Lecture Notes for FY3452 Gravitation and Cosmologyweb.phys.ntnu.no/~mika/skript_grav.pdf · Preface...

Lecture Notes for FY3452

Gravitation and Cosmology

M. Kachelrieß

M. Kachelrieß

Institutt for fysikkNTNU, TrondheimNorwayemail: [email protected]

Watch out for errors, most was written late in the evening.Corrections, feedback and any suggestions always welcome!

Copyright c© M. Kachelrieß 2010–2015.Last up-date May 12, 2015

Contents

1 Special Relativity 9

1.1 Newtonian mechanics and Galilean transformations . . . . . . . . . . . . . . . 91.1.1 Newtonian gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.2 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.3 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.4 The four-velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.5 Energy and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.6 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Hamilton’s principle and the Lagrange function . . . . . . . . . . . . . 181.2.3 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . 201.2.4 Hamilton’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.5 Free relativistic particles . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Classical field theory 25

2.1 Lagrange formalism and path integrals for fields . . . . . . . . . . . . . . . . . 252.1.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.2 Symmetries and Noether’s theorem . . . . . . . . . . . . . . . . . . . . 262.1.3 Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.4 Maxwell field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1.5 Energy-momentum tensor of dust . . . . . . . . . . . . . . . . . . . . . 36

3 Basic differential geometry 37

3.1 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Affine connection and covariant derivative . . . . . . . . . . . . . . . . 413.2.2 Metric connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Riemannian normal coordinates . . . . . . . . . . . . . . . . . . . . . . 453.2.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.5 Vector differential operators . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Schwarzschild solution 49

4.1 Isometries and Killing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Gravitational redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 Orbits of massive particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 Orbits of photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Gravitational lensing 58

4

Contents

6 Black holes 62

6.1 Schwarzschild black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.2 Kerr black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.3 Black holes thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7 Einstein’s field equations 70

7.1 Curvature and the Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . . 70

7.2 Integration and the metric determinant g . . . . . . . . . . . . . . . . . . . . 72

7.3 Einstein-Hilbert action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.3.1 Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.3.2 Dynamical energy-momentum tensor . . . . . . . . . . . . . . . . . . . 75

7.3.3 Cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.3.4 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.4 Alternative theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8 Linearized gravity and gravitational waves 79

8.1 Linearized gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.1.1 Metric perturbations as a tensor field . . . . . . . . . . . . . . . . . . 79

8.1.2 Linearized Einstein equations in vacuum . . . . . . . . . . . . . . . . . 80

8.1.3 Linearized Einstein equations with sources . . . . . . . . . . . . . . . . 81

8.1.4 Polarizations states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.1.5 Detection principle of gravitational waves . . . . . . . . . . . . . . . . 83

8.2 Energy-momentum pseudo-tensor for gravity . . . . . . . . . . . . . . . . . . 84

8.3 Emission of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.3.1 Quadrupol formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.3.2 Energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

9 Cosmological models for an homogeneous, isotropic universe 89

9.1 Friedmann-Robertson-Walker metric for an homogeneous, isotropic universe . 89

9.2 Geometry of the Friedmann-Robertson-Walker metric . . . . . . . . . . . . . 91

9.3 Friedmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9.4 Scale-dependence of different energy forms . . . . . . . . . . . . . . . . . . . . 98

9.5 Cosmological models with one energy component . . . . . . . . . . . . . . . . 99

9.6 The ΛCDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9.7 Determining Λ and the curvature R0 from ρm,0, H0, q0 . . . . . . . . . . . . . 101

9.8 Particle horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

10 Cosmic relics 104

10.1 Time-line of important dates in the early universe . . . . . . . . . . . . . . . 104

10.2 Equilibrium statistical physics in a nut-shell . . . . . . . . . . . . . . . . . . . 106

10.3 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

10.3.1 Equilibrium distributions . . . . . . . . . . . . . . . . . . . . . . . . . 109

10.3.2 Proton-neutron ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

10.3.3 Estimate of helium abundance . . . . . . . . . . . . . . . . . . . . . . 112

10.3.4 Results from detailed calculations . . . . . . . . . . . . . . . . . . . . 112

10.4 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10.4.1 Freeze-out of thermal relic particles . . . . . . . . . . . . . . . . . . . 113

5

Contents

10.4.2 Hot dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11510.4.3 Cold dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

11 Inflation and structure formation 118

11.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11811.1.1 Scalar fields in the expanding universe . . . . . . . . . . . . . . . . . . 11911.1.2 Generation of perturbations . . . . . . . . . . . . . . . . . . . . . . . . 12211.1.3 Models for inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

11.2 Structure formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12411.2.1 Overview and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12411.2.2 Jeans mass of baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . 12511.2.3 Damping scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12611.2.4 Growth of perturbations in an expanding Universe: . . . . . . . . . . . 12711.2.5 Recipes for structure formation . . . . . . . . . . . . . . . . . . . . . . 12811.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6

Preface

These notes summarise the lectures for FY3452 Gravitation and Cosmology I gave in 2009and 2010. Asked to which of the three more advanced topics black holes, gravitational wavesand cosmology more time should be devoted, students in 2009 voted for cosmology, while in2010 black holes and gravitational waves were their favourites. As a result, the notes containprobably slightly more material than manageable in an one semester course.There are various differing sign conventions in general relativity possible – all of them are

in use. One can define these choices as follows

ηab = [S1] (−1,+1,+1,+1)

Rabcd = [S2] (∂cΓ

abd − ∂dΓ

abc + Γa

ecΓebd − Γa

edΓebc) ,

Gab = [S3] 8πG Tab

Rab = [S2][S3]Rbabc (0.1)

Apart from Chapters 5–7 about black holes where we need only the metric (with S1 = 1),we follow the convention of Hobson, Efstathiou and Lasenby ([HEL]). Conventions of otherauthors are summarised in the following table:

HEL dI,R MTW, H W[S1] - - + +[S2] + + + -[S3] - - + -

Some useful books:

H: J. B. Hartle. Gravity: An Introduction to Einstein’s General Relativity (BenjaminCummings)

HEL: Hobson, M.P., Efstathiou, G.P., Lasenby, A.N.: General relativity: an introduction forphysicists. Cambridge University Press 2006. [On a somewhat higher level than Hartle.]

• Robert M. Wald: General Relativity. University of Chicago Press 1986. [Uses a modernmathematical language]

• Landau, Lev D.; Lifshitz, Evgenij M.: Course of theoretical physics 2 - The classicaltheory of fields. Pergamon Press Oxford, 1975.

MTW: Misner, Charles W.; Thorne, Kip S.; Wheeler, John A.: Gravitation. Freeman NewYork, 1998. [Entertaining and nice description of differential geometry - but lengthy.]

• Schutz, Bernard F.: A first course in general relativity. Cambridge Univ. Press, 2004.

• Stephani, Hans: Relativity: an introduction to special and general relativity. CambridgeUniv. Press, 2004.

7

Contents

W: Weinberg, Steven: Gravitation and cosmology. Wiley New York, 1972. [A classics.Many applications; outdated concerning cosmology.]

• Weyl, Hermann: Raum, Zeit, Materie. Springer Berlin, 1918 (Space, Time, Matter,Dover New York, 1952). [The classics.]

Finally: If you find typos (if not, you havn’t read carefully enough), conceptional errors orhave suggestions, send me an email!

List of corrections and changes in 2011:

• Eqs. (1.59,60), (1.115), (3.40), statement after Eq. (3.55) corrected.

• a paragraph about global conservation laws added to Sec. 7.2; few phrases and areference after Eq. (11.26) about particle production in gravitational fields added.


• Eq. (9.3), sign of phase factor in the “Feynman box” on p. 21 corrected, sign in (1.11)corrected. m on the RHS of Eq. (1.49) deleted, index in Eq. (3.49) corrected.

• added motivation to begin of chapter 3, comment on torsion in sec. 3.2.3.


• some indices in box on p.13 corrected, dt in (1.103) deleted. Eqs. (3.10), (3.14) and(7.56-46): indices corrected. p.90: two squares in text “Geometry of FRW spaces”added. Eq. (8.33): prime on the RHS deleted.

• Transition affine → metric connection changed.


• Indices in (0.1) corrected, minus according to the HEL definition of Ricci added to (7.6)and (9.6ff).

• Two examples for the calculation of Christoffel symbols and Ricci tensor added (p.48and p.72).

• coefficient in (8.40) corrected.

• Some corrections in chapter 11; after (11.8) and (11.37).


• Eq. (7.16) corrected.

Thanks to Lars Husdal, Stephan Mertens and Kare Olaussen for pointing out some of theseerrors.

8

1 Special Relativity

1.1 Newtonian mechanics and Galilean transformations

Newton tried to present his mechanics in an axiomatic form. His Lex Prima (or Galileanlaw of inertia) states: Each force-less mass point stays at rest or moves on a straight line at

constant speed.

Despite the axiomatic approach, the properties of the space-time structure were assumed byNewton a priori: “Absolute space” and “absolute time.” Distinguishing between straight andcurved lines requires an affine structure of space, while measuring velocities relies on a metricstructure that allows one to measure distances. Implicitly, Newton assumed an Euclideanstructure for space, and thus the distance between two points (x1, y1, z1) and (x2, y2, z2) in aCartesian coordinate system is

∆S212 = (x1 − x2)

2 + (y1 − y2)2 + (z1 − z2)

2 (1.1)

or for infinitesimal distances

dS2 = dx2 + dy2 + dz2 . (1.2)

In a Cartesian inertial coordinate system, Newton’s lex prima becomes

d2x

dt2=

d2y

dt2=

d2z

dt2= 0 . (1.3)

Most often, we call such a coordinate system just an inertial frame. Newton’s first law is notjust a trivial consequence of its second one, but may be seen as practical definition of thosereference frames for which his following laws are valid.

Since the symmetries of Euclidean space, translations ~a and rotations R, leave Eq. (1.2)and thus also (1.3) invariant, all frames connected by ~x′ = R~x + ~a to an inertial frame areinertial frames too. Additionally, proper Galilean transformations ~x′ = ~x+~vt connect inertialframes moving with relative speed ~v,

~x′ = R~x+ ~vt+ ~a (1.4a)

t′ = t (1.4b)

Eq. (1.4b) corresponds to Newton’s and the “common sense” notation of an absolute time.Taking the time-derivative of Eq.(1.4a), we obtain the classical addition-law for velocities,

~ ′x = ~x+ ~v.

The Principle of Relativity states that identical experiments performed in different inertial

frames give identical results. Galilean transformations keep (1.3) invariant, hence Newton’sfirst law does not allow to distinguish between different inertial frames. Before Einstein, itwas thought that this principle applies only to mechanical experiments. N.B.: Newton himselfinsisted on the concept of “absolute space.”

9


Newton’s Lex Secunda: Observed from an inertial reference frame, the net force on a

particle is proportional to the time rate of change of its linear momentum,

~F =d~p

dt(1.5)

where ~p = mI~v and we call mI the inertial mass.

Newton’s law of motion is invariant against general Galilean transformations for the caseof a system of particles interacting via central forces (as gravitational or Coulomb forces). Ifin such a case ~x(t) is a solution of (1.5), then also ~x′(t) = R~x(t) + ~vt+ ~a.

1.1.1 Newtonian gravity

Newton’s gravitational law as well as the Coulomb law are examples for an instantaneousaction,

~F (~x) =∑

i

Ki~x− ~xi|~x− ~xi|3

, (1.6)

whereKi ∝ mi andKi ∝ qi, respectively. The force ~F (~x, t) depends on the distance ~x(t)−~xi(t)to all sources i (electric charges qi or masses mi) at the same time t, i.e. the force needs notime to be transmitted from ~xi to ~x.

The force law is invariant under changes x′ = Rx + d, if |x′| = |x|. The translation dcould be time-dependent via d = vt. The condition that the norm of the vector ~r is invariant,~x = ~x′, restricts the possible transformation matrices R,

x′Tx′ = xTRTRx = xTx , (1.7)

as RTR = 1. (T denotes the transposed vector or matrix). Hence rotations R and translationsd keep a force law of type (1.6) invariant. In two dimensions, the rotation matrix R can bechosen as

R =

(cosα − sinαsinα cosα

)

. (1.8)

The factor K in Newton’s law is −GmgMg, where we introduced analogue to the electriccharge in the Coulomb law the “charge” mg characterizing the strength of the gravitationalforce between different particles. Surprisingly, one finds mI = mg and we can drop the index.

Since the gravitational field is conservative, we can introduce a potential φ via

~F = −m~∇φ (1.9)

with

φ(x) = − GM

|~x− ~x′| . (1.10)

Analogue to the electric field ~E = −~∇φ we can introduce a gravitational field, ~g = −~∇φ.We then obtain ~∇ · ~g(~x) = −4πGρ(~x) and as Poisson equation,

∆φ(~x) = 4πGρ(~x) , (1.11)

where ρ is the mass density, ρ = dm/d3x.

10


1.1.2 Minkowski space

Comparing the non-relativistic Poisson equation ∆φ = 0 and Maxwell’s wave equation, (∆−(1/c2)∂2t )Aµ = 0 suggests that the latter are invariant under transformations that keep thespace-time interval

∆s2 ≡ c2t2 − ~x2 = c2t′ 2 − ~x′ 2 (1.12)

invariant. This means that both the time and the space difference between two eventsmeasured in the coordinate system K and K ′ may be different, but that the difference∆s2 ≡ c2t2 − ~x2 is the same for all inertial systems. As a result, time and space shouldbe seen not as separate entities but as components of a four-vector x with componentsxa = (ct, ~x), a = 0, 1, 2, 3, that can be mixed by Lorentz transformations.

The distance of infinitesimally close space-time events is also called line-element ds. It isgiven by

ds2 = c2dt2 − dx2 − dy2 − dz2 . (1.13)

More precisely, the line-element ds is defined as norm of the displacement vector

ds = dxµeµ (1.14)

Choosing as basis the coordinate vectors to xa = (ct, ~x), its components are

dsa = dxa = (cdt, d~x) . (1.15)

We compare now our physical requirement on the distance of space-time events, Eq. (1.13),with the general result for the scalar product of two vectors a and b. If these vectors havethe coordinates ai and bi in a certain basis ei, then we can write

a · b = (aiei) · (bjej) = aibj(ei · ej) . (1.16)

Thus we can evaluate the scalar product between any two vectors, if we know the symmetricmatrix g composed of the N2 products of the basis vectors,

gij(x) = ei(x) · ej(x) = gij(x) . (1.17)

This symmetric matrix g is called metric tensor.

Applying this now for the displacement vector, we obtain

ds2 = ds · ds = gijdxidxj

!= c2dt2 − dx2 − dy2 − dz2 . (1.18)

Hence the metric tensor g becomes for the special case of an Cartesian inertial frame diagonalwith elements

gij =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

≡ ηij (1.19)

11


Einstein’s summation convention:

1. Two equal indices, of which one has to be an upper and one an lower index, imply sum-mation. We use Greek letters for indices from one to three, µ = 1, 2, 3, and Latin lettersfor indices from zero to three, i = 0, 1, 2, 3 (but at some places it might be opposite.)Thus

aµbµ =

3∑

µ=1

aµbµ and aib

i =3∑

i=0

aibi .

2. Summation are dummy indices and can be freely exchanged; the free indices of the LHSand RHS of an equation have to agree. Hence

8 = aibi = cijd

ij = cikdik

is okay, whileaµ = bµ or aµ = bν or aii = cijd

ij = wiks

ikl

compares apples to oranges.3. Vector components with lower and upper indices have to be distinguished, because—as

we will see soon—they are connected by the metric tensor as follows:

ai = gijaj .

Since g 6= 1, the components of ai and ai differ in general.

4. Form the differential of a function f(~x). Then

df =∂f

∂xdx+

∂f

∂ydy +

∂f

∂zdz =

∂f

∂xµdxµ ≡ ∂µf dx

µ .

Thus an upper index in the denominator counts as lower index, and vice versa.

Since the metric g is indefinite, the norm√a · a of a vector a can be

a · a > 0 time− like, (1.20)

a · a = 0 nullvector, (1.21)

a · a < 0 space− like. (1.22)

The cone of all null-vectors starting from a point P is called lightcone. The time-like regioninside the light-cone consists of two parts, past and future. Only events inside the past light-cone can influence the physics at point P, while P can influence only his future light-cone.The line x = x(σ) describing the position of an observer is called worldline. To measure

the distance along the worldline it is convenient to introduce the proper-time τ ,

dτ2 = ds2/c2 . (1.23)

Thus τ is real with units of time, measuring the time along the worldline. For instance,

τ12 =

∫ 2

1dτ =

∫ 2

1[dt2 − (dx2 + dy2 + dz2)/c2]1/2 (1.24)

=

∫ 2

1dt [1− (1/c2)((dx/dt)2 + (dy/dt)2 + (dz/dt)2)]1/2 (1.25)

=

∫ 2

1dt [1− v2/c2]1/2 < t2 − t1 . (1.26)

12


This relativistic effect of time dilation is often summarized by “A moving clock runs slower.”

1.1.3 Lorentz transformations

We want to find now the coordinate transformations that keep the space-time interval

∆s2 ≡ c2t2 − ~x2 = c2t′ 2 − ~x′ 2 (1.27)

invariant. This transformations should be similar to the rotation matrices R and are calledLorentz transformations L. If we replace t by it in ∆s2, the difference between two space-timeevents becomes the normal Euclidean distance. Similarly, the identity cos2 α+ sin2 α = 1 foran imaginary angle ψ = iα becomes cosh2 ψ−sinh2 ψ = 1. Thus a close correspondence existsbetween rotations R in Euclidean space that leave ∆~x2 invariant and Lorentz transformationsL that leave ∆s2 invariant. We try therefore as a guess

x = ct′ sinhψ + x′ coshψ , (1.28)

y = y′ , z = z′ , (1.29)

ct = ct′ coshψ + x′ sinhψ . (1.30)

Compared to a normal coordinate rotation in Euclidean space, the trigonometric functionsare replaced by hyperbolic ones.Consider now in system K the origin of the system K ′. Then x′ = 0 and

x = ct′ sinhψ , (1.31)

ct = ct′ coshψ . (1.32)

Dividing the two equations givesx

ct= tanhψ . (1.33)

Since x/t is the relative velocity of the two systems, we have identified the physical meaningof the imaginary “rotation angle ψ” as

tanhψ =v

c≡ β . (1.34)

Using the following identities

coshψ =1

√

1− tanh2 ψ=

1√

1− β2≡ γ (1.35)

sinhψ =tanhψ

√

1− tanh2 ψ=

β√

1− β2= γβ (1.36)

in (1.28) gives the standard form of the Lorentz transformations,

x =x′ + vt′

√

1− (v/c)2= γ(x′ + βct′) (1.37)

ct =ct′ + v/cx′√

1− (v/c)2= γ(ct′ + βx′) . (1.38)

The inverse transformation is obtained by inserting −v and exchanging primed and unprimedquantities. Remarks:

13


• Any other 4-tupel of quantities that transforms as xµ is called (components of a) four-vector.

• We use from now on natural units c = ~ = k = 1, but keep still Newton’s constantG 6= 1.

Relativity of simultaneity Consider in a space-time diagram two frames moving with relativespeed v. Then the time axes are rotated by the angle ψ = arctanh(v). The same holds forlines of constant time. Thus there are sequences of events A,B which one observer (call it1) sees as tA < tB, another one (2) as tB < tA. However, this can happen only if A and B arespace-like separated.

1.1.4 The four-velocity

What is the relativistic generalization of the velocity ~v = d~x/dt? The nominator d~x hasalready the right behavior to become part of a four-vector, if the denominator would beinvariant. We use therefore instead dt the invariant proper time dτ = ds and write

ua =dxa

dτ. (1.39)

The four-velocity is thus the tangent vector to the world-line xa(τ) of a particle. Writtenexplicitly, we have

u0 =dt

dτ=

1√1− v2

= γ (1.40)

and

uµ =dxµ

dτ=

dxµ

dt

dt

dτ=

vi√1− v2

= γvµ . (1.41)

Hence the four-velocity is ua = (γ, γ~v) and its norm is

u · u = γ2 − γ2v2 = γ2(1− v2) = 1 . (1.42)

How to guess physical tensors: We may guess just by considering the different number ofdegrees, which physical 3-dim. quantities should be combined into 4-dim. tensors:

1. A vector a has 4 = 3 + 1 components, i.e. combines a vector ~a and a scalar s, that arerelated by a physical law. For instance current conservation

∂tρ+ ~∇ ·~j = 0

suggest to combine (ρ,~j) = ja and ∂a = (∂t, ~∇) into four-vectors.

2. An anti-symmetric tensor F has 3 + 2 + 1 = 6 components, i.e. combines two vector like~E and ~B.

3. A general tensor F has 16 = 9 + 2 · 3 + 1 components, i.e. combines from a 3-dim. pointof view a 3-dim tensor, 2 vectors and a scalar.

14


1.1.5 Energy and momentum

After having constructed the four-velocity, the simplest guess for the four-momentum is

pa = mua = (γm, γm~v) . (1.43)

For small velocities, v ≪ 1, we obtain

~p =

(

1 +v2

2− . . .

)

m~v (1.44)

p0 = m+mv2

2− . . . = m+ Ekin,nr + . . . (1.45)

Thus we can interpret the components as pα = (E, ~p). The norm follows with (1.42) imme-diately as

p · p = m2 . (1.46)

Solving for the energy, we obtainE =

√

m2 + p2 (1.47)

including the famous E = mc2 as special case for a particle at rest.We postulate now that in relativistic mechanics Newton’s law becomes

f =dp

dτ(1.48)

where we introduced the four-force f .

Pure forces Since both u and p consist of only three independent component, we expectthat this holds true also for the four-force f . We form the scalar product of u and f ,

u · f = u · d(mu)

dτ= u · udm

dτ+mu · du

dτ=

dm

dτ. (1.49)

In the last step we used twice that u · u = 1. Since all electrons ever observed have thesame mass, no force should exist which changes m. As a consequence, we have to ask thatall physical acceptable force-laws satisfy u · f = 0; such forces are called pure forces.As application, we want to generalize the three-dimensional Lorentz force,

~FL = q( ~E + ~v × ~B) . (1.50)

The RHS should become a yet unknown tensor F of rank two contracted with the four-velocity,

fi = qFijuj . (1.51)

The requirement of a pure force means

uifi = qFijuiuj = 0 , (1.52)

and thus the tensor F must be antisymmetric, Fij = −Fji.The force law fi = qFiju

j becomes simplest in a frame with u = (1, 0, 0, 0). Then fi = qFi0

or −~F = −e ~E and −f0 = −qF00 = 0.

15


1.1.6 Doppler effect

We consider as last example how the wave-vector of a photon (ω,~k) transforms under aLorentz transformation, Inserting into the transformation law for any four vector the timecomponent k0 = ω of the four-vector kµ = (ω,~k),

ω′ =ω − V kx

[1− V 2]1/2, (1.53)

where we denote with ω the observed frequency. Setting kx = k cosα = ωc cosα, where α the

angle between the wave vector and the direction of the movement of the source, we obtainthe relativistic Doppler formula

ω

ω′=

[1− V 2

]1/2

1− V cosα. (1.54)

For small relative velocities, v ≪ 1, the quadratic term in the nominator can be neglected,

ω ≈ ω′(1 + V cosα) . (1.55)

Thus one reproduces the non-relativistic Doppler formula where only the radial relative ve-locity enters. For α = 0, photons are emitted in the same direction as the source moves, andthe observed frequency ω is “blue-shifted”.

∗∗∗ Aberration and beaming ∗∗∗ Classical aberration was discovered 1725 by James Bradley:He discovered and later explained an apparent motion1 of celestial objects caused by the finitespeed of light and the motion of the Earth in its orbit around the Sun. Assume e.g. to observea star at the moment when it is right in the zenith. Since the Earth moves, you have to pointthe telescope a bit “ahead” and thus the position of stars changes in the course of one year.To obtain the relativistically correct form for the aberration formula, we rewrite first the

Lorentz transformations for x and t in differential form,

dx = γ(dx′ + βdt′) , (1.56)

dy = dy′ , dz = dz′ , (1.57)

dt = γ(dt′ + βdx′) . (1.58)

Then we obtain the transformation law for velocities ~u as

u‖ =dx‖

dt=

u′‖ + v

1 + v~u′‖/c2

(1.59)

u⊥ =dx⊥dt

=u′⊥

γ(1 + v~u′‖/c2). (1.60)

The so-called aberration formula connects the angle between the velocities ~u and ~u′ in thetwo frames,

tanϑ =u⊥u‖

=u′ sinϑ′

γ(u′ cosϑ′ + v). (1.61)

1This motion depends on the velocity of the observer perpendicular to the line-of-sight to the star, but isindependent of the distance to the star. The apparent motion of stars induced by classical aberration ismuch larger than the parallax due to the motion of the Earth around the Sun.

16

1.2 Classical mechanics

The most interesting case is the aberration of light. We set ϑ′ = π/2, i.e. we ask how lightemitted into one half-sphere appears in a different inertial system. With u = u′ = c it followsthen

tanϑ =c

γv(1.62)

or ϑ ≈ 1/γ for v → c. Thus a fast moving source emits photons mainly in a cone of openingangle ϑ ≈ 1/γ around its forward direction.


We review briefly the Lagrangian and Hamiltonian formulation of classical mechanics.

1.2.1 Calculus of variations

A map F [f(x)] from a certain space of functions f(x) into R is called a functional. We willconsider functionals from the space C2[a : b] of (at least) twice differentiable functions betweenfixed points a and b. Extrema of functionals are obtained by the calculus of variations. Letus consider as functional the action S defined by

S[L(qi, qi)] =

∫ b

adt L(qi, qi, t) , (1.63)

where L is a function of the 2n independent functions qi and qi = dqi/dt as well as of theparameter t. In classical mechanics, we call L the Lagrange function of the system, q are itsgeneralised coordinates, qi the corresponding velocities and t is the time. The extremum ofthis action gives the paths from a to b which are solutions of the equation of motions for thesystem described by L. We discuss in the next section how one derives the correct L given aset of interactions and constraints.The calculus of variations shows how one find those paths that extremize such functionals:

Consider an infinitesimal variation of the path, qi(t) → qi(t) + δqi(t) with δqi(t) = εq(t) thatkeeps the endpoints fixed, but is otherwise arbitrary. The resulting variation of the functionalis

δS =

∫ b

adt δL(qi, qi, t) =

∫ b

adt

(∂L

∂qiδqi +

∂L

∂qiδqi)

. (1.64)

We can eliminate the variation δqi of the velocities, integrating the second term by partsusing δ(qi) = d/dt(δqi),

δS =

∫ b

adt

[∂L

∂qi− d

dt

(∂L

∂qi

)]

δqi +

[∂L

∂qiδqi]b

a

. (1.65)

The boundary term vanishes, because we required that the variations δqi are zero at theendpoints a and b. Since the variations are otherwise arbitrary, the sum in the first brackethas to be zero for an extremal curve, δS = 0. Paths that satisfy δS = 0 are classically allowed.The equations resulting from the condition δS = 0 are called the Euler-Lagrange equationsof the action S,

δS

δqi=∂L

∂qi− d

dt

∂L

∂qi= 0 , (1.66)

17


and give the equations of motion of the system specified by L. Physicists call these equationsoften simply Lagrange equations or, especially in classical mechanics, Lagrange equations ofthe second kind.

Example: Show that the variation δ and the time-derivative d/dt acting on q commute:We consider a variation δq = q − q0 = εη(t) with ε time-independent and denote the original pathwith qo. Then

dq

dt= q = q0 + εη

or

δ(q) ≡ q − q0 = εη =d

dt(δq) ,

since we keep ε constant.

The Lagrangian L is not uniquely fixed: Adding a total time-derivative, L′ = L+df(q, t)/dtdoes not change the resulting Lagrange equations,

S′ = S +

∫ b

adt

df

dt= S + f(q(b), tb)− f(q(a), ta) , (1.67)

since the last two terms vanish varying the action with the restriction of fixed endpoints aand b.

1.2.2 Hamilton’s principle and the Lagrange function

The observation that the solutions of the equation of motions can be obtained as the extremaof an appropriate functional (“the action S”) of the Lagrangian L subject to the conditionsδqi(a) = δqi(b) = 0 is called Hamilton’s principle or the principle of least action. Note thatthe last name is a misnomer, since the the extremum can be also a maximum or saddle-pointof the action.We derive now the Lagrangian L of a free non-relativistic particle from the Galilean principle

of inertia. More precisely, we use that the homogeneity of space and time forbids that Ldepends on ~x and t, while the isotropy of space implies that L depends only on the norm ofthe velocity vector, but not on its direction,

L = L(v2).

Let us consider two inertial frames moving with the infinitesimal velocity ~ε relative to eachother. Then a Galilean transformation connects the velocities measured in the two frames as~v′ = ~v+~ε. The Galilean principle of relativity requires that the laws of motion have the sameform in both frames, and thus the Langrangians can differ only by a total time-derivative.Expanding the difference δL in ε gives with δv2 = 2~v~ε

δL =∂L

∂v2δv2 = 2~v~ε

∂L

∂v2. (1.68)

The difference has to be a total time-derivative. Since v = q, the derivative term ∂L/∂v2 hasto be independent of ~v. Hence, L ∝ v2 and we call the proportionality constant m/2, and thetotal expression kinetic energy T ,

L = T =1

2mv2 . (1.69)

18


Example: Check the relativity principle for finite relative velocities:For

L′ =1

2mv′2 =

1

2m(~v + ~V )2 =

1

2mv2 +m~v · ~V +

1

2mV 2

or

L′ = L+d

dt

(

m~x · ~V +1

2mV 2t

)

.

Thus the difference is indeed a total time derivative.

We can write the velocity with dl2 = dx2 + dy2 + dz2 as

v2 =(dl)2

(dt)2= gik

dxi

dt

dxk

dt, (1.70)

where the quadratic form g is the metric tensor. For instance, in spherical coordinates dl2 =dr2 + r2 sin2 ϑdφ2 + r2dϑ2 and thus

T =1

2m(

r2 + r2 sin2 ϑφ2 + r2ϑ2)

. (1.71)

Choosing the appropriate coordinates, we can account for constraints: The kinetic energy of aparticle moving on sphere with radius R would be simply given by T = m/2R2(sin2 ϑφ2+ ϑ2).For a system of non-interacting particles, L is additive, L =

∑

a12mav

2a. If there are

interactions (assumed for the moment to dependent only on the coordinates), then we subtracta function V (~r1, ~r2, . . .) called potential energy.We can now derive the equations of motions for a system of n interacting particles,

L =∑

a

1

2mav

2a − V (~r1, ~r2, . . . , ~rn) . (1.72)

using the Lagrange equations,

mad~vadt

= − ∂V

∂~xa= ~Fa . (1.73)

We can change from Cartesian coordinates to arbitrary (or “generalized”) coordinated forthe n particles,

xa = fa(q1, . . . , qn), xa =∂fa

∂qkqk . (1.74)

Substituting gives

L =1

2aikq

iqj − V (qi) , (1.75)

where the matrix aik(q) is a quadratic function of the velocities qi that is apart from thefactors ma identical to the metric tensor on the configuration space qn. Finally, we define thecanonically conjugated momentum pi as

pi =∂L

∂qi. (1.76)

A coordinate qi that does not appear explicitly in L is called cyclic. The Lagrange equationsimply then ∂L/∂qi = const., so that the corresponding canonically conjugated momentumpi = ∂L/∂qi is conserved.

19


Feynman’s approach to quantum theory:

The whole information about a quantum mechanical system is contained in its time-evolutionoperator U(t, t′). Its matrix elements K(x′, t′;x, t) (propagator or Green’s function) in thecoordinate basis connect wavefunctions at different times as

ψ(x′, t′) =

∫

d3xK(x′, t′;x, t)ψ(x, t) .

Feynman proposed the following connection between the propagator K and the classical actionS,

K(x′, t′;x, t) = N

∫ q′

q

Dq exp(ıS) ,

where Dq denotes the “integration over all paths.” Hence the difference between the classicaland quantum world is that in the former only paths extremizing the action S are allowed whilein the latter all paths weighted by exp(ıS) contribute.For a readable introduction see R. P. Feynman, A. R. Hibbs: Quantum mechanics and path integrals or R. P.

Feynman (editor: Laurie M. Brown), Feynman’s thesis : a new approach to quantum theory.

1.2.3 Symmetries and conservation laws

Quantities that remain constant during the evolution of a mechanical system are called inte-grals of motions. Seven of them that are connected to the fundamental symmetries of spaceand time are of special importance: These are the conserved quantities energy, momentumand angular momentum.

