework of general relativity were given by Ki-
jowski and Magli ([23], [24]) who presented a gauge-type theory of relativistic elastic
media and a corresponding generalization [25]. The same authors also studied in-
terior solutions of the Einstein field equations in elastic media ([26], [27]).
The recent increasing consideration of relativistic elasticity in the literature shows
1Relativistic elasticity has been treated in the mid-20th century until the early seventies by
many other authors. For further references, see, for example, [19], and for later references see also
[23], [1].
1

the win of recognition and importance of this topic, motivating for a detailed study
of quantities used in this context, the elasticity difference tensor defined in [1] be-
ing one of them. This tensor occurs in the relativistic Hadamard elasticity tensor
and in the Euler equations for elastic matter. However, one can recognize the geo-
metric role of the elasticity difference tensor, since, in principle, it can be used to
understand the influence of the material metric (inheriting elastic properties) on
the curvature of the space-time.
Here, in section 2, general results about relativistic elasticity are presented. In sec-
tion 3, the elasticity difference tensor is analysed and principal directions associated
with this tensor are compared with the eigendirections of the pulled-back material
metric. A specific orthonormal tetrad is introduced to write a general form of the
elasticity difference tensor, which brings in Ricci rotation coefficients used in the
1 + 3 formalism [28] and the linear particle densities.
Finally, in section 4, we apply the results obtained to a static spherically symmetric
space-time and an axially symmetric non-rotating space-time. The software Maple
GRTensor was used to perform some calculations.
2. General results
Let (M, g ) be a space-time manifold, i.e. a 4-dimensional, paracompact, Hausdorff,
smooth manifold endowed with a Lorentz metric gof signature (−,+,+,+), Ubeing
a local chart around a point p∈M. Suppose that Uis filled with a continuum
material. The material space Xis an abstract 3-dimensional manifold, each point
in Xrepresenting an idealized particle of the material. Moreover, the space-time
configuration of the material is described by a mapping
Ψ : U⊂M−→ X ,
which associates to each point pof the space-time the particle ¯pof the material
which coincides with pat a certain time. Therefore Ψ−1( ¯p) represents the flowline
of the particle ¯p. The operators push-forward Ψ∗and pull-back Ψ∗will be used
to take contravariant tensors from Mto Xand covariant tensors from Xto M,
respectively, in the usual way.
If {ξA}(A= 1,2,3) is a coordinate system in Xand {ωa}(a= 0,1,2,3) 2a coor-
dinate system in U⊂M, then the configuration of the material can be described
by the fields ξA=ξA(ωa). The mapping Ψ∗:TpM−→ TΨ(p)Xgives rise to a
(3 ×4) matrix (the relativistic deformation gradient) whose entries are ξA
a=∂ξA
∂ωa.
Assuming that the world-lines of the particles Ψ−1( ¯p) are timelike, the relativis-
tic deformation gradient is required to have maximal rank and the vector fields
ua∈TpM, satisfying uaξB
a= 0, are required to be timelike and future oriented.
The vector field uais the velocity field of the matter and its components obey
uaua=−1, uaξB
a= 0 and u0>0, [23].
One needs to consider, in the material space X, a Riemannian metric ηAB, describ-
ing the “rest frame” space distances between particles calculated in the “locally
relaxed state” or in the “unsheared state” of the material and often taken as the
material metric. These approaches are presented in [23] and in [1], respectively.
2Capital Latin indices A,B,... range from 1 to 3 and denote material indices. Small Latin
indices a,b,... take the values 0,1,2,3 and denote space-time indices.
2

