Properties of mass distributions for discrete cosmology

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Properties of mass distributions for discrete cosmology

Helena Engström

Fysikum

Degree 60 HE credits Theoretical physics 2013

Supervisor: Kjell Rosquist

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This thesis addresses the question of how well the FLRW solution of Einstein’s equations is able to describe the locally inhomogeneous observable universe. The effect of local inhomogeneities on the large scale geometry is called backreaktion and to study this effect we will construct discrete models of the universe containing a small number of black holes and solve the constraint equations without any approximations or averaging. To be able to solve the constraint equations exactly our models will consist of topological 3-spheres and we will find a momentarily static solution for which the equations are greatly simplified.

For the mass configurations considered here we will present the metric intrinsic to the hypersurface and construct and illustrate a cell structure dividing the 3-sphere in different regions and use this structure to discuss symmetries of the spacetimes. We will also present values for the Ricci tensor intrinsic to the hypersurface for points at high symmetry.

In the end we will compare the scale factors of the discrete models with the scale factors of the corresponding continuous FLRW models where we define the scale factors from a calculated value of the volume of the corresponding model.

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Sammanfattning

Det här arbetet undersöker hur väl FLRW-lösningar till Einsteins ekvationer beskriver vårt lokalt inhomogena, observerbara universum. Hur lokala inhomogentiteter påverkar den storskaliga geometrin i universum kallas “backreaktion” och vi kommer att undersöka denna genom att konstruera diskreta modeller där all massa är koncentrerad till ett fåtal svarta hål och lösa tvångsekvationerna utan några approximationer eller medelvärdesprocesser. För att kunna lösa tvångsekvationerna exakt kommer våra modeller att ha en bakgrund av en topologisk 3-sfär och vi kommer att hitta en momentant statisk lösning där tvångsekvationerna förenklas avsevärt och på så sätt blir exakt lösbara.

För massfördelningarna vi undersöker kommer vi att presentera den inre metriken till hyperytan och konstruera och illustrera en cellstruktur som delar upp 3-sfären i olika områden och sedan använda cellstrukturen för att diskutera symmetrier. Vi kommer också att presentera värden för den inre Ricci tensorn i punkter av hög symmetri.

För att kunna jämföra de diskreta modellerna med de kontinuerliga FLRW modellerna kommer vi att beräkna volymen hos de båda och från den beräkna en kvot mellan deras skalfaktorer.

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Chapter 1 Introduction

Cosmology is the study of the origin, the evolution and the fate of the universe. The Standard Model of cosmology, the model that is currently believed to describe the evolution of the universe in a correct way, is the Hot Big Bang model [1]. In this model the universe was enormously compressed when it was just a small fraction of a second old and therefore was very hot and dense. After that, the universe expanded and cooled off and structure in the form of protons and neutrons started to form. As the expansion and cooling went on it allowed for more complex structures like atoms and molecules and later on stars and galaxies.

The gravitational interactions in the standard model are governed by the Friedmann-Lemaitrè- Robertson-Walker (FLRW) model. When observations are interpreted with this model as background one result is the relative abundance of different kinds of matter and energy in the universe. In this thesis we will get an estimate of the backreaction, i.e, how well the FLRW model represents the real universe, by testing the assumptions the model do about matter.

Nearly all cosmological observations that can be done has to be interpreted with respect to a model to be able to say something about the universe. This makes it difficult to study the model itself because it can not be direclty compared with observational data. In this thesis I will test the accuracy of the FLRW model so we first take a look at how the observations are interpreted with this model as background.

Before the observations of the too large redshift of type Ia supernovae in 1998 the matter and energy content in the universe was described by the Standard Cold Dark Matter (SCDM) model. This model contained ordinary matter, dark matter and ordinary energy (radiation). To be able to explain the large redshift with the FLRW model as background the cosmological constant, Λ, a constant allowed in Einstein’s equations has to be non-zero. This new constant can be intrprented as a new kind of energy, called dark energy. The SCDM model was then replaced by the ΛCDM model.

