GENERAL RELATIVITY

AND

COSMOLOGY

Lecture notes

Poul Olesen

The Niels Bohr Institute

Blegdamsvej

DK Copenhagen

Denmark

Autumn

Preface

The following lecture notes on general relativity and cosmology grew out of a one

semester course on these topics and classical gauge theory by Jan Amb jrn and the present

author Subsequently semesters were abandoned and replaced by Blo cks which have

an extension of approximately only two months Therefore the classical eld theory part

which anyhow strongly needed a revision was dropp ed and the general relativity and

cosmology chapters were revised

These lecture notes are introductory and do not in anyway pretend to b e comprehen

sive Several imp ortant topics have b een left out For example gravitational radiation

is not discussed at all There are two reasons for the brevity of the notes the alloted

w as short a couple of months four hours a week and it was hop ed that by making time

the notes equally short there is a bigger chance of getting through that general relativity

and cosmology are exciting sub jects Sometimes the trouble with exp osing the b eauty of

physics is that one has to walk a very long way so people start to feel that they rather

walk in a desert than in a b eautiful garden Students hungry for more comprehensive

studies are referred to the enormous literature

Poul Olesen

Contents

General relativity

The principle of equivalence

Gravitation and geometry

Motion in an arbitrary gravitational eld

The Newton limit

variance The principle of general co

Contravariantand covariant tensors

Dierentiation

A prop erty of the determinant of g

Some sp ecial derivatives

Some applications to physics

Curvature

Parallel transp ort and curvature

Prop erties of the curvature tensor

The energymomentum tensor

Einsteins eld equations for gravitation

The timedep endent spherically symmetric metric

A digression A simpler metho d for computing

The Christoel symbols for the timedep endent spherically symmetric metric

The Ricci tensor

The Schwarzschild solution

Birkho s theorem

The general relativistic Kepler problem

Deection of lightby a massiv e body

Black holes

Kruskal co ordinates

Painleves version of the Schwarzschild metric

Tidal forces and the Riemann tensor

The Tidal force from the Schwarzschild solution

The energymomentum tensor for electromagnetism

The ReissnerNordstrom solution

The spherically symmetric solution in dimensions

Cosmic strings

Cosmology

The cosmological problem

The cosmological standard mo del

CONTENTS

A geometric interpretation of the Rob ertsonWalker metric

Hubbles law

Higher order correction to Hubbles law

Einsteins equations and the Rob ertsonWalker metric

Bang The Big

Existence of the big bang the initial singularity

The age of the Universe according to Big Bang

Discussion of the fate of the Universe

Fitting parameters to observations

The cosmic micro wave radiation background

The matter dominated era

The closed Universe

The at Universe

en universe The op

Inclusion of the cosmological constant

Discussion erse of the life time of the Univ

Causality structure of the big bang The horizon problem

Ination

Observational evidence for the cosmological constant

The end of cosmology

An inhomogeneous univ erse without

Problems

Some constants

Some literature

Chapter

General relativity

The principle of equivalence

Einsteins general theory of relativity is a b eautiful piece of art which connects gravita

tional elds with geometry of space and time and thus provides a scheme in which our

universe can be discussed

Einsteins starting p ointwas the principle of equivalence which can b e understo o d in

the context of Newtons mechanics We have the general equation of motion

F m x

i

where x is the acceleration and m is the inertial mass for a given force the acceleration

i

is smaller the larger the mass is ie the body is more inert the larger m is

i

given by eld the force is In a constant gravitational

F m g

g g

where g and m are constants It is clear that a priori the parameter m is not related to

g g

the inertial mass Newton made exp eriments where the p erio d of oscillation of a p endulum

made up from dierent materials were studied and he found no variation with m m

i g

Later on manyvery precise exp eriments were made which showed that m m to a high

i g

accuracy and this was accepted to such an extent that most text b o oks to day and at

Einsteins time do not b other to put any indices on the masses

Let us consider a constantgravitational eld With m m m one has the equation

i g

of motion

x g

Thus if w e introduce the co ordinate

gt y x

we get

y

Therefore we conclude that an observer living in the ysystem sees no eect of the gravi

tational eld b ecause eq shows that particles move in straight lines as if there was

no force On the other hand eq shows that the observer is freely falling gt is just

GENERAL RELATIVITY

the displacement p ertinent toafreefall All this is true irresp ectiveofany mass b ecause

m m If m m the co ordinate transformation would have to b e replaced by

i g i g

m

g

y x g t

m

i

and hence the y system would dep end on which material we consider through the ratio

m m

g i

When m m the transformation is universal and is easily seen to eliminate

g i

the gravitational eld also if other forces eg electrostatic forces are at work If the

gravitational eld varies in space we can apply the transformation in a suciently

small domain

In Newtonian mechanics we therefore know that an observer in a suciently small

freely falling elevator is unable to detect a gravitational eld Einsteins principle of

ysical phenomena In any arbitrary gravitational generalizes this to any ph equivalence

eld it is p ossible at each spacetime p oint to select lo cally inertial systems

freely falling small elevators such that the laws of physics in these are the

same as in sp ecial relativity

One can use this statement to obtain some insight into the way in which gravity

inuences other physical phenomena by writing down in each of the small elevators some

law of physics and then transform it to a general co ordinate system In the next section

w e shall consider the simplest example namely a particle which is freely falling in an

arbitrary gravitational eld

Some remarks on the history of the Einsteinian version of the equivalence principle

After having nished the sp ecial theory of relativity Einstein thought ab out the prob

lem of how Newton gravity should b e mo died in order to t in with sp ecial relativity At

this p oint Einstein exp erienced what he called the happiest thoughtofmy life namely

that an observer falling from the ro of of a house exp eriences no gravitational eld

Gravitation and geometry

Let us consider a particle which moves under the inuence of a gravitational eld only

Thus in each of the innitely many freely falling systems of inertia we can apply sp ecial

relativity with no forces acting on the particle

In sp ecial relativityan event is describ ed byafourvector y y y where y is the

time Since the elevators are a priori small we need however to consider an innitesimal

four vector dy dy dy The prop er time

d dy dy

is an invariant ie if we make a Lorentz transformation from y to y then

d dy dy dy dy

Most of the historical remarks in these notes are taken from MacTutor History of Mathematics

httpwwwhistorymcsst andrewsacukHistTopicsGeneral relativityhtml where much more infor

mation can b e found

  

Here d means d This convention of leaving out the bracket in the square of innitesimal

quantities will b e used in the following unless it leads to confusion

GRAVITATION AND GEOMETRY

For light d and eq says that the sp eed of light jdydtj is equal to one in

all systems MichelsonMorleys exp eriment The prop er time has the following physical

interpretation Let us consider a clo ckoranyphysical system which sp ecies a time eg

a particle whichdecays with a certain life time whichby denition marks time by small

intervals dt when the clo ck is at rest In the rest system the velo city v dy dy vanishes

Thus

d dy v dy

in the rest system Thus d dy the interval b etween twoticks on the clo ck at rest

oving system In am

v d dy

p

which leads to the formula for the time dilatation dy d v

In four vector notation we write the prop er time as

d dy dy

where for and This convention is called

mostly positive In eq we use the summation convention Whenever an index

o ccurs two times in a pro duct it is to be summed Thus eq means

X X

d dy dy

vitywe should notice that eq is valid in any of the freely obtain the eect of gra To

falling elevators However in general the elevators are dierent in dierent spacetime

points If we denote the spacetime co ordinates in an arbitrary co ordinate system by x

then the y s are functions of the xs

y y x

For an example see the Newtonian case Inserting this in eq we get

y y

d dx dx g xdx dx

x x

where

y y

g x g x

x x

is called the metric tensor

Eq has an almost obvious geometric interpretation In a curved space eg the

surface of a sphere one can introduce lo cal co ordinate systems where Euclidian geometry

is valid in spite of the fact that this geometry is not valid in general in curved space

The lo cal Euclidean geometry corresp onds to eq if we replace by where

is the Kronecker symbol for for d then means

the distance between two p oints computed by the law of Pythagoras Then eq is

ordinates Similarly in Einsteins gravity lo cally the same distance written in arbitrary co

one has pseudoEuclidianMinkowsky geometry due to the principle of equivalence but

the geometry of space is in general nonpseudoEuclidean nonMinkowskian and the

GENERAL RELATIVITY

deviations from at space Minkowski space represent the eects of the gravitational

eld

Some remarks on the history of Einsteins geometrical gravity

According to historians of physics it is not known how Einstein got the idea of relating

gravity with geometry One hypothesis is that he was inspired by a rotating disk Here the

measuring ro ds will b ecome Lorentzcontracted and hence the length of the p eriphery

of a circle will be dierent from radius This means a deviation from Euclidean

geometry In any case in pap ers on gravitation published in he realized that the

Lorentz transformations will not always be applicable in a gravity theory based on the

equivalence principle Spacetime were dynamically inuenced by gravity According to

Kan t Euclidean geometry should be considered as an a priori description the philospher

of space Therefore the notion of space and time as dynamical quantities represented a

strong break with philosophical traditions Kant had not b een right

Einstein now remembered that as a student he had studied Gausss theory of surfaces

To pro ceed Einstein got help with the mathematical formulation of the theory of general

relativity from his former classmate Marcel Grossmann and the latter b eing a professor

of descriptive geometry in Zurich p ointed out the relevance of dierential geometry that

had previously been investigated by anumber of mathematicians Riemann Ricci Levi

Civita Einstein says ab out this p erio d in all my life I have not lab oured so hard

and Ihave b ecome imbued with great resp ect for mathematics the subtler part of which

no w I had in my simplemindedness regarded as pure luxury until

Motion in an arbitrary gravitational eld

We shall now apply the principle of equivalence to see how gravity inuences space in

the simple case where there are no other forces than gravity So let us consider a particle

which moves under the inuence of an arbitrary gravitational eld In the freely falling

system sp ecial relativity applies and we have the equation of motion

d y x

d

The solution to this equation is that y is a linear function of inside a small elevator

where there are no forces This simply means straight line motion inside the elevator in

accordance with the ndings of Gallilei

Using that the y s dep end on x we have

d dy y dx d

d d d x d

y d x y dx dx

x d x x d d

This lo oks somewhat like an equation of motion b ecause of d x d with a force We

can remove the factor multiplying the second derivative of x by the following trick By

the rules of dierentiation we have for for

y x

y x

MOTION IN GRAVITATIONAL FIELD

Thus multiplying eq by x y and summing over we get

dx dx d x

d d d

where

x y

x x y

g We is called the Christoel symbol or the ane connection sometimes denoted f

see that the Christoel symb ol is prop ortional to the gravitational force

So far the metric and have b een expressed in terms of the functional relation

between the lo cal freely falling elevators and the arbitrary system x We shall now show

that the Christoel symbol can be expressed in terms of the metric tensor

dep ends on the second derivatives of y whereas from From eq we see that

eq g dep ends only on the rst derivative Therefore let us dierentiate g by

the denition use of

g y y y y

a

x x x x x x x

From we have

y y y x y

x x x y x x x

where we used the chain rule for dierentiation

x y

y x

Thus we can express the second derivative of y in terms of by means of eq

Using this in eq we get

y y g y y

x x x x x

g g

where we used the expression for the metric Using eq we get

g g g

g

x x x

If one wishes g can b e thoughtofasa matrix One can then consider the inverse

which we denote g

g xg x

The inverse exists

x x

g

y y

b ecause the transformations y x and x y are nonsingular co ordinate transforma

tions Eq now gives the following relation b etween and g

g g g

g

x x x

GENERAL RELATIVITY

The Newton limit

Eqs and determine the motion of a particle in a gravitational eld provided

weknowhow g x dep end on the gravitational eld Later we shall see that the second

derivatives of g are determined through Einsteins eld equations in terms of matter

distributions

At present we shall study a much more mo dest problem namely the Newton limit

where all velocities are small relativeto the velo cityoflight jdxd j and where the

problem is static ie g is timeindep endent To the lowest nontrivial approximation

eq then gives t x

dt d x

d d

Using eq and the fact that all time derivatives vanish we have

g

g

x

Since we are interested in small eects of gravitywe write

g x h x jh j

where h is the correction to the constant metric tensor Then eq gives

h

x

so that since h do es not dep end on time Hence the comp onent of eq

b ecomes simply

d t

d

ie

dt

constant

d

For i i eq b ecomes by use of eq

d x dt

rh x

d d

and b ecause dt is prop ortional to d

d x

rh x

dt

This equation can immediately b e compared to Newtons equation

d x

r x

dt

where x is the gravitational p otential The Newtonian p otential is determined by

Poissons equation

r x G x

THE NEWTON LIMIT



where G i Newtons constantG Gc cmg Forapoint mass one has

the well known result

GM

r

Comparing eqs and we get h constant and requiring that at very

large distances from the pointmass space should be at we get

g x x

Thus we see that suciently close to a point mass spacetime must indeed be slightly

curved

The curvature indicated by eq can be observed Recalling that the prop er

time is the time observed on a freely falling watch we have

d dt g xdx dx g x dt

f al l ing

where the last expression is valid in a gravitational eld where the clo ck is approximately

at rest Thus the time measured in this system in the p oint x is

d

q

dt

g x

This is not in itself an observable eect since all clo cks and physical pro cesses in the p oint

will suer the same eect However we can compare two dierent p oints Here it is

imp ortant to note that if eg an atom emits light in one point this light will travel to

another p oint in a constant time if the metric is time indep endent This follows b ecause

d for light and the line element g x dx dx can be solved for dt dx and

R

subsequently in tegrated over the distance b etween the two p oints dt and the resulting

integrated travel time for light is timeindep endent Therefore it follows that if a wave

length is emitted in time interval dt in one p oint this wave length will b e observed in the

other p oint in the same time interval dt

If the clo ckis aphysical system with a frequency we therefore get dt

v

u

u

g x

t

g x

In the weak eld approximation we then have

x x

which is an observable eect Here is the frequency of light emitted in the p oint

but in accordance with what was said ab ove this is also the frequency when this light is

observed in point provided the metric is timeindep endent Now two sp ectra emitted

from the same type of atom can be identied even if the sp ectral lines are displaced

by the amount so we can compare an atomic sp ectrum emitted from the sun and

ed on earth with the same atomic sp ectrum emitted on earth observ

If we consider light which passes from the sun to the earth we have for the suns

potential

GM







R 

GENERAL RELATIVITY

whereas the earths potential can be ignored relative to In eq G should be



replaced by Gc in ordinary units The frequency of light from the sun is thus shifted

by parts per million relative to light from earthbound sources Taking into account

various other eects the b est exp erimental result is times the predicted value

Another exp eriment made consists in emitting lightfrom a tower of height m

The falling light is then observed on the ground From the frequency shift should

be

cmsec cm

t tom top bo

cmsec



The exp erimental value is



in excellent agreement with the prediction

The principle of general covariance

So far wehave studied the eects of gravityby use of the principle of equivalence according

to which the physics of sp ecial relativityis valid in freely falling lo cal systems of inertia

and the eects of gravity can then be obtained by transforming to an arbitrary system

Such a pro cedure is in general rather complicated

Einstein introduced a new principle whichleadstoamuch more systematic way

of ysics of gravity from the physics without gravity namely the principle obtaining the ph

of general covariance Using his own words well translated to English this principle

states

The general laws of nature are to be expressed by equations which hold

good for all systems of co ordinates that is are covariant ie preserve their

form with resp ect to any substitutions whatever generally covariant

In this connection the laws of nature are to b e understo o d as those which are valid in

sp ecial relativity It then follows that if general covariance is satised then the equivalence

principle is also satised in eachpoint there are freely falling lo cal elevators in which the

laws of nature are those of sp ecial relativit y and from general covariance they are thus

valid laws in all co ordinate systems

We need to sp ecify precisely what we mean by lo cal elevators Clearly they should

not be to o large b ecause then from exp erience with tidal forces we know that gravita

tional eects can be observed on a suciently large scale The equation of motion in a

gravitational eld is given by equation At each point x x we can select the

elevator such that g x at Minkowski space and in order to have lo cal

straightline motion we can select the elevator such that

g x

x

xx

and g given by eq Because of the connection b etween the Cristoel symbol

this ensures that to lowest order near each point of spacetime we have no eect from

gravitational forces However it should be emphasized that second derivatives of g x

