1.1 The Mathematics of General Relativity

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3+1 Approach to

Cosmological Perturbations

Deriving the First Order Scalar Perturbations of the Einstein Field Equations

Kosmologisk störningsräkning utifrån 3+1 formalismen

Härledning av första ordningens skalära störningar av Einsteins fältekvationer

Wilhelm Söderkvist Vermelin

Faculty of Health, Science and Technology Physics, Bachelor Degree Project

15 ECTS credits

Supervisor: Claes Uggla Examiner: Jürgen Fuchs

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Abstract

Experimental data suggest that the universe is homogeneous and isotropic on sufficiently large scales. An exact solution of the Einstein field equations exists for a homogeneous and isotropic universe, also known as a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe. However, this model is only a first approximation since we know that, locally, the universe has anisotropic and inhomogeneous structures such as galaxies and clusters of galaxies. In order to successfully introduce inhomogeneities and anisotropies to the model one uses perturbative methods. In cosmological perturbations the FLRW universe is considered the zeroth order term in a perturbation expansion and perturbation theory is used to derive higher order terms which one tries to match with observations.

In this thesis I present a review of the main concepts of general relativity, discuss the 3+1 formalism which gives us the Einstein field equations in a useful form for the perturbative analysis, and lastly, I derive the first order scalar perturbations of the Einstein field equations.

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Acknowledgements

I would like to thank my supervisor Claes Uggla for his dedication and commitment, for all inspiring and interesting discussions we have had, and for deepening my knowledge in what I feel is one of the most fascinating areas of physics. My thanks to Isabella Danielsson for helping me with illustrations and for always being supportive. I also want to thank my family and friends for supporting me and making me happy.

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1 Introduction

The derivations and discussions in this thesis are based on the ideas and concepts of general relativity. General relativity is a theory of gravitation and describes the interactions of matter and energy with space and time (space-time). Special and general relativity were developed in the early parts of the twentieth century by Albert Einstein.

General relativity is the mathematical framework of modern cosmology. On sufficiently large scales, the universe is homogeneous and isotropic which means that we can model the matter and energy content in the universe as a homogeneous perfect fluid (or several fluids). A universe modeled in this fashion is called a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe and the corresponding Einstein equations can be solved exactly. However, the universe we observe is not homogeneous and isotropic on sufficiently small scales, otherwise we would not observe structures such as stars, galaxies and clusters of galaxies. This shows that FLRW universes cannot be the whole picture and we have to modify the model if we want to incorporate inhomogeneities and anisotropies. A very successful resolution to this is to view the FLRW universe as a zeroth order term in a perturbation expansion and derive higher order perturbation terms using perturbation theory. Perturbation theory is used in many other areas in physics and the idea to employ these techniques in general relativity, and in particular cosmology, originates from the work of Evgeny Lifshitz in 1946 [1]. In this thesis I give a rather extensive discussion of the 3+1 formalism, which gives us powerful tools to perform higher order perturbations. However, I limit myself to first order perturbation calculations in this thesis.

My work is a review of some basic concepts in general relativity, an introduction to the 3+1 formalism, an introduction to perturbation theory in general relativity, and finally we incorporate the 3+1 formalism into perturbation theory and calculate some first order equations.

Throughout the whole thesis, we choose units such that 8πG = c = 1 where G is the gravitational constant and c is the speed of light in vacuum.

1.1 The Mathematics of General Relativity

In some sense, general relativity is a geometrical theory of gravity and several concepts of differential geometry make up the mathematical structure of general relativity. Some of those concepts are manifolds, tensors, curvature, etc. It is important that the reader is familiar with those concepts. Hence, a brief explanation of the most central ideas of general relativity will be given. For a more in-depth discussion, consult [2, 3, 4, 5].

In all derivations and calculations we employ the Einstein summation convention, i.e., if an index appears more than once in an equation summation over that index is implied, e.g.,

S^αµνT_αβµν ≡

3

X

α=0 3

X

µ=0 3

X

ν=0

S^αµνT_αβµν. (1.1)

Indices that are lowercase Greek letters run from 0 to 3 and indices that are Latin lowercase letters run from 1 to 3.

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1.1.1 Manifolds

One of the main ideas of general relativity is that gravity arises from the curvature of space- time, contrary to the Newtonian theory of gravity where gravity is just an intrinsic property of matter that gives rise to a gravitational force. In order to properly speak of “curvature” and

“space-time”, we have to define these concepts mathematically. Firstly, we will address the question “how is space-time described mathematically?”. It turns out that the appropriate mathematical description of space-time is done with manifolds, specifically, space-time can be represented by a Lorentzian manifold. Thus, the notion of manifolds is a centerpiece of the mathematical machinery of general relativity.

Manifolds are one of the most fundamental concepts of mathematics. Manifolds exist in many mathematical structures, but one often works with them implicitly or without knowing it. A differential manifold, M, of dimension n (for our discussion, differential manifolds are the only type of manifolds of interest so the terms “manifold” and “differential manifold” can be used interchangeably) is a topological space that everywhere, locally, looks like Rⁿ. How- ever, globally, manifolds can be curved and have a complicated topology. Maybe somewhat surprisingly, there are many mathematical objects that fits this description, a few examples are

• Rⁿ,

• the 2-sphere, S²,

• the set of all rotations of a rigid body in three dimensions, parametrized by the Euler angles (α, β, γ).

The first example is a somewhat trivial example since Rⁿ both locally and globally looks like Rⁿ because it is Rⁿ. The second example shows that manifolds, though locally resembling Rⁿ, need not globally look like Rⁿ at all. In the third example we see that more abstract mathematical structures that are not usually thought of as geometrical objects can be manifolds.

