Examensarbete
Lanczos Potentials in Perfect Fluid Cosmologies
David Holgersson
Lanczos Potentials in Perfect Fluid Cosmologies
Till¨ampad matematik, Link¨opings tekniska h¨ogskola David Holgersson
LiTH-MAT-EX–04/12–SE
Examensarbete: 20 p Level: D
Supervisor &
Examiner: Fredrik Andersson,
Matematiska institutionen, Till¨ampad matematik, Link¨opings tekniska h¨ogskola Link¨oping: 13 October 2004
Matematiska Institutionen 581 83 LINK ¨OPING SWEDEN 13 October 2004 x x http://www.ep.liu.se/exjobb/mai/2004/tm/012/ LiTH-MAT-EX–04/12–SE
Lanczos Potentials in Perfect Fluid Cosmologies
David Holgersson
We derive the equation linking the Weyl tensor with its Lanczos potential, called the Weyl-Lanczos equation, in 1+3 covariant formalism for perfect fluid Bianchi type I spacetime and find an explicit expression for a Lanczos poten-tial of the Weyl tensor in these spacetimes. To achieve this, we first need to derive the covariant decomposition of the Lanczos potential in this formalism. We also study an example by Novello and Velloso and derive their Lanczos potential in shear-free, irrotational perfect fluid spacetimes from a particular ansatz in 1+3 covariant formalism. The existence of the Lanczos potential is in some ways analogous to the vector potential in electromagnetic theory. Therefore, we also derive the electromagnetic potential equation in 1+3 co-variant formalism for a general spacetime. We give a short description of the necessary tools for these calculations and the cosmological formalism we are using.
Weyl-Lanczos equation, Lanczos potentials, 1+3 covariant formalism, perfect fluid, cosmology, Bianchi type I models, shear-free and irrotational models.
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Abstract
We derive the equation linking the Weyl tensor with its Lanczos potential, called the Weyl-Lanczos equation, in 1+3 covariant formalism for perfect fluid Bianchi type I spacetime and find an explicit expression for a Lanc-zos potential of the Weyl tensor in these spacetimes. To achieve this, we first need to derive the covariant decomposition of the Lanczos potential in this formalism. We also study an example by Novello and Velloso and derive their Lanczos potential in shear-free, irrotational perfect fluid space-times from a particular ansatz in 1+3 covariant formalism. The existence of the Lanczos potential is in some ways analogous to the vector potential in electromagnetic theory. Therefore, we also derive the electromagnetic potential equation in 1+3 covariant formalism for a general spacetime. We give a short description of the necessary tools for these calculations and the cosmological formalism we are using.
Keywords: Weyl-Lanczos equation, Lanczos potentials, 1+3 covariant for-malism, perfect fluid, cosmology, Bianchi type I models, shear-free and irrotational models.
Acknowledgements
I would like to thank my supervisor Fredrik Andersson for helpful suggestions and discussions. I would also like to mention my opponent Jonas Jonsson Holm, who had the strength to proofread the manuscript.
Contents
1 Introduction and Basic Relations 1
1.1 Introduction . . . 1
1.2 Manifolds . . . 2
1.3 Conventions and Notation . . . 3
1.4 Derivative Operators . . . 3
1.5 The Curvature Tensor . . . 4
1.6 The Weyl Tensor . . . 4
1.7 The Electromagnetic Field Tensor . . . 5
1.8 The Lanczos Potential . . . 6
1.9 The Einstein Field Equations . . . 7
2 Geometry of Cosmology Models 9 2.1 Covariant Formalism . . . 9
2.2 Kinematical Quantities . . . 12
2.3 The Source Terms . . . 14
2.4 Covariant Propagation and Constraint equations . . . 15
3 The Maxwell Case 17 3.1 Splitting the Electromagnetic Field . . . 17
3.1.1 Electric Part . . . 19
3.1.2 Magnetic Part . . . 20
3.2 Summing-up . . . 21
4 The Decompositions of the Lanczos Tensor 23 4.1 Splitting the Gravitational Field . . . 23
4.2 The Spacetime Decomposition . . . 24
5 Shearfree and Irrotational Models 27 5.1 Introduction . . . 27
5.2 Constructing the Weyl-Lanczos Equation . . . 28
5.2.1 A-terms . . . 28
5.2.2 C-terms . . . 33
5.3 Summing-up . . . 38
6 Spatially Homogeneous Universes 39
6.1 Bianchi Type I Models . . . 39
6.2 Constructing the Weyl-Lanczos Equation . . . 40
6.2.1 A-terms . . . 40
6.2.2 C-terms . . . 42
6.2.3 S-terms . . . 44
6.2.4 P-terms . . . 45
6.3 Summing-up . . . 46
7 Conclusions and Future Work 49 7.1 Conclusions . . . 49
Chapter 1
Introduction and Basic
Relations
1.1
Introduction
In the lifetime of human civilization we have contact with only a small spactime region of our universe. Even while our telescopes can observe ob-jects remarkably far away by human scales, the information we get is only a portion of our past light cone. Therefore, it is hard to prove theories by ap-pealing only to observation data of objects falling in from space. Theoretical cosmology, where we search for a framework within which to comprehend the information from our observations, started with the cosmology models proposed by A. Einstein and W. de Sitter in 1917, based on Einstein´s the-ory of general relativity. If you check up the word cosmology in Chambers´s Dictionary, it says: ’Cosmology is the science of the universe as a whole’. The realization that space and time considered as a single whole - a four-dimensional manifold called spacetime - is one of the greatest intellectual achievements of the twentieth century.
In modern study of cosmology, several kinds of physics is required. Since the dominant force on the cosmic scale is gravitation, this is the basic ingre-dient. We assume this is given by the Einstein´s theory of general relativity. The matter distrubution of e.g. fluids, gases and fields in spacetime is given by Einstein field equations. In many situations this includes the physics of electromagnetism. On top of this, we sometimes take thermodynamics into consideration and even particle physics.
A cosmological model is a model of our universe which predicts the ob-served properties of the universe, and explains the phenomena in the early universe. A model must also explain why the universe was so homoge-neous and isotropic at the epoch of last scattering of the cosmic microwave background, and how and when inhomogeneities such as galaxies and stars arose. In more restricted sense cosmological models are exact solutions of
the Einstein field equations for a perfect fluid.
The mathematics which is often used to describe curved spacetime is differential geometry. The global geometry of the spacetime is determined by the Riemann curvature tensor, a tensor with rather complicated symme-tries. By decomposing this tensor into simpler parts, the Weyl curvature tensor arises. In 1962 Lanczos introduced a new tensor, now called the Lanczos tensor, which is a potential for the Weyl tensor. To study potential equations can sometimes be easier, or can give additional physical under-standing. In general relativity, the existence of the Lanczos potential is in some ways analogous to the vector potential in electromagnetic theory. Therefore, we first begin with electromagnetism which is much easier and later study gravitation using the Lanczos potential. The aim is to find a procedure to derive the equation linking the Weyl tensor with its Lanczos potential, called the Weyl-Lanczos equation, using a cosmological formal-ism in a cosmology model. In this thesis, we want to go into two examples of such cosmological models. Firstly, we study shear-free and irrotational models (these were also studied by Novello and Velloso, [11], and an explicit example of a Lanczos potential was found; this potential will later act as a check on the Weyl-Lanczos equation that we derive in this thesis), and secondly we will investigate Bianchi type I models.
The following basic relations, covariant descriptions and equations in chapter 1 and 2 are briefly introduced. The reader is assumed to have some knowledge about Einstein´s theory of general relativity and some knowledge about vectors, covectors and tensors. For more thorough introduction to basic relations and cosmology, see [3], [5], [8], [10] and [14].
1.2
Manifolds
A manifold is a set made up of pieces of topological space that looks locally like the Euclidean space Rn, such that these pieces can be ”sewn” together
smoothly. However, this set may have quite different global properties. An example is the surface of a sphere S2such as the Earth, which is not a plane, but small patches of S2are homeomorphic to patches of the Euclidean plane. Each patch is called a coordinate systems or a chart. The local charts can be smoothly ”sewn” together and we can define directions, tangent spaces, and differentiable functions on that manifold. In special relativity and gen-eral relativity, time and three-dimensional space are treated together as a four-dimensional manifold called spacetime. The spacetime geometry is de-termined by this manifold on which is defined a metric of Lorentz signature. The metric tensor, gab, is a tensor of rank 2 that is used to measure
dis-tance and angles in a space or spacetime. A point of spacetime represents an event which has four coordinates. A sequence of events that a particle occupies during its lifetime is represented by a curve called its worldline. A
1.3. Conventions and Notation 3
congruence of worldlines is represented by the 4-velocity field u, which is everywhere tangent to the congruence.
1.3
Conventions and Notation
The spacetime geometry we use is determined by a real 4-dimensional man-ifold on which we define a metric gab of Lorentz signature (− + ++). The
metric can be used to lower indices, in that way a covector is associated to every vector by gbiXi = Xb. The metric is also used to lower indices of
arbitrary tensors, i.e. gbcTac = Tab. The inverse metric gab, is defined by
gabg
bc = δac, where δac is the identity map. In the same way the inverse
metric can be used to raise indices on any tensor. If the tensor Ta is given
the vector Xb as an argument we denote that by letting them have the same
index, T (X) = TaXa = TaXa. The order which Ta and Xa is written is
irrelevant. Also the names of the indices are irrelevant as long as they are the same and have not been used elsewhere. A repeated index, once as lower a index and once as an upper index, is called a dummy index. An index which is used only once is called a free index. The contraction of a tensor is denoted by letting an upper index on a tensor be the same as a lower index, e.g. Tab
ac. If the contraction over those indices makes the tensor vanish, it
is called trace-free, i.e. Ta a= 0.
