Shear-free perfect fluids with a solenoidal electric curvature

0.12 MB·English

by  Norbert Van den Bergh

Shear-free perfect fluids with a solenoidal electric
curvature
Norbert Van den Bergh1, John Carminati2, Hamid Reza
2
1 Karimian1,3 and Peter Huf2
0
2 1 Faculty of Applied Sciences FEA16, Gent University, Galglaan 2, 9000 Gent,

n Belgium
a 2 School of Engineering and Information Technology, Deakin University, Australia
J 3 Faculty of Sciences, Ilam University, Banganjab, Ilam, Iran
4
2 E-mail: [email protected],[email protected],

[email protected]
]
c
q
Abstract. We prove that the vorticity or the expansion vanishes for any shear-free
-
r perfect fluid solution of the Einstein field equations where the pressure satisfies a
g
[ barotropicequationofstate andthe spatialdivergenceof the electric partofthe Weyl

tensor is zero.
2
v
1
4
PACS numbers: 04.20.Jb, 04.40.Nr
2
4
.
1
1. Introduction
0
2
1 There has been much effort devoted to establishing the shear-free fluid conjecture, that
:
v general relativistic, shear-free perfect fluids which obey a barotropic equation of state
i
X p = p(µ) such that p+µ 6= 0, are either non-expanding (θ = 0) or non-rotating (ω = 0).
r
a This conjecture appears as such for the first time in Treciokas and Ellis[21] and has been
established in many special cases, but a general proof or counter-example is still lacking.
In support, the conjecture is known to hold in cases where: p = const (dust with a
cosmological constant)[11, 12, 18]; spatial homogeneity[1, 13]; dp/dµ = 1/3 (incoherent
radiation)[21]; dp/dµ = −1/3[14, 27] or 1/9[26]; ω and u˙ are parallel[8]; vanishing of
the magnetic part H[7] or of the electric part E[14, 9] of the Weyl tensor; θ = θ(µ)[15]
or θ = θ(ω)[9]; Petrov types N[2] and III[3, 4]; there exists a conformal Killing vector
parallel to the fluid flow u[6]. It is noteworthy that there exist Newtonian perfect fluids
with a barotropic equation of state, which are rotating, expanding but non-shearing.
Hence, if true, such behaviour of fluids would be a purely relativistic effect. Recently,
in an attempt to generalize the result established by Collins that H = 0 ⇒ ωθ = 0,
we managed to prove the conjecture for the case when divH = 0 (that is when the the
magnetic part of the Weyl tensor is solenoidal), with a γ-law equation of state[23]. This
result was further generalised in a sense with our being able to establish the conjecture
for the case when divH = 0 and there is a sufficiently generic equation of state[5].
Shearfree perfect fluids 2
In the present article we will focus our attention on generalising the result[14, 9] that
the conjecture holds when the electric part of the Weyl tensor vanishes, by considering
spacetimes for which the electric part is solenoidal. Specifically, we prove
Theorem Consider any shear-free perfect fluid solution of the Einstein field equations
wherethefluidpressuresatisfiesabarotropicequationofstateandthespatialdivergence
of the electric part of the Weyl tensor is zero. Then either the fluid is non-rotating or
non-expanding.