I will not give a rigorous definition of a manifold because it is not crucial to our discussion and the intuitive idea of a manifold is often more helpful than the exact mathematical definition, but it will be useful with a less formal definition, for completeness. In order to define what a manifold is we have to introduce the notion of a chart and a collection of charts called an atlas. A chart is an open subset U ∈ M with a corresponding map from U to an open ball in Rⁿ, i.e., ϕ : U → Rⁿ. An open ball with radius r centered at x₀ ∈ Rⁿ in Rⁿ means simply that it is the set of points that satisfies |x − x0| < r, x ∈ Rⁿ. The map is one-to-one and onto, which means that for all elements in U there is only one corresponding element in Rⁿ and the mapping from U to Rⁿ covers the entire open ball. A C^∞ atlas is a union of charts, {Uα, ϕα} that are smoothly “sewn together”, which is to say that, in all intersections between charts, we require that it is still C^∞. Finally, we say that a C^∞, n-dimensional, manifold M is a set with maximal atlas, i.e., it contains every possible compatible chart.

We require that it has maximal atlas because if we have two equivalent sets with different atlases, they are not taken as different manifolds. The construction of manifolds with charts

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conveys the local homeomorphism to Rⁿ rather clearly. One may ask why we need an atlas to construct the manifold, why not use one chart to describe the entire manifold? The answer is, unsurprisingly, that this is not always possible. A canonical example is the surface of the earth (as a manifestation of S²). It is not possible to find a single coordinate chart that covers the entire surface of the earth because of the coordinate ambiguity that arises at one of the poles, i.e., a chart always leaves at least one uncovered point on S². At least two charts are needed to cover the surface of the earth.

The fact that M is locally homeomorphic to Rⁿ gives us the opportunity to define differentiation on M, but with some caution. Let’s say that we have two manifolds, M and N , of dimension m and n, respectively, and a function f : M → N . To differentiate f we first define the invertible maps ϕ : M → R^m and ψ : N → Rⁿ, see Figure 1.1, and then define

∂f

∂x^µ ≡ ∂

∂x^µ ψ ◦ f ◦ ϕ⁻¹. (1.2)

Of course, it would be far too cumbersome to write derivatives as the right hand side of equation (1.2), but it should be kept in mind that this is what we mean. For brevity we write partial differentiation as

∂_µf ≡ ∂f

∂x^µ. (1.3)

M

ϕ

R^m x^µ

ψ ◦ f ◦ ϕ⁻¹

f N

ψ

Rⁿ

˜ x^µ

Figure 1.1: The mappings associated to f and ∂_µf .

Now we want to impose some structures on our manifolds. We begin by defining vectors and one-forms on the manifold, as elements of a tangent space and a cotangent space at some point p, respectively. It is important that the definition of vectors and one-forms are intrinsic to M itself and do not rely on an embedding in a manifold of higher dimension. With this in mind, it is natural to consider the set of all curves passing trough p and their tangents

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at p, i.e., the directional derivatives at p. We are going to claim that a natural basis for the tangent space, T_p(M), is given by the partial derivatives at p, {∂_µ}. To see that this is sensible we need to establish that the directional derivatives at p form a vector space, that {∂µ} constitute a basis for this vector space, and that the vector space can be identified with T_p(M).

It is obvious that the directional derivatives satisfy most axioms of a vector space (asso- ciativity of addition, commutativity of addition, etc.), but it can be conceptually difficult to realize that the directional derivatives are vectors, i.e., elements of the vector space, if one is used to thinking of vectors as arrows pointing in some direction. The idea stems from the observation that if we consider a function f and the directional derivative operator _dλ^d along a curve parametrized by λ:

d

dλ = ∂x^µ

∂λ ∂_µ (1.4)

and let _dλ^d act on f we get an expansion given by the chain rule df

dλ = ∂x^µ

∂λ ∂_µf. (1.5)

If we regard ^df_dλ as a vector we can view ^∂x_∂λ^µ as the vector components and ∂_µ as the basis vectors. There is one property that a vector space must satisfy and that is that it closes, i.e., the addition of two elements in the vector space is a new element. It is not obvious that the sum of two directional derivative operators are a new directional derivative operator. We can however show that this is the case by requiring that the differential operator we want is linear and obey the Leibniz rule. Consider two differential operators _dλ^d and _dξ^d. We let their sum _dλ^d +_dξ^d act on the product f g where f and g are functions on M:

d dλ + d

dξ

f g = d

dλf g + d

dξf g = df

dλg + dg

dλf + df

dξg + dg dξf =

= df dλ + df

dξ

g + dg dλ+ dg

dξ

f. (1.6)

So we see that the directional derivatives form a vector space and that {∂_µ} is a natural basis for it. We also see that this is the vector space associated with Tp(M) since we constructed the vector space using the directional derivatives at p.

The cotangent space T_p^∗(M) is the space dual to T_p(M), i.e., it is the space of linear maps ω : Tp(M) → R. An element of Tp^∗(M) is called a one-form, often denoted by ω. Similarly to before, we are going to claim that the gradients at p, {dx^µ} constitute an appropriate basis for T_p^∗(M). We see that this is a reasonable claim since

dx^µ∂_ν = ∂x^µ

∂x^ν = δ^µ_ν. (1.7)

An arbitrary one-form ω can be expanded as ω = ω_µdx^µ. For the rest of the discussion, we will use the notation eµ ≡ ∂µ and e^µ ≡ dx^µ, as a reminder that we are dealing with basis vectors and basis one-forms.