The evolution suggested by the Hot Big Bang model makes it possible to divide the evolution of the universe in different eras, dominated by different content. The early universe was hot and dense and it was dominated by radiation in the form of photons and relativistic particles. As the universe cooled and structure was allowed to form it became matter dominated. To explain the large redshift observed in the supernovae with the FLRW model as background the universe has to experience an accelerated expansion at late times. To account for this accelerated expansion the FLRW needs the cosmological constant to be non-zero.

In this thesis we will focus on the matter dominated period assuming that the cosmological constant is zero and develop a way to test some assumptions about the matter present in the FLRW model and how these assumptions effect the scale factor of the universe. A more detailed description of the FLRW model and its assumptions will be discussed in Section 1.2. But before that we have to look at the exact way the gravitational interactions are described, namely Einstein’s theory of relativity. This will be done in Section 1.1. In Section 1.3 I will give the background to our approach and the outline

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of the thesis.

1.1 Einstein’s equations

Einstein’s general theory of relativity describes gravitational interactions, the interactions between massive objects, as interactions between these objects and the spacetime. The spacetime is the three space dimensions and the time dimension combined. In this theory massive objects curve the spacetime and this curvature will affect other objects as they move through the spacetime. The equations governing these interactions are called the Einstein Field Equations (EFE) and can be written as

R^αβ−1

2Rg^αβ= 8πT^αβ. (1.1)

The indices α, β = (0, 1, 2, 3), where zero represents the time-coordinate and the 1, 2, 3 represents the three spatial coordinates. If we write the equations explicit we get one equation for every combination the two indices can take and due to symmetry the result is 10 uniqe equations. The left hand side of (1.1) is the definition of the Einstein tensor, G^αβ, describing the curvature of the spacetime while the right hand side contains the mass and energy in the stress-energy tensor, or energy-momentum tensor, T^αβ. The Ricci tensor, R^αβ, and the Ricci scalar, R, are different measures of curvature and can be derived from the Riemann tensor, Rαβγδ containing all information about the curvature. The EFE can also be written with a constant, Λ, called the cosmological constant, added to the left hand side.

This constant is used in the FLRW model to represent the dark energy. In the following I will assume it is zero. The solution to this system of equations gives the 10 components of the metric for a given stress-energy tensor.

The metric is a relation between infinitesimal intervals in some coordinates chosen to represent the spacetime and a measure of length in the spacetime. Letting ds represent an infinitesimal length in the spacetime this is written as

ds²= gαβdx^αdx^β, (1.2)

where x^α is some chosen coordinate system. Here I have used the Einstein summation convention allowing me to omit the summation sign knowing that if two indices in an expresson is the same, there is always a summation over them. Without the convenion (1.2) would look like

ds²=

3

X

α=0 3

X

β=0

gαβdx^αdx^β.

The EFE are hard to solve even for very simple matter distributions. To understand why we will investigate how the left hand side looks expressed in the metric instead of the Ricci tensor and scalar.

The Ricci tensor and the Ricci scalar are contractions of the Riemann tensor so we will first define the relation between the Riemann tensor and the metric. To do this we use that the covariant derivative, a derivative that is coordinate independent, of a tensor A^α_β can be written as

A^α_β;γ = A^α_β,γ+ Γ^α_µγA^µ_β− Γ^µ_βγA^α_µ, (1.3) where Γ^α_βγis the Christoffel symbol. The comma is shorthand for a partial derivative, A^α_β,γ = ^∂A

α β

∂x^γ . We can express Γ^α_βγ in terms of the metric and its partial derivatives as

Γ^α_βγ= 1

2g^αµ(g_µβ,γ+ g_µγ,β− gγβ,µ) . (1.4) Now we can express the Riemann tensor in terms of the Christoffel symbol and its partial derivatives R^α_βγδ= −2Γ^α_β[γ,δ]+ 2Γ^α_µ[βΓ^µ_γδ], (1.5)

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here the brackets represent antisymmetrization, defined by Γ^α_β[γ,δ]=1

2 Γ^α_βγ,δ− Γ^α_βδ,γ .