Energy The Lagrangian of a closed system depends, because of the homogeneity of time,not on time. Its total time derivative is

dL

dt=∂L

∂qiqi +

∂L

∂qiqi . (1.77)

Replacing ∂L/∂qi by (d/dt)∂L/∂qi, it follows

dL

dt= qi

d

dt

∂L

∂qi+∂L

∂qiqi =

d

dt

(

qi∂L

∂qi

)

. (1.78)

Hence the quantity

E ≡ qi∂L

∂qi− L (1.79)

remains constant during the evolution of a closed system. This holds also more generally, e.g.in the presence of static external fields, as long as the Lagrangian is not time-dependent.

We have still to show that E coincides indeed with the usual definition of energy. Using asL = T (q, q)− U(q), where T is quadratic in the velocities, we have

qi∂L

∂qi= qi

∂T

∂qi= 2T (1.80)

and thus E = 2T − L = T + U .

20


Momentum Homogeneity of space implies that an translation by a constant vector of aclosed system does not change its properties. Thus an infinitesimal translation from ~r to ~r+~εshould not change L. Since velocities are unchanged, we have (summation over a particles)

δL =∑

a

∂L

∂~ra· δ~ra = ~ε ·

∑

a

∂L

∂~ra. (1.81)

The condition δL = 0 is true for arbitrary ~ε, if

∑

a

∂L

∂~ra= 0 . (1.82)

Using again Lagrange’s equations, we obtain

∑

a

d

dt

∂L

∂~va=

d

dt

∑

a

∂L

∂~va= 0 . (1.83)

Hence, in a closed mechanical system the momentum vector of the system

~ptot =∑

a

∂L

∂~va=∑

a

ma~va = const. (1.84)

is conserved.The condition (1.82) signifies with ∂L/∂~ra = −∂U/∂~ra that the sum of forces on all particles

is zero,∑

a~Fa = 0. For the particular case of a two-particle system, ~Fa = −~Fb, we have thus

derived Newton’s third law, the equality of action and reaction.

Isotropy We consider now the consequences of the isotropy of space, i.e. search the conservedquantity that follows from a Lagrangian invariant under rotations. Under an infinitesimalrotation by δ~φ both coordinates and velocities change,

δ~r = δ~φ× ~r , (1.85)

δ~v = δ~φ× ~v . (1.86)

Inserting the expression into

δL =∑

a

(∂L

∂~raδ~ra +

∂L

∂~vaδ~va

)

= 0 (1.87)

gives, using also the definition ~pa = ∂L/∂~va as well as the Lagrange equation ~pa = ∂L/∂~ra,

δL =∑

a

(

~pa · δ~φ× ~ra + ~pa · δ~φ× ~va

)

= 0 . (1.88)

Permuting the factors and extracting δ~φ gives

δ~φ ·∑

a

(

~ra × ~pa + ~va × ~pa

)

= δ~φ · d

dt

∑

a

~ra × ~pa = 0 . (1.89)

Thus the angular momentum

~M =∑

a

~ra × ~pa = const. (1.90)

is conserved.

21


1.2.4 Hamilton’s equations

In the Lagrange formalism, we describe a system by specifying its generalized coordinates andvelocities using the Lagrangian, L = L(qi, qi). An alternative is to use generalized coordinatesand canonically conjugated momenta pi defined as

pi =∂L

∂qi. (1.91)

The passage from qi, qi to qi, pi is a special case of a Legendre transformation. We writefirst the differential of L as

dL =∂L

∂qidqi +

∂L

∂qidqi . (1.92)

Using the definition in Eq. (1.91) and ∂L/∂qi = d/dt(∂L/∂qi) = pi we find

dL = pidqi + pi dq

i . (1.93)

We use the Leibniz rulepi dq

i = d(pi qi)− dpi q

i,

to form a total differential, bringing the d(pq) term on the LHS and reversing the sign,

d(pi qi − L) = −pidqi + qidpi . (1.94)

The function on the LHS to which d is applied is the energy, cf. Eq. (1.79), expressed by piand qi. It is called Hamiltonian or Hamilton function H with

H(qi, pi, t) = piqi − L . (1.95)

If we consider H as function of qi and pi, then its differential is

dH =∂H

∂pidpi +

∂H

∂qidqi . (1.96)

A comparison with Eq. (1.94) gives Hamilton’s equations,

qi =∂H

∂pi, pi = −∂H

∂qi. (1.97)

Consider an observable O = O(x, p, t). Its time-dependence is given by

dO

dt=∂O

∂pipi +

∂O

∂qiqi +

∂O

∂t= −∂O

∂pi

∂H

∂qi+∂O

∂qi∂H

∂pi+∂O

∂t, (1.98)

where we used Hamilton’s equations. If we define the Poisson brackets A,B between twoObservables A and B as

A,B =∂A

∂qi∂B

∂pi− ∂A

∂pi

∂B

∂qi, (1.99)

then we can rewrite Eq. (1.98) as

dO

dt= O,H+ ∂O

∂t. (1.100)

22


Note that we have obtained a formal correspondence between classical and quantum mechan-ics: The time-evolution of an observable O in the Heisenberg picture is given by the sameequation as in classical mechanics, if the Poisson bracket is changed against a commutator.Since the Poisson bracket is antisymmetric, we find

dH

dt=∂H

∂t. (1.101)

Hence the Hamiltonian H is a conserved quantity, if and only if H is time-independent.

1.2.5 Free relativistic particles

We introduced the proper-time τ to measure the time along the worldline of a particle,

τ12 =

∫ 2

1dτ =

∫ 2

1[dt2 − (dx2 + dy2 + dz2)]1/2 (1.102)

=

∫ 2

1[ηµνdx

µdxν ]1/2 . (1.103)

If we use a different parameter σ, e.g. such that σ(τ = 1) = 0 and σ(τ = 2) = 1, then

τ12 =

∫ 1

0dσ

[

ηµνdxµ

dσ

dxν

dσ

]1/2

. (1.104)

Note that τ12 is invariant under a reparameterisation σ′ = f(σ).We check now if the choice L = [ηµνdx

µ/dσ dxν/dσ]1/2 is sensible for a free particle: Lis Lorentz-invariant with xµ = (~x, t) as dynamical variables, while σ plays the role of theparameter time t in the non-relativistic case. The Lagrange equations are

d

dσ

∂L

∂(dxα/dσ)=

∂L

∂xα. (1.105)

Consider e.g. the x1 component, then

d

dσ

∂L

∂(dx1/dσ)=

d

dσ

(1

L

dx1

dσ

)

= 0 . (1.106)

Since L = dτ/dσ, it followsd2x1

dτ2= 0 (1.107)

and the same for the other coordinates.Next we want to add an interaction term with an electromagnetic field. The simplest action

is

Sem = −q∫

dxµAµ(x) = −q∫

dσdxµ

dσAµ(x) . (1.108)

Any candidate for Sem should be invariant under a change of gauge,

Aµ(x) → Aµ(x) + ∂µΛ(x) . (1.109)

This the case, since the induced change in the action,

δΛSem = −q∫ 2

1dσ

dxµ

dσ

∂Λ(x)

∂xµ= −q

∫ 2

1dΛ = −q[Λ(2)− Λ(1)] . (1.110)

23


drops out for fixed endpoints from δS.The Lagrange equations

d

dσ

∂L

∂(dxα/dσ)= − ∂L

∂xα(1.111)

give now with

L = L0 + Lint = m[ηµνdxµ/dσ dxν/dσ]1/2 − q

dxµ

dσAµ(x) . (1.112)

d

dσ

[mdxα/dσ

[ηµνdxµ/dσ dxν/dσ]1/2− qAµ

]

= −q dxλ

dσ

Aλ(x)

∂xα. (1.113)

Performing the differentiation of A(x(σ) with respect to σ and moving it to the RHS, we find

md

dσ

[dxα/dσ

[ηµνdxµ/dσ dxν/dσ]1/2

]

= −q(dxλ

dσ

Aλ

∂xµ− dxλ

dσ

Aµ

∂xλ

)

= −q dxλ

dσFλµ . (1.114)

We choose σ = τ and obtain the Lorentz equation

md2xα

dτ2= −q uλ F α

λ . (1.115)

24

2 Classical field theory

2.1 Lagrange formalism and path integrals for fields

A field is a function of space-time xµ with values in “field space.” The space of values the fieldcan take may be characterized by its transformation properties under Lorentz transformations(scalar φ, vector Aµ, tensor gµν ,. . . or spinor ψa, ψab, . . . fields) and (typically) Lie groups likeR, U(1), SU(n),. . . Thus we have to generalize Hamilton’s principle to a collection of fieldsφa(x

µ), a = 1, . . . , k. To ensure Lorentz invariance, we consider a scalar Lagrange density Lthat may, analogously to L(q, q), depend on the fields and its first derivatives ∂µφa. There isno explicit time-dependence, since “everything” should be explained by the fields and theirinteractions. The Lagrangian L(φa, ∂µφa) is obtained by integrating L over a given spacevolume V .The action S is thus the four-dimensional integral

S[L(φa, ∂µφa)] =∫ b

adt L(φa, ∂µφa) =

∫

Ωd4x L(φa, ∂µφa) , (2.1)

where Ω = V × [ta : tb]. If we choose the Lagrange density L as a Lorentz scalar, we willobtain automatically Lorentz-invariant equations of motions.A variation εφa ≡ δφa of the fields leads to a variation of the action,

δS =

∫

Ωd4x

(∂L∂φa

δφa +∂L

∂(∂µφa)δ(∂µφ

a)

)

, (2.2)

where we have to sum over fields (a = 1, . . . , k) and the Lorentz index µ = 0, . . . , 3. Weeliminate again the variation of the field gradients ∂µφ

a by a partial integration using Gauß’theorem,

δS =

∫

Ωdx4

[∂L∂φa

− ∂µ

(∂L

∂(∂µφa)

)]

δφa = 0 . (2.3)

The boundary term vanishes, since we require that the variation is zero on the boundary ∂Ω.Thus the Lagrange equations for the fields φa are

∂L∂φa

− ∂µ

(∂L

∂(∂µφa)

)

= 0 . (2.4)

If the Lagrange density L is changed by a four–dimensional divergence, the same equationsof motions result.

2.1.1 Conservation laws

Let jµ be a conserved vector field in Minkowski space,

∂µjµ = 0 . (2.5)

25


Thend

dt

∫

Vd3x j0 = −

∫

∂Vd~S ·~j (2.6)

and

Q =

∫

Vd3x j0 (2.7)

is a globally conserved quantity, if there is no outgoing flux ~j through the boundary ∂V . Toshow that Q is a Lorentz invariant quantity, we have to rewrite Eq. (2.7) as a tensor equation.

Consider

Q(t = 0) =

∫

d4x jµ(x)∂µϑ(n · x) (2.8)

with ϑ the step function and n a unit vector in time direction, n · x = x0 = t. Then

Q(t = 0) =

∫

d4x j0(x)∂0ϑ(x0) =

∫

d4x j0(x)δ(x0) =

∫

d3x j0(x) (2.9)

and hence Eqs. (2.7) and (2.8) are equivalent. Since one of them is a tensor equation, Q isLorentz invariant.

In the same way, we can construct in Minkowski space globally conserved quantities Q forconserved tensors: If for instance ∂µT

µν = 0, then

P ν =

∫

d3x T 0ν (2.10)

is a globally conserved vector, and similarly for higher-rank tensors.

2.1.2 Symmetries and Noether’s theorem

Noether’s theorem gives a formal connection between the continuous symmetries of a physicalsystem and the resulting conservation laws. We derive this theorem in two steps, consideringin the first one only internal symmetries.

We assume that our collection of fields φa has a continuous symmetry group. Thus we canconsider an infinitesimal change δφa that keeps L(φa, ∂µφa) invariant,

0 = δL =δLδφa

δφa +δL

δ∂µφaδ∂µφa . (2.11)

Now we exchange δ∂µ against ∂µδ in the second term and use then the Lagrange equations,δL/δφa = ∂µ(δL/δ∂µφa), in the first term. Then we can combine the two terms using theLeibniz rule,

0 = δL = ∂µ

(δL

δ∂µφa

)

δφa +δL

δ∂µφa∂µδφa = ∂µ

(δL

δ∂µφaδφa

)

. (2.12)

Hence the invariance of L under the change δφa implies the existence of a conserved current,∂µJ

µ = 0, with

Jµ1 =

δLδ∂µφa

δφa . (2.13)

26


If the transformation δφa leads to change in L that is a total four-divergence, δL = ∂µKµ,

and boundary terms can be dropped, then the equation of motions are still invariant. Theconserved current is changed to Jµ = δL/δ∂µφa δφa −Kµ.In the second step, we consider a variation of the coordinates, x′µ = xµ + δxµ. Such a

variation implies a change of the fields

φ′a(x′µ) = φa(xµ) + δφa(xµ) (2.14)

and thus also of the Lagrange density. Note that we compare now the field at different points.We can recycle our old result, if we split the total variation δφa(xµ) into a part that comesfrom the changed shape of the field at the same point, φ′a(xµ) − φa(xµ) and the remainder.The first part we know already and we denote it now as δφa(xµ). The reminder we candetermine as follows

δφa(xµ) = φ′a(x′µ)− φa(xµ) = φ′a(x

′µ)− φa(xµ) +

[φ′a(x

′µ)− φ′a(x

′µ)]

(2.15)

= δφa(xµ)−[φ′a(x

′µ)− φ′a(xµ)

](2.16)

= δφa(xµ)−∂φ′a(xµ)

∂xµδxµ = δφa(xµ)−

∂φa(xµ)

∂xµδxµ . (2.17)

In the last line we made first a Taylor expansion and replaced then φ′a by φa, possible forinfinitesimal variations.We consider now the variation of the action S implied by the coordinate change x′µ =

xµ + δxµ. Such a variation implies not only a variation of L but also of the integrationmeasure d4x,

δS =

∫

Ω

[dx4(δL) + (δdx4)L

]. (2.18)

The two integration measures d4x and d4x′ are connected by the Jacobian, i.e. the determinantof the transformation matrix

aij =∂xi

∂xj. (2.19)

Using again that the variation is infinitesimal,

J =

∣∣∣∣

∂xi

∂xj

∣∣∣∣=

1 + ∂δx0

∂x0∂δx0

∂x1 · · ·∂δx1

∂x0 1 + ∂δx1

∂x1

. . . . . .

= 1 +

∂δxµ

∂xµ(2.20)

Inserting first this result and using then (2.17)

δS =

∫

Ωdx4

[

δL+ L ∂δxµ

∂xµ

]

=

∫

Ωdx4

[

δL+∂L∂xµ

δxµ + L ∂δxµ

∂xµ

]

. (2.21)

We combine the last two terms using the Leibniz rule,

δS =

∫

Ωdx4

[

δL+∂

∂xµ(Lδxµ)

]

= 0 . (2.22)

If the system is invariant under the coordinate transformation, the variation of the action iszero, δS = 0, and we found as additional contribution the conserved vector Jµ

2 = Lδxµ.As last step, we replace L and split the total conserved current into two pieces proportional

to δφa and δxµ,

Jµ =∂L

∂(∂µφa)δφa −

[∂L

∂(∂µφa)

∂φa∂xν

− gµνL]

δxν . (2.23)

27


Translations Invariance under translations x′µ = xµ + εµ means φ′a(x′) = φa(x) or δφa = 0.

Hence we obtain a conserved tensor

Tµν =∂L

∂(∂µφa)

∂φa∂xν

− gµνL (2.24)

called canonical energy-momentum (density) tensor. Integrating we obtain as four conservedquantities the energy-momentum of the fields,

P ν =

∫

d3x T 0ν (2.25)

Remarks:

• in general, this procedure leads to Tµν 6= Tνµ. We need however a symmetric tensor assource of gravity.

• can be always symmetrized adding a 4-dim. divergence.

• we will learn later a different method, leading directly to a symmetric energy-momentumtensor (called “dynamical” energy-momentum tensor).

Finally we note that the consequences of Noether’s theorem are different for finite and infinitedimensional symmetries. In the latter case, in general no conserved global charge can bedefined.

Angular momentum If the tensor Tµν is symmetric, we can construct six more conservedquantities. Define

Mµνλ = xν Tµλ − xλ Tµν , (2.26)

then M is conserved with respect to the index µ,

∂µMµνλ = δνµ T

µλ − δλµTµν = T νλ − T λν = 0 , (2.27)

provided that T νλ = T λν . In this case,

Mνµ =

∫

d3xM0νµ = xν T 0µ − xµ T 0ν , (2.28)

is a globally conserved tensor.

Lorentz invariance Rotations and boosts lead to a linear change of coordinates. For in-finitesimal proper Lorentz transformations, Λµ

ν = δµν + ωµν +O(ω2) and

ΛµνΛ

σµ = δσν = (δµν + ωµ

ν)(δσµ + ωσ

µ) = δσν + ωσν + ω σ

ν +O(ω2) . (2.29)

Thus the matrix of infinitesimal generators of Lorentz transformations is antisymmetric,

ωµν = −ωνµ (2.30)

and has six independent elements.

28


The transformed fields φa(x) depends linearly on δωµν and φa(x),

φa(x) = φa(x) +1

2δωµν(I

µν)abφb(x) . (2.31)

The symmetric part of (Iµν)ab does not contribute, because of the antisymmetry of the δωµν .Hence we can choose also the (Iµν)ab as antisymmetric and thus there exists six generators(Iµν)ab corresponding to three boosts and three rotations. The explicit form of the generatorsI (the “matrix representation of the Lorentz group” for spin s) depends on the spin of theconsidered field, as the well-known different transformation properties of scalar (S = 0),spinor (s = 1/2) and vector (s = 1) fields under rotations show.We insert now δxµ = xµ + δωµνxν and δφa = 1

2δωµν(Iµν)abφb(x) into the definition of the

Noether current, Eq. (2.23).

Jµ =∂L

∂(∂µφa)δφa −

[∂L

∂(∂µφa)

∂φa∂xν

− gµνL]

δxν (2.32)

=∂L

∂(∂µφa)

1

2δωµν(Iνλ)abφb(x)− Tµνδω

νλxλ︸︷︷︸

12δωνλ(Tµνxλ−Tµλxµ)

=1

2δωνλMµνλ (2.33)

with the definition

Mµνλ = Tµλxµ − Tµνxλ +∂L

∂(∂µφa)(Iνλ)abφb(x) . (2.34)

2.1.3 Klein-Gordon field

Real field The Klein-Gordon equation as a relativistic wave equation was derived by analogywith the (free) Schrodinger equation,

i∂tψ = H0ψ =p2

2m, (2.35)

and its replacementsE → i∂t ~p→ −i~∇x . (2.36)

With the same replacements, the relativistic E2 = m2 + p2 becomes

(+m2)φ = 0 with = ηµν∂µ∂ν . (2.37)

How do we guess the correct L? The correspondence q ↔ ∂µφ means that the “kinetic”field energy is quadratic in the field derivatives, while mass terms (missing in a nonrelativistictheory) are quadratic in the fields. Thus we try

L =1

2ηµν∂

µφ∂νφ− 1

2m2φ2 (2.38)

With∂

∂φ,α(ηµνφ,µφ,ν) = ηµν

(δαµφ,ν + δαν φ,µ

)= ηανφ,ν + ηµαφ,µ = 2φ,α (2.39)

the Lagrange equation becomes

∂L∂φ

− ∂α

(∂L∂φ,α

)

= −m2φ− ∂α∂αφ = 0 . (2.40)

29


Thus the Lagrange density (2.38) leads to the Klein-Gordon equation. Next we calculate theenergy-momentum tensor,

Tµν =∂L∂φ,µ

φ,ν − ηµνL = φ,µφ,ν − ηµνL . (2.41)

that is already symmetric. The corresponding 00 component is

T 00 = φ,0φ,0 − L =1

2

[

(∂tφ)2 + (∇φ)2 +m2φ2

]

> 0 . (2.42)

Complex field If two field exist with the same mass m, one might wish to combine theminto one complex field,

φ =1√2(φ1 + iφ2) . (2.43)

The resulting Lagrangian density is just the sum,

L = L1 + L2 = ηµν∂µφ∗∂νφ−m2|φ|2 (2.44)

andT 00 = 2φ,0φ

∗,0 − L = |∂tφ|2 + |∇φ|2 +m2|φ|2 > 0 . (2.45)

We consider now plane-wave solutions to the Klein-Gordon equation,

φ = N exp(ikx) (2.46)

If we insert ∂µφ = ikµφ into L, we find L = 0 and thus

T 00 = 2|N |2k0k0 . (2.47)

The other components are necessarily

Tµν = 2|N |2kµkν . (2.48)

Since Tµν is symmetric, we can find a frame in which Tµν is diagonal with T ∝diag(E2, p2x, p

2y, p

2z).

2.1.4 Maxwell field

Should we try to form the Lagrangian density out of the potentials (φ, ~A) or the field-strengths( ~E, ~B) for a (massive) spin-1 particle? In the first case, e.g.

L = ∂µAν∂µAν +m2AµA

µ . (2.49)

This Lagrange density has four degrees of freedom, while a massive/massless spin-1 field hasonly 3/2 dynamical degrees of freedom. Moreover, either the A0 or the spatial components ~Awill give a negative contribution to T 00: We encounter for the first time the general problemthat a field with spin s > 0 has more components that it has physical degrees of freedom. Thefield equations have thus two “senses”: They enforce the correct energy-momentum relationand they act as a constraint to project out the unphysical degrees of freedom.Using the field strengths requires first to form out of ~E and ~B a Lorentz invariant tensor.

We derive the connection of this tensor to the potential A by postulating the coupling jµAµ

between photons and matter.

30


Field tensor We consider a charged particle interacting with an external electromagneticfield A. As Lagrangian for the free particle we use L = −mds or

S0 = −∫ b

ads m (2.50)

Next we want to add an interaction term with an electromagnetic field. The simplest actionis

Sem = −q∫

dxµAµ(x) = −q∫

dσdxµ

dσAµ(x) . (2.51)

Any candidate for Sem should be invariant under a change of gauge,

Aµ(x) → Aµ(x) + ∂µΛ(x) . (2.52)

This the case, since the induced change in the action,

δΛSem = −q∫ 2

1dσ

dxµ

dσ

∂Λ(x)

∂xµ= −q

∫ 2

1dΛ = q[Λ(2)− Λ(1)] (2.53)

drops out from δS for fixed endpoints.With ds =

√dxµdxµ, the variation of the action is

δS = −δ∫ b

a(mds+ eAµdx

µ) = −∫ b

a

(

mdxµδdxµ

ds+ eAµd(δx

µ) + eδAµdxµ

)

. (2.54)

We integrate the first two terms partially,

δS =

∫ b

a

(

md

(dxµds

)

δxµ + eδxµdAµ − eδAµdxµ

)

(2.55)

and use as “always” that the boundary terms vanish. Next we introduce uµ = dxµ/ds anduse

δAµ =∂Aµ

∂xνδxν , dAµ =

∂Aµ

∂xνdxν . (2.56)

Then

δS =

∫ b

a

(

mduµδxµ + e

∂Aµ

∂xνδxµdxν − e

∂Aµ

∂xνδxνdxµ

)

. (2.57)

Finally, we rewrite in the first term duα = duα/ds ds, in the second and third dxα = uαdsand exchange the summation indices µ and ν in the third term. Then

δS =

∫ b

a

[

mduµds

− e

(∂Aν

∂xµ− ∂Aµ

∂xν

)

uν]

δxµds = 0 . (2.58)

For arbitrary variations, the brackets has to be zero and we obtain as equation of motion

mduµds

= fµ = e

(∂Aν

∂xµ− ∂Aµ

∂xν

)

uν ≡ eFµνuν . (2.59)

Comparing this equation with its 3-dimensional correspondence, the equation of motionwith the Lorentz force

m~a = ~FL = q( ~E + ~v × ~B) (2.60)

31


or using the definition F = dA, we can read-off the components of F (the first index denotesraws, the second columns) as

Fµν =

0 −Ex −Ey −Ez

Ex 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

(2.61)

and

Fµν =

0 Ex Ey Ez

−Ex 0 −Bz By

−Ey Bz 0 −Bx

−Ez −By Bx 0

. (2.62)

Connection between 3- and 4-dim. formulation of electrodynamics

The first raw of Fµν = Aν,µ −Aµ,ν reads with Aµ = (φ,−Ak) and ∂ν = ∂/∂xµ = (∂/∂t,∇k) as

F0k = Ek = ∂0Ak − ∂kA0 = −∂Ak

∂t−∇kφ

or ~E = −∂t ~A−∇φ. We go in the opposite direction for ~B = ∇× ~A. In components, we havee.g.

B1 = ∂2A3 − ∂3A

2 = ∂3A2 − ∂2A3 = F32

and similarly for the other components.The force law fµ = eFµνu

ν becomes simplest in a frame with u = (1, 0, 0, 0). Then fµ = eFµ0

or −~F = −e ~E.

Now we can rewrite the Maxwell equations as

∂αFαβ = jβ (2.63)

and

∂αFβγ + ∂βF

γα + ∂γFαβ = 0 . (2.64)

The last equation is completely antisymmetric in all three indices, an contains therefore onlyfour independent equations. It is equivalent to

∂αFαβ = 0 , (2.65)

where

Fαβ =1

2εαβγδFγδ (2.66)

and εαβγδ is the complete antisymmetric tensor in four dimensions,

ε0123 = +1, (2.67)

and all even permutations, −1 for odd permutations and zero otherwise.

32


Current conservation and gauge invariance We take the divergence of Maxwells equation(2.63),

∂ν∂µFµν = ∂νj

ν . (2.68)

Since ∂ν∂µ is symmetric and Fµν antisymmetric, the summation of the two factors has to bezero,

∂ν∂µFµν = −∂ν∂µF νµ = −∂µ∂νF νµ = −∂ν∂µFµν . (2.69)

Thus current conservation,

∂νjν = 0 , (2.70)

follows from the anti-symmetry of F. The latter followed in turn from the assumed gauge-invariant action Sem = −q

∫dxµAµ.

Consider next the transformation of F under a gauge transformation,

Aµ → A′µ = Aµ + ∂µχ . (2.71)

F ′µν = ∂µA

′ν − ∂νA

′µ = Fµν + ∂µ∂νχ− ∂ν∂µχ = Fµν . (2.72)

Thus the gauge invariance of F is again closely connected to the fact that it is an antisymmetrictensor, formed by derivatives of A.

Differential forms:

A surface in R3 can be described at any point either by its two tangent vectors e1 and e2 or by

the normal n. They are connected by a cross product, n = e1 × e2, or in index notation,

ni = εijke1,jek,2 . (2.73)

In four dimensions, the ε tensor defines a map between 1-3 and 2-2 tensors. Since ε is antisym-metric, the symmetric part of tensors would be lost; Hence the map is suited for antisymmetrictensors.Antisymmetric tensors of rank n can be seen also as differential forms: Functions are forms oforder n = 0; differential of functions are an example of order n = 1,

df =∂f

∂xidxi (2.74)

Thus the dxi form a basis, and one can write in general

A = Ai dxi . (2.75)

For n > 1, the basis has to be antisymmetrized,

F =1

2Fµνdx

µ ∧ dxν (2.76)

with dxµ ∧ dxν = −dxν ∧ dxµ. Looking at df , we can define a differentiation of a form ω withcoefficients w and degree n as an operation that increases its degree by one to n+ 1,

dω = dwα,...,βdxn+1 ∧ dxα ∧ . . . ∧ dxβ (2.77)

Thus we have F = dA. Moreover, it follows d2ω = 0 for all forms. Hence a gauge transformationF′ = d(A+ dχ) = F.

33


Wave equation The Maxwell equation (2.63) consists of four equations for the six compo-nents of F. Thus we need either a second equation, i.e. Eq. (2.64), or we should transformEq. (2.63) into an equation for the four components of the four-potential A. Let’s do thelatter and insert the definition of A,

∂µFµν = ∂µ(∂

µAν − ∂νAµ) = Aν − ∂µ∂νAµ = jν . (2.78)

If we choose the Lorentz gauge, ∂µAµ = 0, we obtain the familiar wave equation for photons,

Aν = jν . (2.79)

Lagrange density The free field equation is

∂νFµν = 0 . (2.80)

In order to find L, we multiply by a variation δA that vanishes at the endpoints ta and tb.Then we integrate over Ω = V × [ta : tb], and perform a partial integration,

∫

Ωd4x ∂νF

µν δAµ = −∫

Ωd4x Fµν δ(∂νAµ) = 0 . (2.81)

Next we note that

(Aα,β −Aβ,α)(Aα,β −Aβ,α) = 2(Aα,β −Aβ,α)A

α,β (2.82)

and thus

Fµν δ(∂µAν) =1

2Fµν δFµν . (2.83)

Applying the product rule, we obtain as final result

−1

4δ

∫

Ωd4x FµνF

µν = 0 (2.84)

and

L = −1

4FµνF

µν . (2.85)

Note that we expressed L trough F , but L should be viewed nevertheless as function of A.

Energy-momentum tensor I According to Eq. (2.24) we have

T νµ =

∂Aσ

∂xµ∂L

∂(∂Aσ/∂xν)− δνµL . (2.86)

Since L depends only on the derivatives Aµ,ν , we can use the following short-cut: We know

already that

δL = −1

4δ(FµνF

µν) = Fµν δ(∂νAµ) . (2.87)

Thus∂L

∂(∂Aσ/∂xν)= F σν = −F νσ (2.88)

34


and

T νµ = −∂Aσ

∂xµF νσ +

1

4δνµFστF

στ . (2.89)

Raising the index µ and rearranging σ, we have

Tµν = −∂Aσ

∂xµF ν

σ +1

4ηµνFστF

στ . (2.90)

This result in neither gauge invariant (contains A) nor symmetric. To symmetrize it, weshould add

∂Aµ

∂xσF ν

σ =∂

∂xσ(AµF ν

σ) . (2.91)

The last step is possible for a free electromagnetic field, ∂σFνσ = 0, and shows that we are

allowed to add the LHS. Then the two terms combine to F , and we get

Tµν = −Fµσ F νσ +

1

4ηµνFστF

στ . (2.92)

It is gauge invariant and symmetric. Its trace is zero, Tµµ = 0.

Energy-momentum tensor II We use now an alternative derivation of T, using Maxwell’sequations and Lorentz’s law, Eq. (2.59). Rewriting the latter first as an equation betweenforce density and current density, and inserting then Maxwell’s law gives

fµ = Fµνjν = Fµν

∂F νλ

∂xλ. (2.93)

We use now the product rule to rewrite this as

fµ =∂

∂xλ

(

FµνFνλ)

− F νλ∂Fµν

∂xλ. (2.94)

We now try to rewrite the second term as a symmetric divergence. Starting from

F νλ∂Fµν

∂xλ=

1

2

(

Fνλ∂Fµν

∂xλ+ Fλν

∂Fµλ

∂xν

)

(2.95)

we have exchanged the indices λ and ν in the second term. Then we use first in both factorsof the second term the antisymmetry of F,

=1

2Fνλ

(∂Fµν

∂xλ+∂Fλµ

∂xν

)

(2.96)

then Maxwell (2.64),

= −1

2Fνλ

∂Fνλ

∂xµ(2.97)

= −1

4

∂

∂xµ

(

FνλFνλ)

= −1

4δλµ

∂

∂xλ(FστF

στ ) (2.98)

Combining, we get

fµ =∂

∂xλ

(

FµνFνλ +

1

4δλµ FστF

στ

)

=∂T λµ

∂xλ. (2.99)

Considering the electromagnetic field as an external (i.e. fixed) field, the divergence of itsenergy-momentum tensor corresponds to the four-force density on charges.

35


2.1.5 Energy-momentum tensor of dust

In cosmology, one important contribution to the total energy content of the universe is non-relativistic matter: Galaxies or cold dark matter, traditionally called dust.Consider first how the energy density ρ of dust transforms. An observer moving relative to

the rest frame of dust measures ρ′ = γdm/(γ−1dV ) = γ2ρ. Hence the energy density shouldbe the 00 component of a tensor of rank 2, alas the energy-momentum tensor Tαβ . The onlyrelevant four-vector from which we can built Tαβ is the four-velocity uα of the dust.

Tαβ = ρuαuβ . (2.100)

giving in the rest frame the desired T 00 = ρ.Let us now check the consequences of ∂αT

αβ = 0. We look first at the α = 0 component,

∂tρ+ ~∇ · (ρ~u) = 0 . (2.101)

This corresponds to the mass continuity equation and, because of E = m for dust, at thesame time to energy conservation. Next we consider the α = 1, 2, 3 = i components,

∂t(ρuj) + ∂i(u

iujρ) = 0 (2.102)

or~u∂tρ+ ρ∂t~u+ ~u~∇ · (ρ~u) + (~u · ~∇)~uρ = 0 . (2.103)

Taking the continuity equation into account, we obtain the Euler equation for a force-freefluid without viscosity,

ρ∂t~u+ (~u · ~∇)~uρ = 0 . (2.104)

36

3 Basic differential geometry

We motivate this chapter about differential geometry by giving some arguments why a rela-tivistic theory of gravity should replace Minkowski space by a curved manifold. Let us startby reviewing three basic properties of gravitation.