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1.1.2 Tensors

To proceed, we need to discuss tensors. Tensors are important in general relativity because they can be used to represent physical properties of space-time, such as distance, curvature, energy density and momentum. Mathematically speaking, a tensor is a multilinear function of vectors and one-forms. We say that a (^k_l)-tensor T takes k vectors and l one-forms and maps them onto R. As a result, T can be expanded with respect to the basis of T^p(M) and T_p^∗(M), {e_α} and {e^β}, respectively, in the following way:

T = T^α¹^α²^···α^k_β

1β2···β_le_α₁ ⊗ e_α₂ ⊗ · · · ⊗ e_α_k ⊗ e^β¹ ⊗ e^β² ⊗ · · · ⊗ e^β^l. (1.8) We see that tensors generalizes vectors and one-forms in a straightforward fashion. For example, a (⁰₀)-tensor is a scalar, a (¹₀)-tensor is a vector and a (⁰₁)-tensor is a one-form. We often denote tensors with their components, for notational brevity. We do not necessarily need to contract a (^k_l)-tensor with k vectors and l one-forms. If a (^k_l)-tensor is contracted with k⁰ one-forms and l⁰ vectors the resulting tensor is a ^k−k_l−l0⁰-tensor.

One of the most important tensors in general relativity is the metric tensor, g, with components g_µν. The metric tensor is used in general relativity to generalize the notion of distance. In general relativity we measure distance between two events using the space-time interval ds². The space-time interval can be represented by the metric in the following way:

ds² = g_µνdx^µdx^ν, (1.9)

where the notation dx^µmeans an infinitesimal displacement of x^µ. The metric tensor is useful because it equips our manifold with an inner product, i.e., we can calculate scalar products of tangent vectors using the metric tensor:

u · v ≡ g(u, v) = g_µνu^µv^ν. (1.10) There is no consensus about the sign convention when defining the metric signature but arguably the most common, and the one we choose, is the metric signature (−, +, +, +), in accordance with e.g. [6, 4, 7]. We call a metric with signature (−, +, +, +) Lorentzian.

Hence, it follows that the space-time interval is not always positive. In fact, the sign of ds² has a physical meaning and the different cases have been given names:

• if ds² > 0, we say that the events are space-like separated,

• if ds² = 0, we say that the events are light-like separated,

• if ds² < 0, we say that the events are time-like separated.

The metric can be used to identify the tangent and cotangent spaces by raising and lowering indices on other tensors, e.g.,

S^α_νµ = g_νβS^αβ_µ. (1.11)

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Another important tensor in general relativity is the stress energy tensor T^αβ. The stress energy tensor contains information about the energy and momentum density and the flux of momentum at some point in the space-time. The T_αβ component gives the flux of the αth component of the momentum vector through a surface of constant x^β. In special relativity, conservation of energy and momentum can be stated as

∂_αT^αβ = 0, (1.12)

which, as we will see later, can be extended to general relativity.

Finally, we define totally symmetric and totally anti-symmetric tensors. Given a (⁰_k)- tensor we define a totally symmetric tensor as

A_(α₁···α_k) ≡ 1 k!

X

p

A_p(α₁_)···p(α_k₎ (1.13)

and a totally anti-symmetric tensor as A_[α₁···α_k] ≡ 1

k!

X

p

π_pA_p(α₁_)···p(α_k₎, (1.14)

where p are all the permutations of the indices, and

π_p = +1 if p is an even permutation,

−1 if p is an odd permutation. (1.15) As a further example, consider A^[αβ]λ_(µν):

A^[αβ]λ_(µν)= 1

4[A^αβλ_µν + A^αβλ_νµ− A^βαλ_µν− A^βαλ_νµ]. (1.16) 1.1.3 Curvature

Having constructed vectors, one-forms, and tensors and endowed our manifold M with a metric g_µν we can give a mathematical description of curvature. The notion of curvature starts with the idea of a connection on the manifold. The connection gives us a way to define parallel transport of vectors and covariant derivatives, which is something we need in order to talk about curvature. The covariant derivative generalizes the partial derivatives ∂_µ discussed earlier. In flat spaces we require that the covariant derivative reduces to the partial derivative. The need of a covariant derivative arises from the problem that in some general coordinate system and for manifolds with curvature, the basis vectors themselves change, i.e., have a non-vanishing derivative (for example polar coordinates). We can thus conclude that the form of the covariant derivative should be “partial derivative + derivative of basis vectors”. If we find a collection of numbers such that

∇_αe_β = Γ^µ_αβe_µ, (1.17)

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we can write the covariant derivative of a vector v^α as

∇_βv^α = ∂_βv^α+ v^µΓ^α_µβ, (1.18) where the objects Γ^α_µβare called the Christoffel symbols or connection coefficients. In general relativity, we require that the covariant derivative is compatible with the metric, that is

∇_µg_αβ = 0. (1.19)

It follows from the above relation that the Christoffel symbols can be written in the following way

Γ^α_µν = 1

2g^αβ(∂_νg_µβ + ∂_µg_βν − ∂_βg_µν). (1.20) The covariant derivative of some vector field v along a vector u is denoted by ∇_uv and has components u^µ∇_µv^ν. Parallel transport of v along u is defined by u^µ∇_µv^ν = 0, which captures that v is constant along u. There exists curves on M where v is parallel transported along itself, i.e.,

∇_vv = 0 (1.21)

and those paths are called geodesics. Expressed in components equation (1.21) takes the form

v^µ∇µv^ν = v^µ∂µv^ν + v^µv^αΓ^ν_αµ= 0. (1.22) If we let λ be a parameter of the curve, we have from equations (1.4) and (1.5) that

v^µ= ∂x^µ

∂λ , v^µ∂_µ = ∂x^µ

∂λ ∂_µ= d

dλ. (1.23)

Equations (1.22) and (1.23) can be combined to obtain the geodesic equation d²x^µ

dλ² + Γ^µ_ανdx^α dλ

dx^ν

dλ = 0. (1.24)

Now that we have some sense of geodesics and parallel transport, we can introduce a mathematical description of curvature. Imagine that we have some infinitesimal closed loop on our manifold and parallel transport a vector along that loop, see Figure 1.2. A reasonable measurement of the curvature at the locus of the loop is to consider how much the vector has changed from its original state after one revolution.