The Ricci tensor and the Ricci scalar depends on the metric and its first and second partial derivatives.

The fact that they are contractions of the Riemann tensor together with (1.3) and (1.4) makes it possible to see that the left hand side of the Einstein equations (1.1) can be expressed in terms of the metric and its first and second partial derivatives. This results in a system of 10 coupled, nonlinear, partial differential equations. A system of this kind is very hard to solve. Even without any matter in the universe, finding the metric is not an easy task.

The common way to solve these equations has been to motivate an ansatz for the metric based on symmetry arguments and then calculate the Christoffel symbol with (1.4), calculate the partial derivatives of the Christoffel symbols, use these to get the Riemann tensor through (1.5) and then contract the Riemann tensor to get the Ricci tensor and the Ricci scalar.

1.2 Matter and curvature in the FLRW model

To simplify the procedure of solving the EFE for the universe the FLRW model first assumes that there exists a scale where the matter distribution is homogeneous and isotropic. Homogeneous means that it is the same everywhere and isotropic means that it is the same in all directions. This assumption is supported by observations of Cosmic Microwave Background (CMB) radiation that suggests that the energy density in the early universe is close to exact isotropy and homogeneity. At later times the universe is no longer locally homogeneous and isotropic but should be in a statistical sense.

From the assumption of statistical homogeneity and isotropy the FLRW model defines how the matter distribution should be represented and what form the metric should have for a given homogeneity scale. We will start with the matter representation.

Describing the matter content over an averaging scale will result in a fluid description. The different fluid elements can then interact with one another, change form and so on. If we want the universe to be homogeneous every such fluid element has to have the same density. And to have an isotropic universe adjacent fluid elements have to be at rest with respect to each other.

In the FLRW model the matter description that satisfies these conditions is taken to be the one of a pressureless perfect fluid, also called dust. In this description the different fluid elements do not interact with each other in any way and they are at rest with respect to adjacent fluid elements. The only parameter that describes the matter is the mass density, ρ.

This representation of the matter does not take into account the inhomogeneities that exists on smaller scales. The effect of these inhomogeneities on the large scale geometry is called backreaction [2], which is what we will study in this thesis.

To solve Einstein’s equations the FLRW model restrict the metric to be homogeneous and isotropic.

This ansatz is called the FLRW metric and it has line element ds²= −f (t)dt²+ a²(t)

dr²

1 − kr²+ r²dθ²+ r²sin²θdϕ²

, (1.6)

in spherical coordinates. Here f (t) is a free function and k is a measure of the curvature takeing the values −1, 0 or 1 for a negative, flat and positive curvature respectively. Making this ansatz for the metric reduces the procedure of solving Einstein’s equations for the 10 components of the metric to solve for just a(t), the scale factor.

Imagine that we want to solve the EFE for our universe. And assume that we can find a suitable averaging scale where the matter is homogeneously distributed. Then, on this scale we assume that the metric can be represented by (1.6). This metric describes an average over this particular scale.

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The problem now is that averaging over the metric is not the same as averaging over the curvature, the left hand side of the EFE, because of its non-linearity. This is called the averaging problem [3].

In this thesis I will present a model that handles matter in an exact way. In the end of this work I will compare the scale factor for such a model with the scale factor for a corresponding FLRW model.

In this way we get an estimate on how much the dust approximation (and the averaging problem) affects the size of the universe.

1.3 Constructing a discrete solution

To study the EFE in a more intuitive way one usually splits the equations with respect to an observer, in a space part and a time part. Doing this splitting will result in the constraint equations and the evolution equations. The constraint equations work as initial data for the evolution equations and the task to solve the constraint equations is sometimes called the initial value problem.