TENSORS

are not in general assumed to vanish This amounts to saying that we only assume that

eects of gravity can b e transformed away on a scale which is small relative to the scale

of the gravitational eld

Contravariant and covariant tensors

We shall now present a systematic construction of certain quantities tensors which are

suitable for applying the principle of general covariance For tensors there exist trans

formations when the co ordinates are transformed x x The tensor transformations

comp onents in are linear and homogeneous for the comp onents of the tensors Hence all

the x system vanish if they vanish in the xsystem A law of nature requiring that all

comp onents of a tensor vanish is thus valid in all systems if it is valid in one system

Tensor laws thus follow the principle of general covariance if we ensure that they are valid

in sp ecial relativity The simplest quantity is a scalar quantity which is invariant under

er time is an example of x x Numbers like etc are examples Also the prop

an in variant In general one can also have a scalar eld x dened in each spacetime

point transforming like x x

Another quantity is the co ordinate dierential dx which transforms like

x

dx dx

x

Any quantity which transform like is called a contravariant vector ie U is a

contravariantvector if under a transformation x x one has U U with

x

U x U x

x

It should b e noticed that since d is invariant it follows from that the fourvelocity

dx d dx d dxd isa contravariantvector

In sp ecial relativity x is a vector It is imp ortant to realize that this is not the case in

general relativity where the transformation x x x x is completely arbitrary

In general relativity we can only aord that the co ordinate dierentials dx are vectors

The physical reason for this dierence is as mentioned b efore that in the equivalence

principle we must restrict ourselves to innitesimal elev ators

A covariant vector is dened by

x

A x A x

x

From and we can form a scalar

x x

A x U x A xU xA xU x

x x

We say that by contracting the indices of two vectors we obtain an invariant

From a scalar xwe can form a covariantvector by dierentiation

x x x

x x x

GENERAL RELATIVITY



n



A general tensor can have arbitrarily many indices eg T Its transformation



m



is given by

n m

 

x x x x

n



n



T x T x

m m

 

n m

 

x x x x

A tensor with upstairs as well as downstairs indices is called a mixed tensor The trans

formation law can easily be remembered by noticing that T transforms the same

way as if it had b een a pro duct of n contravariant and m covariant vectors In other

words a pro duct of vectors is a tensor eg A B C transforms as a tensor D etc

etc

The reader should be warned that the summation convention requires some care

ose for example w e have the two relations A B C and D E F What is then Supp

AD Well multiplying the two relations together we apparently get AD B C E F

This expression is however meaningless and hence wrong The summation indices o ccur

four instead of two times Before multiplying A and D together wemust ensure that the

summation indices in A and D are dierent Thus keeping the indices in A we should

use a dierent name for the indices in D eg D E F We then obtain the correct

index occurs only twice expression AD B C E F where each summation

The metric tensor g is a covariant tensor as is easily seen from the denition

y y

g x

x x

y y x x

x x x x

x x

g x

x x

Also g is a contravariant tensor The index of a tensor can be raised or lowered

by means of g or g Eg T g g T By going to a freely falling system we see

that the tensors T and T are the same physical ob ject and T is denoted T The symbol

is easily seen to be a mixed tensor

Often the determinantofg o ccurs

g det g

x x

T

From the transformation law which in matrix form reads g g

x x

where g etc denotes the matrix g and where the sup erscript T denotes the transp ose

of the matrix one obtains

x

g x g x

x

where jxx j is the Jacobi determinant j x x j j detx x j Eq has the

imp ortant consequence that d x dx dx dx dx

q q q

x

g x d x g x d x g x d x invariant

x

where in the rst step we used the usual Jacobi transformation Thus the measure of

p

integration d x is not invariant but should b e multiplied by g

DIFFERENTIATION

Finally let us mention a few simple rules for tensors which follow from the transfor

mation a sum of two tensors is a tensor a pro duct of two tensors is a new tensor

with more indices eg

B A T

and acontraction in a tensor is a new tensor with fewer indices eg

T T

Dierentiation

Wesaw in the last section that dierentiation of a scalar leads to a vector see eq

However in general it is not true that dierentiation of a tensor leads to a new tensor

This is related to the fact that dierentiation is dened by comparing the tensor in

nts and then taking the limit where the p oints approach one another i two dierent po

However a tensor transforms dierently in the two points as can be seen eg from the

transformation law

In the following we shall discuss the concept of dierentiation versus covariance follow

ing Einsteins original pap er To see what happ ens let us consider some tra jectory

x Starting from a scalar x we have

dx x dx

invariant

d x d

where the invariance follows from the fact that d and d are individually invariant

However dx d is a vector and consequently it follows that

x

V x

x

is a covariant vector This statement has already b een checked directly in eq

Now let us dierentiate once more

d dx dx d x

invariant

d x x d d x d

Next let us take the tra jectory x to be a path for a particle which falls freely in an

arbitrary gravitational eld Then we have from the equation of motion

d dx dx

invariant

d x x x d d

Since

dx dx

d d

is a contravariant tensor and since the left hand side of is an invariant it follows

that the quantity in the bracket in eq is a covariant tensor Inserting we

have that

V

V V

x

GENERAL RELATIVITY

is acovariant tensor with indices and V is called the covariant derivativeofV

The result was derived for a tra jectory of a particle falling freely in some gravi

tational eld By choosing all sorts of such elds we can however manage that the curve

is completely arbitrary Hence eq should be valid for x being an arbitrary p oint

in the spacetime continuum Alternatively one can use the connection between

and the metric as well as the fact that the metric g is a tensor to show directly

that is a tensor

tensor V the following quantity If we dierentiate a contravariant

V

V V

x

is a mixed tensor T The concept of covariant dierentiation can be generalized to an

arbitrary tensor One has eg

C

C C C C

x

We leave it to the diligent reader to verify this statement

An imp ortant prop erty of the covariant derivative is that it reduces to the ordinary

derivative in a freely falling lo cal elevator This follows from the denition of lo cality given

in according to which the rst derivatives of the metric tensor can be required to

vanish in a lo cal system Since is directly related to these derivatives through eq

ws that it follo

x j

xx

From this wehavethevery imp ortant conclusion that in spite of the notation mixed

tensor notation it is not true that is a tensor b ecause if it was a tensor then

it follows from that in any point there exists a freely falling elevator where

vanishes and consequently it should vanish in all co ordinate systems Since do es not

vanish in general it follows that it is not a tensor Using the transformation law for the

tensor g and the relation one can easily see that transforms as

x x x x x

x x

x x x x x x

The rst term on the right hand side is what one would get if was a tensor but the

second term shows that transforms in a more complicated way

Because of it follows that in the small elevators the covariant derivative is just

the ordinary derivative This is rather satisfactory since we are supp osed to use the laws

of sp ecial relativityineach elevator and these laws of course contain only the ordinary

derivatives

From the denition of a lo cal elevator ie from the fact that the rst derivatives

of g can be taken to vanish in a lo cal elev ator and from the fact see eq that

g is a tensor it follows that the covariant derivativeofg vanishes

g

Finally let us consider motion along a tra jectory x In this case one can only talk

ab out dierentiation along the curve x We can then pro ject the covariantderivative

to the tangent dx d Eq is then replaced by

dx dV DV

V

D d d

THE DETERMINANT OF g

which is easily shown to be a contravariant vector with index In a lo cal elevator it

reduces to the ordinary derivative dV d

A prop erty of the determinant of g

We shall show an interesting prop erty related to the determinant of g To this end

we need to dierentiate a determinant Let us therefore start by considering an n n

determinant with elements a It is dened by

rs

X

P

n

 

det a a a a

n

n

 

where P is ev en o dd for even o dd p ermutations of the reference sequence n

In the sum we can take an element a outside a bracket and all the terms in

rs

containing the particular factor a can then b e written a A no summation over

rs rs rs

r and s where A is a sum of terms which consist of n factors A is called the

rs rs

enden t complement Because of the construction of the determinant A is indep

rs

of the elements in the r th row and the sth column we have already picked the element

a andeach term in the sum has only one element from the r th row and the sth

rs

column Now each of the n pro ducts in det a contains one and only one element in the

r th row and we have no sum over r

a A a A a A deta

r r r r rn rn

The n complements A A are all indep endent of the elements a a in the

r rn r rn

r th row Thus the quantities A A are unchanged if in det a we interchange the

r rn

elements in the r th row with other elements In particular we can keep A A

r rn

unchanged if we replace the elements in the r th row with the corresp onding elements

a a a in the sth row r s However the new determinant has the value

s s sn

since two rows have the same elements ie

a A a A a A s r

s r s r sn rn

Eqs and can b e used to construct the inverse determinant or matrix Thus

the inverse of

a a a

n

B C

a a a

B C

n

B C

A

a a a

n n nn



is

A A A

 n 

det a det a det a

A A A

B C

  n

B C

det a det a det a

B C

A

A A A

nn

n n

det a det a det a

The correctness of is seen bymultiplying and together by use of

and The result is the unit matrix

Now let us dierentiate det a ie consider the variation det a arising from

Each term in the sum has n factors and the variation is a pro duct of n factors

GENERAL RELATIVITY

multiplied by a according to the usual rule for dierentiation of a pro duct The n

rs

factors do not contain a Thus collecting all terms which are multiplied by a amounts

rs rs

to collecting the terms in the complement A According to this means that

rs

X X



det a A a a det a a

rs rs sr rs

 

where a stands for the elements in the inverse matrix a

sr

From this general result let us return go g With g det g we have g is

symmetric

dg g g dg

w e have g g ie since g was dened as the inverse of g Using

g dg g dg

so has the alternativeform

dg g g dg

From we have

g ln g g

g

x g x x

This prop erty turns out to b e rather useful

Some sp ecial derivatives

In mathematical physics the curl of avector is dened by

V V V V V V

curl V

x x x x x x

For the covariant derivativeofavector V we can similarly dene the curl to b e V V

Using the expression we see that the terms involving the Christoel symbols drop

out and we just have

V V

V V

x x

ie the curl is given in terms of just the ordinary derivatives

The divergence ofavector is given by

V

V V

x

Using the expression we have

g g g g

g g

x x x x

since the rst and the third terms are seen to b e identical byinterchange of the summation

indices and Now we can use to simplify this to

p

ln g g

p

x g x

SOME SPECIAL DERIVATIVES

Eq can then be written in the elegant form

p

V g V

p

g x

p

gd x is invariant Gauss theorem in four dimensions In eq we mentioned that

can then be written

Z Z Z

p p p

d x g V d x g V d g V

x

P

where are surfaces eg for d dx dx dx If these surfaces are taken

o

at innity and if V at innityone has

Z

p

g V d x

For tensors similar simplications o ccur The covariant derivative of a tensor T is given

by

T

T T T

x

Thus the divergence is given by

T

T T T

x

Using we can combine the rst two terms just likewe did in eq

p

T g T T

p

g x

If the tensor is antisymmetric

F F

then the last term in drops out b ecause is symmetric in and and wehave

the simple result

p

F g F

p

g x

The usual Laplace op erator acting on a scalar equals divgrad In our case there is a

similar result The gradient of the scalar S is just S x which when multiplied by the

metric can be contravariant We can then apply to obtain

S

p

g g S

p

g

g x x

x In a lo cal elevator where generalizes the usual dAlembert op erator

g g

reduces to the dAlembertian

GENERAL RELATIVITY

Some applications to physics

Having developed the apparatus of tensors and generalized dierentiation let us enjoy

ourselves with some applications to physics

First let us consider again the problem of a particle which falls in a gravitational

eld g x but this time we apply the principle of general covariance Thus rst we

must know what happ ens in sp ecial relativity Here the particle is completely free Its

fourvelocityisgiven by

dy dy dy

U

d d d

where y are the co ordinates in sp ecial relativity The equation of motion is simply

dU

d

Nowwemust make this equation generally covariant This was done in eq where

we showed that dd should be replaced by DD ie we have

dU DU

U U

D d

This equation is correct because

it is valid in any freely falling lo cal elevator lo cally

it is generally covariant a vector DU D vanishes in one system and hence it

vanishes in all systems

From this example we see that once wehave b een through the somewhat tiresome pro ce

dure of setting up tensor calculus the result can b e derived in a much simpler way

than by applying the equivalence principle directly

To present another example let us consider a small gyroscop e which moves in the

gravitational eld g x In sp ecial relativityitis thus not inuenced byany forces The

gyroscop e taken to be at rest has an angular momentum In the limit where it can be

considered to b e a p ointwesay that it has a spin S The spin is conserved if no external

force acts We can then introduce the four vector S S and hence

dS

d

Also since the gyroscop e is at rest the spin is dened in the rest frame we have

U S

These equations are valid in sp ecial relativity Under the inuence of the gravitational

eld they b ecome

DS dS

U S

D d

S U

SOME APPLICATIONS TO PHYSICS

Eq for example describ es the precession of a gyroscop e carried in a satellite

moving in the gravitational eld from the earth To actually utilize eq we need

however more accurate result for the metric than the one obtained in the Newtonian limit

see eq

Newtons equation can b e generalized to sp ecial relativity

dU

f m

d

where f is called the fourforce In general relativitythis becomes

DU

m F

D

where F is a generalized fourforce

We now turn to electro dynamics The Maxwell equations can be expressed in terms

of an antisymmetric tensor F with F B F B F B B is the magnetic

eld F E F E F E E is the electric eld Furthermore the current

is a fourvector J J where is the charge density and J is the current The

equations

E

J url B div E rE rB c

t

b ecome

F J

x

The other Maxwell equations

B

div B rB r E

t

can be expressed as

F F F

x x x

where the indices are p ermuted cyclically Eqs and are of course valid

in sp ecial relativity as a matter of fact they or rather the equivalent eqs and

are the raison detre for sp ecial relativity

Nowwe can apply our tensor apparatus to obtain the eects of gravitation on electro

magnetism Eq is simple to write in a generally covariantform

F J

Now F is antisymmetric so we can simplify as in eq

p p

g F g J

x

To make generally covariantwe lower the indices

F g g F

GENERAL RELATIVITY

which is still antisymmetric Eq then b ecomes

F F F

where eg

F

F F F

x

Adding the three terms in it is seen that the six terms containing the s cancel

and we thus get the simple result

F F F

x x x

In sp ecial relativity the fourcurrent is conserved It is easy to see that the same must b e

true in general relativity From eq we have

p p

g F g J

x x x

b ecause F is antisymmetric The conservation law

p

g J

x

precisely corresp onds to the vanishing of the fourdivergence of J compare with eq

in generally covariant form

Finally we mention that in sp ecial relativitythe Lorentz force

dp

eE v B

dt

can be written compare with eq

dx

f eF

d

In general relativitythe generalized four force therefore b ecomes

dx

e F F

d

where F is a vector since F is a tensor and dx d isavector

Curvature

The principle of equivalence and the principle of general covariance shows that a gravi

tational eld can be represented by a metric tensor g x which is not everywhere con

stant The opp osite is however not true Supp ose space is just at Minkowski space

and g x everywhere Thus

d dx dx

CURVATURE

We can however always transform the xco ordinate to a dierent system x x and

then

d g x dx dx

x x

g x

x x

In the new system it app ears that the metric do es not represent at space However

it is clear from the derivation that the apparent nonatness is an illusion

The question then arises whether we can somehow avoid being fo oled by someb o dy

who claims to have an interesting gravitational eld g but who has obtained this eld

from at space by applying some complicated mathematical transformation x x To

avoid suchajokewe can notice that in at space if we dierentiate a vector two times the

derivatives commute In curved space derivatives are replaced by covariant derivatives

general do not comm ute To see this let us notice that which in

T

T T T

x

In this equation T is an arbitrary tensor so we can take T V since the covariant

derivative ofavector V is a tensor Thus

V

V V V

x

From this we get is symmetric in and

V V

V V V V

x x

Inserting now

V

V V

x

we obtain the ordinary derivatives of V commute and thus drop out

V V V R

where

R

x x

The quantity R is a mixed tensor b ecause the lefthand side of eq is a tensor

and b ecause V on the righthand side of is a vector R is called the Riemann

Christoel curvature tensor or just the curvature tensor

Now it is clear that nob o dy can fo ol us any longer since if space is at it follows

that the curvature tensor vanishes in the at system However if a tensor vanishes in one

system it is zero in any other system



Such a transformation is very often used eg in quantum mechanics where the hydrogen atom is most