Let ǫABC be the volume form of ηAB , with Ψ∗ǫABC =ǫabc =ǫabcd ud. The particle
density form is nABC =nǫAB C , with nthe particle density yielding the number
of particles in a volume of Xwhen integrated over that volume. One can define,
see [1], a new tensor kAB , which has nABC as its volume form and is conformal to
ηAB :kAB =n2
3ηAB . This tensor will be taken as the material metric in X.
The pull-back of the material metric
kab = Ψ∗kAB =ξA
aξB
bkAB (1)
and the (usual) projection tensor
hab =gab +uaub(2)
are Riemannian metric tensors on the subspace of TpMorthogonal to ua. These
tensors are symmetric and satisfy kab ua= 0 = hab ua.
The state of strain of the material can be measured by the relativistic strain tensor,
according to e.g. [26], [27]:
sab =1
2(hab −ηab) = 1
2(hab −n−2
3kab).(3)
This tensor is also named as constant volume shear tensor (see [20], [1]). The
material is said to be “locally relaxed” at a particular point of space-time if the
material metric and the pro jection tensor agree at that point, i.e. if the strain
tensor vanishes.
When considering elastic matter sources in general relativity, one is confined to a
stress-energy tensor taking the form Tab =−ρgab + 2 ∂ ρ
∂gab =ρuaub+pab , where
pab = 2 ∂ρ
∂gab −ρhab , the energy density being written, for convenience, as ρ=nǫ,ǫ
being the energy per particle.
Choosing an orthonormal tetrad {u, x, y , z}in M, with uin the direction of the
velocity field of the matter and x,y,zspacelike vectors, satisfying the orthogonality
conditions −uaua=xaxa=yaya=zaza= 1, all other inner products being zero,
the space-time metric can be written as
gab =−uaub+hab =−uaub+xaxb+yayb+zazb.(4)
Here we will choose the spacelike vectors of the tetrad along the eigendirections of
ka
b=gackcb , so that
kab =n2
1xaxb+n2
2yayb+n2
3zazb,(5)
where n2
1,n2
2and n2
3are the (positive) eigenvalues of kb
a. The linear particle densities
n1,n2and n3satisfy n=n1n2n3. It should be noticed that those eigenvectors
are automatically orthogonal whenever the eigenvalues referred above are distinct.
However, if the eigenvalues are not all distinct, the eigendirections associated to
the same eigenvalue can (and will) be chosen orthogonal.
It is convenient to consider the spatially projected connection Daacting on an
arbitrary tensor field tb...
c... as follows:
Datb...
c... =hd
ahb
e...hf
c...∇dte...
f... .(6)
Here ∇is the connection associated with gand one has Dahbc = 0. Another
operator ˜
D, such that its action on the same tensor is
˜
Datb...
c... =hd
ahb
e...hf
c... ˜
∇dte...
f... (7)
is also considered. One has
˜
DbXa=DbXa+Sa
bcXc,(8)
3

for any space-time vector field X. The tensor field Sabc is the elasticity difference
tensor introduced by Karlovini and Samuelsson in [1]. This third order tensor can
be written as
Sa
bc =1
2k−am(Dbkmc +Dckmb −Dmkbc ),(9)
where k−am is such that k−amkmb =ha
b. This tensor is used by the same authors
to write the Hadamard elasticty tensor, used to describe elasticity properties in
space-time, and the Euler equations ∇bTab = 0 for elastic matter.
The covariant derivative of the timelike unit vector field ucan be decomposed as
follows
ua;b=−˙uaub+Dbua=−˙uaub+1
3Θhab +σab +ωab,(10)
where ˙uαis the acceleration, σαβ, the symmetric tracefree rate of shear tensor field,
ωαβ , the antisymmetric vorticity tensor field and Θ, the expansion scalar field for
the congruence associated with u.
3. Properties of the Elasticity Difference Tensor
Here we will investigate the algebraic properties of the elasticity difference tensor.
This tensor, important when studying elasticity within the framework of general
relativity, is related to the connection of the space-time, as shown in the previ-
ous section. The following two properties of the elasticity difference tensor are
straightforward:
(i) it is symmetric in the two covariant indices, i. e.
Sa
bc =Sa
cb; (11)
(ii) it is a completely flowline orthogonal tensor field, i.e.
Sa
bcua= 0 = Sa
bcub=Sa
bcuc.(12)
The elasticity difference tensor can be approached using the space-time connection,
as will be shown here.
It is a well known result that the difference between two connections ˜
∇and ∇,
associated with two different metrics ˜gand g, respectively, defined on U, is the
following (1,2) tensor:
Cn
ml =˜
Γn
ml −Γn
ml,(13)
˜
Γnml and Γnml being the Christoffel symbols associated with those two metrics. In
a local chart, this tensor can be written as ([29], [30])
Cn
ml =1
2˜gnp (˜gpm;l+ ˜gpl;m−˜gml;p),(14)
where ˜gnp is such that ˜gnp ˜gpr =δn
rand a semi-colon ; represents the covariant
derivative with respect to g. The difference tensor Cn
ml can be used to write the
difference of the Riemann and the Ricci tensors associated with the two metrics in
the following form (see e.g. [31]):
˜
Ra
bcd −Ra
bcd =−Ca
bd;c+Ca
bc;d−Ca
lcCl
bd +Ca
ldCl
bc (15)
and
˜
Rbd −Rbd =−Ca
bd;a+Ca
ba;d−Ca
laCl
bd +Ca
ldCl
ba.(16)
4