2. Notations and conventions
We shall be examining shear-free, perfect fluid solutions of the Einstein field equations
1
R − Rg = µu u +ph , (1)
ab ab a b ab
2
where u is the future-pointing (time-like) unit tangent vector to the flow, µ and p are
the energy density and pressure of the fluid, respectively, and h = g + u u is the
ab ab a b
projection tensor into the rest space of the observers with 4-velocity u. The vanishing
of the shear can be expressed by
1
u = θh +ω −u˙ u . (2)
a;b ab ab a b
3
Weshall assume familiarity withthenotationandconventions of theorthonormaltetrad
formalism as given by MacCallum [17]. We begin the analysis by choosing, what we
believe to be, a well suited tetrad alignment for the physical problem at hand. This
is the same as in our previous article [23]. First, e and e are aligned with u and ω,
0 3
respectively, such that ω = ωe 6= 0. The relevant variables then become µ, p, θ, ω,
3
u˙ , Ω together with the quantities n and a . Latin indices will be tetrad indices
α α αβ α
taking the values 1, 2, 3, 4. Greek indices take the values 1, 2, 3, while uppercase Latin
indices the values 1, 2 and any expressions involving these have to be read modulo 3 or
2 respectively. The sum of matter density and pressure will be written as
E = µ+p. (3)
Secondly, it is always possible, using the remaining rotational freedom and making use
of the Jacobi identities and field equations, to further specialize the tetrad so as to
achieve
Ω = Ω = Ω +ω = 0, (4)
1 2 3
n = n ≡ n.
11 22
Herewith the tetrad is fixed up to rotations e +ie → eiα(e +ie ) satisfying ∂ α = 0.
1 2 1 2 0
Because of computational advantages, we will be replacing n (α 6= β) and a with the
αβ α
new variables q and r defined by n = (r +q )/2, and a = (r −q )/2. We
α α α−1α+1 α α α α α
Shearfree perfect fluids 3
will also introduce extension variables, z and j which are related to the components of
α
the spatial gradient of the expansion, by
∂ θ = −θ2/3+2ω2 −(µ+3p)/2+j,
0
∂ θ = z , (5)
α α
j being the (3+1) covariant divergence of the acceleration,
j ≡ u˙a = ∂ u˙α +u˙αu˙ −u˙α(r −q ). (6)
;a α α α α
In order to relate the components of E with the spatial gradient of the acceleration,
ab
we will make use of the Ricci identity [16]
E = D u˙ −ω ω +u˙ u˙ , (7)
habi ha bi ha bi ha bi
where S stands for the spatially projected and trace-free part of a tensor S .
habi ab
The complete set of initial equations of the formalism are now the Einstein field
equations and the Jacobi equations, which we present, using the simplifications above,
in the appendix. First notice that the equations (57,60) immediately lead to evolution
equations for the variables r and q ,
α α
