Classical Field Theory

Classical Field Theory

Uwe-Jens Wiese

Institute for Theoretical Physics

Bern University

May 25, 2009

2

Contents

1 Introduction 5

1.1 The Cube of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Electrodynamics and Relativity . . . . . . . . . . . . . . . . . . . . 8

1.4 General Relativity, Cosmology, and Inflation . . . . . . . . . . . . . 9

2 Relativistic Form of Electrodynamics 11

2.1 Euclidean Space and Minkowski Space-Time . . . . . . . . . . . . . 11

2.2 Gradient as a 4-Vector and d’Alembert Operator . . . . . . . . . . 16

2.3 4-Vector Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 4-Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Field Strength Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Inhomogeneous Maxwell Equations . . . . . . . . . . . . . . . . . . 21

2.7 Homogeneous Maxwell Equations . . . . . . . . . . . . . . . . . . . 22

2.8 Space-Time Scalars from Field Strength Tensors . . . . . . . . . . 24

2.9 Transformation of Electromagnetic Fields . . . . . . . . . . . . . . 25

2.10 Electromagnetic Field of a Moving Charge . . . . . . . . . . . . . . 26

2.11 The Relativistic Doppler Effect . . . . . . . . . . . . . . . . . . . . 27

3

4 CONTENTS

2.12 Relativistic Particle in an Electromagnetic Field . . . . . . . . . . 29

2.13 From Point Mechanics to Classical Field Theory . . . . . . . . . . 31

2.14 Action and Euler-Lagrange Equation . . . . . . . . . . . . . . . . . 34

2.15 Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . 35

3 Basics of General Relativity 37

3.1 Spaces of Constant Curvature . . . . . . . . . . . . . . . . . . . . . 37

3.2 Geodesics and Metric Connection . . . . . . . . . . . . . . . . . . . 41

3.3 Parallel Transport and Riemann Tensor . . . . . . . . . . . . . . . 43

3.4 Ricci and Einstein Tensors and Curvature Scalar . . . . . . . . . . 44

3.5 Curved Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Newton-Cartan Curved Space-Time . . . . . . . . . . . . . . . . . 45

3.7 Einstein’s Curved Space-Time . . . . . . . . . . . . . . . . . . . . . 46

3.8 Einstein’s Gravitational Field Equation . . . . . . . . . . . . . . . 47

3.9 The Einstein-Hilbert Action . . . . . . . . . . . . . . . . . . . . . . 48

4 Standard Big Bang Cosmology 49

4.1 The Friedmann-Lemaitre-Robertson-Walker Metric . . . . . . . . . 50

4.2 Cosmological Red-Shift and Hubble Law . . . . . . . . . . . . . . . 51

4.3 Solutions of the Field Equations . . . . . . . . . . . . . . . . . . . 54

4.4 Four Possible Eras in the Late Universe . . . . . . . . . . . . . . . 60

5 The Inflationary Universe 63

5.1 Deficiencies of Standard Cosmology . . . . . . . . . . . . . . . . . . 65

5.2 The Idea of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 The Dynamics of Inflation . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 1

Introduction

In this chapter we give a brief introduction to classical field theory and we relateit to current problems in modern physics.

1.1 The Cube of Physics

In order to orient ourselves in the space of physics theories let us consider whatone might call the “cube of physics”. Classical field theory — the subject of thiscourse — has its well-deserved place in the space of all theories. Theory spacecan be spanned by three axes labelled with the three most fundamental constantsof Nature: Newton’s gravitational constant

G = 6.6720 × 10−11kg−1m3sec−2, (1.1.1)

the velocity of light

c = 2.99792456 × 108m sec−1, (1.1.2)

(which would deserve the name Einstein’s constant), and Planck’s quantum

h = 6.6205 × 10−34kg m2sec−1. (1.1.3)

Actually, it is most convenient to label the three axes by G, 1/c, and h. Fora long time it was not known that light travels at a finite speed or that thereis quantum mechanics. In particular, Newton’s classical mechanics correspondsto c = ∞ ⇒ 1/c = 0 and h = 0, i.e. it is non-relativistic and non-quantum,and it thus takes place along the G-axis. If we also ignore gravity and put

5

6 CHAPTER 1. INTRODUCTION

G = 0 we are at the origin of theory space doing Newton’s classical mechanicsbut only with non-gravitational forces. Of course, Newton realized that in NatureG 6= 0, but he couldn’t take into account 1/c 6= 0 or h 6= 0. Maxwell and theother fathers of electrodynamics had c built into their theory as a fundamentalconstant, and Einstein realized that Newton’s classical mechanics needs to bemodified to his theory of special relativity in order to take into account that1/c 6= 0. This opened a new dimension in theory space which extends Newton’sline of classical mechanics to the plane of relativistic theories. When we takethe limit c → ∞ ⇒ 1/c → 0 special relativity reduces to Newton’s non-relativistic classical mechanics. The fact that special relativity replaced non-relativistic classical mechanics does not mean that Newton was wrong. Indeed, histheory emerges from Einstein’s in the limit c → ∞, i.e. if light would travel withinfinite speed. As far as our everyday experience is concerned this is practicallythe case, and hence for these purposes Newton’s theory is sufficient. There isno need to convince a mechanical engineer to use Einstein’s theory because herairplanes are not designed to reach speeds anywhere close to c. Once specialrelativity was discovered, it became obvious that there must be a theory thattakes into account 1/c 6= 0 and G 6= 0 at the same time. After years of hardwork, Einstein was able to construct this theory — general relativity — whichis a relativistic theory of gravity. The G-1/c-plane contains classical (i.e. non-quantum) relativistic and non-relativistic physics. Classical electrodynamics andgeneral relativity are perfectly consistent with one another. They are the mostfundamental classical field theories and the main subject of this course.

A third dimension in theory space was discovered by Planck who startedquantum mechanics and introduced the fundamental action quantum h. Whenwe put h = 0 quantum physics reduces to classical physics. Again, the existenceof quantum mechanics does not mean that classical mechanics is wrong. It is,however, incomplete and should not be applied to the microscopic quantum world.In fact, classical mechanics is entirely contained within quantum mechanics asthe limit h → 0, just like it is the c → ∞ limit of special relativity. Quantummechanics was first constructed non-relativistically (i.e. by putting 1/c = 0).When we allow h 6= 0 as well as 1/c 6= 0 (but put G = 0) we are doing relativisticquantum physics. This is where the quantum version of classical electrodynamics— quantum electrodynamics (QED) — is located in theory space. Also the entirestandard model of elementary particle physics which includes QED as well as itsanalog for the strong force — quantum chromodynamics (QCD) — is locatedthere. Today we know that there must be a consistent physical theory thatallows G 6= 0, 1/c 6= 0, and h 6= 0 all at the same time. However, this theoryof relativistic quantum gravity has not yet been found, although there are somepromising attempts using string theory.

1.2. FIELD THEORY 7

1.2 Field Theory

Unlike the weak and strong nuclear forces which play a role only at distances asshort as 1 fm = 10−15 m, gravity and electromagnetism manifest themselves atmacroscopic scales. This implies that, while the weak and strong nuclear forcesmust be treated quantum mechanically, gravity and electromagnetism can alreadybe investigated using classical (i.e. non-quantum) physics. The investigation ofclassical electrodynamics has led to the introduction of a powerful fundamen-tal concept — the concept of fields. It has been discovered experimentally that“empty” space is endowed with physical entities that do not manifest themselvesas material particles. Such entities are e.g. the electric and magnetic field. Eachpoint in space and time has two vectors ~E(~x, t) and ~B(~x, t) attached to it. Theelectric and magnetic fields ~E(~x, t) and ~B(~x, t) represent physical degrees of free-dom, 3+3 = 6 for each spatial point ~x. The physical reality of these abstract fieldshas been verified in numerous experiments. We should hence think of “empty”space as a medium with nontrivial physical properties. Maxwell’s equations takethe form

~∇ · ~E(~x, t) = 4πρ(~x, t),

~∇× ~E(~x, t) +1

c∂t

~B(~x, t) = 0,

~∇ · ~B(~x, t) = 0,

~∇× ~B(~x, t) − 1

c∂t

~E(~x, t) =4π

c~j(~x, t). (1.2.1)

They are four coupled partial differential equations that allow us to determinethe electromagnetic fields ~E(~x, t) and ~B(~x, t) once the electric charge and currentdensities ρ(~x, t) and ~j(~x, t) have been specified and some appropriate boundaryand initial conditions have been imposed. These four equations describe all elec-tromagnetic phenomena at the classical level, ranging from Coulomb’s 1/r2 law tothe origin of light. They summarize all results ever obtained over centuries of in-tense experimentation with electromagnetism, and they are the basis of countlesselectronic devices that we use every day. Maxwell’s equations are an enormousachievement of nineteenth century physics. Numerous discoveries of the twenti-eth century would have been completely impossible to make without that priorknowledge.

The field concept has turned out to be of central importance far beyondclassical electrodynamics. Also general relativity — Einstein’s relativistic gen-eralization of Newton’s theory of gravity — is a field theory, and the quantummechanical standard model of elementary particles is a so-called quantum field


theory. In fact, the quantum mechanical version of electrodynamics — quantumelectrodynamics (QED) — is an integral part of the standard model. Applyingthe field concept to the classical dynamics of the electromagnetic field as wellas to gravity will allow us to understand numerous phenomena ranging fromthe origin of light to the cosmological evolution. Furthermore, it will be a firststep towards reaching the frontier of knowledge in fundamental physics which iscurrently defined by the standard model of particle physics.

1.3 Electrodynamics and Relativity

Electrodynamics, the theory of electric and magnetic fields, is also a theory oflight. Light waves, excitations of the electromagnetic field, travel with the velocityc. Remarkably, as was found experimentally by Michelson and Morley in 1895,light always travels with the speed c, independent of the speed of an observer.For example, if a space ship leaves the earth at a speed c/2 and shines light backto the earth, we see the light approaching us with the velocity c, not with c/2.This was correctly described by Einstein in his theory of special relativity in 1905.Nevertheless, as a theory of light, Maxwell’s equations of 1865 had relativity builtin from the start. It was soon realized, e.g. by Lorentz and Poincare, that there is afundamental conflict between electrodynamics and Newton’s classical mechanics.Both theories assume different fundamental structures of space and time and arethus inconsistent. Only after Einstein generalized Newton’s classical mechanicsto his theory of special relativity, this conflict was resolved. In Einstein’s theoryof special relativity space and time are unified to Minkowski’s space-time. Inparticular, Maxwell’s equations are invariant against Lorentz transformations,rotations between spatial and temporal directions in space-time.

A point in space-time x = (~x, t) can be described by a 4-vector

xµ = (x0, x1, x2, x3) = (ct, x, y, z). (1.3.1)

Remarkably, time (multiplied by the velocity of light) acts as a fourth coordinatein space-time. There are two space-time versions of the Nabla operator

∂µ = (1

c

∂

∂t,

∂

∂x,

∂

∂y,

∂

∂z), ∂µ = (

1

c

∂

∂t,− ∂

∂x,− ∂

∂y,− ∂

∂z). (1.3.2)

In the relativistic formulation of classical electrodynamics, the charge densityρ(~x, t) and the current density ~j(~x, t) are combined to a 4-vector

jµ(x) = (cρ(~x, t), jx(~x, t), jy(~x, t), jz(~x, t)). (1.3.3)

1.4. GENERAL RELATIVITY, COSMOLOGY, AND INFLATION 9

Similarly, another 4-vector

Aµ(x) = (Φ(~x, t), Ax(~x, t), Ay(~x, t), Az(~x, t)) (1.3.4)

is constructed from the scalar potential Φ(~x, t) and the vector potential ~A(~x, t).It can be used to construct the so-called field strength tensor

Fµν(x) = ∂µAν(x) − ∂νAµ(x) (1.3.5)

whose components will turn out to be the electric and magnetic fields. Therelativistic formulation of classical electrodynamics gives rise to a very compactform of Maxwell’s equations

∂µFµν(x) =4π

cjν(x). (1.3.6)

Here we have used Einstein’s summation convention according to which repeatedindices are summed over.

1.4 General Relativity, Cosmology, and Inflation

Another fundamental classical field theory is general relativity — Einstein’s rela-tivistic theory of gravity. In this theory space-time itself has non-trivial dynamicalproperties. In particular, it can be curved by matter, radiation, or other forms ofenergy such as e.g. vacuum energy. General relativity is governed by Einstein’sequation of motion

Gµν = 8πGTµν . (1.4.1)

Here Gµν is the so-called Einstein tensor, G is Newton’s constant, and Tµν is theenergy-momentum tensor. The essence of this equation is that mass (and moregenerally any form of energy) is a source of space-time curvature. Applied to theUniverse on the largest scales, general relativity predicts the cosmic evolution.Starting from the big bang, depending on its energy content, the Universe expandsforever or will eventually recollapse. Recent data on supernova explosions and onthe cosmic microwave background radiation indicate that “empty” space containsa uniformly distributed form of energy — so-called vacuum energy (also knownas a cosmological constant or in a more dynamical variant as quintessence). Ifthe Universe is dominated by vacuum energy it will expand forever.

Vacuum energy may also have played an important role immediately after thebig bang. The idea of the inflationary Universe is based on a phase of exponentialexpansion at very early times. The expansion is caused by a scalar field whose


value is slowly rolling to the classical minimum of its potential. The idea of in-flation solves several problems of the standard big bang cosmology. For example,it explains (more or less naturally) why the Universe is old and flat. The courseends with a study of classical scalar field theory in the context of inflation.

Chapter 2

Relativistic Form of

Electrodynamics

The relativistic nature of Maxwell’s equations is not manifest in their originalform. Here we reformulate electrodynamics such that its invariance under Lorentztransformations — i.e. under rotations in Minkowski space-time — becomes man-ifest. This will lead us to a deeper understanding of electromagnetism and willalso pave the way to the discussion of general relativity.

2.1 Euclidean Space and Minkowski Space-Time

Everyday experience confirms that we live in a 3-d Euclidean space. This meansthat space itself is flat (i.e. not curved), homogeneous (i.e. translation invariant),and isotropic (i.e. rotation invariant). The length squared of a vector ~r = (x, y, z)is given by

r2 = |~r|2 = x2 + y2 + z2. (2.1.1)

Obviously, the length is invariant under spatial rotations or reflections, i.e.

r′2

= |~r ′|2 = |~r|2 = r2, (2.1.2)

where~r ′ = O~r. (2.1.3)

Here O is a 3 × 3 orthogonal matrix describing rotations or refections. Theorthogonality condition takes the form

OT O = OOT = 1, (2.1.4)

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12 CHAPTER 2. RELATIVISTIC FORM OF ELECTRODYNAMICS

i.e. the transposed matrix OT is the inverse of the matrix O. A simple example isa rotation by an angle α around the z-axis. In that case the rotation or reflectionmatrix takes the form

O =

cos α sin α 0− sin α cos α 0

0 0 1

. (2.1.5)

The Euclidean distance squared between two points ~ra = (xa, ya, za) and ~rb =(xb, yb, zb) is given by

(∆r)2 = |~ra − ~rb|2 = (xa − xb)2 + (ya − yb)

2 + (za − zb)2. (2.1.6)

This distance is invariant under spatial translations and rotations or reflections,i.e.

(∆r′)2 = |~ra′ − ~rb

′|2 = |~ra − ~rb|2 = (∆r)2, (2.1.7)

where~ra

′ = O~ra + ~d, ~rb′ = O~rb + ~d, (2.1.8)

Here ~d is a displacement vector and O is again an orthogonal rotation or reflectionmatrix.