The projection of the difference tensor orthogonally to uis defined by the expression
ha
nhm
bhl
cCn
ml.(17)
When the connections used to define the difference tensor are associated with the
metrics gab =−uaub+hab and ˜gab =−uaub+kab , then the corresponding difference
tensor, projected according to (17), yields (9) i.e. the elasticity difference tensor
defined in the previous section.
Under this approach, the elasticity difference tensor is the projection, orthogonal
to u, of the difference between two connections, one associated with the space-time
metric and the other with the metric ˜gab =−uaub+kab, where kab is the pull-back
of the material metric kAB .
Calculating the spatially projected versions of equation (15), using (6) and (17),
yields the following expression for the difference of the Riemann tensors:
hf
mhn
ghp
ehq
h[hm
ahb
nhc
phd
q(˜
Ra
bcd −Ra
bcd)]
=−DeSf
gh +DhSf
ge −Sf
keSk
gh +Sf
khSk
ge.(18)
The spatially projection of (16), the difference of the Ricci tensors, can be obtained
analogously by equating the indices a=cin the last expression.
Therefore, these expressions, which contain the elasticity difference tensor, give
the difference between the Riemann and Ricci tensors associated with the metrics
referred to above.
Now we will obtain the tetrad components of the elasticity difference tensor. To
do so, it is more convenient to use the following notation for the orthonormal
tetrad: ea
µ= (ea
0, ea
1, ea
2, ea
3) = (ua, xa, ya, za).Tetrad indices will be represented
by greek letters from the second half or the first half of the alphabet according to
their variation as follows: µ, ν, ρ... = 0 −3 and α, β , γ... = 1 −3. The Einstein
summation convention and the notation for the symmetric part of tensors will only
be applied to coordinate indices, unless otherwise stated. The operation of raising
and lowering tetrad indices will be performed with ηµν =ηµν =diag(−1,1,1,1)
and one has gab =eµaeν b ηµν .
Writing the Ricci rotation coefficients as γµνρ =eµa;bea
νeb
ρ, the tetrad components
of the elasticity difference tensor can be obtained using the standard relationship
Sα
βγ =Sa
bceα
aeb
βec
γ(19)
the result being
Sα
βγ =1
2n2
α
[n2
α−n2
γγα
γβ +n2
α−n2
βγα
βγ +n2
γ−n2
βγα
βγ +Dn(n2
α)en
βδα
γ
+Dp(n2
α)ep
γδα
β−Dl(n2
β)elαδβ γ ].
(20)
An alternative form for the last expression is:
Sα
βγ =1
2[(1 −ǫγα)γα
γβ + (1 −ǫβ α)γα
βγ + (ǫγ α −ǫβα )γα
βγ +mβαδα
γ+mγα δα
β
−mα
βδβγ ǫβα],
(21)
where ǫγα = n2
γ
n2
α!and mα
β=Da(ln n2
β)eaα.
5

The Ricci rotation coefficients, when related to the quantities used in the decom-
position (10), can be split into the set [32]:
γ0α0= ˙uα(22)
γ0αβ =1
3Θδαβ +σαβ −ǫαβγ ωγ(23)
γαβ0=−ǫαβ γ Ωγ(24)
γαβγ =−Aαδβγ +Aβδαγ −1
2(ǫγδα Nδ
β−ǫγδβ Nδ
α+ǫαβδ Nδ
γ).(25)
The quantities Aand Nappear in the decomposition of the spatial commutation
functions Γα
βγ =γα
γβ −γα
βγ , given in [33], where Nis a symmetric object.
The elasticity difference tensor can be expressed using three second order symmetric
tensors, here designated as Mbc
α,α= 1,2,3, as follows:
Sa
bc =Mbc
1
xa+Mbc
2
ya+Mbc
3
za=
3
X
α=1
Mbc
α
ea
α.(26)
Here we will study some properties of the three tensors Mbc
α
in order to understand
until which extent the principal directions of the pulled back material metric remain
privileged directions of the elasticity difference tensor, i.e. of the tensors Mbc
α, by
studying the eigenvalue-eigenvector problem for these second order tensors.
First, we will obtain a general expression for Mbc
α,α= 1,2,3, which depends
explicitely on the orthonormal tetrad vectors, the Ricci rotation coefficients and
the linear particle densities nα. This comes from the contraction of Sabc in (9) with
each one of the spatial tetrad vectors, followed by the use of the relationships (5),
(6) and appropriate simplifications. The final result is
Mbc
α
=um(eαm;(buc)+u(beαc);m) + eα(b;c)−em
αeα(ceαb);m
+γ0αα u(beαc)−γ0α0ubuc
+1
nα
[2nα,(beαc)+ 2nα,mumu(beαc)+nα,m em
αeαbeαc ]
+1
n2
α
{−em
α(eβb eβc nβnβ,m +eγ beγ c nγnγ,m )
+n2
γ[(γ0γα −γαγ 0)u(beγc)+em
α(eγm;(beγ c)−eγ(beγc);m)]
+n2
β[(γ0βα −γαβ0)u(beβ c)+em
α(eβm;(beβ c)−eβ(beβc);m)]},
(27)
where γ6=β6=α, for one pair (β, γ), a comma being used for partial derivatives. It
should be noticed that this expression also contains the non-spatial Ricci rotation
coefficients given in (22), (23) and (24).
Naturally, the expressions obtained for Mbc
αstill satisfy the conditions Mbc
α
ub= 0.
The eigenvalue-eigenvector problem for Mbc
αis quite difficult to solve in general.
However, one can investigate the conditions for the tetrad vectors to be eigenvectors
of those tensors, the results being summarized in the two following theorems.
6