Throughout this thesis we are using geometrised units characterised by c = 1 = 8πGc2 . Consequently, all quantities are given the dimension of a
power of length.
1.4
Derivative Operators
A derivative operator on a manifold is a map wich takes each smooth tensor field type (k, l) to a new smooth tensor field of type (k, l + 1). We call it a covariant derivative and denote it ∇a. Although, this is not a tensor but an
operator, the notation is motivated by the fact that when the connections operates on a tensor, the result is a tensor. It is often notationally convenient to write the connection as ∇aTbc= Tbc;a. We will only consider connections
which are torsion free and compatible with the metric, i.e. [14]
Xa∇aYb− Ya∇aXb = [X, Y ]b, (1.1)
for all vector fields Xa and Ya and
∇agbc= 0. (1.2)
Moreover, we demand that covariant differentation satisfies linearity and the Leibniz rule, i.e.
and
∇a(Ab...c...· Bd...e...) = (∇aAb...c...)Bd...e...+ Ab...c...(∇aBd...e...), (1.4)
for all tensors A and B of type (k, l) and α, β ∈ R We also demand that ∇a
agrees with the usual differential when acting on scalars, i.e. ∇af = (df )a,
for all functions f . It can be shown [14] that there exists a unique derivative operator satisfying all these properties.
1.5
The Curvature Tensor
The curvature tensor generalizes Gauss curvature to higher dimensions. Co-variant differentiation, unlike partial differentiation, is not in general com-mutative. For a vector xa, the commutator of the torsion-free connection
becomes
∇[c∇d]xa= 1 2R
a
bcdxb. (1.5)
By the torsion-freeness of ∇a it follows that the left-hand side of (1.5) is a
tensor, it follows that Ra
bcd is a tensor of type (1,3). It is called the Riemann
tensor. It can be shown by induction, the commutator of an arbitary tensor field, can be expressed in terms of the tensor itself and the Riemann tensor. From [14] we get some useful symmetries of the Riemann tensor:
1. Rabcd= −Rbacd. (1.6)
2. R[abc]d= 0. (1.7)
3. Rabcd= Rcdab. (1.8)
4. The Bianchi identity holds:
∇[aRbc]de= 0. (1.9)
By the antisymmetry properties we can decompose the Riemann tensor into a trace part and a trace free part. The trace of the Riemann tensor defines the Ricci tensor, Rac,
Rac= Rabcb. (1.10)
The scalar curvature, R, also called the Ricci scalar, is defined as the trace of the Ricci tensor:
R= R a
a . (1.11)
1.6
The Weyl Tensor
The trace-free part of the Riemann tensor is called the Weyl conformal tensor. It first makes its appearance for the spacetime dimension four i.e., it is identically zero in lower dimensions. It describes the gravitational field
1.7. The Electromagnetic Field Tensor 5
in a vacuum region of spacetime. Consequently, gravity could not exist (according to general relativity) in a vacuum region if we lived in a universe where spacetime had only three dimensions. The Weyl tensor [9] together with the Ricci tensor holds the same information as the Riemann tensor and is given by Cabcd= Rabcd− 2 n −2(ga[cRd]b− gb[cRd]a) + 2 (n − 1)(n − 2)Rga[cgd]b. (1.12) However, we will only use the Weyl tensor in four dimensions, where
Cabcd= Rabcd−(ga[cRd]b− gb[cRd]a) +
1
3Rga[cgd]b. (1.13) The Weyl tensor has all the algebraic symmetries of the Riemann curvature tensor and in addition it is trace-free. Any tensor with the same algebraic symmetry properties as the Weyl conformal tensor is called a Weyl candidate tensor.
1.7
The Electromagnetic Field Tensor
In electromagnetic theory Maxwell’s equations can be written as first order differential equations in the field tensor Fab. In curved spacetime Maxwell´s
equations are
∇aFab = jb, (1.14)
∇[aFbc] = 0, (1.15)
where jb is the source current. The second Maxwell equation tells us that
there are no magnetic monopoles. By the Poincar´e lemma [14], the second Maxwell equation implies that there exists a vector field Φb such that
Fab = 2∇[aΦb]. (1.16)
However, this prescription is not unique. There are many different potentials which generate the same fields. This is called gauge freedom. Although a gauge transformation alters the potentials, it leaves Fabunchanged. Because
of the arbitrariness in the choice of gauge, we are free to impose an additional condition on Φ to reduce the gauge freedom, which in turn simplifies the problem. The most notable choice is the Lorentz gauge, ∇aΦ
a= 0.
In electromagnetic theory the vector potential was first introduced in order to express the equations of classical electrodynamics in simpler form. In classical physics the only physical effect of an electromagnetic field on a charge is the Lorentz force, and this only exists in regions where the electric or magnetic field is non-vanishing.
1.8
The Lanczos Potential
The Weyl curvature tensor is a four index tensor with rather complicated symmetries, which are similar to the symmetries of Fab. It is therefore
a natural question whether Cabcd can be written as a first derivative of a
potential, analogously to 1.16. It turns out that in four dimensions this is actually the case. The Lanczos potential is a three index tensor with somewhat simpler symmetries. Let Wabcd be an arbitrary Weyl candidate.
If there exists (locally) a tensor Labcwith the symmetries
Labc= L[ab]c, L[abc]= 0, (1.17)
such that Wabcd can be obtained by the formula
Wabcd = 2Lab[c;d]+ 2Lcd[a;b]−
−2g[a[c(Lb]id];i− Lb]i|i|;d]+ Ld]ib];i− Ld]i|i|;b]) +
+4 3g
a
[cgbd]Liji;j, (1.18)
then Labc is called a Lanczos potential for Wabcd [9]. The fact that there
really exists a potential for the Weyl tensor, was proved by Bampi and Cav-iglia [2]. In addition, many explicit examples of the Weyl tensor itself have been found in various geometries. Later we will study one such example in detail, namely a Lanczos potential of the Weyl tensor in certain cosmological spacetimes found by Novello and Velloso [11].
Note that although the Weyl tensor has only 10 degrees of freedom, a tensor Labc which obeys the symmetries (1.17) has 20 independent
compo-nents. Using the Lanczos algebraic gauge
Labb = 0, (1.19)
equation (1.18) can be simplified to
Wabcd= 2Lab[c;d]+ 2Lcd[a;b]−2g[a[c(Lb]id];i+ Ld]ib];i) (1.20)
and the degrees of freedom reduces to 16. The existence of the Lanczos potential plays a similar role in general relativity in the way that the vector potential does in electromagnetic theory. Due to more complicated sym-metries than with Fab, three more terms are necessary in order to give the
RHS the symmetries of an arbitrary Weyl candidate. If we for a second ignore these last three terms, the resemblance between the (1.20) and (1.16) is notable.
From the Lanczos algebraic gauge it can be shown [9] in four dimensions there is a further gauge choice
Labc;c = ξab, (1.21)
where ξab is any antisymmetric tensor. The choice ξab = 0 is called the
Lanczos differential gauge (compare with the Lorentz gauge in section (1.7)). However, not even this gauge choice is sufficient to determine Labc uniquely.
1.9. The Einstein Field Equations 7
1.9
The Einstein Field Equations
The strongest force of nature on large scales is gravity, so the most important part of a physical description of the universe is a theory of gravity. The best candidate we have for this is ’Einstein´s General Theory of Relativity’. In general relativity, spacetime is a four-dimensional manifold on which is defined a Lorentz metric gab. All physical quantities will be modelled by 4D
tensors (or vectors or scalars) and therefore laws of physics will be expressed by 4D tensor equations. Using a variational principle, we can obtain these laws of interaction of geometry and matter (see [14]). What we find is the physical properties of fluids, gases, fields etc. can be described by a 4D 2-index symmetric tensor Tabcalled the stress-energy-momentum tensor. The
matter distribution in spacetime is given by the Einstein field equations, Gab ≡ Rab−
1
2Rgab= Tab, (1.22)
which, because of the ’contracted Bianchi identities’ implies the relation
∇bGab = 0 ⇒ ∇bTab= 0. (1.23)
Together, these statements summarizes the dynamical equations for matter and fields.
The field equations for an arbitrary metric consists 10 non-linear second order differential equations for the metric components. So even in empty space, i.e., vacuum, where the field equations are given by
Gab = 0, (1.24)
Chapter 2
Geometry of Cosmology
Models
In this chapter we give a summary of the geometrical background material that is needed. We focus on the description of spacetime curvature, the kinematics of a fluid and its source terms, and the methods for performing a 1+3-decomposition of the Einstein field equations into propagation and constraint equations.
2.1
Covariant Formalism
We will describe spacetime by covariantly defined variables. We do this be-cause we will have complete coordinate freedom in General Relativity and we can describe physics and geometry by tensor relations and quantites. These will remain valid whatever coordinate system is chosen. We will consider the effect of spacetime curvature on families of timelike curves representing the flowlines of a fluid. Its future-directed 4-velocity, ua, is
ua= dx
a
dτ , (2.1)
where τ is proper time measured along the flowlines
uaua= −1, (2.2)
and generates a splitting of spacetime into ’time’ and ’space’ that we will call 1+3 decomposition.
The acceleration vector
˙ua= ub∇bua, (2.3)
representing the degree to which the matter moves under forces other than gravity plus inertia. The acceleration vanishes for matter in free fall.
