Non-isotropic Cosmology in 1+3-formalism

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Matematiska institutionen

Department of Mathematics

Master’s Thesis

Non-isotropic cosmology in

1+3-formalism

Johan Jönsson

LiTH-MAI-EX--2014/07--SE

Linköping 2014

Department of Mathematics Linköpings universitet SE-581 83 Linköping, Sweden

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Non-isotropic cosmology in

1+3-formalism

Work in cosmology performed at Linköpings Universitet

Johan Jönsson

Examensarbete: 30 hp

Nivå: A

Handledare: Fredrik Andersson

mai_{, Linköpings universitet}

Examinator: Magnus Herberthson

mai, Linköpings universitet

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Avdelning, Institution Division, Department

Department of applied mathematics Department of Mathematics SE-581 83 Linköping Datum Date 2014-12-16 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://liu.diva-portal.org/smash/record.jsf?pid=diva2:780201

ISBN — ISRN

Serietitel och serienummer Title of series, numbering

ISSN —

Titel Title

Icke-isotrop kosmologi i 1+3-formalism Non-isotropic cosmology in 1+3-formalism

Författare Author

Johan Jönsson

Sammanfattning Abstract

Cosmology is an attempt to mathematically describe the behaviour of the universe, the most commonly used models are the Friedmann-Lemaître-Robertson-Walker solutions. These models seem to be accurate for an old universe, which is homogeneous with low anisotropy. However for an earlier universe these models might not be that accurate or even correct. The almost non-existent anisotropy observed today might have played a bigger role in the ear-lier universe. For this reason we will study another model known as Bianchi Type I, where the universe is not necessarily isotropic. We utilize a 1+3-covariant formalism to obtain the equations that determine the behaviour of the universe and then use a tetrad formalism to complement the 1+3-covariant equations. Using these equations we examine the geometry of space-time and its dynamical properties. Finally we briefly discuss the different singularities possible and examine some special cases of geodesic movement.

Nyckelord

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Sammanfattning

Kosmologi är ett försök att beskriva universums beteende med hjälp av matema-tik, den vanligaste modellen är Friedmann-Lemaître-Robertson-Walkers lösning-ar. De här modellerna verkar korrekta för ett gammalt, homogent universum med låg anisotropi. För ett yngre universum är inte de modellerna nödvändigtvis kor-rekta, den låga graden av anisotropi vi ser idag skulle ha kunnat spela en större roll i ett yngre universums utveckling. Därför kommer kommer vi i denna rap-port undersöka en annan modell, kallad Bianchi Typ I där universum inte nöd-vändigtvis är isotropt. Vi använder oss av en 1+3-kovariant formalism för att få fram de ekvationer som styr universums utveckling och en tetradbeskrivning av dessa för att få en fullständig uppsättning ekvationer som beskriver universums utveckling. Med hjälp av dessa ekvationer undersöker vi rumtidens geometri och dynamik. Slutligen diskuterar vi kort vilka typer av singulariteter som kan upp-stå samt undersöker några specialfall av geodetisk rörelse.

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Abstract

Cosmology is an attempt to mathematically describe the behaviour of the uni-verse, the most commonly used models are the Friedmann-Lemaître-Robertson-Walker solutions. These models seem to be accurate for an old universe, which is

homogeneous with low anisotropy. However for an earlier universe these models might not be that accurate or even correct. The almost non-existent anisotropy observed today might have played a bigger role in the earlier universe. For this reason we will study another model known as Bianchi Type I, where the universe is not necessarily isotropic. We utilize a 1+3-covariant formalism to obtain the equations that determine the behaviour of the universe and then use a tetrad formalism to complement the 1+3-covariant equations. Using these equations we examine the geometry of space-time and its dynamical properties. Finally we briefly discuss the different singularities possible and examine some special cases of geodesic movement.

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Acknowledgments

I would like to thank my supervisor, Fredrik Andersson, for his extremely helpful discussions and suggestions.

Linköping, December 2014 Johan Jönsson

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1

Introduction to General Relativity

Einstein formulated the theory of general relativity in 1915 and to this day it is still considered to be the best theory for gravity and matter in the universe. In this chapter, we will give a short review of Einstein’s general relativity. The definitions of vectors and tensors given in this chapter are taken from [8].

1.1 Conventions

In this paper, we will use the Einstein summation convention i.e., if an index is repeated in a term there is an implicit summation over that index, e.g.

TiXi =X

i

TiXi.

We will use geometrized units that sets the speed of light to unity, c = 1, and the gravitational constant, G = _8π1 (this choice will simplify the Einstein field equations).

Space-time and spatial indices a, b, . . . and α, β, . . . will be assumed to run from 0 to 3 and 1 to 3 respectively and will represent components with respect to a general basis. Space-time and spatial indices i, j, . . . and µ, ν, . . . will also run from 0 to 3 and 1 to 3 respectively but represent components with respect to a coordinate basis.

1.2 Manifolds

In general relativity (and in differential geometry) we do not study vector spaces such as Rnbut instead we study what is known as manifolds. A manifold only has to look something like Rnlocally, we make no assumptions about what it looks

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like globally, but locally it is similar to an open subset of Rn. Globally it may have very different properties. An example is the surface of a 2-dimensional sphere (a so called 2-sphere). If we compare the surface of a sphere with R2 we see that they are not similar at all. The surface of a sphere is curved and finite, whereas R2 is flat and infinite. However, if we only look at a small piece of the sphere’s surface it will be similar to a small piece of R2, only slightly curved. In more detail, an n-dimensional, C∞

, real manifold M is a set, together with a collection of subsets Oi, where i belongs to some set I which is not necessarily countable.

The subsets Oi have the following properties:

1. Each point p ∈ M lies in at least one Oi.

2. For each i there is a one-to-one map φi : Oi →Ui ⊂ Rn and Ui is an open

set.

3. If any two of these subsets, Oi and Oj, of M overlap, so that OiT Oj , ∅ _{(where ∅ denotes the empty set), we can consider the map φ}_j◦_φ−1

i (◦

denotes composition) which takes points from φi[OiT Oj] ⊂ Ui ⊂ Rn to

points in φj[OiT Oj] ⊂ Uj ⊂ Rn, i.e. φj◦φ

−₁

i is an ordinary vector-valued

function of n variables. We require that this function be C∞, i.e., infinitely continuously differentiable, and that the subsets φi[OiT Oj] ⊂ Ui ⊂ Rnare

open.

The functions φi are called charts, or coordinate systems. M is also assumed to

satisfy the somewhat technical conditions of being Hausdorff and paracompact [8].

We can now define the concepts of smoothness and differentiability of maps from manifolds into the real numbers. Let M be a manifold with chart maps {_φ_i}_{. Given a map f : M → R, form the map ˜}_{f = f ◦ φ}−1

i , which is an ordinary

function from Rninto the real numbers R. We say that f ∈ C∞(or f ∈ C1, or f is continuous) if ˜f ∈ C∞for all φi (or ˜f ∈ C1, or ˜f is continuous), f and ˜f are often

identified.

It might seem natural to treat manifolds (such as a sphere) as if they were embedded in some larger (flat) vector-space Rn, however in general relativity we do not want to assume more than 4 dimensions. Therefore we study our manifold (which we will later call space-time) from the ”inside”, i.e., we have no reason to add any extra dimensions.