Intrinsic derivatives of arbitrary scalar fields Φ, as derivatives along tetrad vectors,
will be represented ∆eαand defined as:
∆eαΦ = Φ,mem
α,
where a comma is is used for partial derivatives.
Theorem 1. The tetrad vector eαis an eigenvector for M
αiff nαremains invariant
along the two spatial tetrad vectors eβ, such that β6=αi.e. ∆eβ(ln nα) = 0
whenever β6=α.
The corresponding eigenvalue is λ= ∆eα(ln nα).
Proof: In order to solve this eigenvector-eigenvalue equation the following algebraic
conditions are used
Mc
b
α
eb
αeαc =λ, (28)
Mc
b
α
eb
αeβc = 0 (29)
and
Mc
b
α
eb
αeγc = 0,(30)
where γ6=β6=α. Using the orthogonality conditions satisfied by the tetrad
vectors and the properties of the rotation coefficients, namely the fact that they
are anti-symmetric on the first pair of indices, (29) and (30) yield ∆eβ(ln nα) = 0 =
∆eγ(ln nα) so that ∆eβnα= 0 = ∆eγnα. On the other hand from (28) one obtains
λ= ∆eα(ln nα).
It should be noticed that λ= 0 whenever nαremains constant along eα. However
this condition is equivalent to nα=c, with ca constant. In this case, kab =
c2eαaeαb +P
β6=α
n2
βeβa eβb .
Theorem 2. eβis an eigenvector of M
αiff the following conditions are satisfied:
(i) ∆eβ(ln nα) = 0, i.e. nαremains invariant along the direction of eβ;
(ii) γαγβ [n2
α−n2
γ] + γαβγ [n2
α−n2
β] + γβγ α[n2
γ−n2
β] = 0, where γ6=β6=αfor
one pair (β, γ).
The corresponding eigenvalue is λ=−nβ
n2
α∆eαnβ+γαββ (−n2
β
n2
α+ 1).
Proof: Contracting Mc
b
α
eb
β=λec
βwith eαc one obtains ∆eβ(ln nα) = 0. This
condition is satisfied whenever ∆eβnα= 0. The second condition results from
Mc
b
α
eb
βeγc = 0. And contracting Mc
b
α
eb
β=λec
βwith eβc yields the eigenvalue λ. The
used simplifications are based on the orthogonality conditions of the tetrad vectors
and on the properties of the rotation coefficients.
Notice that the two conditions are satisfied if nα=nβ=nγ=c, where cis a
constant. The consequence of this is that λ= 0. In this case, kab =c2xaxb+
c2yayb+c2zazb.
The previous theorems show that strong conditions have to be imposed on nα, for
α= 1,2,3, and the metric in order that the spatial tetrad vectors are principal
directions of M
α,for α= 1,2,3.
7