How you divide the EFE into the constraint equations and the evolution equations is described in Section 1.3.1. The general idea behind how we will construct our solutions as well as the background and outline of this thesis is presented in Section 1.3.2.

1.3.1 3+1 splitting

The idea behind the 3+1 splitting is to split up the spacetime in a space part and a time part. It is useful when we want to study a system in the intuitive way, as a set of initial conditions and its time evolution. In this thesis we will be concerned with the initial conditions, i.e, the universe at a specific time.

When you do the splitting you do it with respect to a family of observers moving in spacetime. For every such family related to a specific four-velocity the splitting will look different. The world lines corresponding to the selected family of observers will then become the time part of the splitting and a three-dimensional surface, called 3-surface, orthogonal to it will be the space part.

If we let u^α be the four-velocity of our observer and h_αβ the three-dimensional metric of the orthogonal 3-surface we can split the four-dimensional spacetime metric g_αβ in space and time by writing

gαβ= hαβ− uαuβ. (1.7)

Splitting the EFE in this way will result in 12 first order evolution equations

˙hαβ− Kαβ = 0 K˙αβ− 2KαµK^µ_β+ KKαβ+⁽³⁾Rαβ = Rαβ, and 4 constraint equations

K²− K_µνK^µν+⁽³⁾R = 2G_µνu^µu^ν (1.8)

Dµ(K_α^µ− δ_α^µK) = Gναu^ν. (1.9)

Here Kαβis the extrinsic curvature, Dµ is the spatial covariant derivative corresponding to the metric h_αβ and a dot represents a time derivative with respect to the four-velocity of the observer

˙h_αβ= h_αβ;γu^γ.

In the constraint equations⁽³⁾R and⁽³⁾Rαβ is the Ricci scalar and the Ricci tensor corresponding to the three-metric hαβ.

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In this thesis we will address the time symmetric initial value problem which is the same as solving the constraint equations on a time symmetric spacelike hypersurface. A surface of this kind is possible if the model has the topology of a 3-sphere, where the evolution of the model is described by an expansion and then a moment of maximal expansion and then contraction. The moment of maximum expansion, i.e, the moment of time symmetry will be the situation we study in this thesis.

As will be explained in Chapter 2 at the moment of time symmetry the extrinsic curvature vanishes.

This together with the fact that we are looking for vacuum solutions greatly simplifies the constraint equations and makes it possible to solve them exactly.

1.3.2 Discrete universes

The exact solutions we will construct and investigate in this thesis consist of simple configurations of Schwarzschild black holes on a background topological 3-sphere. A configuration of this kind was first done in an approximate way by Lindquist and Wheeler, [4]. They called it the Schwarzschild-cell method.

The idea behind the Schwarzschild-cell method is to place black holes in a lattice configuration and then replace the unknown metric in the neighborhood of each black hole with the Schwarzschild metric. This region is then the Schwarzschild-cell and it is approximated with a sphere with the corresponding black hole in the center. They studied lattices containing 5, 8, 16, 24, 120, and 600 black holes corresponding to configurations of the four-dimensional analogues to the platonic solids.

This thesis is a companion work of the paper published by Rosquist, Tavakol, Clifton [5] and for references on the background of our approach see the references in there. In that paper the authors construct exact metrics for the same six mass configurations as Lindquist and Wheeler on a background 3-sphere and compare the scale factor of these model with a corresponding FLRW model. I will do the same in this thesis but for different mass distributions and I will do the comparison in a different way as well.

In the next chapter, Chapter 2, I will define a hypersurface and the properties related to it that we need to derive the constraint equations stated in the previous section. In the end of that chapter we will also discuss the simplifications that occurs on the hypersurface at the moment of time symmetry and introduce a way to generate solutions to the constraint equations..