     

easily solved in spherical co ordinates where d dr r d r sin d The spherical co ordinates

     

x rx x thus lo ok nonat with g g x g x sinx g for

  ij

i j It is clear that this nonat metric do es not representanygravitational eld since we also have

   

d dx dy dz wherex y z are the Cartesian co ordinates

GENERAL RELATIVITY

Figure Parallel transp ort of a vector from P to P through two dierent paths

f

P P P and P P P

f f

From the curvature tensor we can construct a covariant tensor We leaveitto

the hardworking reader to show that it can be written in the form

R g R

g g g g

x x x x x x x x

g

If we go to a lo cal elevator the terms can be brought to vanish and the curvature

tensor is thus lo cally given by the second derivatives of the metric

Parallel transp ort and curvature

In section we discussed the transp ort of vectors which fall freely in a gravitational

eld and which are not inuenced by any forces Thus in any lo cal elevator the vector

S sa y satises the simple equation dS d Here S can be the spin of a small

gyroscop e or the velocityvector The equation satised in an arbitrary system is thus

dS dx

S

d d

The vector S is said to b e parallel transp orted if it satises

Let us now consider the transp ort from a p oint P to a p oint P through two dierent

f

paths namely P P P and P P P as shown in g

f f

The parallelogram has sides a and b which are considered to b e innitesimal In

sp ecial relativity the vector would b e the same in P whether we used the path P P P or

f f

P P P In general relativitywe do not exp ect to obtain the same result since the vector

f

S can exp erience dierentgravitational elds on the two paths

Let us consider the path P P P The change in S can b e computed from

f

S P S P j S P a

where the index on means that is evaluated in P Next we want to go from P to

P In order to apply we need in the point P which is given by

f

j a j j

P



x

THE CURVATURE TENSOR

From we have

S P P P S P j S P b

f P



Inserting and in we get to second order

n o

j a S P j S P a b S P P P S P j

f

x

S P j S P b j S P a b

x

j j S P a b

Furthermore S P can be expressed in terms of S P through so we get

S P P P S P j S P a j S P b

f

j S P a b j j S P a b

x

For the path P P P we can obtain S P P P from the ab ove expression just by

f f

interchange of a and b Subtracting we nd

P a b S S P P P S P P P R S

f f

where R is the curvature tensor dened in eq

From eq we see that as exp ected if there exists a gravitational eld then in

general a vector which is parallel transp orted through two dierent paths to the same

point do es not acquire the same value

If on the other hand we assume that the Riemann tensor vanishes then it follows that

S b ecomes indep endent of the path and only dep ends on the spacetime p oint In other

words S is a eld S x which satises the dierential equation This equation

simply means that the covariant derivative of the eld S x must vanish At one p oin t

we can prescrib e a value of S and the value in any other p oint in a domain where R

vanishes is then obtained byintegrating If the curvature tensor vanishes we can

always dene everywhere inertial co ordinates at co ordinates by parallel transp ort

Thus it is necessary and sucient for a metric g x to be equivalent to a at metric

that R everywhere

Prop erties of the curvature tensor

The covariant form of the curvature tensor has a n umber of symmetries which can

be read o from

R R

R R

R R

R R

R R R p ermuted cyclically

GENERAL RELATIVITY

Eq is a simple consequence of the commutation rule for covariant deriva

tives the left hand side of is clearly antisymmetric in and

We can form the contracted curvature tensor

R g R R

which is called the Ricci tensor Because of it is symmetric

R R

The contraction is essentially unique if wecontracted instead the rst two indices

or the last two indices we get zero b ecause of and Contraction of the other

indices leads again to R b ecause of and

We can make a further contraction in order to get a scalar

R g R g g R

Contraction of and and and gives zero b ecause of antisymmetry in these indices

Apart from the algebraic prop erties mentioned ab ove the curvature tensor also satises

imp ortant dierential identities These are most easily derived from by going to

a lo cal elevator where vanishes Dierentiating R with resp ect to x one obtains

third derivatives on the metric as well as x terms However these terms vanish

in the lo cal system Permuting the indices and cyclically and adding the three

expressions one nds that this sum vanishes In generally co variant language this means

R R R

This equation is valid in the lo cal elevators where the covariant derivatives reduce to

ordinary derivatives and it is generally covariant and hence it holds in all systems

Remembering that the covariant derivative of the metric tensor vanishes we nd by

contracting and in

R R R

Contracting and we get

R R R

and

R R

Again since the covariant derivativeofthe metric vanishes we have from

R g R

It turns out that eq plays a fundamental role in the theory of general relativity

as we shall so on see

THE ENERGYMOMENTUM TENSOR

The energymomentum tensor

We now turn to the essential problem facing us Supp ose we are given some distribution

of matter eg our universe how do we determine the gravitational eld g x So far

we only know how to pro ceed in the Newton limit sect where we found eqs

and

r g x Gx

where x is the mass density in the limit where all velocities are small relative to the

lightvelo city

Eq suggests that we should attempt to understand how x can be gener

cities If w e can nd this we know how to represent alized to arbitrary relativistic velo

some matter distribution eg in our universe in terms of sp ecial relativity

Let us consider a simple situation where wehaveaowofmatter In sp ecial relativity

the relevant quantity is not the mass density but the energy density remember the famous

formula E mc The prop er density x is then dened as the density measured by

an observer moving with the ow The lefthand side of eq suggests that in

general situations it should b e represented by a tensor with two indices this tensor more

should somehow at least contain second derivatives of the metric tensor Thus we should

attempt to generalize the righthand side to a tensor with two indices which for small

velocities reduce to the righthand side of eq In sp ecial relativity the ow of

matter is characterized by its density and the fourvelo city U x of matter at the p oint

x From this we can construct a tensor with two indices

T xx U x U x U dx d

Let us consider T

dx

T

d v

where v is the usual velocity v dxdx and where we used the sp ecial relativistic

relation b etween d dx and v

d dx dx dx v

From eq we see that in the Newton limit v we get T so eq

b ecomes approximately

r g x GT x

Recalling that the mass of a small volume of moving material increases by a factor

p p

v relative to the rest mass and that the volume decreases by a factor v

we see that T in eq is the density measured by a xed observer who sees the

matter passing by with a velocity v Thus T is simply the relativistic energy density

The other comp onents are just

v v

i j

ij ji

T T

v

v

i

i i

T T

v

i

The quantity T is called the energymomentum tensor of sp ecial relativity T is

ij

is the current of momentum the density of momentum as seen by a xed observer and T

GENERAL RELATIVITY

T has the prop erty that for closed systems the energy and momentum are conserved

In dierential form we just have

T v

r

x t v v

which expresses the conservation of a quantity of material with density v moving

with a velocity v x

We also have

T v v v v v v

x x x y x z

x t v x v y v z v

v

x

v rv

x

v t

where we used eq Thus

i i

T v

i

v rv

x v t

since the quantity in the bracket in eq is just the change in the velo city for an

observer following the stream of matter and in the absence of any forces this velocity

cannot change Thus we have by combining and

T

x

and we say that the energymomentum tensor is conserved From it follows that

the total energymomentum P is conserved b ecause implies that

Z

P d x T x t

and hence from

Z Z

T T dP

d x d x

dt t x

where we used that

Z

i

T

d x

i

x

i

if T vanishes at innity From eqs and it is easily veried that P

dened by eq is indeed a fourvector remember that volume d x decreases by a

p

factor v

The energymomentum tensor represents an extremely simple physical system

A somewhat more useful case can be constructed by considering a p erfect uid which

is dened as a uid characterized by a velocity eld v x such that an observer moving

along with the uid sees it as being isotropic around each point Ideal uids are often

used to approximately describ e our universe at large scales much larger than the size

of galaxies and hence the energymomentum distribution of such uids is of particular

vity relevance in the theory of gra

THE ENERGYMOMENTUM TENSOR

To nd the energymomentum tensor we simply use the rest frame where

T

rest

i oi

T T

rest rest

ij ji

T T p

ij

rest rest

The comp onent is the same as in since v and the result for the ij

comp onent expresses the meaning of isotropy Similarly the icomp onents must vanish

due to isotropy In is again the prop er energy density whereas p is called the

pressure

We can now express T in a frame where the uid moves with velo city v x t

T p p U U

To see that T is correct notice that it is a tensor in sp ecial relativity and that it reduces

to in the rest frame Conservation of energy and momentum in the dierential

form for a closed system

T

x

leads to the sp ecial relativistic equations for an ideal uid

v p v

v rv rp v

t p t

We recommend the reader to verify this result

To make the ab ove result generally covariant we pro ceed as done many times b efore

The conservation equation b ecomes replaced by

T

Using eq we get

p

g T T

p

g x

where we can say that the righthand side represents a gravitational force density The

total energymomentum can tentatively b e written down in analogy with

Z

p

P d x g T

which is however not conserved due to the nonvanishing of the righthand side of eq

Physically nonconservation of P is due to the p ossibilityofexchanging energy

and momentum b etween matter and gravitation Also P is not acontravariantvector

The explicit form of the energymomentum tensor for an ideal uid interacting with

gravity is given by

T pg p U U

is a tensor and since it reduces to in sp since T ecial relativity

The energymomentum tensors constructed ab ove are symmetric T T This is

a general prop erty assumed to b e valid in the following

GENERAL RELATIVITY

Einsteins eld equations for gravitation

We are now able to pro ceed to present Einsteins eld equations which were derived

by plausibility arguments in Die Grundlage der allgemeinen Relativitatstheorie

Annalen der Physik Leipzig We return to the Newtonian limit which

can be written as in eq ie

r g x G T x

The lefthand side contains the second derivative of the metric tensor whereas the right

hand side is the comp onent of the energymomentum tensor Thus it is natural to

equation to generalize this

E x G T x

in arbitrary co ordinates where the tensor E should dep end on the metric and its rst

and second derivatives From it is clear that we need to include second derivatives

and the reason we do not include higher derivatives is primarily because of simplicity

However such terms would have tobemultiplied by a constant with dimension of a p ower

of length gx terms would have to be multiplied by some constant with dimension

oking around among the fundamen tal constants of length gx by length etc Lo

of nature there is only one constant with dimension of length the socalled Planck length

Gh

L

P l anck

c

 

Using Gc cmg and h erg s one obtains



L cm

P l anck

It therefore follows that if the constants of dimension length to a p ower are present they

are incredibly small and of no relevance for large scale phenomena in our universe They

could b e relevant in the very early universe which is supp osed to b e very small big bang

Due to the o ccurrence of h it is clear that quantum phenomena must be involved and

hence such terms could be of relevance in quantum gravity However here we shall



pro ceed to consider only scales which are large relativeto cm

The Einstein tensor E xineq thus dep ends on the geometry of spacetime

and should be linear in the second derivatives of g Also it should be a tensor which

represents the genuine physical contents of gravitational elds For this wehave only one

candidate namely the curvature tensor and quantities derived from it Since E has two

indices it should dep end on the contracted curvature tensor Because of the symmetries

of R we saw in sect that there is essentially only one such tensor namely R

Could w e take

E R

The answer is no In sect it was shown that the energymomentum tensor satises

dierential conservation

T



Such terms o ccur in sup erstring theories one of which could be the right theory of quantum

gravity



Mathematicians have shown that the curvature tensor is the only tensor that can be constructed

from the rst and second derivatives of the metric tensor and which is linear in the second derivatives

of the metric tensor

EINSTEINS FIELD EQUATIONS

Going back go eq we therefore also have

E

in general Thus eq is From eq it follows however that R

wrong

However eq shows that the right answer is

E cR g R

which is covariantly conserved Here c is a dimensionless constant which we shall x by

requiring that we get the correct Newton limit In a nonrelativistic system the pressure

terms are much smaller than the energydensity so jT j jT j Eqs and

ij

reduce to

g R G T cR

Since jT j jT j we also have that E is very small so from

ij ij

R g R

ij ij

In the weak eld limit to the rst approximation g Thus

ij ij

R R R R R

ii

ie

R R

Inserting this in we get

E cR R cR

Now R can b e obtained from where the terms are of higher order and hence

can be left out in the Newton limit Thus

g g g g

R

x x x x x x x x

Since all time derivatives vanish we get

g

R R

ij

i j

x x

so from we get

E cR cR R cr g

ii



It is interesting that the rst eld equation that Einstein published actually was precisely

He was aware that it could not b e generally valid

GENERAL RELATIVITY

This shows that we obtain eq provided c Consequently Einsteins eld

equations read

g R G T R

This equation can be written in a dierent form By contraction we get

R R R G T R R

or

R G T

This can b e inserted in eq and we then get

g T R GT

Eq is of course fully equivalentto eq

In empty space T and then gives the vacuum Einstein equation

R

Flat Minkowskispace satises this equation globally and lo cally but in the lo cal case

there are other nontrivial solutions due to the nonlinearity of eq

Eq can be generalized by adding the cosmological term

R g R g G T

This is allowed from the p oint of view of covariant conservation of T since the covariant

derivative of g vanishes However eq do es not satisfy the Newton limit The

cosmological constantmust therefore b e rather small and in this chapter we shall ignore

it However a small which seems to exist has profound cosmological eects as we

shall see later It should be noticed that consistency of the Einstein equation

requires that T is symmetric T T

To conclude this section we now have a nonlinear set of second order dierential

equations whichallows one to compute the gra vitational eld from a given energy

momentum distribution This is precisely the contents of Einsteins general relativity

Remarks on the history of Einsteins eld equations

In Einstein and Grossmann published a pap er where gravitywas for the rst time

describ ed by the metric tensor and the Ricci tensor was introduced as an imp ortanttool

The theory was however not right Also a later pap er from by Einstein contains

a treatise of dierential geometry The contents of this pap er was corrected for technical

variant errors by LeviCivita The aw in Einsteins pap er mainly that the co

conservation of the energymomentum tensor on the righthand side was not correctly

implemented by the geometrical lefthand side as explained ab ove in connection with

was corrected in November by Einstein and by the matematician Hilb ert

who derived the consistent eld equation from a variational principle On November

Einstein submitted a pap er which for the rst time gives the correct eld equation Five

days b efore Hilb ert had actually submitted a pap er which also contained the right eld equation

SPHERICALLY SYMMETRIC METRIC

The timedep endent spherically symmetric met

ric

In order to gain insight in Einsteins equations we shall start by considering a time

dep endent spherically symmetric gravitational eld g x By this we mean that the

metric should be the same when the rectangular co ordinates x x x are sub jected to

a rotation Intuitively this means that the prop er time interval can only dep end on the

following quantities which are rotationally invariant

sin d t r dt xdx r dr dx dr r d

p

where r x y z Thus

d Artdt B rtdr C rtdr dt D rtr d sin d

In general relativity it is imp ortant to realize that the co ordinates are arbitrary Thus

we are free to make transformations x x in In particular we can always

try to select such transformations in a waywhich simplies the expression First

the function D can easily b e removed by intro ducing a new radial variable

q

r r D rt

d then b ecomes dep endent on new functions A B and C which are functions of t

and r Dropping the primes we can write

d Artdt B rtdr C rtdr dt r d sin d

The mixed dr dtterm can also b e removed by resetting our clo cks by use of a new time

co ordinate t which is dened by

C rtdr dt rt rtArtdt

Here rtisanintegrating factor dened such that b ecomes a p erfect dierential

with A t t C t r This requires the integrability condition

rtArt rtC rt

r t

This is a rst order partial dierential equation for since A and C are given Once we

are given at a certain time for all r we can compute it at any other time Using that

C

dt A dt C dtdr dr

A A

the prop er time b ecomes

C

d dt B dr r d sin d

A A



It could happ en that this transformation pro duces a constant r In the following for simplicity

we only consider nontrivial transformations r const turns out to be inconsistent with the Einstein equation

GENERAL RELATIVITY

or renaming these functions

d E rtdt F rtdr r d sin d

This form of the metric is called the standard form of the metric rst derived byWeyl