1.) The idea underlying the equivalence principle emerged in the 16th century, when amongothers Galileo Galilei found experimentally that the acceleration g of a test mass ina gravitational field is universal. Because of this universality, the gravitating massmg = F/g and the inertial mass mi = F/a are identical in classical mechanics, a factthat puzzled already Newton. While mi = mg can be achieved for one material by aconvenient choice of units, there should be in general deviations for test bodies withdiffering compositions.

Knowing more forces, this puzzle becomes even stronger: Contrast the acceleration ofa particle in a gravitational field to the one in a Coulomb field. In the latter case, twoindependent properties of the particle, namely its charge q determining the strength ofthe electric force acting on it and its mass mi, i.e. the inertia of the particle, are neededas input in the equation of motion. In the case of gravity, the “gravitational charge”mg coinicides with the inertial mass mi.

The equivalence of gravitating and inertial masses has been tested already by Newtonand Bessel, comparing the period P of pendula of different materials,

P = 2π

√

mil

mgg, (3.1)

but finding no measurable differences. The first precision experiment giving an upperlimit on deviations from the equivalence principle was performed by Lorand Eotvos in1908 using a torsion balance. Current limits for departures from universal gravitationalattraction for different materials are |∆gi/g| < 10−12.

2.) Newton’s gravitational law postulates as the latter Coulomb law an instantaneous inter-action. Such an interaction is in contradiction to special relativity. Thus, as interactionsof currents with electromagnetic fields replace the Coulomb law, a corresponding de-scription should be found for gravity. Moreover, the equivalence of mass and energyfound in special relativity requires that, in a loose sense, energy not only mass shouldcouple to gravity: Imagine a particle-antiparticle pair falling down a gravitational po-tential well, gaining energy and finally annihilating into two photons moving the gravi-tational potential well outwards. If the two photons would not loose energy climbing upthe gravitational potential well, a perpetuum mobile could be constructed. If all formsof energy act as sources of gravity, then the gravitational field itself is gravitating. Thusthe theory is non-linear and its mathematical structure is much more complicated thanthe one of electrodynamics.

37


3.) Gravity can be switched-off locally, just by cutting the rope of an elevator: Inside afreely falling elevator, one does not feel any gravitational effects except for tidal forces.The latter arise if the gravitational field is non-uniform and tries to stretch the elevator.Inside a sufficiently small freely falling system, also tidal effects plays no role. Thisallows us to perform experiments like the growing of crystalls in “zero-gravity” on theInternational Space Station which is orbiting only at an altitude of 300 km.

Motivated by 2.), Einstein used 1.), the principle of equivalence, and 3.) to derive generalrelativity, a theory that describes the effect of gravity as a deformation of the space-timeknown from special relativity.

In general relativity, the gravitational force of Newton’s theory that accelerates particlesin an Euclidean space is replaced by a curved space-time in which particles move force-freealong geodesic lines. In particular, photons move still as in special relativity along curvessatisfying ds2 = 0, while all effects of gravity are now encoded in the form of the line-elementds. Thus all information about the geometry of a space-time is contained in the metric gµν .

3.1 Tensor algebra

Covariant and contravariant tensors Consider two n dimensional coordinate systems x andx and assume that we can express the xi as functions of the xi,

xi = f(x1, . . . , xn) (3.2)

or more briefly xj = xj(xi). Forming the differentials, we obtain

dxi =∂xi

∂xjdxj . (3.3)

The transformation matrix

aij =∂xi

∂xj(3.4)

is a n × n dimensional matrix with determinant (“Jacobian”) J = det(a). If J 6= 0 in thepoint P , we can invert the transformation,

dxi =∂xi

∂xjdxj = aij dx

j . (3.5)

The transformation matrices are inverse to each other, aijajk = δik. According to the product

rule of determinants, J(a) = 1/J(a).

A contravariant vector X (or contravariant tensor of rank one) has a n-tupel of componentsthat transforms as

Xi =∂xi

∂xjXj . (3.6)

This definition guarantees that the tensor itself is an invariant object, since the transformationof its components is cancelled by the transformation of the basis vectors,

X = Xidxi = Xidx

i = X (3.7)

38

3.1 Tensor algebra

By definition, a scalar field φ remains invariant under a coordinate transformation, i.e.φ(x) = φ(x) at all points. Consider now the derivative of φ,

∂φ(x(x))

∂xi=∂xj

∂xi∂φ

∂xj. (3.8)

This is the inverse transformation matrix and we call a covariant vector (or covariant tensorof rank one) any n-tupel transforming as

Xi =∂xj

∂xiXj . (3.9)

More generally, we call an object T that transforms as

T i,...,nj,...,m =

∂xi

∂xi′. . .

∂xn

∂xn′

︸︷︷︸

n

∂xj′

∂xj. . .

∂xm′

∂xm︸︷︷︸

m

T i′,...,n′

j′,...,m′ (3.10)

a tensor of rank (n,m).

Dual basis We defined earlier gij = ei ·ej . Now we define a dual basis ei with metric gij via

ei · ej = δji . (3.11)

We want to determine the relation of gij with gij . First we set

ei = Aijej , (3.12)

multiply then with ek and obtain

gik = ei · ek = Aijej · ek = Aik . (3.13)

Hence the metric gij maps covariant vectors Xi into contravariants vectors Xi, while gijprovides a map into the opposite direction. In the same way, we can use g to raise and lowerindices of any tensor.Next we multiply ei with ek = gkle

l,

δik = ei · ek = ei · gklel = gklgil (3.14)

or

δik = gklgli . (3.15)

Thus the components of the covariant and the contravariant metric tensors, gij and gij , areinverse matrices of each other.

Example: Spherical coordinates 1:

Calculate for spherical coordinates x = (r, ϑ, φ) in R3,

x′1 = r sinϑ cosφ ,

x′2 = r sinϑ sinφ ,

x′3 = r cosϑ ,

39


the components of gij and gij , and g ≡ det(gij).From ei = ∂x′j/∂xiej , it follows

e1 =xj∂r

e′j = sinϑ cosφ e′1 + sinϑ sinφ e′2 + cosϑe′3 ,

e2 =xj∂ϑ

e′j = r cosϑ cosφ e′1 + r cosϑ sinφ e′2 − r sinϑe′3 ,

e3 =xj∂φ

e′j = −r sinϑ sinφ e′1 + r sinϑ cosφ e′2 .

Since the ei are orthogonal to each other, the matrices gij and gij are diagonal. From the definitiongij = ei · ej one finds gij = diag(1, r2, r2 sin2 ϑ) Inverting gij gives gij = diag(1, r−2, r−2 sin−2 ϑ). Thedeterminant is g = det(gij) = r4 sin2 ϑ. Note that the volume integral in spherical coordinates is givenby

∫

d3x′ =

∫

d3x J =

∫

d3x√g =

∫

drdϑdφ r2 sinϑ ,

since gij =∂xk′

∂xi∂xl′

∂xj g′

kl and thus det(g) = J2 det(g′) = J2 with det(g′) = 1.

3.2 Tensor analysis

Manifolds A manifold M is any set that can be continuously parametrized. The numberof independent parameters needed to specify uniquely any point of M is its dimension, theparameters are called coordinates. Examples are e.g. rigid rotations in R

3 (with 3 Eulerangles, dim = 3) or the phase space (qi, pi) with dim = 2n of classical mechanics.

We require the manifold to be smooth: the transitions from one set of coordinates toanother one, xi = f(xi, . . . , xn), should be C∞. In general, it is impossible to cover all Mwith one coordinate system that is well-defined on all M. (Example are spherical coordinateon a sphere S2, where φ is ill-defined at the poles.) Instead one has to use patches of differentcoordinates that (at least partially) overlap.

Doing analysis on a manifold requires an additional structure that makes it possible tocompare e.g. tangent vectors living in tangent spaces at different points of the manifold: Aprescription is required how a vector should be transported from point P to Q in order tocalculate a derivative. Mathematically, many different schemes are possible (and sensible),but we should require the following:

• A derivative of a tensor should be again a tensor. This may require a modification ofthe usual partial derivative; this modification should however vanish for a flat space.

• The length of a vector should remain constant being transported along the manifold.(Think about the four velocity |u| = 1 or |p| = m.)

• A vector should not be twisted “unnecessarily” being transported along the manifold.

A Riemannian manifold is a differentiable manifold with a symmetric, positive-definitetensor-field gij . Space-time in general relativity is a four-dimensional pseudo-Riemannian(also called Lorentzian) manifold, where the metric has the signature (1,3).

40

3.2 Tensor analysis

3.2.1 Affine connection and covariant derivative

Consider how the partial derivative of a vector field, ∂cXa, transforms under a change of

coordinates,

∂′cX′a =

∂

∂x′c

(∂x′a

∂xbXb

)

=∂xd

∂x′c∂

∂xd

(∂x′a

∂xbXb

)

(3.16)

=∂x′a

∂xb∂xd

∂x′c∂dX

b +∂2x′a

∂xb∂xd∂xd

∂x′c︸︷︷︸

≡−Γabc

Xb . (3.17)

The first term transforms as desired as a tensor of rank (1,1), while the second term—causedby the in general non-linear change of the coordinate basis—destroys the tensorial behavior.If we define a covariant derivative ∇cX

a of a vector Xa by requiring that the result is a tensor,we should set

∇cXa = ∂cX

a + ΓabcX

b . (3.18)

The n3 quantities Γabc (“affine connection”) transform as

Γ′abc =

∂x′a

∂xd∂xe

∂x′b∂xf

∂x′cΓdef +

∂2xd

∂x′b∂x′c∂x′a

∂xd. (3.19)

Using ∇cφ = ∂cφ and requiring that the usual Leibniz rule is valid for φ = XaXa leads to

∇cXa = ∂cXa − ΓbacXb . (3.20)

For a general tensor, the covariant derivative is defined by the same reasoning as

∇cTa...b... = ∂cT

a...b... + Γa

dcTd...b... + . . .− Γd

bcTa...d... − . . . (3.21)

Note that it is the last index of the connection coefficients that is the same as the index ofthe covariant derivative. The plus sign goes together with upper (superscripts), the minuswith lower indices.From the transformation law (3.19) it is clear that the inhomogeneous term disappears for

an antisymmetric combination of the connection coefficients Γ in the lower indices. Thus thiscombination forms a tensor, called torsion,

T abc = Γa

bc − Γacb . (3.22)

We consider only symmetric connections, Γabc = Γa

cb, or torsionless manifolds. We will justifythis choice later, when we consider the geodesic motion of a classical particle.

Parallel transport We say a tensor T is parallel transported along the curve x(σ), if itscomponents T a...

b... stay constant. In flat space, this means simply

d

dσT a...b... =

dxc

dσ∂cT

a...b... = 0 . (3.23)

In curved space, we have to replace the normal derivative by a covariant one. We define thedirectional covariant derivative along x(σ) as

D

dσ=

dxc

dσ∇c . (3.24)

41


Then a tensor is parallel transported along the curve x(σ), if

D

dσT a...b... =

dxc

dσ∇cT

a...b... = 0 . (3.25)

3.2.2 Metric connection

The covariant derivative defined with the help of the affine connection guarantees that thederivative maps tensors into tensors. However, relations like ds2 = gikdx

idxj or gikpipj = m2

become invariant under parallel transport only, if the metric tensor is covariantly constant,

∇cgab = ∇cgab = 0 . (3.26)

A connection satisfying Eq. (3.26) is called metric compatible and leaves lengths and anglesinvariant under parallel transport. This requirement guarantees that we can introduce lo-cally in the whole space-time Cartesian inertial coordinate systems where the laws of specialrelativity are valid. Moreover, these local inertial systems can be consistently connected byparallel transport using an affine connection satisfying the constraint 3.26.Now we want to build in this constraint into a new, more specific definition of the covariant

derivative: We consider again how a vector V and its components V a = ea · V transformunder a coordinate change. The derivative of a vector V transforms as a tensor,

∂aV → ∂aV =∂xb

∂xa∂bV , (3.27)

since V is an invariant object. If we consider however its components V a = ea ·V, then themoving coordinate basis in curved space-time, ∂ae

b 6= 0, introduces an additional term

∂aVb = eb · (∂aV) +V · (∂aeb) (3.28)

in the derivative ∂aVb. The first term eb ·(∂aV) transforms as a tensor, since both eb and ∂aV

are tensors. This implies that the combination of the two remaining terms has to transformas tensor too, which we define as (new) covariant derivative

∇aVb ≡ eb · (∂aV) = ∂aV

b −V · (∂aeb) . (3.29)

The first relation tells us that we can view the covariant derivative ∇aVb as the projection of

∂aV onto the direction eb.If we expand now the partial derivative of the basis vectors as a linear combination of the

basis vectors,∂le

k = −Γklje

j , and ek,l = Γjklej , (3.30)

and call the coefficients connection coefficients, our two definitions of the covariant derivativeseem to agree. However, we did not differentiate in (3.28) the “dot”, i.e. the scalar product.As a consequence, the conncetion defined by (3.30) will be compatible to the metric, whilefor a general affine connection the covariant derivative in (3.29) would contain an additionalterm proportional to ∇ag

bc.Now we differentiate the definition of the metric tensor, gab = ea · eb, with respect to xc,

∂cgab = (∂cea) · eb + ea · (∂ceb) = Γdaced · eb + eaΓ

dbced = (3.31)

= Γdacgdb + Γd

bcgad . (3.32)

42

3.2 Tensor analysis

We obtain two equivalent expression by a cyclic permutation of the indices a, b, c,

∂bgca = Γdcbgda + Γd

abgcd︸︷︷︸

(3.33)

∂agbc = Γdbagdc︸︷︷︸

+Γdcagbd . (3.34)

We add the first two terms and subtract the last one. Using additionally the symmetriesΓabc = Γa

cb and gab = gba, the underlined terms cancel, and dividing by two we obtain

1

2(∂cgab + ∂bgac − ∂agbc) = Γd

cbgad . (3.35)

Multiplying by gea and relabeling indices gives as final result

Γabc = abc ≡ 1

2gad(∂bgdc + ∂cgbd − ∂dgbc) . (3.36)

This equation defines the Christoffel symbols abc (aka Levi-Civita connection aka Rieman-nian connection): It is the unique connection on a Riemannian manifold which is metriccompatible and torsion-free (i.e. symmetric). Admitting torsion, on the RHS of Eq. (3.36)three permutations of the torsion tensor T a

bc would appear. Such a connection would be stillbe a metric connection, but not torsion-free.We now check our claim that the connection (3.36) is metric compatible. First, we define1

Γabc = gadΓdbc . (3.37)

Thus Γabc is symmetric in the last two indices. Then it follows

Γabc =1

2(∂bgac + ∂cgba − ∂agbc) . (3.38)

Adding 2Γabc and 2Γbac gives

2(Γabc + Γbac) = ∂bgac + ∂cgba − ∂agbc (3.39)

+ ∂agbc + ∂cgab − ∂bgac = 2∂cgab (3.40)

or∂cgab = Γabc + Γbac . (3.41)

Applying the general rule for covariant derivatives, Eq. (3.21), to the metric,

∇cgab = ∂cgab − Γdacgdb − Γd

bcgad = ∂cgab − Γacb − Γbca (3.42)

and inserting Eq. (3.41) shows that

∇cgab = ∇cgab = 0 . (3.43)

Hence ∇a commutes with contracting indices,

∇c(XaXa) = ∇c(gabX

aXb) = gab∇c(XaXb) (3.44)

1We showed that g can be used to raise or to lower tensor indices, but Γ is not a tensor.

43


and “conserves” the norm of vectors. (Exercise: Repeat these steps including torsion.)

Since we can choose for a flat space an Cartesian coordinate system, the connection coef-ficients are zero and thus ∇a = ∂a. This suggests as general rule that physical laws valid inMinkowski space hold in general relativity, if one replace ordinary derivatives by covariantones and ηij by gij .

We derive finally a formula for the contracted connection coefficients that will be lateruseful. We start from the definition,

Γaab =

1

2gad(∂agdb + ∂bgad − ∂dgab) =

1

2gad∂bgad ., (3.45)

where we exchanged the indices a and d in the last term.

Since ∂cg = ggab∂cgab (cf. Lemma below), we have

∂cg = ggab(Γabc + Γbac) = g(Γbbc + Γa

ac) = 2gΓaac . (3.46)

Thus the contraction of the connection coefficients is given by

Γaab =

1

2g−1∂bg =

1

2∂b ln |g| =

1√

|g|∂b√

|g| (3.47)

These expressions are useful to compute the contracted covariant derivative in expressionslike

∇aXa = ∂aX

ab + ΓacaX

c = ∂aXa +

1√

|g|(∂a√

|g|)Xa =1√

|g|∂a(√

|g|Xa) . (3.48)

Lemma: Laplace’s expansion of the determinant of an nn square matrix A expresses the determinanta as a sum of n determinants of (n− 1)(n− 1) sub-matrices B of A (the “cofactors”),

a = det(A) =n∑

j=1

aijBij

Differentiating with respect to aij , we get

∂a

∂aij= Bij ,

since aij does not occur in any of the cofactors Bij . If the aij depend on x, differentiating with respectto xk gives

∂a

∂xk=

∂a

∂aij

∂aij∂xk

= Bij

∂aij∂xk

= aA−1ij

∂aij∂xk

.

(The inverse A−1 of the matrix A is given by A−1 = B/a.)Applying this to the metric tensor, we obtain

∂cg = ggab∂cgab .

44

3.2 Tensor analysis

3.2.3 Riemannian normal coordinates

In a (pseudo-) Riemannian manifold, one can find in each point P a coordinate system, called(Riemannian) normal or geodesic coordinates, with the following properties,

g′ab(P ) = ηab (3.49)

∂c′g′ab(P ) = 0 (3.50)

Γ′abc(P ) = 0 (3.51)

We proof it by construction. We choose new coordinates x′a centered at P ,

x′a = xa − xaP +1

2Γabc(x

b − xbP )(xc − xcP ) (3.52)

Here Γabc are the connection coefficients in P calculated in the original coordinates xa. We

differentiate∂x′a

∂xd= δad + Γa

db(xb − xbP )) . (3.53)

Hence ∂x′a/∂xd = δad at the point P . Differentiating again,

∂2x′a

∂xd∂xe= Γa

dbδbe = Γa

de . (3.54)

Inserting these results into the transformation law (3.19) of the connection coefficients,where we swap in the second term derivatives of x and x′,

Γ′abc =

∂x′a

∂xd∂xf

∂x′b∂xg

∂x′cΓd

fg −∂2x′a

∂xd∂xf∂xd

∂x′b∂xf

∂x′c(3.55)

gives

Γ′abc = δad δ

fb δ

gc Γ

dfg − Γa

df δdb δ

fc = Γa

bc − Γabc (3.56)

or

Γ′ade(P ) = 0 . (3.57)

Thus we have found a coordinate system with vanishing connection coefficients at P . By alinear transformation (that does not affect ∂gab) we can bring finally gab into the form ηab:As required by the equivalence principle, we can introduce in each point P of space-time afree-falling coordinate system in which physics is described by the known physical laws in theabsence of gravity.

Note that the introduction of Riemannian normal coordinates is in general only possible, ifthe connection is symmetric: Since the anti-symmetric part of the connection coefficients, thetorsion, transforms as a tensor, it can not be eliminated by a coordinate change. This impliesnot necessarily a contradiction to the equivalence principle, as long as the torsion is properlygenerated by source terms in the equation of motions of the matter fields. In particular, thespin current of fermions leads to non-zero torsion. As the elementary spins in macroscopicbodys cancel, torsion is in all relevant astrophysical and cosmological applications negligible.This justifies our choice of a symmetric connection.

45


3.2.4 Geodesics

A geodesic curve is the shortest or longest curve between two points on a manifold. Sucha curve extremizes the action S(L) of a free particle, L = gabx

axb, (setting m = 2 andx = dx/dσ), along the path xa(σ). The parameter σ plays the role of time t in the non-relativistic case, while t become part of the coordinates. The Lagrange equations are

d

dσ

∂L

∂(xc)− ∂L

∂xc= 0 (3.58)

Only g depends on x and thus ∂L/∂xc = gab,cxaxb. With ∂xa/∂xb = δab we obtain

gab,cxaxb = 2

d

dσ(gacx

a) = 2(gac,bxaxb + gacx

a) (3.59)

or

gacxa +

1

2(2gac,b − gab,c)x

axb = 0 (3.60)

Next we rewrite the second term as

2gca,bxaxb = (gca,b + gcb,a)x

axb (3.61)

multiply everything by gdc and obtain

xd +1

2gdc(gab,c + gac,b − gab,c)x

axb = 0 . (3.62)

We recognize the definition of the Levi-Civita connection and rewrite the equation of ageodesics as

xc + Γcabx

axb = 0 . (3.63)

The connection entering the equation for an extremal curve is the Levi-Civita connection,because we used the Lagrangian of a classical spinless particle.This result justifies the use of a torsionless connection which is metric compatible: Although

a star consists of a collection of individual particles carrying spin ~si, its total spin sums up tozero,

∑

i ~si ≈, 0, because the ~si are uncorrelated. Thus we can describe macrosopic matter ingeneral relativity as a a classical spinless point particle (or fluid, if extended). In such a case,only the symmetric part of the connection influences the geodesic motion of the consideredsystem.In an inertial system, an alternative definition of a geodesics as the “straightest line” is

the path generated by propagating its tangent vector parallel to itself. The tangent vectoralong the path x(σ) is ua = dxa/dσ. Then the requirement (3.25) of parallel transport for ua

becomesD

dσ

dxc

dσ=

d2xc

dσ2+ Γc

ab

dxa

dσ

dxb

dσ= 0 . (3.64)

Introducing the short-hand xa = dxa/dσ, we obtain again the geodesic equation in its usualform,

xc + Γcabx

axb = 0 . (3.65)

(But note that for any connection other than the Levi-Civita connection the two definitionswill disagree.)

46

3.2 Tensor analysis

Example: Sphere S2. Calculate the Christoffel symbols of the two-dimensional unit sphere S2.The line-element of the two-dimensional unit sphere S2 is given by ds2 = dϑ2 + sin2 ϑdφ2. A fasteralternative to the definition (3.36) of the Christoffel coefficients is the use of the geodesic equation:From the Lagrange function L = gabx

axb = ϑ2 + sin2 ϑφ2 we find

∂L

∂φ= 0 ,

d

dt

∂L

∂φ=

d

dt(2 sin2 ϑφ) = 2 sin2 ϑφ+ 4 cosϑ sinϑϑφ

∂L

∂ϑ= 2 cosϑ sinϑφ2 ,

d

dt

∂L

∂ϑ=

d

dt(2ϑ) = 2ϑ

and thus the Lagrange equations are

φ+ 2 cotϑϑφ = 0 and ϑ− cosϑ sinϑφ2 = 0 .

Comparing with the geodesic equation xκ+Γκµν x

µxν = 0, we can read off the non-vanishing Christoffel

symbols as Γφϑφ = Γφ

φϑ = cotϑ and Γϑφφ = − cosϑ sinϑ. (Note that 2 cotϑ = Γφ

ϑφ + Γφφϑ.)

3.2.5 Vector differential operators

The gradient of a scalar function φ has as components

∇iφ = ∂iφ = gij∂jφ = gij∂φ

∂xj. (3.66)

The covariant divergence of a vector field with components Xi is

∇iXi = ∂iX

i + ΓikiX

k =1√

|g|∂a(√

|g|Xa) . (3.67)

The Laplace operator ∆ ≡ ∇a∇a of a scalar is

∆φ = ∇a∇aφ =1√

|g|∂a(√

|g|gab∂bφ) . (3.68)

Example: Spherical coordinates 3:

Calculate for spherical coordinates x = (r, ϑ, φ) in R3 the gradient, divergence, and the Laplace

operator. Note that one uses normally normalized unit vectors in case of a diagonal metric: thiscorresponds to a rescaling of vector components V i → V i/

√gii (no summation in i) or basis vectors.

We express the gradient of a scalar function f first as

∂if ei = gij∂f

∂xjei =

∂f

∂rer +

1

r2∂f

∂φeφ +

1

r2 sin2 ϑ

∂f

∂ϑeϑ

and rescale then the basis, e∗i = ei/√gii, or e∗r = er, e

∗

φ = reφ, and e∗ϑ = r sinϑeϑ. In this new(“physical”) basis, the gradient is given by

∂if e∗i =∂f

∂re∗r +

1

r

∂f

∂ϑe∗ϑ +

1

r sinϑ

∂f

∂φe∗φ .

47


The covariant divergence of a vector field with rescaled components Xi/√gii is with

√g = r2 sinϑ

given by

∇iXi =

1√

|g|∂i(√

|g|Xi) =1

r2 sinϑ

(∂(r2 sinϑXr)

∂r+∂(r2 sinϑXϑ)

r∂ϑ+∂(r2 sinϑXφ)

r sinϑ∂φ

)

=1

r2∂(r2Xr)

∂r+

1

r sinϑ

∂(sinϑXϑ)

∂ϑ+

1

r sinϑ

∂Xφ

∂φ

=

(∂

∂r+

2

r

)

Xr +

(∂

∂ϑ+

cotϑ

r

)

Xφ +1

r sinϑ

∂Xφ

∂φ.

48

4 Schwarzschild solution

4.1 Isometries and Killing vectors

An isometry is a distance conserving mapping between different spaces. In the case of a Rie-mannian manifold, the transformed metric gij(x) has to be the same function of its argumentx as the original metric gij(x) of its argument x,

gij(x) = gij(x) . (4.1)

Note the difference to the definition of a scalar, φ(x) = φ(x). In the latter case, we requirethat a scalar field has the same value at a point P which in turn changes coordinates from xto x.The metric transforms under an arbitrary transformation as

gij(x) =∂xi

∂xk∂xj

∂xlgkl(x) (4.2)

or

gij(x) =∂xi

∂xk∂xj

∂xlgkl(x) . (4.3)

For an isometry,

gij(x) = gij(x) =∂xi

∂xk∂xj

∂xlgkl(x) . (4.4)

We consider now the effect of an infinitesimal coordinate transformation,

xi = xi + εξ(xk) , ε≪ 1 (4.5)

on the metric,

gij(x) =∂xi

∂xk∂xj

∂xlgkl(x)

= (δik + εξi,k)(δjl + εξj,l)g

kl(x)

= gij(x) + ε(ξi,j + ξj,i) +O(ε2) . (4.6)

In order to check the condition (4.4), we have to know g(x) as function of x. A Taylorexpansion gives

gij(x) = gij(x+ εξ) = gij(x) + εξk∂kgij(x) +O(ε2) . (4.7)

Setting equal Eqs. (4.6) and (4.7),

gij(x) + ε(ξi,j + ξj,i) = gij(x) + εξk∂kgij(x) , (4.8)

we see that the metric is kept invariant, if

ξi,j + ξj,i − ξk∂kgij = 0 (4.9)

49


orgjk∂kξ

i + gik∂kξj − ξk∂kg

ij = 0 . (4.10)

Expressing partial derivatives as covariant ones, the terms containing connection coefficientscancel and we obtain the Killing equations

∇iξj +∇jξi = 0 . (4.11)

The vectors ξ are called Killing vectors of the metric. Moving along a Killing vector field, themetric is kept invariant.Since Eq. (4.11) is tensor equation, the previous Eq. (4.10) is also invariant under arbi-

trary coordinate transformations, although it contains only partial derivatives. This suggeststhat one can introduce the derivative of an arbitrary tensor along a vector field, called Liederivative, without the need for a connection.

Example: Killing vectors of R3:

Choosing Cartesian coordinates, dl2 = dx2 +dy2 +dz2, makes it obvious that translations correspondto Killing vectors ξ1 = (1, 0, 0), ξ2 = (0, 1, 0), and ξ3 = (0, 0, 1). We find the Killing vectors describingrotational symmetry by writing for an infinitesimal rotation around, e.g., the z axis,

x′ = cosαx− sinαy ≈ x− αy ,

y′ = sinαx+ cosαy ≈ y + αx ,

z′ = z .

Hence ξz = (−y, x, 0) and the other two follow by cyclic permutation. One of them, ξz, we could

have also identified by rewriting the line-element in spherical coordinates and noting that dl does not

contain φ dependent terms.

Conserved quantities along geodesics Assume that the metric is independent from onecoordinate, e.g. x1. Then there exists a corresponding Killing vector, ξ = (1, 0, 0, 0), andx1 is a cyclic coordinate, ∂L/∂x1 = 0. With L = dτ/dσ, the resulting conserved quantity∂L/∂x1 = const. can be written as

∂L

∂x1= −g1β

dxβ

Ldσ= −g1β

dxβ

dτ= −ξ · u . (4.12)

Hence the quantity ξ · u is conserved along the solutions xa(σ) of the Lagrange equation, i.e.along geodetics.

4.2 Schwarzschild metric

The metric1 outside of a radial-symmetric mass distribution is given in Schwarzschild coor-dinates as

ds2 =dr2

1− 2Mr

+ r2(dϑ2 + sin2 ϑdφ2)− dt2(

1− 2M

r

)

. (4.13)

Its main properties are1We use in the two chapter about black holes the metric with signature +,+,+,−, following Hartle.

50

4.3 Gravitational redshift

• symmetries: The metric is time-independent and spherically symmetric. Hence two(out of the four) Killing vectors are ξ = (1, 0, 0, 0) and η = (0, 0, 0, 1), where we ordercoordinates as t, r, φ, ϑ.

• asymptotically flat: we recover Minkowski space for M/r → ∞.

• the metric is diagonal.

• potential singularities at r = 2M and r = 0. The radius 2M is called Schwarzschildradius and has the value

RS =2GM

c2= 3km

M

M⊙(4.14)

• At Rs, the coordinate t becomes space-like, while r becomes time-like.

4.3 Gravitational redshift

An observer with four-velocity uobs measures the frequency

ω = E = −p · uobs (4.15)

of a photon with four-momentum p. For an observer at rest,

uobs · uobs = −1 = gtt(ut)2 . (4.16)

Hence

uobs = (1− 2M/r)−1/2ξ . (4.17)

Inserting this into (4.15), we find for the frequency measured by an observer at position r,

ω(r) = −(1− 2M/r)−1/2ξ · p . (4.18)

Since ξ · p is conserved and ω∞ = −ξ · p, we obtain

ω∞ = ω(r)

√

1− 2M

r. (4.19)

Thus a photon climbing out of the potential wall of the mass M looses energy, in agreementwith the principle of equivalence. The information sent towards an observer at infinity by aspaceship falling towards r = 2M will be more and more redshifted, with ω → 0 for r → 2M .This indicates that r = 2M is an event horizon hiding all processes inside from the outside.

If M/r ≪ 1, we can expand the square root. Inserting also G and c, we find

ω∞ ≈ ω(r)

(

1− GM

rc2

)

= ω(r)

(

1− VNc2

)

, (4.20)

where VN is the Newtonian potential.

51


4.4 Orbits of massive particles

Radial equation and effective potential for massive particles

Spherically symmetry means that the movement of a test particle is contained in a plane. Wechoose ϑ = π/2 and uϑ = 0. We replace in the normalization condition u · u = −1 writtenout for the Schwarzschild metric,

−1 = −(

1− 2M

r

)(dt

dτ

)2

+

(

1− 2M

r

)−1(dr

dτ

)2

+ r2(dφ

dτ

)2

, (4.21)

the velocities ut and ur by the conserved quantities

e ≡ −ξ · u =

(

1− 2M

r

)dt

dτ(4.22)

l ≡ η · u = r2 sinϑ2dφ

dτ. (4.23)

Setting then A = 1− 2M/r, we find

−1 = −e2

A+

1

A

(dr

dτ

)2

+l2

r2. (4.24)

We want to rewrite this equation in a form similar to the energy equation in the Newtoniancase. Multiplying by A/2 makes the dr/dτ term similar to a kinetic energy term. Bringingalso all constant terms on the LHS and calling them E ≡ (e2 − 1)/2, we obtain

E ≡ e2 − 1

2=

1

2

(dr

dτ

)2

+ Veff (4.25)

with

Veff = −Mr

+l2

2r2− Ml2

r3= V0 −

Ml2

r3. (4.26)

Hence the energy of a test particle in the Schwarzschild metric can be, as in the Newtoniancase, divided into kinetic energy and potential energy. The latter contains the additional termMl2/r3, suppressed by 1/c2, that becomes important at small r.The asymptotic behavior of Veff for r → 0 and r → ∞ is

Veff(r → ∞) → −Mr

and Veff(r → 0) → −Ml2

r3, (4.27)

while the potential at the Schwarschild radius, V (2M) = 1/2, is independent of M .We determine the extrema of Veff by solving dVeff/dr = 0 and find

r1,2 =l2

2M

[

1±√

1− 12M2/l2]

(4.28)

Hence the potential has no extrema for l/M >√12 and is always negative: A particle can

reach r = 0 for small enough but finite angular momentum, in contrast to the Newtoniancase. By the same argument, there exists a last stable orbit at r = 6M , when the two extremar1 and r2 coincide for l/M =

√12.