1.3 Tensors and Vectors

We will start by discussing the notion of vectors and we will use the same defini-tions as [8]. On a manifold M, letF be the collection of C∞ functions from M into the real numbers, R. A tangent vector v at a point p ∈ M is defined as a map v:F → R that is both linear and obeys the Leibniz’ rule;

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1.3 Tensors and Vectors 3 v(f g) = f (p)v(g) + g(p)v(f ), ∀f , g ∈F . (1.2) Using this definition, the collection of all tangent vectors at a point p ∈ M, Vp,

has the structure of a vector space under the addition law (v1+ v2)(f ) = v1(f ) +

v2(f ) and the scalar multiplication law (av)(f ) = av(f ). Theorem 2.2.1 in [8,

chapter 2.2] says that if n is the dimension of the manifold M, then dim Vp = n.

In theorem 2.2.1 a basis for Vp, {Xi}, is introduced with

X_i(f ) = ∂ ∂xi(f ◦ φ −1₎ _φ(p) . (1.3)

The basis {Xi}is called a coordinate-basis and is often simply denoted {∂/∂xi}.

With this coordinate basis we introduce the notation v(f ) = viXi(f ) = vi ∂f

∂xi, (1.4)

where vi_{is the i:th component of v in the basis {∂/∂x}i}_{, i.e. v = v}i_X

i = vi ∂_∂xi. By

choosing a different chart φ0 _{we would end up with a different coordinate basis}

{_X0

j}. Using the chain rule we can express the old basis vectors {Xi}in terms of

the new basis {X0_j}

X_i = ∂x 0_j ∂xi _φ(p) X0_j,

where x0j denotes the j:th component of the map φ0◦_φ−1_{. We can now find the} components v0j of a vector v in the new basis in terms of the components vi in the old basis

v0i = vj∂x

0_i

∂xj. (1.5)

This is known as the vector transformation law.

Now that we have defined what we mean by vectors we can introduce dual vectors (or covectors). Let V∗p be the collection of linear maps f : Vp → R. V∗p

is called the dual vector space to Vpand the elements of V∗p are called dual

vec-tors (or covecvec-tors). If {v1, . . . , vn}are a basis for Vp then we can define elements

v1∗, . . . , vn∗∈_V∗_p_by

vi∗(vj) = δij, (1.6)

where δij is the Kronecker delta with δij = 1 if i = j and 0 otherwise. The

covectors {vi∗}_{will form a basis for the dual space V}∗_p_{, called the dual basis of} {_v₁_{, . . . , v}_n}_{. Components of covectors are transformed from one coordinate base} to another according to

v0_i = vj ∂xj

∂x0_i. (1.7)

We can now introduce the notion of a tensor. A tensor, T , of type (k, l) is a mul-tilinear map T : V∗p×. . . × V ∗ p | {z } k times ×_V_p×_{. . . × V}_p | {z } l times

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takes k covectors, l vectors and produces a number. To denote the components of

T relative to some basis, {vi}_{we use the notation T}i1...ik_j

1...jl. The collection of all

tensors of type (k, l) is writtenT (k, l). A tensor field is an assignment of a tensor

T over Vpat each point p ∈ M.

We will now give two useful operators on tensors, contraction and outer prod-uct. Contraction on the i:th covector slot and the j:th vector slot indices is a map

C :T (k, l) → T (k − 1, l − 1), the components of a contraction (in the basis {vi}) is Ti1...ik−1

j1...kl−1= T

i1...n...ik−1

j1...n...jl−1. (1.8)

The outer product of two tensors, T and U , of type (k, l) and (k0, l0) respectively is a tensor of type (k + k0, l + l0). The components of the outer product (in the basis {_v_i}_{) is} Si1...ik+k0 j1...jl+l0 = T i1...ik j1...jlU ik+1...ik+k0 jl+1...jl+l0. (1.9)

In general the components of a tensor of type (k, l) transform according to

T0i1...ik j1...jl = T m1...mk n1...nl ∂x0i1 ∂xm1 . . . ∂xnl ∂x0_j_l, (1.10)

this is known as the tensor transformation law.

A vector (or tangent) field, v, is an assignment of a tangent vector v|p ∈ Vp

at each point p ∈ M, in addition v is smooth if, for any smooth (C∞) function

f , v(f ) is smooth. Similarly a covariant vector field, w, is smooth if the function

w(v) is smooth, for any smooth vector field v. In the same way a tensor field, T , of type (k, l) is smooth if, for all smooth covariant vector fields w1_{, . . . , w}k_{and all}

smooth contravariant vector fields v1. . . vl, the function T (w1, . . . , wk; v1, . . . , vl)

is smooth.

For a more detailed treatment of vectors and tensors see e.g. [8]

1.3.1 Symmetric and Anti-symmetric

A (2, 0) tensor, T is said to be anti-symmetric if Tij = −Tji and symmetric if

Tij = Tji. The symmetric part of T is defined as

T(ij)= 1 2

Tij+ Tji.

In a similar way we define the anti-symmetric (skew-symmetric) part of a tensor as T[ij]= 1 2 Tij−_Tji_. Note that Tij = T(ij)+ T[ij], (1.11)

for all tensors T of type (2, 0).

From these definitions we see that for a symmetric tensor Sij S(ij)= Sij

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1.4 The Metric Tensor 5

S[ij]= 0 and for an anti-symmetric tensor Aij

A(ij)= 0

A[ij]= Aij.

These operators can be generalised to tensors of higher rank

T(i1i2...in)₌ 1 n! X π Tiπ(1)...iπ(n)_, _(1.12) T[i1i2...in]₌ 1 n! X π δπTiπ(1)...iπ(n), (1.13)

where the sum is taken over all permutations π of 1, . . . , n and δπis +1 for even

permutations and −1 for odd permutations.

If, for some reason, we want to exclude an index (or several indices) from sym-metrization or anti-symsym-metrization we will wrap them with vertical bars, e.g.,

T[a|b|c]= 1 2

Tabc−_Tcba

means anti-symmetrization over the indices a and c, the index b has been ex-cluded by the vertical bars.

1.4 The Metric Tensor

We will now define the metric tensor, g, with which we can define concepts such as distance and length of a vector. The metric tensor is a symmetric, covariant tensor field of rank two that is non-degenerate, i.e.,

gijuivj = 0, ∀vj∈V ⇔ ui = 0.

A general line element is defined:

ds2 = gijdxidxj,

where dxirepresents an infinitesimal change of the xicoordinate. Using the met-ric tensor we can define the dot product (or scalar product) between two vectors Xand Y,

X · Y= gijXiYj, (1.14)

and the square of the length (or norm) of a vector

X2= gij(x)XiXj.

If X2> 0 for all vectors Xi , 0 then the metric is said to be positive definite, and if X2 < 0 for all vectors Xi , 0 it is said to be negative definite. Otherwise the metric is called indefinite.

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Two vectors, X and Y, are said to be orthogonal if

gijXiYj= 0.