It should be noted that the transformations used ab ove do not involve the angles and

therefore the spherical symmetry is still manifest

The metric tensor thus has the form

g F g r g r sin g E

rr tt

tt rr

g g g g





r sin

F r E

whereas the determinant g is

g r EF sin

The invariantvolume elementisthus

q

E rtF rt dr d d dt r sin

If E and F were constants this would just be the usual volume element in spherical

co ordinates

A digression A simpler metho d for computing

In order to write down the Einstein eld equations we need rst to know the Christoel

symbols for the metric One metho d is to use eq directlyie to use

g g g

g

x x x

This is a straightforward but tedious metho d A somewhat simpler way consists in ob

taining the s from a variational principle

First let us consider some functional F F x x where x dx d and

demand that the tra jectory x x is obtained by requiring the minimum variation

of the integral

Z

F x x d

ie

Z Z

F x x d F x x d

Z

F F

x x d

x x

Now x d x d Requiring that the variations vanish at the ends of the interval of

integration we obtain by a partial integration

Z Z

d F F

x d Fd

x d x

COMPUTING

or the EulerLagrange equation

F d F

d x x

Now we can easily check that if we take

dx dx

F g x

d d

then the EulerLagrange equation b ecomes

d dx dx dx g x

g x

d d x d d

Doing the dierentiation on the lefthand side we get

g dx dx d x g dx dx

g

x d d d x d d

so after multiplication with g we get

g g d x dx dx

g

d x x d d

g g d x g dx dx

g

d x x x d d

where in the last step we used that the names of the summation indices are arbitrary and

dx dx is symmetric in and From and we see that this is precisely

the equation of motion for a freely falling particle in a gravitational eld sp ecied by the

metric g x

As a sideremark we mention that use of the EulerLagrange equation also

shows that can be obtained from the variational principle

s

Z

dx dx

g x d

d d

We leave it to the reader to verify this statement Eq shows that freely falling

particles follow a curve with the shortest distance since the square ro ot times d in

R

is the distance ie can be written d The curve followed by a

freely falling particle is therefore called a geo desic

To return to the main issue of obtaining the Christoel symbols we see that eq

or

d x dx dx

d d d

allows one to obtain the s from a variational principle using the EulerLagrange equa

tions with F given by In our case the line element therefore leads

to the variational principle

Z

h i

E t F r r r sin d

where t dtd r dr d d d and dd

GENERAL RELATIVITY

The Christoel symbols for the timedep endent

spherically symmetric metric

Letusstartby applying eq to the case where x in the EulerLagrange equation

is the time In that case gives from

E d F

E tt r

d t t

Performing the dierentiation on the lefthand side remember that E dep ends on r and

t which in turn dep ends on

E E E F

E t t tr t r

t r t t

or

E E F

t t tr r

E t E r E t

From this we get by comparison with the following s remember that mixed

terms like t r occur twice in

E

t

tt

E t

E

t t

rt tr

E r

F

t

rr

E t

Next let us use the EulerLagrange equation for the problem when wevary

x r

d E F

F r t r r r sin

d r r

Pro ceeding as b efore we p erform the dd on the lefthand side and obtain

r F F E r

tr r t sin r

F t F r F r F F

Comparing with we see that

F

r

F

r r

tr rt rr

F t

F r



r

r r sin E

r r

tt

F r F F

Pro ceeding now with the variable we get since E and F are indep endent of

d

r r sin cos

d

which gives

r sin cos r

THE RICCI TENSOR

ie

sin cos

r r

r

Finally we need only to vary

d

r sin

d

or

cos

r

r sin

Hence the only remaining nonvanishing Christoel symbols b ecome

cos

r r

r sin

The reader can now make a psychological exp eriment where heshe compares the time it

takes to obtain the s by means of the variational principle and by the direct application

of the expression This would also provide a check of the correctness of the results

and

The Ricci tensor

In order to study the Einstein equation or we need to know the Ricci

tensor R From eqs we have

ln g ln g

R

x x x x

Inserting the s found in the last section as well as g given by eq we get the

following nonvanishing comp onents of the Ricci tensor

E E E F

R

rr

E r E r EF r r

F F E F F

rF r E t E t t EF t

r F r E

R

F F r EF r

E E F E

R

tt

F r F r r rF r

E F F E F

EF r F t F t EF t t

F

R

tr

rF t

R sin R

GENERAL RELATIVITY

Thus given some energymomentum tensor T we can in principle obtain second order

dierential equation for the metric functions E and F by use of eg eq

g T R G T

For example

F

G T

tr

rF t

etc etc

The Schwarzschild solution

We shall now consider the case where space is empty except for a mass M situated at

r Thus except for r we must satisfy the vacuum Einstein equations R

Using we see that this implies that the timedep endence drops out From R

tr

we get that F is indep endent of time and it is then easily checked that all the time

derivatives drop out of the other R s From the last eq it follows that we only

need to consider R R R It is seen that R and R contain rather similar

rr tt rr tt

terms Therefore w e form

R F E R

tt rr

F E rF F r E r

This gives

ln F ln E

r r

or

E rt F r f t

We now imp ose the boundary condition that the metric should b ecome at

Minkowski at innity so the function f t in is xed to be for r and

hence also for all other r s Thus

E r

F r

Now we can take R and R to vanish b ecause of R then vanishes auto

rr tt

matically Using the result we get

dE r

R E r r

dr

d E r dE r

R

rr

E r dr rE r dr

d R r

rE r dr

Thus if R then R is automatically zero Hence from R we get

rr

d

rE r dr

BIRKHOFFS THEOREM

so

rE r r constantr C

r and F r in terms of the constant C which From we then know the functions E

can be xed by means of the results g GM r obtained in the Newton limit

see eqs and

Thus from we get C MG and hence

MG

E r

r

F r

MG

r

and the metric b ecomes

MG dr

r d sin d d dt

MG

r

r

This is the Schwarzschild solution The radial variable r has the prop erty that the solution

is asymptotically at Furthermore the solution is singular at r as one would exp ect

b ecause of the presence of the mass This singularity can be shown to be physical since

eg the scalar R R is singular only at r For an elab oration of this feature

see sections and

Remarks on the history of the Schwarzschild solution

Karl Schwarzschild was a wellknown astronomer who had wide interests in all

branches of physics and mathematics When the war started in August he entered

military service where he calculated missile tra jectories in France and later in Russia

Here he w orked out his solution explained ab ove of the newly prop osed Einstein eld

equation He send his pap er to Einstein who replied I had not exp ected that one

could formulate the exact solution of the problem in such a simple way Schwarzschild

contracted an illness while in Russia After having returned to Berlin in March he

died two months later at the age of years

Birkho s theorem

We saw from R that the function F is timeindep endent and all timederivatives

tr

disapp eared In principle the function E could dep end on time

MG

E rtf t

r

but f tcan be removed by a redenition of time t t where

Z

q

t

t f t dt

Thus the Schwarzschild metric always results and we have Birkho s theorem

saying that a spherically symmetric gravitational eld in empty space must b e static with

a metric given by the Schwarzschild solution

GENERAL RELATIVITY

This theorem is similar to Newtons result that the gravitational eld outside a spher

ically symmetric mass distribution b ehaves af if the mass was concentrated in the centre

It is a priori far from obvious that the similar result applies in general relativity where

the b o dy needs not b e static A nonstatic b o dy can in general emit gravitational waves

However Birkho s theorem shows that no gravitational radiation can escap e from a

spherical b o dy into empty spaces

Birkho s theorem can also b e applied if wehave a spherical cavity in a spherical mass

distribution In this case there is no mass at the center and hence GM and the

metric is just at Minkowski Thus if eg the universe was a spherical mass distribution

eect in a spherical cavity the metric would be at and gravitation would have no

The general relativistic Kepler problem

We shall now discuss some implications of the Schwarzschild solution for the solar

system In it app ears that there is a singularity for a radius r MG which

is called the Schwarzschild radius We shall discuss this phenomena later Here we only

remark that for the sun M G km which is deep in the solar interior Hence this



apparent singularity is not relevant for the solar system since the solution is only

valid outside the massive b o dies for the earth the Schwarzschild radius is cm

In eqs and we have written down the equations of

enden t and using the result motion Using that E and F can be taken to be timeindep

we get

d

r r sin cos

d

d

r sin

d

d MG

t

d r

where the dot denotes dierentiation with resp ect to We also have an equation for r

which we replace by the line element

MG r

t r sin

MG

r

r

Here we assumed that the particle is not a photon since for a photon d We shall

return to the photon case later

We thus have four equations for the four functions t r so we can

study planetary motion by solving these equations A great simplication o ccurs b ecause

this motion lies in a plane It is easy to see that if wex at some initial value of then

will b e xed to the same value for all other s Taking eg

for some and for theneq implies that this remains true for

all Eq can then be integrated

r H

THE KEPLER PROBLEM

where H is a constant of integration Also eq gives

MG

t L

r

where L is another constant Substituting these results into we get

r H L

MG MG

r

r r

In applications to orbital motion we are interested in the shap e of the orbit and wewould

therefore liketoknow r as a function of instead of We have

r dr

r

d

From we thus have

H

r r r

r

Inserting this in we get

MG H MG H

r L

r r r r

which is a dierential equation for r r

In the classical Kepler problem it is convenient to intro duce the new variable

u

r

It turns out that this substitution is also convenient in general relativity From

we have r u u and eq gives

MGu L H u H u MGu

which can be solved for u

L MG

u u u MGu

H H

This gives the exact result

Z

u

du

q



L  MG

u

u u MGu

 

H H

The integral is of the elliptic type and gives u so the inverse function

provides us with u u r However in practice this elliptic integral is not

very useful Although it can easily b e evaluated numerically using eg Mathematica the



One can express u or r as functions of byinverting the integral by means of Weierstrass p

MG M G

function const where the Weirstrass invariants are given by g g

 

r   H

M G M G L

For more discussion of this solution see E T Whittaker A treatise on the

 H H

analytical dynamics of particles and rigid bodiesCambridge University Press pages

GENERAL RELATIVITY

main work is to express the constants L H in terms of quantities that are measured by

the astronomers

To orient ourselves in this mess let us notice that the term MGu under the square



is the ro ot is exp ected to b e very small relativetoeg the u term since MGu

Schwarzschild radius divided by the average planetary radius As a matter of fact if

we did not have this term the result would b e the Newtonian result except that

the constant H in would b e related to the usual time not the prop er time This

suggests that we expand the integral approximately around the Newtonian solution

To do this systematically we notice that in the motion around the sun the orbit reaches

amaximum minimum distance called the aphelia p erihelia where

r r r with dr d ie for u u we also have dud Eq then

 

gives

L

u



H MGu H



which determines the integration constants L and H in terms of u which are known



from astronomical observations This is a third order equation To simplify the analysis

we expand to second order

L

MGu M G u u



 

H H

Subtracting these two equations for u and u we get



L

u u u u MG MGu u

  

H

or

u u L



MG

H MGu u



The result can be written





Z



r

du L

p

r r u



H MGu H

r

MGu



where is measured from the p erihelia To pro ceed we expand

MGu M G u

MGu

so

L L MGL M G L

u u u

H MGu H H H H

C u uu u



The last form follows from the fact that the left hand side is approximately a second order

p olynomial in u which vanishes for u u according to eq Consider the term



linear in u in with co ecient C u u It should equal MG L H according



to the left hand side of eq Hence

u u L



C u u MG



H MGu u



THE KEPLER PROBLEM

where eq has b een used Thus

C MGu u



Inserting this in we get



Z

MGu du

r

q

r r MG





r r



u uu u

r





p

where the constant factor outside the integral comes from expanding C The integral

can be p erformed trivially by making the substitution

u u u u sin u

 

r

and we get

cos r r MG MG



r r r r

 

Now when r go es from r to r it follows from that go es from to



The total charge in per revolution is thus jr r j For an ellipse



this equals so the p erihelia precesses an amount where

jr r j



Thus

MG radiansrevolution

r r



For Mercury this gives per revolution In each century Mercury makes

revolutions and observations go back to From general relativitywe thus have

p er century

whereas observational data give

p er century

obs

This is the most imp ortant exp erimental verication of general relativity since it is sen

sitive to the second order expansion of g It should be pointed out that due to p ertur

bations from other planets and due to the rotation of the earth Newtonian gravity also

gives a precession of Mercury

Newton

whereas the actual observation gives

Tot

obs

TOT

so the value is obtained as the dierence One

obs N ew ton

obs

might feel unhappy ab out the subtraction of two large numb ers b ecause small systematic

errors may inuence quite considerably Some discussion of this point has b een

dy has found any convincing evidence against the made in the literature However nob o

result

GENERAL RELATIVITY

Deection of light by a massive body

So far wehave considered the gravitational eects for a massivebody moving around the

sun Now we wish to nd these eects for light which is characterized by d ie

velocity of lightis always one in the freely falling elevators Therefore we cannot divide

by d as we did in the previous section

Let us consider the free motion along a curve describ ed by x x p where p is

principle leads to some parameter Pro ceeding as in section the equivalence

dx dx d x

dp dp dp

since light moves in a straight line in the freely falling elevators Changing to another

parameter p q with p pq and q q p we obtain

d x d q dx dx dx dq

dp dq dq dq dq dp

Thus we see that we get again the geo desic equation of motion if and only if d q dp

ie if q is a linear function of p It therefore follows that the form of the geo desic

equation remains only if we make linear transformations of the parameter so we can

take d constant dp and use p as another parameter For a photon one then has

d dp ie the constant of prop ortionalityvanishes The equations of motion in the

last section should therefore be replaced by similar equations with replaced by p

In most equations this change only leads to trivial alterations However eq

now b ecomes

dr

MG dt d d

dp

sin r

MG

r dp dp dp

r

since d Thus intro ducing again integration constants as in eqs and

with ddp and t dtdp resp ectively we get

H H MG

L r

r r r

Pro ceeding as in sect we get the solution

Z

r

du

q

r r



L

r



u MGu



H

which is again a elliptic integral which should be expanded

Let us consider a lightray passing the sun The radial variable r originates at the sun

and measures the distance to the light ray At the point of closest approach r r one

has dr d r has its minimum so from we get

MG H

L

r r

DEFLECTION OF LIGHT

Thus

Z

r

du

q

r

u MGu u MGu

o

Z

r

u u du

q

MG

u u

u u

Using

u u u

u

u u u u

we get

Z

r

du MGu

q

r MGu

u

u u

u

v

s

u

u

u u uu



t

A

sin MGu

u u uu

Now r u go es from to r and then go es from r to The total change in is

o

jr j b ecause of the symmetry of the problem Half of the b ending takes place

b efore the passage of r r and the other half after the passage of this point If there

was no deection this would b e so the deection is jr j Hence

from we get

MG

r

For the sun we get with r the solar radius km

A recent observational value is in go o d agreement with The quantity

was rst observed in and the excellent agreement with brought Einsteins

general relativityinto the newspap ers However is in general very dicult to measure

with high precision The precession of p erihelia in sect is far more accurate b oth

from an exp erimental and a theoretical p ointof view

Instead of using the sun as the lense one can obtain much more sp ectacular eects by

use of galactic lenses We shall not discuss the details but the principle is quite simple

e consider a source which is p ositioned in suchaway that it lies on an straight Supp ose w

line passing through the center of a galaxy to an observer Just like in the case of the

sun light passing to the observer will b e b end by the galaxy Because of the very sp ecial

geometry there is cylindrical symmetry Thus if we take the sourcegalaxyobserver line

to b e the z axis the b ending will result in an eect which is rotationally invariant in the

x y plane Therefore the observer will see a circle where the maximal intensity is

around the galaxy Such a circle is called an Einstein Ring since it was rst predicted

by Einstein himself If the geometry is not so symmetric the ring degenerates to one or

e b een neatly observed more arcs These phenomena hav

GENERAL RELATIVITY

Black holes

As already mentioned the Schwarzschild solution lo oks singular for r MG In our

solar system this seems not to be relevant However it could be that somewhere in our

universe there exists an ob ject which is so small that r MG could be reached The

question is then what happ ens at the Schwarzschild radius

We know from Birkho s theorem shown in sect that outside a p ossibly time

us consider dep endent spherically symmetric ob ject Schwarzschilds solution is valid Let

radial motion where r and t are functions of prop er time Since we have from

and that H and

L

t

MG

r

MG

r L

r

Now let us consider a particle originating at large distance r where space is at

Minkowski and sp ecial relativity is valid In the asymptotic system let the clo ck be

chosen such that dtd for r Then L and hence

t

MG

r

MG

r

r

Now we have two times the prop er time is the one measured by an observer falling

freely in the gravitational eld and the time t is the time measured by an observer at

rest at large distances where space is at From the second equation we get since

p

p

r r MG r decreases with

p

r r

MG

with r r for The solution is valid as long as we are outside the b o dy

so if the radius of the b o dy is less than MGitfollows from that the freely falling

observer passes the Schwarzschild radius without observing any singularity If the mass

is just a p oint mass this observer reaches r in a nite prop er time

p

r

r

MG

Incidentally it should b e noticed that this conclusion could have b een reached without any

calculations since the second equation is the same as one would obtain in Newton

gravity why except that there prop er time is replaced by Newton time However

mathematics do es not care ab out the time variable it cares only ab out the dierential

equation so from everyday intuition Newton gravity we know that the point r

would b e reached in a nite time

Let us now consider this motion from the p oint of view of the observer at rest at large

distances from the massive ob ject From we have

s

MG r MG dr

dt r r t

BLACK HOLES

This dierential equation can b e solved exactly without any problem However it is more

instructive to consider the two cases where r is much larger than the Schwarzschild radius

or where r MG In the rst case wecanuseM Gr and hence b ecomes

approximately the same as the prop er time result Hence at large distances there is

approximately no dierence between the prop er time and the asymptotic time which is

of course quite reasonable For r MG the situation is quite dierent Here eq

gives to a go o d approximation

s

MG dr

r MG r MG

dt r r MG

with the solution

t



MG

r MG constant e

Thus we see that from the point of view of the asymptotic observer the Schwarzschild

radius is never reached it takes an innite time to get there

This feature can be put into p ersp ective if we think of the asymptotic observer as a

human b eing on this earth who observes a distant spherically symmetric b o dy by sending

a test body towards it and compare this with the situation as seen by an observer inside

the test body Seen from earth it takes an innite time for the test body to get to the

ens Schwarzschild radius and during this time the entire evolution of the universe happ