3∂ r = −z −θ(u˙ +r ), (8)
0 α α α α
3∂ q = z +θ(u˙ +q ), (9)
0 α α α α
while (50) and the (0α) field equations (62-64) give us the spatial derivatives of ω,
2
∂ ω = z −ω(q +2u˙ ), (10)
1 2 1 1
3
2
∂ ω = − z +ω(r −2u˙ ), (11)
2 1 2 2
3
∂ ω = ω(u˙ +r −q ), (12)
3 3 3 3
together with the algebraic restriction
2
n = z . (13)
33 3
3ω
The evolution equation for n follows from (58):
θ
∂ n = − n. (14)
0
3
Actingwiththecommutators[∂ , ∂ ]and[∂ , ∂ ]onthepressureandusing(56)together
0 α 1 2
with the conservation laws
∂ µ = −Eθ, ∂ p = −Eu˙ , (15)
0 α α
leads to a first set of evolution equations for the acceleration and vorticity:
∂ u˙ = p′z −Gθu˙ ,
0 α α α
1 (16)
∂ ω = ωθ(−2+3p′),
0
3
where we have defined
p′′ 1
G ≡ E −p′ + . (17)
p′ 3
Shearfree perfect fluids 4
The spatial derivatives of the acceleration can be obtained from (7), using (6):
1 1
∂ u˙ = − ω2 + j −u˙ q +u˙ r −u˙2 +E (18)
A A 3 3 A+1 A+1 A−1 A−1 A AA
2 1
∂ u˙ = ω2+ j −u˙ q +u˙ r −u˙2 +E (19)
3 3 3 3 1 1 2 2 3 33
1
∂ u˙ = −p′ωθ+q u˙ + n u˙ −u˙ u˙ +E (20)
1 2 2 1 33 3 1 2 12
2
1
∂ u˙ = p′ωθ−r u˙ − n u˙ −u˙ u˙ +E (21)
2 1 1 2 33 3 1 2 12
2
1
∂ u˙ = − u˙ n −r u˙ −u˙ u˙ +E (22)
1 3 2 33 3 1 1 3 13
2
1
∂ u˙ = u˙ n +q u˙ −u˙ u˙ +E (23)
2 3 1 33 3 2 2 3 23
2
1
∂ u˙ = − u˙ n +nu˙ +q u˙ −u˙ u˙ +E (24)
3 1 2 33 2 1 3 1 3 13
2
1
∂ u˙ = u˙ n −nu˙ −r u˙ −u˙ u˙ +E . (25)
3 2 1 33 1 2 3 2 3 23
2
Next we act with the [∂ , ∂ ] commutators on ω and θ and use the propagation of (13)
0 α
along u in order to obtain expressions for the evolution of z along e and for the spatial
α 0
gradient of j:
1 1
∂ z = θ(−1+p′)z − ω(−1+9p′)z + θω(9G−2)u˙ (26)
0 1 1 2 2
2 2
1 1
∂ z = θ(−1+p′)z + ω(−1+9p′)z − θω(9G−2)u˙ (27)
0 2 2 1 1
2 2
∂ z = θ(−1+p′)z , (28)
0 3 3
1 1 E
∂ j = p′θz − ω(27p′+13)z + u˙ (18ω2+θ2 −3j −3µ)− u˙
1 1 2 1 1
6 3 2p′
1
+ θω(9G−2)u˙ +4ω2q (29)
2 1
2
1 1 E
∂ j = p′θz + ω(27p′+13)z + u˙ (18ω2 +θ2 −3j −3µ)− u˙
2 2 1 2 2
6 3 2p′
1
− θω(9G−2)u˙ −4ω2r (30)
1 2
2
1 E
∂ j = p′θz + u˙ (θ2 −18ω2 −3j −3µ)− u˙ −4(r −q )ω2. (31)
3 3 3 3 3 3
3 2p′
Now we may evaluate P [∂ , ∂ ]u˙ , using (6) and (18-19), which leads to an expression
α 0 α α
for the evolution of j in terms of ∂ z . With the aid of the field equations (68-70), their
α α
propagation along u and the use of the [∂ , ∂ ] commutators on r ,q ,u˙ (excluding
0 α α α α
r , q , u˙ ) together with the [∂ , ∂ ] commutators on ω and Jacobi equation (53) ,
1 2 3 1 2
one obtains algebraic expressions for the directional derivative ∂ z , as well as evolution
3 3
equations for j as
θ(G′E +p′ −2Gp′)
∂ j = u˙2 +(1−2G)(z u˙ +z u˙ +z u˙ )
0 1 1 2 2 3 3
p′
Shearfree perfect fluids 5
1
− θ(1+3G)j −p′θ(1−9p′)ω2 (32)
3
θ(G′E −2Gp′) 2G
∂ z = (u˙2 −3u˙2)− (z u˙ +z u˙ −2z u˙ )−2z u˙