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(0, 0)

A^µ

(δa, 0)

B^ν

(δa, δb)

A^µ (0, δb)

B^ν

Figure 1.2: Parallel transport around an infinitesimal loop on M.

Parallel transporting v along this loop amounts to the following transformation

δv^α = [(δa)A^µ∇_µ(δb)B^ν∇_ν − (δb)B^ν∇_ν(δa)A^µ∇_µ]v^α = (1.25)

= (δa)(δb)A^µB^ν[∇_µ∇_ν − ∇_ν∇_µ]v^α =

= (δa)(δb)A^µB^ν[∇_µ, ∇_ν]v^α. (1.26)

There is a tensor that governs this transformation, in the following way

δv^α = (δa)(δb)A^µB^νR^α_βµνv^β, (1.27) and we can identify

[∇_µ, ∇_ν]v^α ≡ R^α_βµνv^β. (1.28) R^α_βµν is a (¹₃)-tensor and is known as the Riemann tensor or the curvature tensor. As we have tried to state, the Riemann tensor encodes information about the curvature of M. If the manifold is flat (i.e., the manifold has no curvature) we have that R^α_βµν = 0. The Riemann tensor obeys some important tensor identities, namely, anti-symmetry when exchanging the first and second pair of indices and symmetry when exchanging both pairs:

g_αλR^λ_βµν = R_αβµν = −R_βαµν = −R_αβνµ = R_µναβ, R_αβµν+ R_ανβµ+ R_αµνβ = 0. (1.29) The contraction of the Riemann tensor on the first and third index is another tensor called the Ricci tensor

R^µ_αµν ≡ R_αβ = R_βα. (1.30)

We can also consider the contraction of the Ricci tensor:

R^µ_µ = g^µνR_µν ≡ R, (1.31)

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which is called the Ricci scalar. Both these definitions will be important when formulating Einstein’s equations. Lastly, another very important identity that the Riemann tensor obeys is the Bianchi identity:

∇_λR_αβµν+ ∇_νR_αβλµ+ ∇_µR_αβνλ = 0. (1.32) An important form of the Bianchi identity can be found when contracting (1.32) twice and using identities (1.29), (1.30) and (1.31):

0 = g^βνg^µα(∇_λR_αβµν + ∇_νR_αβλµ+ ∇_µR_αβνλ) =

= g^βνg^µα(∇_λR_αβµν − ∇_νR_αβµλ− ∇_µR_βανλ) =

= g^βν(∇_λR^µ_βµν

| {z }

≡R_βν

−∇_νR^µ_βµλ

| {z }

≡R_βλ

−∇^αR_βανλ) =

= ∇_λR − ∇^βR_βλ− ∇^αR^ν_ανλ

| {z }

≡R_αλ

= ∇_λR − 2∇^ρR_ρλ ⇒

⇒ ∇^ρR_ρλ = 1

2∇_λR. (1.33)

Equation (1.33) leads us to define a new tensor, the Einstein tensor, G_αβ ≡ R_αβ− 1

2g_αβR (1.34)

with the property

∇^αGαβ = 0. (1.35)

1.2 Einstein’s Equations

The formulation of Einstein’s equations begins by the observation that the physical laws governing falling objects contained in a small enough region of space-time reduce to those of special relativity, that is, one cannot distinguish a gravitational acceleration from a uniform acceleration caused by something else. This is called the equivalence principle. We add the restriction “in a small enough region of space-time” because if the region is too big gravitational tidal forces may give a possibility to detect the gravitational field. The equivalence principle gives rise to the idea that gravity should not be considered a force but a manifestation of the curvature of space-time. It is often said that matter and energy tells space-time how to curve and space-time tells matter how to move. In this spirit, it is tempting to write an equation in this fashion (see equation (1.12))

R_αβ = κ²T_αβ, (1.36)

however, local energy and momentum conservation must be satisfied in the following manner

∇_µT^µν = 0. (1.37)

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As we have seen in equation (1.33), it is clearly not generally true that ∇µR^µν = 0. On the other hand, the Einstein tensor obeys this condition, which leads us to believe that

G_αβ = κ²T_αβ. (1.38)

This guess happens to be true. The constant κ² = 8πG/c⁴ where G is Newton’s gravitational constant while c is the speed of light. Here we choose units such that 8πG = c = 1 which results in

G_αβ = T_αβ. (1.39)

The above equations are called the Einstein field equations. An alternative equivalent way of writing the equations is

R_αβ = T_αβ −1

2g_αβT, (1.40)

where T ≡ T^µ_µ is the trace of the stress energy tensor. It will turn out later that we need to add a constant to this equation called the cosmological constant, which will be discussed later. Equation (1.39) then reads

G_αβ + Λg_αβ = T_αβ. (1.41)

1.2.1 Source Terms

As previously stated, the sources of a gravitational field are matter and energy and information about the matter and energy are encoded in the stress energy tensor T_µν. Since matter and energy comes in different forms (solids, fluids, radiation, etc.) we have to specify the stress energy tensor further, i.e., we have to find the stress energy tensor for the particular system we are interested in. In cosmology, we are primarily interested in two types of source terms of the gravitational field, a perfect fluid and a scalar field.

A perfect fluid is defined as a gas of non-interacting particles. A perfect fluid has no heat conduction, no viscosity and is characterized by its rest frame (mass and energy) density ρ and its isotropic pressure p. The stress energy tensor of a perfect fluid then takes the form

T_µν = (ρ + p)u_µu_ν + pg_µν, (1.42) where u^µis the four velocity of the fluid. Perfect fluids are quite useful in cosmology and can be used to model matter on large scales of the universe with a barotropic equation of state, i.e., p = p(ρ). In the most conventional barotropic models we have p = wρ where w = const., e.g.,

w ≤ 0 dust,

1

3ρ radiation. (1.43)

We discuss perfect fluids in more detail in the section about cosmology.