However the observer in the test bod y reaches the Schwarzschild radius in a nite time

on his watch and then he pro ceeds He thus sees what happ ens after t on our

clo ck

A body which collapses to a sphere with radius equal to the Schwarzschild radius is

called a black hole The reason for this is that no particles including photons can be

emitted from the surface Togive a full discussion of black holes one needs also to solv e

the Einstein equations inside the mass distribution which turns out to be a relatively

complicated task which we shall not attempt However the blackness is clear from

since if we reverse the time direction it follows that a particle emitted from the

surface r MG would require an innite time t to reach us For light d so radial

motion of light satises

MG dr

dt

MG

r

r

ie

MG dr

r MG

dt r

so the velo cityoflight seen from the asymptotic system approaches zero as r MG

Comparing with eq we see that if light should reach us from the Schwarzschild

radius an innite time would be needed so this can never happ en The body is black

Thus if a body shrinks to its Schwarzschild radius it disapp ears from our view Since

g is zero for r MG it follows from the discussion that there is

warzschild an innite red shift Therefore the shrinking body fades out of sight The Sch

radius is called the horizon of the black hole for obviuos reasons

Do black holes exist The stability of a star is determined by the balance between

the pull of gravity and the pressure from radiation emitted by nuclear fusion When the

light nuclei have been used up the fusion ceases and the gravitational pull may win

GENERAL RELATIVITY

This requires masses larger than the mass of our sun which will not end up as a black

hole In some cases it can happ en that the radius asymptotically approaches MG from

ab ove and a black hole can b e formed There is nowmuch observational evidence for the

existence of black holes For example one can observe a star whichmoves around a p oint

following Keplers laws However in the p oint nothing is observed This p oint is therefore

exp ected to b e a black hole Also Xray radiation from falling charged particles has b een

observed Quite recently it seems as if the existence of the horizon has b een directly

observed near the black hole candidate Cygnus XR One tracks the ultraviolet emission

from hot clumps of gas circling XR In two cases the signature of the emission dims

rapidly b efore disapp earing as it dips b elow the horizon The light dims as it is stretched

accordance with what wesawabove We refer to by gravitytoeverlonger wavelengts in

the literature for a discussion of these in teresting questions

We end by some historical remarks Black Holes were predicted from Newtonian

gravity by Laplace r in in a pap er entitled Pro of of the theorem that the

attractive force of a heavenly b o dy could b e so large that light could not ow out of it

The argument is quite simple Let us consider light as consisting of particles with mass

ultimately you can take the limit since the result is indep endentof In mo dern

terminology we would call the photon mass At the time when Laplace did his work

and there were the comp eting theories of light namely that it has a wave nature Fresnel

others or it has a particle nature Newton We know now that bo th pictures are right

in a duality sense Anyhow consider a spherical body with mass M and let us try to

sho ot a particle with mass away from this body The condition for escap e is that the

kinetic energy exceeds the gravitational energy so

v GMr

Thus after division by we see that the critical radius is given by

r GM v

crit

Here Laplace mentions that v is the velo city on the surface of the b o dy In general it can

dep end on r but if we assume that v c which we have taken to be one the critical

radius is just the Schwarzschild radius Thus Laplace correctly predicted the existence of

black holes which is a most remarkable historical fact

Kruskal co ordinates

The Schwarzschild radius app ears as a singularity in the metric To investigate whether

this is a real singularity or whether it has b een induced b y our choice of co ordi

nates we should compute some invariant quantity One obvious choice is given by

R R invariant The diligent reader can verify that for the Schwarzschild so

lution one gets R R M G r Thus this invariant is completely regular for

r MG and is only singular for r This singularityis physically reasonable since

int mass is situated at r the po

Motivated by the remarks ab oveonemay then ask if it is p ossible to nd new co ordi

nates without any singularity at r MG but singular for r in this connection it

should b e remembered that in general relativity the co ordinates are arbitrary These new

KRUSKAL COORDINATES

co ordinates were found by Kruskal in and we shall motivate them in the following

Since the angular co ordinates are not changed we shall consider only the quantity

MG dr

C B

ds dt

A

MG

r

r

dr MG dr

dt dt

MG MG

r

r r

It is natural to try to absorb the singular factor asso ciated with dr by a new variable

To this end we notice that

dr r

d r MG ln

MG

MG

r

where we have taken rMG We then have

MG r r

ds dt dr MGd ln dt dr MGd ln

r MG MG

This suggest that we intro duce new variables v by



t r r

ln v



MG MG

t r r

ln v

MG MG

From eq it follows that the tra jectory of lightmoving radially in a Schwarzschild

black hole is given by v or v The metric in the new variables b ecomes



MG

ds M G dv dv



r

This form of the metric still has a oneway membrane for r MG b ecause of the bracket

on the right hand side Toremove this factor by another change of co ordinates we notice

that from it can b e written in the two forms

tr tr

r

v  v

 

MG MG

e e e e

MG

Multiplying these two expressions together gives

r

r

v v 

 

MG

e e

MG

Thus b ecomes

r 

M G

v v 

 

MG 

ds dv dv e e



r

Thus the unwanted factor M Gr can be removed by the substitution



v





dv ds e

 

GENERAL RELATIVITY

which amounts to

v lns

 

with use

r s

tMG rMG

e s s e



MG s



The metric then b ecomes

rMG

ds M G e ds ds



r

which is singular only at r in agreement with the fact that the Riemann tensor is

singular only in this point In this metric r should be considered as a function of s



Usually this metric is written in terms of still other variables

s t r s r t



The metric then takes the nal form

rMG

d M G e dt dr r r t d sin d

r

where

r

rMG

t r e all r

MG

t t t t

tanh coth for rMG for rMG

r MG r MG

as is seen from eq Here we have left it as an excersise for the reader to do the

ab ove calculations for the case r MG It turns out that the nal result is the one

exhibited in Alternatively we can continue the expression analytically

from rMG to rMG

In the Kruskal form of the metric the variable r in front of the angular terms

is an implicit function of t r given by The same is true for the factor in front

of dt dr in

From we see that the lines of constant rt are formed by a set of hyp erb olas

intersected by the lines t r tanhtMG in the t r plane The singular p oint r

is mapp ed to the hyperb ola t r and the horizon is mapp ed to the degenerate

hyperb ola t r consisting of t wo straight lines It should b e noticed that has

two solutions for given t r corresp onding to the two branches of the hyperb olas These

two solutions are b ounded by the r hyp erb ola t r The solutions lead to the

same metric One class of solutions can b e identied with the standard asymptotically at

universe introduced originally in the discussion of the Schwarzschild solution The second

class of solutions corresp ond to the same metric and is a ghost universe These two

universes are related through the Kruskal variables and they are said to b e connected

by a Schwarzschild throat This connection is not physical since once a test body is

the horizon it will always hit the point mass inside

Let us consider a light ray travelling radially in the r co ordinate From we

see that it has the equation of motion jdr dt j corresp onding to a velo city of light

equal one If we study this in the old co ordinates rt we see from that the light

PAINLEVE COORDINATE

ray starts at some nite t with r MG then travels towards r MG corresp onding

to t and crosses the line t and go es into the interior of the horizon In this

continued motion r decreases and most remarkably t also decreases This means that t

is not a go o d time variable inside the horizon

If the ray is emitted from the inside of the horizon it will travel through increasing r s

but t is decreasing When the ray crosses the horizon t The motion measured

rev ersed in the asymptotic time t is thus time

Painleves version of the Schwarzschild metric

The Kruskal form of the Schwarzschild metric discussed in the previous section is

somewhat complicated Here we shall present a much simpler transformation of the

Schwarzschild co ordinates which has no singularityfor r MG It was originally found

by Painleve who used these co ordinates to criticize Einstein gravity for allowing

ordinates b eeing the singularities to come and go As we know this is a result of the co

arbitrary in general relativity so one ask for some scalar quantity x where a singularity

cannot be transformed away due to x x under x x Such a scalar was given

in the b eginning of the last section R R M G r This shows that

there is no singularityin r MG

To go back to Painleve we start from

MG dr

d dt r d

MG

r

r

where d represents the angular part of the Schwartzschild metric which is not changed

in the following A new time t is introduced by the equation

s

MG dr

dt dt

MG

r

r

The variable r is unchanged Integrating eq we obtain

p

p

p

r MG

p

MGr MG ln t t

p

r MG

where rMG For rMG a similar real expression is valid

Inserting eq in the metric we get after some simple algebra

s

MG MG

d dt dr dt dr r d

r r

Thus we see that the dr term in the new co ordinates is not asso ciated with a singularity

in accordance with the fact that r MG is not a singularityof the theory

We still have g for r GM so at the horizon light is innitely redshifted

Thus some charged matter falling towards the center and therefore emitting radiation will

b e observed to pro duce ever increasing wavelenghts and will ultimately b e unobservable

From eq or

s

MG r

t t

MG

r

r

GENERAL RELATIVITY

where a dot means derivative with resp ect to prop er time we obtain by use of eq

MG

t

MG MG

r

r r

This shows that for a freely falling test person the prop er time observed by him equiva

lence principle is equal to the Painleve time t plus a p ossible constant t const

Measured on a star clo ck t the observer will reach the horizon in a nite time

however if he communicates with us by means of radiation this will get out of range for

our detectors at large distance when r MG

Tidal forces and the Riemann tensor

So far we have discussed a point observer falling towards the black hole However if

the observer has an extension tidal forces will o ccur In this section we shall study this

eect in details Let us consider the separation between two nearby geo desics ie the

tra jectories of two freely falling observers

x and x x

These two co ordinates can be thought of as the head and feets resp ectively of a freely

falling observer assuming that this observer is quite elastic so that head and feets can

freely Let us consider the covariant derivativeofx where this derivativewas move

introduced in eq ie

D x d x dx

xx

D d d

This is a vector since the co ordinate dierence x is a vector We want to nd the

tidal force so inspired by Newtonean mechanics we take the second derivative of the

co ordinate dierence ie

D D dx D x d x

xx

D D D d d

To simplify life we go to a freely falling elevator where the Christoel symbols but not

their derivatives vanish in x so

D d x d x D x d dx dx dx

xx x

D D d d d d x d d

The two neighbouring geo desics head and feets satisfy

d x dx dx d x

x

d d d d

and

d x x dx x dx x

x x

d d d

x

x d x dx dx d

x O x

d d x d d

TIDAL FORCE FROM SCHWARZSCHILD

in the freely falling elevator in x Subtracting the two equations we get

x

dx dx d x

x

d x d d

Inserting this in eq we obtain

D x dx dx dx D dx

x R x

D D x x d d d d

Here we used the expression for the Riemann tensor in freely falling co ordinates However

eq is valid in any co ordinate system since the left hand side is a vector and the

pro duct of the last three factors on the right is a tensor Hence the rst factor on the

rightmust b e the Riemann tensor in any frame of reference

The result is thus that the tidal force on an extended observer ie the second deriva

tive of the dierence between head and feets is given by the Riemann tensor

We can actually demonstrate that the acceleration can be interpreted as the

fouracceleration of a particle moving along the geo desic x x as seen by an observer

the head of the feets seen from moving along the geo desic x so it is the acceleration

First wehave to realize that a freely falling observer will claim that a vector eld V

say which is dened along his geo desic path of motion is constant if it is carried into

itself by parallel transp ort ie if remember that the DD derivativeisjustthe dd

derivative in the freely falling system lo cated at x

DV

D

Thus in forming time derivatives shehe will compare the actual change V with the

change for a parallel transp orted vector and thus consider the derivative

V V

DV

parallel transp ort

d D

Now given the p osition vector x of the neighb ouring particle the distance to the feets

seen from the head he will dene the velo city vector the velo city with which the feets

move away

D x

V

D

and by use of the acceleration of feets seen from head is therefore precisely

DV D D x

a

D D D

The ab ove result is completely general In the next section we shall see what

happ ens to the tidal force for a Schwarzschild black hole

The Tidal force from the Schwarzschild solu

tion

Let us consider the metric given bytheSchwarzschild solution An observer is in free fall

towards the center following a geo desic t t andr r in the standard co ordinates

GENERAL RELATIVITY

At time heshe passes the p oint P t r The co ordinates in the freely falling

system around P are denoted t r The freely falling co ordinates should b e Lorentzian

according to the principle of equivalence Using that the metric transforms as a tensor

we have the conditions

r t

P g P P g P g P

tt rr t t

t t

t t r r

g P P P g P P P g P

t r tt rr

t r t r

and

t r

g P P g P P g P

r r tt rr

r r

The metric on the left hand side refers to the freely falling Lorentzian lo cal system and

the metric g is the Schwarzschild metric For later use we shall not use the explicit

form of this metric until the very end of this section Thus most of the following results

are valid for any spherically symmetric metric

To solve these equations we intro duce the quantities

q q

t t

P g P P g P

tt tt

t r

and

q q

r r

P g P P g P

rr rr

t r

From eqs these quantities should satisfy

These equations are precisely the dening equations of the Lorentz transformation see

almost any b o ok on sp ecial relativity or solve directly so we can solveby taking

v v

where

p

v

Here v is a parameter restricted by v

Next we can insert this solution for the s in eqs and to obtain

t v t

q q

P P

t r

g P g P

tt tt

r v r

q q

P P

t r

g P g P

rr rr

So far v is a free parameter and we now ask for the physical meaning of it The falling

observer is at rest in his own freely falling system so his actual tra jectory satises

dt dx

and for

d d

TIDAL FORCE FROM SCHWARZSCHILD

Transforming these conditions to the rt co ordinates we nd

t dt dt

q

d t d

g P

tt

and

dr dt r v

q

d t d

g P

rr

An observer at rest at the point P ie remaining at the value r r at all times

sees the freely falling observer move a distance use the dr MGrterm in the

Schwarzschild metric

q

dr

dl g P d v d

rr

d

where we used eq For similar reasons this takes the time

q

dT dt g P d

tt

So in other words this entirely xed observer sees the nearby falling observer his feets

pass him with a velocity dl dT v v Thus the parameter v can be interpreted

as the velocity of the freely falling observer as seen by an observer who remains xed at

r r So v is the velocityofthe feets as seen from the head

We shall now use the general formula to an observer falling radially in the

spherically symmetric metric In the observ ers freely falling co ordinates the acceleration

between head and feets is

dx dx

a R x

d d

This expression simplies considerably since obviously the observer is at rest in his freely

falling system so dx d for Also dx d is a unit vector and g

so it follows that dt d Therefore

a R x

tt

Now let us take the observer to be entirely in the radial direction with the distance h

between head and feet We then have

a h R

tr t

If we are only interested in the tidal acceleration between head and feet w e have r

r

and consequently we only need to evaluate R

tr t

To do this we rst use that the primed system is at freely falling elevator in P so

x x x x

r

R P R P R P

tr t

rtrt

r t r t

x x x x

Here we also used the transformation formula for the Riemann tensor From eqs

and the transformation co ecients can be evaluated

r r t t

r

R P R v R R

tr t tr tr tr tr tr tr

r t r t

GENERAL RELATIVITY

Now

r

R P g P R P

rtrt rr tr t

by use of the diagonal Schwarzschild metric The Riemann tensor can be evaluated by

use of the list of nonvanishing Christoel symb ols in Section

r r

tt tt

r r r t r r

R

tr t

tr t tt r tt tr rr

r r

r

Here we used that since F t Therefore we nd

tr

E E E E

r

R

tr t

r F r F r E r F r

It should b e noticed that this formula is valid in any spherically symmetric metric