Given a metric g there exists an orthonormal basis {v1, v2, . . . , vn}of the tangent

space Vp such that gij(va)i(vb)j = 0 if a , b and gij(va)i(vb)j = ±1 if a = b, where

we use a and b as labels for the different basis vectors and (va)i is the i:th

com-ponent of basis vector va. The signature of the metric is the number of vectors

with gij(va)i(va)j = −1 (no summation over a) and the number of vectors with gij(va)i(va)j = 1 (no summation over a). A four dimensional manifold with

met-ric with signature ( − + + + ) (meaning one vector with negative square length negative and three with positive), which is what we will be using, is called a space-time (or Lorentzian manifold).

Since gij is non-degenerate, which implies det gij , 0, there exists an inverse metric gijgiven by the relation

gijgjk= δik,

where δikis the Kronecker delta.

The metric tensor can be used to raise or lower indices on other tensors, e.g.,

Tij= gikTkj

and

Tij = gjkTik.

1.5 Trace of a Tensor

The trace of a (2, 0) tensor T can be written, using the metric tensor Tr(T ) = gijTij = Tii.

1.6 Decomposition of a Tensor

Using these operations we can decompose any tensor of rank 2 into a symmetric and trace-free part, an anti-symmetric part and a part containing the trace of the tensor. Let Vij = T(ij)− 1 nT g ij_, Aij = T[ij]

where T = gijTij is the trace of the tensor Tij, Vij is the symmetric trace-free

part of Tij _{and A}ij _{is the anti-symmetric part of T}ij_{. This allows us to write} Tij = Vij+ Aij+ 1

nT g

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1.7 Covariant Derivative 7 Thus we have decomposed the tensor T into a symmetric trace-free part, an anti-symmetric part and a part containing the trace of T . This holds for all tensors T of type (2,0).

1.7 Covariant Derivative

The covariant derivative, ∇i, is an operator and not a tensor but when it operates

on a tensor the result is again a tensor. We require that ∇iis a linear operator that

satisfies Leibniz’ rule

∇_i_(αAj..._k..._{+ βB}j..._k..._{) = α∇}_i_Aj..._k..._{+ β∇}_i_Bj..._k..._, _(1.16) where α, β are constants, and

∇_i_(Aj..._k..._Bl..._m..._{) = (∇}_i_Aj..._k..._)Bl..._m..._{+ A}j..._k..._(∇_i_Bl..._m..._). _(1.17) We also require that when acting upon a real-valued function the covariant deriva-tive agrees with the gradient, i.e.,

∇_i_{f = ∂i}_{f , ∀f .}

Following (1.16), (1.17) the covariant derivative can be written ∇_k_Xi _{= ∂}_k_Xi_{+ C}i_jk_Xj_,

where the tensor Cijk is called an affine connection. We have a great deal of

freedom in determining C, we will require that C be torsion-free, i.e., Cijk = Cikj

and if we also require that the covariant derivative be metric, i.e.,

∇_i_gjk _{= 0,} _(1.18)

we change the notation of Cijk to Γijk and call it the Christoffel symbol. The

Christoffel symbols are uniquely determined by gij:

Γi_jk = 1 2g il ∂jglk+ ∂kglj−∂lgjk , as proved in [2], [8].

1.7.1 Hypersurface Orthogonality

Given a vector field u, we want to know if there exists a family of hypersurfaces S such that u is orthogonal to any S ∈ S. This requirement can be expressed using the covariant derivative as (see [8])

u[i∇juk]= 0. (1.19)

This means that u must be proportional to the gradient of some smooth scalar field, i.e.,

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where f and s are smooth scalar fields. In the special case that

∇_[i_u_j]_{= 0,} _(1.21)

umust be identically equal to the gradient of some smooth scalar field s so that

ui = −∇is, (1.22)

i.e., in this case the scalar field f can be set to unity. This can be compared to the well known result from vector calculus that a vector field with zero rotation can locally be expressed as the gradient of a scalar field.

1.8 The Curvature Tensor

The Riemann tensor is one way of describing curvature in a Lorentzian manifold, in some sense it measures the deviation of the manifold from Euclidean space (Rn_{). The Riemann tensor can be defined as the commutator}

[∇i, ∇j]Ak = ∇i∇jAk− ∇j∇iAk= RijklAl, (1.23)

i.e., the Riemann tensor is the commutator of the covariant derivative acting upon any covector and itself. For a proof that [∇i, ∇j]Ak is a tensor, see [8, chapter 3].

A different way to express the Riemann tensor is by using the Christoffel sym-bols (see [8]):

Rijkl = ∂jΓlik−∂iΓljk+ ΓmikΓlmj− ΓmjkΓlmi.

Because of its relation to the second derivative it is easy to see that the Riemann curvature tensor is anti-symmetric in the first pair of indices, meaning R[ij]kl =

Rijkl. If we use the metric to lower the first index we get some useful symmetries:

R[ijk]l = 0 ⇔ Rijkl+ Rkijl+ Rjkil = 0 (1.24)

and

Rijkl = −Rjikl= −Rijlk= Rklij, (1.25)

hence the Riemann curvature tensor is anti-symmetric also in the second pair of indices (as long as ∇kgij = 0) and symmetric under interchange of the first

and second pair of indices. We also get the so called Bianchi identities for the Riemann tensor:

∇_i_Rlmjk_{+ ∇}_k_Rlmij_{+ ∇}_j_Rlmki _{= 0.} _(1.26)

We also define the Ricci tensor

Rij = Rkikj = gklRlikj, (1.27)

and the Ricci scalar

R = gijRij. (1.28)

From these two we then define the Einstein tensor

Gij = Rij−

1

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1.9 The Geodesic Equation 9 This tensor in turn satisfies the contracted Bianchi identity (for more information see, e.g. [8])

∇_j_Gij _{= 0.} _(1.30)

1.8.1 The Weyl Tensor

In four dimensions the information contained in the Riemann tensor can also be obtained from the Ricci tensor Rij and from the so called Weyl tensor, which in

four dimensions is given by

Cijkl = Rijkl+ gi[lRk]j+ gj[kRl]i+

1

3gi[kgl]jR. (1.31) The Weyl tensor has the same symmetries as the Riemann tensor, and one addi-tional symmetry

Cijil= 0. (1.32)

Using the symmetries of (1.25) this means that the Weyl tensor is trace-free over any pair of indices.

1.9 The Geodesic Equation

In curved geometry the notion of a ”straight line” needs to be modified. We there-fore introduce geodesics. A particle that moves freely, without being acted upon by external forces, will always move along a geodesic.

1.9.1 Parallel Transport

Take a smooth curve, γ, defined on a manifold M, let v be a vector defined in a point p ∈ γ. If we move v from the point p along the curve γ, we will end up with a new vector v0

on γ. By parallel transport we ensure that the new vector, v0, is ’as parallel to the original v as possible’. Given a manifold M and a smooth curve C with parametrization xi = xi(s) in some coordinate system xi, the vector

ui = dx_dsi is said to be a tangent vector of C. A vector vi, given at each point on C, is said to be parallel transported along C if

ui∇_i_vj _{= 0} _(1.33)

along the curve. A geodesic is a curve whose tangent vector, ui = dx_dτi, is parallel transported along itself,

ui∇_i_uj_{= 0,}

and τ is, in this case, called an affine parameter (see [8]) of the geodesic. This equation can also be written

d2_xi dτ2 + Γ i jk dxj dτ dxk dτ = 0,

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by applying the Euler-Lagrange equations d dτ ∂L ∂ ˙xi ! − ∂L ∂xi = 0 to the Lagrangian L = gij˙xi ˙xj, (1.34) where ˙xi =dx_dτi.