In Chapter 3 we first construct a metric that solves the constraint equations and that corresponds to a 3-sphere populated by black holes. We also calculate the location of the horizons for the black holes in our four models.

In Chapter 4 the properties of the chosen mass configurations will be discussed. For each configuration we will define a cell structure and for the three simplest configurations we will illustrate this structure by means of a stereographic projection. From this cell structure I will also find points in the models where a number of planes of reflection symmetry intersect and for this point calculate the intrinsic Ricci tensor present in the constraint equations.

Chapter 5 handles the comparison between our discrete models and the continuous FLRW model.

There is no obvious way to define a scale factor in our discrete models, the value of the scale factor differs from place to place. To define a length we replace the black holes with stars which allow us to calculate a volume of the universe and from that define the scale factor.

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Chapter 2 Hypersurfaces

In this chapter I will present the theory of hypersurfaces that is needed for our model of the universe.

The goal of this chapter is to derive the constraint equations presented in Section 1.3.1, these equations are also called the Gauss-Codazzi equations. In the end of this chapter we will also discuss methods used to solve these equations in the specific case we are interested in. In the 3+1 splitting the hypersurface of concern is spacelike, hence I will restrict myself to spacelike hypersurfaces.

The first two sections, 2.1 and 2.2, follow the notation and outline of the chapter on hypersurfaces in Poisson’s book [6]. For the derivations done in the latter part of this chapter the notation and outline of [7] and [8] is used.

2.1 Defining the hypersurface

2.1.1 Defining equations

A hypersurface is a three-dimensional submanifold in a four-dimensional spacetime. You can select your hypersurface, let us call it Σ, in two ways. Either by putting a restriction on the coordinates,

ϕ(x^α) = 0, (2.1)

or by giving parametric equations of the form

x^α= x^α(y^a), (2.2)

where x^α, α = (0, 1, 2, 3), are the coordinates of the four-dimensional spacetime and y^a, a = (1, 2, 3), the coordinates on our hypersurface. To illustrate this we can take a look at the 3-sphere. If we use (2.1) the 3-sphere is defined by

ϕ(x, y, z, w) = x²+ y²+ z²+ w²− R²= 0, and if we use (2.2) we get

x(χ, θ, ϕ) = R sin χ sin θ cos ϕ

y(χ, θ, ϕ) = R sin χ sin θ sin ϕ (2.3)

z(χ, θ, ϕ) = R sin χ cos θ w(χ, θ, ϕ) = R cos χ,

where R is the radius of the 3-sphere and χ, θ, ϕ are hyperspherical coordinates where χ, θ ∈]0, π[ and ϕ ∈ [0, 2π[.

These two ways to define the hypersurface are of course mathematically equivalent.

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2.1.2 Normal and tangent

To describe our hypersurface and how it is embedded in the four-dimensional spacetime we need to define a normal and a tangent. We do this from our defining equations (2.1) and (2.2).

Since we are restricting ourselves to spacelike hypersurfaces we know that our normal is timelike, hence we would like to define a unit normal nα which satisfies nαn^α= −1.

The vector ϕ,αis normal to the hypersurface because the value of ϕ changes only in the direction orthogonal to Σ. We can then write the normal as

n_α= ϕ,α

| g^µνϕ,µϕ,ν|¹², (2.4)

where g^µν is the metric of the four-dimensional spacetime and the overall sign is chosen so that n⁰= _dτ^dt > 0, where τ is the proper time.

From (2.2) we can define the vectors

e^α_a = ∂x^α

∂y^a. (2.5)

They describe how coordinates restricted to the hypersurface change when moving along the hypersurface thus they are tangents to curves contained in Σ. They are also a basis on Σ and can be used to project down tensors to the hypersurface, changing the coordinates from x^α to y^a.