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A scalar field φ has a stress energy tensor of the form [8]

T_µν = ∇_µφ∇_νφ − 1

2∇_λφ∇^λφ + V (φ)

g_µν, (1.44)

where the scalar field potential V (φ) is specific to the system under consideration. A special case that is of interest in cosmology is when ∇_µφ is time-like and we can define a vector field u normal to the levels surfaces of φ, i.e., surfaces of φ = const. in the following way

u^µ= ∇^µφ

p−(∇_νφ)(∇^νφ). (1.45)

If we denote

ρ = −1

2∇_µφ∇^µφ + V (φ), p = −1

2∇_µφ∇^µφ − V (φ), (1.46) we see that equation (1.44) takes the algebraic form of the stress energy tensor for a perfect fluid, equation (1.42). Demanding local energy-momentum conservation (∇_νT^µν = 0), equation (1.44) gives, the Klein-Gordon equation

∇µ∇^µφ − V⁰(φ) = 0. (1.47)

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2 The 3+1 Formalism

The 3+1 formalism relies on the splitting of space-time into space + time, i.e., slicing space- time, equipped with a Lorentzian metric (with signature (−, +, +, +)), into hypersurfaces and treating time as a separate coordinate. We require that the hypersurfaces are space- like so that we can induce a Riemannian metric (with signature (+, +, +)) on them. The requirement that the hypersurfaces are space-like means that we have a globally hyperbolic space-time, which we assume is true for any physically realizable system (a globally hyperbolic space-time has no closed time-like curves, i.e, you cannot travel to the past).

There are several reasons as to why the 3+1 formalism is of interest in general relativity.

The 3+1 formalism is useful when we want to rewrite Einstein’s equations as an initial value problem, i.e., a Cauchy problem. There are also several advantages of having a 3+1 approach to cosmological perturbation theory, namely

• the close connection to initial data problems in general relativity,

• 3+1 is the dominating approach in numerical relativity,

• a 3+1 space-time splitting is closely related to Newtonian gravity and is useful when connecting relativistic dynamics with Newtonian examinations of large scale structure forming,

• it simplifies writing the main kinematic objects in general relativity in a form that is suitable for performing perturbations of Robertson-Walker geometries to arbitrary order.

In this discussion of the 3+1 formalism we mainly follow [9] and [10].

2.1 Notation and Framework

We assume that we have a space-time characterized by (M, g), where M is a 4-dimensional differentiable manifold (C^∞) and g is a Lorentzian metric. We also assume that (M, g) is time-orientable, i.e., it is possible to continuously divide (M, g) into the past and the future. We denote the connection on M by ∇. It should be noted that we will introduce other connections so we call ∇ the “space-time connection” to distinguish it from other connections. As stated earlier, the inner product on M of any two vectors u and v is given by

u, v ∈ T_p(M), u · v = g(u, v) = g_µνu^µv^ν. (2.1) Similarly, the action of one-forms on vectors will be denoted as

ω ∈ T_p^∗(M), v ∈ T_p(M), h ω, v i = ω_αv^α. (2.2)

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2.2 Hypersurfaces

The hypersurfaces we will discuss are 3-dimensional surfaces embedded in a 4-dimensional space. Hypersurfaces are of interest to us because in the framework of the 3+1 formalism they will be introduced as surfaces of local simultaneity of a certain type of observer in the space-time. In this section a brief explanation of hypersurfaces is given and the mathematical tools needed to make a 3+1 splitting of space-time is presented.

2.2.1 Hypersurfaces Embedded in Space-time

A hypersurface is a 3-dimensional manifold which we will denote by ˆΣ. The image of ˆΣ by an embedding Φ in M will be called Σ, i.e., the embedding Φ is such that

Σ = Φ( ˆΣ). (2.3)

The embedding (2.3) is one-to-one and such that Φ and Φ⁻¹ is continuous. This assures that Σ does not self-intersect and can be described as a surface of some constant value of a scalar field t in M:

∀p ∈ M, p ∈ Σ ⇐⇒ t(p) = 0. (2.4)

We can relate the tangent spaces of ˆΣ and M at p with the push-forward mapping Φ∗: Φ∗ : Tp( ˆΣ) −→ Tp(M)

v 7−→ Φ∗v. (2.5)

Similarly, the pull-back mapping Φ^∗ relates the cotangent spaces of ˆΣ and M in the following fashion:

Φ^∗ : T_p^∗(M) −→ T_p^∗( ˆΣ)

ω 7−→ Φ^∗ω. (2.6)

Recall that one-forms ω are linear functions of vectors v and the equation (2.6) can be generalized to pull-backs of multilinear functions of vectors. An important example of this is the induced metric on Σ denoted by h_ij:

h ≡ Φ^∗g. (2.7)

From the definition above, we see that the components of h is

h_ij = g_ij. (2.8)

Since we have the restriction that the hypersurfaces are space-like, the signature of h_ij is (+, +, +).

A normal vector to Σ can be defined using the scalar field t and considering Σ as a level surface of t. The gradient ~∇t (an arrow indicate contravariant components, i.e., in the present case ~∇t has components ∇^µt) is normal to Σ in the sense that

g_µνv^ν∇^µt = v^µ∇_µt = 0, v ∈ T_p(Σ). (2.9)

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We define the unit normal vector to Σ as

n ≡ 1

p− ~∇t · ~∇t

∇t~ (2.10)

which has the components,

n^µ = ∇^µt

p−(∇^µt)(∇_µt). (2.11)

The restriction that the hypersurfaces are space-like gives that ~∇t and n are time-like and that n · n = n^µn_µ= −1.