1.10 Einstein’s Field Equations

The best candidate we have for a realistic description of the matter distribution in the universe are the Einstein Field Equations (from here on simply ’EFE’);

Gij+ gijΛ= Rij −

1

2gijR + gijΛ= Tij, (1.35)

where Tij is the energy-momentum tensor and Λ is the cosmological constant,

responsible for the acceleration of the expansion of the universe [4, 7]. This is the basic relation linking the matter distribution in the universe to the geometry of space-time, which in turn determines the motion of the matter.

The EFE, together with the contracted Bianchi identity, imply conservation of energy-momentum,

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2

Introduction to Cosmology

Cosmology is the study of the universe, its origins and its evolution. In order to describe the universe in a scientific manner we need a cosmological model, which has to be realistic and must therefore agree with observations.

In order to gain any information from a cosmological model it is necessary that its description be rather simple, thus it needs symmetries or other special properties.

2.1 1+3-Formalism

We will decompose the EFE in a 1+3-formalism using propagation and constraint equations (the definitions in this chapter are taken from [3]).

2.1.1 The 4-velocity of Matter

Let M be a manifold and let O be an open subset of M. A congruence,[8], in O is a family of curves such that through each point p ∈ O there passes exactly one curve in this family.

We assume that there exists a timelike vector field u, describing the average motion of the matter in the universe. It follows (see [8]) that there exists a con-gruence of timelike curves with u as tangents. The 4-velocity of the matter in the universe is

ui =dx

i

dτ, uiu

i _{= −1} _(2.1)

where τ is the proper time, measured along these curves. Using ui _{we can define}

a unique projection tensor

hij = gij+ uiuj ⇒ hikhkj = hij, hii = 3, hijuj = 0 (2.2) 11

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projecting orthogonally to ui. For the spaces orthogonal to ui, hij will act as a

metric, i.e., if a tensor Tij is orthogonal to ui, Tijui = Tijuj = 0, we can use hij to

raise and lower the indices of the tensor, e.g.,

Tijuj= 0 ⇒ Tij= hkjTik.

Using the 4-dimensional volume element ηijkl= η[ijkl], η0123=

q

|_{det g}_ij|_{, let us} define a volume element for the spaces orthogonal to ui:

ηijk= ulηijkl ⇒ ηijk= η[ijk], ηijkuk= 0. (2.3)

We also define two derivatives: the covariant time derivative along the fundamen-tal world lines

˙

Tij = uk∇_k_Tij_, _(2.4)

and the orthogonally projected covariant derivative ˜

∇_k_Tij_{= h}i_l_hj_m_hn_k∇_n_Tlm_, _(2.5) i.e., the projection of the covariant derivative orthogonal to ui_.

We use angle brackets to denote the orthogonal projection of vectors and the projected trace-free part of 2-tensors

vhii= hijvj, T h_iji = h(ikhj)l− 1 3h ij_h kl Tkl. (2.6)

We also use angle brackets to denote the orthogonal projection of covariant time derivatives along ui ˙vhii= hij˙vj, T˙ h_iji = [h(ikhj)l− 1 3h ij_h kl] ˙Tkl. (2.7)

2.1.2 The Energy-Momentum Tensor

We split the energy-momentum tensor, Tij, into different parts, orthogonal or

parallel to ui_. Tij = gikgjlTkl = −_u_i_uk_{+ h}_ik −_u_j_ul_{+ h}_jl_T_kl = uiujukulTkl−ujulhikTkl−uiukhjlTkl+ hikhjlTkl = µuiuj+ qiuj+ uiqj+ phij+ πij, (2.8) where qiui = 0, πii = 0, πij = π(ij), πijuj = 0.

These quantities all have physical interpretations; µ = Tijuiuj is the relativistic

energy density relative to ui, qi = −Tjkujhkiis the relativistic momentum density,

and also the energy flux relative to ui, p = 1₃Tijhij is the isotropic pressure and πij = Tklhkh_ihl_jiis the trace-free anisotropic pressure (also called the stress).

Equations of state which relates these quantities to each other are also needed. We will discuss this more shortly.

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2.1 1+3-Formalism 13

2.1.3 Kinematical Quantities

We will now split the covariant derivative of ui_{into different parts, just as we did}

for the energy-momentum tensor, Tij, above.

∇_i_uj _{= −u}_i_uj_˙ _{+ ˜}∇_i_uj_{= −u}_i_uj_˙ ₊1

3θhij+ σab+ ωab, (2.9) where ˙ui = uj∇_juiis the relativistic acceleration, representing the effect of forces other than gravity (and inertia) acting upon the matter; we have further split the part orthogonal to ui, ˜∇_i_uj, into different parts with different symmetry proper-ties, just as in chapter 1.6. θ = ˜∇_i_ui_{, the trace of the orthogonally projected} co-variant derivative, is the rate of volume expansion for the universe. σij = ˜∇h_iu_jiis

a symmetric, trace-free (i.e., σij = σ(ij), σijuj = 0, σii = 0) tensor describing the

rate of shear; and ωab = ˜∇[iuj]is the anti-symmetric vorticity tensor, describing

the rotation of the matter relative to a non-rotating frame (a so called Fermi-propagated frame).

2.1.4 The Weyl Tensor

We will now discuss the Weyl curvature tensor in the 1+3-covariant description, but first a very short detour to motivate our future actions.

The Maxwell Tensor

In a way similar to the energy-momentum tensor the Maxwell tensor, Fij, can be

split relative to ui into electric and magnetic parts,

Fij = gikgjlFkl = (uiuk+ hik)(ujul+ hjl)Fkl = . . . = 2u[iEj]−ηijkHk, (2.10) where Ei = Fijuj ⇒ Eiui = 0, (2.11) and Hi = 1 2ηijkF jk _⇒ _Hi_ui _{= 0.} _(2.12)

Turning our attention once again to the Weyl curvature tensor we can split it in a very similar way, thus obtaining the ’electric’ and ’magnetic’ parts of the Weyl curvature tensor: Cijkl = 4u[iu[kEj]l]+ 4h[i[kEj]l]+ 2ηijmu[kHl]m+ 2ηklmu[iHj]m, (2.13) where Eij = Cikjlukul ⇒ Eii = 0, Eij = E(ij), Eijuj = 0, (2.14) and Hij = 1 2ηilmC lm jkuk ⇒ Hii = 0, Hij = H(ij), Hijuj = 0. (2.15)

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Together with the Ricci tensor Rij (which can be determined at each point using

the Einstein’s Field Equations (1.35)) we can now describe the curvature of space time in this 1+3-decomposed way. We can also calculate the Riemann curvature tensor in 1+3-decomposed form, taken from [3],

Rijkl = R ij P kl+ R ij I kl+ R ij E kl+ R ij H kl Rij_{P kl}= 2 3(µ + 3p − 2Λ)u [i_u [khj]l]+ 2 3(µ + Λ)h i [khjl], Rij_{I kl}= −2u[ihj][kql]−2u[kh[il]qj]−2u[iu[kπj]l]+ 2h[i[kπj]l], Rij_{E kl}= 4u[iu[kEj]l]+ 4h[i[kEj]l], Rij_{H kl} = 2ηijmu[kHl]m+ 2ηklmu[iHj]m. (2.16)

Worth noting is that the Weyl tensor satisfies Cijkl = R ij E kl+ R

ij H kl.