Relativistic theories such as Einstein’s special relativity or Maxwell’s electro-dynamics are invariant not only under spatial translations and rotations but alsounder Lorentz transformations that rotate spatial and temporal coordinates intoone another. The resulting enlarged invariance is known as Lorentz invariance(against space-time rotations) or as Poincare invariance against space-time rota-tions and translations. Minkowski was first to realize that in relativistic theoriesspace and time (which are separate entities in Newtonian mechanics) are natu-rally united to space-time. A point in Minkowski space-time is described by fourcoordinates — one for time and three for space — which form a so-called 4-vector

xµ = (x0, x1, x2, x3) = (ct, x, y, z). (2.1.9)

In particular, the time t (multiplied by the velocity of light c) plays the role ofthe zeroth component of the 4-vector. Minkowski’s space-time does not haveEuclidean geometry. In particular, the length squared of the 4-vector xµ is givenby

s2 = (x0)2 − (x1)2 − (x2)2 − (x3)2, (2.1.10)

and may thus be negative. Besides the so-called contra-variant vector xµ it isuseful to introduce an equivalent form

xµ = (x0, x1, x2, x3) = (ct,−x,−y,−z), (2.1.11)

2.1. EUCLIDEAN SPACE AND MINKOWSKI SPACE-TIME 13

the so-called co-variant 4-vector. Both the co- and the contra-variant 4-vectorscontain the same physical information. Their components are simply related by

x0 = x0 = ct, x1 = −x1 = −x, x2 = −x2 = −y, x3 = −x3 = −z. (2.1.12)

The length squared of the 4-vector can then be written as

3∑

µ=0

xµxµ = (x0)2 − (x1)2 − (x2)2 − (x3)2 = s2. (2.1.13)

Instead of always writing sums over space-time indices µ explicitly, Einstein hasintroduced a summation convention according to which repeated indices (one co-and one contra-variant index) will automatically be summed. Using Einstein’ssummation convention the above equation simply takes the form

xµxµ = (x0)2 − (x1)2 − (x2)2 − (x3)2 = s2. (2.1.14)

The summation over µ is no longer written explicitly, but is still implicitly under-stood, because the index µ occurs twice (once as a co- and once as a contra-variantone).

The norm of a 4-vector induces a corresponding metric via

(x0)2 − (x1)2 − (x2)2 − (x3)2 =3∑

µ=0

3∑

ν=0

gµνxµxν = gµνxµxν . (2.1.15)

In the last step, we have again used Einstein’s summation convention and havedropped the explicit sums over the repeated indices µ and ν. The metric tensorg with the elements gµν is a 4 × 4 matrix given by

g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (2.1.16)

The metric tensor can also be used to relate co- and contra-variant 4-vectors bylowering a contra-variant index, i.e.

xµ = gµνxν =

3∑

ν=0

gµνxν . (2.1.17)

Again, the repeated index ν is summed over, while the unrepeated index µ is notsummed. Let us also introduce the inverse metric g−1 with the components gµν


which is given by

g−1 =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

, (2.1.18)

and which obeys

gg−1 = 1. (2.1.19)

In components this relation takes the form

gµνgνρ =

3∑

ν=0

gµνgνρ = δ ρµ , (2.1.20)

where δ ρµ is just the Kronecker symbol, i.e. it represents the matrix elements of

the unit matrix 1. The inverse metric can now be used to raise co-variant indices,e.g.

xµ = gµνxν =

3∑

ν=0

gµνxν . (2.1.21)

Let us ask under what kind of rotations the length squared of a 4-vector isinvariant. The rotated 4-vector can be written as

x′µ = Λµνx

ν , (2.1.22)

where Λ is a 4 × 4 space-time rotation matrix. Similarly, we obtain

x′µ = gµνx′ν = gµνΛν

ρxρ. (2.1.23)

The length squared of x′µ is then given by

s′2

= x′µx′µ = gµνΛν

ρxρΛµ

σxσ. (2.1.24)

It is invariant under space-time rotations, i.e. s′2 = s2, only if

gµνΛνρΛ

µσ = gρσ. (2.1.25)

This can be rewritten as

ΛT µσ gµνΛν

ρ = gTσρ, (2.1.26)

or equivalently as the matrix multiplication

ΛT gΛ = gT = g. (2.1.27)

2.1. EUCLIDEAN SPACE AND MINKOWSKI SPACE-TIME 15

This condition is the Minkowski space-time analog of the Euclidean space condi-tion OT O = 1 for orthogonal spatial rotations. One now obtains

x′µ = gµνΛν

ρxρ = gµνΛν

ρgρσxσ = [gΛg−1] σ

µ xσ = xσ[gΛg−1]Tσ

µ

= xσΛ−1σµ. (2.1.28)

Here we have used eq.(2.1.27) which leads to

[gΛg−1]T = g−1ΛT g = Λ−1. (2.1.29)

Finally, as a consequence of eq.(2.1.28) we obtain

xν = x′µΛµ

ν . (2.1.30)

Space-time rotations which obey eq.(2.1.27) are known as Lorentz transfor-mations. A simple example of a Lorentz transformation is a rotation betweentime and the x-direction, which leaves the y- and z-coordinates unchanged. Thistransformation is given by

Λ =

cosh α − sinhα 0 0− sinhα cosh α 0 0

0 0 1 00 0 0 1

. (2.1.31)

In components, this takes the form

ct′ = ct cosh α − x sinh α, x′ = x cosh α − ct sinh α, y′ = y, z′ = z. (2.1.32)

Introducing

β =v

c, γ =

1√1 − β2

=1√

1 − v2/c2(2.1.33)

and identifying

cosh α = γ, sinhα =√

cosh2 α − 1 =v/c√

1 − v2/c2= βγ, (2.1.34)

one indeed arrives at the familiar Lorentz transformation

t′ =t − xv/c2

√1 − v2/c2

, x′ =x − vt√1 − v2/c2

, y′ = y, z′ = z. (2.1.35)

Lorentz transformations form a group known as the Lorentz group.


The distance squared in space-time between two 4-vectors xµa and xµ

b is givenby

(∆s)2 = (x0a − x0

b)2 − (x1

a − x1b)

2 − (x2a − x2

b)2 − (x3

a − x3b)

2, (2.1.36)

and may again be negative. This distance is invariant under both Lorentz trans-formations Λ and space-time translations dµ, i.e. (∆s′)2 = (∆s)2, with

x′aµ

= Λµνx

νa + dµ, x′

bµ

= Λµνx

νb + dµ. (2.1.37)

Lorentz transformations and space-time translations again form a group — theso-called Poincare group — which contains the Lorentz group as a subgroup.

2.2 Gradient as a 4-Vector and d’Alembert Operator

In order to do field theory in manifestly Lorentz-invariant form, we also need tocombine temporal and spatial derivatives to a 4-vector. Let us introduce

∂µ = (∂0, ∂1, ∂2, ∂3) =

(1

c

∂

∂t,− ∂

∂x,− ∂

∂y,− ∂

∂z

)=

(∂

∂x0,

∂

∂x1,

∂

∂x2,

∂

∂x3

).

(2.2.1)How does this object transform under Lorentz transformations? Using eq.(2.1.30)one obtains

∂′µ =∂

∂x′µ

=∂xν

∂x′µ

∂

∂xν= Λµ

ν∂ν . (2.2.2)

This shows that ∂µ indeed transforms as a contra-variant 4-vector. Similarly, onecan define the co-variant 4-vector

∂µ = (∂0, ∂1, ∂2, ∂3) =

(1

c

∂

∂t,

∂

∂x,

∂

∂y,

∂

∂z

)=

(∂

∂x0,

∂

∂x1,

∂

∂x2,

∂

∂x3

). (2.2.3)

One can form the scalar product of the co- and contra-variant derivative 4-vectors. In this way one obtains a second derivative operator which transformsas a space-time scalar, i.e. it is invariant under Lorentz transformations. ThisMinkowski space-time analog of the Laplace operator in Euclidean space is knownas the d’Alembert operator and is given by

= ∂µ∂µ =1

c2

∂

∂t2− ∂

∂x2− ∂

∂y2− ∂

∂z2. (2.2.4)

2.3. 4-VECTOR CURRENT 17

2.3 4-Vector Current

Let us now begin to express electrodynamics in manifestly Lorentz co-variantform. We proceed step by step and begin with the charge and current densitiesρ(~r, t) and ~j(~r, t). The corresponding continuity equation which expresses chargeconservation takes the form

∂tρ(~r, t) + ~∇ ·~j(~r, t) = 0. (2.3.1)

Since temporal and spatial derivatives are combined to a gradient 4-vector, it isnatural to combine ρ(~r, t) and ~j(~r, t) to the current 4-vector

jµ(x) = (cρ(~r, t), jx(~r, t), jy(~r, t), jz(~r, t)). (2.3.2)

Here we have introduced x = (~r, t) as a short-hand notation for a point inspace-time. It goes without saying that this should not be confused with thex-component of the spatial vector ~r. In Lorentz-invariant form the continuityequation thus takes the form

∂µjµ(x) =1

c∂tcρ(~r, t) + ∂xjx(~r, t) + ∂yjy(~r, t) + ∂zjz(~r, t) = 0. (2.3.3)

Here we have combined the co-variant 4-vector ∂µ and the contra-variant 4-vectorjµ(x) to the Lorentz-scalar zero. The Lorentz invariance of the continuity equa-tion implies that charge conservation is valid in any inertial frame, independentof the motion of an observer.

Of course, the charge and current densities themselves are dependent on thereference frame in which they are considered. For example, let us consider auniform static charge distribution ρ(~r, t) = ρ. How is this charge distributiondescribed by an observer in another reference frame moving with velocity v inthe x-direction? In order to answer this question, we simply perform the corre-sponding Lorentz transformation

j′µ

= Λµνj

ν . (2.3.4)

In components this equation takes the form

cρ′

j′xj′yj′z

=

cosh α − sinhα 0 0− sinhα cosh α 0 0

0 0 1 00 0 0 1

cρ000

, (2.3.5)


such that

ρ′ = ρ cosh α =ρ√

1 − v2/c2, j′x = −cρ sinhα = − vρ√

1 − v2/c2= −vρ′,

j′y = j′z = 0. (2.3.6)

Of course, it is no surprise that the moving observer sees the static charge dis-tribution as a current flowing opposite to his direction of motion. It may be lessobvious why he sees an enhanced charge density ρ′ = γρ. Doesn’t this meanthat the moving observer sees a larger total charge, which would violate chargeconservation? We should not forget that a moving observer sees a given lengthLx contracted to

L′x =

√1 − v2

c2Lx =

Lx

γ(2.3.7)

in the direction of motion, while transverse lengths remain the same, i.e. L′y = Ly,

L′z = Lz. Hence, the total charge contained in the volume V ′ = L′

xL′yL

′z is given

byQ′ = ρ′V ′ = γρL′

xL′yL

′z = ρLxLyLz = ρV = Q, (2.3.8)

and is hence indeed independent of the reference frame.

If a general (non-uniform and non-static) charge and current density is trans-formed into another reference frame, one must also transform the space-timepoint x at which the density is evaluated, i.e.

j′µ(x′) = Λµ

νjν(x) = Λµ

νjν(Λ−1x′). (2.3.9)

2.4 4-Vector Potential

From the scalar potential Φ(~r, t) and the vector potential ~A(~r, t) one can constructanother 4-vector field

Aµ(x) = (Φ(~r, t), Ax(~r, t), Ay(~r, t), Az(~r, t)). (2.4.1)

Under Lorentz transformations it again transforms as

A′µ(x′) = ΛµνAν(Λ−1x′). (2.4.2)

Scalar and vector potentials transform non-trivially under gauge transformations

ϕΦ(~r, t) = Φ(~r, t) +1

c∂tϕ(~r, t), ϕ ~A(~r, t) = ~A(~r, t) − ~∇ϕ(~r, t). (2.4.3)

2.4. 4-VECTOR POTENTIAL 19

In 4-vector notation this relation takes the form

ϕAµ(x) = Aµ(x) + ∂µϕ(x). (2.4.4)

Here ϕ(x) is an arbitrary space-time-dependent gauge transformation function.This function is a space-time scalar, i.e. under Lorentz transformations it trans-forms as

ϕ′(x′) = ϕ(Λ−1x′). (2.4.5)

The Lorenz gauge fixing condition

1

c∂tΦ(~r, t) + ~∇ · ~A(~r, t) = 0 (2.4.6)

can be re-expressed as∂µAµ(x) = 0. (2.4.7)

In the Lorenz gauge the wave equations take the form

1

c2∂2

t Φ(~r, t) − ∆Φ(~r, t) = 4πρ(~r, t), (2.4.8)

1

c2∂2

t~A(~r, t) − ∆ ~A(~r, t) =

4π

c~j(~r, t), (2.4.9)

which can be combined to

Aµ(x) =4π

cjµ(x). (2.4.10)

A plane wave solution of the wave equation in the vacuum takes the form

Aµ(x) = Cµ exp(ikνxν) = Cµ exp(i(ωt − ~k · ~r)), (2.4.11)

with the co- and contra-variant wave 4-vectors

kµ = (k0, k1, k2, k3) = (ω

c,−kx,−ky,−kz),

kµ = (k0, k1, k2, k3) = (ω

c, kx, ky, kz). (2.4.12)

Inserting the plane wave ansatz into the Lorenz gauge condition one obtains

∂µAµ(x) = ikµCµ exp(ikνxν) = 0 ⇒ kµCµ = 0. (2.4.13)

Similarly, inserting the ansatz in the wave equation in the vacuum one finds

Aµ(x) = −kρkρCµ exp(ikνxν) = 0 ⇒ kρk

ρ =ω2

c2− |~k|2 = 0 ⇒ ω = |~k|c.

(2.4.14)The 4-vector kµ has norm zero, i.e. it is a so-called null-vector.


2.5 Field Strength Tensor

It may not be entirely obvious how to express the electric and magnetic fields~E(~r, t) and ~B(~r, t) in relativistic form. We need to use

~E(~r, t) = −~∇Φ(~r, t) − 1

c∂t

~A(~r, t), ~B(~r, t) = ~∇× ~A(~r, t). (2.5.1)

Obviously, ~E(~r, t) and ~B(~r, t) are constructed form the 4-vectors ∂µ and Aµ(x).The scalar product of these two 4-vectors

∂µAµ(x) =1

c∂tΦ(~r, t) + ~∇ · ~A(~r, t) (2.5.2)

appears in the Lorenz gauge fixing condition but does not yield the electric ormagnetic field. The 4-vectors ∂µ and Aµ(x) can also be combined to the sym-metric tensor

Dµν = ∂µAν(x) + ∂νAµ(x), (2.5.3)

as well as to the anti-symmetric tensor

Fµν = ∂µAν(x) − ∂νAµ(x). (2.5.4)

Under gauge transformations these tensors transform as

ϕDµν = ∂µ ϕAν(x) + ∂ν ϕAµ(x)

= ∂µAν(x) + ∂µ∂νϕ(x) + ∂νAµ(x) + ∂ν∂µϕ(x)

= Dµν + 2∂µ∂νϕ(x),ϕFµν = ∂µ ϕAν(x) − ∂ν ϕAµ(x)

= ∂µAν(x) + ∂µ∂νϕ(x) − ∂νAµ(x) − ∂ν∂µϕ(x)

= Fµν , (2.5.5)

i.e. the anti-symmetric tensor Fµν is gauge invariant, while the symmetric tensorDµν is not. As a consequence, it does not play any particular role in electrody-namics. Since the electromagnetic fields are gauge invariant, we expect them to

2.6. INHOMOGENEOUS MAXWELL EQUATIONS 21

be related to Fµν . Let us consider the various components of this tensor

F 01(x) = ∂0A1(x) − ∂1A0(x) =1

c∂tAx(~r, t) + ∂xΦ(~r, t) = −Ex(~r, t),

F 02(x) = ∂0A2(x) − ∂2A0(x) =1

c∂tAy(~r, t) + ∂yΦ(~r, t) = −Ey(~r, t),

F 03(x) = ∂0A3(x) − ∂3A0(x) =1

c∂tAz(~r, t) + ∂zΦ(~r, t) = −Ez(~r, t),

F 12(x) = ∂1A2(x) − ∂2A1(x) = −∂xAy(~r, t) + ∂yAx(~r, t) = −Bz(~r, t),

F 23(x) = ∂2A3(x) − ∂3A2(x) = −∂yAz(~r, t) + ∂zAy(~r, t) = −Bx(~r, t),

F 31(x) = ∂3A1(x) − ∂1A3(x) = −∂zAx(~r, t) + ∂xAz(~r, t) = −By(~r, t).