However, the conditions to have eαas eigenvector of M
αseem less restrictive then
the conditions for eβ, for all values of β6=α, to be eigenvector of the same tensor
M
α, since these involve not only intrinsic derivatives of the scalar fields but also
rotation coefficients of the metric. Furthermore, for eαto be an eigenvector of
M
αonly conditions on nαhave to be satisfied, namely that nαremains constant
along the directions of eβfor all values of β6=α(in which case the eigenvalue
corresponding to eαdepends only on nα). On the other hand, the conditions
imposed for eβfor all β6=αto be eigenvectors of M
αalso involve nβfor all β6=α.
Next we will use the previous theorems to establish the conditions for eα, with
α= 1,2,3 to be an eigenvector of the three tensors M
1, M
2, M
3simultaneously, the
results being:
(i) ∆eβ(ln nα) = 0,
(ii) ∆eα(ln nβ) = 0,
(iii) γαβγ [n2
α−n2
β] + γαγβ [n2
γ−n2
α] + γβγ α[n2
β−n2
γ] = 0,
for all values of βand γsuch that β6=γ6=α.
These conditions must be satisfied for all values of β6=α. It is not easy to find
the general solution to these equations, however one can say that, in general, the
principal directions of the pulled back material metric kare not, in general, the
principal directions of the three tensors M
1,M
2and M
3. It should be noticed that
the (mathematical) solution corresponding to n1=n2=n3= const.is not an
interesting result from the physical point of view.
As a special case, we now consider that all eigenvalues of kabare equal, i.e.
n1=n2=n3=n1
3.(31)
Therefore, kab =n2
3hab, so that these tensors are conformally related. In physical
terms, this corresponds to the unsheared state described in [1]: the energy per
particle, ǫ, has a minimum under variations of gAB such that nis held fixed. The
above theorems in this section simplify significantly in this case, as can easily be
proved using (31) in those theorems. For completeness, we give the expressions for
the elasticity difference tensor and the tensors M
αin this special case:
Sa
bc =1
3
1
nδa
cDbn+δa
bDcn−hadhbc Ddn,
Mbc
α=1
3
1
n(eαcn,b +eαb n,c + (eαcub+eαb uc)∆e0n−hbc∆eαn).
4. Examples
Here, examples concerning the static spherically symmetric case and an axially
symmetric, non-rotating metric are presented, where we apply the analysis devel-
oped in the last section. The main problem when dealing with examples lies in the
difficulties of finding an orthonormal tetrad for the space-time metric such that the
corresponding spacelike vectors are precisely the principal directions of the pulled
back material metric. However, in the examples presented, this difficulty was over-
come.
8

4.1. The static spherically symmetric case. In this section we analyse the
elasticity difference tensor and corresponding eigendirections for the static spher-
ically symmetric metric, due to its significance on modelling neutron stars. The
metric regarded here can be thought of as the interior metric of a non rotating star
composed by an elastic material.
For a static spherically symmetric spacetime the line-element can be written as
ds2=−e2ν(r)dt2+e2λ(r)dr2+r2dθ2+r2sin2θdφ2,(32)
where the coordinates ωa={t, r, θ, φ}are, respectively, the time coordinate, the
radial coordinate, the axial coordinate and the azimuthal coordinate. Choosing the
basis one-forms ua= (−eν(r),0,0,0), xa= (0, eλ(r),0,0), ya= (0,0, r, 0) and za=
(0,0,0, r sin θ) for the orthonormal tetrad, the metric is given by gab =−uaub+
xaxb+yayb+zazband hab =xaxb+yayb+zazbdefines the corresponding pro jection
tensor. Using this tetrad, the pulled-back material metric becomes
kab =n2
1xaxb+n2
2yayb+n2
2zazb,(33)
where we have chosen n3=n2since for this material distribution khas only two
different eigenvalues.
Let ξA={˜r, ˜
θ, ˜
φ}be the coordinate system in the material space X. Since the
space-time is static and spherically symmetric, ˜rcan only depend on rand one
can take ˜
θ=θand ˜
φ=φso that the configuration of the material is entirely
described by the material radius ˜r(r). Moreover, the only non-zero components of
the deformation gradient are dξ1
dω1=d˜r
dr ,dξ2
dω2= 1 and dξ3
dω3= 1.
In Xthe material metric is kAB = ˜xA˜xB+ ˜yA˜yB+ ˜zA˜zB, with ˜xA=e˜
λd˜rA,
˜yA= ˜rd˜
θAand ˜zA= ˜rsin ˜
θd ˜
φA, and where ˜
λ=λ(˜r).The pull-back of the material
metric is then
ka
b=gackcb =gac (ξC
cξB
bkCB ) = d˜r
dr 2
e2˜
λ−2λδa
1δ1
b+˜r2
r2δa
2δ2
b+˜r2
r2δa
3δ3
b,(34)
Comparing (33) and (34) it is simple to obtain the following values for the linear
particle densities (all positive), which are found to depend on ronly:
n1=n1(r) = d˜r
dr e˜
λ−λ(35)
n2=n2(r) = n3(r) = ˜r
r(36)
The non-zero components of the strain tensor (3), when written as functions of the
quantities nα, are
srr =1
2e2λ(1 −n−2
3n2
1)
sθθ =1
2r2(1 −n−2
3n2
2)
sφφ =1
2r2sin2θ(1 −n−2
3n2
2)
Using the expressions obtained for the nαone can find that the condition for this
tensor to vanish identically is that ˜r=ce±R1
reλ−˜
λdr.
9