2.1.5 Auxiliary Quantities

We will now define some useful quantities to simplify our lives a bit further on. We begin with the vorticity vector

ωi = 1 2η ijk_ωjk _⇒ _ωi_ui _{= 0.} _(2.17) We also define ω2= 1 2ωijω ij _≥_{0, σ}2₌1 2σijσ ij _≥_0. _(2.18)

The ’curl’ of a symmetric, trace-free tensor T , orthogonal to ui so that Tij =

T(ij), Tii = 0, Tijui = 0, is defined in the following way

(curl T )ij = ηklhi∇˜_k_T_lji_. _(2.19)

2.1.6 Propagation and Constraint Equations

We get three sets of equations describing the dynamical evolution of our model and the matter in it.

Ricci Identities

The first set of equations come from applying the Ricci identities (1.23) to the vector field ui

2∇[i∇j]uk = Rijklul. (2.20)

Using (2.9) and the EFE (1.35) we get, after separating the equations into trace, trace-free symmetric, anti-symmetric and parallel parts, three propagation equa-tions and three constraint equaequa-tions.

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2.1 1+3-Formalism 15 To illustrate how to do this, we will now derive the so called Raychaudhuri equation1. We begin by studying the right hand side of (2.20)

Rijklul = R ij P klu l_{+ R}ij I klu l_{+ R}ij E klu l_{+ R}ij H klu l_.

Studying each term separately gives

Rij_{P kl}ul =2 3(µ + 3p − 2Λ)u [i_u [khi]l]ul+ 2 3(µ + Λ)h i [khjl]ul =1 3(µ + 3p − 2Λ)(−u [i_ul_hj] kul) = 1 3(µ + 3p − 2Λ)u [i_hj] k, (2.21) Rij_{I kl}ul = −2u[ihj][kql]ul−2u[kh[il]qj]ul−2u[iu[kπj]l]ul+ 2h[i[kπj]l]ul = ulh[ikqj]ul+ u[iulπj]kul = −h[ikqj]−u[iπj]k, (2.22) Rij_{E kl}ul = 4u[iu[kEj]l]ul+ 4h[i[kEj]l]ul = −2u[iulEj]kul = 2u[iEj]k (2.23) and

Rij_{H kl}ul = 2ηijmu[kHl]mul+ 2ηklmu[iHj]mul = −ηijmulHkmul = ηijmHkm. (2.24)

The left hand side of (2.20) can be written 2∇[i∇j]uk= (∇i∇j− ∇j∇i)uk = ∇i(−uju˙k+ θ 3hj k_{+ σ} jk+ ωjk) − ∇j(−uiu˙k+ θ 3hi k_{+ σ} ik+ ωik) = (uiu˙j− θ 3hij−σij−ωij) ˙u k₋_u j∇iu˙k+ ∇i( θ 3hj k_{+ σ} jk+ ωjk) −_(u_j_u_˙_i−θ 3hji −σji−ωji) ˙u k_{+ u} i∇ju˙k− ∇j( θ 3hi k_{+ σ} ik+ ωik). (2.25) We multiply both sides of (2.20) with ui and obtain, for the right hand side

uiRijklul = uiRI ijklul+ uiRP ijklul+ uiRE ijklul+ uiRH ijklul = 1 6(µ + 3p − 2Λ)u i_(u ihjk−ujhik) − 1 2u i_(h ikqj−hjkqi) − 1 2u i_(u iπjk−ujπik) + ui(uiEjk−ujEik) + uiηijmHkm = −1 6hj k_{(µ + 3p − 2Λ) +}1 2πj k₋_E jk. (2.26)

1_{Named after Amal Kumar Raychaudhuri (1923 - 2005), a physicist from Bangladesh that taught}

himself differential geometry and the theory of general relativity. Shortly after his death a docu-mentary about his career was completed by Vigyan Prasar, an autonomous organisation under the Indian Department of Science and Technology, and the Inter-University Centre for Astronomy and Astrophysics.

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For the left hand side we get 2ui∇_[i∇_j]_uk _{= u}i

uiuj˙ − θ

3hij−σij−ωij ˙ uk−_ui_uj∇_i_u_˙k + ui∇_i _θ 3hj k_{+ σ} jk+ ωjk −_ui ujui˙ − θ

3hji −σji−ωji ˙ uk + uiui∇_j_u_˙k−_ui∇_j _θ 3hi k_{+ σ} ik+ ωik = − ˙uju˙k−uiuj∇iu˙k+ ui∇i _θ 3hj k_{+ u}i_∇ iσjk+ ui∇iωjk−uiuju˙iu˙k − ∇_j_u_˙k−_ui∇_j _θ 3hi k₋_ui_∇ jσik−ui∇jωik = − ˙uju˙k−ujui∇iu˙k+ 1 3u i_∇ i θhjk + ui∇_i_σ_jk_{+ u}i∇_i_ω_jk −_ui_u_˙_i_u_j_u_˙k− ∇_j_u_˙k−1 3u i_∇ j θhik −_ui∇_j_σ_ik−_ui∇_j_ω_ik_. (2.27) Applying Leibniz’ rule to ui∇_i_uj_uj_{= u}i∇_i_{(−1) = 0 we get}

0 = ui∇_i_(u_j_uj_{) = u}j_uj_˙ _{+ u}_j_u_˙j_{= 2u}_j_u_˙j_, _(2.28) which allows us to remove one term from (2.27)

2ui∇_[i∇_j]_uk _{= − ˙}_uj_u_˙k−_uj_ui∇_i_u_˙k₊ 1 3u i_∇ i θhjk+ ui∇_i_σjk_{+ u}i∇_i_ωjk − ∇_j_u_˙k−1 3u i_∇ j θhik −_ui∇_j_σ_ik−_ui∇_j_ω_ik_. (2.29)

To obtain the Raychaudhuri equation we take the trace of (2.29) and (2.26). Equa-tion (2.26) becomes uiRijjlul = − 1 6hj j_{(µ + 3p − 2Λ) +} 1 2πj j₋_E jj = − 1 2(µ + 3p − 2Λ). (2.30) Before we look at equation (2.29) we need to make use of Leibniz’ rule for a few quantities. Firstly 0 = ∇j(uiσij) = ∇j(ui)σij+ ui∇jσij = (−uju˙i+ 1 3θhj i_{+ σ} ji+ ωji)σij+ ui∇jσij = 2σ2+ ui∇_j_σij_, (2.31) since uiσij = 0, hijσij = σii = 0 and ωij is anti-symmetric and σij is symmetric

(the contracted product of a symmetric tensor and an anti-symmetric tensor must be zero). Secondly 0 = ∇j(uiωij) = ∇j(ui)ωij+ ui∇jωij= (−uju˙i+ 1 3θhj i _{+ σ} ji+ ωji)ωij+ ui∇jωij = −2ω2+ ui∇_jωij_, (2.32)