(2.5.6)

Hence, the anti-symmetric tensor indeed contains the electric and magnetic fieldsas

Fµν(x) =

0 −Ex(~r, t) −Ey(~r, t) −Ez(~r, t)Ex(~r, t) 0 −Bz(~r, t) By(~r, t)Ey(~r, t) Bz(~r, t) 0 −Bx(~r, t)Ez(~r, t) −By(~r, t) Bx(~r, t) 0

. (2.5.7)

The co-variant components of this tensor are given by

Fµν(x) =

0 Ex(~r, t) Ey(~r, t) Ez(~r, t)−Ex(~r, t) 0 −Bz(~r, t) By(~r, t)−Ey(~r, t) Bz(~r, t) 0 −Bx(~r, t)−Ez(~r, t) −By(~r, t) Bx(~r, t) 0

. (2.5.8)

2.6 Inhomogeneous Maxwell Equations

Let us first consider the inhomogeneous Maxwell equations

~∇ · ~E(~r, t) = 4πρ(~r, t), ~∇× ~B(~r, t) − 1

c∂t

~E(~r, t) =4π

c~j(~r, t). (2.6.1)

These are four equations with the components of the 4-vector current jµ(x) on theright-hand side. Hence, on the left-hand side there must also be a 4-vector. Theleft-hand side consists of derivatives, i.e. of components of the gradient 4-vectors∂µ, and of the electromagnetic fields, i.e. of the components of the field strengthtensor Fµν . Hence, the 4-vector ∂µ and the tensor Fµν on the left-hand side mustbe combined to another 4-vector. This can be achieved by forming ∂µFµν(x) and


thus by contracting (i.e. by summing) one co- and one contra-variant index. Thevarious components of this object take the form

∂µFµ0(x) = ∂xEx(~r, t) + ∂yEy(~r, t) + ∂zEz(~r, t)

= ~∇ · ~E(~r, t) = 4πρ(~r, t),

∂µFµ1(x) = −1

c∂tEx(~r, t) + ∂yBz(~r, t) − ∂zBy(~r, t)

= [~∇× ~B]x(~r, t) − 1

c∂tEx(~r, t) =

4π

cjx(~r, t),

∂µFµ2(x) = −1

c∂tEy(~r, t) − ∂xBz(~r, t) + ∂zBx(~r, t)

= [~∇× ~B]y(~r, t) −1

c∂tEy(~r, t) =

4π

cjy(~r, t),

∂µFµ3(x) = −1

c∂tEz(~r, t) + ∂xBy(~r, t) − ∂yBx(~r, t)

= [~∇× ~B]z(~r, t) −1

c∂tEz(~r, t) =

4π

cjz(~r, t). (2.6.2)

Indeed, these equations can be summarized as

∂µFµν(x) =4π

cjν(x). (2.6.3)

At this point the usefulness of the compact 4-dimensional notation should beobvious.

Inserting eq.(2.5.4) into the inhomogeneous Maxwell equations we obtain

∂µFµν(x) = ∂µ(∂µAµ(x)−∂νAµ(x)) = Aν(x)−∂ν∂µAµ(x) =4π

cjν(x). (2.6.4)

If the 4-vector potential obeys the Lorenz gauge fixing condition ∂µAµ(x) = 0,this is nothing but the wave equation eq.(2.4.10).

2.7 Homogeneous Maxwell Equations

How can we express the homogeneous Maxwell equations

~∇ · ~B(~r, t) = 0, ~∇× ~E(~r, t) +1

c∂t

~B(~r, t) = 0 (2.7.1)

in 4-dimensional form? Except for the vanishing right-hand side, they look verysimilar to the inhomogeneous equations. All we need to do is to substitute ~E(~r, t)

2.7. HOMOGENEOUS MAXWELL EQUATIONS 23

by − ~B(~r, t) and ~B(~r, t) by ~E(~r, t). Such a substitution is known as a dualitytransformation. Under this operation the field strength tensor turns into thedual tensor

Fµν(x) =

0 Bx(~r, t) By(~r, t) Bz(~r, t)−Bx(~r, t) 0 −Ez(~r, t) Ey(~r, t)−By(~r, t) Ez(~r, t) 0 −Ex(~r, t)−Bz(~r, t) −Ey(~r, t) Ex(~r, t) 0

, (2.7.2)

and the homogeneous Maxwell equations can thus be expressed as

∂µFµν(x) = 0. (2.7.3)

The co-variant components of the dual field strength tensor take the form

Fµν(x) =

0 −Bx(~r, t) −By(~r, t) −Bz(~r, t)Bx(~r, t) 0 −Ez(~r, t) Ey(~r, t)By(~r, t) Ez(~r, t) 0 −Ex(~r, t)Bz(~r, t) −Ey(~r, t) Ex(~r, t) 0

. (2.7.4)

It is obvious that the field strength tensor Fµν(x) and its dual Fµν(x) consist

of the same components, namely of the electric and magnetic fields ~E(~r, t) and~B(~r, t). Hence, there must be a relation between the two tensors. This relationtakes the form

Fµν(x) =1

2ǫµνρσF ρσ(x). (2.7.5)

Here ǫµνρσ is a totally anti-symmetric tensor with components 0 or ±1. If anyof the indices µ, ν, ρ, and σ are equal, the value of ǫµνρσ is zero. Only if allindices are different, the value of ǫµνρσ is non-zero. The value is ǫµνρσ = 1 ifµνρσ is an even permutation of 0123 (i.e. it requires an even number of indexpair permutations to turn µνρσ into 0123). Similarly, ǫµνρσ = −1 if µνρσ is anodd permutation of 0123. For example, we obtain

F01(x) =1

2ǫ01ρσF ρσ(x) =

1

2

(ǫ0123F

23(x) + ǫ0132F32(x)

)

= F 23(x) = −Bx(~r, t), (2.7.6)

as well as

F12(x) =1

2ǫ12ρσF ρσ(x) =

1

2

(ǫ1203F

03(x) + ǫ1230F30(x)

)

= F 03(x) = −Ez(~r, t), (2.7.7)


Inserting eq.(2.7.5) into the homogeneous Maxwell equations (2.7.3) one ob-tains

∂µFµν(x) =1

2ǫµνρσ∂µF ρσ(x) =

1

2ǫµνρσ∂µ (∂ρAσ(x) − ∂σAρ(x)) = 0. (2.7.8)

Due to the anti-symmetry of ǫµνρσ and the commutativity of the derivatives ∂µ

and ∂ρ, this equation is automatically satisfied. This is no surprise, because theoriginal Maxwell equations were also automatically satisfied by the introductionof the scalar and vector potentials Φ(~r, t) and ~A(~r, t). The homogeneous Maxwellequations can alternatively be expressed as

∂µF ρσ + ∂ρF σµ + ∂σFµρ = 0. (2.7.9)

Indeed, multiplying this relation with εµνρσ and applying cyclic permutations tothe indices µ, ρ, and σ, one again arrives at eq.(2.7.8).

2.8 Space-Time Scalars from Field Strength Tensors

Which scalar quantities can be formed by combining the field strength tensorsFµν(x) and Fµν(x)? First, we can construct the combination

Fµν(x)Fµν(x) = 2(

~B(~r, t)2 − ~E(~r, t)2)

, (2.8.1)

which will later turn out to be the Lagrange density of electrodynamics. Thenwe can construct

Fµν(x)Fµν(x) = 2(

~E(~r, t)2 − ~B(~r, t)2)

, (2.8.2)

which is thus the same up to a minus-sign. While the electromagnetic fieldsthemselves are obviously not Lorentz-invariant, the difference of their magnitudessquared is.

One can also mix the field strength tensor with its dual and one then obtains

Fµν(x)Fµν(x) = 4~E(~r, t) · ~B(~r, t). (2.8.3)

Similarly, one obtains

Fµν(x)Fµν(x) = 4~E(~r, t) · ~B(~r, t), (2.8.4)

which is thus equivalent. Interestingly, the projection of the electric on the mag-netic field ~E(~r, t) · ~B(~r, t) is also Lorentz invariant, i.e. it has the same value inall inertial frames.

2.9. TRANSFORMATION OF ELECTROMAGNETIC FIELDS 25

2.9 Transformation of Electromagnetic Fields

Since they form the components of the field strength tensors, it is obvious thatthe electromagnetic fields ~E(~r, t) and ~B(~r, t) are not Lorentz-invariant, i.e. theydepend on the motion of the observer. Under a Lorentz transformation the fieldstrength tensor transforms as

F ′µν(x′) = Λµ

ρΛνσF ρσ(x) = Λµ

ρΛνσF ρσ(Λ−1x′). (2.9.1)

This can be rewritten in matrix form as

F ′µν(x′) = Λµ

ρFρσ(Λ−1x′)ΛT ν

σ ⇒ F ′(x′) = ΛF (Λ−1x′)ΛT . (2.9.2)

Under the particular Lorentz transformation of eq.(2.1.31) the fields thus trans-form as

0 −E′x −E′

y −E′z

E′x 0 −B′

z B′y

E′y B′

z 0 −B′x

E′z −B′

y B′x 0

=

C −S 0 0−S C 0 00 0 1 00 0 0 1

0 −Ex −Ey −Ez

Ex 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

C −S 0 0−S C 0 00 0 1 00 0 0 1

=

C −S 0 0−S C 0 00 0 1 00 0 0 1

ExS −ExC −Ey −Ez

ExC −ExS −Bz By

EyC − BzS −EyS + BzC 0 −Bx

EzC + ByS −EzS − ByC Bx 0

=

0 −Ex −CEy + SBx −CEz − SBy

Ex 0 SEy − CBz SEz + CBy

EyC − BzS −EyS + BzC 0 −Bx

EzC + ByS −EzS − ByC Bx 0

. (2.9.3)

Here C = cosh α = γ and S = sinh α = βγ and we have used C2 − S2 = 1. Incomponents we thus obtain

E′x = Ex, E′

y =Ey − Bzv/c√

1 − v2/c2, E′

z =Ez + Byv/c√

1 − v2/c2,

B′x = Bx, B′

y =By + Ezv/c√

1 − v2/c2, B′

z =Bz − Eyv/c√

1 − v2/c2. (2.9.4)

Interestingly, the longitudinal components Ex and Bx remain the same in aninertial frame moving with velocity ~v = (v, 0, 0) in the x-direction, while thetransverse components change.


2.10 Electromagnetic Field of a Moving Charge

Let us consider a static point charge q in its rest frame. The correspondingelectromagnetic field is then given by

~E(~r, t) = q~r

|~r|3 , ~B(~r, t) = 0. (2.10.1)

What is the electromagnetic field seen by a moving observer? Or equivalently,what is the field of a charge moving in the opposite direction? Let us performa Lorentz transformation in order to determine the field seen by the movingobserver. First of all, we have

t =t′ + x′v/c2

√1 − v2/c2

, x =x′ + vt′√1 − v2/c2

, y = y′, z = z′, (2.10.2)

such that

|~r|2 = x2 + y2 + z2 =(x′ + vt′)2

1 − v2/c2+ y′

2+ z′

2. (2.10.3)

The moving observer sees the charge at the position (−vt′, 0, 0) such that thedistance vector from the charge to the point ~r ′ = (x′, y′, z′) is given by

~R ′ = (x′ + vt′, y′, z′), (2.10.4)

with the length squared

R′2 = (x′ + vt′)2 + y′2+ z′

2, (2.10.5)

such that

|~r|2 =R′2 − d2v2/c2

1 − v2/c2, (2.10.6)

2.11. THE RELATIVISTIC DOPPLER EFFECT 27

where d2 = y′2 + z′2 = y2 + z2 is the transverse distance squared. Let us nowlook at the various components of the electromagnetic field. One obtains

E′x(~r ′, t′) = Ex(~r, t) = q

x′ + vt′√1 − v2/c2|~r|3

= q(x′ + vt′)(1 − v2/c2)√

R′2 − d2v2/c23 ,

E′y(~r

′, t′) =Ey(~r, t)√1 − v2/c2

= qy′√

1 − v2/c2|~r|3= q

y′(1 − v2/c2)√

R′2 − d2v2/c23 ,

E′z(~r

′, t′) =Ez(~r, t)√1 − v2/c2

= qz′√

1 − v2/c2|~r|3= q

z′(1 − v2/c2)√

R′2 − d2v2/c23 ,

B′x(~r ′, t′) = Bx(~r, t) = 0,

B′y(~r

′, t′) =Ez(~r, t)v/c√

1 − v2/c2= q

z′v/c√1 − v2/c2|~r|3

=qv

c

z′(1 − v2/c2)√

R′2 − d2v2/c23 ,

B′z(~r

′, t′) = − Ey(~r, t)v/c√1 − v2/c2

= −qy′v/c√

1 − v2/c2|~r|3= −qv

c

y′(1 − v2/c2)√

R′2 − d2v2/c23 .

(2.10.7)

Hence, using ~v ′ = (−v, 0, 0), we can write

~E′(~r ′, t′) = q~R′(1 − v2/c2)

√R′2 − d2v2/c2

3 , ~B′(~r ′, t′) =q~v ′

c×

~R′(1 − v2/c2)√

R′2 − d2v2/c23 . (2.10.8)

Since a moving charge generates a current, it should not be surprising that themoving observer also sees a magnetic field. To the leading order of the non-relativistic expansion, i.e. for sufficiently small velocity v such that v2/c2 is neg-ligible, one finds

~E′(~r ′, t′) = q~R ′

R′3, ~B′(~r ′, t′) =

q~v ′

c×

~R ′

R′3. (2.10.9)

2.11 The Relativistic Doppler Effect

Let us now consider the special case of a plane wave propagating in the x-directiondescribed by the wave 4-vector

kµ = (ω

c, k, 0, 0) = (k, k, 0, 0). (2.11.1)


Let us assume that the corresponding electric field is pointing in the y-direction,i.e. ~E = (0, E, 0). The magnetic field is then pointing in the z-direction and isgiven by ~B = (0, 0, E). After the Lorentz transformation we have

k′µ = Λµνk

ν = ((C − S)k, (C − S)k, 0, 0)

=1 − v/c√1 − v2/c2

(k, k, 0, 0) =

√c − v

c + v(k, k, 0, 0). (2.11.2)

Similarly, the electric field is now given by

E′x = 0, E′

y =E − Ev/c√1 − v2/c2

= E

√c − v

c + v, E′

z = 0,

B′x = 0, B′

y = 0, B′z =

E − Ev/c√1 − v2/c2

= E

√c − v

c + v. (2.11.3)

Up to an overall rescaling by a factor of√

(c − v)/(c + v) the vectors ~k, ~E, and ~Bremain unchanged. The scaling factor is related to the relativistic Doppler effect.In particular, the observer following the electromagnetic wave with velocity v seesthe frequency

ω′ = |~k ′|c =

√c − v

c + v|~k|c =

√c − v

c + vω, (2.11.4)

which is thus smaller than the frequency ω observed in the rest frame.