Calculating the quantities given in (10) one obtains
Θ = 0
˙ua=0, e2νdν
dr ,0,0
σab :σ12 =1
2e4νdν
dr =σ21
ωab :ω12 =e2νdν
dr +1
2e4νdν
dr
ω21 =−ω12,
where the remaining components of σab and ωab vanish.
The non-zero components of the elasticity difference tensor Sa
bc are:
Sr
rr =1
n1
dn1
dr
Sθ
θr =1
n2
dn2
dr
Sφ
φr =1
n2
dn2
dr
Sr
θθ =re−2λ−re−2λn2
2
n2
1
−e−2λr2n2
n2
1
dn2
dr
Sr
φφ =e−2λrsin2−e−2λrsin2θn2
2
n2
1
−e−2λr2sin2θn2
n2
1
dn2
dr .
Since Sa
bc =Sa
cb, there are only seven non-zero components for this tensor on the
coordinate system chosen above.
Again, using (35) and (36)one obtains that:
(i) the components Sθθr and Sφ
φr are zero whenever the function ˜ris of the form
˜r=c1r, where c1is a constant;
(ii) Srrr is zero whenever ˜r=c2+c3Reλ−˜
λdr;
(iii) the components Srθθ and Srφφ are zero whenever ˜r=c4eRe−2˜
λ+2λ
rdr.
The second order symmetric tensors M
α, for α= 1,2,3 have the following non-zero
components:
Mrr
1=eλ
n1
dn1
dr
Mθθ
1=e−λr−e−λrn2
2
n2
1
−e−λr2n2
n2
1
dn2
dr
Mφφ
1
=e−λrsin2θ−e−λrsin2θn2
2
n2
1
−e−λr2sin2θn2
n2
1
dn2
dr
Mrθ
2=Mθr
2=r
n2
dn2
dr
Mrφ
3
=Mφr
3
=rsinθ
n2
dn2
dr
The eigenvalues and eigenvectors of these tensors are presented in tables 1, 2 and
3, being then compared with the eigendirections of the material metric.
10

Table 1 - Eigenvectors and eigenvalues for M
1
Eigenvectors Eigenvalues
x µ1=e−λ
n1
dn1
dr
y µ2=e−λ
r−e−λ
r
n2
2
n2
1
−e−λn2
n2
1
dn2
dr
z µ3=e−λ
r−e−λ
r
n2
2
n2
1
−e−λn2
n2
1
dn2
dr
Notice that, in the present example, M
1maintains the eigenvectors of k, namely x,
yand z, the two last ones being associated with the same eigenvalue. Therefore
the canonical form for M
1is Mbc
1=µ1xbxc+µ2(ybyc+zbzc), where µ1and µ2are
the eigenvalues corresponding to xand y(≡z), respectively.
Table 2 - Eigenvectors and eigenvalues for M
2
Eigenvectors Eigenvalues
x+y µ4=e−λ
n2
dn2
dr
x−y µ5=−e−λ
n2
dn2
dr
z µ6= 0
In this case, only the eigenvector zof kremains as eigenvector, however the corre-
sponding eigenvalue being zero. The other two eigenvectors are x+yand x−yso
that the canonical form for M
2can be expressed as Mbc
2= 2µ4(xbyc+ybxc), where
µ4=e−λ1
˜r
d˜r
dr −1
r.
Table 3 - Eigenvectors and eigenvalues of M
3
Eigenvectors Eigenvalues
x+z µ7=e−λ
n2
dn2
dr
x−z µ8=−e−λ
n2
dn2
dr
y µ9= 0
Comparing M
2and M
3, it is easy to see that the role of zand yis interchanged. The
eigenvalues of M
2are equal to the eigenvalues of M
3and the canonical form of this
tensor field can be written as Mbc
3= 2µ7(xbzc+zbxc), where µ7=e−λ1
˜r
d˜r
dr −1
r.
It should be noticed that the case n2constant is not interesting to analyze, since
this corresponds to the vanishing of the tensors M
2and M
3.
x,yand zwould only remain eigenvectors for M
2and M
3if ˜rwould be of the form:
˜r=cr, in which case M
2and M
3were reduced to a zero tensor.
11