Given ua, the projection tensor h
ab is defined by
hab= gab+ uaub, (2.4)
which at each point projects into the 3-space orthogonal to ua. One may
think of hab as the metric of the restspaces of observers moving with
4-velocity ua. It follows that hachbc = hab, (2.5) haa = gachca = gac(gca+ ucua) = gacgca+ gacucua= δaa+ uaua = 4 − 1 = 3 (2.6) habub = gabub+ uaubub = ua− ua= 0. (2.7)
In particular we note that if
Tabub = 0, (2.8)
then we can raise the second index with hab instead of gab, since
hbcTac = (gbc+ ubuc)Tac= Tab. (2.9)
A volume element for the restspaces is also defined as
ηabc= udηdabc, (2.10)
where ηabcd is the totally antisymmetric spacetime permutation tensor. It
follows that
η[abc] = 1
6(ηabc− ηacb+ ηcab− ηcba+ ηbca− ηbac)
= 1
6(ηabc+ ηabc+ ηabc+ ηabc+ ηabc+ ηabc)
= ηabc, (2.11)
ηabcuc = 0. (2.12)
A useful relation between this and the metric is (see [12]). ηabcdηpqrd= −6ga[pg
q b g
r]
d , (2.13)
Using this we obtain
ηabeηcde = ugηgabeuhηhcde= uguh(−6)gg[hgacg d] b
= −6uguh(−ugu[h+ hg[h)(−uauc+ hac)(−u d] b + h d] b ) = ... = hachbd− hadhbc= 2ha[chbd]= 2h[achb]d. (2.14)
2.1. Covariant Formalism 11
In a similiar way we can also show that
ηabcηdef = 6h[adhbehc]f,
ηadeηcde = 2hac,
ηabcηabc = 6. (2.15)
Two derivatives are also defined. The first is the time derivative along the fundamental worldlines, where for any tensor T
˙
Tabcd= ue∇eTabcd. (2.16)
We also introduce e∇a, the covariant derivative operator orthogonal to ua,
by totally projecting the covariant derivative of any tensor T as e
∇eTabcd= hafhbghichjdhke∇kTf gij (2.17)
We use angle brackets to denote orthogonal projection of vectors and the projected symmetric trace-free part of tensors, i.e.
vhai = habvb (2.18) and Thabi = (h(achb)d− 1 3h abh cd)Tcd. (2.19)
Especially, we will see projected derivatives of the type e ∇haubi = (ha(chbd)− 1 3habh cd) e∇ (cud)= = ha(chbd)∇cud− 1 3habh cd∇ cud. (2.20)
Furthermore, we define the ’curl’ of vectors and 2-tensors as
(curl v)a= ηabc∇ebvc, (2.21)
and
2.2
Kinematical Quantities
Any spatial tensor Tab = hachbdTcd may be covariantly decomposed into its
symmetric and antisymmetric parts as: Tab = T[ab]+ T(ab) = ηabcTc+ Thabi+ 1 3hcdT cdh ab, (2.23)
where Thabi= h(achb)dTcd−13hcdTcdhab is the projected symmetric tracefree
part, and the spatial vector Ta is equivalent to the antisymmetric part:
T[ab] = ηabcTc ⇔ Ta=
1 2ηabcT
bc. (2.24)
The kinematic variables are established by decomposing the covariant derivative of the fluid 4-velocity, ua, into its irreducible parts:
∇aub = −ua˙ub+ e∇aub= −ua˙ub+
1
3Θhab+ σab+ ωab. (2.25) The rate of rotation of neighbouring curves is given by the vorticity tensor ωab, while the rate of change of their separation is given by the expansion
tensor Θab and the rate of change of volume is given by taking the trace
of Θab. Equation (2.25) is a particular case of (2.23) and we define the
following quantities as
1. the symmetric tensor of expansion:
Θab= hcahdb∇(duc)= e∇(aub), (2.26)
which describes the rate of change of the separation of neighbouring curves. The scalar volume expansion, Θ, is defined as
Θ = Θabhab = ∇buahab = e∇aua, (2.27)
which describes the rate of change of volume is given by taking the trace of Θab.
2. the symmetric traceless shear tensor, σab = Θab−
1
3habΘ = e∇haubi (2.28)
⇒ σab = σ(ab), σabub= 0, σaa = 0,
which describes the rate of distorsion of the matter flow. 3. the antisymmetric vorticity tensor,
ωab = hcahdb∇[cud]= e∇[aub], (2.29)
2.2. Kinematical Quantities 13
which describes the rate of rotation of neighbouring curves1. The vorticity
vector is defined as ωa = 1 2η abcω bc, (2.30) ⇒ ωaua= 0, ωabωb = 0.
Further, we derive the covariant derivative of hab, which will be useful
later. Remember gab = −uaub+ hab. The result of the action of ∇a on hbc
then becomes ∇ahbc = ∇a(gbc+ ubuc) = uc∇aub+ ub∇auc = uc(−ua˙ub+ 1 3Θhab+ σab+ ωab) +ub(−ua˙uc+ 1 3Θhac+ σac+ ωac) = −2uau(b˙uc)+ 2
3Θha(buc)+ 2σa(buc)+ 2ωa(buc), (2.31) where we used (2.25). We also need to derive an expression of the derivative of ηabc. The mathematical trick here is to start with the following product
and apply Leibniz´rule to this product: 1 6ηef g∇a(ηbcdη ef g) = 1 6ηef g(η ef g∇ aηbcd+ ηbcd∇aηef g). (2.32)
Consider the LHS of equation (2.32). Using definition (2.13), we rewrite the LHS as LHS = 1 6ηef g∇a(ηbcdη ef g) = η ef g∇a(h[behcfhd]g) = 3ηef gh[behcf∇|a|hd]g = 3ηef gh[behcf∇|a|(ud]ug) = 3ηg[bcud]∇aug. (2.33)
Splitting up the covariant derivative of ug, the LHS of (2.32) becomes
LHS = 3ηg[bcud](−ua˙ug+
1 3Θha
g+ σ
ag+ ωag)
= −3ua˙ugηg[bcud]+ Θηa[bcud]+ 3σagηg[bcud]
+3ωagηg[bcud]. (2.34)
We next consider the RHS of (2.32). Rewriting this we get RHS = 1 6ηef g(η ef g∇ aηbcd+ ηbcd∇aηef g) = ∇aηbcd+ 1 6ηbcdηef g∇aη ef g = ∇aηbcd+ 1 12ηbcd∇a(ηef gη ef g) = ∇aηbcd+ 1 12ηbcd∇a(6) = ∇aηbcd+ 0 = ∇aηbcd. (2.35)
Substituting (2.34) and (2.35) back into equation (2.32), we can write the final expression of the derivative of ηabc as
∇aηbcd = −3ua˙ugηg[bcud]+ Θηa[bcud+ 3σagηg[bcud]
+3ωagηg[bcud]. (2.36)
2.3
The Source Terms
We use a decomposition of the energy-momentum tensor Tab with respect
to the timelike vector field uain the form
Tab = µuaub+ uaqb+ phab+ πab, (2.37)
where µ = Tabuaub is the relativistic energy density relative to ua,
qa = −Tbcubhca is the relativistic momentum density, which is also the
energy flux relative to ua, p = 13Tabhab is the isotropic pressure, and πab =
hacTcd− 13hab(hcdTcd) is the trace-free anisotropic pressure. Our attention
will be focussed on perfect fluids, which are characterized by three quantities: their 4-velocity ua; their proper energy density field µ; and their scalar
pressure field p. This suggests we take the energy-momentum tensor for a perfect fluid to be of the form
Tab = µuaub+ phab, uaua= −1, (2.38)
so that
qa= 0, πab= 0 (2.39)
In addition, one often imposes a particular relation between p and µ, called an equation of state. However, in this thesis we will not specify any partic-ular equation of state.
In section 1.9 we imposed the relation
2.4. Covariant Propagation and Constraint equations 15
This relation is also an important self-consistency check of Einstein´s equa-tion because this equaequa-tion alone implies the geodesic hypothesis that the world lines of test bodies are geodesics of the spacetime metric. For example (see [5]), the fluid equation of motion implied by ∇bTab = 0 for a
distribu-tion of pressure free matter (’dust’ or ’Cold Dark Matter’), Tab= µuaub, is
given by
0 = ∇bTab= ∇b(µuaub) (2.41)
Applying Leibniz rule to this product we get
(∇bµ)uaub+ µ(∇bua)ub+ µua∇bub= 0 (2.42)
We next contract this equation with ua,
−(∇bµ)ub+ µubua∇bua− µ∇bub = 0 (2.43)
which makes the second term vanish, leaving −ub∇bµ − µ∇bub = 0.
Substituting this result back in (2.42) and dividing by µ 6= 0, we get
ub∇bua= 0, (2.44)
which tells us that the individual dust particles moves on geodesics.