In contrast to non-relativistic physics, in relativistic physics there is also atransverse Doppler effect. In order to understand this, let us now consider aplane wave propagating in the z-direction described by the wave 4-vector

kµ = (ω

c, 0, 0, k) = (k, 0, 0, k). (2.11.5)

After the Lorentz transformation we now have

k′µ = Λµνk

ν = (Ck,−Sk, 0, k), (2.11.6)

such that

ω′ = Cω =ω√

1 − v2/c2. (2.11.7)

Hence, if one moves perpendicular to the direction of propagation of an electro-magnetic wave, one sees a larger frequency than an observer at rest. It shouldbe noted that angles are also frame-dependent. When we say that the observermoves perpendicular to the propagation direction of the electromagnetic wave,we make this statement in the frame of the observer at rest.

2.12. RELATIVISTIC PARTICLE IN AN ELECTROMAGNETIC FIELD 29

2.12 Relativistic Particle in an Electromagnetic Field

In non-relativistic physics Netwon’s equation and the Lorentz force determinethe motion of a charged particle in an electromagnetic field through Newton’sequation of motion

m~a(t) = q

[~E(~r(t), t) +

~v(t)

c× ~B(~r(t), t)

]. (2.12.1)

What is the correct relativistic version of the equation of motion? In order toanswer this question, let us first consider a free relativistic particle of mass m.The corresponding action is given by

S[~r(t)] = −∫

dt mc2√

1 − ~v(t)2/c2, (2.12.2)

which is just the rest energy mc2 times the length of the particle’s world line.The Lagrange function hence takes the form

L = −mc2√

1 − ~v(t)2/c2, (2.12.3)

and the momentum canonically conjugate to the coordinate ~r(t) is given by

pi =∂L

∂vi(t)=

mvi(t)√1 − ~v(t)2/c2

. (2.12.4)

The corresponding Hamilton function is then given by

H = ~p · ~v − L =m~v(t)2√

1 − ~v(t)2/c2+ mc2

√1 − ~v(t)2/c2

=mc2

√1 − ~v(t)2/c2

=√

(mc2)2 + (~pc)2 (2.12.5)

in agreement with the energy-momentum relation for a free relativistic particle.

The action of a charge and current distribution in an external electromagneticfield is given by

S[j(x), A(x)] = −1

c

∫d4x Aµ(x)jµ(x)

= −∫

dtd3x [ρ(~x, t)Φ(~x, t) − 1

c~j(~x, t) · ~A(~x, t)]. (2.12.6)

The charge and current densities of a single charged particle are given by

ρ(~x, t) = qδ(~x − ~r(t)), ~j(~x, t) = q~v(t)δ(~x − ~r(t)). (2.12.7)


It is easy to show that charge is conserved, i.e.

∂tρ(~x, t) + ~∇ ·~j(~x, t) = 0. (2.12.8)

Inserting eq.(2.12.7) into eq.(2.12.6), and integrating over space, for the actionone obtains

S[~r(t), A(~r(t))] = −∫

dt

[qΦ(~r(t), t) − q

~v(t)

c· ~A(~r(t), t)

]. (2.12.9)

This action is invariant under gauge transformations because∫

dt

[ϕΦ(~r(t), t) − ~v(t)

c·ϕ ~A(~r(t), t)

]=

∫dt

[Φ(~r(t), t) +

1

c∂tϕ(~r(t), t) − ~v(t)

c· ( ~A(~r(t), t) − ~∇ϕ(~r(t), t))

]=

∫dt

[Φ(~r(t), t) − ~v(t)

c· ~A(~r(t), t) +

1

c

d

dtϕ(~r(t), t)

], (2.12.10)

and because the total derivative

d

dtϕ(~r(t), t) = ∂tϕ(~r(t), t) + ~v · ~∇ϕ(~r(t), t), (2.12.11)

integrates to zero as long as ϕ(~r(t), t) vanishes in the infinite past and future.Identifying the total Lagrange function as

L = −mc2√

1 − ~v(t)2/c2 − qΦ(~r(t), t) + q~v(t)

c· ~A(~r(t), t), (2.12.12)

it is straightforward to derive the equation as the Euler-Lagrange equation

d

dt

∂L

∂vi(t)− ∂L

∂ri= 0, (2.12.13)

and one obtains

d

dt

(m~v(t)√

1 − ~v(t)2/c2

)= q

[~E(~r(t), t) +

~v(t)

c× ~B(~r(t), t)

]. (2.12.14)

The theory can also be formulated in terms of a classical Hamilton function

H = ~p(t) · ~v(t) − L, (2.12.15)

where ~p again is the momentum canonically conjugate to the coordinate ~r. Onenow finds

m~v(t)√1 − ~v(t)2/c2

= ~p(t) − q

c~A(~r(t), t), (2.12.16)

2.13. FROM POINT MECHANICS TO CLASSICAL FIELD THEORY 31

and thus one obtains

H =

√(mc2)2 +

[~p(t)c − q ~A(~r(t), t)

]2+ qΦ(~r(t), t). (2.12.17)

This is indeed consistent because

vi(t) =dri(t)

dt=

∂H

∂pi(t)=

[pi(t)c − qAi(~r(t), t)] c√(mc2)2 +

[~p(t)c − q ~A(~r(t), t)

]2 . (2.12.18)

The other equation of motion is

dpi(t)

dt= − ∂H

∂ri(t). (2.12.19)

It is straightforward to show that these equations of motion are again equivalentto eq.(2.12.14).

2.13 From Point Mechanics to Classical Field Theory

Before we formulate electrodynamics in the Lagrangian formalism, let us comparea generic classical field theory with point mechanics. Point mechanics describesthe dynamics of classical non-relativistic point particles. The coordinates of theparticles represent a finite number of degrees of freedom. In the simplest case —a single particle moving in one spatial dimension — we are dealing with a singledegree of freedom: the x-coordinate of the particle. The dynamics of a particle ofmass m moving in an external potential V (x) is described by Newton’s equation

m∂2t x = ma = F (x) = −dV (x)

dx. (2.13.1)

Once the initial conditions are specified, this ordinary second order differentialequation determines the particle’s path x(t), i.e. its position as a function of time.Newton’s equation results from the variational principle to minimize the action

S[x] =

∫dt L(x, ∂tx), (2.13.2)

over the space of all paths x(t). The action is a functional (a function whoseargument is itself a function) that results from the time integral of the Lagrangefunction

L(x, ∂tx) =m

2(∂tx)2 − V (x). (2.13.3)


The Euler-Lagrange equation

∂t∂L

∂(∂tx)− ∂L

∂x= 0, (2.13.4)

is nothing but Newton’s equation.

Classical field theories are a generalization of point mechanics to systems withinfinitely many degrees of freedom — a given number for each space point ~x. Inthis case, the degrees of freedom are the field values φ(~x), where φ is some genericfield. In case of a neutral scalar field, φ is simply a real number representing onedegree of freedom per space point. A charged scalar field, on the other hand, isdescribed by a complex number and hence represents two degrees of freedom perspace point. The scalar Higgs field φa(~x) (with a ∈ 1, 2) in the standard modelof particle physics, for example, is a complex doublet, i.e. it has four real degreesof freedom per space point. An Abelian gauge field Ai(~x) (with a spatial directionindex i ∈ 1, 2, 3) — for example, the vector potential in electrodynamics — is aneutral vector field with 3 real degrees of freedom per space point. One of thesedegrees of freedom is redundant due to the U(1)em gauge symmetry. Hence, anAbelian gauge field has two physical degrees of freedom per space point whichcorrespond to the two polarization states of the massless photon. The time-component A0(~x) does not represent a physical degree of freedom. It is just aLagrange multiplier field that enforces the Gauss law. A non-Abelian gauge fieldAa

i (~x) is charged and has an additional index a. For example, the gluon fieldin chromodynamics with a color index a ∈ 1, 2, ..., 8 represents 2 × 8 = 16physical degrees of freedom per space point, again because of some redundancydue to the SU(3)c color gauge symmetry. The field that represents the W - and Z-bosons in the standard model has an index a ∈ 1, 2, 3 and transforms under thegauge group SU(2)L. Thus, it represents 2 × 3 = 6 physical degrees of freedom.However, in contrast to the photon, the W - and Z-bosons are massive due to theso-called Higgs mechanism and have three (not just two) polarization states. Theextra degree of freedom is provided by the Higgs field.

The analog of Newton’s equation in field theory is the classical field equation ofmotion. For example, for a neutral scalar field this is the Klein-Gordon equation

∂µ∂µφ = −dV (φ)

dφ. (2.13.5)

Again, after specifying appropriate initial conditions it determines the classicalfield configuration φ(x), i.e. the values of the field φ at all space-time pointsx = (ct, ~x). Hence, the role of time in point mechanics is played by space-time infield theory, and the role of the point particle coordinates is now played by the

2.13. FROM POINT MECHANICS TO CLASSICAL FIELD THEORY 33

field values. As before, the classical equation of motion results from minimizingthe action

S[φ] =

∫d4x L(φ, ∂µφ). (2.13.6)

The integral over time in eq.(2.13.2) is now replaced by an integral over space-time and the Lagrange function of point mechanics gets replaced by the Lagrangedensity function (or Lagrangian)

L(φ, ∂µφ) =1

2∂µφ∂µφ − V (φ). (2.13.7)

A simple massive free field theory has the potential

V (φ) =M2

2φ2. (2.13.8)

Here M is the mass of the scalar field. Note that the mass term correspondsto a harmonic oscillator potential in the point mechanics analog. As before, theEuler-Lagrange equation

∂µ∂L

∂(∂µφ)− ∂L

∂φ= 0, (2.13.9)

is the classical equation of motion, in this case the Klein-Gordon equation

(∂µ∂µ + M2)φ = 0. (2.13.10)

In point mechanics the momentum canonically conjugate to the coordinate xis given by

px =∂L

∂(∂tx), (2.13.11)

and the corresponding Hamilton function is given by

H = px∂tx − L. (2.13.12)

Similarly, in classical field theory the momentum canonically conjugate to thefield φ is

pφ =∂L

∂(∂tφ)=

1

c2∂tφ. (2.13.13)

The energy density is described by the Hamilton density or Hamiltonian

H = pφ∂tφ − L =1

2

(1

c2∂tφ∂tφ + ~∇φ · ~∇φ

)+ V (φ)

=1

2

(c2pφpφ + ~∇φ · ~∇φ

)+ V (φ). (2.13.14)


It is interesting to consider the so-called energy-momentum tensor (also knownas the stress-energy tensor)

T µν = ∂µφ∂νφ − Lgµν , (2.13.15)

which represents the currents corresponding to a set of four conserved charges,i.e.

∂µT µν = 0. (2.13.16)

The time-time-component of the energy-momentum tensor is the energy density

T 00 = ∂0φ∂0φ − L = H. (2.13.17)

Similarly, the space-time components of the energy-momentum tensor representthe momentum density

T i0 = ∂iφ∂0φ. (2.13.18)

The continuity equation that describes energy conservation is given by

∂0T00 + ∂iT

i0 = 0. (2.13.19)

The analogies between point mechanics and field theory are summarized intable 2.1.

2.14 Action and Euler-Lagrange Equation

The Lagrangian for the electromagnetic field interacting with a charge and currentdistribution jµ is given by

L = − 1

16πFµνFµν − 1

cAµjµ. (2.14.1)

The corresponding action is obtained by integrating the Lagrangian over space-time, i.e.

S[A] =

∫dt d3x L =

∫d4x

(− 1

16πFµνFµν − 1

cAµjµ

). (2.14.2)

The action can be viewed as a functional (i.e. a function of a function) of theelectromagnetic 4-vector potential Aµ. The Euler-Lagrange equation of motionresulting from the principle of least action now takes the form

∂µ∂L

∂∂µAν− ∂L

∂Aν=

1

4π∂µFµν − 1

cjν = 0. (2.14.3)

2.15. ENERGY-MOMENTUM TENSOR 35

Point Mechanics Field Theory

time t space-time x = (ct, ~x)

particle coordinate x field value φ

particle path x(t) field configuration φ(x)

action S[x] =∫

dt L(x, ∂tx) action S[φ] =∫

d4x L(φ, ∂µφ)

Lagrange function LagrangianL(x, ∂tx) = m

2 (∂tx)2 − V (x) L(φ, ∂µφ) = 12∂µφ∂µφ − V (φ)

equation of motion field equation

∂t∂L

∂(∂tx) − ∂L∂x = 0 ∂µ

∂L∂(∂µφ) − ∂L

∂φ = 0

Newton’s equation Klein-Gordon equation

∂2t x = −dV (x)

dx ∂µ∂µφ = −dV (φ)dφ

kinetic energy m2 (∂tx)2 kinetic energy 1

2∂µφ∂µφ

harmonic oscillator potential m2 ω2x2 mass term M2

2 φ2

conjugate momentum px = ∂L∂(∂tx) conjugate momentum pφ = ∂L

∂(∂tφ)

Hamilton function H = px∂tx − L Hamiltonian H = pφ∂tφ −L

Table 2.1: The dictionary that translates point mechanics into the language of

field theory.

Indeed, this yields just the inhomogeneous Maxwell equations

∂µFµν =4π

cjν . (2.14.4)

The homogeneous Maxwell equations are automatically satisfied as a consequenceof Fµν = ∂µAν − ∂νAµ.

2.15 Energy-Momentum Tensor

Let us consider the energy-momentum tensor of the free electromagnetic field(i.e. in the absence of charges and currents)

T µν = − 1

4πFµρF ν

ρ − Lgµν , (2.15.1)

which obeys the continuity equation

∂µT µν = 0. (2.15.2)


The time-time component of the energy-momentum tensor is given by

T 00 = − 1

4πF 0ρF 0

ρ +1

16πFρσF ρσg00 =

1

4π~E2 +

1

8π( ~B2 − ~E2)

=1

8π( ~E2 + ~B2) = u. (2.15.3)

Here u is the well-known energy density of the electromagnetic field. The space-time components of the energy-momentum tensor take the form

T i0 = − 1

4πF iρF 0

ρ +1

16πFρσF ρσgi0 =

1

4πεijkEjBk =

1

4π( ~E × ~B)i =

1

cSi.

(2.15.4)Here εijk is the completely anti-symmetric Levi-Civita tensor and

~S =c

4π~E × ~B (2.15.5)

is the Poynting vector which is known to represent the momentum density of theelectromagnetic field. The continuity equation that represents energy conserva-tion takes the form

∂µT µ0 = ∂0T00 + ∂iT

i0 =1

c(∂tu + ~∇ · ~S) = 0. (2.15.6)

Chapter 3

Basics of General Relativity

General relativity is a theory of gravity, a generalization of Newton’s theory. Forweak gravitational fields and for processes slow compared to the velocity of light,it reduces to Newton’s theory of gravity. General relativity is a classical theory,which has strongly resisted quantization, just like Einstein himself had problemswith quantum mechanics. General relativity is physics in curved space-time. Thegravitational “field” is represented by the metric of space-time, and by relatedquantities like the connection, Riemann’s curvature tensor, the Ricci tensor andthe curvature scalar. Matter fields are the sources that curve space-time, andin turn the dynamics of matter is influenced by the curvature itself. In orderto understand the cosmological evolution we don’t need too much of generalrelativity, but we must understand its basic principles, and we must get used toRiemann’s geometry. First, we will talk just about space (not yet about space-time), and indeed curved spaces play a role in the physics of the early Universe,although our space is rather flat.

3.1 Spaces of Constant Curvature

In 1854 in Gottingen (Germany) Riemann formulated the theory that is usedto describe the geometry of curved spaces. Half a century later, Einstein usedRiemann’s mathematical framework to describe curved space-time.