The orbits can be classified according the relative size of E and Veff for a given l:

52

4.4 Orbits of massive particles

-0.6

-0.4

-0.2

0

0.2

0.4

0 2 4 6 8 10 12 14

Vef

f

r/M

l=6M

l=4M

l=3.7M

l=2M

-0.1

-0.05

0

0.05

0.1

0 2 4 6 8 10 12 14

Vef

f

r/M

l=4.6M

l=4M

l=3.7M

l=2M

Figure 4.1: The effective potential Veff for various values of l/M as function of distance r/M ,for two different scales.

• Bound orbit exists for E < 0. Two circular orbits, one stable at the minimum of Veff andan unstable one at the maximum of Veff ; orbits that oscillate between the two turningpoints.

• Scattering orbit exists for E > 0: If E > maxVeff, the particle hits after a finite timethe singularity r = 0. For 0 < E < maxVeff, the particle turns at E = maxVeff andescapes to r → ∞.

Radial infall

We consider the free fall of a particle that is at rest at infinity, dt/dτ = 1, E = 0 and l = 0.The radial equation (4.25) simplifies to

1

2

(dr

dτ

)2

=M

r(4.29)

and can be integrated by separation of variables,

∫ 0

rdrr1/2 =

√2M

∫ τ

τ∗

dτ (4.30)

with the result2

3r3/2 =

√2M(τ∗ − τ) . (4.31)

Hence a freely falling particle needs only a finite proper time to fall from finite r to r = 0. Inparticular, it passes the Schwarzschild radius 2M in finite proper time.

We can answer the same question using the coordinate time t by combining Eqs. (4.22) and(4.29),

dt

dr= −

(2M

r

)−1/2(

1− 2M

r

)−1

. (4.32)

53


Integrating gives

t =

∫

dr

(2M

r

)−1/2(

1− 2M

r

)−1

= t′ + 2M

−2

3

( r

2M

)3/2− 2

( r

2M

)1/2+ ln

∣∣∣∣∣

√

r/2M + 1√

r/2M − 1

∣∣∣∣∣

(4.33)

→ ∞ for r → 2M .

Since the coordinate time t is the proper time for an observer at infinity, a freely fallingparticle reaches the Schwarzschild radius r = 2M only for t→ ∞ for such an observer.

The last result can be derived immediately for light-rays. Choosing a light-ray in radialdirection with dφ = dϑ = 0, the metric (4.13) simplifies with ds2 = 0 to

dr

dt= 1− 2M

r. (4.34)

Thus light travelling towards the star, as seen from the outside, will travel slower and sloweras it comes closer to the Schwarzschild radius r = 2M . The coordinate time is ∝ ln |1−2M/r|and thus for an observer at infinity the signal will reach r = 2M again only asymptoticallyfor t→ ∞.

Perihelion precession

We recall first the derivation of the law of motion r = r(φ) in the Newtonian case. We solvethe Lagrange equations for L = (1/2)m(r2 + r2φ2)−GMm/r, obtaining

r2φ = l , (4.35)

r =l

r2− GM

r. (4.36)

We eliminate t bydr

dt=

dr

dφ

dφ

dt=

dr

dφ

l

r2≡ r′

l

r2(4.37)

and introduce u = 1/r,

u′′ + u =GM

l2. (4.38)

The solution follows as

u =GM

l2(1 + e cosφ) . (4.39)

We redo the same steps, starting from Eq. (4.25) for the Schwarzschild metric,

r2 +l2

r2= e2 − 1 +

2M

r− 2Ml2

r3. (4.40)

We eliminate first t and introduce u = 1/r,

(u′)2 + u2 =e2 − 1

l2+

2Mu

l2+ 2Mu3 . (4.41)

54

4.5 Orbits of photons

We can transform this into a linear differential equation differentiating with respect to φ.Thereby we eliminate also the constant (e2 − 1)/l2, and dividing2 by u′ it follows

u′′ + u =M

l2+ 3Mu2 =

GM

l2+

3GM

c2u2 . (4.42)

In the last step we reintroduced c and G. Hence we see that the Newtonian limit correspondsto c → ∞ (“instantaneous interactions”) or v/c → 0 (“static limit”). The latter statementbecomes clear, if one uses the virial theorem: GMu = GM/r ∼ v2.

In most situations, the relativistic correction is tiny. We use therefore perturbation theoryto determine an approximate solution of (4.42), u = u0 + δu, where u0 is the Newtoniansolution. Inserting u into (4.42), we obtain neglecting suppressed terms

(δu)′′ + δu =3(GM)3

c2l4(1 + 2e cosφ+ e2 cos2 φ) . (4.43)

Its solution is

δu =3(GM)3

c2l4

[

1 + eφ sinφ+ e2(1

2− 1

6cos(2φ)

)]

. (4.44)

While the first and third term lead only to extremely tiny changes in the orbital parameters,the second term is linear in φ and its effects accumulate therefore with time. Thus we includeonly δu = Aeφ sinφ in the approximate solution,

u =GM

l2[1 + e(cosφ+ α sinφ)] =

GM

l2[1 + e cos(φ(1− α))] . (4.45)

Hence the period is 2π/(1− α), and the ellipse processes with

∆φ =6πGM

a(1− e2)c2. (4.46)


We repeat the discussion of geodesics for massive particle for massless ones by changingu · u = −1 into u · u = 0 and by using an affine parameter λ instead of the proper-time.Reordering gives

1

b2≡ e2

l2=

1

l2

(dr

dλ

)2

+Weff (4.47)

with the impact parameter b = |l/e| and

Weff =1

r2

(

1− 2M

r

)

. (4.48)

The maximum of Weff is at 3M with height 1/27M2. For impact parameters b > 27M ,photon orbits have a turning point and photons escape to infinity. For b < 27M , they hitr = 0, while for b = 27M a (unstable) circular orbit is possible.

2The case u′ = 0 corresponds to radial infall treated in the previous section.

55


Light deflection

We transform Eq. (4.47) as in the m > 0 case into a differential equation for u(φ). For smalldeflections, we use again perturbation theory. In zeroth order in v/c, we can set the RHS of

u′′ + u =3GM

c2u2 (4.49)

to zero. The solution u0 is a straight line,

u0 =sinφ

b. (4.50)

Inserting u = u0 + δu gives

(δu)′′ + δu =3GM

c2sin2 φ

b2. (4.51)

A particular solution is

δu =3GM

2c2b2(1 + 1/3 cos(2φ)) . (4.52)

Thus the complete approximate solution is

u = u0 + δu =sinφ

b+

3GM

2c2b2(1 + 1/3 cos(2φ)) . (4.53)

Considering the limit r → ∞ or u→ 0 of this equation gives half of the deflection angle of alight-ray with impact parameter b to a point mass M ,

∆φ =4GM

c2b. (4.54)

For a numerical estimate, we use a light-ray grazing the solar surface, b = R⊙,

∆φ⊙ =4GM⊙

c2R⊙=

2Rs

R⊙≈ 10−5 ≈ 2′′ . (4.55)

Shapiro effect – radar echos Shapiro suggested to use the time-delay of a radar signal astest of general relativity. Suppose we send a radar signal from the Earth to Venus where itis reflected back to Earth. The point r0 of closest approach to the Sun is characterized bydr/dt|r0 = 0.

Rewriting Eq. (4.47) as

r2 +l2

r2

(

1− 2M

r

)

= e2 (4.56)

and introducing the Killing vector e in r2,

r2 =

(dr

dt

dt

dλ

)2

=e2

1− 2M/r

(dr

dt

)2

, (4.57)

we find1

(1− 2M/r)3

(dr

dt

)2

+l2

e2r2− 1

1− 2M/r= 0 . (4.58)

56


We now evaluate this equation at the point of closest approach, i.e. for dr/dt|r0 = 0,

l2

e2=

r201− 2M/r

, (4.59)

and use this equation to eliminate l2/e2 in (4.58). Then we obtain

dr

dt=

1

1− 2M/r

[

1− r20(1− 2M/r)

r2(1− 2M/r0)

]1/2

(4.60)

or

t(r, r0) =

∫ r

r0

dr

1− 2M/r

[

1− r20(1− 2M/r)

r2(1− 2M/r0)

]−1/2

. (4.61)

Next we expand this expression in M/r ≪ 1,

t(r, r0) =

∫ r

r0

drr

(r2 − r20)1/2

[

1− 2M)

r+

Mr0r(r + r0)

]

(4.62)

=(r2 − r20)

1/2

c+

2GM

c2ln

[

r + (r2 − r20)1/2

r0

]

+GM

c2

(r − r0r + r0

)1/2

, (4.63)

where we restored also G and c in the last step. The first term corresponds to straight linepropagation and thus the excess time ∆t is given by the second and third term. Finally, wecan use that the orbits both of Earth and Venus are much more distant from the Sun thanthe point of closest approach, rE , rV ≫ r0. Hence we obtain for the time delay

∆t =4GM

c3

[

ln

(4rErVr20

)

+ 1

]

. (4.64)

57

5 Gravitational lensing

One distinguishes three different cases of gravitational lensing, depending on the strength ofthe lensing effect:

1. Strong lensing occurs when the lens is very massive and the source is close to it: Inthis case light can take different paths to the observer and more than one image ofthe source will appear, either as multiple images or deformed arcs of a source. In theextreme case that a point-like source, lens and observer are aligned the image forms an“Einstein ring”.

2. Weak Lensing: In many cases the lens is not strong enough to form multiple images orarcs. However, the source can still be distorted and its image may be both stretched(shear) and magnified (convergence). If all sources were well known in size and shape,one could just use the shear and convergence to deduce the properties of the lens.

3. Microlensing: One observes only the usual point-like image of the source. However,the additional light bent towards the observer leads to brightening of the source. Thusmicrolensing is only observable as a transient phenomenon, when the lens crosses ap-proximately the axis observer-source.

Lens equation We consider the simplest case of a point-like mass M , the lens, between theobserver O and the source S as shown in Fig. 5.1. The angle β denotes the (unobservable)angle between the true position of the source and the direction to the lens, while ϑ± are theangles between the image positions and the source. The corresponding distances DOS, DOL,and DLS are also depicted in Fig. 5.1 and, since DOS+DLS = DOL does not hold in cosmology,we keep all three distances. Finally, the impact parameter b is as usual the smallest distancebetween the light-ray and the lens.Then the lens equation in the “thin lens” (b ≪ Di) and weak deflection (α ≪ 1) limit

follows from AS + SB = AB as

ϑDos = αDls + βDos . (5.1)

The thin lens approximation implies ϑ ≪ 1, and since β < ϑ, also β is small. Solving for band inserting for the deflection angle α = 4GM/(c2b) as well as b = ϑDol, we find first

β = ϑ− 4GM

c2Dls

DosDol

1

ϑ. (5.2)

Multiplying by ϑ, we obtain then a quadratic equation,

ϑ2 − βϑ− ϑE = 0 , (5.3)

where we introduced the Einstein angle

ϑE =

(

2RSDls

DosDol

)1/2

. (5.4)

58

⑦L

⑦S

β

ϑ+

ϑ−

α

Dsl Dlo

Dso

A

B

Figure 5.1: The source S that is off the optical axis OL by the angle β appears as two imageson opposite sides from the optical axis OL. The two images are separated by theangles ϑ± from the optical axis O.

The two images of the source are deflected by the angles

ϑ± =1

2[β ± (β2 + 4ϑ2E)

1/2] (5.5)

from the line-of-sight to the lens. If we do not know the lens location, measuring the separationof two lenses images, ϑ++ϑ−, provides only an upper bound on the lens mass. If observer, lensand source are aligned, then symmetry implies that ϑ+ = ϑ− = ϑE , i.e. the image becomesa circle with radius ϑE. Deviations from this perfectly symmetric situations break the circleinto arcs as shown in an image of the galaxy cluster Abell 2218 in Fig. 5.2.For a numerical estimate of the Einstein angle in case of a stellar object in our own galaxy,

we set M =M⊙ and Dls/Dos ≈ 1/2 and obtain

ϑE = 0.64′′ × 10−4

(M

M⊙

Dol

10 kpc

)1/2

. (5.6)

The numerical value of order 10−4 of an arc-second for the deflection led to the name “mi-crolensing.”

Magnification Without scattering or absorption of photons, the conservation of photonnumber implies that the intensity along the trajectory of a light-ray stays constant, 1

2

h3f(x,p) =

dN

d3x d3p=

dN

dA dt dΩ dE=

I

hcp2. (5.7)

In particular, we found that the observed intensity I equals the surface brightness B of thesource: The F ∝ 1/r2 law follows since the solid angle dΩ seen by a detector of size Adecreases as 1/r2.

1We define here intensity as connected to the energy flux F , while often the particle flux is used.

59

5 Gravitational lensing

Figure 5.2: Gravitational lensing of the galaxy cluster Abell 2218.

Gravity can affect this result in two ways: First, gravity can redshift the frequency ofphotons, νsr = νobs(1 + z). This can be either the gravitational redshift as in Sec. (4.3)or a cosmological redshift due to the expansion of the universe (that will be discussed inSec. 9.2). Thus the intensity Iobs at the observed frequency photons νobs is the emittedintensity evaluated at νobs(1 + z) and reduced by (1 + z)3,

Iobs(νobs) =I(νsr)

(1 + z)3. (5.8)

In both cases, this redshift depends only on the initial and the final point of the photontrajectory, but not on the actual path in-between. Thus the redshift cancels if one considersthe relative magnification of a source by gravitational lensing.

Second, gravitational lensing affects the solid angle the source is seen in a detector of fixedsize. As a result, the apparent brightness of a source increases proportionally to the increaseof the visible solid angle, if the source cannot be resolved as a extended object (cf. Sec...).Hence we can compute the magnification of a source by calculating the ratio of the solid anglevisible without and with lensing.

In Fig. 5.3, we sketch how the two lensed images are stretched: An infinitesimal smallsurface element 2π sinβdβdφ ≈ 2πβdβdφ of the unlensed source becomes in the lense plane2πϑ±dϑ±dφ. Thus the images are tangentially stretched by ϑ±/β, while the radial size ischanged by dϑ±/dβ. Thus the magnification a± of the source is

a± =ϑ±dϑ±βdβ

. (5.9)

Differentiating Eq. (5.5) gives

dϑ±dβ

=1

2

[

1± β

(β2 + 4ϑ2E)1/2

]

(5.10)

60

ϑ−

ϑ+β

Figure 5.3: The effect of gravitational lensing on the shape of an extended source: The surfaceelement 2πβdβdφ of the unlensed image at position β is transformed into the twolensed images of size 2πϑ±dϑ±dφ at position ϑ±.

and thus

atot = a+ + a− =x2 + 2

x(x2 + 4)1/2> 1 (5.11)

with x = β/ϑE. For large separation x, the magnification atot goes to one, while the mag-nification diverges for x → 0 as atot ∼ 1/x: In this limit we would receive light from aninfinite number of images on the Einstein circle. Physically, the approximation of a pointsource breaks down when x reaches the extension of the source. Since atot is larger than one,gravitational lensing always increases the total flux observed from a lensed source, facilitatingthe observation of very faint objects. As compensation, the source appears slightly dimmedto all those observers who do not see the source lensed.Two important applications of gravitational lensing are the search for dark matter in the

form of black holes or brown dwarfs in our own galaxy by microlensing and the determinationof the value of the cosmological constant by weak lensing observations.In microlensing experiments that have tried to detect dark matter in the form of MACHOs

(black holes, brown dwarfs,. . . ) one observed stars of the LMC. If a MACHO with speedv ≈ 220 km/s moves through the line-of-sight of a monitored star, its light-curve is magnifiedtemporally. If v is the perpendicular velocity of the source,

β(t) =

[

β20 +v2

D2ol

(t− t0)2

]1/2

(5.12)

The magnification a(t) is symmetric around t0 and its shape can be determined insertingtypical values for Dol, the MACHO mass.

61

6 Black holes

6.1 Schwarzschild black holes

Eddington-Finkelstein coordinates The elements grr and gtt of the metric tensor changesign at r = 2M . Thus t is only for r > 2M time-like, becoming space-like for r < 2M andvice versa for r. Geodesics cover only the region r > 2M within finite coordinate time t:Integrating Eq. (4.34), we find

t = +r + 2M ln∣∣∣r

2M− 1∣∣∣+ q outgoing rays , (6.1)

t = −r − 2M ln∣∣∣r

2M− 1∣∣∣+ p ingoing rays , (6.2)

for radial light-rays, where q and p are integration constants. We extend now theSchwarzschild metric by using the integration constant of an infalling light-ray as a newvariable, the “advanced time parameter p”,

p = t+ r + 2M ln |r/2M − 1| . (6.3)

Forming the differential,

dp = dt+ dr +( r

2M− 1)−1

dr = dt+

(

1− 2M

r

)−1

dr , (6.4)

we can eliminate dt from the Schwarzschild metric and find

ds2 = −(

1− 2M

r

)

dp2 + 2dpdr + r2dΩ . (6.5)

This metric is regular at 2M and valid for all r > 0. For a discussion, see the book.

Kruskal coordinates Amanifold is calledmaximal, if all geodesics can be extended to infinitevalues of their affine parameters (in both directions) or end at a singularity. A manifold iscalled geodescially complete, if all geodesics can be extended and do not end at a singularity.A trivial example for a geodescially complete manifold is Minkowski space.The steps to find the maximal extension of the Schwarzschild metric, the Kruskal solution,

can be motivated as follows: Use first not only the advanced time parameter p but also theretarded time parameter q and eliminate both t and r as independent variables. Find newcoordinates for which the metric is conformally flat, i.e. a metric which is Minkowskian apartfrom an overall factor Ω,

ds2 = gabdxadxb = Ω2(x)ηabdx

adxb . (6.6)

Moreover, eliminate singular points at r > 0.

62

6.2 Kerr black holes

The first step gives

ds2 = −(

1− 2M

r

)

dpdq + r2dΩ , (6.7)

where r is considered to be a function of p and q,

1

2(p− q) = r + 2M ln

∣∣∣r

2M− 1∣∣∣ . (6.8)

We consider now only the metric ds2 of the two-dimensional p, q space and try to find coor-dinates that exhibit explicitly conformal flatness of this space. Defining

t =1

2(p+ q) , r =

1

2(p− q) , (6.9)

gives

ds2 = −(

1− 2M

r

)

(dt2 − dr2) . (6.10)

We succeeded to find an explicitly conformal metric, but the conformal factor 1 − 2M/ris singular. Hence we should try different coordinates p and q that eliminate the singularityat r = 2M , but keep the conformal structure. A transformation of the type p = p(p) andq = q(q) will do the job, because

ds2 = −(

1− 2M

r

)dp

dp

dq

dqdpdq (6.11)

has the same structure as Eq. (6.7). Kruskal suggested

p = exp[p/(4M)] , q = − exp[−q/(4M)] , (6.12)

which gives

ds2 = −32M3

rexp

(

− r

2M

)

dpdq . (6.13)

The standard form of the Kruskal coordinates follows by defining a time-like variable V anda space-like variable U by

V =1

2(p+ q) , U =

1

2(p− q) (6.14)

as

ds2 = −32M3

rexp

(

− r

2M

)

(dV 2 − dU2) + r2dΩ . (6.15)


The metric outside of a rotating black hole is using Boyer-Lindquist coordinates given by

ds2 = −(

1− 2Mr

ρ2

)

dt2 − 4Mar sin2 ϑ

ρ2dφdt+

ρ2

∆dr2 + ρ2dϑ2

+

(

r2 + a2 +2Mra2 sin2 ϑ

ρ2

)

sin2 ϑdφ2 , (6.16)

with

a =L

M, ρ2 = r2 + a2 cos2 ϑ , ∆ = r2 − 2Mr + a2 . (6.17)

Its main properties are

63

6 Black holes

• Symmetries: The metric is time-independent and axially symmetric. Hence two Killingvectors are ξ = (1, 0, 0, 0) and η = (0, 0, 0, 1), where we order coordinates as t, r, ϑ, φ.

• The metric is asymptotically flat.

• Potential singularities at ρ = 0 and ∆ = 0.

• The weak-field limit shows that L is the angular momentum of the rotating black hole.

• The presence of the mixed term gtφ means that infalling particles (and thus space-time)is dragged around the rotating black hole.

Orbits in the equatorial plane ϑ = π/2 could be derived in the same way as for theSchwarzschild case, for ϑ 6= π/2 the discussion becomes much more involved.

Singularity The singularity at ρ2 = 0 = r2 + a2 cosϑ2 corresponds to

r = 0 and ϑ = π/2 . (6.18)

We would expect that r = 0 is compatible with all ϑ values. To understand this point, weconsider the M → 0 limit of (6.16) keeping a = J/M fixed,

ds2 = −dt2 +ρ

r2 + a2dr2 + r2dϑ2 + (r2 + a2) sin2 ϑdφ2 = −dt2 + dx2 + dy2 + dz2 . (6.19)

The comparison with the Minkowksi metric shows that

x =√

r2 + a2 sinϑ cosφ ,

y =√

r2 + a2 sinϑ sinφ ,

z = r cosϑ . (6.20)

Hence the singularity at r = 0 and ϑ = π/2 corresponds to a ring of radius a in the equatorialplane z = 0 of the Kerr black hole.

Horizons We define an event horizon as a three-dimensional hypersurface, f(xa) = 0, thatis null. Hence we require that at each point of such a surface a null tangent vector n existsthat is orthogonal to two space-like tangent vectors. From

0 = nana = gabnanb (6.21)

we see that ds = 0 along the horizon. Hence the (future) light cones are tangents to thehorizon, and the worldlines of all particles lie entirely on one side of the surface.In a stationary, axisymmetric space-time the general equation of a surface, f(xa) = 0,

becomes f(r, ϑ) = 0. The condition for a null surface becomes

0 = gab(∂af)(∂bf) = grr(∂rf)2 + gϑϑ(∂ϑf)

2 . (6.22)

For a special set of coordinates, the condition for a surface may be even simpler, f(r) = 0 or

0 = grr(∂rf)2 . (6.23)

Then the condition defining a horizons becomes simply grr = 0 or grr = 1/grr = ∞.

64


This is the case of the surface defined by the coordinate singularity ∆ = r2−2Mr+a2 = 0that depends only on r,

r± =M ±√

M2 − a2 . (6.24)

Hence, r− and r+ define indeed an inner and outer horizon around a Kerr black hole.(For an interpretation see the space-time diagram 6.1 that uses coordinates of the advancedEddington-Finkelstein type.)

We can determine the null tangent vector checking the condition n ·n = 0 inserting r+ anddr = 0 into the metric,

0 = gtt(nt)2 + 2gtφn

tbφ + gφφ(nφ)2 + gϑϑ(n

ϑ)2 (6.25)

or (fastest using Eq. (6.40))

(2Mr+ sinϑ

ρ2+

)2(

nφ − a

2Mr+nt)2

+ ρ2+(nϑ)2 = 0 . (6.26)

Hence nϑ = 0 and

na = C(1, 0, 0, a/(2Mr+)) (6.27)

with C as an arbitrary constant. The rotation velocity of the black hole is defined as therotation velocity of the light-rays forming the horizon. With

ω =dφ

dt=

dφ

dλ

dλ

dt=nφ

nt(6.28)

we obtain

ωH =a

2Mr+. (6.29)

The surface A of the (outer) horizon follows from inserting r+ together with dr = dt = 0into the metric,

ds2 = ρ2+dϑ2 +

(

r2+ + a2 +2Mr+a

2 sin2 ϑ

ρ2+

)

sin2 ϑdφ2 , (6.30)

Using r2± + a2 = 2Mr±, it follows

ds2 = ρ2+dϑ2 +

(2Mr+ρ+

)2

sin2 ϑdφ2 . (6.31)

Hence√g2 = 2Mr+ sinϑ and integration gives the area of the horizon as

A = 8πMr+ = 8πM(M +√

M2 − a2) . (6.32)

Note that the area depends on the angular momentum of the black hole that can in turnbe manipulated by dropping material into the hole: The horizon area A for fixed mass Mbecomes maximal for a non-rotating black hole, and decreasing to A = 8πM2 for a maximallyrotating one with a = M . For a > M , the metric component grr = ∆ gas no real zero andthus no event horizon exists.

65

6 Black holes

Dragging of inertial frames The Kerr metric is a special case of a metric with gtφ 6= 0, butgµν independent of φ. In this case, the φ component of momentum of a particle is conservedalong its geodesics,

pφ =∂L

∂φ= const. (6.33)

Consider now a particle with pr = pϑ = 0. Then

pφ = gφtpt + gφφpφ (6.34)

pt = gttpt + gφtpφ . (6.35)

For a massive particle p = mu. Thus dividing pφ and pt gives specialising to a particle withzero angular momentum, pφ = 0,

pφ

pt=

dφ

dt=gφt

gtt≡ ω(r, φ) . (6.36)

This relation holds also for a photon. Thus both massive and massless particles with zeroangular momentum falling into a Kerr black hole will acquire a non-zero angular rotationvelocity ω as seen by an observer from infinity.The allowed range of values for Ω = dφ/dt can be derived from u·u = −1 with ur = uϑ = 0,

gtt(ut)2 + 2gtφu

tuφ + gφφ(uφ)2 = −1 . (6.37)

Factoring out (ut)2, the maximal and minimal values of Ω follow when the bracket vanishes,

(ut)2[gtt + 2gtφ Ω+ gφφΩ

2]= −1 , (6.38)

and the four-velocity cannot be longer normalized.

Ω± = − gtφgφφ

±√(gtφgφφ

)2

− gttgφφ

. (6.39)

The allowed range of values, Ωmin ≤ Ω ≤ Ωmax, shrinks to a single value when

(gtφgφφ

)2

= ω2 =gttgφφ

. (6.40)

This happens at the horizon r+ and defines the rotation velocity of the black hole. In thecase of a Kerr black hole, one obtains

ωH =a

2Mr+. (6.41)

Ergosphere We consider a light-ray with dϑ = dr = 0. Then the line-element becomes

gttdt2 + 2gtφdtdφ+ gφφdφ

2 = 0 . (6.42)

Dividing by dt2 we find

φ± = − gtφgφφ

±√(gtφgφφ

)2

− gttgφφ

(6.43)

66


Figure 6.1: Space-time diagram in advanced Eddington-Finkelstein coordinates for a Kerrblack hole with a < M . Between the two horizons r− < r < r+, light cones areoriented towards r−, particles have to cross r−. Inside the inner horizon, geodesicsare possible that do not reach r = 0 in finite time. The behavior for r → 0 (andϑ 6= π/2) suggests that one can extend the space-time to r < 0.

For gtt = 0, we find for the two possible light-rays

φ+ = −2gtφgφφ

, φ− = 0 . (6.44)

Hence, the rotating black hole drags space-time at gtt = 0 so strongly that even a photon canonly co-rotate. Similarly, this condition specifies a surface inside which no stationary observersare possible: The normalization condition u ·u = −1 is inconsistent with ua = (1, 0, 0, 0) andgtt > 0.Solving

gtt = 1− 2Mr

ρ2= 0 (6.45)

we find the position of the two stationary limit surfaces at

r1/2 =M ±√

M2 − a cosϑ . (6.46)

The ergosphere is the space bounded by these two surfaces.

Extension of the Kerr metric The behavior of geodesics for r → 0 (and ϑ 6= π/2) suggeststhat one can extend the space-time to r < 0. For r → −∞, the extension becomes asymptot-ically flat, i.e. there exists a second Minkowski space that is connected to ours via the Kerrblack hole. Since for negative r, ∆ is always positive, ∆ = r2− 2Mr+ a2 > 0, the singularity

67

6 Black holes

is not protected by an event horizon in the “other” Minkowksi space. Moreover, there existclosed time-like curves: Consider a curve depending only on φ in the equatorial plane, theline-element for small, negative r is

ds2 =

(

r2 + a2 +2Ma2

r

)

dφ2 ∼ 2Ma2

rdφ2 < 0 (6.47)

time-like.The cosmic censorship hypothesis postulates that singularities formed in gravitational col-

lapse are always covered by event horizons. Thus we are in the “r > 0” Minkowski space ofall Kerr black holes – or the r < 0 is simply a mathematical artefact of a highly symmetricalmanifold, not showing up in real physical situations.

6.3 Black holes thermodynamics

Classically, the area of a black hole can only increase with time. The only other quantity inphysics with the same property is the entropy, dS ≥ 0. This suggests a connection betweenthe black hole area and its entropy. To derive this relation, we apply the first law of thermo-dynamics dU = TdS − pdV + . . . to a black hole. Its internal energy U is given by U =Mc2

and thusdU = dMc2

!= TdS − ~ωd~L , (6.48)

where ~ωd~L denotes the mechanical work done on a rotating macroscopic body.Our experience with the thermodynamics of non-gravitating systems suggests that the

entropy is an extensive quantity and thus proportional to the volume, S ∝ V . We sketchnow one argument, that shows that S ∝ A in the case of a black hole. We introduce the“rationalized area” α = A/4π = 2Mr+. The parameters describing a Kerr black hole are itsmass M and its angular momentum L and thus α = α(M,L). We form the differential andfind √

M2 − a2

2αdα = dM +

~a

αd~L = dM + ~ωd~L . (6.49)

The factor in front of dα is positive, as its interpretation as temperature requires. Ourconfidence in the correspondence between the first law of black hole thermodynamics, (6.49),and the usual first law of thermodynamics is strengthened by the fact that for a charged blackhole the term Φdq appears on the RHS, i.e. the normal term representing the work done byadding the charge dq to the black hole with surface potential Φ.We identify

TdS =

√M2 − a2

2αdα (6.50)

and thus S = f(A). The validity of the area theorem requires that f is a linear function,the proportionality coefficient between S and A can be only determined by calculating thetemperature of black hole. Hawking could show 1974 that a black hole in vacuum emitsblack-body radiation (“Hawking radiation”) with temperature

T =2√M2 − a2

A(6.51)

and thus

S =kc3

4~GA =

A

4L2Pl

. (6.52)

68

6.3 Black holes thermodynamics

Thus a black hole surrounded by a cooler medium emits radiation and heats up the envi-ronment. The entropy of a black hole is large, because its basic unit of entropy, 4L2

Pl, is sotiny.

Evaporation of Black Holes The radiation of a BH is a quantum process, as indicatedby the presence of Planck’s constant ~ in Eqs. (6.51) and (6.52). We can understand thebasic mechanism and the magnitude of the black hole temperature by considering a quantumfluctuation near its event horizon at RS : Heisenberg’s uncertainty principle allows the creationof a virtual electron-positron pair with total energy E for the time ∆t <∼ ~/E. This pair canbecome real, if the gravitational force acting on them during the time ∆t can supply theenergy E. More precisely, the work is done by the tidal force

dF =∂F

∂rdr = −GME/c2

2r3dr , (6.53)

because the electron-positron pair is freely falling.After the time ∆t, the pair is separated by the distance ∆r = c∆t ∼ c~/E. The energy

gain by the tidal force isF∆r ∼ (E/c2)GM/r3(∆r)2 . (6.54)

If we require that F∆r > E and identify the energy E of the emitted pair with the temperatureT of the black hole, we find

kT ∼ E <∼ ~(GM/r3)1/2 . (6.55)

The emission probability is maximal close to the event horizon, r = RS , and thus we obtainindeed kT ∼ E ∼ ~c3/(GM). This result supports the idea that the entropy of a black holeis indeed non-extensive and proportional to the surface of its event horizon. Since entropycounts the possible micro-states of a system, any promising attempt to combine gravity andquantum mechanics should provide a microscopic picture for these results.

69

7 Einstein’s field equations

7.1 Curvature and the Riemann tensor

We are looking for an invariant characterisation of an manifold curved by gravity. As thediscussion of normal coordinates showed, the first derivatives of the metric can be (at onepoint) always chosen to be zero. Hence this quantity will contain second derivatives of themetric, i.e. first derivatives of the Christoffel symbols.

The commutator of covariant derivatives will in general not vanish,

(∇a∇b −∇b∇a)Tc...d... = [∇a,∇b]T

c...d... 6= 0 , (7.1)

(think at the parallel transport from A first along ea, then along eb to B and then back to Aalong −ea and −eb on a sphere), is obviously a tensor and contains second derivatives of themetric.