2.2.2 Intrinsic curvature

The intrinsic curvature of a hypersurface Σ is the corresponding curvature of Σ to that of the “ordinary” curvature of M described by R^α_βµν. Since the hypersurface is a Riemannian manifold, it has a torsion free connection which we will denote by D with the property

Dh = 0, (2.12)

where h is the induced metric on Σ, cf. equation (2.7). The Riemann tensor of (Σ, h) describes what we call the intrinsic curvature Σ. The components of this Riemann tensor are given by

v ∈ T_p(Σ), (D_iD_j− D_jD_i)v^k =³R^k_lijv^l, (2.13) where we write a superscript “³ ” to remind us that this Riemann tensor belongs to (Σ, h).

The corresponding Ricci tensor and Ricci scalar are given in the same way as before:

3R_ij ≡³R^k_ikj, ³R ≡ h^{ij 3}R_ij. (2.14)

3R is often called the Gaussian curvature of (Σ, h).

2.2.3 Extrinsic Curvature

The extrinsic curvature of a hypersurface is a measurement of how the hypersurface bends and curves with respect to the manifold in which it is embedded. In our case, the extrinsic curvature of Σ depends on how it curves in M. More specifically stated, the extrinsic curvature is related to the change of direction of n as one travels on Σ. We can describe this mathematically by associating to each tangent vector of Σ the change of the normal vector along the tangent vector:

v ∈ T_p(Σ), χ : v → ∇_vn, (2.15)

where χ(v) is called the Weingarten map or the shape-operator. We see that χ(v) ∈ Tp(Σ) since

n · χ(v) = n · ∇_vn = 1

2∇_v(n · n) = 0. (2.16)

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An important property of the Weingarten map is that it is self-adjoint in the following manner

u · χ(v) = v · χ(u). (2.17)

The Weingarten map being self-adjoint assures that the tensor K defined by

K(u, v) ≡ −u · χ(v) = −u · ∇_vn, u, v ∈ T_p(Σ) (2.18) is symmetric with respect to its arguments, i.e., K(u, v) = K(v, u). Equation (2.18) gives a definition of what we call the extrinsic curvature tensor of Σ, which has the following components:

K_ij ≡ K(e_i, e_j) = −(∇_βn_α)(e_i)^α(e_j)^β. (2.19) 2.2.4 The Orthogonal Projector

At each point p in space-time, the tangent space of M can be decomposed in the following way

T_p(M) = T_p(Σ) ⊕ Vect(n), (2.20)

where Vect(n) is the 1-dimensional subspace of M spanned by n. This decomposition relies on the condition that our hyperspaces are space-like. An orthogonal projector onto T_p(Σ) associated to this decomposition can be defined as

~h : Tp(M) −→ Tp(Σ)

v 7−→ v + (v · n) n, (2.21)

where the components of ~h are

h^α_β = δ^α_β+ n^αn_β. (2.22)

Two obvious but important properties of ~h is h(n) = 0,~

h(v) = v,~ ∀ v ∈ T_p(Σ). (2.23)

The usefulness of (2.21) comes from the fact that it gives us a mapping T_p(M) → T_p(Σ).

Recall that the push-forward, equation (2.5), only provided a mapping T_p(Σ) → T_p(M) and the pull-back, equation (2.6), provided the mapping T_p^∗(M) → T_p^∗(Σ). We also see that the orthogonal projector makes it possible to define a map such that

~h^∗ : T_p^∗(Σ) −→ T_p^∗(M)

ω 7−→ ω + h ω, n i n, (2.24)

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where n is the covector dual to n. The components of the map (2.24) is obtained by using equation (2.22)

h^α_βω_α = δ^α_βω_α+ n^αn_βω_α = ω_β + n^αω_αn_β. (2.25) We see that acting with the orthogonal projector on a one-form in the cotangent space of Σ gives it a component orthogonal to T_p^∗(Σ) and it becomes a one-form of T_p^∗(M). The map (2.24) can be extended to act on multilinear forms and an important case of this when we act with it on the induced metric h:

h ≡ ~h^∗h = g + n ⊗ n. (2.26)

We denote this operator by the same symbol as the induced metric because it is a bilinear form on Σ and serves the same role as the induced metric. Expressed in components, equation (2.26) is given by

h_αβ = g_αβ + n_αn_β. (2.27)

We will use equation (2.27) extensively in a later section to obtain 3+1 compositions of tensors.

The extrinsic curvature tensor K is a priori a bilinear form on Σ and we can use the map (2.24) to extend it to M:

K ≡ ~h^∗K. (2.28)

We can extend this operation to any (^k_l)-tensor T on Σ that we may want to extend to M, denoted as ~h^∗T . The component of such an operation is given by

(~h^∗T )^α¹^···α^k_β

1···β_l = h^α¹_µ₁· · · h^α^k_µ_kh^ν¹_β

1· · · h^ν^l_β

lT^µ¹^···µ^l_ν₁_···ν_k. (2.29) In particular, the covariant derivative of T on Σ, DT can be written as

DT = ~h^∗∇T (2.30)

and thus has the components D_ρT^α¹^···α^k_β

1···β_l = h^α¹_µ₁· · · h^α^k_µ_kh^ν¹_β

1· · · h^ν^l_β

lh^σ_ρ∇_σT^µ¹^···µ^l_ν₁_···ν_k. (2.31) 2.2.5 The Relation between K and ∇n

Since the unit normal vector n has the property n · n = −1 it is reasonable to interpret it as the 4-velocity of some observer. The corresponding 4-acceleration would then be written as

a ≡ ∇_nn. (2.32)

Note that a ∈ T_p(Σ), since n · a = n · ∇_nn = (1/2)∇_n(n · n) = 0.