Let us begin with something very simple, namely with two-dimensional spaces,and let us first discuss the simplest 2-d space, the plane R

2. We use Cartesiancoordinates and characterize a point by x = (x1, x2). The distance between two

37

38 CHAPTER 3. BASICS OF GENERAL RELATIVITY

infinitesimally close points is computed following Pythagoras

(ds)2 = (dx1)2 + (dx2)2 = gij(x)dxidxj . (3.1.1)

We have just introduced a metric

g(x) =

(g11(x) g12(x)g21(x) g22(x)

)=

(1 00 1

). (3.1.2)

The metric is independent of x, which is a special case, because R2 is flat. Also

the metric is symmetric, i.e.

gji(x) = gij(x), (3.1.3)

which is generally the case.

Coordinates, however, are not physical. We can choose other ones if we wantto. Sometimes it is useful to work with curve-linear coordinates, for example withpolar coordinates. We then have

x1 = r cos ϕ, x2 = r sin ϕ, (3.1.4)

such that

dx1 = cos ϕ dr − r sinϕ dϕ, dx2 = sinϕ dr + r cos ϕ dϕ, (3.1.5)

and hence(ds)2 = (dr)2 + r2(dϕ)2. (3.1.6)

Now we have another metric

g(x) =

(grr(r, ϕ) grϕ(r, ϕ)gϕr(r, ϕ) gϕϕ(r, ϕ)

)=

(1 00 r2

), (3.1.7)

which does depend on x (namely on r). The metric depends on the choice ofcoordinates. Of course, the space is still the same. In particular, even a flat spacemay have an x-dependent metric. Hence, curve-linear coordinates certainly donot imply a curved space.

Let us consider something a bit more interesting, a curved space — the sphereS2. We can easily visualize the sphere S2 by embedding it in the flat space R

3.However, two-dimensional inhabitants of the surface of the sphere could calculatejust as we do — embeddings are not necessary. For 3-d people like ourselves it isparticularly easy, because we just write

x1 = R sin θ cos ϕ, x2 = R sin θ sin ϕ, x3 = R cos θ, (3.1.8)

3.1. SPACES OF CONSTANT CURVATURE 39

and therefore

(ds)2 = (dx1)2 + (dx2)2 + (dx3)2

= R2(cos θ cos ϕ dθ − sin θ sin ϕ dϕ)2

+ R2(cos θ sin ϕ dθ − sin θ cos ϕ dϕ)2 + R2 sin2 θ dθ2

= R2(dθ2 + sin2 θdϕ2). (3.1.9)

The metric is thus given by

g(x) = R2

(1 00 sin2 θ

). (3.1.10)

Of course, we can again introduce other coordinates. The following choice mayseem unmotivated, but it will be useful later. We introduce

ρ = sin θ, dρ = cos θ dθ, (3.1.11)

and obtain

(ds)2 = R2(dρ2

1 − ρ2+ ρ2dϕ2). (3.1.12)

Thus the new metric takes the form

g(x) = R2

( 11−ρ2 0

0 ρ2

). (3.1.13)

The sphere has a constant positive curvature. Let us now consider a space withconstant negative curvature — the hyperbolically curved surface H2. We can nolonger embed this surface in R

3, and we may thus feel a bit uncomfortable. Forthe 2-d people on the surface of S2 it does not make a difference, and from nowon also we must rely just on the mathematics. A possible metric of H2 is

(ds)2 = R2(dθ2 + sinh2 θ dϕ2), (3.1.14)

where now θ ∈ [0,∞]. We can embed H2 in a 3-d Minkowski-space by writing

(ds)2 = (dx1)2 + (dx2)2 − (dx3)2, (3.1.15)

if we identify

x1 = R sinh θ cos ϕ, x2 = R sinh θ sin ϕ, x3 = R cosh θ, (3.1.16)

because we then have

(ds)2 = R2(cosh θ cos ϕ dθ − sinh θ sin ϕ dϕ)2

+ R2(cosh θ sinϕ dθ − sinh θ cos ϕ dϕ)2 − R2 sinh2 θ dθ2

= R2(dθ2 + sinh2 θ dϕ2). (3.1.17)


We can still try to visualize H2, when we realize that

(x1)2 + (x2)2 − (x3)2 = R2(sinh2 θ − cosh2 θ) = −R2. (3.1.18)

Interpreting

(x3)2 = (x1)2 + (x2)2 + R2 (3.1.19)

correctly is, however, delicate, because we still are in Minkowski-space, and wemay hence still feel a bit uncomfortable. Let us again introduce new coordinates

ρ = sinh θ, dρ = cosh θ dθ, (3.1.20)

such that the metric then is

(ds)2 = R2(dρ2

1 + ρ2+ ρ2dϕ2). (3.1.21)

We can now write the metrics for R2, S2 and H2 in a common form

(ds)2 = R2(dρ2

1 − kρ2+ ρ2dϕ2). (3.1.22)

Here k = 0, 1,−1 for R2, S2 and H2, respectively. For R

2 the introductionof r = Rρ is a bit artificial. We have introduced a scale factor R, while ρ isdimensionless.

We now have a list of all two-dimensional spaces of constant curvature. Thislist is complete, at least as long as we restrict ourselves to local properties. How-ever, we can still have various global topologies. For example, we can roll up theplane to a cylinder by identifying appropriate points. Our 2-d people don’t no-tice the difference (the cylinder is as flat as the plane), unless they travel aroundtheir entire Universe (global topology). We can also roll up the cylinder to atorus T 2, even though we cannot visualize this in R

3 without extra curvature.The torus is finite, just like the sphere, but as flat as the plane. Similarly, thereare spaces that look like H2 locally, but are finite. Spaces that are locally like S2

are always finite. Still, we can also change the global topology of S2, for exampleby identifying antipodal points.

All we said about two-dimensional spaces of constant curvature generalizestrivially to three-dimensional spaces of constant curvature. The general form ofthe metric then is

(ds)2 = R2(dρ2

1 − kρ2+ ρ2(dθ2 + sin2 θ dϕ2)). (3.1.23)

3.2. GEODESICS AND METRIC CONNECTION 41

When we investigate the Universe on the largest scales, we find that it is ratherhomogeneous and isotropic (galaxy distribution, cosmic background radiation).Since the observable Universe is homogeneous and isotropic, and since we assumethat we don’t inhabit a special place in it, we conclude that the whole Universe isa three-dimensional space of constant curvature. Note that we do not talk aboutspace-time yet — just about 3-d space. We have investigated various metrics,and we found that the metric depends on the coordinate system. Coordinates,however, are not physical. When we want to answer physical questions (like“what is the shortest path from A to B ?”, or “how strongly curved is space ?”)we need more of Riemann’s beautiful mathematics.

3.2 Geodesics and Metric Connection

What is the shortest path from A to B ? We are looking for a geodesic (and thatis exactly what particles in space-time do as well). Let us consider an arbitrary(d-dimensional) space with a metric, and a curve x(λ) connecting two points Aand B, i.e. x(0) = A, x(1) = B. The length S of the curve is then given by

S =

∫ 1

0dλ (gij(x)xixj)1/2, x =

dx

dλ. (3.2.1)

The integral is reparameterization invariant, i.e. defining

x′ =dx

dλ′=

dx

dλ

dλ

dλ′= x

dλ

dλ′, (3.2.2)

implies

S′ =

∫ 1

0dλ′ (gij(x)x′ix′j)1/2 =

∫ 1

0dλ

dλ′

dλ(gij(x)xi dλ

dλ′xj dλ

dλ′)1/2 = S. (3.2.3)

We are looking for a curve of shortest length — i.e. we are trying to solve avariational problem — and we can use what we know from classical mechanics.The classical action

S =

∫ t1

t0

dt L(x, x) (3.2.4)

is minimal for the classical path that solves the Euler-Lagrange equation

d

dt

∂L

∂x− ∂L

∂x= 0. (3.2.5)

Applying this to our geodesics problem implies

d

dλ

1

2(gij(x)xixj)−1/2 2gklx

l − 1

2(gij(x)xixj)−1/2 ∂kglm(x)xlxm = 0. (3.2.6)


We now use the reparameterization invariance, and parameterize the curve by itsarc-length s

ds = (gij(x)xixj)1/2dλ, (3.2.7)

such that

d

ds(gkl

dxl

ds) − 1

2∂kglm

dxl

ds

dxm

ds= 0 ⇒

gkld2xl

ds2+ ∂igkl

dxi

ds

dxl

ds− 1

2∂kglm

dxl

ds

dxm

ds= 0. (3.2.8)

We introduce the inverse metric

gijgjk = δik, (3.2.9)

and we use the symmetry of the metric to write

d2xl

ds2+

1

2glk(∂igkj + ∂jgki − ∂kgij)

dxi

ds

dxj

ds= 0. (3.2.10)

The new object deserves its own name. It is called the metric connection, or justthe connection

Γlij =

1

2glk(∂igkj + ∂jgki − ∂kgij). (3.2.11)

The Γlij are also known as Christoffel symbols. Using the connection, the equation

for the geodesic takes the form

d2xl

ds2+ Γl

ij

dxi

ds

dxj

ds= 0. (3.2.12)

Let us consider a trivial example — geodesics in R2. We know that these are

straight lines

xl = als + bl ⇒ d2xl

ds2= 0 ⇒ Γl

ij = 0. (3.2.13)

All coefficients of the connection vanish. Now we go to polar coordinates

x1 = r cos ϕ, x2 = r sin ϕ ⇒d2r

ds2cos ϕ − 2

dr

dssinϕ

dϕ

ds− r cos ϕ(

dϕ

ds)2 − r sin ϕ

d2ϕ

ds2= 0,

d2r

ds2sin ϕ + 2

dr

dscos ϕ

dϕ

ds− r sin ϕ(

dϕ

ds)2 + r cos ϕ

d2ϕ

ds2= 0 ⇒

d2r

ds2− r(

dϕ

ds)2 = 0, 2

dr

ds

dϕ

ds+ r

d2ϕ

ds2= 0 ⇒

Γrϕϕ = −r, Γϕ

rϕ =1

r. (3.2.14)

3.3. PARALLEL TRANSPORT AND RIEMANN TENSOR 43

All other connection coefficients vanish. As expected, the connection dependson the choice of coordinates. Hence, it is useful to look for a “good” coordinatesystem, before one begins to compute the connection coefficients. In our example,the calculation in Cartesian coordinates was much simpler.

3.3 Parallel Transport and Riemann Tensor

The connection is also used to define parallel transport of a vector along somecurve. The equation

vl(λ) + Γlijv

i(λ)xj = 0 (3.3.1)

yields the vector vl(λ) parallel transported along the curve x(λ) for a given initialvl(0). Next we introduce a covariant derivative matrix

Dlij = δl

i∂j + Γlij, (3.3.2)

and we write

xjDjv(x(λ)) = xj(δli∂j + Γl

ij)vi(x(λ))

= xj∂jvl(x(λ)) + Γl

ijvi(x(λ))xj

= vl(λ) + Γlijv

i(λ)xj = 0. (3.3.3)

In particular, we can view the equation for a geodesic as a parallel transportequation for the tangent unit vector ti(s) = dxi/ds

dtl(s)

ds+ Γl

ijti(s)

dxj

ds= 0. (3.3.4)

Parallel transport in curved space is qualitatively different from the one in flatspace. Let us consider any closed curve in R

2. When we choose Cartesian coordi-nates, all connection coefficients vanish, and a vector parallel transported alongthe curve returns to its initial orientation. On the other hand, when we do thesame on the surface of S2, the vector does not return to its initial orientation.

Let us consider parallel transport around an infinitesimal surface element. We


compare the results along two paths

[Dj ,Dk] = DjDk − DkDj

= DlmjD

mik − Dl

mkDmij

= (δlm∂j + Γl

mj)(δmi ∂k + Γm

ik) − (δlm∂k + Γl

mk)(δmi ∂j + Γm

ij )

= δli∂j∂k + ∂jΓ

lik + Γl

ik∂j + Γlij∂k + Γl

mjΓmik

− δli∂k∂j − ∂kΓ

lij − Γl

ij∂k − Γlik∂j − Γl

mkΓmij

= ∂jΓlik − ∂kΓ

lij + Γl

mjΓmik − Γl

mkΓmij = Rl

ijk. (3.3.5)

We have just introduced Riemann’s curvature tensor

Rlijk = ∂jΓ

lik − ∂kΓ

lij + Γl

mjΓmik − Γl

mkΓmij . (3.3.6)

In flat space the curvature tensor vanishes. This follows immediately when wechoose Cartesian coordinates, for which all connection coefficients are zero.

3.4 Ricci and Einstein Tensors and Curvature Scalar

Via index contraction we can obtain other curvature tensors from the Riemanntensor. The Ricci tensor, for example, is given by

Rik = Rjijk, (3.4.1)

and the curvature scalar is defined as

R = Rii = gkiRik. (3.4.2)

Finally, the Einstein tensor is given by

Gik = Rik − 1

2gikR. (3.4.3)

We now have discussed enough Riemann geometry in order to attack generalrelativity. What was geometry of space so far, will then turn into the geometro-dynamics of space-time.

3.5 Curved Space-Time

We have learned something about curved spaces, and we have discussed the basicsof Riemann’s geometry. Let us now apply these tools to curved space-time. From

3.6. NEWTON-CARTAN CURVED SPACE-TIME 45

special relativity we know Minkowski-space — flat space-time. When we chooseCartesian coordinates its metric takes the form

(ds)2 = (dt)2 − (dx1)2 − (dx2)2 − (dx3)2. (3.5.1)

From now on we will work in natural units in which time is measured in units oflength by setting the velocity of light to c = 1. Denoting t = x0 we can write

(ds)2 = gµνdxµdxν , g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (3.5.2)

We can discuss curved space-times when we allow more general (space-time-dependent) metrics. We can then use Riemann’s geometry, define the connection

Γρµν =

1

2gρσ(∂µgσν + ∂νgσµ − ∂σgµν), (3.5.3)

and write down the equation for a geodesic

d2xρ

ds2+ Γρ

µν

dxµ

ds

dxν

ds= 0. (3.5.4)

3.6 Newton-Cartan Curved Space-Time

Before we turn to Einstein’s theory of gravity, let us look back at Newton’s goodold theory. There space and time are well separated, space is flat and time isabsolute. Let us take a mass distribution ρ(y) and let us compute its gravitationalpotential

Φ(x) = G

∫dy

ρ(y)

|x − y| , ∆Φ(x) = 4πGρ(x). (3.6.1)

A particle of mass m moving in this potential follows Newton’s equation of motion

md2xl

dt2= F l = −m

∂Φ

∂xl⇒ d2xl

dt2+

∂Φ

∂xl= 0. (3.6.2)

The particle’s path is a curved trajectory in flat space, with the particle movingalong it as absolute time passes. Let us reinterpret this in terms of a curvedNewton space-time (as it was first done by Cartan in 1923). Then absolute timewould be another coordinate. We also introduce the proper time s of the particlewhich it reads off from a comoving watch. Since Newton’s time is absolute, we


simply have s = t. We can therefore write Newton’s equations of motion (witht = x0) in the form

d2xl

ds2+

∂Φ

dxl

dx0

ds

dx0

ds= 0, (3.6.3)

which we can identify as the equation for a geodesic in a curved space-time. Theconnection coefficients then are

Γl00 =

∂Φ

∂xl, l ∈ 1, 2, 3, (3.6.4)

and all other coefficients vanish. This connection cannot be derived from anunderlying metric, however, it still defines a meaningful equation for geodesicsin a curved Newton-Cartan space-time. We can also write down the Riemanntensor

Rσµνρ = ∂νΓ

σµρ − ∂ρΓ

σµν + Γσ

λνΓλµρ − Γσ

λρΓλµν . (3.6.5)

The only non-vanishing components are

Rl0i0 = −Rl

00i = ∂iΓl00 =

∂2Φ

∂xi∂xl. (3.6.6)

The Ricci tensor is then given by

Rµρ = Rνµνρ ⇒ R00 = Ri

0i0 =∂2Φ

∂xi2= ∆Φ = 4πGρ, (3.6.7)

and again all other components are zero. We cannot build the curvature scalaror the Einstein tensor in this case, because we don’t have a metric. The aboveresults can be interpreted as follows. The ordinary curved trajectories in flatspace are “straight” lines (geodesics) in the curved Newton-Cartan space-time,and the mass density ρ is responsible for the curvature (Ricci tensor). From thispoint it is not far to go to Einstein’s general relativity.