2.1.3 The induced metric

To complete our definition of the hypersurface we would like to define a metric intrinsic to Σ. The natural way to do this is to restrict our line element to curves contained in Σ. For a displacement within Σ we have

ds²_Σ = g_αβdx^αdx^β

= g_αβ ∂x^α

∂y^ady^a ∂x^β

∂y^bdy^b

(2.6)

= habdy^ady^b, where

hab= gαβe^α_ae^β_b, (2.7)

is the induced metric also called the first fundamental form, of the hypersurface. It is important to have in mind that while hab behaves as a tensor under transformations in the intrinsic coordinates y^a → y^a⁰ it behaves as a scalar under transformations in the spacetime coordinates x^α→ x^α⁰. This is the case for all tensors expressed in the intrinsic coordinates and we will call them three-tensors.

Later on in this chapter we will also need an expression for the inverse metric and we will now show that

g^αβ= −n^αn^β+ h^abe^α_ae^β_b, (2.8) where h^ab is the inverse of the induced metric, is the desired expression.

The vectors e^α_a and n^α can be thought of as a basis in the four-dimensional spacetime, i.e, every tensor in that spacetime can be fully expressed by these vectors. So, if we can show that the inverse metric (2.8) gives the same value for the inner product between the basis vectors as the metric gαβ, we have shown that it is the inverse.

With the usual metric we have the inner products

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g_αβe^α_an^β = n^αe_αa= 0 gαβn^αn^β = n^αnα= −1

gαβe^α_ae^β_b = e^α_aeαb= hab. and with the inverse metric we get

g^αβeαanβ = −n^αn^βeαanβ+ h^cde^α_ce^β_deαanβ= 0 g^αβnαnβ = −n^αn^βnαnβ+ h^cde^α_ce^β_dnαnβ= −1 g^αβeαaeβb = −n^αn^βeαaeβb+ h^cde^α_ce^β_deαaeβb= hab.

In the first line both terms become zero because they both involve the product n^αeαa = 0. In the second line the fist term becomes -1 because n^αnα = −1 and the second term vanishes. In the third line the first term vanishes and the second term becomes habbecause e^α_aeαb= haband hach^cd= δ^d_a.

2.2 Projections of the spacetime derivative

In this section we will define the intrinsic covariant derivative as the projection of the spacetime covariant derivative down to the hypersurface. We will also define the extrinsic curvature from the spacetime covariant derivative projected along the normal.

2.2.1 The projector

The basis vectors e^α_a can be used to project a general tensor field down to the hypersurface, but this changes the spacetime coordinates x^αto the intrinsic coordinates y^a. In the next section we will need a projector that does not change the coordinates. So, before we define an intrinsic derivative I would like to define the projector h^αβ = h^abe^α_ae^β_b. It performs contractions with the basis but without the coordinate change. We apply the projector with one index down because then h_β^αh_γ^β = h_γ^α, as we demand of a projector.

By using (2.8) we can express the projector as

h^αβ= h^abe^α_ae^β_b = g^αβ+ n^αn^β. (2.9) This relation will be used frequently throughout this chapter as a way to rewrite the projector in terms of the spacetime metric and vice versa. It is important to have in mind that the projector is orthogonal to the normal by construction, i.e that h^α_βnα= (g^α_β+ n^αnβ)nα= nβ− nβ= 0.

2.2.2 Intrinsic covariant derivative

In this section we will derive a covariant derivative on the hypersurface. For simplicity we restrict ourselves to tangent vector fields A^α, where

A^α= A^ae^α_a, A^αnα= 0, Aa= Aαe^α_a.