For the special case of a vector Xa we obtain with

∇cXa = ∂cX

a + ΓabcX

b (7.2)

first

∇d∇cXa = ∂d(∂cX

a + ΓabcX

b) + Γaed(∂cX

e + ΓebcX

b)− Γecd(∂eX

a + ΓabeX

b) (7.3)

and the same expression with d↔ c,

∇c∇dXa = ∂c(∂dX

a + ΓabdX

b) + Γaec(∂dX

e + ΓebdX

b)− Γedc(∂eX

a + ΓabeX

b) (7.4)

We subtract the two equations, using that ∂c∂d = ∂d∂c and Γabc = Γa

cb,

[∇c,∇d]Xa = [∂cΓ

abd − ∂dΓ

abc + Γa

ecΓebd − Γa

edΓebc]X

b ≡ RabcdX

b . (7.5)

The tensor Rabcd is called Riemann or curvature tensor. Contracting the indices a and c, we

obtain the Ricci tensor

Rab = Rcacb = ∂cΓ

cab − ∂bΓ

cac + Γc

abΓdcd − Γd

bcΓcad . (7.6)

A further contraction gives the curvature scalar,

R = Rabgab . (7.7)

70

7.1 Curvature and the Riemann tensor

Symmetry properties Inserting the definition of the Christoffel symbols and using normalcoordinates, the Riemann tensor becomes

Rabcd =1

2∂d∂bgac + ∂c∂agbd − ∂d∂agbc − ∂c∂bgad . (7.8)

The tensor is antisymmetric in the indices c ↔ d, antisymmetric in a ↔ b and symmetricagainst an exchange of the index pairs (ab) ↔ (cd). Moreover, there exists one algebraicidentity,

Rabcd +Radbc +Racdb = 0 . (7.9)

Since each pair of indices (ab) and (cd) can take six values, we can combine the antisym-metrized components of R[ab][cd] in a symmetric six-dimensional matrix. The number ofindependent components of this matrix is thus for d = 4 space-time dimensions

n× (n+ 1)

2− 1 =

6× 7

2− 1 = 20 ,

where we accounted also for the constraint (7.9). In general, the number n of independentcomponents is in d space-time dimensions given by n = d2(d2 − 1)/12, while the number mof field equations is m = d(d+ 1)/2. Thus we find

d 1 2 3 4n 0 1 6 20m - 3 6 10

This implies that an one-dimensional manifold is always flat (ask yourself why?). Moreover,the number of independent components of the Riemann tensor is smaller or equals the numberof field equations for d = 2 and d = 3. Hence the Riemann tensor vanishes in empty space, ifd = 2, 3. Starting from d = 4, already an empty space can be curved.The Bianchi identity is a differential constraint,

∇eRabcd +∇cRabde +∇dRabec = 0 , (7.10)

that is checked again simplest using normal coordinates. In the context of general relativ-ity, the Bianchi identities are necessary consequence of the Einstein-Hilbert action and therequirement of general covariance.

Example: Sphere S2. Calculate the Ricci tensor Rαβ and the scalar curvature R of the two-dimensional unit sphere S2.We have already determined the non-vanishing Christoffel symbols of the sphere S2 as Γφ

ϑφ = Γφφϑ =

cotϑ and Γϑφφ = − cosϑ sinϑ. We will show later that the Ricci tensor of a maximally symmetric

space as a sphere satisfies Rab = Kgab. Since the metric is diagonal, the non-diagonal elements of theRicci tensor are zero too, Rφϑ = Rϑφ = 0. We calculate with

Rab = Rcacb = ∂cΓ

cab − ∂bΓ

cac + Γc

abΓdcd − Γd

bcΓcad

the ϑϑ component, obtaining

Rϑϑ = 0− ∂ϑ(Γφϑφ + Γϑ

ϑϑ) + 0− ΓdϑcΓ

cϑd = 0 + ∂ϑ cotϑ− Γφ

ϑφΓφϑφ

= 0− ∂ϑ cotϑ− cot2 ϑ = 1 .

71


From Rab = Kgab, we find Rϑϑ = Kgϑϑ and thus K = 1. Hence Rφφ = gφφ = sin2 ϑ.The scalar curvature is (diagonal metric with gφφ = 1/ sin2 ϑ and gϑϑ = 1)

R = gabRab = gφφRφφ + gϑϑRϑϑ =1

sin2 ϑsin2 ϑ+ 1× 1 = 2 .

Note that our definition of the Ricci tensor guaranties that the curvature of a sphere is also positive,

if we consider it as subspace of a four-dimensional space-time.

7.2 Integration and the metric determinant g

In special relativity, Lorentz transformations left the volume element d4x invariant, d4x′ =dt′d2x′⊥dx

′‖ = (γdt)d2x⊥(dx‖/γ) = dx4. We allow now for arbitrary coordinate transforma-

tion for which the Jacobi determinant can deviate from one.

Variation of the metric determinant g We consider a variation of a matrixM with elementsmij(x) under an infinitesimal change of the coordinates, δxa = εxa,

δ ln detM ≡ ln det(M + δM)− ln det(M)

= lndet(M + δM)

det(M)

= ln det[M−1(M + δM)] = ln det[1 +M−1δM ] =

= ln[1 + Tr(M−1δM)] +O(ε2) = Tr(M−1δM) +O(ε2) . (7.11)

In the last step, we used ln(1 + x) = x+O(x2). Hence

Tr(M−1 ∂

∂xµM) =

∂

∂xµln det(M) . (7.12)

Useful formula for derivatives Applied to derivatives of√

|g|, we obtain

1

2gad∂bgad =

1

2∂b ln g =

1√

|g|∂b(√

|g| (7.13)

while we find for contracted Christoffel symbols

Γaab =

1

2gad(∂agdb + ∂bgad − ∂dgab) =

1

2gad∂bgad =

1

2∂b ln g =

1√

|g|∂b(√

|g| . (7.14)

Next we consider the derivative of vector fields,

∇aXa = ∂aX

ab + ΓacaX

c = ∂aXa +

1√

|g|(∂a√

|g|)Xa =1√

|g|∂a(√

|g|Xa) , (7.15)

and of anti-symmetric tensors of rank 2,

∇µAµν = ∂µA

µν + ΓµλµA

λν + ΓνλµA

µλ =1√

|g|∂µ(√

|g|Aµν) . (7.16)

72

7.3 Einstein-Hilbert action

In the latter case, the third term ΓνλµA

µλ vanishes because of the antisymmetry of Aµλ sothat we could combine the first two as in the vector case. This generalises to completelyanti-symmetric tensors of all orders. For a symmetric tensor, we find

∇aSab = ∂aS

ab + ΓacaS

cb + ΓbcaS

ac =1√

|g|∂a(√

|g|Sab) + ΓbcaS

ac . (7.17)

We can express Γbca as derivative of the metric tensor,

∇aSab =

1√

|g|∂a(√

|g|Sab)−

1

2(∂bgca)S

ac . (7.18)

Thus we can perform the covariant derivative of Sab without the need to know the Christoffelsymbols.

Global conservation laws An immediate consequence of Eq. (7.15) is a covariant form ofGauß’ theorem for vector fields. In particular, we can conclude from local current conser-vation, ∇aj

a = 0, the existence of a globally conserved charge. If the conserved current ja

vanishes at infinity, then we obtain also in a general space-time∫

Ωd4x√

|g| ∇aja =

∫

Ωd4x∂a(

√

|g|ja) = 0 . (7.19)

For a non-zero current, the volume integral over the charge density j0 remains constant,∫

Ωd4x√

|g| ∇aja =

∫

V (t2)d3x√

|g|j0 −∫

V (t2)d3x√

|g|j0 = 0 . (7.20)

Thus the conservation of Noether charges of internal symmetries as the electric charge, baryonnumber, etc., is not affected by an expanding universe.Next we consider the energy-momentum tensor as example for a locally conserved symmet-

ric tensors of rank two. Now, the second term in Eq. (7.18) prevents us to convert the localconservation law into a global one. If the space-time admits however a Killing field ξ, thenwe can form the vector field P a = T abξb with

∇aPa = ∇a(T

abξb) = ξb∇aTab + T ab∇aξb = 0 . (7.21)

Here, the first term vanishes since T ab is conserved and the second because T ab is symmetric,while ∇aξb is antisymmetric. Therefore the vector field P a = T abξb is also conserved, ∇aP

a =0, and we obtain thus the conservation of the component of the energy-momentum vector indirection ξ.In summary, global energy conservation requires the existence of a time-like Killing vector

field. Moving along such a Killing field, the metric would be invariant. Since we expect in anexpanding universe a time-dependence of the metric, a time-like Killing vector field does notexist and the energy contained in a “comoving” volume changes with time.


Our main guide in choosing the appropriate action for the gravitational field is simplicity. ALagrange density has mass dimension four (or length−4) such that the action is dimensionless.

73


In the case of gravity, we have to account for the dimensionfull coupling, Newton’s constantG, and require therefore that the Lagrange density without coupling has mass dimension two.Among the possible terms we can select are

L =√

|g|

Λ + bR+ c∇a∇bRab + d(∇a∇bR

ab)2 + . . .

+f(R) + . . .

(7.22)

Choosing only the first term, a constant, will not give dynamical equations. The next simplestpossibility is to pick out only the second term, as it was done originally by Hilbert. We willsee later that, if we do no include a constant term Λ in the gravitational action, it will pop upon the matter side. Thus we include Λ right from the start and define as the Einstein-HilbertLagrange density for the gravitational field

LEH =√

|g|(R− 2Λ) . (7.23)

We derive the resulting field equations for the metric tensor gab directly from the actionprinciple

δSEH = δ

∫

Ωd4x√

|g|(R− 2Λ) = δ

∫

Ωd4x√

|g| (gabRab − 2Λ) = 0 . (7.24)

We allow for variations of the metric gab restricted by the condition that the variation of gaband its first derivatives vanish on the boundary ∂Ω.

δSEH =

∫

Ωd4x

√

|g| gabδRab +√

|g|Rabδgab + (R− 2Λ) δ

√

|g|

. (7.25)

Our task is to rewrite the first and third term as variations of δgab or to show that they areequivalent to boundary terms.

Let us start with the first term. Choosing normal coordinates, the Ricci tensor at theconsidered point P becomes

Rab = ∂cΓcab − ∂bΓ

cac . (7.26)

Hence

gabδRab = gab(∂cδΓcab − ∂bδΓ

cac) = gab∂cδΓ

cab − gac∂cδΓ

bab , (7.27)

where we exchanged the indices b and c in the last term. Since ∂cgab = 0 at P , we can rewritethe expression as

gabδRab = ∂c(gabδΓc

ab − gacδΓbab) = ∂cX

c . (7.28)

The quantity X is a vector, since the difference of two connection coefficients transforms as atensor. Replacing in Eq. (7.28) the partial derivative by a covariant one promotes it thereforein a valid tensor equation,

gabδRab =1√

|g|∂a(√

|g|Xa) . (7.29)

Thus this term corresponds to a surface term that vanishes.

Next we rewrite the third term using

δ√

|g| = 1

2√

|g|δ|g| = 1

2

√

|g| gabδgab = −1

2

√

|g| gabδgab (7.30)

74


and obtain

δSEH =

∫

Ωd4x√

|g|

Rab −1

2gabR+ Λ gab

δgab = 0 . (7.31)

Hence the metric tensor fulfils in vacuum the equation

1√

|g|δSEHδgab

= Rab −1

2Rgab + Λgab ≡ Gab + Λgab = 0 , (7.32)

where we introduced the Einstein tensor Gab. The constant Λ is called cosmological constant.

7.3.1 Einstein equations

We consider now the combined action of gravity and matter, as the sum of the Einstein-Hilbert Lagrange density LEH/2κ and the Lagrange density Lm including all relevant matterfields,

L =1

2κLEH + Lm =

1

2κ

√

|g|(R− 2Λ) + Lm . (7.33)

In Lm, the effects of gravity are accounted for by the replacements ∂a, ηab → ∇a, gab,while we have to adjust later the constant κ such that we reproduce Newtonian dynamicsin the weak-field limit. We expect that the source of the gravitational field is the energy-momentum tensor. More precisely, the Einstein tensor (“geometry”) should be determined bythe matter, G = −κT. Since we know already the result of the variation of SEH, we concludethat the variation of Sm should give

2√

|g|δSmδgab

= Tab . (7.34)

The tensor Tab defined by this equation is called dynamical energy-momentum tensor . In orderto show that this rather bold definition makes sense, we have to prove that ∇a(δSm/δg

ab) = 0and we have to convince ourselves that this definition reproduces the standard results we knowalready. Einstein’s field equation follows then as

Gab + Λgab = −κTab . (7.35)

7.3.2 Dynamical energy-momentum tensor

We start by proving that the dynamical energy-momentum tensor defined by by Eq. (7.34) isconserved. We consider the change of the matter action under variations of the metric,

δSm =1

2

∫

Ωd4x√

|g| Tabδgab = −1

2

∫

Ωd4x√

|g| T abδgab . (7.36)

We allow infinitesimal but otherwise arbitrary coordinate transformations,

xi = xi + ξ(xk) . (7.37)

For the resulting change in the metric δgab we can use Eqs. (4.6) and (4.7),

δgij = ∇iξj +∇jξi . (7.38)

75


We use that T ab is symmetric and that general covariance guarantees that δSm = 0 for acoordinate transformation,

δSm = −∫

Ωd4x√

|g| T ab∇aξb = 0 . (7.39)

Next we apply the product rule,

δSm = −∫

Ωd4x√

|g| (∇aTab)ξb +

∫

Ωd4x√

|g| ∇a(Tabξb) = 0 . (7.40)

The second term is a four-divergence and thus a boundary term that we can neglect. Theremaining first term vanishes for arbitrary ξ, if the energy-momentum tensor is conserved,

∇aTab = 0 . (7.41)

Hence the local conservation of energy-momentum is a consequence of the general covarianceof the gravitational field equations, in the same way as current conservation follows fromgauge invariance in electromagnetism.We now evaluate the dynamical energy-momentum tensor for the examples of the Klein-

Gordon and the photon field. We rewrite Eq. (2.38) including a potential V (φ) (that couldbe also a mass term, V (φ) = m2φ2/2) as

L =1

2gab∇aφ∇bφ− V (φ) . (7.42)

(Remember that ∇aφ = ∂aφ for a scalar field.) Varying the action gives

δSKG =1

2

∫

Ωd4x

√

|g|∇aφ∇bφ δgab + [gab∇aφ∇bφ− 2V (φ)]δ

√

|g|

=

∫

Ωd4x√

|g|δgab1

2∇aφ∇bφ− 1

2gabL

. (7.43)

and thus

Tab =2√

|g|δSmδgab

= ∇aφ∇bφ− gabL . (7.44)

Next we consider the free electromagnetic action,

δSem = −1

4

∫

Ωd4x√

|g|FabFab = −1

4

∫

Ωd4x√

|g|gacgbdFabFcd , (7.45)

remember that F does not depend on g and get

δSem = −1

4

∫

Ωd4x

(δ√

|g|)FcdFcd +

√

|g|δ(gacgbd)FabFcd

= −1

4

∫

Ωd4x√

|g|δgab

−1

2gabFcdF

cd + 2gcdFacFbd

. (7.46)

Hence the dynamical energy-momentum tensor is

Tab = −FacFc

b +1

4gabFcdF

cd . (7.47)

76


7.3.3 Cosmological constant

To understand better the meaning of the constant Λ, we ask now if one of know energy-matter tensors could mimick a term gabΛ. First we consider a scalar field. The constancy ofΛ requires clearly ∇aφ = 0 and thus

Tab = gabV0(φ) , (7.48)

where V0 is the minimum of the potential V (φ). Hence a scalar field with a non-zero minimumof its potential acts as a cosmological constant.

Next we consider a perfect fluid described by the two parameters density ρ and pres-sure P . We know already that for a pressureless fluid T ab = ρuaub and we expectT ab = diagρ, P, P, P for a perfect fluid in a specific frame. Hence

T ab = (ρ+ P )uaub − Pgab , (7.49)

and a fluid with P = −ρ, i.e. marginally fulfilling the strong energy condition, has the sameproperty as a cosmological constant.

Is it possible to distinguish a term like Tab = gabV0(φ) in Sm from a non-zero Λ in SEH? Inprinciple yes, since a cosmological constant fulfils P = −ρ exactly and independently of allexternal parameters like temperature or density. The latter change with time in the universeand therefore there may be detectable differences to a fluid with P = P (ρ, T, . . .) and a scalarfield with V = V (ρ, T, . . .), even if they mimick today very well a cosmological constant withP = −ρ.

7.3.4 Equations of motion

We show now that Einstein’s equation imply that particles move along geodesics. By analogywith a pressureless fluid, T ab = ρuaub, we postulate

T ab(x) =m√

|g|

∫

dτdxa

dτ

dxb

dτδ(4)(x− x(τ)) (7.50)

for a point-particle moving along x(τ) with proper time τ . Inserting this into

∇aTab = ∂aT

ab + ΓacaT

cb + ΓbcaT

ac =1√

|g|∂a(√

|g|T ab) + ΓbcaT

ac = 0 (7.51)

gives∫

dτ xaxb∂

∂xaδ(4)(x− x(τ)) + Γb

ca

∫

dτ xaxcδ(4)(x− x(τ)) = 0 . (7.52)

We can replace ∂/∂xa = −∂/∂xa acting on δ(4)(x− x(τ) and use moreover

xa∂

∂xaδ(4)(x− x(τ)) =

d

dτδ(4)(x− x(τ)) (7.53)

to obtain

−∫

dτ dxad

dτδ(4)(x− x(τ)) + Γb

ca

∫

dτ xaxcδ(4)(x− x(τ)) = 0 . (7.54)

77


Integrating the first term by parts we obtain

∫

dτ(

xa + Γbcax

axc)

δ(4)(x− x(τ)) = 0 . (7.55)

The integral vanishes only, when the wordline fulfils the geodesic equation. Hence Einstein’sequation implies already the equation of motion, in contrast to linear theories as the Maxwellequations, where the Lorentz force law has to be postulated separately.

7.4 Alternative theories

Tensor-scalar theories The field equations for a purely scalar theory of gravity would be

φ = −4πGT aa . (7.56)

It predicts no coupling between photons and gravitation, since T aa = 0 for the electromagnetic

field. A purely vector theory for gravity fails, since it predicts not attraction but repulsionfor two masses.However, it may well be that gravity is a mixture of scalar, vector and tensor exchange,

dominated by the later. An important example for a tensor-scalar theory is the Brans-Dicketheory. Here one use gab to describe gravitational interactions but assumes that the strength,G, is determined by a scalar field φ,

φ = −4πG(TM )aa

φ(Gab + Λgab) = −8π(TMab + T φ

ab) . . (7.57)

78

8 Linearized gravity and gravitational waves

8.1 Linearized gravity

8.1.1 Metric perturbations as a tensor field

We are looking for small perturbations hab around the Minkowski1 metric ηab,

gab = ηab + hab , hab ≪ 1 . (8.1)

These perturbations may be caused either by the propagation of gravitational waves or by thegravitational potential of a star. In the first case, current experiments show that we shouldnot hope for h larger than O(h) ∼ 10−22. Keeping only terms linear in h is therefore anexcellent approximation. Choosing in the second case as application the final phase of thespiral-in of a neutron star binary system, deviations from Newtonian limit can become large.Hence one needs a systematic “post-Newtonian” expansion or even a numerical analysis todescribe properly such cases.

We choose a Cartesian coordinate system xa and ask ourselves which transformations arecompatible with the splitting (8.1) of the metric. If we consider global (i.e. space-time inde-pendent) Lorentz transformations Λb

a, then x′a = Λa

bxb. The metric tensor transform as

g′ab = ΛcaΛ

dbgcd = Λc

aΛdb(ηcd + hcd) = ηab + Λc

aΛdbhcd = η′ab + Λc

aΛdbhcd . (8.2)

Since h′ab = ΛcaΛ

dbhcd, we see that (global) Lorentz transformations respect the splitting (8.1).

Thus the perturbation hab transforms as a rank-2 tensor on Minkowski space. We can viewtherefore hab as a symmetric rank-2 tensor field defined on Minkowski space that satisfiesthe linearized Einstein equations, similar as the photon field is a rank-1 tensor field fulfillingMaxwell’s equations.

The splitting (8.1) is not invariant under general coordinate transformations. Restrictingourselves to infinitesimal ones, xi = xi + εξ(xk), the Killing equation simplifies to

h′ab = hab − ∂aξb − ∂bξa , (8.3)

because the term ξc∂chab is quadratic in the small quantities h and εξ and can be neglected.

It is more fruitful to view this equation not as a coordinate but as a gauge transformation:Both h′ab and hab describe the same physical situation, since the (linearized) Einstein equationsdo not fix uniquely hab for a given source.

Comparison with electromagnetism The Maxwell equation (2.63) consists of four equationsfor the six components of F. Thus we need either a second equation, i.e. Eq. (2.64), or we

1The same analysis could be performed for small perturbations around an arbitrary metric g(0)ab , adding

however considerable technical complexity.

79


Aa = ja wave equation hab = T ab

∂aAa = 0 gauge condition ∂ah

ab = 0transverse polarization transverse, traceless

Aa(x) =∫d3x′ J

a(tr,~x′)|~x−~x′| solution hab(x) =

∫d3x′ T

ab(tr,~x′)|~x−~x′|

Lem = −23 dαd

α energy loss Lgr = −15

...I αβ

...Iαβ

Table 8.1: Comparison of basic formulas for electromagnetic and gravitational radiation.

should transform Eq. (2.63) into an equation for the four components of the four-potentialA. Inserting the definition of A gives

∂aFab = ∂a(∂

aAb − ∂bAa) = Ab − ∂a∂bAa = jb . (8.4)

Gauge transformations of the photon field,

Ai(x) → Ai(x) + ∂iΛ(x) . (8.5)

do affect neither the fieldstrength tensor nor the equation of motions. Thus we are free tochoose the Lorentz gauge, ∂aA

a = 0, obtaining the familiar wave equation for photons,

Aa = ja . (8.6)

Note that we can still perform gauge transformations of the type Ai(x) → Ai(x) + Λ(x), ifΛ(x) satisfies Λ = 0.

8.1.2 Linearized Einstein equations in vacuum

From ∂aηbc = 0 and the definition

Γabc =

1

2gad(∂bgdc + ∂cgbd − ∂dgbc) (8.7)

we find for the change of the connection linear in h

δΓabc =

1

2ηad(∂bhdc + ∂chbd − ∂dhbc) =

1

2(∂bh

ac + ∂ch

ab − ∂ahbc) . (8.8)

Here we used η to raise indices which is allowed in linear approximation. Remembering thedefinition of the Riemann tensor,

Rabcd = ∂cΓ

abd − ∂dΓ

abc + Γa

ecΓebd − Γa

edΓebc , (8.9)

we see that we can neglect the terms quadratic in the connection terms. Thus we find for thechange

δRabcd = ∂cδΓ

abd − ∂dδΓ

abc

=1

2∂c∂bhad + ∂c∂dh

ab − ∂c∂

ahbd − (∂d∂bhac + ∂d∂ch

ab − ∂d∂

ahbc)

=1

2∂c∂bhad + ∂d∂

ahbc − ∂c∂ahbd − ∂d∂bh

ac . (8.10)

80


The change in the Ricci tensor follows by contracting a and c,

−δRbd = δRcbcd =

1

2∂c∂bhcd + ∂d∂

chbc)− ∂c∂chbd − ∂d∂bh

cc . (8.11)

Next we introduce h ≡ hcc, = ∂c∂c, and relabel the indices,

−δRab =1

2∂a∂chcb + ∂b∂ch

ca −hab − ∂a∂bh . (8.12)

We now rewrite all terms apart from hab as derivatives of the vector

Va = ∂chca −

1

2∂ah , (8.13)

obtaining

−δRab =1

2−hab + ∂aVb + ∂bVa . (8.14)

Looking back at the properties of hab under gauge transformations, Eq. (8.3), we see that wecan gauge away the second and third term. Thus the linearized Einstein equation in vacuumbecomes simply

hab = 0 (8.15)

if the harmonic gauge2

Va = ∂chca −

1

2∂ah = 0 , (8.16)

is chosen. Thus the familiar wave equation holds for all ten independent components ofhab, and the perturbations propagate with the speed of light c. Inserting plane waves hab =εab exp(ikx) into the wave equation, one finds immediately that k is a null vector.

Alternative form of the Einstein equation We can rewrite the Einstein equation such thatthe only geometrical term on the LHS is the Ricci tensor. Because of

R aa − 1

2g aa (R− 2Λ) = R− 2(R− 2Λ) = −R+ 4Λ = −κT a

a (8.17)

we can perform with T ≡ T aa the replacement R = 4Λ − κT in the Einstein equation and

obtain

Rab = −κ(Tab −1

2gabT ) + gabΛ . (8.18)

Thus an empty universe with Λ = 0 is characterized by a vanishing Ricci tensor Rab = 0.

8.1.3 Linearized Einstein equations with sources

We found 2δRab = hab. By contraction follows 2δR = h. Combining both terms gives

(

hab −1

2ηabh

)

= 2(δRab −1

2ηabδR)

= −2κδTab . (8.19)

2Alternatively, this gauge is called Hilbert, Loren(t)z, de Donder,. . . , gauge.

81


Since we assumed an empty universe in zeroth order, δTab is the complete contribution to theenergy-momentum tensor. We omit therefore in the following the δ in δTab.We introduce as useful short-hand notation the “trace-reversed” amplitude as

hab ≡ hab −1

2ηabh . (8.20)

The harmonic gauge condition becomes then

∂ahab = 0 (8.21)

and the linearized Einstein equation in the harmonic gauge follow as

hab = −2κTab . (8.22)

Because of ¯hab = hab and Eq. (8.18), we can rewrite this wave equation also as

hab = −2κTab (8.23)

with the trace-reversed energy-momentum tensor Tab.

Newtonian limit The Newtonian limit corresponds to v/c → 0 and thus the only non-zeroelement of the energy-momentum tensor becomes T tt = ρ. We compare the metric

ds2 = (1 + 2Φ)dt2 − (1− 2Φ)(dx2 + dy2 + dz2

)(8.24)

to Eq. (8.1) and find as metric perturbations

htt = 2Φ hij = 2δijΦ hti = 0 . (8.25)

In the static limit → −∆ and v = 0, and thus

−∆(h00 −1

2η00h) = −4∆Φ = −2κρ . (8.26)

Hence the linearized Einstein equation has the same form as the Newtonian Poisson equation,and the constant κ equals κ = 8πG.

8.1.4 Polarizations states

TT gauge We consider a plane wave hab = εab exp(ikx). The symmetric matrix εab iscalled polarization tensor. Its ten independent components are constrained both by the waveequation and the gauge condition ∂ahab = 0.Even after fixing the harmonic gauge ∂ahab = 0, we can still add four function ξa with

ξa = 0. We can choose them such that four components of hab vanish. In the transverse

traceless (TT) gauge, one sets (α = 1, 2, 3)

h0α = 0, h = 0 . (8.27)

The harmonic gauge condition becomes Va = ∂bhba or

V0 = ∂bhb0 = ∂0h

00 = −iωε00eikx = 0 , (8.28)

Vα = ∂bhbα = ∂βh

βα = −ikβεαβeikx = 0 . (8.29)

82


Thus ε00 = 0 and the polarization tensor is transverse, kβεαβ = 0. If we choose the plane

wave propagating in z direction, ~k = k~ez, the z raw and column of the polarization tensorvanishes too. Accounting for h = 0 and εab = εba, only two independent elements are left,

ε =

0 0 0 00 ε11 ε12 00 ε12 −ε11 00 0 0 0

. (8.30)

Thus a gravitational wave—or in a particle picture the graviton—has two polarization states,as expected for a massless particle.In general, one can construct the polarization tensor in TT gauge by setting first the

non-transverse part to zero and then subtracting the trace. The resulting two independentelements are (again for ~k = k~ez) then ε11 = 1/2(εxx − εyy) and ε12.

Helicity We determine now how a metric perturbation hab transforms under a rotation withthe angle α. We choose the wave propagating in z direction, ~k = k~ez, the TT gauge, and therotation in the xy plane. Then the general Lorentz transformation Λ becomes

Λ =

1 0 0 00 cosα sinα 00 − sinα cosα 00 0 0 1

. (8.31)

Since ~k = k~ez and thus Λ ba kb = ka, the rotation affects only the polarization tensor. We

rewrite ε′ab = Λ caΛ

db εcd in matrix notation,

ε′ = ΛεΛt . (8.32)

It is sufficient in TT gauge to perform the calculation for the xy sub-matrices. The resultafter introducing circular polarization states ε± = ε11 ± iε12 is

ε′± = exp(2iα)ε± . (8.33)

Any plane wave ψ which is transformed into ψ′ = eihαψ by a rotation of an angle α aroundthe propagation axis is said to have helicity h. Thus gravitational waves have helicity two.Doing the same calculation in an arbitrary gauge, one finds that the remaining, unphysicaldegrees of freedom transform as helicity one and zero.

8.1.5 Detection principle of gravitational waves

Consider the effect of a gravitational wave on a free test particle that is initially at rest,ua = (1, 0, 0, 0). As long as the particle is at rest, the geodesic equation simplifies to ua = Γa

00.The four relevant Christoffel symbols are in linearized approximation, cf. Eq. (8.8),

Γa00 =

1

2(∂0h

a0 + ∂0h

a0 − ∂ah00) . (8.34)

We are free to choose the TT gauge in which all component of hab appearing on the RHSare zero. Hence the acceleration of the test particle is zero and its coordinate position isunaffected by the gravitational wave. (TT gauge defines a “comoving” coordinate system.)

83


The physical distance l of e.g. two test-particles is given by integrating

dl2 = gαβdξαdξβ = (hαβ − δαβ)dξ

αdξβ , (8.35)

where gαβ is the spatial part of the metric and dξ the spatial coordinate distance betweeninfinitesimal separated test particles. Hence the passage of a periodic gravitational wave,hab ∝ cos(ωt), results in a periodic change of the separation of freely moving test particles.The relative size of this change, ∆L/L, is given by the amplitude h of the gravitational wave.

8.2 Energy-momentum pseudo-tensor for gravity

We consider again the splitting (8.1) of the metric, but we take into account now terms ofsecond order in hab. We rewrite the Einstein equation by bringing the Einstein tensor on theRHS and adding the linearized Einstein equation,

R(1)ab − 1

2R(1) ηab = −κTab +

(

−Rab +1

2Rgab +R

(1)ab − 1

2R(1) ηab

)

. (8.36)

The LHS of this equation is the usual gravitational wave equation, while the RHS now includesas source not only matter but also the gravitational field itself. It is therefore natural to define

R(1)ab − 1

2R(1) ηab = −κ (Tab + tab) (8.37)

with tab as the energy-momentum pseudo-tensor for gravity. If we expand all quantities,

gab = ηab + h(1)ab + h

(2)ab +O(h3) , Rab = R

(1)ab +R

(2)ab +O(h3) , (8.38)

we can set, assuming hab ≪ 1, Rab − R(1)ab = R

(2)ab + O(h3), etc. Hence we find as energy-

momentum pseudo-tensor for the metric perturbations h(1)ab at O(h3)

tab = −1

κ

(

R(2)ab − 1

2R(2) ηab

)

. (8.39)

This tensor is symmetric, quadratic in h(1)ab and conserved because of the Bianchi identity.

However, tab is not gauge-invariant, since it can be made at each point identically to zeroby a suitable coordinate transformation. In the case of gravitational waves we may expectthat averaging tab over a volume large compared to the wave-length considered solves thisproblem. Moreover, such an averaging simplifies the calculation of tab, since all terms oddin kx cancel. Nevertheless, the calculation is messy, but gives a simple result. We will use ashort-cut via the following two digressions. . .