The tetrad components of the elasticity difference tensor can directly be obtained
from (21):
S111 =e−λ1
n1
dn1
dr
S221 =e−λ1
n2
dn2
dr
S331 =e−λ1
n2
dn2
dr
S122 =e−λ1
r−e−λ1
r
n2
2
n2
1
−e−λn2
n2
1
dn2
dr
S133 =e−λ1
r−e−λ1
r
n2
2
n2
1
−e−λn2
n2
1
dn2
dr .
The expressions for the Ricci rotation coefficients are
γ122 =e−λ
r
γ133 =e−λ
r
γ233 =cos θ
rsin θ.
4.2. The axially symmetric non-rotating case. First, consider an elastic, ax-
ially symmetric, uniformly rotating body in interaction with its gravitational field.
The exterior of the body may be described by the following metric, [27],
ds2=−e2νdt2+e2µdr2+e2µdz2+e2ψ(dφ −ωdt)2,(37)
where ν, ψ, ω , µ are scalar fields depending on rand z.
Assume that the material metric is flat. Introducing in Xcylindrical coordinates
ξA={R, ζ, Φ}, then the material metric takes the form:
ds2=dR2+dζ2+R2dΦ2,(38)
where the parameters R,ζdepend on rand z, Φ being Φ(t, r, z, φ) = φ−Ω(r, z)t.
Now, consider the limiting case of an axially symmetric non-rotating elastic system
for which the space-time metric is given by
ds2=−e2νdt2+e2µdr2+e2µdz2+e2ψdφ2.(39)
This metric is obtained from (37), when ω= 0 and the angular velocity Ω = 0.
Imposing R=R(r), ζ=zand gab =gab(r), one obtains a further reduction to
cylindrical symmetry. This reduction is considered in [27].
So, the space-time metric we will work with is given by (39), where ν, µ, ψ depend
on ronly, and it can be written as gab =−uaub+xaxb+yayb+zazb, where
ua= (−eν(r),0,0,0), xa= (0, eµ,0,0), ya= (0,0, eµ(r),0) and za= (0,0,0, eψ(r)).
The space-time coordinates are ωa={t, r, z, φ}.
In Xthe material metric kAB is given by kAB = ˜xA˜xB+ ˜yA˜yB+ ˜zA˜zB, where
˜xA=dRA, ˜yA=dzAand ˜zA=RdφA. The relativistic deformation gradient has
12

the following non-zero components dξ1
dω2=dR
dr ,dξ2
dω1= 1 and dξ3
dω3= 1. Calculating
the pull-back of the material metric one obtains
ka
b=gackcb =gac ξC
cξB
bkCB =e−2µδa
1δ1
b+dR
dr 2
e−2µδa
2δ2
b+R2e−2ψδa
3δ3
b.(40)
The corresponding line-element can be expressed as
ds2=dr2+dR
dr dz2+R2dφ2.(41)
On the other hand, the material metric in the space-time Mis given by
kab =n2
1xaxb+n2
2yayb+n2
3zazb.(42)
Comparing (40) with (42) one concludes that the linear particle densities (all posi-
tive) are expressed as
n1=n1(r) = e−µ(43)
n2=n2(r) = e−µdR
dr (44)
n3=n3(r) = Re−ψ.(45)
The strain tensor (3) is composed of the following components
srr =1
2e2µ(1 −n−2
3n2
1)
szz =1
2e2µ(1 −n−2
3n2
2)
sφφ =1
2e2ψ(1 −n−2
3n2
3)
The strain tensor vanishes if the condition R(r) = r=eψ−µis satisfied.
Calculating the quantities given in (10) one obtains
Θ = 0
˙ua=0, e2νdν
dr ,0,0
σab :σ12 =1
2e4νdν
dr =σ21
ωab :ω12 =e2νdν
dr +1
2e4νdν
dr
ω21 =−ω12,
where the remaining components of σab and ωab vanish.
13