We define the intrinsic covariant derivative as a projection of the spacetime covariant derivative,

Aa|b= Aα;βe^α_ae^β_b, (2.10)

or in the spacetime coordinates

A_α|β= h^µ_αh^ν_βAµ;ν. (2.11)

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We would now like to show that (2.10) is nothing but the covariant derivative of Aa defined as usual with a Levi-Civita connection Γ^a_bccompatible with the intrinsic metric hab. We start from (2.10)

A_α;βe^α_ae^β_b = (A_αe^α_a)_;βe^β_b − Aαe^β_be^α_a;β

= Aa,βe^β_b − eaγ;βe^β_bA^ce^γ_c

= ∂Aa

∂x^β

∂y^b − e^γ_ceaγ;βe^β_bA^c

= Aa,b− ΓcabA^c, where we in the last step have defined

Γ_cab= e^γ_ce_aγ;βe^β_b, (2.12)

and we end up with the familiar expression

A_a|b= A_a,b− Γ^c_abA_c, for the covariant derivative.

The last thing we have to show is that the connection defined in (2.12) is compatible with the induced metric, i.e that the covariant derivative of the metric vanishes. We start by expressing it in the spacetime coordinates

h_ab|c = hαβ;γe^α_ae^β_be^γ_c

= (g_αβ+ n_αn_β)_;γe^α_ae^β_be^γ_c

= (n_α;γn_β+ n_αn_β;γ)e^α_ae^β_be^γ_c

= 0,

where we have used the expression for the projector (2.9) and in the last step that n_αe^α_a = 0.

It follows from the compability with the metric that we can express the connection as Γ_cab=1

2(h_ca,b+ h_cb,a− h_ab,c) . (2.13)

2.2.3 Extrinsic curvature

If the tangent component of A^α_;βe^β_b is the covariant derivative with respect to the metric on the hypersurface, what is the normal component? To find the answer we start by expressing A^α_;βe^β_b with the metric as

A^α_;βe^β_b = g^α_µA^µ_;βe^β_b = (−n^αn_µ+ h^ame^α_ae_mµ)A^µ_;βe^β_b

= −(n_µA^µ_;βe^β_b)n^α+ h^am(A_µ;βe^µ_me^β_b)e^α_a, (2.14) where we have used the last part of (2.9) to express the metric. If we now use the definition of intrinsic derivative, (2.10), and the fact that A^µ is orthogonal to n^µ we can write

A^α_;βe^β_b = (nµ;βA^µe^β_b)n^α+ h^amA_m|be^α_a

= A^a_|be^α_a + A^a(nµ;βe^µ_ae^β_b)n^α. (2.15)

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The first part of (2.15) is the tangent component and the second part is the normal component of the derivative.

We now introduce the three-tensor

Kab= −nµ;βe^µ_ae^β_b, (2.16)

or in the spacetime coordinates

Kαγ= −nµ;βh^µ_αh^β_γ. (2.17)

This tensor is called the extrinsic curvature or the second fundamental form, of the hypersurface.

With it we can write the covariant derivative of the tangent vector field A^αas A^α_;βe^β_b = A^a_|be^α_a − A^aK_abn^α.

The extrinsic curvature is a measure of how much normal vectors to the hypersurface differs at neighboring points, i.e, it describes how the hypersurface is embedded in the four-dimensional spacetime. We will use the extrinsic curvature frequently in the rest of this chapter so we will show that it is symmetric and that it is related to the Lie derivative of the spacetime metric.

To show that the extrinsic curvature is symmetric we will use that the spacetime covariant derivative is torsion-free. In this case we have

f;αβ− f;βα= 0, (2.18)

where f is a smooth scalar function. Now we recall the definition of the normal (2.4) and rewrite it as nα= gϕ;α, where g is the normalization factor. We can replace the derivative in (2.4) with a covariant derivative because ϕ is a scalar field. Combining this with (2.18) we get

n_α;β− nβ;α = 2n_[α;β]= 0,

i.e, that the antisymmetric part of n_α;β is zero. Looking at the definition of the extrinsic curvature (2.16) or (2.17) it follows that it has to be symmetric.