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Explicit calculation of the extrinsic curvature extended to M, given by equation (2.28), results in

K(u, v) = K(~h(u), ~h(v)) = −~h(u) · ∇~h(v)n = −[u + (u · n) n] · ∇_{v+(v·n) n}n =

= −[u + (u · n) n] · [∇vn + (v · n)∇nn] = −u · ∇vn − (v · n)(u · a) =

= ∇n(u, v) − h v, n ih u, a i. (2.33)

Since equation (2.33) is valid for any pair of vectors u and v, we can reason that

∇n = −K − a ⊗ n, (2.34)

which in components are

∇_αn_β = −K_αβ− n_αa_β. (2.35)

Equations (2.34) and (2.35) establishes the relationship between K and ∇n. By taking the trace of (2.34) with respect to g (contracting (2.35) with g^αβ) we obtain

g^αβ∇_αn_β = −g^αβK_αβ− g^αβa_αn_β = −K^α_α =⇒ K = −∇ · n, (2.36) where we denote the trace K ≡ K^α_α.

2.3 Foliations

In the previous section we discussed the properties and mathematical structure of a single hypersurface Σ. Now we consider a continuous set of hypersurfaces Σ_t that together covers the entire manifold M. The foliation of M requires that we have a globally hyperbolic space- time and as we have discussed earlier, this restriction still comprises most physical situations in cosmology or astrophysics that we are interested in anyway.

2.3.1 Definition of a Foliation

We maintain the idea that each hypersurface Σ_t is a level surface of a scalar field t which is regular, which assures that the hypersurfaces are non-intersecting, i.e.,

∀ t ∈ R, Σ_t≡ {p ∈ M, ˆt(p) = t} (2.37) and

Σ_t∩ Σ_t⁰ = ∅ for t 6= t⁰. (2.38) Each hypersurface Σ_t of the foliation is called a leaf or a slice. We assume that the foliation covers all of M, see Figure 2.1a:

M = [

t∈R

Σt. (2.39)

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2.4 Foliation Kinematics

The kinematics of the foliation are determined by the 3-dimensional leaves Σ_t and the infinitesimal neighboring leaf Σt+dtand the 4-dimensional space that fills the space between the leaves. In [6] and [10] it is discussed what is needed to give the foliation structure “rigidity”.

What is needed can be summarized in the following way:

• The metric h_ij on the space-time slices in order to measure proper distances.

• The lapse of proper time between the slices given by dτ = N dt, where N is the lapse function.

• The relative velocity of observers traveling normal to the leaves (Eulerian observers) and the lines corresponding to constant spatial coordinates xⁱ, given by xⁱ_t+dt = xⁱ_t− βⁱdt, where β is known as the shift vector.

See Figure 2.1b. It is worthwhile to discuss some of these concepts in a bit more detail.

2.4.1 Lapse Function

We have seen before that the unit normal vector to Σt n, defined in equation (2.10), is time-like and have the property n · n = −1. We may write

n = −N ~∇t, (2.40)

or in components

n^µ = −N ∇^µt, (2.41)

(a) The foliation of M. (b) The shift vector and lapse function.

Figure 2.1: 3-dimensional representations of the foliation, normal vector, lapse and shift vector.

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where N is known as the “lapse function”, coined by Wheeler [6], and given by

N ≡ 1

p− ~∇t · ~∇t

= 1

p−(∇^µt)(∇_µt). (2.42)

This ensures that n is a unit vector. The lapse function carries information about the proper distance between the leaves Σ_t and Σ_t+dt.

2.4.2 Eulerian Observers

The property of the normal vector n that n · n = −1 suggests that we may view n as the 4-velocity of some observer. These observers are called Eulerian observers and their world lines are always perpendicular to the foliation leaves. This means that the leaves are the surfaces of simultaneity for these observers. Eulerian observers should be contrasted with Lagrangian observers, which are observers that are moving with respect to some reference frame in space-time along a time-like congruence. Lagrangian observers are often called comoving observers.

The 4-acceleration of a set of Eulerian observers is defined by

a ≡ ∇_nn. (2.43)

We can also express the acceleration in terms of the spatial connection D and the lapse N in the following way

a_α= n^µ∇_µn_α = −n^µ∇_µ(N ∇_αt) = −n^µ∇_µN ∇_αt − n^µN ∇_µ∇_αt

| {z }

∇α∇µt

=

= 1

Nn_αn^µ∇_µN − N n^µ∇_α

−1 Nn_µ

= 1

Nn_αn^µ∇_µN − 1

Nn^µ(∇_αN )n_µ+ n^µ∇_αn_µ

| {z }

=0

=

= 1

Nn_αn^µ∇_µN − 1

N(∇_αN ) n_µn^µ

| {z }

=−1

= 1

N(∇_αN + n_αn^µ∇_µN ) =

= 1

N(δ_α^µ+ n_αn^µ)∇_µN = 1

Nh_α^µ∇_µN = 1

ND_αN = D_αln N. (2.44)

Thus, we have that the acceleration for Eulerian observers can be written as

a = D ln N. (2.45)

2.4.3 Shift Vector

The foliation of space-time is in general such that the normal unit vector n does not coincide with e₀ =: t. This only happens if the lines xⁱ = const. are perpendicular to the leaves Σ_t, which of course is not always the case. The shift vector β relates the coordinates to the normal vector of Σ_t in the following way

t = N n + β, (2.46)

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which has components

t^µ = N n^µ+ β^µ. (2.47)

As we saw earlier, we need the shift vector to describe the motion of Eulerian observers in space-time with respect to the foliation and the coordinates we have imposed on M. From the construction of the shift vector we see that it obeys

n · β = 0, (2.48)

that is, β is tangent to Σ_t.