3.7 Einstein’s Curved Space-Time

The existence of a metric is essential for Einstein’s theory. The equation of motionfor a massive particle that feels gravity only (free falling particle) is the equationfor a geodesic

d2xρ

ds2+ Γρ

µν

dxµ

ds

dxν

ds= 0. (3.7.1)

Here we have parameterized the particle’s trajectory by its arc-length — theproper time s. Introducing the four-velocity

uµ =dxµ

ds(3.7.2)

3.8. EINSTEIN’S GRAVITATIONAL FIELD EQUATION 47

we can writeduρ

ds+ Γρ

µνuµuν = 0. (3.7.3)

The four-velocity is a tangent unit-vector

u2 = uµuµ = gµνuµuν = gµνdxµ

ds

dxν

ds=

(ds)2

(ds)2= 1. (3.7.4)

The equation of motion for massless particles (photons, light rays) is simply

(ds)2 = 0, (3.7.5)

which is also known as a null-geodesic.

3.8 Einstein’s Gravitational Field Equation

We still need an equation that determines the curvature of space-time based onthe mass distribution. This is Einstein’s field equation

Gµν = Rµν − 1

2gµνR = 8πGTµν . (3.8.1)

Here Tµν is the energy-momentum tensor of matter (particles and fields). Hence,it is not just mass, but any form of energy and momentum, that determinesthe curvature of space-time. In order to achieve a static Universe, Einstein hadmodified his equation to

Gµν = 8πGTµν + gµνΛ. (3.8.2)

After he learned of Edwin Hubble’s observation of the expanding Universe, Ein-stein has been quoted as considering the introduction of the cosmological con-stant Λ his biggest blunder. Today there is indeed observational evidence foran extremely small but non-zero cosmological constant, which causes the cosmicexpansion to accelerate. A non-zero value of Λ implies that “empty” space pro-duces a gravitational effect due to vacuum energy. Since Λ represents a form ofenergy, it should indeed be considered part of the energy-momentum tensor. Itis presently not understood why the cosmological constant is so incredibly small.This is the so-called cosmological constant problem, one of the greatest puzzlesin modern physics.


3.9 The Einstein-Hilbert Action

In 1915 there was a fierce competition between Einstein and Hilbert about findingthe correct equations for general relativity, i.e. the relativistic theory of gravity.Five days before Einstein had published his field equations, Hilbert had publishedan action from which the field equations follow by a variational principle. Al-though the field equations that Hilbert had published turned out to be incorrect,his action indeed yields Einstein’s correct field equations. The Einstein-Hilbertaction is a functional of the metric g given by

S[g] =

∫d4x

√−detg L. (3.9.1)

HeredV = d4x

√−detg (3.9.2)

is the invariant volume element of space-time and detg < 0 is the determinant ofthe metric. The Lagrangian takes the form

L =1

16πGR + Lm, (3.9.3)

where R is the scalar curvature and Lm is the Lagrangian of “matter”, i.e. anynon-gravitational form of energy (including fields of all kinds). Einstein’s fieldequations result as the Euler-Lagrange equations associated with the Einstein-Hilbert action upon variation with respect to the metric. The correspondingderivation is somewhat tedious and goes beyond the scope of this course. Thevariation of

√−detg R with respect to gµν yields 1

16πGGµν while

1√−detg

∂(√

−detg Lm

)

∂gµν=

∂Lm

∂gµν− 1

2gµνL = −1

2Tµν . (3.9.4)

In the last step we have used

1√−detg

∂√−detg

∂gµν= −1

2gµν . (3.9.5)

Chapter 4

Standard Big Bang Cosmology

We now have the tools that we need to deal with curved space-time, and we havethe dynamical equations for the motion of massive particles and of light rays. Alsowe have Einstein’s field equations, which determine the space-time curvature fromthe energy-momentum distribution of particles and fields. Now we want to applygeneral relativity to the Universe as a whole. We are interested only in the largestscales, not in local structures like galaxy-clusters, galaxies, or even single stars, orplanetary systems. Of course, all these objects curve space-time locally (in caseof black holes even very strongly). However, we will average over all the localstructures, and we assume that the whole Universe is filled homogeneously andisotropically by a gas of matter, whose “molecules” are, for example, the galaxies.At first sight, this idealization may seem very drastic. On the other hand, weknow from hydrodynamics that a continuum description of gases works very well,although they have a very discontinuous structure at molecular scales. At theend it is a question for observations, how homogeneous and isotropic our Universereally is. The observed galaxy distribution is indeed rather homogeneous andisotropic when one averages out the structures of galaxy-clusters and individualgalaxies. Also the cosmic background radiation is extremely isotropic, whichindicates that space was homogeneous and isotropic immediately after the BigBang.

The standard cosmological model assumes that the Universe is a homoge-neous and isotropic three-dimensional space of constant curvature, however, witha time-dependent scale parameter R. The corresponding metric is the so-calledFriedmann-Lemaitre-Robertson-Walker (FLRW) metric. The Einstein field equa-tions then determine the dynamics of the Universe (expansion), when one assumesa certain energy-momentum tensor of matter. Before we study the dynamics of

49

50 CHAPTER 4. STANDARD BIG BANG COSMOLOGY

space-time in detail, we will investigate the kinematics of the FLRW metric, i.e.we will play free falling observer and we will investigate the horizon.

4.1 The Friedmann-Lemaitre-Robertson-Walker

Metric

Let us use some observational facts to obtain a useful ansatz for the metric of ourUniverse on the largest scales. Spatial homogeneity and isotropy lead us to thespaces of constant curvature. An experiment that we repeat every day tells usthat we are interested in three-dimensional spaces. Thus we have the candidatesR

3, S3 and H3 eventually endowed with some non-trivial global topology. Thespatial part of the metric is then given by

R2(dρ2

1 − kρ2+ ρ2(dθ2 + sin2 θdϕ2)), k = 0,±1, (4.1.1)

where R is a scale parameter (of dimension length). We now use Hubble’s obser-vation that our Universe is expanding, and thus allow R to be time-dependent.Altogether, we make the FLRW ansatz

ds2 = dt2 − R(t)2(dρ2

1 − kρ2+ ρ2(dθ2 + sin2 θdϕ2)). (4.1.2)

We have used a special space-time coordinate system, which clearly manifeststhe symmetries of our ansatz. Indeed, we are in the comoving coordinate systemof the “galaxy-gas”, and only in this system the Universe appears homogeneousand isotropic.

We know how to compute the connection coefficients, the Riemann tensor, theRicci tensor, the curvature scalar and the Einstein tensor. It is quite tedious tocompute these quantities for the three-dimensional spaces of constant curvature.Since we have done this exercise in two-dimensions (in a homework), we simply

4.2. COSMOLOGICAL RED-SHIFT AND HUBBLE LAW 51

quote the results

Γlij =

1

2glk(∂jgki + ∂igkj − ∂kgij),

Γ0ij = −R

Rgij , Γi

0j =R

Rδij ,

Rij = −(R

R+ 2

R2

R2+ 2

k

R2)gij , R00 = −3

R

R,

R = −6(R

R+

R2

R2+

k

R2),

Gij = (2R

R+

R2

R2+

k

R2)gij , G00 = 3(

R2

R2+

k

R2). (4.1.3)

All other components vanish.

4.2 Cosmological Red-Shift and Hubble Law

Let us now follow a light ray (photon) through the Universe. We ignore allinteractions with matter (interstellar gas, plasma) and consider only gravitationaleffects. Hence, we are looking for a null-geodesic

ds2 = 0. (4.2.1)

Since the Universe is homogeneous and isotropic, we can limit ourselves to motionalong the ρ-direction, and put θ = ϕ = 0. We then have

ds2 = dt2 − R(t)2dρ2

1 − kρ2= 0 ⇒ dt

R(t)= ± dρ√

1 − kρ2. (4.2.2)

We consider a light signal, that has been emitted at time t = 0 (say in the momentof the big bang) at the space-point with dimensionless coordinate ρ = ρH . Weobserve the signal today (at time t) at the point ρ = 0, such that

∫ t

0

dt′

R(t′)=

∫ ρH

0

dρ√1 − kρ2

. (4.2.3)

Today, what is the distance to the point, at which the light signal was emittedoriginally? Even though the Universe has expanded in the mean time (R(t) hasincreased) the dimensionless coordinate of that point still is ρH . The distance to


that point is given by

dH(t) =

∫ 1

0dλ (−gij x

ixj)1/2 =

∫ ρH

0dρ√

−gρρ =

∫ ρH

0dρ

R(t)√1 − kρ2

= R(t)

∫ t

0

dt′

R(t′). (4.2.4)

The distance to the horizon is determined by the behavior of R(t) close to thebig bang t = 0. Let us assume that R(t) ∝ tn. Then, for n < 1,

dH(t) ∝ tn∫ t

0

dt′

t′n= [

tn

(1 − n)t′n−1]t0 =

t

1 − n, (4.2.5)

i.e. the distance is finite. For n ≥ 1, on the other hand, the distance to thehorizon is infinite. In a radiation dominated Universe, as it existed when thecosmic background radiation was generated, one has n = 1/2. Now the Universeis dominated by non-relativistic matter and we have n = 2/3. In both cases, thedistance to the horizon is indeed finite.

Now let us consider the motion of a massive particle along a geodesic

duρ

ds+ Γρ

µνuµuν = 0, (4.2.6)

and let us concentrate on the zero-component of this equation

du0

ds+ Γ0

µνuµuν = 0, (4.2.7)

In the FLRW metric one has

Γ0ij = −R

Rgij , (4.2.8)

and we obtaindu0

ds− R

Rgiju

iuj =du0

ds+

R

R|~u|2 = 0. (4.2.9)

The four-velocity is a tangent unit-vector

(u0)2 − |~u|2 = 1 ⇒ u0du0 − |~u|d|~u| = 0 ⇒1

u0

d|~u|ds

+R

R|~u| =

ds

dx0

d|~u|ds

+R

R|~u| = |~u| + R

R|~u| = 0 ⇒

|~u||~u| = −R

R, (4.2.10)

such that|~u| ∝ R−1. (4.2.11)

4.2. COSMOLOGICAL RED-SHIFT AND HUBBLE LAW 53

The four-momentum is given by

pµ = muµ, (4.2.12)

which implies that the three-momentum |~p| ∝ R−1 is red-shifted. The ordinarythree-velocity is

vi =dxi

dt=

dxi

ds

ds

dx0=

ui

u0⇒ |~v|2 =

|~u|2(u0)2

=|~u|2

1 + |~u|2 ⇒ |~u|2 =|~v|2

1 − |~v|2 .

(4.2.13)Hence, for small |~v| we have

|~v| ∝ R−1, (4.2.14)

i.e. in an expanding Universe a free falling observer will ultimately come to restin the cosmic rest frame of matter.

Let us also investigate the red-shift of light signals. We consider a signalemitted at time t1 and coordinate ρ1, and observed at time t0 at coordinateρ = 0, such that

ds2 = 0 ⇒ dt

R(t)= ± dρ√

1 − kρ2⇒

∫ t0

t1

dt

R(t)=

∫ ρ1

0

dρ√1 − kρ2

. (4.2.15)

Let us assume that a wave-maximum was emitted at t1, and that the next wave-maximum was emitted at t1 + δt1. That maximum will then be observed at timet0 + δt0, such that ∫ t0+δt0

t1+δt1

dt

R(t)=

∫ ρ1

0

dρ√1 − kρ2

. (4.2.16)

This immediately implies

∫ t0+δt0

t0

dt

R(t)=

∫ t1+δt1

t1

dt

R(t)⇒ δt0

R(t0)=

δt1R(t1)

. (4.2.17)

The wave-length λ is inversely proportional to the frequency, and hence propor-tional to δt, such that

λ(t0)

R(t0)=

λ(t1)

R(t1)⇒ λ(t) ∝ R(t). (4.2.18)

The wave-lengths of light in the Universe are stretched together with space itself.In an expanding Universe one hence finds a red-shift z with

1 + z =λ(t0)

λ(t1)=

R(t0)

R(t1). (4.2.19)


Let us introduce the Hubble parameter

H(t) =R(t)

R(t), (4.2.20)

which measures the strength of the expansion, and let us Taylor expand aboutthe present epoch

R(t) = R(t0) + (t − t0)R(t0) ⇒ R(t)

R(t0)= 1 + (t − t0)H(t0), (4.2.21)

and hence

1 + z =R(t0)

R(t1)= 1 − (t1 − t0)H(t0) ⇒ z = (t0 − t1)H(t0). (4.2.22)

What is the distance to the emitting galaxy? We find

d =

∫ 1

0dλ (−gij x

ixj)1/2 =

∫ ρ1

0dρ√

−gρρ = R(t0)

∫ ρ1

0

dρ√1 − kρ2

= R(t0)

∫ t0

t1

dt′

R(t′)= t0 − t1, (4.2.23)

such thatz = dH(t0). (4.2.24)

This is Hubble’s law: the observed red-shift of light emitted from a distant galaxyis proportional to the distance of the galaxy. To derive Hubble’s law we havemade a power series expansion about the present epoch. Consequently, there willbe corrections to Hubble’s law at early times. It is interesting that the leadingorder calculation is independent of the details of the dynamics of the Universe(radiation- or matter-domination, curved or flat, open or closed).

4.3 Solutions of the Field Equations

The Einstein field equation takes the form

Gµν = 8πGTµν . (4.3.1)

In the FLRW metric, the non-vanishing components of the Einstein tensor are

Gii = (2R

R+

R2

R2+

k

R2)gii, G00 = 3(

R2

R2+

k

R2). (4.3.2)

4.3. SOLUTIONS OF THE FIELD EQUATIONS 55

Consequently, the energy-momentum tensor of matter must be diagonal as well.Let us consider the energy-momentum tensor of an ideal gas

Tµν = (ρ + p)uµuν − pgµν . (4.3.3)

Here ρ is the energy density and p is the pressure of the gas, and uµ is the four-velocity field. The tensor Tµν is diagonal only if uµ has a non-vanishing zerothcomponent only, i.e. if u0 = 1. This means that, in the coordinate system of theFLRW metric, the gas of matter is at rest, and simply follows the expansion ofthe Universe (cosmic rest frame). Thus, we have

Tii = −pgii, T00 = ρ. (4.3.4)

Density and pressure are temperature-dependent functions that are connectedvia an equation of state. From the space and time components we obtain twoequations, the Friedmann equation

3(R2

R2+

k

R2) = 8πGρ, (4.3.5)

as well as

2R

R+

R2

R2+

k

R2= −8πGp. (4.3.6)

Hence, one obtains

d

dt(8πGρR3) + 8πGp

d

dtR3 =

d

dt(3RR2 + 3kR) − (2

R

R+

R2

R2+

k

R2)3R2R =

3R3 + 6RRR + 3kR − 6RRR − 3R3 − 3kR = 0 ⇒d

dt(ρR3) = −p

d

dtR3. (4.3.7)

This equation can be identified as the first law of thermodynamics (energy con-servation)

dE = −pdV, (4.3.8)

where E = ρR3 is the rest energy of a given volume R3. Let us form a linearcombination of the two original equations

6R

R+ 3

R2

R2+ 3

k

R2− 3(

R2

R2+

k

R2) = −8πG(3p + ρ) ⇒

R

R= −4πG

3(3p + ρ). (4.3.9)


For the matter that we find on earth, one has 3p + ρ > 0 and hence R < 0.Observations of the rotation curves of galaxies indicate that also so-called darkmatter must exist in the Universe. One speculates that the dark matter mayconsist of heavy non-relativistic weakly interacting massive elementary particles(so-called WIMPs). Ordinary matter together with non-relativistic (so-calledcold) dark matter would lead to a decelerated expansion. If 3p+ρ > 0 was realizedalso in the past, there necessarily must have been a big bang, not earlier than theHubble time H(t0)

−1. Observations of very distant supernovae have revealed thatthe expansion of the Universe is actually accelerating. This implies that 3p+ρ <0. No ordinary form of matter obeys such an equation of state, which wouldrequire a negative pressure. However, vacuum energy (e.g. Einstein’s cosmologicalconstant or quintessence — a dynamical form of so-called dark energy) indeedhas a negative pressure p = −ρ such that 3p + ρ = −2ρ < 0.