2.4
Covariant Propagation and Constraint
equa-tions
In a spacetime with a timelike congruence u, the information contained in the Einstein field equations can be displayed by expressing the Ricci identities as applied to u,
2∇[c∇d]ua= Rabcdub, (2.45)
and the Bianchi identities (1.9), or equivalently ∇dCabcd = −∇[aRb]c−
1
6gc[a∇b]R, (2.46)
in terms of the kinematic quantities (2.25) and the electric and magnetic parts of the Weyl tensor (4.1) and (4.2). In both cases the Ricci tensor is expressed in terms of the stress energy tensor (2.37) using the Einstein field equations (1.22). By separating out the parallell part into a trace part, a symmetric tracefree, and antisymmetric parts and the perpendicular part similarly, we obtain a set of evolution equations. The result of decomposing the Ricci identities is three propagation and three constraint equations. The propagation equations are:
1. The Raychaudhuri equation ˙ Θ − e∇a˙ua= − 1 3Θ 2+ ( ˙u a˙ua) − 2σ2+ 2ω2− 1 2(µ + 3p), (2.47)
which is the basic equation of gravitational attraction2. 2. The vorticity propagation equation
˙ωhai−1 2η abc∇e b˙uc= − 2 3Θω a+ σa bωb. (2.48)
3. The shear propagation equation ˙σhabi− e∇ha˙ubi = −2 3Θσ ab+ ˙uha˙ubi− σha cσbic− ωhaωbi −(Eab−1 2π ab). (2.49)
The constraint equations are, 1. The (0α)-equation 0 = e∇bσab− 2 3∇e aΘ + ηabc[ e∇ bωc+ 2 ˙ubωc] + qa. (2.50)
2. The vorticity divergence identity
0 = e∇aωa− ˙uaωa. (2.51)
3. The Hab-equation
0 = Hab+ 2 ˙uhaωbi+ e∇haωbi−(curl σ)ab, (2.52) where the ’curl’ of the shear is (curl σ)ab = ηcdha∇e
cσbid. For the complete
sets of equations, see [6].
Chapter 3
The Maxwell Case
In this section we shall give a formulation of the electromagnetic theory in a general spacetime using the ideas for performing a 1+3 decomposition of spacetime.
3.1
Splitting the Electromagnetic Field
The electromagnetic field is represented by the antisymmetric Maxwell ten-sor Fab. This splits into an electric and a magnetic 4-vector relative to ua,
respectively defined by Ea = Fabub, (3.1) Ha = 1 2η bc a Fbc. (3.2) It follows that Eaua= 0, (3.3) and Haua= 0. (3.4)
In special relativity, when spacetime is the manifold R4with a flat metric of Lorentz signature (− + ++), the electromagnetic field vectors E and B are written in terms of the potentials as
Ea = −∂aφ − ˙Aa (3.5)
Ha = curl Aa (3.6)
This is how we are used to see the field vectors. However, in curved spacetime the situation will be different. We remember
Fab = 2∇[aΦb]. (3.7)
We first consider the electromagnetic field tensor on the left-hand side of eq.(3.7). To split Fab, we put the previously defined projection tensor hab,
into the equation and obtain
LHS: Fab = gacgbdFcd= (−uauc+ hac)(−ubud+ hbd)Fcd
= uaubucudFcd− uauchbdFcd− hacubudFcd
+hachbdFcd. (3.8)
Using Ea= ubFab so that Eaua= 0 we get
Fab = uahbdEd− ubhacEc+ hachbdFcd
= uaEb− ubEa+ hachbdFcd
= 2u[aEb]+ h[achb]dFcd. (3.9)
Using the relation h[ach
b]d = 12ηabeηcde, derived from the previous section,
we get Fab = 2u[aEb]+ 1 2ηabeη cdeF cd, (3.10) where Ha= 12ηabcFbc, so that Fab= 2u[aEb]+ ηabeHe. (3.11)
Consider the right-hand side of (3.7). We wish now to split the potential into φ and Aa defined by
φ= uaΦ
a Aa= habΦb,
where φ is the projection of Φa parallel to the velocity vector ua and Aa is
the projection of Φbin the 3-space orthogonal to uaat each point. It follows
that Φa= gabΦb= (−uaub+ hab)Φb = −uaφ+ Aa, (3.12) so Aa= Φa+ uaφ φ= Φaua ⇒ uaAa= 0
Using this, the right-hand side of (3.7) becomes
RHS: 2∇[aΦb] = ∇aΦb− ∇bΦa= ∇a(gbcΦc) − ∇b(gacΦc) = ∇a (−ubuc+ hbc)Φc − ∇b (−uauc+ hac)Φc = −∇a(ubφ) + ∇aAb+ ∇b(uaφ) − ∇bAa = −2∇[a(ub]φ) + 2∇[aAb] (3.13)
Using Leibniz property on the first term and split the covariant derivative of ub into its irreducible parts, we obtain
2∇[aΦb] = −2u[b∇a]φ −2φ(−u[a˙ub]+1
3Θh[ab]+ σ[ab]+ ω[ab]) +2∇[aAb]
3.1. Splitting the Electromagnetic Field 19
Contribution from the expansion of the fluid, 13Θhab, and the symmetric
shear tensor, σab, vanishes since we take the anti-symmetric part. We may
write equation (3.7) as
2u[aEb]+ ηabeHe= −2u[b∇a]φ+ 2φu[a˙ub]−2φωab+ 2∇[aAb]. (3.15)
In the following sections we will decompose this equation into one equation for Ea and one for Ha.
3.1.1 Electric Part
We next contract both sides of equation (3.15) with ub to obtain Ea = ub(−ub∇aφ+ ua∇bφ) −2φub(−1 2ua˙ub+ 1 2ub˙ua+ ωab) + u b∇ aAb− ub∇bAa
= ∇aφ+ uaub∇bφ+ φuaub˙ub+ φ ˙ua−2φubωab
+ub∇aAb− ub∇bAa. (3.16)
The third term vanishes by applying Leibniz rule on the product ub˙u b, im-plying, ub˙ub = ubua∇aub = 1 2u a∇ a(ubub) = 1 2u a∇ a(−1) = 0. (3.17)
The contraction also makes the fifth term vanish, leaving
Ea= ∇aφ+ uaφ˙+ φ ˙ua+ ub∇aAb− ub∇bAa. (3.18)
Our expression for Eacan be split further. Applying Leibniz property
”back-wards” on ub∇ aAb gives ub∇aAb = ∇a(u| {z }bAb =0 ) − Ab∇aub = Abua˙ub− 1 3ΘAa− A bσ ab− Abωab. (3.19)
Expanding ∇aφ and ub∇bAa gives
∇aφ = gab∇bφ= (−uaub+ hab)∇bφ
= −uaφ˙+ e∇aφ, (3.20)
and
ub∇bAa = A˙a= gabA˙b= (−uaub+ hab) ˙Ab
Using eq.(3.19), (3.20) and (3.21), the final expression for the electric part of Fab can be written as
Ea = −uaφ˙+ e∇aφ+ uaφ˙ + φ ˙ua+ Abua˙ub−
1 3ΘAa −Abσab− Abωab+ uaubA˙b− ˙Ahai = ∇eaφ+ φ ˙ua− ˙Ahai− 1 3ΘAa− A bσ ab− Abωab, (3.22) since Abua˙ub+ uaubA˙b = −uaubA˙b+ uaubA˙b = 0. (3.23) 3.1.2 Magnetic Part
Multiplying equation (3.11) and (3.15) with ηabc gives
ηabcFab = 2Hc (3.24)
and
ηabc∇[aAb] = ηabc(−ub∇aφ+ ua∇bφ)
−2φηabc(−1 2ua˙ub+ 1 2ub˙ua+ ωab) + 2η abc∇ [aAb] = 2ηabc∇aAb−4φωc, (3.25) since ηabcu a= 0. Expansion gives 2ηabc∇aAb = 2ηabcgadgbe∇dAe
= 2ηabc(−uaud+ had)(−ube+ hbe)∇dAe
= 2ηabchadhbe∇dAe= 2ηabc∇eaAb. (3.26)
Thus,
Hc= ηabc∇eaAb−2φωc. (3.27)
Rewriting the first term, we get
Ha = (curl A)a−2φωa, (3.28)
3.2. Summing-up 21
3.2
Summing-up
We have derived an expression of the electric and magnetic part of the elec-tromagnetic field tensor in a 1+3 decomposition of spacetime. The electric and magnetic part can in curved spacetimes be written as
Ea= e∇aφ − ˙A<a>+ φ ˙ua− 1 3ΘAa− A bσ ab− Abωab (3.29) and Ha= (curl A)a−2φωa. (3.30)
In analogy with this case, we will in the coming chapters derive the Weyl-Lanczos equation for two cosmological models.
Chapter 4
The Decompositions of the
Lanczos Tensor
In this chapter we will decompose the Lanczos tensor into simpler parts, as we did with the Maxwell tensor in the previous chapter.
4.1
Splitting the Gravitational Field
In analogy with electromagnetic theory, the Weyl conformal curvature tensor Cabcd is split relative to ua into an electric and a magnetic part according
to [8]
Eac = Cabcdubud, (4.1)
Hac = ∗Cabcdubud, (4.2)
where the dual is defined by
∗C abcd = 1 2ηab efC ef cd. (4.3)
These tensors are symmetric and trace-free,
Eab = E(ab), Eaa= 0, (4.4)
and
Hab= H(ab). Haa= 0 (4.5)
They also satisfy
Eabub = Habub= 0. (4.6)
These tensors represent the free gravitational field, enabling long-range grav-itational effects, such as gravity waves and tidal forces. The Weyl tensor can be reconstructed according to [8]
Cabcd = 4u[au[cEb]d]+ 4h[a[cEb]d]+ 2ηabeu[cHd]e+ 2ηcdeu[aHb]e. (4.7)
Also notice that Cabcd= 0 ⇔ Eab = 0 Hab= 0 (4.8)
4.2
The Spacetime Decomposition
The splitting of the vector potential (3.12) in electromagnetic theory involve a straightforward application of the 1+3 splitting of vector Φa. For the 1+3
splitting of the Lanczos tensor, we need the generalisation to rank-3 tensors of the decomposition (2.23). By expanding Labc to performing the 1+3
decomposion of spacetime and use the symmetry properties,
Labc= L[ab]c, L[abc]= 0, Labb = 0, (4.9)
we get
Labc = gadgbegcfLdef
= (−uaud+ had)(−ubue+ hbe)(−ucuf + hcf)Ldef
= −uaubucudueufLdef | {z } =0 +uaubuduehcfLdef | {z } =0 +uaucudufhbeLdef− uaudhbehcfLdef +ubucueufhadLdef− ubuehadhcfLdef
−ucufhadhbeLdef+ hadhbehcfLdef. (4.10)
Let
Aa= udufLdaf, Bab = udhbfLdaf, Nab= 12ηdeaLdeb,
so that
Aaua= 0, Babub = 0, Nabua= 0.