The non-zero components of the elasticity difference tensor are
Sr
rr =1
n1
dn1
dr
Sz
zr =1
n2
dn2
dr
Sφ
φr =1
n3
dn3
dr
Sr
zz =dµ
dr −n2
2
n2
1
dµ
dr −n2
n2
1
dn2
dr
Sr
φφ =e−2ψ−2µdψ
dr −n2
3
n2
1
dψ
dr −n3
n2
1
dn3
dr .
It can be observed that only seven components of the elasticity difference tensor
are non-zero.
Using the expressions (43), (44) and (45) one can conclude that:
(i) Srrr is zero whenever µ(r) = c, where cis a constant;
(ii) Szzr is zero whenever R(r) = c1+c2Reµ(r)dr;
(iii) Sφ
φr is zero whenever R(r) = c3eψ(r);
(iv) Srzz is zero whenever R(r) = ±Rp2µ(r) + c4dr +c5;
(v) Srφφ is zero whenever R(r) = ±q2Re2ψ
e2µ
dψ
dr dr +c6.
The second-order tensors M
1,M
2and M
3have the following non-zero components:
Mrr
1=eµ1
n1
dn1
dr
Mzz
1=eµ dµ
dr −n2
2
n2
1
dµ
dr
−n2
n2
1
dn2
dr !
Mφφ
1
=e2ψ−µdψ
dr −n2
3
n2
1
dψ
dr −n3
n2
1
dn3
dr 
Mrz
2=Mzr
2=eµ1
n2
dn2
dr
Mrφ
3
=Mφr
3
=eψ1
n3
dn3
dr .
The next three tables contain the eigenvalues and eigenvectors for these tensors,
which are then compared with the eigenvectors of the pulled-back material metric.
Table 1 - Eigenvectors and eigenvalues for M
1
Eigenvectors Eigenvalues
x λ1=e−µ1
n1
dn1
dr
y λ2=e−µdµ
dr −n2
2
n2
1
dµ
dr −n2
n2
1
dn2
dr 
z λ3=e−µdψ
dr −n2
3
n2
1
dµ
dr −n3
n2
1
dn3
dr 
One can observe that the eigendirections x,yand zof kare also eigenvectors for
the tensor M
1and the eigenvectors are associated with different eigenvalues. The
canonical form for M
1can be written as Mbc
1=λ1xbxc+λ2ybyc+λ3zbzc.
14

Table 2 - Eigenvectors and eigenvalues for M
2
Eigenvectors Eigenvalues
x+y λ4=e−µ1
n2
dn2
dr 
x−y λ5=−e−µ1
n2
dn2
dr 
z λ6= 0
M
2inherits only the eigenvector zof k, which corresponds to a zero eigenvalue. The
other two eigenvectors of M
2are linear combinations of xand y:x+yand x−y,
whose corresponding eigenvalues are symmetric in sign. The canonical form for M
2
can be written as Mbc
2= 2λ4(xbyc+ybxc), where λ4=d2R
dr2
dR
dr
−dµ
dr e−µ.
Table 3 - Eigenvectors and eigenvalues for M
3
Eigenvectors Eigenvalues
x+z λ7=e−µ1
n3
dn3
dr 
x−z λ8=−e−µ1
n3
dn3
dr 
y λ9= 0
M
3inherits the eigenvalue yof k, which is associated with the eigenvalue zero.
The other two eigenvectors of M
3are linear combinations of xand z:x+zand
x−z. These two eigenvectors are associated with sign symmetric eigenvalues.
The canonical form for M
3can be written as Mbc
3= 2λ7(xbzc+zbxc), where λ7=
1
R
dR
dr −dψ
dr e−µ.
xand ywould only be eigenvectors for M
2if R(r) would be of the form R(r) =
c1+Reµdrc2, but in this case M
2would vanish. xand zwould only be eigenvectors
for M
3if R(r) would be of the form R(r) = c3eψand this would reduce M
3to a zero
tensor.
One can observe that the role that yand n2play for the tensor M
2is the same that
zand n3play for M
3. That is, the results for M
2and M
3are very similar, only y
and n2are substituted by zand n3, respectively.
The tetrad components of the elasticity difference tensor obtained from (21) and
the expressions for the Ricci rotation coefficients are listed below:
S111 =e−µ1
n1
dn1
dr
S221 =e−µ1
n2
dn2
dr
S331 =e−µ1
n3
dn3
dr
S122 =e−µdµ
dr −e−µn2
2
n2
1
dµ
dr −e−µn2
n2
1
dn2
dr
S133 =e−µdψ
dr −e−µn2
3
n2
1
dψ
dr −e−µn3
n2
1
dn3
dr .
15

The expressions for the Ricci coefficients are
γ122 =
dµ
dr
eµ
γ133 =
dψ
dr
eµ.
5. Acknowledgements
The authors would like to thank L. Samuelsson for many valuable discussions on
this work.
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16

∗Departamento de Matem´
atica para a Ciˆ
encia e Tecnologia, Universidade do Minho,
4800 058 Guimar˜
aes, Portugal
E-mail address:evaz@mct.uminho.pt
♮Departamento de Matem´
atica para a Ciˆ
encia e Tecnologia, Universidade do Minho,
4800 058 Guimar˜
aes, Portugal
E-mail address:ireneb@mct.uminho.pt
17