2.4.4 Normal Evolution Vector

Let us define the normal evolution vector as follows

m ≡ N n. (2.49)

m is thereby a time-like vector with the properties

m · m = −N², and ∇_mt = m^µ∇_µt = 1. (2.50) The usefulness of the normal evolution vector is that it Lie drags the hypersurfaces Σt, i.e., it carries Σ_tto the neighboring leaf Σ_t+dt. As a consequence, we have that the Lie derivative along m, defined in Appendix A.1, of any vector tangent to Σ_t will remain tangent:

∀v ∈ T_p(M), £_mv ∈ T_p(M). (2.51)

2.4.5 Gradients of m and n

From equations (2.34) and (2.45) we obtain

∇n = −K − n ⊗ D ln N, (2.52)

which has components

∇_αn_β = −K_αβ − n_αD_βln N. (2.53) The gradient of m is obtained using that

∇m = ∇(N n) = ∇N ⊗ n + N ∇n. (2.54)

Combining the above relations we get

∇m = −N K − N n ⊗ D ln N + ∇N ⊗ n = −N K − n ⊗ DN + ∇N ⊗ n, (2.55) which has components

∇_αm_β = −N K_αβ− n_αD_βN + (∇_αN )n_β, (2.56) or with one index raised:

∇_αm^β = −N K_α^β − n_αD^βN + (∇_αN )n^β. (2.57)

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2.4.6 Evolution of The 3-metric

The evolution of the 3-metric, h_µν is naturally given by the Lie derivative along m. We exploit the fact that the Lie derivatives can be expressed using any torsion free connection and thus use (A.7) and (2.56) to calculate

£mhαβ = m^µ∇µhαβ + hµβ∇αm^µ+ hαµ∇βm^µ =

= N n^µ∇_µ(n_αn_β) − h_µβ(N K^µ_α+ n_αD^µN − (∇_αN )n^µ) −

− h_αµ(K^µ_β+ n_βD^µN − (∇_βN )n^µ) =

= N ( n^µ∇_µn_α

| {z }

=N⁻¹DαN

n_β + n_α n^µ∇_µn_β

| {z }

=N⁻¹DβN

) − N K_βα− n_αD_βN − N K_αβ − n_βD_αN =

= −2N K_αβ. (2.58)

Thus

£_mh = −2N K. (2.59)

2.5 3+1 Splitting of the Metric

The metric on M with respect to some coordinate system {e_α}, is given by the expansion

g = g_αβe^α⊗ e^β. (2.60)

We can obtain each component by calculating

g_αβ = g(e_α, e_β). (2.61)

Using that e₀ = ∂_t and equation (2.46), the g₀₀-component thus becomes

g₀₀ = g(e₀, e₀) = −N²+ β_kβ^k. (2.62) The g_i0-component is

gi0 = g(ei, e0) = ei· (N n + β) = βi, (2.63) by virtue of e_i· n = 0 since e_i ∈ T_p(Σ_t). For symmetry reasons, g_i0 = g_0i. Also, since e_i is tangent to Σt we have

g_ij = g(e_i, e_j) = h_ij, (2.64) where h_ij is the metric on Σ_t. As a consequence, g_µν is in matrix form:

g_µν =−N²+ β_kβ^k β_j β_i h_ij

, (2.65)

while the inverse metric g^µν is

g^µν = 1 N²

−1 β^j

βⁱ N²h^ij − βⁱβ^j

. (2.66)

Note that, in general, g^ij 6= h^ij.

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2.6 Various 3+1 Splittings of the Riemann Tensor

When considering the splittings and projections of the Riemann tensor R^α_βµν associated with (M, g) we obtain results that will be useful when deriving the 3+1 splittings of the Einstein tensor, for a more in depth discussion consult [9].

2.6.1 Gauss Relation

One obtains the Gauss relation when considering a total projection of the Riemann tensor onto the hypersurface, Σt, by means of acting on every index with h^µ_ν:

h^µ_αh^ν_βh^γ_ρh^σ_δR^ρ_σµν =³R^γ_δαβ+ 2K^γ_[αK_β]δ. (2.67) We can contract the above relation on indices γ and α and obtain

h^µ_αh^ν_βR_µν+ h_αµn^νh^ρ_βR^µ_νρσ =³R_αβ + KK_αβ− K_αµK^µ_β, (2.68) called the contracted Gauss relation. Lastly, by taking the trace with respect to h we obtain the scalar Gauss relation

R + 2R_µνn^µn^ν =³R + K² − K_ijK^ij. (2.69) 2.6.2 Codazzi Relation

We apply the Riemann tensor to the unit normal vector n^γ and use the Ricci identity (1.28):

(∇_α∇_β− ∇_β∇_α)n^γ = R^γ_µαβn^µ. (2.70) As before we consider a projection of the above relation onto Σtwhich once again is performed using the orthogonal projector on each index

h^µ_αh^ν_βh^γ_ρR^ρ_σµνn^ν = h^µ_αh^ν_βh^γ_ρ(∇_µ∇_ν− ∇_ν∇_µ)n^ρ. (2.71) By using equation (2.34) and other relations discussed earlier to rewrite the above relation, called the Codazzi relation, we obtain

h^γ_ρn^σh^µ_αh^ν_βR^ρ_σµν = 2D_[βK^γ_α]. (2.72) Contracting the above relation on the indices γ and α we obtain

h^γ_αn^νR_µν = D_αK − D_µK^µ_α, (2.73) which is called the contracted Codazzi relation.

1.1 The Mathematics of General Relativity

3+1 Approach to

Cosmological Perturbations

Contents

1 Introduction

1.1 The Mathematics of General Relativity

1.2 Einstein’s Equations

2 The 3+1 Formalism

2.1 Notation and Framework

2.2 Hypersurfaces

2.3 Foliations

2.4 Foliation Kinematics

2.5 3+1 Splitting of the Metric

2.6 Various 3+1 Splittings of the Riemann Tensor