By putting these definitions into the RHS of (4.10), we obtain Labc = uauchbeAe− uahbeBec+ ubuchadAd+ ubhadBdc − ucufhadhbeLdef+ hcfhadhbeLdef = uaucAb− uaBbc− ubucAa+ ubBac − ucufha[dh e] b Ldef+ h f c ha[dh e] b Ldef. (4.11)
The relation (2.14) applied on the last two terms in the last equality gives Labc = 2ucu[aAb]−2u[aBb]c−
1 2ucu
fη
abgηdegLdef
+1 2h
f
c ηabgηdegLdef
= 2ucu[aAb]−2u[aBb]c− ucufηabgNgf
4.2. The Spacetime Decomposition 25
Next we decompose Nab by putting
Pa= Nabub, Paua= 0,
Qab= hbcNac, Qabua= 0, Qabub = 0,
so that (4.12) becomes
Labc= 2ucu[aAb]−2u[aBb]c− ηabducPd+ ηabdQdc. (4.13)
From Bab and Qab, we firstly define
Cab = B(ab)− 1 3habD, (4.14) and Sab= Q(ab)− 1 3habJ, (4.15)
where D = Baa and J = Qaa, which are the symmetric trace-free part of
Bab respective Qab. Secondly, we define
Mab = B[ab] (4.16)
and
Tab= Q[ab] (4.17)
which are the antisymmetric part of Bab and Qab respectively. Using this,
the Lanczos tensor can be written as
Labc = 2u[aAb]uc−2u[aMb]c−2u[aCb]c− 2 3u[ahb]cD − ηabducPd+ ηabdTdc+ ηabdSdc+ 1 3ηabcJ. (4.18) This is the full 1+3 splitting of the Lanczos tensor. By using the symmetry properties for the Lanczos tensor and the Lanczos algebraic gauge, we will now simplify this expression. By taking the totally anti-symmetric part of Labc, the components 2u[aAb]uc, 2u[aCb]c and 23u[ahb]cD will vanish, leaving
0 = L[abc] = −2u[aMbc]− u[cηab]dPd− η[abdTc]d+ η[abdSc]d +1
3η[abc]J. (4.19)
Contracting with uc gives
0 = −2 3u cu cMab− 1 3u cu cηabdPd = 2 3Mab+ 1 3ηabdP d, ⇔2Mab = −ηabdPd. (4.20)
Equation (4.19) multiplied with ηabc gives 0 = −ηabcηabdTcd+ ηabcηabdScd+ 1 3η abcη abcJ, = −2hcdTcd+ 2hcdScd+ 2J = 2J (4.21)
which means that
J = 0. (4.22) Thus, Labc = 2u[aAb]uc+ u[aηb]cdPd−2u[aCb]c− 2 3Du[ahb]c −ucηabdPd+ ηabdTdc+ ηabdSdc. (4.23)
We now use the gauge choice, L b
ab = 0, which is called the Lanczos algebraic
gauge. This will in the end simplify our expression for Labc even more.
0 = Labb = uaAbub− ubAaub+ 1 2uaηb b dPd− 1 2ubηa b dPd − uaCbb+ ubCab− 1 3Duahb b+1 3Dubha b − ubηabdPd+ ηabdTdb+ ηabdSdb = Aa− Dua+ ηabdTdb (4.24)
Contracting this with ua gives
0 = uaLabb = uaAa+ D + uaηabdTdb (4.25)
which means that
D = 0, (4.26)
and consequently,
Aa= −ηabdTdb. (4.27)
We solve for Tbc by multiplying with ηabc.
ηabcAa = −ηabcηaedTde = −2he[bh c] d T de = −2Tcb ⇔ Tbc = 1 2ηabcA a (4.28)
Using this we can write the term η d
ab Tdc from (4.23) expressed by ηabdTdc = 1 2ηab dη cedAe= 1 2(hachbe− haehbc)A e = 1 2(hacAb− hbcAa) = −A[ahb]c (4.29) Thus, the final expression for Lanczos potential becomes
Labc = 2u[aAb]uc− A[ahb]c−2u[aCb]c+ ηabdSdc
Chapter 5
Shearfree and Irrotational
Models
5.1
Introduction
To test our formalism, we consider an already known case first investigated by Novello and Velloso [11]. Novello Velloso says, if in a perfect fluid space-time there is a space-timelike vector field which is shear-free and irrotational,
σab = 0, ωab = 0, (5.1)
then a Lanczos potential1 is given by
Labc=
4
3u[a˙ub]uc− 2
3˙u[ahb]c. (5.2)
Comparing this potential with (4.30), we see that Novello and Velloso´s potential corresponds to the choice P = S = C = 0 and A = 23˙ua. In
order to check the power of the 1+3 formalism, we will rederive Novello and Velloso´s Lanczos potential by considering the ansatz
Labc= 2u[aAb]uc− A[ahb]c−2u[aCb]c. (5.3)
We will translate the Weyl-Lanczos equation into 1+3 formalism and then split the equation into an electric and a magnetic part. We can then com-pare our equations with the evolution equations which are contained in the Einstein´s field equations. If we can identify the quantities, then we have an explicit form of Lanczos potential which also will agree with Novello and Velloso´s potential. This will be like an exercise for what is to come.
Since we have no shear and no vorticity, we write the covariant derivative of ua as
∇aub = −ua˙ub+
1
3Θhab. (5.4)
1In [11] Novello and Velloso do not use the Lanczos algebraic gauge, hence they give
the Lanczos potential as Labc = 2 ˙u[aub]uc which differs from our form, only by a gauge
transformation.
5.2
Constructing the Weyl-Lanczos Equation
Since Lanczos equation is linear, we can simplify all calculations by consider-ing the contribution of A and C separately, and then add the results. Again, we remind ourselves that the Weyl-Lanczos equation in Lanczos algebraic gauge can be written as
Cabcd= 2Lab[c;d]+ 2Lcd[a;b]−2g[a[c(Lb]id];i+ Ld]ib];i) (5.5)
5.2.1 A-terms
Starting with Labc only expressed in terms of A, we raise indices a and b,
Labc= 2u[aAb]uc− A[ahb]c. (5.6)
The action of ∇d on Labc and Leibniz rule applied on this gives
∇dLabc = 2∇d(u[aAb]uc) − ∇d(A[ahb]c) =
= 2u[aAb]∇duc+ 2ucu[a∇dA]+ 2A[buc∇dua]
−h[bc∇dAa]− A[a∇dhb]c. (5.7)
Using (2.25) and (2.31) gives us ∇dLabc = 2u[aAb](−ud˙uc+
1
3Θhdc) + 2ucu
[a∇ dAb]
+2ucA[b(−ud˙ua]+
1 3Θhd
a]) − h[b
c∇dAa]
−A[aub](−ud˙uc+
1 3Θhdc) − ucA [a(−u d˙ub]+ 1 3Θhd b])
= −3ud˙ucu[aAb]−3ucud˙u[aAb]+ Θuchd[aAb]
+Θhdcu[aAb]+ 2ucu[a∇dAb]− h[bc∇dAa]. (5.8)
The Weyl-Lanczos equation is constructed from four terms, see (1.20). To obtain the first, we antisymmetrize over c and d,
2∇[dLabc] = 6u[c˙ud]u[aAb]+ 2Θu[chd][aAb]
+4u[cu[a∇d]Ab]+ 2h[a[c∇d]Ab], (5.9) and the second term is obtained by switching the indices a to c and b to d raising and lowering the respective index pairs.