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2.1 1+3-Formalism 17 since uiωij = 0 and ωii = 0. We also need that

∇_j_(h_ij_{) = ∇}_j_(u_i_uj_{+ g}_ij_{) = ∇}_j_(u_i_uj_{) = u}j∇_j_ui_{+ u}_i∇_j_uj = ˙ui+ ui(−uju˙j+ θ 3hj j_{+ σ} jj+ ωjj) = ˙ui+ uiθ, (2.33) and ∇_i_u_˙i _{= g}l_i_g_ki∇_l_u_˙k _{= (−u}l_u_i_{+ h}l_i_)(−u_k_ui_{+ h}_ki_)∇_l_u_˙k = uiukului∇lu˙k−uluihki∇lu˙k−ukuihli∇lu˙k+ ˜∇iu˙i = −ukul∇lu˙k+ ˜∇iu˙i. (2.34)

Putting all of the above into (2.29) we get 2ui∇_[i∇_j]_uj _{= − ˙}_u_j_u_˙j−_u_j_ui∇_i_u_˙j₊ 1 3u i_∇ i θhjj + ui∇_i_σ_jj_{+ u}i∇_i_ω_jj − ∇_j_u_˙j−1 3u i_∇ j

θhij−_ui∇_j_σij−_ui∇_jωij = − ˙uju˙j−ujui∇iu˙j+ ˙θ + ujui∇iu˙j−∇˜iu˙i −1 3θu i_{( ˙}_u i+ uiθ) + 2σ2−2ω2 = − ˙uju˙j−∇˜_j_u_˙j_{+ ˙}_{θ +}θ 2 3 + 2σ 2₋_2ω2_, (2.35) remembering that hii = 3.

Finally, putting left hand side (2.35) equal to right hand side (2.30), and mov-ing some terms around, we get

The Raychaudhuri equation ˙

θ − ˜∇_i_u_˙i _{= −}1 3θ

2_{+ ˙}_ui_u_˙i₋_2σ2_{+ 2ω}2₋1

2(µ + 3p) + Λ, (2.36) which describes the basic gravitational attraction. Since θ determines the expansion of the universe, ˙θ measures the rate of change in the expansion

of the universe. The Raychaudhuri equation shows that a positive cosmo-logical constant will counteract the attractive forces of gravity, i.e., it will help to speed up the expansion of the universe by increasing ˙θ. The shear

of the matter in the universe, σ , will help gravity, i.e., decrease ˙θ and thus

it will help to slow down the expansion of the universe. The rotation of the matter, ω, will also counteract the attractive forces of gravity. The acceler-ation and the spatial inhomogeneity of the acceleracceler-ation of the matter will counteract gravity, and finally the term µ + 3p can be seen as the active grav-itational mass density, the term driving the attractiveness of gravity. All other propagation and constraint equations in this chapter are taken from [3].

The vorticity propagation equation ˙ ωhii−1 2η ijk_∇_˜ ju˙k= − 2 3θω i_{+ σ}i jωj, (2.37)

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relating the change in the rotation of the universe to the expansion and shear of the universe.

The shear propagation equation ˙ σhiji−∇˜hi_u_˙ji_{= −}2 3θσ ij _{+ ˙}_uh_i ˙ uji−_σhi_k_σjik−_ωhi_ωji− Eij− 1 2π ij_{. (2.38)}

This shows how the ’electric’ part of the Weyl curvature tensor, Eij, in-duces shear which then, through the vorticity propagation and Raychaud-huri equations, alter the dynamics of the matter expansion.

We also get a set of constraint equations: The (0α)-equation

0 = ˜∇_j_σij− 2 3∇˜

i_{θ + η}ijk_{[ ˜}_∇_jωk_{+ 2 ˙}_ujωk_{] + q}i_, _(2.39)

relating the relativistic momentum density to the spatial inhomogeneity of the expansion θ and shear σij.

The vorticity divergence identity

0 = ˜∇_i_ωi−_u_˙_i_ωi_. _(2.40) The Hij-equation

0 = Hij+ 2 ˙uhiωji+ ˜∇hi_ωji−_{(curl σ )}ij_, _(2.41) showing how the ’magnetic’ part of the Weyl curvature tensor comes from the curl of the shear and the change in vorticity.

The Contracted Bianchi Identity

The second set of equations comes from the contracted Bianchi identity (1.30) which, together with the EFE (1.35), imply the conservation of energy-momentum (1.36). Projecting parallelly and orthogonally to ui we get the propagation equa-tions,

the energy conservation equation ˙

µ + ˜∇_i_qi _{= −θ(µ + p) − 2( ˙}_ui_qi_{) − σ}_ij_πij _(2.42) and

the momentum conservation equation ˙qhii+ ˜∇i_{p + ˜}∇_j_πij _{= −}4

3θq

i₋_σi

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2.1 1+3-Formalism 19

The Bianchi Identity

The third set of equations we obtain by splitting Rijkl into the Ricci tensor, Rij,

and the Weyl curvature tensor, Cijkl. Using the 1+3-decomposition of those

quan-tities and Einstein’s field equations (1.35), the Bianchi identity (1.26) gives us two propagation equations and two constraint equations. The propagation equations are:

the Eij-propagation equation ( ˙Ehiji+1 2π˙ h_iji ) − (curl H)ij+ 1 2∇˜ h_i qji= −1 2(µ + p)σ ij ₋_θ(Eij ₊1 6π ij₎ + 3σhik(Ejik−1 6π jik_{) − ˙}_uh_i qji + ηklhi[2 ˙ukHlji+ ωk(Ekji+ 1 2πl ji_)] (2.44) and the Hij-propagation equation

˙ Hhiji+ (curl E)ij−1 2(curl π) ij _{= −θH}ij_{+ 3σ}h_i kHjik+ 3 2ω h_i qji −_ηklhi_{[2 ˙}_u_k_E_lji−1 2σ ji kql−ωkHlji]. (2.45) We also get 2 constraint equations:

the (div E)-equation 0 = ∇˜_j_(Eij ₊1 2π ij_{) −}1 3∇˜ i_{µ +}1 3θq i₋ 1 2σ i jqj−3ωjHij −_ηijk_[σ_jl_Hkl −3 2ωjqk], (2.46)

and the (div H)-equation

0 = ∇˜_j_Hij_{+ (µ + p)ω}i_{+ 3ω}_j_(Eij−1 6π ij₎ + ηijk[1 2∇˜jqk+ σjl(Ek l₊1 2πk l_)]. (2.47)

2.1.7 Equations of State

As mentioned above, in order to have physics play a role in our mathematical model we use certain equations of state to relate the different terms of the decom-posed energy-momentum tensor, Tij.

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Perfect Fluid

If we assume that we have no energy flux relative to ui, i.e., qi = 0, and that we have no anisotropic pressure in the fluid so that πij = 0, the energy-momentum

tensor becomes

Tij = µuiuj+ phij, (2.48)

i.e., the matter in the universe is described by a perfect fluid and ui_{becomes the}

4-velocity of a comoving observer.