We can read the Friedmann equation as an equation for the Hubble parameter

H2 +k

R2=

8πG

3ρ ⇒ k

H2R2=

8πGρ

3H2− 1 = Ω − 1. (4.3.10)

We have introduced the parameter

Ω =ρ

ρc, ρc =

3H2

8πG. (4.3.11)

Of course, H2R2 ≥ 0, such that we obtain a relation between the critical densityof matter ρc (that depends on time via H) and the curvature k of space. Fork = 1 space has positive curvature (S3) and ρ > ρc. For k = −1 (H3), on theother hand, ρ < ρc. Finally, if the density is critical (ρ = ρc) — i.e. if Ω = 1 —space is flat (R3).

First, let us consider a matter dominated Universe — a system of non-relativistic matter clustered in galaxies, which are far away from each other.Then p/ρ ≤ 10−6, such that we can neglect the pressure and put p = 0. Usingthe first law of thermodynamics one finds

ρR3 =3

4πM, (4.3.12)

where M is the conserved total mass contained in a sphere of radius R. Insertingthis in the Friedmann equation one obtains

3

(R2

R2+

k

R2

)= 6

GM

R3⇒ R2

2− GM

R= −k

2. (4.3.13)


The last equation can be interpreted as the equation of motion of a “point parti-cle” (of unit mass) at position R(t) in an attractive external potential −GM/Rwith total “energy” −k/2. We then obtain

R =

√2MG

R− k ⇒

∫ t

0dt′ =

∫ R(t)

0

dR√2MG

R − k. (4.3.14)

In flat space (k = 0) we find

t =

∫ R(t)

0dR

√R

2MG=

2

3

√R(t)3

2MG⇒ R(t) = (

9

2MGt2)1/3 ∝ t2/3. (4.3.15)

The same behavior follows for small R, even in the cases k = ±1.

Let us now consider a negatively curved space (k = −1) for large values of R

t =

∫ R(t)

0dR ⇒ R(t) = t. (4.3.16)

Such a Universe expands forever, faster than a flat space. For a positively curvedspace, on the other hand, one has

R2 =2MG

R− 1 = 0 ⇒ Rmax = 2MG, (4.3.17)

i.e. the scale parameter reaches a maximal value Rmax, and then the Universecontracts ending in a big crunch.

At very early times the Universe was filled with a gas of hot relativistic par-ticles — it was radiation dominated. Then the equation of state is

p =1

3ρ, (4.3.18)

and we obtain

d

dt(ρR3) = −p

d

dtR3 = −1

3ρ3R2R = −ρR2R ⇒

ρR3 + 3ρR2R + ρR2R = ρR3 + 4ρR2R = 0, (4.3.19)

such thatd

dt(ρR4) = ρR4 + 4ρR3R = 0. (4.3.20)

Hence, we can writeρR4 = N, (4.3.21)


with N being a constant which is proportional to the number of particles in therelativistic gas. By again inserting this in the Friedmann equation we now obtain

3

(R2

R2+

k

R2

)=

8πGN

R4⇒ R2

2− 4πGN

3R2= −k

2, (4.3.22)

which can be interpreted in the “point particle” analogy as motion in the attrac-tive potential −4πGN/3R2 such that

R =

√8πGN

3R2− k. (4.3.23)

The Universe is radiation dominated at early times. Hence, we can assume thatR is small, such that we can neglect k and obtain

t =

∫ R(t)

0dR R

√3

8πGN=

1

2R(t)2

3

8πGN⇒ R(t) ∝ t1/2. (4.3.24)

Let us also consider an empty Universe, however, with a cosmological constantΛ. Then we have Tµν = 0 and thus

Gµν = 8πGTµν + gµνΛ = gµνΛ. (4.3.25)

In the ideal gas parametrization of the energy-momentum tensor this correspondsto

ρ =Λ

8πG, p = −ρ = − Λ

8πG. (4.3.26)

Remarkably, in contrast to a relativistic or non-relativistic gas, a positive cos-mological constant (i.e. vacuum energy) exerts a negative pressure. As we willsee, this leads to an accelerated expansion of the Universe that counteracts thedeceleration due to matter. Again, by insertion into the Friedmann equation wenow obtain

3

(R2

R2+

k

R2

)= Λ ⇒ R2

2− ΛR2

6= −k

2, (4.3.27)

which implies

R =

√Λ

3R2 − k ⇒ t =

∫ R(t)

R(0)

dR√Λ3 R2 − k

. (4.3.28)

First, let us consider a flat Universe (k = 0), such that

t =

∫ R(t)

R(0)

dR√Λ3 R

=

√3

Λ[log R]

R(t)R(0) ⇒ R(t) = R(0) exp(

√Λ

3t). (4.3.29)


In this case, the Hubble parameter is

H(t) =R(t)

R(t)=

√Λ

3, (4.3.30)

and thus constant in time. Exponential expansion due to vacuum energy isexactly what happens in the inflationary Universe.

How does the cosmological constant affect the expansion of the Universe to-day? For a matter dominated Universe with cosmological constant we write

2R

R+

R2

R2+

k

R2= Λ, 3(

R2

R2+

k

R2) = 8πGρ + Λ ⇒

d

dt(8πGρR3) = 3

d

dt(R2R + kR) − Λ

d

dtR3

= 3(2RRR + R3 + kR) − 3ΛR2R = 0 ⇒

ρR3 =3

4πM ⇒ 3

(R2

R2+

k

R2

)=

6GM

R3+ Λ ⇒

1

2R2 − Λ

6R2 − GM

R= −k

2. (4.3.31)

To get an overview over all possible solutions, we can again interpret the equationin the point particle analogy now with the potential

V (R) = −Λ

6R2 − GM

R, (4.3.32)

and with total energy −k/2. For k = 0,−1 the Universe expands forever. At theend the (Λ/3)R2 term dominates, and the Universe expands exponentially. Fork = 1 the expansion comes to an end (R = 0) if

Λ

3R2 +

2GM

R= 1. (4.3.33)

We obtain a static Universe, if at the same time R = 0, such that

Λ

3=

MG

R3⇒ Λ

3R2 +

2Λ

3R2 = 1 ⇒ R =

1√Λ

. (4.3.34)

The density in such an Einstein Universe is

ρ =3

4π

M

R3=

1

4πGΛ. (4.3.35)

The static Einstein Universe exists for a critical cosmological constant Λc =1/(3GM)2. For Λ > Λc the Universe expands forever, if it does so at present.


For Λ < Λc there are two cases. For R > 0 and large R the Universe expandsforever, and there was a minimal radius in the past. For R > 0 and small R thereis also a solution with a maximal radius and a subsequent contraction that endsin a big crunch.

4.4 Four Possible Eras in the Late Universe

It is interesting to ask what may happen to the Universe in the distant future,even if we won’t be able to test these ideas in observations. Whatever we predictwill be based on our incomplete knowledge about particle physics (and physics ingeneral) today. Thus, many of the following considerations should be consideredhighly speculative and may have to be revised as our knowledge progresses.

Following Adams and Laughlin, we may divide the future somewhat arbitrar-ily into four possible eras. During the stelliferous era that started with galaxyformation, stars dominate the Universe. Stars burn out after a while, and starformation will come to an end at about 1014 years after the bang. By then,the stellar objects have turned into brown dwarfs, white dwarfs, neutron stars,and an occasional black hole. White dwarfs and neutron stars are supported bydegeneracy pressure of electrons and neutrons, respectively. When these objectsdominate the Universe, we enter the degenerate era, which may last until about1040 years after the bang (depending on appropriate assumptions about the pro-ton life-time). At that time, all protons may have decayed, and thus all stellarobjects have disappeared. As a result, we enter the black hole era, which endswhen the last biggest black hole in the Universe has evaporated at about 10100

years. After that, in the dark era, the Universe consists of photons, as well assome electrons and positrons, which ultimately annihilate into photons at about10140 years after the bang. If nothing else happens by then, the Universe will bea very dull place, unpleasant to live in for even the most patient creatures.

Let us start discussing the stelliferous era, which we have just entered. Thecrucial observation is that, while massive stars dominate the Universe now, theywill soon run out of fuel. Depending on their mass, they will either blow upin a supernova, leaving behind a neutron star, or they will become a red giant,and then end as a white dwarf. Less massive stars, on the other hand, burn thenuclear fuel at a much more moderate rate. These objects therefore live muchlonger, up to 1013 years, and finally also become white dwarfs (without goingthrough a red giant phase). There are also brown dwarfs, objects which are notsufficiently massive to ignite a nuclear fire in their cores. They remain unchanged

4.4. FOUR POSSIBLE ERAS IN THE LATE UNIVERSE 61

during the stelliferous era.

The fate of the earth is decided when our sun enters the red giant phasein about five billion years. Most likely the earth will be swallowed and grilledwithin a time of about 50 years. Still, there is a chance that the sun looses enoughmaterial before going into the red giant phase, thus shifting the planets to furtheraway orbits. This may save the earth for a long time to come.

Star formation happens in the gaseous clouds within galaxies. However, thesupply of material is finite, and after a while there will be nothing left to makestars. One can estimate that the last star will form at about 1014 years after thebang, and it will soon be burned out. This is when the stelliferous era comes toan end.

At the end of the stelliferous era we are left with a system of many brownand white dwarfs, and an occasional neutron star or black hole, and we now enterthe degenerate era. The above stellar objects are still bound inside galaxies, butgalaxies will now die, because over time the stellar objects evaporate at a timescale of 1020 years, or fall into the black hole at the center of the galaxy. Still,during this era, occasionally a luminous star may be formed in a collision of brownor white dwarfs. One can estimate the collision rate for such processes, and onefinds that a typical galaxy will then contain about 50 luminous stars, comparedto 1010 today. At about 1030 years after the bang, all galaxies will have turnedinto a system of black holes and single evaporated stars.

The degenerate era comes to an end when the evaporated stellar objectsoutside black holes disappear due to proton decay. Depending on one’s favoriteGUT theory, this may happen at different times, e.g. 1037 years after the bang.When a proton decays, it may turn into a pion and a positron. The positronannihilates with an electron and produces two photons, and the pion also decaysinto two photons. At the end we loose a proton and an electron, and we getfour photons. This means that the stars still shine, however, with a luminosityof about 10−24 of the sun. The stellar temperature then is about 0.06 K. Onesuch star can power no more than a couple of light bulbs. Before a white dwarfdisappears completely, its electron degeneracy is lifted, because the mass gets toosmall. Similarly, a neutron star will turn into a white dwarf, when sufficientlymany neutrons have decayed. Also planets — e.g. the earth if it survived the sun’sred giant phase — will disappear via proton decay and shine by emitting photons.The luminosity, however, is ridiculously small, about 0.4 mW. Depending on theproton life-time, the degenerate era may come to an end at about 1040 years afterthe bang. Based on the standard model of particle physics alone (i.e. withoutusing GUT theories) the proton would live much longer, about 10172 years.


At the end of the degenerate era we enter the black hole era, during whichblack holes coalesce and finally disappear by emitting Hawking radiation at about10100 years after the big bang. We then enter the dark era, in which the uni-verse consists of just photons, and a few electrons and positrons. Electrons andpositrons form positronium states — typically of the horizon scale, and cascadedown to the ground state, from which they soon annihilate and become photons.Then, at an estimated 10140 years after the big bang, not much can happen,because there is nothing left that would interact with anything else.

Of course, for the Universe to survive until 10140 years after the bang, wehave assumed that the Universe is not closed. Otherwise, the fun might end in abig crunch as early as 1011 years after the bang. Even if the Universe is flat inthe part that we can observe now, as it expands a large density fluctuation mayenter the horizon later, and close it. One can use inflation to make the Universehomogeneous on scales even larger than the ones we care about today. In orderto guarantee survival of the Universe for about 10100 years, one needs about 130e-foldings, which may be possible with sufficiently elaborate inflationary modelbuilding.

Also, any amount of vacuum energy today will dominate the Universe inthe future. This should lead to another inflationary epoch. Thus, in order to gothrough the previously discussed eras, we must assume an even tinier cosmologicalconstant than what is actually observed. Otherwise, stellar objects loose causalcontact before they disappear via proton decay. One would see stars disappearthrough the horizon, and it gets dark much before the dark era.

An even more drastic effect may be due to vacuum tunneling. What if westill live in the wrong vacuum? This is quite possible, since today the vacuumenergy is not exactly zero. Then a bubble of true vacuum may nucleate anytime anywhere in the Universe. Such a bubble will grow with the velocity oflight, turning the whole Universe into the true vacuum. For us this would bedesasterous, because nothing in the old Universe will survive this change. Alsosuch an event is impossible to predict, and could basically happen tomorrow.I guess we should all feel much saver, when some clever string theorist finallyproves that the true vacuum is SU(3) ⊗ SU(2) ⊗ U(1) invariant.

Much of this last section is speculative, and — in any case — not testable inobservations. Many of the above ideas may have to be revised, as our knowledgeincreases. Still, whatever we may learn in the future, it is likely that the Universewill remain an interesting and lively place for a very long time. We should doour best to make sure that the same remains true for the earth.

Chapter 5

The Inflationary Universe

The standard model of cosmology, i.e. a Friedmann Universe with Robertson-Walker metric which is first radiation and then matter or vacuum dominated, isvery successful in describing phenomena in the early Universe. The synthesis oflight nuclei at about 1 sec to 1 min after the big bang provides the earliest test ofthe standard model of cosmology. Also the explanation of the cosmic backgroundradiation should be viewed as a big success. The standard model of cosmologytogether with the standard model of particle physics and its extensions (e.g.grand unified theories (GUT)) lead to interesting phenomena in the extremelyearly Universe, like phase transitions in the strong and electroweak sector, orthe creation of magnetic monopoles at a GUT phase transition. Although thestandard model of particle physics agrees very well with experiments, from atheoretical point of view it is not completely satisfactory due to its large numberof parameters. Also the standard model of cosmology has some problems. On theone hand, we know that due to the classical treatment of gravity it cannot be validaround the Planck time. On the other hand, it contains several parameters in theform of initial conditions, which must be fine-tuned in order to explain the largeage of our Universe. Such fine-tuning appears unnatural to most cosmologists.The Robertson-Walker ansatz for the metric was motivated by observation (e.g.by the observed isotropy of the cosmic background radiation). In the frameworkof the standard model, it remains an open question why our Universe is isotropicand homogeneous on such large scales. On the other hand, on smaller (but stillvery large) scales we observe a lot of structure, like galaxies, galaxy clusters,super-clusters as well as large voids. In the framework of the standard model itis unclear how these deviations from homogeneity have evolved from small initialfluctuations. Finally, GUTs — which can at least qualitatively account for the

63

64 CHAPTER 5. THE INFLATIONARY UNIVERSE

baryon asymmetry — unavoidably lead to monopole creation, but monopoleshave not been observed. All these puzzles find an answer in Alan Guth’s idea ofthe inflationary Universe.