3 3 p′(1+3p′) 3 1+3p′ 1 1 2 2 3 3 3 3
2G+9p′2 −6p′ +1 θ(−2+3G)
+ ω2θ + E +r z −q z (33)
33 2 2 1 1
1+3p′ 1+3p′
(G′ ≡ dG and u˙2 ≡ u˙2+u˙2+u˙2). Also the solutions of equations (53) and (65) are given
dµ 1 2 3
by
θ(2Gp′ −G′E) 2G
∂ q = (u˙2 −3u˙2)+ (z u˙ +z u˙ −2z u˙ )
2 1 3ωp′(1+3p′) 3 3ω(1+3p′) 1 1 2 2 3 3
r −3q +3u˙ θ(−2+3G) 9p′2 +2G−9p′
3 3 3
+ z − E − θω
3 33
3ω 3ω(1+3p′) 3(1+3p′)
1
− (r z −q z )+(r +q )r +n(q +r )−E (34)
2 2 1 1 1 1 2 3 3 12
3ω
and
θ(2Gp′ −G′E) 2G
∂ r = (u˙2 +u˙2 −2u˙2)+ (z u˙ +z u˙ −2z u˙ )
1 2 3ωp′(1+3p′) 1 2 3 3ω(1+3p′) 1 1 2 2 3 3
3r −q +3u˙ θ(−2+3G) 9p′2 +2G−9p′
3 3 3
+ z − E − θω
3 33
3ω 3ω(1+3p′) 3(1+3p′)
1
− (r z −q z )−(r +q )q −n(q +r )+E . (35)
2 2 1 1 2 2 1 3 3 12
3ω
From the Ricci identity [16], H = 2u˙ ω +D ω , the components of the (trace-free)
habi ha bi ha bi
magnetic part of the Weyl curvature are given by
H = −ω(u˙ +r ), H = −ω(u˙ −q )
11 3 3 22 3 3
H = z /3−ωq , H = −z /3+ωr ,H = 0. (36)
13 2 1 23 1 2 12
At this stage we introduce the condition that E is solenoidal, divE = 0, which by the
Bianchi identity[16], (divE) = 1D µ−3ωbH reduces to
a 3 a ab
Eu˙ +3p′ω(z −3ωq ) = 0, (37)
1 2 1
Eu˙ −3p′ω(z −3ωr ) = 0, (38)
2 1 2
(E +18ω2p′)u˙ −9ω2p′(−r +q ) = 0. (39)
3 3 3
Apart from µωθE 6= 0 we shall also assume that p′ 6= 0,±1/3,1/9, u˙2+u˙2 6= 0, E 6= 0,
1 2 ab
H 6= 0,θ not just a function of either µ or ω, so as to avoid duplication of known
ab
results. Propagating equations (37) and (38) gives then
3p′[2E −9(1−3p′)ω2]z −θ[2E(6p′ −2)+81Gp′ω2]u˙ = 0, (40)
A A
These two equations show that z is parallel to u˙ , unless
A A
2E −9(1−3p′)ω2 = 2E(6p′−2)+81Gp′ω2 = 0. (41)
Shearfree perfect fluids 6
3. u˙ parallel to z
A A
Taking the evolution of u˙ z −u˙ z = 0 along u, this results in
2 1 1 2
27p′(3G+3p′ −1)ω2 −E(−1+27Gp′ +9p′ −54p′2) = 0.
If the coefficient of ω2 were zero, the latter equation would give
E(−1+9p′)2 = 0.
Hence, thevorticity isafunctionofthematterdensity andonehas∂ µ∂ ω−∂ ω∂ µ = 0,
1 2 2 1
which reduces to (u˙ z +u˙ z )E = 0. It follows that u˙ = u˙ = 0, which is impossible.
1 1 2 2 1 2
4. u˙ not parallel to z
A A
Eliminating ω from (41) fixes the equation of state to be
9Gp′ −18p′2 +9p′ −1 = 0. (42)
Note that a γ-law equation of state is then only possible when p′ = −1/3 or 1/9, so
henceforth we will exclude all sub-cases in which p′ is constant. By (41) the vorticity is
again a function of matter density. It follows that
∂ µ∂ ω −∂ ω∂ µ = 0,
A 3 A 3
which after eliminating q , r and q using equations (37), (38) and (39) yield
1 2 3
E(−3ωu˙ +z )u˙ = 0.
A A+1 3
It follows that u˙ is zero, as otherwise, propagation of equation (41) along e would
3 A
imply E(−1+3p′)u˙ = 0 and again u˙ = 0.