Digression I: Graviton propagator In order to read off the graviton propagator, we have toexpand the Einstein-Hilbert action to O(h3). A lengthy but straightforward calculation givesin the harmonic gauge

SEH =1

32πG

∫

d4x

[1

2(∂ahbc)

2 − 1

4(∂ah)

2

]

. (8.40)

84

8.2 Energy-momentum pseudo-tensor for gravity

To read off the propagator, we should bring this expression by partial integrations in the form12h

abPabcdhcd. We find

SEH =1

32πG

∫

d4x1

2hab

1

2[ηabηcd − ηacηbd + ηadηbd]

︸︷︷︸

Pabcd

hcd , (8.41)

and the Feynman propagator of the graviton in momentum space follows as

DabcdF (k) =

P abcd

k2 + iε. (8.42)

Specializing (8.40) to the TT gauge, we obtain

SEH =1

32πG

∫

d4x1

2(∂ahβγ)

2 . (8.43)

Digression II: Averaged stress tensor of a scalar field We calculate first the dynamicalenergy-momentum tensor for a scalar field with potential V (φ),

L =1

2gµν∂µφ∂νφ− V (φ) . (8.44)

Varying the action gives

δSKG =1

2

∫

Ωd4x

√

|g|∂aφ∂bφ δgab + [gab∂aφ∂bφ− 2V (φ)]δ√

|g|

=

∫

Ωd4x√

|g|δgab1

2∂aφ∂bφ− 1

2gabL

. (8.45)

and thus

Tab =2√

|g|δSmδgab

= ∂aφ∂bφ− gabL . (8.46)

We consider now a free field, i.e. set now V (φ) = 0, and take the average over a volume Ωlarge compared to the typical wavelength of the field,

〈Tab〉 =1

Ω

∫

d4x Tab = 〈∂aφ∂bφ〉 − ηab〈(∂cφ)2〉 . (8.47)

Performing a partial integration of the second term, we can drop the surface term, and usethen the equation of motions,

〈(∂cφ)2〉 = −〈φφ〉 = 0 . (8.48)

Hence 〈Tab〉 = 〈∂aφ∂bφ〉. Comparing now SKG and SEH in the TT gauge suggests that theaveraged energy-momentum pseudo-tensor of the gravitational field is given in this gauge by

〈Tab〉 =1

32πG〈∂ahβγ∂bhβγ〉 . (8.49)

This suggestion is less bold than it looks like: All ten independent components of hab satisfy(in the linear limit) the Klein-Gordon equation. In the TT gauge, we have eliminated allunphysical degrees of freedom. The only difference to the scalar case are the indices βγ thatjust regulate how the field hab (not T ab) transforms under Lorentz transformations.

85


8.3 Emission of gravitational waves

At present, gravitational waves have not been measured yet. However, the observation ofclose neutron star-neutron star binaries shows that such systems loose energy, leading to ashrinkage of their orbit with time, consistent with the prediction for the energy loss by theemission of gravitational waves.The steps in deriving this energy loss formula are similar to the corresponding derivation

for the dipole emission formula of electromagnetic radiation. Step one, the derivation of theGreen’s function for the wave equation (8.22) is exactly the same, after having fixed thegauge freedom. In the second step, we have to connect the amplitude of the field at largedistances (“in the wave zone”) to the source, i.e. the current ja and the energy-momentumtensor T ab, respectively. Finally, we use the connection between the field and its (pseudo)energy-momentum tensor (T ab

em or tab) to derive the energy flux through a sphere around thesource.

8.3.1 Quadrupol formula

Gravitational waves in the linearized approximation fulfil the superposition principle. Hence,if the solution for a point source is known,

xG(x− x′) = δ(x− x′) , (8.50)

the general solution can be obtained by integrating the Green’s function over the sources,

hab(x) = −2κ

∫

d4x′G(x− x′)Tab(x′) . (8.51)

The Green’s function G(x−x′) is not completely specified by Eq. (8.50): We can add solutionsof the homogeneous wave equation and we have to specify how the poles of G(x − x′) aretreated. In classical physics, one chooses the retarded Green’s function G(x− x′) defined by

G(x− x′) = − 1

4π|~x− ~x′|δ[|~x− ~x′| − (t− t′)]ϑ(t− t′) , (8.52)

picking up the contributions along the past light-cone.Inserting the retarded Green’s function into Eq. (8.51), we can perform the time integral

using the delta function and obtain

hab(x) = 4G

∫

d3x′Tab(t− |~x− ~x′|, ~x′)

|~x− ~x′| . (8.53)

The retarded time tr ≡ t− |~x− ~x′| denotes the emission time tr of a signal emitted at ~x′ thatreaches ~x at time t propagating with the speed of light.We perform now a Fourier transformation from time to angular frequency,

hab(ω, ~x) =1√2π

∫

dt eiωthab(t, ~x) =4G√2π

∫

dt

∫

d3x′ eiωtTab(tr, ~x

′)

|~x− ~x′| . (8.54)

Next we change from the integration variable t to tr,

hab(ω, ~x) =4G√2π

∫

dtr

∫

d3x′ eiωtreiω|~x−~x′|Tab(tr, ~x′)

|~x− ~x′| (8.55)

86

8.3 Emission of gravitational waves

and introduce the Fourier transformed Tab(ω, ~x′),

hab(ω, ~x) = 4G

∫

d3x′ eiω|~x−~x′|Tab(ω, ~x′)

|~x− ~x′| . (8.56)

We proceed using the same approximations as in electrodynamics: We restrict ourselves toslowly moving, compact sources observed in the wave zone and choose the coordinate systemsuch that |~x′| ≪ |~x|. Then most radiation is emitted at frequencies such that |~x−~x′| ≈ r andthus

hab(ω, ~x) = 4Geiωr

r

∫

d3x′ Tab(ω, ~x′) . (8.57)

Next we use (flat-space) energy-momentum conservation,

∂

∂tT 00 +

∂

∂xβT 0β = 0 , (8.58a)

∂

∂tTα0 +

∂

∂xβTαβ = 0 . (8.58b)

We differentiate (8.58a) with respect to time, and use then (8.58b),

∂2

∂t2T 00 = − ∂2

∂xβ∂tT 0β =

∂2

∂xα∂xβTαβ . (8.59)

Multiplying with xαxβ and integrating gives thus

d2

dt2

∫

d3x xαxβT 00 =

∫

d3x xαxβ∂2

∂xρxσT ρσ = 2

∫

d3x Tαβ . (8.60)

Here we dropped also a surface term, assuming that the source is compact. The harmonicgauge condition ∂ahab = 0 implies in Fourier space

h0b(ω, ~x) =i

ωhαb(ω, ~x) . (8.61)

Hence we need to calculate only the space-like components of hab(ω, ~x). We define asquadrupole moment of the source energy-momentum tensor

Iαβ =

∫

d3x xαxβT 00(x) . (8.62)

Then we can rewrite the solution for h0b(ω, ~x) as

hαβ(ω, ~x) = −4Gω2 eiωr

rIαβ(ω) . (8.63)

Fourier-transforming back to time, the quadrupole formula for the emission of gravitationalwaves results,

hαβ(t, ~x) =2G

rIαβ(tr) . (8.64)

87


Since hab is traceless, the trace of Iab does not produce gravitational waves: It is connectedto the dipole moment and its time derivative vanishes because of conservation of linear mo-mentum. Thus it is more convenient to replace Iab by the reduced (trace-less, irreducible)quadrupole moment

Qαβ =

∫

d3x

[

xαxβ − 1

3δαβr2

]

T 00(x) . (8.65)

Our derivation neglected perturbations of flat space and seems therefore not applicable to aself-gravitating system. However, our final result depends only on the motion of the particles,not how it is produced. An analysis at next order in perturbation theory shows indeed thatour result applies to self-gravitating systems like binary stars.Note the following peculiarity of a gravitational wave experiment: Such an experiment

measures the amplitude hab ∝ 1/r of a metric perturbation, while the sensitivity of all otherexperiments (light, neutrinos, cosmic rays, . . . ) is proportional to the energy flux ∝ 1/r2

of radiation. Increasing thus the sensitivity of a gravitational wave detector by a factor 10increases the number of potential sources by a factor 1000, in contrast to a factor 103/2 forother detectors.One may wonder if this behavior contradicts the fact that also the energy-flux of a gravi-

tational wave follows as 1/r2 law. However, the energy dissipated from a gravitational wavecrossing the Earth (including our experimental set-up) is extremely tiny, while the energy-density of gravitational wave with amplitude as small as h ∼ 10−22 is surprisingly large (checkit e.g. with (8.68)!).

8.3.2 Energy loss

We evaluate now Eq. (8.49) for a plane-wave

hβγ = Aβγ cos(kx) . (8.66)

with amplitudes (chosen to be real) Aβγ . Using 〈sin2(kx)〉 = 1/2, we obtain

〈tab〉 =1

64πGkakbAβγA

βγ . (8.67)

The energy-flux F , i.e. the energy crossing an unit area per unit time, in the direction ~n isin general F = ct0knk. For a plane-wave with wave-vector k

F = ct0iki =c5

64πGk0kikiAβγA

βγ = ct00 , (8.68)

where we used k0 = −kiki. Thus we got the reasonable result that the energy-flux is simplythe energy-density t00 multiplied with the wave-speed c.In the case of a spherical wave emitted from the origin, we choose ~n = ~er. Then

F(~er) =1

64πG〈t0ini〉 =

1

64πG〈(∂thβγ)(~n · ~∇)hβγ〉 = 1

64πG〈(∂thβγ)(∂rhβγ)〉 . (8.69)

Inserting the quadrupole formula, one finds

Lgr = −dE

dt= −

∫

dΩr2F(~er) =G

5

...Qαβ

...Q

αβ. (8.70)

88

9 Cosmological models for an homogeneous,

isotropic universe

9.1 Friedmann-Robertson-Walker metric for an homogeneous,

isotropic universe

Einstein’s cosmological principle Einstein postulated that the Universe is homogeneous andisotropic at each moment of its evolution. Note that a space isotropic around at least twopoints is also homogeneous, while a homogeneous space is not necessarily isotropic. The CMBprovides excellent evidence that the universe is isotropic around us. Baring suggestions thatwe live at a special place, the universe is also homogeneous.

Weyl’s postulate In 1923, Hermann Weyl postulated the existence of a privileged classof observers in the universe, namely those following the “average” motion of galaxies. Hepostulated that these observers follow time-like geodesics that never intersect. They mayhowever diverge from a point in the (finite or infinite) past or converge towards such a pointin the future.

Weyl’s postulate implies that we can find coordinates such that galaxies are at rest. Thesecoordinates are called comoving coordinates and can be constructed as follows: One choosesfirst a space-like hypersurface. Through each point in this hypersurface lies a unique worldlineof a privileged observer. We choose the coordinate time such that it agrees with the proper-time of all observers, g00 = 1, and the spatial coordinate vectors such that they are constantand lie in the tangent space T at this point. Then ua = δa0 and for n ∈ T it follows na = (0, ~n)and

0 = uana = gabu

anb = g0βnβ . (9.1)

Since n is arbitrary, it follows g0β = 0. Hence as a consequence of Weyl’s postulate we maychoose the metric as

ds2 = dt2 − dl2 = dt2 − gαβdxαdxβ . (9.2)

The cosmological principle constrains further the form of dl2: Homogeneity requires thatthe gαβ can depend on time only via a common factor S(t), while isotropy requires that only

~x · ~x, ~dx · ~x, and ~dx · ~dx enter dl2. Hence

dl2 = C(r)(~x · ~dx)2 +D(r)( ~dx · ~dx)2 = C(r)r2dr2 +D(r)[dr2 + r2dϑ2 + r2 sin2 ϑdφ2] (9.3)

We can eliminate the functionD(r) by the rescaling r2 → Dr2. Thus the line-element becomes

dl2 = S(t)[B(r)dr2 + r2dΩ

](9.4)

with dΩ = dϑ2 + sin2 ϑdφ2, while B(r) is a function that we have still to specify.

89

9 Cosmological models for an homogeneous, isotropic universe

Maximally symmetric spaces are spaces with constant curvature. Hence the Riemann ten-sor of such spaces can depend only on the metric tensor and a constant K specifying thecurvature. The only form that respects the (anti-)symmetries of the Riemann tensor is

Rabcd = K(gacgbd − gadgbc) . (9.5)

Contracting Rabcd with gac, we obtain in three dimensions for the Ricci tensor

Rbd = gacRabcd = Kgac(gacgbd − gadgbc) = K(3gbd − gbd) = 2Kgbd . (9.6)

A final contraction gives as curvature R of a three-dimensional maximally symmetric space

R = gabRab = 2Kδaa = 6K . (9.7)

A comparison of Eq. (9.6) with the Ricci tensor for the metric (9.4) will fix the still unknownfunction B(r). We proceed in the standard way: Calculation of the Christoffel symbols withthe help of the geodesic equations, then use of the definition (7.6) for the Ricci tensor,

Rrr =1

rB

dB

dr= 2Kgrr = 2KB (9.8)

Rϑϑ = 1 +r

2B2

dB

dr− 1

B= 2Kgϑϑ = 2Kr2 . (9.9)

(The φφ equation contains no additional information.) Integration of (9.8) gives

B =1

A−Kr2(9.10)

with A as integration constant. Inserting the result into (9.9) determines A as A = 1. Thuswe have determined the line-element of a maximally symmetric 3-space with curvature K as

dl2 =dr2

1−Kr2+ r2(sin2 ϑdφ2 + dϑ2) . (9.11)

Going over to the full four-dimensional line-element, we rescale for K 6= 0 the r coordinateby r → |K|1/2r. Then we absorb the factor 1/|K| in front of dl2 by defining the scale factorR(t) as

R(t) =

S(t)/|K|1/2, K 6= 0

S(t) K = 0(9.12)

As result we obtain the Friedmann-Robertson-Walker (FRW) metric for an homogeneous,isotropic universe

ds2 = dt2 −R2(t)

[dr2

1− kr2+ r2(sin2 ϑdφ2 + dϑ2)

]

(9.13)

with k = ±1 (positive/negative curvature) or k = 0 (flat three-dimensional space). Finally,we give two alternatives forms of the FRW metric that are also often used. The first one usesthe conformal time dη = dt/R,

ds2 = R2(η)[dη2 − dl2)

](9.14)

90

9.2 Geometry of the Friedmann-Robertson-Walker metric

and gives for k = 0 a conformally flat metric. In the second one, one introduces r = sinχ fork = 1. Then dr = cosχdχ = (1− r2)1/2dχ and

ds2 = dt2 −R2(t)[dχ2 + S2(χ)(sin2 ϑdφ2 + dϑ2)

](9.15)

with S(χ) = sinχ = r. Defining

S(χ) =

sinχ for k = 1 ,

χ for k = 0 ,

sinhχ for k = −1 .

(9.16)

the metric (9.15) is valid for all three values of k.


Geometry of the FRW spaces Let us consider a sphere of fixed radius at fixed time, dr =dt = 0. The line-element ds2 simplifies then to R2(t)r2(sin2 ϑdφ2 + dϑ2), which is the usualline-element of a sphere S2 with radius rR(t). Thus the area of the sphere is A = 4π(rR(t))2 =4π[S(χ)R(t)]2 and the circumference of a circle is L = 2πrR(t), while rR(t) has the physicalmeaning of a length.

By contrast, the radial distance between two points (r, ϑ, φ) and (r + dr, ϑ, φ) is dl =R(t)dr/

√1− kr2. Thus the radius of a sphere centered at r = 0 is

l = R(t)

∫ r

0

dr′√1− kr′ 2

= R(t)×

arcsin(r) for k = 1 ,r for k = 0 ,arcsinh(r) for k = −1 .

(9.17)

Using χ as coordinate, the same result follows immediately

l = R(t)

∫ χ(r)

0dχ = R(t)χ . (9.18)

Hence for k = 0, i.e. a flat space, one obtains the usual result L/l = 2π, while for k = 1(spherical geometry) L/l = 2πr/ arcsin(r) < 2π and for k = −1 (hyperbolic geometry)L/l = 2πr/arcsinh(r) > 2π.

For k = 0 and k = −1, l is unbounded, while for k = +1 there exists a maximal distancelmax(t). Hence the first two case correspond to open spaces with an infinite volume, while thelatter is a closed space with finite volume.

Hubble’s law Hubble found empirically that the spectral lines of “distant” galaxies areredshifted, z = ∆λ/λ0 > 1, with a rate proportional to their distance d,

cz = H0d . (9.19)

If this redshift is interpreted as Doppler effect, z = ∆λ/λ0 = vr/c, then the recession velocityof galaxies follows as

v = H0d . (9.20)

91


O

G

d

d′′

d′

Figure 9.1: An observer at position d′ sees the galaxy G recessing with the speedH(d− d′) = Hd′′, if the Hubble relation is linear.

The restriction “distant galaxies” means more precisely that H0d ≫ vpec ∼ few × 100 km/s.In other words, the peculiar motion of galaxies caused by the gravitational attraction ofnearby galaxy clusters should be small compared to the Hubble flow H0d. Note that theinterpretation of v as recession velocity is problematic. The validity of such an interpretationis certainly limited to v ≪ c.The parameter H0 is called Hubble constant and has the value H0 ≈ 71+4

−3 km/s/Mpc. Wewill see soon that the Hubble law Eq. (9.20) is an approximation valid for z ≪ 1. In general,the Hubble constant is not constant but depends on time, H = H(t), and we will call ittherefore Hubble parameter for t 6= t0.We can derive Hubble’s law by a Taylor expansion of R(t),

R(t) = R(t0) + (t− t0)R(t0) +1

2(t− t0)

2R(t0) + . . . (9.21)

= R(t0)

[

1 + (t− t0)H0 −1

2(t− t0)

2q0H20 + . . .

]

, (9.22)

where

H0 ≡R(t0)

R(t0)and q0 ≡ −R(t0)R(t0)

R2(t0)(9.23)

is called deceleration parameter: If the expansion is slowing down, R < 0 and q0 > 0.Hubble’s law follows now as an an approximation for small redshift: For not too large

time-differences, we can use the expansion Eq. (9.21) and write

1− z ≈ 1

1 + z=R(t)

R0≈ 1 + (t− t0)H0 . (9.24)

Hence Hubble’s law, z = (t0−t)H0 = d/cH0, is valid as long as z ≈ H0(t0−t) ≪ 1. Deviationsfrom its linear form arises for z >∼ 1 and can be used to determine q0.

Hubble’s law as consequence of homogeneity Consider Hubble’s law as a vector equationwith us at the center of the coordinate system,

v = Hd . (9.25)

92


galaxy, r = 0

t1 + δt1

t1

t2

t2 + δt2

observer, r

Figure 9.2: World lines of a galaxy emitting light and an observer at comoving coordinatesr = 0 and r, respectively.

What sees a different observer at position d′? He has the velocity v′ = Hd′ relative to us.We are assuming that velocities are small and thus

v′′ ≡ v − v′ = H(d− d′) = Hd′′ , (9.26)

where v′′ and d′′ denote the position relative to the new observer. A linear relation be-tween v and d as Hubble law is the only relation compatible with homogeneity and thus the“cosmological principle”.

Lemaıtre’s redshift formula

A light-ray propagates with v = c or ds2 = 0. Assuming a galaxy at r = 0 and an observerat r, i.e. light rays with dφ = dϑ = 0, we rewrite the FRW metric as

dt

R=

dr√1− kr2

. (9.27)

We integrate this expression between the emission and absorption times t1 and t2 of the firstlight-ray,

∫ t2

t1

dt

R=

∫ r

0

dr√1− kr2

(9.28)

and between t1 + δt1 and t2 + δt2 for the second light-ray (see also Fig. 9.2),

∫ t2+δt2

t1+δt1

dt

R=

∫ r

0

dr√1− kr2

. (9.29)

The RHS’s are the same and thus we can equate the LHS’s,

∫ t2

t1

dt

R=

∫ t2+δt2

t1+δt1

dt

R. (9.30)

93


We change the integration limits, subtracting the common interval [t1 + δt1 : t2] and obtain

∫ t1+δt1

t1

dt

R=

∫ t2+δt2

t2

dt

R. (9.31)

Now we choose the time intervals δti as the time between two wave crests separated by thewave lengths λi of an electromagnetic wave. Since these time intervals are extremely shortcompared to cosmological times, δti = λi/c≪ ti, we can assume R(t) as constant performingthe integrals and obtain

δt1R1

=δt2R2

orλ1R1

=λ2R2

. (9.32)

The redshift z of an object is defined as the relative change in the wavelength between emissionand detection,

z =λ2 − λ1λ1

=λ2λ1

− 1 (9.33)

or

1 + z =λ2λ1

=R2

R1. (9.34)

This result is intuitively understandable, since the expansion of the universe stretches alllengths including the wave-length of a photon. For a massless particle like the photon, ν = cλand E = cp, and thus its frequency (energy) and its wave-length (momentum) are affected inthe same way. By contrast, the energy of a non-relativistic particle with E ≈ mc2 is nearlyfixed.A similar calculation as for the photon can be done for massive particles. Since the geodesic

equation for massive particles leads to a more involved calculation, we use in this case howevera different approach. We consider two comoving observer separated by the proper distanceδl. A massive particle with velocity v needs the time δt = δl/v to travel from observer oneto observer two. The relative velocity of the two observer is

δu =R

Rδl =

R

Rvδt = v

δR

R. (9.35)

Since we assume that the two observes are separated only infinitesimally, we can use theaddition law for velocities from special relativity for the calculation of the velocity v′ measuredby the second observer,

v′ =v − δu

1− vδu= v − (1− v2)δu+O(δu2) = v − (1− v2)v

δR

R. (9.36)

Introducing δv = v − v′, we obtain

δv

v(1− v2)=δR

R. (9.37)

and integrating this equation results in

p =mv√1− v2

=const.

R. (9.38)

Thus not the energy but the momentum p = ~/λ of massive particles is red-shifted: Thekinetic energy of massive particles goes quadratically to zero, and hence peculiar velocitiesrelative to the Hubble flow are strongly damped by the expansion of the universe.

94


Luminosity and angular diameter distance

In an expanding universe, the distance to an object depends on the expansion history, thebehaviour of the scale factor R(t), between the time of emission t of the observed light andits reception at t0. From the metric (9.15) we can define the (radial) coordinate distance

χ =

∫ t0

t

dt

R(t)(9.39)

as well as the proper distance d = gχχχ = R(t)χ. The proper distance is however only fora static metric a measurable quantity and cosmologists use therefore other, operationallydefined measures for the distance. The two most important examples are the luminosity andthe angular diameter distances.

Luminosity distance The luminosity distance dL is defined such, that the inverse-square lawbetween luminosity L of a source at distance d and the received energy flux F is valid,

dL =

(L

4πF

)1/2

. (9.40)

Assume now that a (isotropically emitting) source with luminosity L(t) and comoving coor-dinate χ is observed at t0 by an observer at O. The cut at O through the forward light coneof the source emitted at te defines a sphere S2 with proper area

A = 4πR2(t0)S2(χ) . (9.41)

Two additional effects are that the frequency of a single photon is redshifted, ν0 = νe/(1+ z),and that the arrival rate of photons is reduced by the same factor due to time-dilation. Hencethe received flux is

F(t0) =1

(1 + z)2L(te)

4πR20S

2(χ)(9.42)

and the luminosity distance in a FRW universe follows as

dL = (1 + z)R0S(χ) . (9.43)

Note that dL depends via χ on the expansion history of the universe between te and t0.

Observable are not the coordinates χ or r, but the redshift z of a galaxy. Differentiating1 + z = R0/R(t), we obtain

dz = −R0

R2dR = −R0

R2

dR

dtdt = −(1 + z)Hdt (9.44)

or

t0 − t =

∫ t0

tdt =

∫ 0

z

dz

H(z)(1 + z). (9.45)

Inserting the relation (9.44) into Eq. (9.39), we find the coordinate χ of a galaxy at redshiftz as

χ =

∫ t0

t

dt

R(t)=

1

R0

∫ z

0

dz

H(z)(9.46)

95


For small redshift z ≪ 1, we can use the expansion (9.22)

χ =

∫ t0

t

dt

R0[1− (t− t0)H0 + . . .]−1 (9.47)

≈ 1

R0[(t− t0) +

1

2(t− t0)

2H0 + . . .] =1

R0H0[z − 1

2(1 + q0)z

2 + . . .] (9.48)

In practise, one observes only the luminosity within a certain frequency range instead of thetotal (or bolometric) luminosity. A correction for this effect requires the knowledge of theintrinsic source spectrum.

Angular diameter distance Instead of basing a distance measurement on standard candles,one may use standard rods with know proper length l whose angular diameter ∆ϑ can beobserved. Then we define the angular diameter distance as

dA =l

∆ϑ. (9.49)

dA =R0S(χ)

1 + z. (9.50)

Thus at small distances, z ≪ 1, the two definitions agree by construction, while for largeredshift the differences increase as (1 + z)2.

9.3 Friedmann equations

The FRW metric together with a perfect fluid as energy-momentum tensor gives for thetime-time component of the Einstein equation

R = −4πG

3(ρ+ 3P )R , (9.51)

for the space-time components

RR+ 2R2 + 2K = 4πG(ρ− P ) , (9.52)

and 0 = 0 for the space-space components. Eliminating R and showing explicitly the con-tribution of a cosmological constant to the energy density ρ, the usual Friedmann equationfollows as

H2 ≡(

R

R

)2

=8π

3Gρ− k

R2+

Λ

3. (9.53)

while the “acceleration equation” is

R

R=

Λ

3− 4πG

3(ρ+ 3P ) . (9.54)

This equation determines the (de-) acceleration of the Universe as function of its matter andenergy content. “Normal” matter is characterized by ρ > 0 and P ≥ 0. Thus a static solution

96

9.3 Friedmann equations

is impossible for a universe with Λ = 0. Such a universe is decelerating and since today R > 0,R was always negative and there was a “big bang”.We define the critical density ρcr as the density for which the spatial geometry of the

universe is flat. From k = 0, it follows

ρcr =3H2

0

8πG(9.55)

and thus ρcr is uniquely fixed by the value of H0. One “hides” this dependence by introducingh,

H0 = 100h km/(sMpc) .

Then one can express the critical density as function of h,

ρcr = 2.77× 1011h2M⊙/Mpc3 = 1.88× 10−29h2g/cm3 = 1.05× 10−5h2GeV/cm3 .

Thus a flat universe with H0 = 100hkm/s/Mpc requires an energy density of ∼ 10 protonsper cubic meter. We define the abundance Ωi of the different players in cosmology as theirenergy density relative to ρcr, Ωi = ρ/ρcr.In the following, we will often include Λ as other contributions to the energy density ρ via

8π

3GρΛ =

Λ

3. (9.56)

Thereby one recognizes also that the cosmological constant acts as a constant energy density

ρΛ =Λ

8πGor ΩΛ =

Λ

3H20

. (9.57)

We can understand better the physical properties of the cosmological constant by replacingΛ by (8πG)ρΛ. Now we can compare the effect of normal matter and of the Λ term on theacceleration,

R

R=

8πG

3ρΛ − 4πG

3(ρ+ 3P ) (9.58)

Thus Λ is equivalent to matter with an E.o.S. wΛ = P/ρ = −1. This property can be checkedusing only thermodynamics: With P = −(∂U/∂V )S and UΛ = ρΛV , it follows P = −ρ.The borderline between an accelerating and decelerating universe is given by ρ = −3P or

w = −1/3. The condition ρ < −3P violates the so-called strong energy condition for “normal”matter in equilibrium. An accelerating universe requires therefore a positive cosmologicalconstant or a dominating form of matter that is not in equilibrium.Note that the energy contribution of relativistic matter, photons and possibly neutrinos,

is today much smaller than the one of non-relativistic matter (stars and cold dark matter).Thus the pressure term in the acceleration equation can be neglected at the present epoch.Measuring R/R, R/R and ρ fixes therefore the geometry of the universe.

Thermodynamics The first law of thermodynamics becomes for a perfect fluid with dS = 0simply

dU = TdS − PdV = −PdV (9.59)

ord(ρR3) = −Pd(R3) . (9.60)

97


Dividing by dt,Rρ+ 3(ρ+ P )R = 0 , (9.61)

we obtain our old result,ρ = −3(ρ+ P )H . (9.62)

This result could be also derived from ∇aTab = 0. Moreover, the three equations are not

independent.

9.4 Scale-dependence of different energy forms

The dependence of different energy forms as function of the scale factor R can derived fromenergy conservation, dU = −PdV , if an E.o.S. P = P (ρ) = wρ is specified. For w = const.,it follows

d(ρR3) = −3PR2dR (9.63)

or eliminating Pdρ

dRR3 + 3ρR2 = −3wρR2 . (9.64)

Separating the variables,

−3(1 + w)dR

R=

dρ

ρ, (9.65)

we can integrate and obtain

ρ ∝ R−3(1+w) =

R−3 for matter (w = 0) ,R−4 for radiation (w = 1/3) ,const. for Λ (w = −1) .

(9.66)

This result can be understood also from heuristic arguments:

• (Non-relativistic) matter means that kT ≪ m. Thus ρ = nm ≫ nT = P and non-relativistic matter is pressure-less, w = 0. The mass m is constant and n ∝ 1/R3, henceρ is just diluted by the expansion of the universe, ρ ∝ 1/R3.

• Radiation is not only diluted but the energy of each single photon is additionally red-shifted, E ∝ 1/R. Thus the energy density of radiation scales as ∝ 1/R4. Alternatively,one can use that ρ = aT 4 and T ∝ 〈E〉 ∝ 1/R.

• Cosmological constant Λ: From 8π3 Gρλ = Λ

3 one obtains that the cosmological constantacts as an energy density ρλ = Λ

8πG that is constant in time, independent from a possibleexpansion or contraction of the universe.

• Note that the scaling of the different energy forms is very different. It is thereforesurprising that “just today”, the energy in matter and due to the cosmological constantis of the same order (“coincidence problem”).

Let us rewrite the Friedmann equation for the present epoch as

k

R20

= H20

(8πG

3H20

ρ0 +Λ

3H20

− 1

)

= H20 (Ωtot,0 − 1) . (9.67)

We express the curvature term for arbitrary times through Ωtot,0 and the redshift z as

k

R2=

k

R20

(1 + z)2 = H20 (Ωtot,0 − 1)(1 + z)2 . (9.68)

98

9.5 Cosmological models with one energy component

Dividing the Friedmann equation (9.53) by H20 = 8πGρcr/3, we obtain

H2(z)

H20

=∑

i

Ωi(z)− (Ωtot,0 − 1)(1 + z)2

= Ωrad,0(1 + z)4 +Ωm,0(1 + z)3 +ΩΛ − (Ωtot,0 − 1)(1 + z)2 (9.69)

This expression allows us to calculate the age of the universe (9.45), distances (9.43), etc. fora given cosmological model, i.e. specifying the energy content Ωi,0 and the Hubble parameterH0 at the present epoch.

9.5 Cosmological models with one energy component

We consider a flat universe, k = 0, with one dominating energy component with E.o.S w =P/ρ = const.. With ρ = ρcr (R/R0)

−3(1+w), the Friedmann equation becomes

R2 =8π

3GρR2 = H2

0R3+3w0 R−(1+3w) , (9.70)

where we inserted the definition of ρcr = 3H20/(8πG). Separating variables we obtain

R−(3+3w)/20

∫ R0

0dR R(1+3w)/2 = H0

∫ t0

0dt = t0H0 (9.71)

and hence the age of the Universe follows as

t0H0 =2

3 + 3w=

2/3 for matter (w = 0) ,1/2 for radiation (w = 1/3) ,→ ∞ for Λ (w = −1) .

(9.72)

Models with w > −1 needed a finite time to expand from the initial singularity R(t = 0) = 0to the current size R0, while a Universe with only a Λ has no “beginning”.In models with a hot big-bang, ρ, T → ∞ for t → 0, and we should expect that classical

gravity breaks down at some moment t∗. As long as R ∝ tα with α < 1, most time elapsedduring the last fractions of t0H0. Hence our result for the age of the universe does not dependon unknown physics close to the big-bang as long as w > −1/3.If we integrate (9.71) to the arbitrary time t, we obtain the time-dependence of the scale

factor,

R(t) ∝ t2/(3+3w) =

t2/3 for matter (w = 0) ,

t1/2 for radiation (w = 1/3) ,exp(t) for Λ (w = −1) .

(9.73)

Age problem of the universe. The age of a matter-dominated universe is (expanded aroundΩ0 = 1)

t0 =2

3H0

[

1− 1

5(Ω0 − 1) + . . .

]

. (9.74)

Globular cluster ages require t0 ≥ 13 Gyr. Using Ω0 = 1 leads to H0 ≤ 2/3 × 13Gyr =1/19.5Gyr or h ≤ 0.50. Thus a flat universe with t0 = 13 Gyr without cosmological constantrequires a too small value of H0. Choosing Ωm ≈ 0.3 increases the age by just 14%.

99


0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

t 0H0

Ωm

Ωm+ΩΛ=1

open

Figure 9.3: The product t0H0 for an open universe containing only matter (dotted blue line)and for a flat cosmological model with ΩΛ +Ωm = 1 (solid red line).

We derive the age t0 of a flat Universe with Ωm +ΩΛ = 1 in the next section as

3t0H0

2=

1√ΩΛ

ln1 +

√ΩΛ√

1− ΩΛ. (9.75)

Requiring H0 ≥ 65 km/s/Mpc and t0 ≥ 13 Gyr means that the function on the RHS shouldbe larger than 3× 13Gyr× 0.65/(2× 9.8Gyr) ≈ 1.3 or ΩΛ ≥ 0.55.