One then assumes that e.g. at a first order GUT phase transition a scalarfield remained in the symmetric phase, i.e. at an unstable minimum of thetemperature-dependent effective potential, while the true minimum correspondsto the broken phase. The energy of the scalar field then acts as a cosmologicalconstant, which leads to an exponential, inflationary expansion of the Universe.In this way any effect of earlier initial conditions is eliminated, and the Universebecomes homogeneous and isotropic on the largest scales. Eventually presentmagnetic monopoles get extremely diluted, and are effectively eliminated fromthe Universe. Furthermore, the quantum fluctuations of the scalar field give riseto initial inhomogeneities, which may have evolved into the structures observedtoday. The exponential expansion leads to a tremendous increase of the scale fac-tor, and hence to a flat space. This implies Ω = 1, and hence requires a sufficientamount of vacuum energy.

When Ω = 1, this also explains the large age of our Universe (compared to thePlanck time) and hence an important condition for our own existence. Hence, theinflationary Universe allows us to avoid the anthropic principle, which explainsthe large age of the Universe by the mere fact that we exist: we could not havedeveloped in a short-lived Universe. Alan Guth’s original idea has been modifiedby Linde who has introduced the scenario of chaotic inflation. One then assumesthat initially the scalar field assumes some chaotic random values, and henceonly certain regions of space undergo inflation. In this way different parts ofthe Universe with possibly different vacuum structure evolve. It should then beviewed as a historical fact, that in our local vacuum bubble the symmetry wasbroken to the SU(3)⊗SU(2)⊗U(1) symmetry of the standard model of particlephysics and not to something else. Linde still uses the anthropic principle toexplain why this is so: in another vacuum we could not have evolved.

Since at present the Universe does no longer expand exponentially, inflation(if it ever took place) must have come to an end. This must have been associatedwith the scalar field “rolling” down to the stable minimum of broken symmetry.In this process an enormous amount of latent heat is released, which manifestsitself in an enormous entropy generation. At present that entropy resides in thecosmic background radiation. The idea of inflation thus naturally leads to aUniverse similar to the one we live in: old, flat and homogeneous and isotropic.Still, inflation should not yet be viewed as a completely established fact. It is avery attractive idea that still needs to be formulated in a realistic particle physics

5.1. DEFICIENCIES OF STANDARD COSMOLOGY 65

framework, and it needs to be tested further in observations. A strong indicationfor an inflationary epoch is the verification of Ω = 1. Still, inflation does not solveall problems of cosmology. In particular, the question of the cosmological constantremains unsolved: why is the vacuum energy so small? To answer this questionone probably needs a quantum theory of gravity, one of the great challenges intheoretical physics.

5.1 Deficiencies of Standard Cosmology

Although the standard model of cosmology is very successful in many respects,it leaves some important questions unanswered. First, there is the question ofthe age of the Universe and, related to that, its flatness, which is also connectedwith the enormous entropy in the cosmic background radiation. To understandthe age problem, let us consider the Friedmann equation

R2

R2+

k

R2=

8π

3Gρ ⇒ k

H2R2=

8πGρ

3H2− 1 =

ρ

ρc− 1 = Ω − 1, (5.1.1)

where H = R/R is the Hubble parameter, and

ρc =3H2

8πG(5.1.2)

is the critical density. It is an observational fact, that our Universe has an energydensity close to the critical one, i.e.

Ω ≈ 1. (5.1.3)

What does this mean for earlier times? As long as the Universe was matterdominated, one has

R(t) ∝ t2/3 ⇒ H(t) =R(t)

R(t)=

2

3t, (5.1.4)

and thus

Ω(t) − 1 =k

H(t)2R(t)2∝ t2

t4/3= t2/3. (5.1.5)

The Universe started to be matter dominated at an age of 105 years, and today itis about 1010 years old. Hence, when the cosmic background radiation decoupled,we had

Ω(t) − 1 = (105

1010)2/3 ≈ 10−3, (5.1.6)


i.e. the density was even closer to the critical one than it is today. At earliertimes the Universe was radiation dominated with

R(t) ∝ t1/2 ⇒ H(t) =R(t)

R(t)=

1

2t, (5.1.7)

and hence

Ω(t) − 1 =k

H(t)2R(t)2∝ t2

t= t. (5.1.8)

At the time when nucleosynthesis started, i.e. about 1 sec after the bang, onefinds

Ω(t) − 1 = 10−3 10−7

105= 10−15, (5.1.9)

which again means that the density was extremely close to critical. Going furtherback in time, e.g. to a GUT phase transition at about 10−34 sec after the bang,one obtains

Ω(t) − 1 = 10−1510−34 = 10−49, (5.1.10)

and if one goes back to the Planck time, 10−44 sec, one even finds

Ω(t) − 1 = 10−1510−44 = 10−59. (5.1.11)

At first sight, today a value of Ω close to one seems not unnatural, but whohas fine-tuned the energy density to sixty decimal places when the Universe wascreated at the Planck time?

This problem can also be expressed in terms of the entropy. In the standardcosmology the total entropy S in a comoving volume of size R(t)3 is conserved.In the radiation dominated epoch the energy density is

ρ =π2

30g∗T

4, (5.1.12)

and the entropy density is

s =2π2

45g∗T

3 =S

R3. (5.1.13)

At the Planck time the temperature is equal to the Planck mass, i.e. T = mP ≈1019 GeV, such that then

Ω − 1

Ω=

k

H2R2

3H2

8πGρ=

3k

8πGρR2=

3k

8πGρ(s

S)2/3

=3k

8π

30

π2g∗(2π2

45g∗)

2/3 1

Gm2P

S−2/3 ≈ 0.1S−2/3. (5.1.14)

5.1. DEFICIENCIES OF STANDARD COSMOLOGY 67

Using Ω − 1 ≈ 10−59 one obtains the enormous entropy S ≈ 1087 per comovingvolume. Today this entropy is contained in the cosmic background radiation.

Not only the enormous entropy of the cosmic background radiation cannotbe understood in the framework of the standard cosmology. Also its isotropypresents a puzzle. The observed isotropy of the cosmic background radiationmotivated the Robertson-Walker ansatz for the metric, but one does not un-derstand where the isotropy comes from. In a radiation or matter dominatedRobertson-Walker space-time the distance to the horizon

dH(t) = R(t)

∫ t

0

dt′

R(t′)(5.1.15)

is finite. In a matter dominated Universe, e.g. one has dH(t) ∝ 3t. This is themaximal distance between two events that are causally connected. When matterand radiation decoupled at about td = 105 years after the bang, physical processescould establish thermal equilibrium over scales of about

dH(td) ∝ 3td. (5.1.16)

Today, photons of the cosmic background radiation reach us from different direc-tions. Their regions of generation at time td cannot have been in causal contact,and still the photons have almost exactly the same temperature.

Let us consider a photon that was emitted at time td at the point ρ, and thatreaches us today at time t0 = 1010 years at ρ = 0. We then have

ds2 = dt2 − R(t)2dρ2

1 − kρ2= 0 ⇒ dt

R(t)= ± dρ√

1 − kρ2, (5.1.17)

and thus ∫ t0

td

dt′

R(t′)=

∫ ρ

0

dρ′√1 − kρ2

. (5.1.18)

A second photon reaches us today from exactly the opposite direction. Thecoordinate difference between the two points of creation of the photons is therefore2ρ. At the time td of their creation, this corresponded to a physical distance

d(td) = 2

∫ ρ

0dρ√

−gρρ = 2

∫ ρ

0dρ

R(td)√1 − kρ2

= 2R(td)

∫ t0

td

dt′

R(t′)

= 2dH(td)

∫ t0

td

dt′

R(t′)/

∫ td

0

dt′

R(t′). (5.1.19)


Since the decoupling of the photons the Universe was matter dominated, i.e.R(t) ∝ t2/3. Thus we obtain

d(td)

dH(td)= 2(3t

1/30 − 3t

1/3d )/3t

1/3d

= 2(t0td

)1/3 − 2 ≈ 2(1010

105)1/3 ≈ 40. (5.1.20)

This implies that the points at which the two photons were created were sepa-rated by 40 causality-lengths, and still the photons have the same temperature.Who has thermalized the two causally disconnected regions at extremely similartemperatures?

A GUT phase transition would unavoidably lead to the creation of magneticmonopoles via the so-called Kibble mechanism. The contribution of monopolesto the energy density leads to Ω ≈ 1011 in complete disagreement with observa-tion. How can we get rid of the unwanted monopoles, without discarding GUTsaltogether? After all, GUT are quite useful, because they may explain the baryonasymmetry of the Universe?

5.2 The Idea of Inflation

The idea of the inflationary Universe is to use vacuum energy (a non-zero cosmo-logical constant) to blow up the size of the Universe exponentially. When vacuumenergy dominates the Friedmann equation takes the form

R(t)2

R(t)2=

Λ

3⇒ R(t) = R(t0) exp(

√Λ

3(t − t0)). (5.2.1)

When the inflationary period lasts for a time ∆t the scale factor of the Universeincreases by

Z =R(t0 + ∆t)

R(t0)= exp(

√Λ

3∆t). (5.2.2)

The inflationary period ends when the vacuum energy goes to zero, because somescalar field (e.g. of a GUT) rolls down to the stable minimum of a low energybroken phase. In that moment latent heat is released, and the Universe (now inthe low energy broken phase) is reheated up to Tc. This process is accompaniedby entropy and particle creation. The entropy increase is given by

S(t0 + ∆t)

S(t0)= Z3 = exp(

√3Λ∆t). (5.2.3)

5.3. THE DYNAMICS OF INFLATION 69

Assuming that the entropy before inflation was not unnaturally large (e.g. S(t0) ≈1), in order to explain S ≈ 1087 we need an inflation factor

Z = 1029 ⇒√

Λ

3∆t = 67. (5.2.4)

An exponential expansion can also solve the horizon problem, because wethen have

R(t0 + ∆t)

∫ t0+∆t

t0

dt′

R(t′)=

R(t0 + ∆t)

R(t0)

3

Λ[1 − exp(−

√Λ

3∆t)]

≈ Z

√3

Λ=

Z

ln Z∆t. (5.2.5)

For comparison, a radiation dominated Universe (with R(t) ∝ t1/2) has

R(∆t)

∫ ∆t

0

dt′

R(t′)= 2∆t. (5.2.6)

Thus, the horizon problem is improved by a factor

Z

2 ln Z= 1027, (5.2.7)

although a factor 40 would be sufficient to solve it.

Also the monopole problem is solved by inflation, because monopoles createdbefore or during inflation are diluted by a factor Z3. In this way the monopolecontribution to the critical density is reduced to

ΩM = 1011Z−3 = 10−76. (5.2.8)

Hence, both the monopole and the horizon problem automatically disappear,once the flatness (or entropy) problem is solved.

5.3 The Dynamics of Inflation

Let us consider a scalar field φ (e.g. in a GUT) that obtains a vacuum expectationvalue. The potential of the scalar field may be given by

V (φ) =M2

2φ2 +

λ

4!φ4. (5.3.1)


Let us assume that at time t0 the system is in the symmetric phase with φ = 0.Afterwards it slowly rolls down to the broken symmetry minimum with |φ| = v.The Lagrange density of the scalar field is given by

L(φ, ∂µφ) =1

2∂µφ∂µφ − V (φ), (5.3.2)

and the corresponding energy momentum tensor is

Tµν = ∂µφ∂νφ − Lgµν . (5.3.3)

Assuming that φ is spatially constant, but time-dependent, for a Robertson-Walker metric one obtains

T00 =1

2φ2 + V (φ) = ρ,

Tii = −(1

2φ2 − V (φ))gii = −pgii ⇒ p =

1

2φ2 − V (φ). (5.3.4)

As usual, we have identified density and pressure. The Einstein field equation

Gµν = 8πGTµν , (5.3.5)

then takes the following form

3(R2

R2+

k

R2) = 8πG(

1

2φ2 + V (φ)),

2R

R+

R2

R2+

k

R2= −8πG(

1

2φ2 − V (φ)). (5.3.6)

We now identify the effective cosmological constant as

Λ = 8πGV (φ). (5.3.7)

The typical scale for the potential V (φ) is the GUT scale 1014 GeV. Hence, wecan assume V (φ) = (1014GeV)4 and thus

√Λ

3=

√8π

3

V (0)

m2P

=

√8π

3

1014

10191014GeV ≈ 109GeV. (5.3.8)

In order to reach a sufficient amount of inflation, we need it to last for at least

∆t = 67

√3

Λ≈ 10−7GeV−1 ≈ 10−32sec. (5.3.9)

This is about 100-1000 times the age of the Universe at the time of inflation.

5.3. THE DYNAMICS OF INFLATION 71

Is it possible to delay the transition to the broken phase with φ = v for sucha long time? To answer that question, we consider the equation of motion for thescalar field

φ + 3R

Rφ +

∂V

∂φ= 0. (5.3.10)

The term 3(R/R)φ, which deviates from the ordinary Klein-Gordon equation,results from the red-shift of the momentum of the scalar field. For a long-lastinginflationary period we need a small φ. Ignoring φ results in

φ = − R

3R

∂V

∂φ. (5.3.11)

In the Friedmann equation (with k = 0) we can furthermore neglect φ2 comparedto V (φ) such that

R

R=

√8πG

3V (φ) ⇒ φ = − 1√

24πGV (φ)

∂V

∂φ. (5.3.12)

The duration of inflation is now given by

∆t =

∫ t0+∆t

t0

dt′ =

∫ φ(t+∆t)

φ(t0)dφ

1

φ=

∫ v

0dφ√

24πGV (φ)(∂V

∂φ)−1

=√

24πtP

∫ v

0dφ√

V (φ)(∂V

∂φ)−1. (5.3.13)

We parameterize the potential as

V (φ) = λ(φ2 − v2)2, (5.3.14)

and we obtain∫ v

0dφ√

V (φ)(∂V

∂φ)−1 =

∫ v

0dφ

√λ

φ2 − v2

2λ(φ2 − v2)2φ=

1

4√

λ

∫ v

0

dφ

φ= ∞.

(5.3.15)We need to distort φ slightly from its unstable value φ = 0. For example, we canput it to φ = φ0 initially, and obtain

∆t =

√3π

2λtP ln(

v

φ0). (5.3.16)

The value of v is determined, for example, by the GUT scale, and λ ≈ 1. In orderto reach

∆t ≈ 10−32sec ≈ 1012tP , (5.3.17)


one needs an extremely small initial scalar field value

φ0 ≈ v exp(−1012). (5.3.18)

In order to last sufficiently long, the initial conditions for inflation must be ad-justed extremely carefully. Hence, unfortunately, we have traded the fine-tuningproblem for Ω for another one — the fine-tuning of φ0. For other forms of thescalar potential the new fine-tuning problem may be less severe, but this partic-ular scenario of inflation does not look very natural.

At the end of the inflationary period the scalar field approaches its vacuumvalue by oscillating around it. The oscillations are damped by the couplings toother fields. This leads to a release of the latent heat stored in the scalar field.The generated entropy reheats the Universe to the critical temperature Tc of theGUT transition.

New data to be obtained by the Planck satellite are expected to further testthe intriguing idea of cosmic inflation. The chances seem reasonably good thatinflation will become part of a future standard model of cosmology.

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