A A
When u˙ is zero, the expressions (16), (22), (23), (19) and (13) show that
3
z = 0, n = 0, E = u˙ r , E = −u˙ q ,
3 33 13 1 3 23 2 3
1
E = − (2ω2 +j)+u˙ q −u˙ r .
33 1 1 2 2
3
Moreover, the conditions divE = 0, (37-39) can now be rewritten as follows
Eu˙ +3ωp′(z −3ωq ) = 0,
1 2 1
Eu˙ −3ωp′(z −3ωr ) = 0, (43)
2 1 2
p′(q −r )ω2 = 0.
3 3
Propagating equation (41)(a) along e and e , using equations (43), yields
1 2
E(−u˙ +3p′q ) = 0, (44)
1 1
E(u˙ +3p′r ) = 0. (45)
2 2
Again propagating the latter equations along e and e respectively and eliminating E
1 2 12
shows then
9θp′2(3p′ −1)(18p′2 −15p′ +1)j +9p′2(3p′ −1)(36p′2−9p′ +5)(u˙ z +u˙ z )
1 1 2 2
−9θ(3p′ −1)(162p′4 +108p′3 −162p′2 +21p′−1)(u˙2 +u˙2)
1 2
+6p′3θ(27p′2 −18p′ +7)E = 0. (46)
Shearfree perfect fluids 7
Now we use z = 0 and equations (41-44), (45) and (43) to simplify equation (33 ) to
3
(3p′ −1)(162p′4+108p′3 −162p′2 +21p′ −1)(u˙2+u˙2)−6(27p′2 −18p′ +7)Ep′3
1 2
−9p′2(3p′ −1)(18p′2−15p′ +1)j = 0. (47)
This can now be combined with (46) to give
p′2(3p′ −1)(36p′2 −9p′ +5)(z u˙ +z u˙ ) = 0.
1 1 2 2
It follows that u˙ is orthogonal to z , after which propagation of equation (46) along
A A
e gives
0
2(1296p′4−2106p′3 +522p′2 +9p′ −5)(3p′ −1)2(u˙2 +u˙2)
1 2
−24p′2E(p′ +1)(81p′3 −207p′2 +93p′ −7) = 0. (48)
Taking the time derivative of the latter equation, eliminating j,z using equation (46)
1
and z u˙ +z u˙ = 0, results in a new equation between p,p′ and u˙2 +u˙2. Eliminating
2 2 1 1 1 2
the acceleration between the latter pair eventually results in
Ep′3(3p′ −1)(18p′2 −15p′ +1)
×
1296p′4−2106p′3 +522p′2 +9p′ −5
(26244p′6−59778p′5 −106515p′4 +156249p′3−43065p′2+3267p′ −26) = 0,
showing that p′ has to be a constant, in contradiction with 42 and our assumptions
p 6= −1/3,1/9.
5. Discussion
Newtonian homogeneous cosmologies are known[19] to provide counter-examples to the
Newtonian shear-freefluidconjecture. Inthese modelsthespatial divergence ofthetidal
field (the Newtonian analogue of E) automatically vanishes, such that, when searching
forpossible counter-examples to therelativistic shear-free fluidconjecture, itistempting
to look for candidates within the class of perfect fluids having a solenoidal electric
part of the Weyl tensor. The result obtained in this paper shows that this attempt
remarkably (or luckily —depending on one’s attitudes towards the conjecture) fails:
shear-free perfect fluids obeying a barotropic equation of state and having divE = 0 are
all non-rotating or non-expanding, in contrast to their Newtonian brethren. Whether
the conjecture in its full generality is true or false still remains an intriguing issue.