If we use E F which is valid for the Schwarzschild metric we obtain

E

r

R E

tr t

r

Using this in and we get the result

E

r

R

tr t

r

Here we used that g E Also from eq we get the nal result

rr

h E

r

a

r

Now we can sp ecify to the Schwarzschild metric by taking E MGr so eq

gives

MGh

r

a

r

Thus in some arbitrary p oint r this is the acceleration that an observer will see hisher

feets moveaway assuming that the feets fall rst towards r It is of some interest that

this is also the result obtained in Newton gravity The acceleration of the head is MGr

and the feets have acceleration MGr h Therefore the relative acceleration is

M Ghr where we expanded in the assumed small quantity hr

The result is in units where c In standard units we should rewrite

as

M Ghc

r

a

r

It should be noticed that this acceleration do es not behave in a sp ecial way at the

Schwarzschild radius and the only singularity o ccurs at r as exp ected Note that

r

a do es not dep end on the relativevelo city v of head and feets

If we take M to b e the solar mass then if r is the Schwarzschild radius km

and if h m then

h m

a c msec msec

r m

Schwarzschild

Actually our sun will not end up as a black hole Instead the mass should b e somewhat larger p erhaps

a few solar masses but this do es not change the disasterous order of magnitude estimate essentially

THE EM TENSOR FOR ELECTROMAGNETISM

This is a rather disastous acceleration

However if we take one of the large black holes observed in the center of several

galaxies like the Milky Way with masses of order solar masses the acceleration

would only b e



a h msec

where h is measured in meters Thus large black holes are quite friendly

The energymomentum tensor for electromag

netism

In this section we shall construct the energymomentum tensor for electromagnetism as

int mass with a preliminary for nding the solution of the Einstein equations for a po

charge the ReissnerNordstrom solution We start by considering sp ecial relativity so

the metric tensor is just We assume that the reader is familiar from the course on

electromagnetism with the energy density

E B T

and the Poynting vector

i

T E B i

i

We have used that the Poynting vector is the momentum density and hence is to be

i

identied with T

We start by reminding the reader that the antisymmetric electromagnetic eld tensor

F can b e written

E E E

B C

E B B

B C

F B C

A

E B B

E B B

Let us notice that

i i ij

F F F F F F F F E B

i i ij

Then we can write

F F T E

Now we can write E in afancy way

E F F

and collecting results we thus have

T F F F F

Next consider the Poynting vector We have

T E B E B F F F F F F

GENERAL RELATIVITY

This obviously generalizes to

i i

T F F

We can combine eqs and to form the quantity

F F T F F

which is a tensor in sp ecial relativity In general relativity one therefore has from the

principle of covariance

T F F g g F F

We see that this energymomentum tensor is symmetric as it should be since it is the

source on the right hand side of the Einstein equations

In a domain where there is no current J thisenergymomentum tensor is conserved

in sp ecial relativity and in general relativity it is covariantly conserved If currents and

charges are present only the total energymomentum tensor is conserved The total T

is given by the expression ab ove plus the energymomentum tensor of the charge

current carrying particles We leave it to the reader to work out the details Problem

only need this tensor outside a pointcharge since we shal

The ReissnerNordstrom solution

In the case of the Schwarzschild solution we could take advantage of the fact that the

energymomentum tensor vanishes outside the spherically symmetric mass This simplies

the mathematics considerably since we could solve the empty space Einstein equations and

take into account the mass by suitable b oundary conditions at innity In the following

we shall solve a somewhat more complicated problem where we have a point mass with

mass M and a charge q Since the energymomentum tensor of electromagnetism involves

longrange forces we cannot use the empty space Einstein equations any longer The

relevant electromagnetic energymomentum tensor is given by eq and can be

the form written in

F F T F F

As in the case of the Schwarzschild solution the obvious spherical symmetry of the prob

lem allows us to write the metric in the form

d Artdt B rtdr r d sin d

ij

Since we have a static p oint charge there is no magnetic eld so F i j

The electric eld must be radial b ecause of the spherical symmetry and hence the only

r

non vanishing contravariant eld is F E E Similarly the only covariant eld is

r

r

e then easily obtain the F g g F g g F ArtB rtE From w

r r rr

energymomentum tensor

T for

r

T T ArtB rt E rt

r

T T ArtB rt E rt

THE REISSNERNORDSTROM SOLUTION

In particular we notice that the trace of the energymomentum tensor vanishes

T

Away from the point charge it follows from that

p

g F

x

Since g ArtB rtr sin from this leads to the two equations

q

r

ArtB rt F

x

and

q

r

ArtB rtF r

r

r r

Wetake the electric eld to b e timeindep endent ie F F r so from we

obtain

ArtB rt time indep endent f r

From we then obtain

const const

q

E

r

r f r

r ArtB rt

From the results transcrib ed to the metric by the replacements E A

t

and F B and from T or T we have

tr

r

Brt

rB rt t

so B rt is time indep endent From it then follows that the function Art is

also time indep endent ie

Art Ar and B rtB r

To pro ceed we consider the Einstein equations remember that the trace of the energy

momentum tensor vanishes see

t t r r

R GT and R GT

t t r r

t r t r

Using T T we therefore have R R From this means

t r t r

ln Ar Ar Br ln B r

t r

R R

t r

rAr B r r rB r r rB r r r

This assumption is not necessary The reader can show that the electric eld is timeindep endentby

doing the following calculations with an a priori time dep endent electric eld and show that the result

is the same as we shall obtain In analogy with Birkho s theorem one nds that it is only consistent

with the Einstein equations to have ArtAr f t but f t can b e absorb ed by a redenition of t

The electric eld is then constant in time

GENERAL RELATIVITY

This dierential equation obviously has the solution ln B r ln Ar const The

constant can b e determined from the b oundary condition that space is at at innity ie

Ar B r for r Therefore we have

B r Ar

as in the Schwarzschild case The electric eld in therefore simplies

q const

E

r

r r

Here weidentied the constant with the electric charge This is seen to corresp ond to the

standard denition of the charge by means of Gauss law

To nd the so far unknown function B r in the metric we need the Einstein equation

R GT GE In using this equation the actual long range character of

the energymomentum tensor will play a crucial role We have

rAr q

R G

r r r

where we used Equation leads to the simple dierential equation

rAr q

G

r r

with the solution

C q

Ar G

B r r r

Here C is an constant of integration which we determine by requiring that we get the

Schwarzschild solution for the case q Thus C MG and the metric

b ecomes

MG Gq dr

d dt r d sin d



Gq

MG

r r



r r

which is the ReissnerNordstrom solution

The metric displays the imp ortant feature that in Einsteins theory of gravity

an electric charge inuences the gravitational behavior A similar feature is not true in

Newtons gravity which is completely decoupled from electromagnetism It should be

observed that the gravitational eect of the charge decreases faster at large r than the

purely gravitational eect However for small r the situation is just the opp osite namely

that the electric contribution r dominates over pure gravity r with several

e shall see in the following interesting consequences as w

The solution has the feature that if the charge is large enough there is no

horizon In this case the singularity at r is called a naked singularity This feature

can be seen by observing that the quantity

MG Gq

Ar

r r

THE REISSNERNORDSTROM SOLUTION

in the metric go es to innity for r and approaches for r In between these

values the function Ar has a minimum Requiring that A r we nd the value of r

corresp onding to the minimum

q

r

min

M

In this p oint A has the value

GM

Ar

min

q

The minimum is p ositiveif

q

G

M

Thus if this condition is satised there is no horizon For a proton one has

q

GM

so a proton do es not have a horizon However this argument should not be taken to o

seriously since a proton cannot be descib ed by classical physics

If the condition is violated ie if the charge is not so large wehave a horizon

like in the Schwarzschild case This distance where the horizon is passed is given by the

equation Ar which has the solutions

s

q

A

r MG



M G

A freely falling observer seen from an observer at large distance passes out of view at

r r which is the horizon This follows b ecause the metric for photons d

in radial motion const and const leads to the velocity of light

MG Gq dr

for r r

dt r r

as seen by an observer far away from the charged mass For a material particle d

we get the equations of motion letting the clo cks run identically at innite distance

MG Gq

t

r r

and directly from the metric

MG r r Gq r r

 

r

r r r r

The last equation tells us how a freely falling observer using prop er time sees the radial

fall Like in the Schwarzschild case nothing sp ecial happ ens in passing the horizon r

However the rest of the fall is very dierent because at the distance

q r r



r

c

M r r



GENERAL RELATIVITY

the velo city r vanishes This means that the electric contribution to the metric overwhelms

the pure gravity contribution and the combined eect app ears repulsive Thus although

gravity is attractive outside the horizon we see that this do es not hold inside Therefore

it is not always true that gravity is attractive

This distance is below r and r as is easily seen from the last form of eq



This means that an observer can pass through the original asymptotically at

region rMG to the region r r r and further into the region r r r where

  c

the motion comes to rest The singularityat r is therefore never reached

ens near r let us nd the acceleration b y dierentiating To study what happ

c

with resp ect to the prop er time On b oth sides we get a factor r and disregarding the

pointwhere r is exactly equal to r we get by dividing by r

c

r r r

c 

r

r r

We see that at large distances the acceleration is negative as exp ected in a free fall

Coming from large distances we see that r increases until r r Then r decreases

c

and r for r r Then r actually b ecomes p ositive corresp onding to acceleration in

c

the direction away from the singularity at r When r approaches r corresp onding

c

p ossibilty to r the acceleration has the p ositivevalue r r r r The exciting



c

therefore exists that the observer can reemerge into a region with r r However

since this observer has passed out of view the rst time she went through r r and

this to ok an innite co ordinate time t the region with r r into which she reemerges

is an asymptotically at region which is dierent from the asymptotically at region

in which this observer started the fall This observer has passed through a wormhole

pro duced b y the charge Alternatively two observers living in dierent asymptotically

at universes can fall freely and meet at r They can compare their informations but

c

they cannot communicate back to their original universes since they both went out of

sight for observers situated in the original universes Unfortunately there is apparently

no way in which an observer can travel from the original asymptotically at universe to

another one and then back to the original universe to inform ab out what was seen on the

other side of the w ormhole

It should be mentioned that the ReissnerNordstrom metric actually has an innite

number of asymptotically at spaces with the same metric This can be seen from the

following gedankenexperiment First let the freely falling observer fall from some distance

outside the horizon In a nite prop er time she will reach r r and then move out

c

into another asymptotically at space As mentioned b efore this space must b e dierent

from the original one since she passed out of view after an innite co ordinate time t

likeinthe Schwarzschild case as seen from the observers living in the rst asymptotically

at space In the new space her velo city decreases as she mo ves outward in r and by

means of a suitable ro cket she can turn around and fall again Rep eating this pro cedure

in sucient nite prop er time she will travel to an arbitrarily large numb er of dierent

asymptotically at universes Thus there exists an innite number of such universes

For the ReissnerNordstrom solution co ordinates analogous to the Kruskal co ordinates

have b een found These are more complicated due to the existence of the two horizons

r and two sets of co ordinates are actually needed



We end the discussion of the charged metric by calculating the tidal force in the

ReissnerNordstrom solution The rst change we have to make is notational replacing

THE SPHERICALLY SYMMETRIC SOLUTION IN DIMENSIONS

E and F by A and B Using that A B like in Schwarzschild we can use equation

which now reads

MGh h A r

c

r

a

r r r

Here we inserted the explicit form for Ar and introduced the critical distance given in

eq We should note that this result has no analogy in Newton gravity which is

uninuenced by the charge

The result diers from the corresp onding Schwarzschild result by not

b eing positive denite For r r the acceleration vanishes and for r b elow r it

c c

b ecomes negative so the feet will now accelerate towards the head Presumably the

ecome somewhat seasick The fall will con tinue from r to r where observer will b

c c

the motion stops as mentioned b efore In r the negative acceleration acieves the value

c

M Ghr

c

The spherically symmetric solution in di

mensions

Einsteins eld equations lo ok the same way in all dimensions It is therefore of interest

to show that the solutions can dier remarkably by going to other dimensions than

Consequently let us lo ok for a solution analogous to the Schwarzschild solution in two

space and one time dimension It turns out that this result is quite interesting from the

ntof view of principles and is also quite dierent from the Schwarzschild solution so i po

forthisreason we shall give the derivation We start from the metric

p

 

r x y

d dt e dr r d dt e dx dy

Here we used the notation r x y For a static metric the most general rotationally

invariant dimensional metric is Ar dr B r dr r d as can b e seen as in section

q

R

AB dr

one obtains the spatial part of Intro ducing the new variable r exp

r B

Using the EulerLagrange variational principle we easily obtain

d d d

t e x x y e e y x y e

d d x d y

The last equation follows by symmetry from the second by observing that the metric

is symmetric in x and y Hence all results derived from this metric must be

symmetric under interchange of x and y The Christoel symb ols can then b e easily read

o by p erforming the dierentiations For example the second equation leads to

x x y x y

x x y

In this way we obtain

x x t x

yy xy tt xx

x x y

GENERAL RELATIVITY

The remaining symb ols followby the symmetry between x and y

y y y

xx xy yy

y y x

Also for the logarithm of the metric tensor we have

q

log g x y r

We are now in the p osition to compute the Ricci tensor R for whichwehave the general

formula

log g log g

R

x x x x

We then obtain

R R i x y

tt ti

Also

xx

R

xx

x x xx

x x x

x y

xx xx

x y x y

xx x xy x xx xx

x x y x y

q

R x y

yy

x y

where the last equation follows by symmetry ie without doing any calculations After

afew calculations we also nd

x y

xy xy

x y

R

xy

x y xy xy

xy x y x y

From these equations it now follows that

p

q

 

x y xx yy 

r x y R R g R g R e

tt xx yy

where the op erator multiplying exp in the last equation is the at Laplacian acting

on an r dep endent function

Let us consider a point mass M at rest in the point r The corresp onding energy

density is given by

p

M x g

Here x x y is the two dimensional delta function Thus is valid

r

only in the x y co ordinates where g e The reason for the identication in eq

is that

Z

p

d x g M

The Einstein equation

R g R G T

p

The reason for the g in is that the density should be a scalar The delta function

R

  

satises d xF x x F for some arbitrary function F x However d x is not invariant but

p p

 

d x g is Therefore x g is invariant A similar result is valid in higher dimensions

THE SPHERICALLY SYMMETRIC SOLUTION IN DIMENSIONS

with T and all other comp onents T therefore gives by contraction

R G T

therefore reduces to use that in the x y co ordinates g e

r r GM x

It is easy to solve eq if we use that

r log r x

which shows that log r is the Greens function for the two dimensional Laplace op erator

log r trivially satises eq Eq can be shown by noticing that for r

In the neighbo urhood of r we have

Z I I Z

r

d x r log r dn rlog r dn d

r

as claimed in eq The Einstein equation now has the solution

r GM log r

The metric thus b ecomes

GM

d dt r dr r d

Here r o ccuring in the power is measured in some arbitrary units

The surprising feature concerning the metric is that it can be transformed to

a simpler form lo oking like at space We can intro duce anew radial variable r by

GM GM

dr r dr or r r GM

By this substitution w e get

d dt dr GM r d dt dr r d

where we intro duced the rescaled angle by

GM

y It is imp ortant to notice that although this metric lo oks at the angle is restricted b