9.6 The ΛCDM model

We consider a flat Universe containing as its only two components pressure-less matter anda cosmological constant, Ωm + ΩΛ = 1. Then the curvature term in the Friedmann equationand the pressure term in the deceleration equation play no role and we can hope to solvethese equations for a(t). Multiplying the deceleration equation (9.54) by two and adding itto the Friedmann equation (9.53), we eliminate ρm,

2a

a+

(a

a

)2

= Λ . (9.76)

(We denote the scale factor in this section with a.) Next we rewrite first the LHS and thenthe RHS as total time derivatives: With

d

dt(aa2) = a3 + 2aaa = aa2

[(a

a

)2

+ 2a

a

]

, (9.77)

we obtaind

dt(aa2) = aa2Λ =

1

3

d

dt(a3)Λ . (9.78)

Integrating is now trivial,

aa2 =Λ

3a3 + C . (9.79)

100

9.7 Determining Λ and the curvature R0 from ρm,0, H0, q0

The constant C can be determined most easily by setting a(t0) = 1 and comparing theFriedmann equation (9.53) with (9.79) for t = t0 as C = 8πGρm,0/3.Next we introduce the new variable x = a3/2. Then

da

dt=

dx

dt

da

dx=

dx

dt

2x−1/3

3, (9.80)

and we obtain as new differential equation

x2 − Λx2/4 + C/3 = 0 . (9.81)

Inserting the solution x(t) = A sinh(√Λt/2) of the homogeneous equation fixes the constant

A as A =√

3C/Λ. We can express A also by the current values of Ωi as A = Ωm/ΩΛ =(1− ΩΛ)/ΩΛ. Hence the time-dependence of the scale factor is

a(t) = A1/3 sinh2/3(√3Λt/2) . (9.82)

The time-scale of the expansion is set by tΛ = 2/√3Λ.

The present age t0 of the universe follows by setting a(t0) = 1 as

t0 = tΛarctanh(√

ΩΛ) . (9.83)

The deceleration parameter q = −a/aH2 is an important quantity for observational testsof the ΛCDM model. We calculate first the Hubble parameter

H(t) =a

a=

2

3tΛcoth(t/tΛ) (9.84)

and then

q(t) =1

2[1− 3 tanh2(t/tΛ) . (9.85)

The limiting behavior of q corresponds with q = 1/2 for t → 0 and q = −1 for t → ∞as expected to the one of a flat Ωm = 1 and a ΩΛ = 1 universe. More interesting is thetransition region and, as shown in Fig. 9.4, the transition from a decelerating to an acceleratinguniverse happens for ΩΛ = 0.7 at t ≈ 0.55t0. This can easily converted to redshift, z∗ =a(t0)/a(t∗)− 1 ≈ 0.7, that is directly measured in supernova observations.

9.7 Determining Λ and the curvature R0 from ρm,0, H0, q0

General discussion: We apply now the Friedmann and the acceleration equation to thepresent time. Thus R0 = R0H0, R = −q0H2

0R0 and we can neglect the pressure term inEq. (9.54),

R0

R0= −q0H2

0 =Λ

3− 4πG

3ρm,0 . (9.86)

Thus we can determine the value of the cosmological constant from the observables ρm,0, H0

and q0 viaΛ = 4πGρm,0 − 3q0H

20 . (9.87)

Solving next the Friedmann equation (9.53) for k/R20,

k

R20

=8πG

3ρm,0 +

Λ

3−H2

0 , (9.88)

101


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.5 1 1.5 2

q

t/t0

Ω=0.9

Ω=0.1

Figure 9.4: The deceleration parameter q as function of t/t0 for the ΛCDM model and variousvalues for ΩΛ (0.1, 0.3, 0.5, 0.7 and 0.9 from the top to the bottom).

we write ρm,0 = Ωmρcr and insert Eq. (9.87) for Λ. Then we obtain for the curvature term

k

R20

=H2

0

2(3Ωm − 2q0 − 2) . (9.89)

Hence the sign of 3Ωm − 2q0 − 2 decides about the sign of k and thus the curvature of theuniverse. For a universe without cosmological constant, Λ = 0, equation (9.87) gives Ωm = 2q0and thus

k = −1 ⇔ Ωm < 1 ⇔ q0 < 1/2 ,

k = 0 ⇔ Ωm = 1 ⇔ q0 = 1/2 , (9.90)

k = +1 ⇔ Ωm > 1 ⇔ q0 > 1/2 .

For a flat universe with Λ = 0, ρm,0 = ρcr and k = 0,

0 = 4πG3H2

0

8πG+H2

0 (q0 − 1) = H20

(3

2+ q0 − 1

)

, (9.91)

and thus q0 = 1/2. In this special case, q0 < 1/2 means k = −1 and thus an infinite spacewith negative curvature, while a finite space with positive curvature has q > 1/2.

Example: Comparison with observations: Use the Friedmann equations applied to the presenttime to derive central values of Λ and k,R0 from the observables H0 ≈ (71 ± 4) km/s/Mpc andρ0 = (0.27± 0.04)ρcr, and q0 = −0.6. Discuss the allowed range and significance of the values.We evaluate first

H20 ≈

(7.1× 106cm

s 3.1× 1024cm

)2

≈ 5.2× 10−36s−2 .

102

9.8 Particle horizons

The value of the cosmological constant Λ follows as

Λ = 4πGρm,0 − 3q0H20 = 3H2

0

(ρ

2ρcr− 3q0

)

≈ 3H20 × (

1

2× 0.27 + 0.6) ≈ 0.73× 3H2

0

or ΩΛ = 0.73.The curvature radius R follows as

k

R20

= 4πGρm,0 −H20 (q0 + 1) = 3H2

0

(ρ

2ρcr− q0 + 1

3

)

(9.92)

= 3H20 (0.135± 0.02− 0.4/3) = 3H2

0 (0.002± 0.02) (9.93)

thus a flat universe (k = 0) is consistent with observations.

9.8 Particle horizons

The particle horizon lH is defined as distance out to which one can observe a particle byexchange of a light signal, i.e. it is the border of the region causally connected to the observer.Without expansion, lH = ct0, where t0 is the age of the universe. In an expanding universe,the path the light has to travel will be stretched, dlH = R0/R(t)cdt, and thus

lH = cR0

∫dt′

R(t′)

For a matter- or radiation-dominated universe R(t) = R0(t/t0)α with α = 2/3 and 1/2,

respectively. Both models start with an initial singularity t = 0, and thus

lH(t0) = c

∫ t0

0dt

(t

t0

)−α

=ct0

1− α.

The ratiolH(t)

R(t)∝ t

tα∝ t1−α

gives the fraction of the Hubble horizon that was causally connected at time t < t0. Since0 < α < 1, this fraction decreases going back in time.For an universe dominated by a cosmological constant Λ > 0, R(t) = R0 exp(

√

Λ/3t) =R0 exp(Ht) and thus

lH(t2) = cR0

∫ t0

t

dt′

exp(Ht′)=cR0

H[exp(−Ht)− exp(−Ht0)]

With R(t0) = R0 and thus t0 = 0,

lH(t)

R0=

c

H[exp(−Ht)− 1] .

Since t < t0 = 0, the expression in the bracket is always larger than one and the causallyconnected region is larger than the Hubble horizon. If exponential expansion would havepersisted for all times, then lH(t) → ∞ for t → −∞ and thus the whole universe would becausally connected.

103

10 Cosmic relics

10.1 Time-line of important dates in the early universe

Different energy form today. Let us summarize the relative importance of the various energyforms today. The critical density ρcr = 3H2

0/(8πG) has with h = 0.7 today the numericalvalue ρcr ≈ 7.3 × 10−6 GeV/cm3. This would corresponds to roughly 8 protons per cubicmeter. However, main player today is the cosmological constant with ΩΛ ≈ 0.73. Next comes(pressure-less) matter with Ωm ≈ 0.27 that consists mostly of non-baryonic dark matter,while only Ωb = 4% of the total energy density of the universe consists of matter that weknow. The energy density of cosmic microwave background (CMB) photons with temperatureT = 2.7K = 2.3× 10−4 eV is ργ = aT 4 = 0.4 eV/cm3 or Ωγ ≈ 5× 10−5.

The contribution of the three neutrino flavors to the energy density depends on the unknownabsolute neutrino mass scale, 5 × 10−5 <∼ Ων <∼ 0.05. The lower bound corresponds to three(effectively) massless neutrinos, the upper to one massive neutrino flavor with mν ∼ 0.3 eV.

Different energy forms as function of time The scaling of Ωi with redshift z, 1 + z =R0/R(t) is given by

H2(z)/H20 = Ωm,0(1 + z)3 +Ωrad,0(1 + z)4 +ΩΛ − (Ωtot,0 − 1)(1 + z)2

︸︷︷︸

≈ 0

. (10.1)

Thus the relative importance of the different energy forms changes: Going back in time, oneenters first the matter-dominated and then the radiation-dominated epoch.

The cosmic triangle shown in Fig. 10.1 illustrates the evolution in time of the various energycomponents and the resulting coincidence problem: Any universe with a non-zero positivecosmological constant will be driven with time to a fix-point with Ωm,Ωk → 0. The onlyother non-evolving state is a flat universe containing only matter—however, this solution isunstable. Hence, the question arises why we live in an epoch where all energy componentshave comparable size.

Temperature increase as T ∼ 1/R has three main effects: Firstly, bound states like atomsand nuclei are dissolved when the temperature reaches their binding energy, T >∼ Eb. Secondly,particles with mass mX can be produced, when T >∼ 2mX , in reactions like γγ → XX. Thusthe early Universe consists of a plasma containing more and more heavier particles that are inthermal equilibrium. Finally, most reaction rates Γ = nσv increase faster than the expansion

rate of the universe for t → 0, since n ∝ T 3 for relativistic particles, while H ∝ ρ1/2rad ∝ T 2.

Therefore, reactions that have became ineffective today were important in the early Universe.

Matter-radiation equilibrium zeq: The density of matter decreases slower than the energydensity of radiation. Going backward in time, there will be therefore a time when the density

104

10.1 Time-line of important dates in the early universe

OPEN

FLAT

CLOSED

1.0

0.0

0.0

1.0

0.0

2.0

1.0

-0.5

Ωk

Ωm

ΩΛ

0.5

0.5

1.5

1.5

0.5

OCDM

SCDM

ΛCDM

Figure 10.1: The cosmic triangle showing the time evolution of the various energycomponents.

of matter and radiation were equal. Before that time with redshift zeq, the universe wasradiation-dominated ,

Ωrad,0(1 + zeq)4 = Ωm,0(1 + zeq)

3 (10.2)

or

zeq =Ωm,0

Ωrad,0− 1 ≈ 5400 . (10.3)

This time is important, because i) the time-dependence of the scale factor changes fromR ∝ t2/3 for a matter to R ∝ t1/2 for a radiation dominated universe, ii) the E.o.S. and thus thespeed of sound changed from w ≈ 1/3, v2s = (∂P/∂ρ)S = c2/3 to w ≈ 0, v2s = 5kT/(3m) ≪ c2.The latter quantity determines the Jeans length and thus which structures in the Universecan collapse.

Recombination zrec: Today, most hydrogen and helium in the interstellar and intergalacticmedium is neutral. Increasing the temperature, the fraction of ions and free electron increases,i.e. the reaction H+γ ↔ H++e− that is mainly controlled by the factor exp(−Eb/kT ) will beshifted to the right. By definition, we call recombination the time when 50% of all atoms areionized. A naive estimate gives kT ∼ Eb ≈ 13.6 eV≈ 160.000K or zrec = 60.000. However,there are many more photons than hydrogen atoms, and therefore recombination happenslatter: A more detailed calculation gives zrec ∼ 1000.Since the interaction probability of photons with neutral hydrogen is much smaller than withelectrons and protons, recombination marks the time when the Universe became transparentto light.

105

10 Cosmic relics

Big Bang Nucleosynthesis At Tns ∼ ∆ ≡ mn −mp ≈ 1.3 MeV or t ∼ 1 s, part of protonsand neutrons forms nuclei, mainly 4He. As in the case of recombination, the large numberof photons delays nucleosynthesis relative to the estimate Tns ≈ ∆ to Tns ≈ 0.1MeV.

Quark-hadron or QCD transition Above T ∼ mπ ∼ 100 MeV, hadrons like protons, neu-trons or pions dissolve into their fundamental constituents, quarks q and gluons g.

Baryogenesis All the matter observed in the Universe consists of matter (protons and elec-trons), and not of anti-matter (anti-protons an positrons). Thus the baryon-to-photon ratiois

η =nb − nbnγ

=nbnγ

=Ωbρcr/mN

2ζ(3)T 3γ /π

2≈ 7× 10−10 . (10.4)

The early plasma of quarks q and anti-quarks q contained a tiny surplus of quarks. After allanti-matter annihilated with matter, only the small surplus of matter remained. The tinyasymmetry can be explained by interactions in the early Universe that were not completelysymmetric with respect to an exchange of matter-antimatter.

10.2 Equilibrium statistical physics in a nut-shell

The distribution function f(p) of a free gas of fermions or bosons in kinetic equilibrium areare

f(p) =1

exp[β(E − µ)]± 1(10.5)

where β = 1/T denotes the inverse temperature, E =√

m2 + p2, and +1 refers to fermionsand -1 to bosons, respectively. As we will see later, photons as massless particles stay alsoin an expanding universe in equilibrium and may serve therefore as a thermal bath for otherparticles. A species X stays in kinetic equilibrium, if e.g. in the reaction X + γ → X + γ theenergy exchange with photons is fast enough.

The chemical potential µ is the average energy needed, if an additional particles is added,dU =

∑

i µdNi. If µ is zero, If the species X is also in chemical equilibrium with other species,e.g. via the reaction X + X ↔ γ + γ with photons, then their chemical potentials are relatedby µX + µX = 2µγ = 0.

The number density n, energy density ρ and pressure P of a species X follows as

n =g

(2π)3

∫

d3p f(p) , (10.6)

ρ =g

(2π)3

∫

d3p Ef(p) , (10.7)

P =g

(2π)3

∫

d3pp2

3Ef(p) . (10.8)

The factor g takes into account the internal degrees of freedom like spin or color. Thus for aphoton, a massless spin-1 particle g = 2, for an electron g = 4, etc.

106

10.2 Equilibrium statistical physics in a nut-shell

Derivation of the pressure integral for free quantum gas:Comparing the 1. law of thermodynamics, dU = TdS − PdV , with the total differentialdU = (∂U/∂S)V dS + (∂U/∂V )SdV gives P = −(∂U/∂V )S .Since U = V

∫Ef(p) and S ∝ ln(V f(p)), differentiating U keeping S constant means P =

−V∫(∂E/∂V )f(p).

We write ∂E/∂V = (∂E/∂p)(∂p/∂L)(∂L/∂V ). To evaluate this we note that ∂E/∂p = p/E,that from V = L3 it follows ∂L/∂V = 1/(3L2) and that finally the quantization conditions offree particles, pk = 2πk/L implies ∂p/∂L = −p/L. Combined this gives ∂E/∂V = −p2/(3EV ).

In the non-relativistic limit T ≪ m, eβ(m−µ) ≫ 1 and thus differences between bosons andfermions disappear,

n =g

2π2e−β(m−µ)

∫ ∞

0dp p2e−β p2

2m = g

(mT

2π

)3/2

exp[−β(m− µ)] , (10.9)

ρ = mn , (10.10)

P = nT ≪ ρ . (10.11)

These expressions correspond to the classical Maxwell-Boltzmann statistics1. The number ofnon-relativistic particles is exponentially suppressed, if their chemical potential is small. Sincethe number of protons per photons is indeed very small in the universe, cf. Eq. (10.4), andtherefore also the number of electron (the universe should be neutral), the chemical potentialµ can be neglected in cosmology at least for protons and electron.

In the relativistic limit T ≫ m with T ≫ µ all properties of a gas are determined by itstemperature T ,

n =g T 3

2π2

∫ ∞

0dx

x2

ex ± 1= ε1

ζ(3)

π2gT 3 , (10.12)

ρ =g T 4

2π2

∫ ∞

0dx

x3

ex ± 1= ε2

π2

30gT 4 , (10.13)

P = ρ/3 , (10.14)

where for bosons ε1 = ε2 = 1 and for fermions ε1 = 3/4 and ε2 = 7/8, respectively.

Since the energy density and the pressure of non-relativistic species is exponentially sup-pressed, the total energy density and the pressure of all species present in the universe canbe well-approximated including only relativistic ones,

ρrad =π2

30g∗T

4 , (10.15)

Prad = ρrad/3 =π2

90g∗T

4 , (10.16)

where

g∗ =∑

bosons

gi

(TiT

)4

+7

8

∑

fermions

gi

(TiT

)4

. (10.17)

Here we took into account that the temperature of different particle species can differ.

1Integrals of the type∫

∞

0dxx2ne−ax2

can be reduced to a Gaussian integral by differentiating with respectto the parameter a.

107

10 Cosmic relics

Entropy Rewriting the first law of thermodynamics, dU = TdS − PdV , as

dS =dU

T+P

TdV =

d(V ρ)

T+p

TdV =

V

T

dρ

dTdT +

ρ+ P

TdV (10.18)

and comparing this expression with the total differential dS(T, V ), one obtains

∂S

∂V T=ρ+ P

T. (10.19)

Since the RHS is independent of V for constant T , we can integrate and obtain

S =ρ+ P

TV + f(T ) . (10.20)

The integration constant f(T ) has to vanish to ensure that S is an extensive variable, S ∝ V .The total entropy density s ≡ S/V of the universe can again approximated by the rela-

tivistic species,

s =2π2

45g∗ST

3 , (10.21)

where now

g∗,S =∑

bosons

gi

(TiT

)3

+7

8

∑

fermions

gi

(TiT

)3

. (10.22)

The entropy S is an important quantity because it is conserved during the evolution of theuniverse. Conservation of S implies that S ∝ g∗,SR

3T 3 = const. and thus the temperatureof the Universe evolves as

T ∝ g−1/3∗,S R−1 . (10.23)

When g∗ is constant, the temperature T ∝ 1/R. Consider now the case that a particlespecies, e.g. electrons, becomes non-relativistic at T ∼ me. Then the particles annihilate,e++e− → γγ, and its entropy is transferred to photons. Formally, g∗,S decreases and thereforethe temperature decreases for a short period less slowly than T ∝ 1/R.Since s ∝ R−3 and also the net number of particles with a conserved charge, e.g. nB ≡

nB − nB ∝ R−3 if baryon number B is conserved, the ratio nB/s remains constant.

Relativistic degrees of freedom. To obtain the number of relativistic degrees of freedomg∗ in the universe as function of T , we have to know the degrees of freedom of the variousparticle species:

• The spin degrees of freedom of massive particles with spin s are 2s+1, and of neutrinos1, where we count particles and anti-particles separately. Massless bosons like photonsand gravitons are their own anti-particle and have 2 spin states.

• Below TQCD ∼ 250 MeV strongly interacting particles are bound in hadrons, whileabove TQCD free quarks and gluons exist.

• Quarks have as additional label 3 colors, there are eight gluons.

• We assume that all species have the same temperature and approximate their contribu-tion to g∗ by a step function ϑ(T −m).

Using the “Particle Data Book” to find the masses of the various particles, we can constructg∗ as function of T as shown in table 10.1.

108

10.3 Big Bang Nucleosynthesis

Temperature new particles 4∆g∗ 4g∗

T < me γ + νi 4× (2 + 3× 2× 7/8) 29me < T < mµ e± 14 43mµ < T < mπ µ± 14 57mπ < T < Tc π±, π0 12 69Tc < T < ms u, u, d, d, g 6× 14 + 4× 8× 2− 12 205ms < T < mc s, s 3× 14 = 42 247mc < T < mτ c, c 42 289mτ < T < mb τ± 14 303mb < T < mW,Z b, b 42 345mW,Z < T <? W±, Z 4× 3× 3 = 36 381

Table 10.1: The number of relativistic degrees of freedom g∗ present in the universe as func-tion of its temperature.


Nuclear reactions in stars are supposed to produce all the observed heavier elements. However,stellar reaction can explain at most a fraction of 5% of 4He, while the production of the weaklybound deuterium and Lithium-7 in stars is impossible. Thus the light elements up to Li-7 areprimordial: Y (D) = few×10−5, Y ( 3H) = few×10−5 Y (4He) ≈ 0.25, Y ( 7Li) ≈ (1−2)×10−7.Observational challenge is to find as ”old” stars/gas clouds as possible and then to extrapolateback to primordial values.

Estimate of 4He production by stars:The binding energy of 4He is Eb = 28.3 MeV. If 1/4 of all nucleons were fussed into 4He duringt ∼ 10 Gyr, the luminosity-mass ratio would be

L

Mb

=1

4

Eb

4mpt= 5

erg

g s≈ 2.5

L⊙

M⊙

.

The observed luminosity-mass ratio is however only LMb

≤ 0.05L⊙/M⊙. Assuming a roughly

constant luminosity of stars, they can produce only 0.05/2.5 ≈ 2% of the observed 4He.

Big Bang Nucleosynthesis (BBN) is controlled by two parameters: The mass differencebetween protons and neutrons, ∆ ≡ mn −mp ≈ 1.3 MeV and the freeze-out temperature Tfof reaction converting protons into neutrons and vice versa.

10.3.1 Equilibrium distributions

In the non-relativistic limit T ≪ m, the number density of the nuclear species with massnumber A and charge Z is

nA = gA

(mAT

2π

)3/2

exp[β(µA −mA)] . (10.24)

109

10 Cosmic relics

In chemical equilibrium, µA = Zµp + (A − Z)µn and we can eliminate µA by inserting theequivalent expression of (10.24) for protons and neutrons,

exp(βµA) = exp[β(Zµp + (A− Z)µn)] =nZp n

A−Zn

2A

(2π

mNT

)3A/2

exp[β(Zmp + (A− Z)mn)] .

(10.25)Here and in the following we can set in the pre-factors mp ≈ mn ≈ mN and mA ≈ AmN ,keeping the exact masses only in the exponentials. Inserting this expression for exp(βµA)together with the definition of the binding energy of a nucleus, BA = Zmp+(A−Z)mn−mA,we obtain

nA = gA

(2π

mNT

)3(A−1)/2 A3/2

2AnZp n

A−Zn exp(βBA) . (10.26)

The mass fraction XA contributed by a nuclear species is

XA =AnAnB

with nB = np + nn +∑

i

AinAiand

∑

i

Xi = 1 . (10.27)

With nZp nA−Zn /nN = XZ

p XA−Zn nA−1

N and η ∝ T 3 and thus nA−1B ∝ ηA−1T 3(A−1), we have

XA ∝(T

mN

)3(A−1)/2

ηA−1XZp X

A−Zn exp(βBA) . (10.28)

The fact that η ≪ 1, i.e. that the number of photons per baryon is extremely large, meansthat nuclei with A > 1 are much less abundant and that nucleosynthesis takes place laterthan naively expected. Let us consider the particular case of deuterium in Eq. (10.28),

XD

XpXn=

24ζ(3)√π

(T

mN

)3/2

η exp(βBD) (10.29)

with BD = 2.23 MeV. The start of nucleosynthesis could be defined approximately by thecondition XD/(XpXn) = 1, or T ≈ 0.1MeV according to the left panel in Fig. 10.2. The rightpanel of the same figure shows the results, if the equations (10.28) together with

∑

iXi = 1are solved for the lightest and stablest nuclei. Now it becomes clear that in thermal equi-librium between 0.1 <∼ T <∼ 0.2MeV essentially all free neutrons will bind to 4He. For lowtemperatures one cannot expect that the true abundance follows the equilibrium abundance,Eq. (10.28), shown in Fig. 10.2. First, in the expanding universe the weak reactions thatconvert protons and nucleons will freeze out as soon as their rate drops below the expansionrate of the universe. This effect will discussed in the following in more detail. Second, theCoulomb barrier will prevent the production of nuclei with Z ≫ 1. Third, neutrons are notstable and decay.

10.3.2 Proton-neutron ratio

Gamov criterion The interaction depth τ = nlσ gives the probability that a test particleinteracts with cross section σ in a slab of length l filled with targets of density n. If τ ≫ 1,interactions are efficient and the test particle is in thermal equilibrium with the surrounding.We can apply the same criteria to the Universe: We say a particle species A is in thermalequilibrium, as long as τ = nlσ = nσvt≫ 1. The time t corresponds to the typical time-scale

110


1e-15

1e-10

1e-05

1

100000

1e+10

1e+15

1e+20

0.1 1 10 100

XD

/(X

p X

n)

T/MeV

1e-25

1e-20

1e-15

1e-10

1e-05

1

1

XA

T/MeV

n=p

D

3He

4He

12C

Figure 10.2: Relative equilibrium abundance XD/(XpXn) of deuterium as function of tem-perature T (left) and equilibrium mass fractions of nucleons, D, 3He, 2He and12C (right).

for the expansion of the universe, τ = (R/R)−1 = H−1. Note that this is also the typicaltime-scale for changes in the temperature T . Thus we can rewrite this condition as

Γ ≡ nσv ≫ H . (10.30)

A particle species ”goes out of equilibrium” when its interaction rate Γ becomes smaller thanthe expansion rate H of the universe.

Decoupling of neutrinos The interaction rates of neutrinos in processes like n↔ p+e−+νeor e+e− ↔ νν is σ ∼ G2

FE2. If we approximate the energy of all particle species by their

temperature T , their velocity by c and their density by n ∼ T 3, then the interaction rate ofweak processes is

Γ ≈ 〈vσnν〉 ≈ G2FT

5 (10.31)

The early universe is radiation-dominated with ρrad ∝ 1/R4, H = 1/(2t) and negligible cur-vature k/R2. Thus the Friedmann equation simplifies to H2 = (8π/3)Gρ with ρ = g∗π

2/30T 4,or

1

2t= H = 1.66

√g∗

T 2

MPl. (10.32)

Here, we introduced also the Planck mass MPl = 1/√GN ≈ 1.2 × 1019 GeV. Requiring

Γ(Tfr) = H(Tfr) gives as freeze-out temperature Tfr of weak processes

Tfr ≈(1.66

√g∗

G2FMPl

)1/3

≈ 1MeV (10.33)

with g∗ = 10.75. The relation between time and temperature follows as

t

s=

2.4√g∗

(MeV

T

)2

. (10.34)

Thus the time-sequence is as follows

111

10 Cosmic relics

• at Tfr ≈ 1 MeV: the neutron-proton ratio freezes-in and can be approximated by theratio of their equilibrium distribution in the non-relativistic limit.

• as the universe cools down from Tfr to Tns, neutrons decay with half-live τn ≈ 886 s.

• at Tns, practically all neutrons are bound to 4He, with only small admixture of otherelements.

Proton-neutron ratio Above Tf , reactions like νe + n ↔ p + e− keep nucleons in thermalequilibrium. As we have seen, Tf ∼ 1 MeV and thus we can treat nucleons in the non-relativistic limit. Then their relative abundance is given by the Boltzmann factor exp (−∆/T )for T >∼ Tf with ∆ = mN −mP = 1.29MeV for the mass difference of neutrons and protons.Hence for Tf ,

nnnp

∣∣∣∣t=tfr

= exp

(

−∆

Tf

)

≈ 1

6. (10.35)

As the universe cools down to Tns, neutrons decay with half-live τn ≈ 886 s,

nnnp

∣∣∣∣t=tns

≈ 1

6exp

(

− tnsτn

)

≈ 2

15. (10.36)

10.3.3 Estimate of helium abundance

The synthesis if 4He proceeds though a chain of reactions, pn → dγ, dp → 3Heγ, d 3He →4Hep. Let’s assume that 4He formation takes place instantaneously. Moreover, we assumethat all neutrons are bound in 4He. We need two neutrons to form one helium atom, n(4He) =nn/2, and thus

Y (4He) ≡ M(4He)

Mtot=

4mN × nn/2

mN (np + nn)=

2nn/np1 + nn/np

=4

17∼ 0.235 (10.37)

Our naive estimate not too far away from Y ∼ 0.245.The dependence of Y ( 4He) on the input physics is rather remarkable.• The helium abundance dependence exponentially on ∆ and Tf :

– The mass difference ∆ depends on both electromagnetic and strong interactions.BBN tests therefore the time-dependence of fundamental interaction expected e.g.in string theories.

– The freeze-out temperature Tfr depends on number of relativistic degrees of freedomg∗ and restricts thereby additional light particle.

– a non-zero chemical potential of neutrinos.

• A weaker dependence on start of nucleosynthesis Tns and thus ηb or Ωb.

10.3.4 Results from detailed calculations

Detailed calculations predict not only the relative amount of light elements produced, butalso their absolute amount as function of e.g. the baryon-photon ratio η. Requiring that therelative fraction of helium-4, deuterium and lithium-7 compared to hydrogen is consistent withobservation allows one to determine η or equivalently the baryon content, Ωbh

2 = 0.019±0.001.Although the binding energy per nucleon of Carbon-12 and Oxygen-16 is higher than the of4He, they are not produced: at time of 4He production Coulomb barrier prevents alreadyfusion. Also, stable element with A = 5 is missing.

112

10.4 Dark matter

Figure 10.3: Abundances of light-elements as function of η (left) and of the number of lightneutrino species (right).

10.4 Dark matter

10.4.1 Freeze-out of thermal relic particles

When the number density nX of a particle species X is not changed by interactions, then it isdiluted just by the expansion of space, nX ∝ R−3. It is convenient to account for this trivialexpansion effect by dividing nX through the entropy density s ∝ R−3, i.e. to use the quantityY = n/s. We first consider again the equilibrium distribution Yeq for µX = 0,

Yeq =nXs

=

452π4

(π8

)1/2 gXg∗S

x3/2 exp(−x) = 0.145 gXg∗S

x3/2 exp(−x) for x≫ 3,45ζ(3)2π4

εgXg∗S

= 0.278 εgXg∗S

for x≪ 3(10.38)

where x = T/m and geff = 3/4 (geff = 1) for fermions (bosons). If the particle X is in chemicalequilibrium, its abundance is determined for T ≫ m by its contribution to the total number ofdegrees of freedom of the plasma, while Yeq is exponentially suppressed for T ≪ m (assumingµX = 0). In an expanding universe, one may expect that the reaction rate Γ for processeslike γγ ↔ XX drops below the expansion rate H mainly for two reasons: i) Cross sectionsmay depend on energy as, e.g., weak processes σ ∝ s ∝ T 2 for s <∼ m2

W , ii) the density nXdecreases at least as n ∝ T 3. Around the freeze-out time xf , the true abundance Y startsto deviate from the equilibrium abundance Yeq and becomes constant, Y (x) ≈ Yeq(xf) forx >∼ xf . This behavior is illustrated in Fig. 10.5.

Boltzmann equation When the number N = nV of a particle species is not changed byinteractions, then the expansion of the Universe dilutes their number density as n ∝ R−3.The corresponding change in time is connected with the expansion rate of the universe, theHubble parameter H = R/R, as

dn

dt=

dn

dR

dR

dt= −3n

R

R= −3Hn . (10.39)

113

10 Cosmic relics

1e-20

1e-15

1e-10

1e-05

1

1 10 100 1000

Y(x)

/Y(x

=0)

x = m/T

Y∞

increasing σ: ↓

↓ xf

Figure 10.4: Illustration of the freeze-out process.

Additionally, there might be production and annihilation processes. While the annihilationrate βn2 = 〈σannv〉 n2 has to be proportional to n2, we allow for an arbitrary function asproduction rate ψ,

dn

dt= −3Hn− βn2 + ψ . (10.40)

In a static Universe, dn/dt = 0 defines equilibrium distributions neq. Detailed balance requiresthat the number of X particles produced in reactions like e+e− → XX is in equilibrium equalto the number that is destroyed in XX → e+e−, or βn2eq = ψeq. Since the reaction partners(like the electrons in our example) are assumed to be in equilibrium, we can replace ψ = ψeq

by βn2eq and obtaindn

dt= −3Hn− 〈σannv〉(n2 − n2eq) . (10.41)

This equation together with the initial condition n ≈ neq for T → ∞ determines n(t) for agiven annihilation cross section σann.Next we rewrite the evolution equation for n(t) using the dimensionless variables Y and x.