Shearfree perfect fluids 8
6. Appendix A
Commutator relations, using σ = 0:
αβ
[∂ ,∂ ] = u˙ ∂ −θ ∂ +(ω +Ω )∂ −(ω +Ω )∂
0 1 1 0 1 1 3 3 2 2 2 3
[∂ ,∂ ] = u˙ ∂ −(ω +Ω )∂ −θ ∂ +(ω +Ω )∂
0 2 2 0 3 3 1 2 2 1 1 3
[∂ ,∂ ] = u˙ ∂ +(ω +Ω )∂ −θ ∂ −(ω +Ω )∂
0 3 3 0 2 2 1 3 3 1 1 2
(49)
[∂ ,∂ ] = −2ω ∂ +q ∂ +r ∂ +n ∂
1 2 3 0 2 1 1 2 33 3
[∂ ,∂ ] = −2ω ∂ +q ∂ +r ∂ +n ∂
2 3 1 0 3 2 2 3 11 1
[∂ ,∂ ] = −2ω ∂ +r ∂ +q ∂ +n ∂
3 1 2 0 3 1 1 3 22 2
Using in addition the simplifications ω = ω = Ω = Ω = 0, Ω = −ω = −ω,
1 2 1 2 3 3
n = n = n one obtains the:
11 22
Jacobi equations:
∂ ω = ω(u˙ +r −q ) (50)
3 3 3 3
∂ n+∂ r +∂ q −n(r −q )−r r +q q = 0 (51)
1 2 3 3 2 1 1 3 2 3 2
∂ n+∂ r +∂ q −n(r −nq )−r r +q q = 0 (52)
2 3 1 1 3 2 2 3 1 3 1
2
∂ r +∂ q +∂ n − θω −r r +q q −n (r −q ) = 0 (53)
1 2 2 1 3 33 2 1 2 1 33 3 3
3
∂ u˙ −∂ u˙ −nu˙ −q u˙ −r u˙ = 0 (54)
2 3 3 2 1 3 2 2 3
∂ u˙ −∂ u˙ −nu˙ −r u˙ −q u˙ = 0 (55)
3 1 1 3 2 3 1 1 3
4
2∂ ω +∂ u˙ −∂ u˙ −u˙ q −u˙ r −u˙ n + θω = 0 (56)
0 1 2 2 1 1 2 2 1 3 33
3
1 θ
∂ (r −q )− ∂ θ+z + (r −q +2u˙ ) = 0 (57)
0 α α α α α α α
3 3
3∂ n+3∂ ω +nθ −3ω(u˙ +r −q ) = 0 (58)
0 3 3 3 3
3∂ n +3∂ ω +n θ−3ω(u˙ +r −q ) = 0 (59)
0 33 3 33 3 3 3
1
∂ (r +q )+ θ(r +q ) = 0 (60)
0 α α α α
3
Einstein equations:
1 1
∂ θ = − θ2 +2ω2 − (µ+3p)+j (61)
0
3 2
2/3z +∂ ω −ω(r −2u˙ ) = 0 (62)
1 2 2 2
2/3z −∂ ω −ω(q +2u˙ ) = 0 (63)
2 1 1 1
2/3z −ωn = 0 (64)
3 33
−∂ r +∂ q −r r −q q −2r q −2r n+n (r +q )−2q n =
1 2 2 1 1 2 1 2 2 1 3 33 3 3 3
−∂ u˙ −∂ u˙ −2u˙ u˙ +u˙ q −u˙ r (65)
1 2 2 1 1 2 1 2 2 1
Shearfree perfect fluids 9
−∂ r +∂ q +∂ n−∂ n −n(r −q )−2q n −r r −q q −2r q =
2 3 3 2 1 1 33 1 1 1 33 2 3 2 3 3 2
−∂ u˙ −∂ u˙ +u˙ (n −n)−2u˙ u˙ +u˙ q −u˙ r (66)
2 3 3 2 1 33 2 3 2 3 3 2
∂ q −∂ r +∂ n −∂ n−r (2n −n)−nq −r r −q q −2r q =
1 3 3 1 2 33 2 2 33 2 1 3 1 3 1 3
−∂ u˙ −∂ u˙ −u˙ (2u˙ +r )+u˙ (n−n )+u˙ q (67)
1 3 3 1 1 3 3 2 33 3 1
1
−∂ r +∂ q +∂ q −∂ r +q2 +r2 + n2 −nn +r2 +q2 −r q −r q =
1 1 1 1 2 2 3 3 2 3 2 33 33 1 1 2 2 3 3
2 8 1
θ2 + ω2 − (2µ−j)−∂ u˙ −u˙2 +u˙ r −u˙ q (68)
9 3 3 1 1 1 3 3 2 2
1
−∂ r +∂ q −∂ r +∂ q +q2 +r2 + n2 −nn +r2 +q2 −r q −r q =