2∇[bLcda] = 6u[a˙ub]u[cAd]+ 2Θu[ahb][cAd]
5.2. Constructing the Weyl-Lanczos Equation 29
To obtain the third and fourth terms in (1.20) we start all over from Lbid. We
let the derivative ∇i act on Lbid and derive the third term in an analogous
way:
∇iLbid = −3ui˙udu[bAi]−3udui˙u[bAi]+ Θudhi[bAi]
+Θhidu[bAi]+ 2udu[b∇iAi]− h[id∇iAb] = −3 2ui˙udu bAi+3 2ui˙udu iAb−3 2udui˙u bAi+3 2udui˙u iAb +1 2Θudhi bAi−1 2Θudhi iAb+1 2Θhidu bAi−1 2Θhidu iAb +udub∇iAi− udui∇iAb− 1 2h i d∇iAb+ 1 2h b d∇iAi = −3 2˙udA b− u dA˙b−ΘudAb+ 1 2Θu bA d+ udub∇iAi −1 2h i d∇iAb+ 1 2h b d∇iAi. (5.11)
Multiplying by 2gca and antisymmetrizing over the indices ab and cd gives
us
2g[c[a∇|i|Lb]id] = 2(−u[cu[a+ h[c[a) −3 2˙udA b]− u d]A˙b]−ΘudAb] +1 2Θu b]A d]+ ud]ub]∇iAi− 1 2h i d]∇iAb]+ 1 2h b] d]∇iAi = 3u[cu[a˙ud]Ab]−3h[c[a˙ud]Ab]−2h[c[aud]A˙b] −2Θh[c[aud]Ab]+ Θh[c[aub]Ad] +h[c[aud]ub]∇iAi+ u[cu[ah|i|d]∇iAb] −h[c[ah|i|d]∇iAb]+ h[c[ahb]d]∇iAi. (5.12)
By renaming and moving the indices, the fourth and last term in (1.20) become
2g[c[a∇|i|Ld]ib] = 3u[cu[a˙ub]Ad]−3h[c[a˙ub]Ad]−2h[c[aub]A˙d]
−2Θh[c[aub]Ad]+ Θh[c[aud]Ab]
+h[c[aub]ud]∇iAi+ u[cu[ahb]i∇iAd]
−h[c[ahb]i∇|i|Ad]+ h[c[ahd]b]∇iAi. (5.13)
Using (5.9)-(5.13) in (1.20), the expression of Weyl-Lanczos equation, so far only containing A, can be written as
Cabcd = 2∇[dLabc]+ 2∇[bLcda]−2g[a[aLb]id];i−2g[c[aLd]ib];i
= 3u[c˙ud]u[aAb]+ 3u[a˙ub]u[cAd]+ 3h[c[a˙ud]Ab] +3h[c[a˙ub]Ad]+ 2h[c[aud]A˙b]+ 2h[c[aub]A˙d]
−Θh[c[aud]Ab]−Θh[c[aub]Ad]+ 4u[cu[a∇d]Ab] +4u[au[c∇b]Ad]+ 2h[a[c∇d]Ab]+ 2h[c[a∇b]Ad] −2h[c[aud]ub]∇iAi−2h[c[ahd]b]∇iAi
−u[cu[ah|i|d]∇iAb]− u[cu[ah|i|b]∇iAd]
+h[c[ah|i|d]∇iAb]+ h[c[ah|i|b]∇iAd]. (5.14)
Electric Part of A-terms
To split the Weyl-Lanczos equation into electric part we use the defintition (4.1):
Eac= Cabcdubud (5.15)
Multiplying (5.14) with ud gives
udCabcd = 3 2˙ucu [aAb]+3 2u [a˙ub]A c− hc[aA˙b]+ hc[aub]A˙dud +1 2Θhc [aAb]+ 2u cu[aA˙b]+ 2u[a∇cAb]+ 2udu[auc∇b]Ad +2u[a∇b]Ac+ h[acA˙b]+ udhc[a∇b]Ad+ hc[aub]∇iAi −1 2u [ah|i| c∇iAb]− 1 2ucu du[ah ib]∇iAd− 1 2u [ah ib]∇iAc +1 2u dh c[ahib]∇iAd. (5.16)
To obtain the electric part we multiply (5.16) with ub.
ubudCabcd = 3 4˙ucA a+3 4˙u aA c− 1 2hc aA˙bu b− 1 2hc aA˙ dud +ucuaA˙bub+ ucA˙a+ uaub∇cAb+ ∇cAa +uaucA˙dud+ ucud∇aAd+ uaA˙c+ ∇aAc +1 2h a cA˙bub+ 1 2hc aA˙ dud− 1 2hc a∇ iAi −1 4u au bhic∇iAb− 1 4h i c∇iAa− 1 4ucu dh ia∇iAd −1 4hi a∇iA c. (5.17)
To make this equation easier to work with we lower the index a and rewrite the equation as Eac = 3 2˙u(aAc)+ 2u(aA˙c)+ 2∇(aAc)−2uaucA b˙u b −2u(aAb∇c)ub− 1 2hca∇ iA i+ 1 2A bu (ahic)∇iub −1 2hi(a∇ iA c), (5.18)
5.2. Constructing the Weyl-Lanczos Equation 31
where we have used that ub∇
cAb = −Ab∇cub. We expand the covariant
derivatives of ua, Eac = 3 2˙u(aAc)+ 2u(aA˙c)+ 2∇(aAc)−2uaucA b˙u b
−ucAb(−ua˙ub+
1 3Θhab) − uaA b(−u c˙ub+ 1 3Θhcb) −1 2hca∇ iA i+ 1 4uah i cAb(−ui˙ub+ 1 3Θhib) −1 2hi(a∇ iA c)+ 1 4uch i aAd(−ui˙ud+ 1 3Θhid) = 3 2˙u(aAc)+ 2u(aA˙c)+ 2∇(aAc)− 1 3ΘucAa −1 3ΘuaAc− 1 2hca∇ iA i− 1 2hi(a∇ iA c)+ 1 12ΘuaAc + 1 12ΘucAa. = 3 2˙u(aAc)+ 2u(aA˙c)− 2 3Θu(aAc)+ 1 6Θu(aAc) +2∇(aAc)−1 2hca∇ iA i− 1 2hi(a∇ iA c). (5.19)
Let us consider each term which still includes a covariant derivative. Using the metric tensor, we can raise or lower free indices of the covariant derivate of A, expand by putting gab = −uaub+ hab into the equation and obtain the
fully orthogonal projected covariant derivative e∇. However, terms which are not orthogonal to A will appear and may seem to complicate the expression for Eac. In the end, all these terms will cancel. The main-key to get rid of
these terms is to use the Leibniz rule (1.4) ”backwards” as we did in (3.19) in the Maxwell case. Starting with the first covariant derivative in the RHS of equation (5.19), we obtain 2∇(aAc) = 2gabgcd∇(bAd)= 2(−uaub+ hab)(−ucud+ hcd)∇(bAd) = 2uaubucud∇bAd−2uaubhcd∇(bAd)−2ucudhab∇(bAd) +2habhcd∇(bAd) = 2uaucudA˙d− uahcdA˙d− uaubhcd∇dAb− ucudhab∇bAd −uchabA˙b+ 2 e∇(aAc) = 2uaucudA˙d− uahcdA˙d+ uaAbhcd∇dub+ ucAdhab∇bud −uchabA˙b+ 2 e∇(aAc) = 2uaucudA˙d− uahcdA˙d+ uaAbhcd(−ud˙ub+ 1 3Θhd b) +ucAdhab(−ub˙ud+ 1 3Θhb d) − u chabA˙b+ 2 e∇(aAc) = −2u(aA˙c)− ua˙ubAbhcdud+ 1 3ΘuaAbhc dh db
−uc˙udAdhabub+ 1 3ΘucAdha bh bd+ 2 e∇(aAc) = −2u(aA˙c)+2 3Θu(aAc)+ 2 e∇(aAc). (5.20)
By using the same procedure the second covariant derivative term of equa-tion (5.19) becomes −1 2hac∇ iA i = − 1 2hacg bd∇ bAd = − 1 2hac(−u bud+ hbd)∇ bAd = 1 2hacu bA˙ b− 1 2hac∇e bA b. (5.21)
The last covariant derivative of equation (5.19) becomes −1 2hi(a∇ iA c) = − 1 4hia∇ iA c− 1 4hic∇ iA a= − 1 4hiagc b∇iA b− 1 4hicga b∇iA b = −1 4hia(−ucu b+ h cb)∇iAb− 1 4hic(−uau b+ h ab)∇iAb) = 1 4hiaucu b∇iA b− 1 4hiahc b∇iA b+ 1 4hicuau b∇iA b −1 4hicha b∇iA b = −1 4hiaucAb∇ iub−1 4∇eaAc− 1 4hicuaAb∇ iub−1 4∇ecAa = −1 4hiaucAb(−u i˙ub+1 3Θh ib) − 1 2∇e(aAc) −1 4hicuaAb(−u i˙ub+1 3Θh ib) = −1 6Θu(aAc)− 1 2∇e(aAc) (5.22)
Collecting all terms (5.20)-(5.22) into equation (5.19), we obtain Eac = 3 2˙u(aAc)+ 2u(aA˙c)−2u(aA˙c)+ 2 3Θu(aAc)+ 2 e∇(aAc) −2 3Θu(aAc)+ 1 2hacu bA˙ b− 1 2hac∇e bA b− 1 6Θu(aAc) −1 2∇e(aAc)+ 1 6Θu(aAc) = 3 2A(a˙uc)− 1 2hac˙u bA b+ 3 2∇e(aAc)− 1 2hac∇e bA b. (5.23)
By comparison with (2.20) these terms will be recognized as projected sym-metric trace-free parts
3 2A(a˙uc)− 1 2hac˙u bA b= 3 2Aha˙uci (5.24) and 3 2∇e(aAc)− 1 2hac∇e bA b= 3 2∇ehaAci. (5.25)
5.2. Constructing the Weyl-Lanczos Equation 33
Thus, the electric part of the Weyl tensor in terms of A becomes Eac =
3
2Aha˙uci+ 3
2∇ehaAci (5.26)
Magnetic Part of A-terms
If in a given spacetime there is a field of observers ua which is shearfree
and irrotational, the magnetic part of the Weyl tensor vanishes for these observers,
Hab= 0. (5.27)
The proof of this is trivial (use eq.(2.52)). To determine if this condition on the magnetic part gives rise to any restrictions on the Lanczos potential, we use definition (4.2). Starting with the previously derived expression (5.16) and multiply with 12ηabe, we obtain
0 = Hec= 1 2ηabe 1 2Θhc [aAb]+ udh c[a∇b]Ad+ 1 2u dh c[ahib]∇iAd , (5.28) where we have used that ηabeua = 0. The ’Leibniz backwards’-trick then
yields 0 = 1 2ηabe 1 2Θhc [aAb]− A dh[ac(−ub]˙ud+ 1 3Θh b]d) −1 2Adhc [ah ib](−ui˙ud+ 1 3Θh id) = 1 2ηabe 1 2Θhc [aAb]+ A dhc[aub]˙ud− 1 3ΘA [bh ca] +1 2Adhc [ah ib]ui˙ud− 1 6A ih e[ahib] = 1 2ηabe 1 2Θhc [aAb]−1 3ΘA [bh ca]− 1 6ΘA [bh ca] ≡ 0, (5.29)
which is identically zero. Every term multiplied with ηabewhich is zero have
been removed during the calculations.