Changing from n = sY to Y we can eliminate the 3Hn term,

dn

dt= −3Hn+ s

dY

dt. (10.42)

With (2t)−2 = H2 ∝ ρ ∝ T 4 ∝ x−4 or t = t∗x2, we obtain

dY

dx= −sx

H〈σannv〉

(Y 2 − Y 2

eq

). (10.43)

Finally we recast the Boltzmann equation in a form that makes our intuitive Gamov criterionexplicit,

x

Yeq

dY

dx= −ΓA

H

[(Y

Yeq

)2

− 1

]

(10.44)

114

10.4 Dark matter

Figure 10.5: Illustration of the freeze-out process. The quantity Y = nX/s is nX divided bythe entropy density s ∝ R−3 to scale out the trivial effect of expansion.

with ΓA = neq〈σannv〉: The relative change of Y is controlled by the factor ΓA/H timesthe deviation from equilibrium. The evolution of Y = nX/s is shown schematically inFig. 10.5: As the universe expands and cools down, nX decreases at least as R−3. Therefore,the annihilation rate ∝ n2 quenches and the abundance “freezes-out:” The reaction rates arenot longer sufficient to keep the particle in equilibrium and the ratio nX/s stays constant.

For the discussion of approximate solutions to this equation, it is convenient to distinguishaccording to the freeze-out temperature: hot dark matter (HDM) with xf ≪ 3, cold darkmatter (CDM) with xf ≫ 3 and the intermediate case of warm dark matter with xf ∼ 3.

10.4.2 Hot dark matter

For xf ≪ 3, freeze-out occurs when the particle is still relativistic and Yeq is not changingwith time. The asymptotic value of Y , Y (x → ∞) ≡ Y∞, is just the equilibrium value atfreeze-out,

Y∞ = Yeq(xf) = 0.278geffg∗S

, (10.45)

where the only temperature-dependence is contained in g∗S . The number density today isthen

n0 = s0Y∞ = 2970Y∞cm−3 = 825geffg∗S

cm−3 . (10.46)

The numerical value of s0 used will be discussed in the next paragraph. Although a HDMparticle was relativistic at freeze-out, it is today non-relativistic if its mass m is m ≫ 3K ≈0.2meV. In this case its energy density is simply ρ0 = ms0Y∞ and its abundance Ωh2 = ρ0/ρcror

Ωh2 = 7.8× 10−2 m

eV

geffg∗S

. (10.47)

Hence HDM particles heavier than O(100eV) overclose the universe.

115

10 Cosmic relics

10.4.3 Cold dark matter

Abundance of CDM For CDM with xf ≪ 3, freeze-out occurs when the particles are alreadynon-relativistic and Yeq is exponentially changing with time. Thus the main problem isto find xf , for late times we use again Y (x → ∞) ≡ Y∞ ≈ Y (xf), i.e. the equilibriumvalue at freeze-out. We parametrize the temperature-dependence of cross section as 〈σann〉 =σ0(T/m)n = σ0/x

n. For simplicity, we consider only the most relevant case for CDM, n = 0or s-wave annihilation. Then the Gamov criterion becomes with H = 1.66

√g∗ T

2/MPl andΓA = neq〈σannv〉,

g

(mTf2π

)3/2

exp(−m/Tf)σ0 = 1.66√g∗

T 2f

MPl(10.48)

or

x−1/2f exp(xf) = 0.038

g√g∗MPlmσ0 ≡ C . (10.49)

To obtain an approximate solution, we neglect first in

lnC = −1

2lnxf + xf (10.50)

the slowly varying term lnxf . Inserting next xf ≈ lnC into Eq. (10.50) to improve theapproximation gives then

xf = lnC +1

2ln(lnC) . (10.51)

The relic abundance for CDM follows from n(xf) = 1.66√g∗ T

2f /(σ0MPl) and n0 =

n(xf)[R(xf)/R0]3 = n(xf)[g∗,f/g∗,0][T0/T (xf)]

3 as

ρ0 = mn0 ≈ 10xfT

30√

g∗,fσ0MPl(10.52)

or

ΩXh2 =

mn0ρcr

≈ 4× 10−39cm2

σ0xf (10.53)

Thus the abundance of a CDM particle is inverse proportionally to its annihilation crosssection, since a more strongly interacting particle stays longer in equilibrium. Note that theabundance depends only logarithmically on the mass m via Eq. (10.51) and implicitly viag∗,f on the freeze-out temperature Tf . Typical values of xf found numerically for weaklyinteracting massive particles (WIMPs) are xf ∼ 20. Partial-wave unitarity bounds σann asσann ≤ c/m2. Requiring Ω < 0.3 leads to m < 20 − 50 TeV. This bounds the mass of anystable particle that was once in thermal equilibrium.

Baryon abundance from freeze-out:We can calculate the expected baryon abundance for a zero chemical potential using the formulasderived above. Nucleon interact via pions; their annihilation cross section can be approximatedas 〈σv〉 ≈ m−2

π . With C ≈ 2× 1019, it follows xf ≈ 44, Tf ∼ 22MeV and Y∞ = 7× 10−20.The observed baryon abundance is much larger and can be not explained as a usual freeze-outprocess.

116

10.4 Dark matter

Figure 10.6: Particles proposed as DM particle with Ω ∼ 1, the expected size of their crosssection and their mass.

Cold dark matter candidates

A particle suitable as CDM candidate should interact according Eq. (10.53) with σ ∼10−37 cm2. It is surprising that the numerical values of T0 and MPl conspire in Eq. (10.53) tolead to numerical value of σ0 typical for weak interactions. Cold dark matter particles withmasses around the weak scale and interaction strengths around the weak scale were dubbed“WIMP”. An obvious candidate was a heavy neutrino, mν ∼ 10GeV, excluded early by directDM searches, neutrino mass limits, and accelerator searches. Presently, the candidate withmost supporters is the lightest supersymmetric particle (LSP). Depending on the details ofthe theory, it could be a neutralino (most favorable for detection) or other options. Themass range open of thermal CDM particles is rather narrow: If it is too light, it becomes awarm or hot dark matter particle. If it is too heavy, it overcloses the universe. There existshowever also the possibility that DM was never in thermal equilibrium. Two examples arethe axion (a particle proposed to solve the CP problem of QCD) and superheavy particle(generically produced at the end of inflation). An overview of different CDM candidates isgiven in Fig. 10.6.

117

11 Inflation and structure formation

11.1 Inflation

Shortcomings of the standard big-bang model

• Causality or horizon problem: why are even causally disconnected regions of the universehomogeneous, as we discussed for CMB?

The horizon grows like t, but the scale factor in radiation or matter dominated epochonly as t2/3 or t1/2, respectively. Thus for any scale l contained today completely insidethe horizon, there exists a time t < t0 where it crossed the horizon. A solution to thehorizon problem requires that R grows faster than the horizon t. Since R ∝ t2/[3(1+w],we need w < −1/3 or (q < 0, accelerated expansion of the universe).

• Flatness problem: the curvature term in the Friedmann equation is k/R2. Thus thisterm decreases slower than matter (∝ 1/R3) or radiation (1/R4), but faster than vacuumenergy. Let us rewrite the Friedmann equation as

k

R2= H2

(8πG

3H2ρ+

Λ

3H2− 1

)

= H2 (Ωtot − 1) . (11.1)

The LHS scales as (1 + z)2, the Hubble parameter for MD as (1 + z)3 and for RDas (1 + z)4. General relativity is supposed to be valid until the energy scale MPl.Most of time was RD, so we can estimate 1 + zPl = (t0/tPl)

1/2 ∼ 1030 (tPl ∼ 10−43s).Thus if today |Ωtot − 1| <∼ 1%, then the deviation had too be extremely small at tPl,|Ωtot − 1| <∼ 10−2/(1 + zPl)

2 ≈ 10−62!

Taking the time-derivative of

|Ωtot − 1| = |k|H2R2

=|k|R2

(11.2)

gives

d

dt|Ωtot − 1| = d

dt

|k|R2

= −2|k|RR3

< 0 (11.3)

for R > 0. Thus Ωtot − 1 increases if the universe decelerates, i.e. R decreases (radia-tion/matter dominates), and decreases if the universe accelerates , i.e. R increases (orvacuum energy dominates). Thus again q < 0 (or w < −1/3) is needed.

• The standard big-bang model contains no source for the initial fluctuations required forstructure formation.

118

11.1 Inflation

Solution by inflation Inflation is a modification of the standard big-bang model where aphase of accelerated expansion in the very early universe is introduced. For the expansion afield called inflaton with E.o.S w < −1/3 is responsible. We discuss briefly how the inflationsolves the short-comings of standard big-bang model for the special case w = −1:

• Horizon problem: In contrast to the radiation or matter-dominated phase, the scalefactor grows during inflation faster than the horizon scale, R(t2)/R(t1) = exp[(t2 −t1)H] ≫ t2/t1. Thus one can blow-up a small, at time t1 causally connected region, tosuperhorizon scales.

• Flatness problem: During inflation R = HR, R = R0 exp(Ht) and thus

Ωtot − 1 =k

R2∝ exp(−2Ht) . (11.4)

Thus Ωtot − 1 drives exponentially towards zero.

• Inflation blows-up quantum fluctuation to astronomical scales, generating initial fluc-tuation without scale, P0(k) = kns with ns ≈ 1, as required by observations.

11.1.1 Scalar fields in the expanding universe

Equation of state We consider a scalar field, Eq. (2.38), including a potential V (φ),

L =1

2gµν∇µφ∇νφ− V (φ) , (11.5)

that could be also a mass term, V (φ) = m2φ2/2). We remember first the expressions for theenergy-density ρ = T 00 and the pressure P ,

ρ =1

2φ2 + V , P =

1

2φ2 − V (11.6)

and as equation of state

w =P

ρ=φ2 − 2V (φ)

φ2 + 2V (φ)∈ [−1 : 1] . (11.7)

Thus a classical scalar field may act as dark energy, w < 0, leading to an accelerated expansionof the Universe. A necessary condition is that the field is “slowly rolling”, i.e. that its kineticenergy is smaller than its potential energy, φ2/2 < V (φ).

Field equation in a FRW background We use Eq. (2.38) including a potential V (φ) (thatcould be also a mass term, V (φ) = m2φ2/2),

L =1

2gµν∇µφ∇νφ− V (φ) , (11.8)

to derive the equations of motions for a scalar field in a flat FRW metric, gab =diag(1,−a2,−a2,−a2), gab = diag(1,−a−2,−a−2,−a−2), and

√

|g| = a3. Varying the ac-tion

SKG =

∫

Ωd4x a3

1

2φ2 − 1

2a2(∇φ)2 − V (φ)

(11.9)

119


gives

δSKG =

∫

Ωd4x a3

φδφ− 1

a2(∇φ) · δ(∇φ)− V ′δφ

=

∫

Ωd4x

− d

dt(a3φ) + a∇2φ− a3V ′

δφ

=

∫

Ωd4x a3

−φ− 3Hφ+1

a2∇2φ− V ′

δφ!= 0 . (11.10)

Thus the field equation for a Klein-Gordon field in a FRW background is

φ+ 3Hφ− 1

a2∇2φ+ V ′ = 0 . (11.11)

The term 3Hφ acts in an expanding universe as a friction term for the oscillating φ field.Moreover, the gradient of φ is also suppressed for increasing a; this term can be thereforeoften neglected in an expanding universe.

Number of e-foldings and slow roll conditions We can integrate R = RH for an arbitrarytime-evolution of H,

R(t) = R(t0) exp

(∫

dtH(t)

)

. (11.12)

If we define the number N of e-foldings as N = ln(R2/R1), then

N = lnR2

R1=

∫

dt H(t) =

∫dφ

φH(t) . (11.13)

With φ+ 3Hφ+ V ′ = 0 or

φ = − φ+ V ′

3H≈ − V ′

3H(11.14)

and the Friedmann equation H2 = 8πGV/3 it follows

N =

∫

dφ3H2

V ′=

∫

dφ8πGV

V ′≫ 1 . (11.15)

Successful inflation requires N >∼ 40 and thus

ε ≡ 1

2

(V ′

8πGV

)2

≪ 1 . (11.16)

Additionally to large V and a flat slope V ′, the potential energy can dominate only, if |φ| ≪|V ′|. Then the field equation reduces to V ′ ≈ −3Hφ, or after differentiating to V ′′φ ≈ −3Hφ.Thus another condition for inflation is

1 ≫ |φ||V ′| ≈

|V ′′φ||3HV ′| ≈

|V ′′|24πGV

(11.17)

and one defines as second slow-roll condition

η ≡ V ′′

8πGV≪ 1 . (11.18)

Hence inflation requires large V , a flat slope V ′ and small curvature V ′′ of the potential.

120

11.1 Inflation

Solutions of the KG field equation in a FRW background Next we want to rewrite the KGequation as the one for an harmonic oscillator with a time-dependent oscillation frequency.We introduce first the conformal time dη = dt/a,

φ =dφ

dt=

dφ

dη

dη

dt=

1

aφ′ , (11.19)

φ =1

a

d

dη

(1

aφ′)

=1

a2φ′′ − a′

a3φ′ , (11.20)

and express also the Hubble parameter as function of η,

H =a

a=a′

a2≡ H

a. (11.21)

Inserting these expressions into Eq. (11.11) and multiplying with a2 gives

φ′′ + 2Hφ′ −∇2φ+ V ′ = 0 . (11.22)

Performing then a Fourier transformation,

φ(~x, t) =∑

k

φk(t)ei~x~k , (11.23)

and using as potential a mass term, V ′ = m2φ, we obtain

φ′′k + 2Hφ′k + (k2 +m2a2)φk = 0 . (11.24)

Finally, we can eliminate the friction term 2Hφ′k by introducing φk(η) = uk(η)/a. Then aharmonic oscillator equation for uk,

u′′k + ω2kuk = 0 , (11.25)

with the time-dependent frequency

ω2k(η) = k2 +m2a2 − a′′

a(11.26)

results. You can check that the action for the field u using conformal coordinates η, ~x ismathematically equivalent to the one of a scalar field in Minkowski space with time-dependentmass m2

eff(η) = m2a2 − a′′/a. This time-dependence appears, because the gravitational fieldcan perform work on the field u. Alternatively, we could show that “the” vacuum at differenttimes η is not the same, because we compare the vacuum for fields with different effectivemasses, leading to particle production. For an excellent introduction into this subject see thebook by V. F. Mukhanov and S. Winitzki, “Introduction to quantum fields in gravity;” for afree pdf file of the draft version see http://sites.google.com/site/winitzki/.We consider now as two limiting cases the short and the long-wavelength limit. In the

first case, k2 +m2a2 ≫ a′′

a , the field equation is conformally equivalent to the one in normalMinkowski space, with solution

uk(η, ~x) =1√2k

(Ake−ikx +Ake

ikx) . (11.27)

121


In the opposite limit, a′′uk = au′′k, with the solution φk = const. The complete solution isgiven by Hankel functions H3/2(η),

uk(η) = Ake−ikx

(

1− i

kη

)

+Bkeikx

(

1 +i

kη

)

. (11.28)

Modes outside the horizon are frozen in with amplitude

|φk| =∣∣∣uka

∣∣∣ =

H√2k3

. (11.29)

11.1.2 Generation of perturbations

We treated the scalar field driving inflation, the “inflaton”, as a classical field. As everysystem it is subject to quantum fluctuations. These fluctuations are of the order δφ ∼ H, i.e.the same for all Fourier modes φk. Thus a generic prediction of inflation is an “scale-invariant”spectrum of primordial perturbations, P0(k) ∝ kns with ns ≈ 1. Primordial perturbationswith such a spectrum are indeed required to explain the CMB anisotropies.

We consider fluctuations of the inflaton field φ around its classical average value,

φ(~x, t) = φ0(t) + δφ(~x, t) . (11.30)

Inserting this into the field equation (11.11) gives six terms. We evaluate first the potentialterm, assuming that the potential has its minimum for φ = φ0 = 0. Then

V (φ) = V (0) +1

2V ′′φ2 +O(φ3) (11.31)

and we see that the second derivative of the potential acts as a effective mass term, m2eff = V ′′

for the φ field. Thus

V (φ+ δφ) = V ′(φ0) + V ′′(φ0)δφ = V ′(φ0) +m2φδφ . (11.32)

Taking into account that the classical term φ0 satisfies separately the field equation (11.11)gives as equation for the fluctuations

[∂2

∂t2− 1

a2∇2 + 3H

∂

∂t+m2

φ

]

δφ = 0 . (11.33)

We perform next a Fourier expansion of the fluctuations,

δφ(~x, t) =∑

k

φk(t)ei~x~k , (11.34)

with k as comoving wave-number. Since the proper distance varies as a~x, the momentum is~p = ~k/a.

φk + 3Hφk +

(k2

a2+m2

φ

)

φk = 0 . (11.35)

Comparing this equation with (11.11), we see that the fluctuations obey basically the sameequation as the average field. The only difference is the effective mass term.

122

11.1 Inflation

Going over to conformal time

φ′′k + 2Hφ′k + k2φk = 0 , (11.36)

and to φk(η) = uk(η)/a gives

u′′k + 2Hu′k +

(

k2 − a′′

a

)

uk = 0 . (11.37)

Combining a = 1/(Hη2) and a′′ = −2/(Hη3) or

a′′

a=

2

η(11.38)

gives

u′′k +

(

k2 − 2

η

)

uk = 0 . (11.39)

Hence fluctuations satisfy also

uk(η) = Ake−ikx

(

1− i

kη

)

+Bkeikx

(

1 +i

kη

)

. (11.40)

Power spectrum of perturbations The two-point correlation function of the field φ is

〈φ(~x′, t′)φ(~x, t)〉 =∑

k

〈φ(~x′, t′)|k〉〈k|φ(~x, t)〉 =∫

d3k

(2π)3|φk|2 eik(x

′−x) . (11.41)

We introduce spherical coordinates in Fourier space and choose x = x′,

〈φ2(~x, t)〉 =∫

4πk2dk

(2π)3|φk|2 =

∫

dkk2

2π2|φk|2

︸︷︷︸

≡P (k)

=

∫dk

k∆2

φ(k) . (11.42)

The functions P (k) is the power spectrum, but often one calls also ∆2φ(k) with the same name.

The spectrum of fluctuations ∆2φ(k) outside of the horizon is

∆2φ(k) =

k3

2π2|φk|2 =

H2

4π2(11.43)

Hence, the power-spectrum of superhorizon fluctuations is independent of the wave-numberin the approximation that H is constant during inflation. The total area below the function∆2

φ(k) = const. plotted versus ln(k) gives 〈φ2(~x, t)〉, as shown by the last part of Eq. (11.42).

Hence a spectrum with ∆2φ(k) = const. contains the some amount of fluctuation on all angular

scales. Such a spectrum of fluctuations is called a Harisson-Zel’dovich spectrum, and isproduced by inflation in the limit of infinitely slow-rolling of the inflaton.

Fluctuations in the inflaton field, φ = φ0+δφ, lead to fluctuations in the energy-momentumtensor T ab = T ab

0 + δT ab, and thus to metric perturbations gab = gab0 + δgab. These metricperturbations hab affect in turn all matter fields present.

123


V

0 MP λ M

4MP

P− 1/4

ϕ

OFC

hot universe

reheating

inflation

C

OF

?

t0

a

tP

–43 sec –35 sec t0

17 sect~ 10 ~ 10~ 10

Figure 11.1: Left: A slowly rolling scalar field as model for inflation. Right: The evolutionof the scale factor R including an inflationary phase in the early universe.

11.1.3 Models for inflation

Inflation has to start and to stop (“graceful exit problem”). In order to start inflation, theinflaton has to be displaced from its equilibrium position.Original idea of Guth: Symmetry restoration at a first order (discontinuous) phase transi-tion, bubble creation or 2.order. Latent heat of phase transition is used to reheat universe(expansion lead to cool, empty state!) and to create particles. Too inhomogenous.Modern ideas: Chaotic inflation: quantum fluctuation in a patch of the universe. Field rollsback, inflation ends when φ back in minimum. Oscillates around minimum, coupling to otherparticles leads to particle production.If the coupling to other particles is “large”, then (instantaneous) reheating aT 4

rh = V . Gener-ically, the coupling should be small. Delay leads to aT 4

rh = V (R/R′)3.

11.2 Structure formation

11.2.1 Overview and data

• Structure formation operates via gravitational instability, but needs as starting point aseed of primordial fluctuations (generated in inflation)

• Growth of structure is inhibited by many factors, e.g. pressure.The distance travelled by a freely falling particle is R ∼ gt2/2 with g = GM/R2; ort ∼

√

R3/GM ∼√

1/Gρ. Thus τff ∼ 1/√Gρ.

Pressure can balance gravity, if τff >∼ λ/vs. This defines a critical length (“Jeans length”)

λ ∼ vs√Gρ

,

below which pressure can counteract density perturbations (resulting in acoustic oscil-lations), above the density perturbation grows. Shows already that structure formationis sensitive to E.o.S. (compare e.g. radiation with v2s = 1/3 with baryonic matterv2s = 5T/(3m)).

124


• If growth of perturbation leads to Ω ≥ 1 in a region, the region decouples from theHubble expansion and collapses.

• Assume ρ = ρm + ργ . If perturbations in ρ are adiabatic, i.e. the entropy per baryon isconserved, δ(ρm/s) = 0 or δ ln(ρm/T

3) = 0, then δ ln ρm − 3δ lnT = 0 or

δρmρm

= 3δT

T.

[Another possibility would be δρ = 0 or δρm = −δργ = −4aT 3δT = −4ργδT/T and4δT/T = −δρm/ργ = −(ρm/ργ) (δρm/ρm). In the radiation epoch ρm/ργ ≪ 1 andtemperature fluctuations are suppressed.]

⇒ Temperature fluctuation in CMB at z ≈ 1100 and matter fluctuation today 0 ≤ z <∼ 5have the same origin, if primordial fluctuations are adiabatic.

• Basics of structure formation:assume initial fluctuations and examine how they are transformed by gravitational in-stability, interactions and free-streaming of different particle species

• comparison with observations via i) power-spectrum P (k) = |δk|2, where

δk ∝∫

d3x e−ikxδ(x) ∝ kns with δ(x) ≡ ρ(x)− ρ

ρ

or ii) correlation function∫d3x n(x)n(x+ x0) or normalized

ξ(x0) ≡1/V

∫d3x n(x)n(x+ x0)

(1/V∫d3x n(x)n(x+ x0))2

− 1

The correlation function is the Fourier-transform of the power spectrum.Typical ξ ≈ (r/r0)

γ with γ ∼ 1.8 for 0.1 <∼ r <∼ 10Mpc.

• An example of the status in 1995 is shown in Fig. 11.2. The field is driven by atremendous growth of data:galaxy catalogues: Hubble ’32: 1250, Abell ’58: 2712 cluster, 2dF : 250.000, SDSS (-’08): 106.CMB experiments: ’65 detection, COBE ’92: anisotropies, towards ’00: first peak, . . .N-body simulations: Peeble ’70: N = 100, Efstahiou, Eastwood ’81: N = 20.000, 2005:Virgo: Millenium simulation N = 106.

11.2.2 Jeans mass of baryons

Consider mixture of radiation and non-relativistic nucleons after e+e− annihilations, i.e. T ≈0.5 MeV. With ρ = ρm + ργ and P ≈ Pγ = ργ/3, we have

vs =

(∂ρ

∂P

)1/2

S

=1√3

(

1 +∂ρm∂ργ S

)−1/2

=1√3

(

1 +3ρm4ργ

)−1/2

(11.44)

where we used δρmρm

= 3 δTT =

3δργ4ργ

.

For t≪ teq, the adiabatic sound speed is close to vs = 1/√3, while vs = 0.76/

√3 for t = teq.

125


CDM n = 1

HDM n = 1 MDM

n = 1

0.001 0.01 0.1 1 10

k ( h Mpc )-1

5

P (

k )

( h

Mpc

)-3

3

10

410

1000

100

10

1

0.1

TCDM n = .8

COBE

d ( h Mpc )-1

1000 100 10 1Microwave Background Superclusters Clusters Galaxies

Figure 11.2: Comparison of the predicted power spectrum normalized to COBE data in severalmodels popular around ’95 with observations: HDM (Ων = 1), CDM (Ωm = 1)and MDM (Ωm = 0.8, Ων = 0.2).

The Jeans mass of baryons is close to the horizon size until recombination. Then vs drops tothe value for a mono-atomic gas, v2s = 5Tb

3m , where m ∼ mH ∼ 1 GeV.

The total mass MJ contained within a sphere of radius λJ/2 = π/kJ is

MJ =4π

3

(π

kJ

)3

ρ0 =π5/2

6

v3s

G3/2ρ1/20

(11.45)

is called the Jeans mass. It is unchanged by the expansion of the universe, since the wave-number kJ ∝ R and ρ0 ∝ 1/R3.

Let us compare the Jeans mass just before and after recombination,

MJ(zeq,>) =π5/2

6

v3sG3/2ρ1/2

∼ 1015(Ωh2

)−2M⊙ (11.46)

and

MJ(zeq,<) ∼ 105(Ωh2

)−1/2M⊙ (11.47)

The Jeans mass of baryons does not coincide with the observed mass of galaxies, neither fitsthe corresponding length scale the break in the power spectrum around k ≈ 0.04h/Mpc.

11.2.3 Damping scales

Collisional or Silk damping Consider which fluctuations are damped by dissipative pro-cesses, e.g. by Thomson scattering. (The mean free path of photons is always much largerthan the one of electrons, lγ = (neσT )

−1 ≫ le = (nγσT )−1, since nγ ∼ 1010ne. Thus photon

diffusion is much more important then electron diffusion.)

126


Figure 11.3: Acoustic baryon oscillation in the correlation function of galaxies with largeredshift of the SDSS, astro-ph/0501171.

A sound wave with wavelength λ can be damped if the diffusion time τdiff is smaller thanthe Hubble time tH . Estimate τdiff by a random-walk with N = λ2/l2int steps, each with sizelint = 1/neσth,

τdiff = Nlint = λ2/lint < tH . (11.48)

Thus the damping scale is λD = (lintlH)1/2. If there would be a baryon-dominated epoch,then ne ∝ ρ and ρ ∝ 1/t2, hence lintlH ∝ ρ−1−1/2 and λD ∝ ρ−3/4. Finally, the correspondingmass scale is MD ∝ λ3ρ ∝ ρ−9/4ρ ∝ ρ−5/4. Numerically,

MD = 1012(Ωh2)−5/4M⊙ (11.49)

and the corresponding length scale is (taking into account Ωb ≈ 0.04 < Ωm ≈ 0.27 andh ≈ 0.7)

λD = 3.5(Ωm/Ωb)1/2(Ωh2)−3/4Mpc ≈ 40Mpc . (11.50)

Thus the Silk scale λD has the right numerical value to explain the break in the powerspectrum at k ≈ 0.04h/Mpc. Fluctuations are damped on scales λ >∼ λD, the stronger thelarger Ωb. Since MD ≫ MJ , acoustic oscillation should be visible for k >∼ kD in the powerspectrum of galaxies. First evidence was found last years, cf. Fig. 11.3.

11.2.4 Growth of perturbations in an expanding Universe:

We restrict ourselves to the simplest case: perturbations in a pressure-less, expanding medium.Starting from a homogeneous universe, we add matter inside a sphere of radius R, ρ→ ρ(1+δ).Then the acceleration on the surface of this sphere is

R

R= −4π

3Gρδ . (11.51)

The time evolution of the mass density is

ρ(t) = ρ(t)[1 + δ(t)] = ρ0/a3(t)[1 + δ(t)] (11.52)

127


and thus mass conservation

M =4π

3ρ[1 + δ]R3 = const. (11.53)

impliesR(t) ∝ a(t)[1 + δ]−1/3 . (11.54)

Expanding for δ ≪ 1 and differentiating gives

R

R=a

a− 2

3

a

aδ − 1

3δ. (11.55)

Combined with (11.51) we obtain

δ + 2Hδ − 4πGρδ = 0. (11.56)

For a matter-dominated universe, H = 2/(3t) and ρ = 1/(6πGt2). Inserting the trial solutionδ ∝ tα gives

α(α− 1)tα−2 +4

3αtα−2 − 2

3tα−2 = 0 (11.57)

or

α2 +1

3α− 2

3= 0 (11.58)

and finally α = −1 and 2/3. Thus the general solution δ(t) = At−1 + Bt2/3 consists of adecaying mode δ ∝ 1/t and a mode growing like δ ∝ t2/3 ∝ R.During the radiation-dominated epoch, with δγ = 0, one can neglect the term 4πGρδ. WithH = 1/(2t)

δ +1

tδ = 0 (11.59)

with solution δ(t) = δ(ti)[1 + a ln(t/ti)]. Thus perturbations do not grow until zeq.

Non-linear regime N-body simulations are mainly used to study structure formation on thesmallest scale, e.g. the dark matter profile of a galaxy.

11.2.5 Recipes for structure formation

Summary of different effects

• On sub-horizon scales our Newtonian analysis applies. During the radiation-dominatedepoch, perturbations do not grow. During the matter-dominated epoch, perturbationsgrow on scales larger than the Jeans scale as δ ∝ t2/3 ∝ R. Perturbations on smallerscales oscillate as acoustic waves.

• Before recombination, baryons are tightly coupled to radiation. The baryon Jeans scaleis of order of the horizon size. After recombination, it drops by a factor 1010.

• Silk damping reduces power on scales smaller than 40 Mpc.

• Free-streaming of HDM suppresses exponentially suppress power on scales smaller thanfew Mpc (for Ων = 1).

• CDM with baryons would be affected only by Silk damping.

128


0.01 0.10k (h Mpc

−1)

103

104

105

Pg(

k) (h

−3 M

pc3 )

Ων = 0.05

Ων = 0.01

Ων = 0

⇒∑mν <∼ 2.2 eV at 95% C.L.

Figure 11.4: Neutrino mass limits from the 2dF galaxy survey: For Ων >∼ 0.05 there is tooless power on scales smaller than (or since normalization is arbitrary) slope toosteep).

Recipe

• The connection between the initial perturbation spectrum Pi(k) = |δk,i|2 and the ob-served power spectrum P (k) today is formally given by the transfer function T (k),

P (k) = T 2(k)Pi(k). (11.60)

• Inflation predicts that an initial perturbation spectrum Pi(k) ∝ kns with ns ≈ 1, gen-erally adiabatic ones.

• Normalize Pi(k) to the COBE data.

• Choose a set of cosmological parameters h,ΩCDM ,Ωb,ΩΛ,Ων , ns.

• Calculate T (k).

• Fix a prescription to convert ρ ≈ ρCDM with ρb measured in observation (“bias”).

• Derive statistical quantities to be compared to observations; perform a likelihood anal-ysis.

11.2.6 Results

• The three models without cosmological constant shown in Fig. 11.2 all fail.

129


• The exponential suppression on small scales typical for HDM is not observed, can beused to derive limit on Ων <∼ 0.05 or

∑mνi

<∼ 2.2 eV.

• Acoustic baryon oscillations are only a tiny sub-dominant effect, but are now observed,cf. Fig. 11.3.

130

Index

aberrationrelativistic, 17

beaming, 17Bianchi identity, 71black hole

entropy, 68ergosphere, 66Kerr, 63Schwarzschild, 62

blackbody radiationcosmic, see cosmic microwave back-

groundBoltzmann equation, 112

conformal time, 89coordinate

cyclic, 20coordinates

Boyer-Lindquist, 63comoving, 88Eddington-Finkelstein, 62Kruskal, 62Riemannian normal, 46Schwarzschild, 50

cosmic censorship, 68cosmological constant, 95, 96cosmological principle, 88critical density, 96curvature scalar, 70

dark mattercold, 114hot, 114

distanceangular diameter, 95luminosity, 94

Einstein angle, 58energy-momentum tensor

dynamical, 75gravity, 83

equilibriumchemical, 105kinetic, 105

equivalence principle, 38, 46ergosphere, 66event horizon, 64

forcepure, 16

Friedmann equation, 95

Gamov criterion, 109gauge

harmonic, 80Lorentz, 35transverse traceless, 81

Hamilton’s principle, 19Harisson-Zel’dovich spectrum, 122Hawking radiation, 68Helicity, 82Hubble parameter, 91Hubble’s law, 90

inflation, 117isometry, 49

Jeans length, 104

Killing equation, 50Killing vector, 49Klein-Gordon equation

in a FRW background, 119

ΛCDM model, 99Lemaıtre’s redshift formula, 92lens equation, 58

metric

131

Index

Friedmann-Robertson-Walker, 89

parallel transport, 42power spectrum, 122principle of equivalence, 39

quadrupole formula, 86

recombination, 104Ricci tensor, 70

Schwarzschild metric, 50Shapiro effect, 56spaces

maximally symmetric, 89

Weyl’s postulate, 88

132

Lecture Notes for FY3452 Gravitation and Cosmologyweb.phys.ntnu.no/~mika/skript_grav.pdf · Preface...

Documents

Transcript of Lecture Notes for FY3452 Gravitation and Cosmologyweb.phys.ntnu.no/~mika/skript_grav.pdf · Preface...