2 2 2 2 1 1 3 3 3 1 2 33 33 2 2 1 1 3 3
2 8 1
θ2 + ω2 − (2µ−j)−∂ u˙ −u˙2 +u˙ r −u˙ q (69)
9 3 3 2 2 2 1 1 3 3
1
−∂ r +∂ q +∂ q −∂ r +q2 +r2 − n2 +r2 +q2 −r q −r q =
3 3 3 3 1 1 2 2 1 2 2 33 3 3 1 1 2 2
2 1
θ2 + (2ω2 −2µ+j)−∂ u˙ −u˙2 +u˙ r −u˙ q (70)
9 3 3 3 3 2 2 1 1
References
[1] Banerji S (1968) Prog.Theor. Phys. 39 365
[2] Carminati J (1990) J. Math. Phys. 31 2434
[3] Carminati J and Cyganowski S (1996) Class. Quantum Grav 13 1805
[4] Carminati J and Cyganowski S (1997) Class. Quantum Grav 14 1167
[5] Carminati J, Karimian H R, Van den Bergh N and Vu K T (2009) Class. Quantum Grav. 26
195002
[6] Coley A A (1991) Class. Quantum Grav. 8 955
[7] Collins C B (1984) J. Math. Phys. 25 995
[8] Collins C B (1986) Can. J. Phys. 64, 191
[9] CyganowskiS and Carminati J (2000) Gen. Rel. Grav. 32, 221
[10] Czapor S R, McLenaghan R G and Carminati J (1992). Gen. Rel. Grav. 24 911
[11] Ellis G F R (1967) J. Math. Phys. 8 1171
[12] G¨odel K (1949) Rev. Mod. Phys. 21 447
[13] King A R and Ellis G F R (1973) Commun. Math. Phys. 31 209
[14] Lang J M (1993) Contributions to the study of general relativistic shear-free perfect fluids Ph. D.
thesis, University of Waterloo, Canada
[15] Lang J M and Collins C B (1988) Gen. Rel. Grav. 20 683
[16] Maartens R and Bassett B A (1998) Class. Quantum Grav. 15 705
[17] MacCallum M A H 1971 Cosmological Models from a Geometric Point of View (Cargese) vol 6
(New York:Gordon and Breach) p 61
[18] Schu¨cking E (1957) Naturwissenschaften 19 507
[19] Senovilla J M M, Sopuerta C F, Szekeres P (1998) Gen. rel. Grav. 30 3
[20] Sopuerta C F (1998) Class. Quantum Grav. 15 1043
[21] Treciokas R and Ellis G F R (1971) Commun. Math. Phys. 23 1
[22] Van den Bergh N, (1988) Class. Quantum Grav. 5, L169
[23] Van den Bergh N, Carminati J and Karimian H R (2007) Class. Quantum Grav. 24 1
[24] Vu K T and Carminati J (2003). Gen. Rel. Grav. 35 263
[25] Vu K T and Carminati J (2008). Preprint.
[26] Van den Bergh N (1999) Class. Quantum Grav. 16 117
[27] White A J and Collins C B (1984) J. Math. Phys. 25 332