GM

since the original standard angle is of course restricted to b e between and

Due to the form of the metric it is clear that in any point t r there are

no p ointwise eects of gravity Thus there are no lo cal gravitational elds

However due to the fact that the variation of the angle is restricted to b e less than

there are global gravitational eects This is illustrated in gure with GM

The two points P and P are in the same distance from the origin In between these

there is a forbidden region A space with a forbidden region of ints for po

GENERAL RELATIVITY

Figure

this typ e is called conical The two points are therefore the same point So an observer

can be in P and P in the following this point is called PP The same applies to

a source of light In the conical co ordinates light follows a straight line Thus if light

originates in P P it can move along two straight lines which crosses one another in

a single p oint There then app ears a double picture of the source in PP In the r

co ordinates a lightray coming from far away and going far away in the opp osite direction

corresp onds to a change in polar angle but the physical angle in the original

standard co ordinates is just

GM

We reemphasize that this solution is valid in two space and one time dimensions Thus

it is of course not directly p ossible to do an exp eriment testing the strange b ehaviour

found ab ove However in the next section we shall see that conical spaces can exist even

in our threeone dimensional space

Cosmic strings

Let us now consider dimensions where we take the metric to b e

r

d dt dz e dx dy with r x y

The x y part of the metric is thus the same as the metric studied in the previous

section The metric tensor g having one or both indices equal to z is thus trivial The

calculation of the Ricci tensor R therefore pro ceeds like in the previous section Using

and we get for the tensor

G R g R

that

G R g R R

tt tt tt

COSMIC STRINGS

Also it is easily shown from the results and that

G G G R

xx yy zz

while all other comp onents of G vanish From we see that

 r

R e r r with r x y

The conclusion is thus that the tensor G vanishes except for

G G R

zz tt

The metric is thus consistent with an energymomentum tensor of the form

T T T for all other and

tt zz

Furthermore we take the nonvanishing comp onents of T to be prop ortional to a two

dimensional delta function

T T x

p

tt zz

g

The constant has dimension energylength and is called the string tension The ob ject

describ ed by the energy momentum tensor and is called a cosmic string

It is a line along the z axis with supp ort only in the point x y The pressure

is negative since T by denition is p ositive We shall later see that it is p ossible to

tt

construct such an ob ject in certain eld theories In this case the cosmic string is a

defect i e an ob ject which contains false vacuum giving rize to the negative pressure

Pro ceeding as in the previous section we can now solve the Einstein equations

r G log r

The metric can no w b e written in the two co ordinate systems

G

d dt dz r dr r d dt dz dr r d

where r and are dened by

G

r r G G

Wethus see that again there app ears a conical space where the angle has a range which

is less than

The conical space leads to the o ccurence of double pictures of a given light source

placed b ehind the cosmic string seen from an observer in front of the string This is

shown in gures and

In g we see the situation from the p ointofview of the rco ordinates so the

lightrays emitted from the source Q eg a quasar b end towards the observer O In g

we see the situation from the point of view of the conical co ordinates r where

we get lightmoves in straightlines From g

tan l tanad l where a G

GENERAL RELATIVITY

r



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 

 





 

 

 



 

 

 



 

 

 





 

 



 

 

 

  

   

   

  

   

   

  

  

   

  

   

   

  

  

   

  

   

  

  

   

  

   

   

 

   

  

   

   

 

   

  

   

   

  

  

  

   

   

  

  

  

   

   

  

  

  

   

  

   

   

 

   

  

   

  

  

  

  

   

 

   

  

  

   

 

   

 

   

  

  

   

 

   

  

  

  

   

 

   

 

   

   

 

   

 

   

 

  

  

   

  

   



   

  

   

  

  

   

  

   

 

  

  

   



   

  

   

 

   

 

   



   

  

   



   

 

   

  

  

 

   

 

  

  

   

 

  

 

  

 

 



  

 

 

 

 

 

  



 

  

 

 



  

 

 



  

  



 

 

  



 

  



  



  



 

 

 

  



  



 

 

 

 

 

 



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Figure

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  Q

 u

 



















































































































































































































Figure

Here is the angular separation b etween the two pictures and d is the distance from the

observer to the string and l is the distance from the string to the source Qwhich app ears

to be in two points in the conical co ordinates but these points are to be identied In

practice the quantity G is small relativetoone and hence we can expand eq

Gld l

Atypical cosmic string is generated in the Grand Unication characterized by having

the strengths of weak electromagnetic and strong interactions equal which has an

energy scale of order GeV leading to GeV The string tension then has

the value

g cm tonsm elephantsm



in ordinary units From this leads to an angular separation of order

ie aproximately arc seconds This is clearly within the p ossibilty for observation

A cosmic string is of course an extended ob ject and hence should be observed by a

concentration of many double pictures along the string which itself cannot be observed

directly Also the cosmic string do es not have to b e a straight line but can have the form

of a curve moving around in spacetime So far there is no unambiguous observational

evidence for cosmic strings but there exist some observations of unusual concentration of

double pictures along rather crumbled curves These pictures are candidates for cosmic strings

Chapter

Cosmology

The cosmological problem

Sp eculations ab out the nature of the Universe form an integral part of the history of

mankind Various mythologies and religions have several scenarios to oer In mo dern

physics the canonical scenario is based on astronomical observations describ ed in terms of

Einsteins general relativity In physics it is always imp ortant to x a relevant scale If we

wish to describ e the solar system there is no need to b other ab out the prop erties of the

Universe at intergalactic scales If on the other hand we wish to understand the large

scale structure of the Universe we do not need to b e concerned with small details like our

solar system Then we can consider even the galaxies to be p oint particles No w it

turns out that galaxies tend to cluster so we have also clusters of galaxies which on

avery large scale may b e considered to b e particles There are also clusters of clusters of

galaxies sup erclusters which when viewed at a large scale are also particles This

amount to building up the Universe from particles with diameter lightyears

This situation is somewhat similar to considering a gas consisting of a large number of

particles where we do not keep track of the individual particles but rather consider the

large scale statistical prop erties It therefore also follows that in the large scale view of

the universe we cannot assign any center Every poi nt is equivalenttoany other p oint

in the universe This means a radical deviation from the medieval idea that the earth is

the center of the universe One can say that since it turned out that we do not live in the

center of the universe we then assume that nob o dy else has this honor

The idea that all p oints in our large scale universe are equivalent is called the cos

mological principle It means that the universe is assumed to be homogeneous and

isotropic b ecause for us and hence for everyb o dy else the universe app ears to be ap

proximately isotropic around any poi nt

One of the rst prop onents of this picture of the universe was Giordano Bruno

who went beyond the Cop ernican helio centric theory which still maintained a

universe with a sphere of xed stars Instead Bruno made the suggestion that the universe

is innite and the stars are solar systems likeourown



With masses M



  

With masses M



  

With masses M





When Bruno refused to retract his views he was sentenced to death He addressed his judges saying

Perhaps your fear in passing judgement on me is greater than mine in receiving it

COSMOLOGY



Figure Gaussian co ordinates for a comoving system S and S are three dimensional

spaces hypersurfaces

The extent to which the cosmological principle is right is not precisely known In

principle one could measure the energymomentum tensor T x in all p oints of the

universe and then solve the Einstein equations However astronomy do es certainly not

provide data enough for obtaining T x Therefore the cosmological principle should

b e considered as a rst aid for interpreting the observations rather than as providing an

extremely precise description of the universe

The cosmological standard mo del

del not to be confused with the standard mo del in The cosmological standard mo

particle physics is based on the cosmological principle ie a homogeneous and isotropic

universe as well as the assumed existence of a cosmic standard time This time is

intimately connected to the evolution of the universe For example the temp erature is

b elieved to decrease as the universe evolves and the cosmic standard time could then b e

taken as a function of the temp erature which increases monotonically as the temp erature

decreases Instead of temp erature one could use other scalars the main p oint b eing that

the cosmic time t increases when the universe evolves The existence of t is taken to mean

that at a given xed value of t the matter of the universe is at rest in three dimensional

space just like in the formulation of the cosmological principle this means in reality that

on the a verage the matter of the universe is at rest in the three dimensional space As

time t increases the matter of the universe follows this increasing t This co ordinate

system is therefore called the comoving system

We can describ e this comoving system by means of socalled Gaussian co ordinates

where we consider a geo desic along which time increases see g

The co ordinates x x x remain constant along any geo desic p erp endicular to the

initial surface S Along such a geo desic wehave d dt and hence g In order

o

lines are orthogonal to the dimensional hypersurface it is necessary that any that the t

four vector abc is orthogonal to the vector which is tangent to the tline

so in general we need g g g on the hyp ersurface The line elementthus has

THE STANDARD MODEL

the form

i j

d dt g dx dx i j

ij

i

x S

Consider the equation for a geo desic

dx d x dx

d d d

When d dt and the co ordinates x x x are constants along the geo desic we deduce

i

that for i This implies from that

g g

i

i

t x

and since g on any geo desic we obtain

g

i

t

along the geo desic tlines Thus the elements g g g of the metric tensor which are

zero on the initial surface S remain zero on all other surfaces Thus the metric for a

comoving system has the form

i j

d dt g t x x x dx dx

ij

over all space Eq is thus a consequence of the assumed existence of a cosmic

standard time

The picture of the development of the universe presented ab oveisthus that at a given

cosmic time t the galaxies clusters of galaxies etc are on the average at rest Thus

we can construct the threedimensional space co ordinates by putting numb ers x y z on

each galaxy As time increases the co ordinates remain the same but all distances can be

scaled with a timedep endent scale factor

Now let us consider an arbitrary but xed point in space around which we take

spherical co ordinates r The requirement of isotropy in this point means that the

line element has the form

d dt f t r dr f t r r d sin d

since the only rotational invariantatthispoint are t rdr and dr r d sin d

The functions f and f only dep end on t and r since a andor dep endence of f

and f would destroy rotational invariance Since we are using comoving co ordinates eq

implies that there cannot be a dr dtterm in eq even though it is rotational

invariant Thus eq is the most general form of the line element

es by the only thing that happ ens is that distances Nowwemust ensure that as time go

are scaled Thus the galaxies are dragging along the co ordinate mesh so each galaxy

keeps the same co ordinates The galaxies do move but the distance b etween the galaxies

move to o so the co ordinates remain the same This means that in rand remain

the same and the only thing which can happ en is that the overall scale changes If at

a given but arbitrary time we consider two points r and r with d

COSMOLOGY

d at b oth p oints then the ratio of the line elements f t r dr and f t r dr must

remain the same at all times since only the overall scale can change ie f t r f t r

is indep endent of time and equal some function F r r ie

f t r f t r F r r

Taking r to b e xed this implies

f t r f t Lr

ie f factorizes The same can be shown for the function f take eg dr d and

d at the two p oints Thus we are led to the line element

d dt f t Lr dr g t H r r d sin d

This line element clearly has the prop erty that the co ordinates remain the same and the

q

only thing which happ ens is that lengths are scaled Intro ducing r H r r we get the

slightly simpler form for the line element

d dt f t Lr dr g t r d sin d

where we have dropp ed the prime on r

Eq now represents a line element which is consistent with the standard cos

mological mo del It should be remembered however that in eq we selected an

arbitrary but xed point as the origin of the spherical co ordinates Later on w e have to

ensure that this choice of an arbitrary origin agrees with the assumption of a homogeneous

space

Wecannow compute the Christoel symb ols using the variational principle discussed

in section

Z

dt d dr d

A

d f tLr g tr sin

d d d d

After straightforward calculations we get

fL

t t t

r g r sin g

rr

rg rg

L

r r r

sin

rr

L fL fL

f

r r

rt tr

f

g

t t r r

r g

sin cos

g

t t

r r

r g

cot

THE STANDARD MODEL

The Ricci tensor can now be obtained from

ln g ln g

R

x x x x

For reasons which shall so on b e clear we start by computing R

tr

g f

R

tr

rg rf

where the dot indicates the time derivative The Einstein equations read

R GT g T g

The energymomentum tensor has the form of a p erfect uid see sect G since this

means an isotropic and homogeneous universe if the pressure p and the density are only

functions of tie

T pt g pttU U

where U is given in comoving co ordinates by

t i

U U

The last equation follows b ecause matter is at rest Thus R must vanish since T and

tr tr

g both vanish Hence from

tr

g f

g f

so we get g f up to a constant which can always b e absorb ed by trivial redenitions

The line elementisthus from

h i

d dt f t Lr dr r d sin d

r

Inserting f g the Christoel symbols simplify somewhat for example b ecomes

time indep endent etc and the Ricci tensor is more easily evaluated

L Lf Lf

R

rr

rL f

r rL r f

f R

L L f

R sin R

f f

R

tt

f f

R for

where prime denotes the derivative with resp ect to r Since the metric is diagonal we

can easily construct the mixed Ricci tensor

L f f

r

R R

rr r

fL rf L f f

COSMOLOGY

L f f

R R

r f r f r fL rf L f f

R R R

r f sin

f f

t

R

t

f f

R for

The curvature scalar is then trivially obtained

f L

R

f r f L rf L

r

Now isotropy clearly requires that R R R since if this is not the case there

r

would b e one or more preferred direction This also follows from the energymomentum

r

tensor of course Consulting eqs we see that R R is a nontrivial

r

requirement

is homogeneous in space so it must not dep end on We also need to ensure that R

any other variable but time Again consulting we see that in general this is not

true but has to be imp osed b y hand This is due to the fact that originally we imp osed

r

isotropy relativeto a denite p oint We can repair this by requiring see R

r

L

k

rL

where k is a constant and see R

L

k

r r L rL

r

The constants on the righthand side have to be the same because of R R Eq

r

is trivially solved

L

c kr

where c is a constant Inserting this in eq we get

c

k k

r

ie c Thus we have found the explicit form of the spatial dep endence of the line

element ie

dr

d dt at r d sin d

kr

where we have replaced f t by at for reasons which will b e clear later

From eq we see that the space part of the Ricci tensor

k k

ki ij

k l k l

R

li kj ij kl ij

j k

x x

i ik

R g R

j kj

A GEOMETRIC INTERPRETATION

r k

indicates is given by L r a L ka for R Here the in front of R and

ij

lm r

that these quantities are computed in dimensions time xed Thus the Ricci scalar

in dimensions is

k

R

at

The Gaussian curvature K in dimensions is given by R K so

k

K

at

This means that at is the radius of curvature This suggests that the metric

which is called the Rob ertsonWalker metric has a geometric interpretation which we

shall discuss in the next section

A geometric interpretation of the Rob ertson

Walker metric

The metric in comoving co ordinates has a trivial time part whereas the space part is

nontrivial It is of interest that the threedimensional part of the metric has a

simple geometric interpretation Let us consider threespace as emb edded in a ctitious

four dimensional space where the fourth co ordinate is called z The metric is assumed

to b e just Euclidean so the elementoflength ds is given by Pythagoras theorem

ds dx dz

Let us consider ahyp ersphere

x z R

where R is the radius Dierentiating we get

dz z dz dx

so we can eliminate dz from

dx

ds dx

z

In this relation z can be eliminated by use of

dx

ds dx

R x

Intro ducing p olar co ordinates we can write this as

r dr

ds dr r d sin d

R r



Readers who are not familiar with dierential geometry in three dimensions can skip these remarks

since in the next section we shall derive the result from simple considerations

COSMOLOGY

ie

R dr

r d sin d ds

R r

If we measure the distance r relativetothe radius R we obtain with rR

d

ds R d sin d

If for a xed time we put at R then we see that the metric is precisely the

space part of with k

The curvature of the hypersphere is always R large radius means small curvature

and vice versa in agreementwithintuition Thus at a xed time this agrees with

The interpretation of the Rob ertsonWalker metric given ab ove has a particularly nice

interpretation in terms of the framework of comoving co ordinates One can picture co

moving co ordinates the following way Let us imagine that space is lled with a dense

c k and cloud ideal uid of freely falling particles Each particle carries a small clo

is given a xed set of spatial co ordinates which are determined at a certain time say

t The spacetime co ordinate t x of any event are dened by taking x to be the

co ordinate of the particle which is just passing by where the event o ccur and t as the

ordinate mesh time read o from the particles clo ck Alternatively one can say that a co

is b eing dragged along by the cloud of particles with time dened by clo cks xed on

the mesh The existence of a cosmic standard time which is assumed in the standard

mo del means that we can choose t in suchaway that all particles are at rest at that

time It then follows as in that this remains true at all times

In cosmology we can use the galaxies as the points dragging along the co ordinate