Theγ-law

For a perfect fluid, one way to describe the relation between µ and p is with a so called γ-law:

p = (γ − 1)µ, ˙γ = 0. (2.49)

In order to remove unphysical solutions to the EFE we require that one, or several, energy conditions, such as the dominant energy condition µ + p > 0 and µ > 0 or the strong energy condition µ > 0, µ + p > 0 and µ + 3p > 0 hold.

By choosing different values of γ we get different types of solutions, for in-stance if γ = 1 we get p = 0 and Tij = µuiuj, a so called dust model. In a dust

model the radiation pressure p vanishes and the universe is dominated by the matter. This is a good model for an old universe. By choosing γ = 4/3 we get

p = µ/3 and Tij = µ(uiuj+ 13hij), a radiation dominated universe, thought to be

a good model for a young universe. Our universe is currently a mixture of radi-ation and matter, however there is no guarantee that by choosing a γ between 1 and 4/3 we will end up with a good approximation of our current universe. A

γ-law is a very simple model that does not necessarily have to correspond to an

accurate description of the universe.

2.2 The Friedmann-Lemaître-Robertson-Walker

Solutions

The simplest cosmological models assume that the universe is (spatially) homo-geneous, put simply this means that at any time every part of space should look like any other part, i.e., there are no special places in space.

We will refer to these space-times as Friedmann-Lemaître-Robertson-Walker-space-times (FLRW-Friedmann-Lemaître-Robertson-Walker-space-times).

2.2.1 Spatial Homogeneity

We assume there exists a foliation of space-time, i.e., space-time can locally be described as a disjoint union of homogeneous three-dimensional hypersurfaces,

M=[

S∈S S,

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2.2 The Friedmann-Lemaître-Robertson-Walker Solutions 21 where S denotes the family of hypersurfaces. This subdivision of space-time is schematically shown in figure 2.1. In any one of these hypersurfaces S ∈ S, every

Figure 2.1:Schematic figure of the foliation of space time

point should look like any other. More precisely, for any p, q ∈ S there should exist an isometry (see [8]) f on S with f (p) = q. A space-time satisfying these conditions is said to be spatially homogeneous.

2.2.2 Global Isotropy

Furthermore we assume that these hypersurfaces are globally isotropic i.e., they are isotropic (spherically symmetric) about every point, meaning that there can be no special directions in space. Global isotropy actually implies that the uni-verse must be spatially homogeneous.

Because of global isotropy, the matter flow ui must be orthogonal to every surface S ∈ S. Therefore there exists a scalar field t, constant on each hyper-surface S ∈ S, which we can use to label every hyper-surface S ∈ S, i.e., with St0 we

mean the hypersurface S ∈ S where t = t0. If we use this scalar field t to define

our time-coordinate, at any particular moment in time the universe has e.g., the same density everywhere. This is what is known as cosmological time. We will do a more detailed definition of cosmological time in chapter 4.

2.2.3 The Energy-Momentum Tensor of FLRW-space-times

We use a perfect fluid description, with a γ-law, of the matter in the universe, meaning that Tij = µuiuj + phij, where p = µ(γ − 1), with the dominant energy

condition, µ + p > 0. Finally we assume that the EFE with a cosmological constant hold, i.e. Gij+ gijΛ= Tij.

In these space-times the quantities ˙u, ω, σ , H and E must all vanish (= 0).

In addition, all spatial derivatives (the ˜∇_{operator) of the quantities described in} this chapter must also vanish.

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2.2.4 Curvature and the Friedmann Equation

The assumptions we have made lead to three possibilities for the curvature of the universe ([8]):

• Positive curvature. In these models space can be described as a 3-sphere , which means that space is finite.

• Zero curvature. Here space is flat and therefore infinite.

• Negative curvature. Space can be described as a three-dimensional hyper-boloid, which is infinite.

Measurements indicate that space is very close to being flat. As the flat case is also the easiest and, as we shall see, most natural to generalize we will focus on this case from now on. We introduce Cartesian coordinates (x, y, z) on these flat spaces giving us the metric (see [8])

ds2= gijdxidxj = −dt2+ S(t)2

dx2+ dy2+ dz2,

where S(t) is the average length scale, determined by the Friedmann equation (see [3]) 3S˙ 2 S2 = M S3γ + Λ,

in which M is a constant and µ = _SM3γ. By choosing values for γ, Λ and translating

the time-function t so that S(0) = 0 whenever possible, we obtain different solu-tions to the Friedmann equation. We will present the solusolu-tions for γ = 1 (a dust model), γ = 4/3 (a radiation dominated model) with Λ = 0 and Λ > 0. Measure-ments have shown that Λ > 0 seems to best describe the current universe (see e.g. [4, 7]), however the case Λ = 0 is significantly simpler and therefore also worth studying.

• γ = 1, Λ = 0

This case produces the solution

S(t) = _3M 4 1/3 t2/3. (2.50) • γ = 1, Λ > 0

This case gives the solution

S(t) = _M 2Λ 1/3 cosh √ 3Λt − 11/3. (2.51)

If we study the asymptotic behaviour of this solution we get:

As t → ∞ we get S(t) ∝_2ΛM1/3 e √

Λ 3 t

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2.2 The Friedmann-Lemaître-Robertson-Walker Solutions 23

As t → 0 we get S(t) ∝_2ΛM1/31 +3Λ₂ t2−₁1/3₌3M

4

1/3

t2/3, which is exactly what we found for Λ = 0.

• γ = 4/3, Λ = 0

For a radiation dominated universe without cosmological constant we get

S(t) = _4M 3 1/4 t1/2. (2.52) • γ = 4/3, Λ > 0

By adding a positive cosmological constant we get the solution

S(t) = ₁ Λ 1/4     sinh r 4Λ 3 t       1/2 . (2.53)

Again, studying the asymptotic behaviour of the solution we get:

As t → ∞ we get S(t) ∝ _Λ11/4 e √

Λ 3 t

2 , which is the same general

be-haviour as for γ = 1. As t → 0 we get S(t) ∝_Λ11/4 q 4Λ 3 t 1/2 =4₃1/4t1/2, which is, up to a multiplicative constant, the same as we found for Λ = 0.

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3

Tetrad Description

While the 1+3-covariant equations give an intuitive understanding of the defined quantities and their geometrical significance, they do not guarantee the existence of a metric or a connection which solves the EFE. For this we need another way of describing the dynamics of our cosmological model, to achieve this we turn to a tetrad description of our cosmological model. This tetrad description will con-tain, as a subset, component versions of all the equations from the 1+3-covariant description. Chapter 3 in [3] gives a more detailed overview of the tetrad descrip-tion in cosmology, for a more general descripdescrip-tion of tetrads, see e.g. [8, chapter 3]

3.1 General Tetrad Formalism

Usually we use a coordinate basis to describe the geometry of space-time and the basis vectors are not guaranteed to be orthogonal (in general they are not). A tetrad however is a set of four mutually orthogonal vector fields, {ea}a=0,1,2,3,

normalized so that

gij(ea)i(eb)j = ηab, (3.1)

with ηab= diag(−1, +1, +1, +1), the Minkowski metric, and we can define the dual

tetrad (ea)i in the following way

(ea)i(eb)i = δab ⇔ (ea)i(ea)j= δij. (3.2)

The basis vectors (ea)i can be written in terms of a local coordinate basis using

the components of the tetrad vectors (ea)i:

e_a(f ) = (ea)i ∂f ∂xi, (ea) i _{= e} a xi. (3.3) 25

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We can express any tensor using our tetrad vectors in the following way (recall that Greek indices take the values 1, 2, 3)

Tij = Tab(ea)i(eb)j =

T00(e0)i(e0)j+ T0β(e0)i(eβ)j+ Tα0(eα)i(e0)j+ Tαβ(eα)i(eβ)j.