5.2.2 C-terms
In an analogous way, we will derive the Weyl-Lanczos equation in terms of C. Consider Labcexpressed only in terms of C. By raising the indices a and
b, we get
Applying the derivative operator ∇d on Labc and then using Leibniz and
splitting up the covariant derivative gives
∇dLabc = −2∇d(u[aCb]c) = −2u[a∇dCb]c−2C[bc∇dua]
= −2u[a∇dCb]c−2C[bc(−ud˙ua]+
1 3Θhd a]) = −2u[a∇dCb]c+ 2C[bcud˙ua]− 2 3Θhd [aCb] c. (5.31)
To obtain the first term in (1.20), we antisymmetrize over the indices c and d,
2∇[dLabc]= 4u[a∇[cCb]d]+ 4C[b[cud]˙ua]− 4 3Θh[d
[aCb]
c]. (5.32)
Switch the indices a to c and b to d, then raise ab and lower cd and the second term becomes
2∇[bLcda]= 4u[c∇[aCd]b]+ 4C[d[aub]˙uc]−
4 3Θh
[b
[cCd]a]. (5.33)
To obtain the third and fourth term in (1.20), we start all over and reshuffle the indices to get Lbid. The tensor resulting from the action of ∇i on Lbid is
∇iLbid = −2u[b∇iCi]d+ 2C[idui˙ub]− 2 3Θhi [bCi] d = −ub∇iCid+ ˙Cbd− 1 3Θhi bCi d+ ΘCbd = −ub∇iCid+ ˙Cbd+ 2 3ΘC b d. (5.34)
Multiplying with gca and antisymmetrizing over the indices ab and cd
re-spectively gives the third term,
2g[c[aLb]id];i = 2(−u[cu[a+ h[c[a)(−ub]∇|i|Cid]+ ˙Cb]d]+
2 3ΘC b] d]) = −2h[c[aub]∇|i|Cid]−2u[cu[aC˙b]d]+ 2h[c[aC˙b]d] −4 3Θu[cu [aCb] d]+ 4 3Θh[c [aCb] d]. (5.35)
By renaming and moving the indices, the last term becomes
2g[c[aLb]id];i = −2h[c[aud]∇|i|Cib]−2u[cu[aC˙d]b]+ 2h[c[aC˙d]b]
−4 3Θu[cu [aC d]b]+ 4 3Θh[c [aC d]b]. (5.36)
5.2. Constructing the Weyl-Lanczos Equation 35
The complete Weyl-Lanczos equation for the C-terms is then Cabcd = 2∇[dLabc]+ 2∇[bLcda]−2g[a[aLb]id];i−2g[c[aLd]ib];i
= −4u[a∇[dCb]c]+ C[b[cud]˙ua]−4 3Θh[d [aCb] c] −4u[c∇[bCd]a]+ C[d[aub]˙uc]−4 3Θh [b [cCd]a] +2h[c[aub]∇|i|Cid]+ 2u[cu[aC˙b]d]−2h[c[aC˙b]d] +4 3Θu[cu [aCb] d]− 4 3Θh[c [aCb] d]+ 2h[c[aud]∇|i|Cib] +2u[cu[aC˙d]b]−2h[c[aC˙d]b]+4 3Θu[cu [aC d]b] −4 3Θh[c [aC d]b] = 4C[b[cud]˙ua]+ 4C[d[aub]˙uc]+ 4u[cu[aC˙b]d]−4h[c[aC˙d]b] +8 3Θu[cu [aC d]b]−4u[a∇[dCb]c]−4u[c∇[bCd]a] +2h[c[aub]∇|i|Cid]+ 2h[c[aud]∇|i|Cib]. (5.37)
Electric Part of C-terms
To obtain the electric part of the Weyl-Lanczos, we first multiply eq.(5.37) with ud: udCabcd = −2C[bc˙ua]+ 2ucudu[aC˙b]d+ 2u[aC˙b]c −2udhc[aC˙db]+ 4 3Θu [aC cb]−2u[aC˙b]c +2udu[a∇cCb]d−2ucud∇[bCda]−2∇[bCca] +udhc[aub]∇iCid− hc[a∇|i|Cib] (5.38)
Secondly, we multiply with ub to obtain
ubudCabcd = ubucuduaC˙bd+ uducC˙ad− ubudhcaC˙db +ubudua∇cCbd+ ud∇cCad− uducC˙da +ubuduc∇aCdb− ˙Cca+ ub∇aCcb +2 3ΘCc a− 1 2u dh ca∇iCid− 1 2ubhc a∇iC ib. (5.39)
This is the electric part of Weyl-Lanczos equation only including C. We next lower the index a and use Leibniz rule on terms including a derivative to obtain the derivatives of ua. Those terms which contain contractions of ua
definition (5.4) and simplify. These steps are written as Eac = ubuducuaC˙bd− ubudhcaC˙db+ ubudua∇cCbd +ud∇cCad+ ubuduc∇aCdb− ˙Cca+ ub∇aCcb +2 3ΘCca− 1 2u dh ca∇iCid− 1 2ubhca∇ iC ib = −Cbduducua˙ub+ Cdbudhca˙ub− Cbdudua∇cub −Cad∇cud− Cdbuduc∇aub− ˙Cca− Ccb∇aub +2 3ΘCca+ 1 2C i dhca∇iud+ 1 2Ci bh ca∇iub = −Cad(−uc˙ud+ 1 3Θhc d) − ˙C ca− Ccb(−ua˙ub+ 1 3Θhab) +2 3ΘCca+ C i dhca(−ui˙ud+ 1 3Θhi d) = Caduc˙ud− 1 3ΘCac− ˙Cca− 1 3ΘCca+ 2 3ΘCca +Ccbua˙ub+ 1 3Θhcahi dCi d = Caduc˙ud− ˙Cca+ Ccbua˙ub+ 1 3Θhcahi dCi d, (5.40)
where we in the final equality have used that Cab is defined to be
trace-free. Consider ˙Cca and expand this by putting gab = −uaub+ hab into the
equation. We obtain
Eac = Caduc˙ud− gcdgabC˙db+ Ccbua˙ub
= Caduc˙ud−(−ucud+ hcd)(−uaub+ hab) ˙Cdb+ Ccbua˙ub
= −uducC˙ad− uaubucudC˙db+ ucudhabC˙db
+uaubhcdC˙db− habhcdC˙db− ubuaC˙cb
= −uducC˙ad+ uaubucCdb˙ud+ uchabUdC˙db
+uahcdubC˙db− habhcdC˙db− ubuaC˙cb, (5.41)
where we applied Leibniz rule ’backwards’ on the second term in the final equality. The second term vanishes by orthogonality. Note that, by also using ’Leibniz’ on the third and fourth ˙C-terms, it is easy to see that using hcd to lower the indices on these terms is allowed. Further, we find
Eac = −uducC˙ad+ ucudC˙da+ uaubC˙cb− ˙Chcai− ubuaC˙cb
= − ˙Chcai, (5.42)
which is the fully orthogonally projected derivative. Magnetic Part of C-terms
Finally, we determine the magnetic part of Weyl-Lanczos only expressed in terms of C using (4.2). Starting from our ansatz, Labc= −2u[aCb]c, and use
5.2. Constructing the Weyl-Lanczos Equation 37
previous calculations all the way to equation (5.38), we multiply this with
1
2ηabe and use that ηabeua= 0. The derivation follows:
0 = Hec = 1 2ηabe −2C [b c˙ua]−2udhc[aC˙db]−2uduc∇[bCda] −2∇[bCca]− hc[a∇iC|i|b]
= ηabe − C[bc˙ua]+ hc[aCdb]˙ud+ ucCd[a(−ub]˙ud+
1 3Θh
b]d)
−∇[bCca]− hc[a∇|i|Cib]
= ηabe − C[bc˙ua]+ hc[aCdb]˙ud− ucCd[aub˙ud+
1 3ΘucC [ab] −gbfgcggah∇[fCgh]− 1 2g i fgbghc[a∇|f |Cg]i
= ηabe − C[bc˙ua]+ hc[aCdb]˙ud− h[bfha]h(−ucug+ hcg)∇[fCh]g
−1 2(−u
iu
f + hif)hbghc[a∇|f |Cg]i
= ηabe − C[bc˙ua]+ hc[aCdb]˙ud− ucCg[a∇b]ug− e∇[bCca]
−1 2ufhc [aCb] i∇fui− 1 2hc [a∇e|i|Cb] i
= ηabe − C[bc˙ua]+ hc[aCdb]˙ud− ucCg[a(−ub][u˙ g +1 3Θh b]g) − e∇[bCca]− 1 2hc [aCb] i˙ui− 1 2hc [a∇e|i|Cb] i = ηabe − C[bc˙ua]+ 1 2hc [aC db]˙ud− e∇[bCca] −1 2hc [a∇e|i|Cb] i (5.43) Since Hec= Hce, it is allowed to symmetrize over indices ec. It follows,
0 = ηab(eCc)a˙ub+ 1 2η(ec)bC b i˙ui+ ηab(e∇eaCc)b+ 1 2η(ec)a∇e iCb i. (5.44)
Due to symmetries with C, we can change the parenthesis to
0 = ηabheCcia˙ub+ ηabhe∇eaCcib. (5.45) This means that the equation Hab= 0 gives rise to restrictions on Cab.