Now to the relation with the Lie derivative of the spacetime metric. We can define the Lie derivative of the tensor Tαγ along a vector u^α in terms of covariant derivatives as

LuTαγ = Tαγ;βu^β+ u^β_;αTβγ+ u^β_;γTαβ. (2.19) From this we can write down the Lie derivative of the spacetime metric along the normal

Lngαβ= n^µ_;αgµβ+ n^µ_;βgαµ= nβ;α+ nα;β= 2n_(α;β). (2.20) So, we can write

Kab= −n_(α;β)e^α_ae^β_b = −1

2Ln(gαβ)e^α_ae^β_b, (2.21) or in spacetime coordinates

Kµν = −n_(α;β)h^α_µh^β_ν= −1

2Ln(gαβ)h^α_µh^β_ν. (2.22)

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2.3 Gauss-Codazzi equations

In this section I will derive the Gauss equation and the Codazzi equation. These two equations relates projections of the four-dimensional Riemann tensor to quantities related to the hypersurface, like the extrinsic curvature and the three-dimensional Riemann tensor. There is a third projection of the four-dimensional Riemann tensor which gives rise to the Ricci equation. If you are interested in its derivation you find it in [7]. The Gauss-Codazzi equations become the constraint equations when the four-dimensional Riemann tensor is eliminated through the Einstein equations. This will be done in Section 2.3.3.

I will do all calculations in this section in the spacetime coordinates because I find this derivation more intuitive.

To avoid confusing the Riemann tensors with one another I give the four-dimensional one a hat, and as usual it is defined as

V^α_;βγ− V^α_;γβ = ˆR^α_µβγV^µ, (2.23) where V^µ is a general vector.

2.3.1 Gauss equation

The Gauss equation is a relation between the three-dimensional and the four-dimensional Riemann tensors. The intrinsic Riemann tensor is defined in the same way as the four-dimensional one,

A^α_|βγ− A^α_|γβ= R^α_µβγA^µ, (2.24)

with the intrinsic covariant derivative instead of the one used in (2.23). Here A^α is again a tensor tangent to the hypersurface.

If we want to relate the two Riemann tensors it seems like a good idea to start by relating the covariant derivative on the hypersurface with the covariant derivative in the spacetime. We start with the intrinsic one and find

A^α_|β = h^α_µh^ν_βA^µ_;ν = h^ν_β(g^α_µ+ n^αnµ)A^µ_;ν

= h^ν_βA^α_;ν− h^ν_βn^αA^µnµ;ν= h^ν_βA^α_;ν− n^αA^σh^ν_βh^µ_σnµ;ν (2.25)

= h^ν_βA^α_;ν+ n^αA^σK_βσ.

In this derivation we have used that A^αn_α= 0 by definition, and hence A^α_;βn_α= −A^αn_α;β as well as the definition of the extrinsic curvature (2.17).

We can now use this to define A^α_|βγ in terms of the four-dimensional derivative,

A^α_|βγ = h^ξ_γh^α_σh^ρ_β(h^σ_µh^ν_ρA^µ_;ν);ξ

= h^ξ_γh^α_σh^ρ_β(h^σ_µ;ξh^ν_ρA^µ_;ν+ h^σ_µh^ν_ρ;ξA^µ_;ν+ h^σ_µh^ν_ρA^µ_;νξ) =

= h^α_µh^ν_βh^ξ_γA^µ_;νξ− h^ν_βK^α_γnµA^µ_;ν − h^α_µKβγn^νA^µ_;ν. (2.26) Here we have also used that

h^β_αh^δ_γh^µ_δ;β= h^β_αh^δ_γ(g^µ_δ+ n^µnδ);β= −Kαγn^µ.

Now we have what we need to construct the three-dimensional Riemann tensor. For simplicity I will antisymmetrize over one term from (2.26) at a time. We start with the first one

Properties of mass distributions for discrete cosmology

Properties of mass distributions for discrete cosmology

Contents

Chapter 1

Introduction

Chapter 2

Hypersurfaces