(3.4) In this way the metric components in tetrad formalism become

gab= gij(ea)i(eb)j = ea·eb= ηab. (3.5)

We can also extract the coordinate components of the metric evaluated in a point using the inverse equation

gij = ηab(ea)i(eb)j. (3.6)

Using this basis of orthonormal vectors we can raise and lower indices using the metric gab = ηab and its inverse gab = ηab. Now that we can express the metric

both in tetrad components and coordinate components we can make certain we avoid some ambiguity in the future. Consider (ea)i, this can either be interpreted

as gij(ea)j, i.e., we lower the coordinate component index. It could also mean gab(eb)i, i.e., we lower the tetrad index instead. In fact, it does not matter, the two

versions are equal

gij(ea)j = (eb)i(ec)jgbc(ea)j= δac(eb)igbc= gab(eb)i. (3.7)

By choosing (e0)i = ui and using the fact that the shear tensor σij is orthogonal

to ui i.e., σijui = 0 we obtain

0 = σijuj = σ00(e0)i(e0)j(e0)j+ σ0β(e0)i(eβ)j(e0)j

+ σα0(eα)i(e0)j(e0)j+ σαβ(eα)i(eβ)j(e0)j= −σ00(e0)i −σα0(eα)i,

(3.8)

which in turn means that σ00 = 0 and σα0= 0. Doing the same thing with σijui

we get σ0β= 0, thus

σij = σαβ(eα)i(eβ)j. (3.9)

Of course, any tensor which is orthogonal to ui _{can be rewritten in a similar way.}

This is rather useful for our projection operator hij_{since it means that}

hαβ= hij(eα)i(eβ)j = ηαβ, (3.10)

i.e., hαβ = 1 if α = β and 0 otherwise.

Next we define the ’Ricci rotation coefficients’

ωcab= (ec)j(eb)i∇i(ea)j, (3.11)

not to be confused with the vorticity tensor ωij or vorticity vector ωi. This nota-tion is unfortunate but also standard, and the risk of confusing the different ω:s should be small.

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3.1 General Tetrad Formalism 27 Using the Ricci rotation coefficients we can write the covariant derivative in tetrad formulation

∇_c_T_ab_{= (e}_a₎i_(e_b₎j_(e_c₎k∇_k_T_ij_{= (e}_a₎i_(e_b₎j_(e_c₎k∇_k_(T_de_(ed₎_i_(ee₎_j₎ = (ea)i(eb)j(ec)k(ed)i(ee)j∇kTde+ (ea)i(eb)j(ec)k(ee)jTde∇k(ed)i

+ (ea)i(eb)j(ec)k(ed)iTde∇k(ee)j.

Note that (ea)i(ed)i = δad, which gives

∇_cTab _{= δ}_ad_δbe_(e_c₎k∇_k_Tde_{+ δ}_be_(e_a₎i_(e_c₎k_Tde∇_k_(ed₎_i_{+ δ}_ad_(e_b₎j_(e_c₎k_Tde∇_k_(ee₎_j = (ec)k∇kTab+ (ea)i(ec)kTdb∇k(ed)i+ (eb)j(ec)kTad∇k(ed)j.

In order to get to our definition of the Ricci coefficients we note that (ea)i∇k(ed)i = ∇k (ea)i(ed)i −_(ed₎_i∇_k_(e_a₎i _{= −(e}d₎_i∇_k_(e_a₎i_. _(3.12) Thus we get ∇_c_T_ab_{= (e}_c₎k∇_k_T_ab_{+ (e}_a₎i_(e_c₎k_T_db∇_k_(ed₎_i_{+ (e}_b₎j_(e_c₎k_T_ad∇_k_(ed₎_j = (ec)k∇kTab−(ed)i(ec)kTdb∇k(ea)i −(ed)j(ec)kTad∇k(eb)j

= (ec)k∇kTab−ωdacTdb−ωdbcTad.

(3.13)

If we apply this to the metric we get,

0 = ∇cgab= (ec)k∇kgab−ωdacgdb−ωdbcgad= −ωabc−ωbac ⇒

ωabc= ω[ab]c, (3.14)

hence the Ricci rotation coefficients are anti-symmetric in the first pair of indices (as opposed to the Christoffel symbols which are symmetric).

We can calculate the tetrad components of the Riemann tensor in a straight-forward (but time consuming) way. We start with

Rabcd = Rijkl(ea)i(eb)j(ec)k(ed)l = (ea)i(eb)j(ec)k(∇i∇j− ∇j∇i)(ed)k. (3.15)

A rather long derivation follows, which is very similar to how we obtained (3.13). The resulting formula, also derived in [3], is

Rabcd = ec(ωabd) − ed(ωabc) + ωaecωebd−ωaedωebc+ ωabeωecd−ωabeωedc, (3.16)

and a contraction gives the tetrad components of the Ricci tensor (and hence the EFE)

Rbd = ea(ωabd) − ed(ωaba) + ωaeaωebd−ωadeωeba= Tbd−

1

2T gbd+ Λgbd, (3.17) where T = Taa= gabTabis the trace of Tab.

Non-isotropic Cosmology in 1+3-formalism

Matematiska institutionen

Department of Mathematics

Master’s Thesis

Non-isotropic cosmology in

1+3-formalism

Johan Jönsson

Non-isotropic cosmology in

1+3-formalism

Work in cosmology performed at Linköpings Universitet

Sammanfattning

Abstract

Acknowledgments

Contents

1

Introduction to General Relativity

1.1

Conventions

1.2

Manifolds

1.3

Tensors and Vectors

1.3.1

Symmetric and Anti-symmetric

1.4

The Metric Tensor

1.5

Trace of a Tensor

1.6

Decomposition of a Tensor

1.7

Covariant Derivative

1.7.1

Hypersurface Orthogonality

1.8

The Curvature Tensor

1.8.1

The Weyl Tensor

1.9

The Geodesic Equation

1.9.1

Parallel Transport

1.10

Einstein’s Field Equations

2

Introduction to Cosmology

2.1

1+3-Formalism

2.1.1

The 4-velocity of Matter

2.1.2

The Energy-Momentum Tensor

2.1.3

Kinematical Quantities

2.1.4

The Weyl Tensor

2.1.5

Auxiliary Quantities

2.1.6

Propagation and Constraint Equations

2.1.7

Equations of State

2.2

The Friedmann-Lemaître-Robertson-Walker

Solutions

2.2.1

Spatial Homogeneity

2.2.2

Global Isotropy

2.2.3

The Energy-Momentum Tensor of FLRW-space-times

2.2.4

Curvature and the Friedmann Equation

3

Tetrad Description

3.1

General Tetrad Formalism