5.3
Summing-up
The purpose of this chapter is to check our Lanczos potential in an already known case by deriving the Weyl-Lanczos equation from a particular ansatz (P = S = 0) of Labc and make sure its electric and magnetic part
satis-fies Einstein´s equations with a perfect fluid as its source. We will now find an explicit form of a Lanczos potential in a shear-free and irrotational model. By collecting the results from (5.26) and (5.42), we can obtain the final expression for the electric part as
Eab=
3
2Aha˙ubi+ 3
2∇ehaAbi− ˙Chbai. (5.46) The investigation of the magnetic part gave
0 = Hab = ηcdhaCbic˙ud+ ηcdha∇ecCbid. (5.47)
We now make the choice A = 23˙u and C = 0. Then equation (5.47) is identically satisfied and the RHS of (5.46) becomes
3
2Aha˙ubi+ 3
2∇ehaAbi− ˙Chbai= ˙uha˙ubi+ e∇ha˙ubi. (5.48) However, in these spacetimes equation (2.49) becomes
Eab = ˙uha˙ubi+ e∇ha˙ubi. (5.49)
Hence,
3
2Aha˙ubi+ 3
2∇ehaAbi− ˙Chbai = Eab, (5.50) so equation (5.46) is satisfied. It follows that our choice A = 23˙u, C = 0, i.e.
Labc=
4
3u[a˙ub]uc− 2
3˙u[ahb]c, (5.51)
defines a Lanczos potential for the Weyl tensor in this class of spacetimes, in agreement with Novello and Velloso´s result. The derivation in this chapter of a Lanczos potential is easier to follow than the proof for Novello and Velloso´s ’ad hoc’-potential in [11].
Chapter 6
Spatially Homogeneous
Universes
These models are some of the major models of theoretical cosmology, because they express mathematically the idea of the ’cosmological principle’: all points of space at the same time are equivalent to each other.
6.1
Bianchi Type I Models
The anisotropic case is the family of Bianchi universes. The simplest class is the Bianchi Type I family, where the three-dimensional spatial space is the rest-space of the matter. They are spatially homogeneous and the fluid flow, which are orthogonal to these homogeneous surfaces, is a geodesic and irrotational. Thus, these models obey the restrictions
˙ua= ωab= 0, (6.1)
which implies the covariant derivative of ua as ∇aub =
1
3Θhab+ σab, (6.2)
where σab obeys the symmetries
σ(ab)= σab, σabub = 0, σaa= 0. (6.3)
From the Gauss equation and the Ricci identities for ua, the Ricci tensor of
the metric of the orthogonal 3-spaces is given by [8]
3R
ab = − ˙σhabi−Θσab+ e∇ha˙ubi+ πab+
1 3hab
3R, (6.4)
where the Ricci scalar is
3R= 2µ −2
3Θ
2+ 2σ2. (6.5)
3R
ab is related to Eab via (2.49) and show how the matter tensor determines
the 3-space average curvature. In these models 3Rab = 0, which implies 3R= 0.
We can find a tetrad so that the covariant equations obeyed by these models can be expressed in tetrad equations1. From this, it follows that Hab = 0 in these models also.
In general, Bianchi I models are described by solutions (see [13]) which have a ’cigar’-like singular behaviour for t = 0, meaning: a small spatial region which is spherical at some time becomes a very long, thin, elongated ellipsoid, and in the limiting case t = 0 a straight line. There is also a special case when out of a sphere, a strongly flattened, rotating ellipsoid first is formed and finally a ’pancake’ singularity as we follow its history backwards in time.
6.2
Constructing the Weyl-Lanczos Equation
We consider the contributions of A, C, S and P separately. The whole Weyl-Lanczos equation would be far from easy to calculate at once. The same procedure are followed as in the previous case, and therefore we will skip some details. Once again we remind ourselves that the Weyl-Lanczos equation in Lanczos algebraic gauge can be written as
Cabcd = 2Lab[c;d]+ 2Lcd[a;b]−2g[a[c(Lb]id];i+ Ld]ib];i). (6.6)
6.2.1 A-terms
The decomposed Lanczos tensor only expressed in terms of A becomes
Labc= 2u[aAb]uc− A[ahb]c. (6.7)
By putting (6.7) into the terms of (6.6) one by one, we obtain the following equations: 2∇[dLabc] = 4u[cu[a∇d]Ab]−2h[b[c∇d]Aa]+ 2ΘA[bu[chd]a] +6A[bu[cσd]a], (6.8) 2∇[bLcda] = 4u[au[c∇b]Ad]−2h[d[a∇b]Ac]+ 2ΘA[du[ahb]c] +6A[du[aσb]c], (6.9) 2g[c[a∇|i|Lb]id] = u[cu[ahb]d]∇iAi+ h[c[ahb]d]∇iAi+ u[cu[ah|i|d]∇iAb]
−h[c[ah|i|d]∇iAb]−2h[c[aud]A˙b]+ Θh[c[aub]Ad]
6.2. Constructing the Weyl-Lanczos Equation 41
−2Θh[c[aAb]ud]+ 3h[c[aub]σd]iAi+ 3h[c[aud]σib]Ai
+3h[c[aud]σib]Ai,
(6.10) 2g[c[a∇|i|Ld]ib] = u[cu[ahd]b]∇iAi+ h[c[ahb]b]∇iAi+ u[cu[ah|i|b]∇iAd]
−h[c[ah|i|b]∇|i|Ad]−2h[c[aub]A˙d]+ Θh[c[aud]Ab]
−2Θh[c[aub]Ad]+ 3h[c[aub]σd]iAi +3h[c[aud]σib]Ai.
(6.11)
Collecting all terms (6.8) to (6.11) into equation (6.6), the Weyl-Lanczos equation only containing A-terms becomes
Cabcd = 4u[cu[a∇d]Ab]+ 4u[au[c∇b]Ad]+ 2h[a[c∇d]Ab]+ 2h[c[a∇b]Ad]
−2h[c[aub]ud]∇iAi−2h[c[ahb]d]∇iAi− u[cu[ah|i|d]∇iAb]
+h[c[ah|i|d]∇iAb]− u[cu[ah|i|b]∇|i|Ad]+ h[c[ah|i|b]∇|i|Ad]
+2h[c[aud]A˙b]+ 2h[c[aub]A˙d]−Θh[c[aAd]ub]−Θh[c[auc]Ab] −6σ[c[aud]Ab]−6σ[a[cub]Ad]−6h[c[aσd]iub]Ai
−6h[c[aσb]iud]Ai. (6.12)
Electric Part
We next contract equation (6.12) with ud and then ub to obtain the electric
part written as Eac = 2∇(aAc)+ 2ubu(a∇c)Ab− 1 2hca∇iA i −1 2hi(a∇ iA c)− 1 2u(ahc)iu b∇iA b+ 2uaucudA˙d +2u(aA˙c). (6.13)
By decomposing the derivatives into their irreducible parts, all terms which are not orthogonal to uawill cancel. The final expression for E
ab becomes Eab= 3 2∇e(aAb)− 1 2hab∇iA i = 3 2∇ehaAbi. (6.14)
Magnetic Part
If we contract equation (6.12) with ud and then 12ηabe, we get the magnetic
part written as Hec = 1 2ηabeu dCab cd = 1 2ηabe 1 2ΘA [bh ca]+ 3hc[aA|i|σib]+ 3A[bσca] +hc[a(∇b]Ad)ud+ 1 2hc [ah ib](∇iAd)ud . (6.15)
Further decomposition gives the expression, Hec= 3 2ηabeσc [aAb]+ 3 4ηaecσ abA b. (6.16)
Because the first term contains two antisymmetric factors, we can remove the ’antisymmetrization-sign’ over indices a and b. We now symmetrize over indices ec and obtain the magnetic part written as
Hec= H(ec)= 3 2ηab(eσc) aAb= 3 2ηabheσci aAb, (6.17)
where we have changed the parenthesis in the last equality because
3
2ηab(eσc)aAb is trace-free over ec.
6.2.2 C-terms
The part of the Lanczos tensor containing only C-terms is given by
Labc= −2u[aCb]c. (6.18)
Then, the four terms in (6.6) become
2∇[dLabc] = −4u[a∇[dCb]c]−4 3ΘC [b [chd]a] −4C[b[cσd]a], (6.19) 2∇[bLcda] = −4u[c∇[bCd]a]− 4 3ΘC[d [ahb] c] −4C[d[aσb]c], (6.20) 2g[c[a∇|i|Lb]d] = −2h[c[aub]∇|i|Cid]−2u[cu[aC˙b]d] +2h[c[aC˙b]d]−4 3Θu[cu [aCb] d] +4 3Θh[c [aCb] d]+ 2u[cu[aC|i|d]σib]