mesh The comoving tra jectories are then just the paths of the galaxies If we do that

the threedimensional pro jection of the hypersphere can then be regarded as a

ballo on where at a given time the galaxies are marked by dots on the threesurface As

the ballo on is inated the dots move but the distance b etween the dots will movetooso

each dot keeps the same co ordinates

In this picture one can say that R is the radius of the universe since from it is

p

clear that x cannot exceed R Thus in eq is always less than one

If k in is negative we cannot use the hypersphere Instead we can consider

the surface

x z R ds dx dz

which is called a pseudohypersphere In this case space is obviously unlimited and

genuinely nonEuclidean GaussBolyaiLobachevski geometry Pro ceeding as b efore

one obtains the Rob ertsonWalker metric with k In this case R do es not

have the interpretation as the radius of the universe but it still sets the scale and is

therefore called the cosmic scale factor The curvature is everywhere negative and is

given by R

x again with a d The case k corresp onds to a completely at space with ds

timedep endent scale factor at

HUBBLES LAW

Hubbles law

We shall nowinvestigate the frequency of light emitted by some distant ob ject in a universe

describ ed by the Rob ertsonWalker metric Since lighthas d we get for radial

motion from

dr

dt at

kr

From this we get for a light wave leaving a galaxy at r r at time t t and arriving

at our galaxy r at time t t



sin r k

Z Z

t r



dt dr

p

r k

at

t

kr





sinh r k

In comoving co ordinates galaxies have constant co ordinates r only the scale

changes Thus r is time indep endent We wish to relate to the frequency of

light so let the p erio ds be t and t as measured by us and on the emitting galaxy

resp ectively Thus

Z Z

t t t

dt dt

at at

t t t

  



so using that t is very small sec for a typical lightsignalwe get

t t

at at



where we used the rather safe approximation at sec at ie the universe



evolves very little in sec Eq then gives for the frequencies

t at

t at

This shows that the wave length shifts like the scale factor The wave length is stretched

or shortened at the same rate as the universe expands or contracts

Conventionally this relations is expressed in terms of the redshift parameter z

at at at

z

at at

where is the wave length and where we used

o

From eq we see that since is the light observed by us after a long travel

o

from the galaxy at r r then if z ie red shift or if z blue shift

it follows that at at red shift or at at blue shift Obviously t t in

e observe the light later than it was emitted Hence wehave the result b oth cases since w

that an expanding contracting universe is characterized by a red shift blue

shift of the sp ectral lines For increasing t t we exp ect that at will increase

decrease more and more and hence z will increase decrease more and more so the

larger smaller the value of z is the farther is the ob ject

COSMOLOGY

Supp ose that the emitting galaxy is relatively close to us We can then expand eq

and obtain to lowest order

a t

t t z

at

where t t is suciently small Now in the rst approximation t t is the distance

measured by an astronomer remember the velo cityoflight is one as is clear intuitively

as well as from eq whichtolowest order implies that t t at r and for

the metric we know that at r is the physical distance for r small relative to

the scale parameter Thus can be written

a t

z H L H

at

where L is the distance measured by the astronomer If L is not small one needs to go

back to the relation

Let us notice that for r small it follows from the Rob ertsonWalker line element that

a t r is approximately the radial velocity of the galaxy and since t t at r

r

eq can be written

z v H L

r

This maybe interpreted as a frequency shift due to the Doppler eect Such an interpre

tation should however not be taken to o seriously since light is certainly inuenced by

ation of eq shows the gravitational elds as the deriv

The relation is the famous Hubble law Observationally it is found

that there is a red shift which increases with the distance of the galaxy The Hubble

constant H actually dep ends on time Due to new observations it is b ecoming b etter

known

years

H

Hubbles discovery of a roughly linear relation between velocities and distances had a

profound eect on cosmology and in a sense it marks the b eginning of mo dern cosmology

A very short biography of Edwin Hubble

Hubble had an undergraduate degree in astronomy and mathematics However he

then to ok a legal degree and practized as an attorney in Kentucky where he joined its

bar in He served in the rst world war where he got the rank of ma jor He was

ed with law and went back to his studies in astronomy In he began to work at bor

Mt Wilson Observatory in California where he would work for the rest of his life He was

researching nebulae fuzzy patches of lightinthe sky In he announced the discovery

of a Cepheid or variable star in the Andromeda Nebulae Since the work of Henrietta

Leavitt had made it p ossible to calculate the distance to Cepheids he calculated that this

Cepheid was much further away than anyone had thought and that therefore the nebulae

was not a gaseous cloud inside our galaxy like so many nebulae but in fact a galaxy

elieved of stars just like the Milky Way Only much further away Until now p eople b

that the only thing existing ouside the Milky Way were the Magellanic Clouds The



Most of this material is taken from wwwedwinhubblecom Check also hubble biography in go ogle

where much more material can b e found

HIGHER ORDER HUBBLE LAW

universe was much bigger than had been previously presumed By observing redshifts

in the wavelengths of light emitted by the galaxies he saw that galaxies were moving

away from each other at a rate constant to the distance between them Hubbles Law

The further away they were the faster they receded This led to the calculation of the

point where the expansion b egan and supp orted the big bang theory When Einstein

w as static and he introduced prop osed his theory of gravity he thought that the universe

the cosmological constant to accomo date this Following Hubbles discoveries Einstein is

quoted as having said that second guessing his original ndings ie the eld equations

without naturally leading to a nonstatic universe was the biggest blunder of his life

He visited Hubble to thank him in

Higher order correction to Hubbles law

In recent years astronomical observations have b ecome very precise and the next order

term in the Hubble law b ecomes imp ortant as we shall discuss later To obtain this

expansion we should return to exact eq So let us start by the second order Taylor

expansion



H a t

tt at a t t t a t t t q tt

at at at at

where we expanded around t and where

a t at a t

q H

at a t

The quantity q is called the deceleration parameter Going back to the exact equation

we therefore see that it can be expanded as

at

z H t H q t t t t

at

To lowest order t is the distance between the source and the observer However the

universe is not static so to b e more accurate we should takeinto account the metric of the

universe Since lighthas d we have for radial motion that the comoving distance is

Z

t

dt

at

t



This is the constant distance which ts with the emission and observation times t and

t However the physical distance is the comoving distance multiplied by the scale factor

ie

Z

t

dt

L at

at

t



By use of the expansion we see that

L t H t O t

COSMOLOGY

The correction is consistent with intuition If the universe expands H then the

physical distance is also expanded relative to the naive distance t Similarly if the

universe contracts The distance then b ecomes shorter than naively exp ected

We can now reexpress z given by in terms of L instead of t

L H q z H L

Later we shall see that the second term in this expression has played an imp ortant role

in mo dern cosmology

Einsteins equations and the Rob ertsonWalker

metric

So far the scale factor at has not b een determined as a function of cosmic time To

do that we shall return to the Einstein equations following from eqs and

From by use of the comoving four velocity we get

t

T T p

t

r

T T p

r

with the and comp onents given by the same expression as Using the results

or for the function Lr we then get f t at

a

t

R

t

a

a a k

r

R R R

r

a a a

so the Einstein equations read

a G pa

aa a k G pa

In these equations we did not include the cosmological constant In mo dern cosmology

this is unreasonable since pla ys an imp ortant role so we shall nowshow that there is a

simple way to include the cosmological constant in the pressure and density To see this

write the Einstein equation with the cosmological constant in the form

R R GT GT

where the new energymomentum tensor is given by

pt ptt U U T T

G

with

tt ptpt

G G

EINSTEINS EQUATIONS

Thus wesee that the cosmological constant can be included in the Einstein equations just

by changing the meaning of the energy density and the pressure in accordance with eq

For we see that the energy is always p ositive whereas the pressure can

be negative In particular if the cosmological constant dominates the pressure b ecomes

negative The physical meaning of is related to vacuum since do es not refer to

any state of matter Thus it is an inevitable consequence of having a vacuum energy

momentum tensor T vac g G note that this is the only tensor that can

es not contain anything except be formed in a vacuum since by denition this state do

virtualit y that the pressure is negative if the cosmological constant is p ositive which is

what is observed Here we should remember that g

In the following we shall leave out the tilde in the density and pressure The eect of

constant can then always be obtained in a simple way from eq the cosmological

Later we discuss the imp ortant eects of the cosmological constant and we shall then

return to the substitution

In addition to the Einstein equation wealsohave to ensure that the energymomentum

tensor is covariantly conserved Using eq this means

p p

T g g pU U p U U

p

x g x

In comoving co ordinates we have from that Eq is then trivial for

r remember that due to the homogeneous space p and can only dep end on

time not on space and we have

dp d

p

g p

p

dt g dt

The determinantforthe Rob ertsonWalker metric is given by

r sin

g at

kr

In eq only the time dep endence of g is relevant and we get

h i

dpt d

at at ptt

dt dt

Eq can also b e written

d

a pa

da

If an equation of state p p is given eq can be solved for a and the

evolution can then be computed from and For example if the pressure is

negligible p then



at p

If one has extreme relativistic particles like photons the universe is radiation dominated

and with p



at p

COSMOLOGY

We notice that the Einstein equations and can be simplied by eliminating

a Thus inserting in we get

G

a a k

which is called the Friedmann equation

The functional equations in the standard mo del are thus the equation of state p p

the energy conservation and the Friedmann equation The last two equations

are easy to remember the quantity a is prop ortional to the volume of the universe

for k and M a is prop ortional to the mass of the universe Eq can

then be written multiply by da dM pdV which expresses energy balance during

the evolution since dM is the change in energy balanced by pdV which is the work done

against the pressure forces The Friedmann equation has an almost Newtonian

interpretation multiply b y

a k

a G

a

and the rst term is the Newtonian kinetic energy of a test particle with unit mass and

with co ordinate prop ortional to the scale factor the second term G mass of the universe

a is the Newtonian p otential The sum of the kinetic energy and the p otential

energy is then the constant total energywhich is negative p ositive for k k

and vanishes for k These considerations should be taken purely as an aid for the

memory

The Big Bang

If the pressure p is p ositive many features of the standard mo del can be understo o d

without sp ecication of the equation of state p p This is in particular true in the

early erse where it will turn out that the cosmological constant is not of imp ortance univ

To pro ceed we need to use the Friedmann equation ie

G a k

a a

and the energy conservation ie

d

a pa

da

It is also convenient to use eq

a G pa

which is a consequence of eqs and We can now qualitatively derive the

following results



It is easy to show that eqs and are implied byeqs and The reason we

only havetwo indep endent equations is that T is already contained in R g R



THE BIG BANG

Figure Plot of at with a and a

Existence of the big bang the initial singularity

If p or more generally if p is p ositive it follows from that aa

which means that gravity is attractive Since we observe red shifts and not blue shifts

it follows from the discussion in section that at present the Hubble constant is

p ositive so aa Thus in a plot of at versus t the velo city a is p ositive but the

acceleration a is negative and we have a situation like the one shown in g It is

seen that sometime in the past the function a t has reached zero

Conventionally we take this time to be t Clearly a vanishing scale factor

means that the Rob ertsonWalker metric b ecomes singular since the threedimensional

comoving space shrinks to a single p oint as t This singularity is the famous big

bang Such singularities are known to o ccur generically in most spacetimes under very

general assumptions so it is not just a feature of the Rob ertsonWalk er metric The

o ccurrence of the singularity means that the universe starts as an explosion and at

increases until it reaches its present value

The age of the Universe according to Big Bang

The history of the present universe is thus that it starts with a and at increases

until it reaches its present value at where t is the age of the universe

o

As mentioned in the last subsection a when p Thus we have that a is

decreasing and therefore is smaller to day than it was in the past

a t a t for tt

Using this inequalitywe obtain an upp er limit for the ageofthe universe

Z Z

at at

da da at

t

a a t a t H t



See eg SW Hawking and GFR EllisThe largescale structure of spacetime Cambridge Univ

Press

COSMOLOGY

where H t is the Hubble constant at present see eq From eq we see

that the upp er limit on the age of the universe is between and billion years

yr t

This limit is actually an imp ortant feature of the big bang mo del since there are arguments

based on geology stellar evolution and the nuclear abundances which put lower b ounds

on the age of the universe in a waywhich is indep endent of the big bang mo del We shall

return to this p oint later

Discussion of the fate of the Universe

Having lo oked at the past let us now turn to the future Supp ose atcontinues to grow

and ultimately at From eq we see that if p the quantity a is

and it follows that constant or decreasing Therefore we must have a for a

a k at least as fast as a since a at least likea Thus it is only p ossible

for at to go to innity if k or k For k we thus get for the op en

universe

at t for t and k

for the at universe For k we have similarly

at slower than t for t and k

For k it is imp ossible for at to go to innity and from we see that a t

b ecomes zero when at has its maximum value a where a G Since a

max

max

is always negative the scale parameter will then drop down again with a negativ e and

blueshifted sp ectral lines and reach at in a nite time t This is the closed

universe

It therefore would b e quite interesting to know the sign of k to see whether the universe

expands forever or whether we shall b e heading for the disastrous future singularity We

shall discuss this in the next section

Fitting parameters to observations

The general relativistic equations and are incomplete without the sp eci

cation of an equation of state p p In the standard mo del two such equations are

used namely for relativistic matter relev ant for the early universe and for nonrelativistic

matter relevant for the present universe

Most of the matter galaxies and clusters of galaxies observed to day app ear to be

nonrelativistic with kinetic energy much smaller than the rest energy In the kinetic

p

where is the ro otmeansquare theory of gases here applied to galaxies p

average velocity of the gas molecules the galaxies Reintro ducing the velo city of light

the pressure in the energymomentum tensor enters like pc and hence pc c

and for galaxies observed now c Hence eq gives with p

B

a

FITTING PARAMETERS

where B is a constant of integration Eq means that the energy content of a

comoving volume a remains constant when atchanges

In the early universe one usually assumes the existence of a hot gas of extreme rela

tivistic particles with the kinetic energy much larger than the rest energy For a relativistic

ideal gas one has

p

and eq gives

D

a

where D is a constant of integration The a behavior is related to the fact that

relativistically the energy in a comoving volume red shifts like a while the volume

still scales as a

Now at very early times matter was very hot andasthe universe expands and co ols

the kinetic energy of the matter redshifts and ultimately it b ecomes nonrelativistic

We shall try to t the various parameters to observations In order to t the crucial

parameter k op en at closed universe for k k k resp ectively let us

use eq in the form

a G k G

H

crit

a a

If then the expansion is driven entirely by and k In eq H is the

cr it

value of the Hubble constant at time t From

h

H h

yr

to day we get



h gcm

crit

Intro ducing eq b ecomes

cr it

k G

H

crit

a

so corresp ond to k k k resp ectively

Eq has a simple intuitive app eal Supp ose is very large Then we have a

lot of energy with a lot of gravitational attraction which will counteract the expansion

making a closed universe Similarlyif is very small there is very little counteraction

to expansion and the universe is op en

Since is given in terms of the Hubble constant and since could in principle

cr it

y measuring the mass density in the universe the Friedmann equation b e determined b

allows a determination of the sign of k and hence the fate of the universe

If one considers luminous matter stars one nds

lum

lum

crit

For a relativistic particle one has energy jmomentumj the factor arises b ecause statistically

particles can move in three directions

COSMOLOGY

so However one can also estimate the masses of dynamical systems by asking

for the gravitational forces necessary to explain the observed motions

For a system with spherical symmetry one can use Keplers third law GM r

where M is the mass interior to the orbit of an ob ject a star with velocity and orbital

radius r By studying the orbital motion of stars in a galaxy at the radius where light has

essentially disapp eared this radius is called the Holmb erg radius one can measure the

mass of the luminous material and one gets However studies of orbits of stars in

spiral galaxies for elliptic galaxies similar results do not exist p erhaps b ecause of the lack

yond the Holm b erg radius have of test b o dies ie stars outside luminous matter be

p

r but instead stays revealed the interesting and stunning result that does notgoas

approximately constant This phenomenon is called at rotation curves and it means

that the mass M increases linearly with r even though there is essentially no luminous

matter to observe This indicate that a large part of all matter is dark For galaxies

one gets

dyn

dyn

crit

so The discrepancy between and gives rise to the search theoreti

cally and exp erimentally for dark matter We shall however not enter this fascinating

sub ject

From geology stellar evolution and evolution of nuclear abundances one can make

arguments whichputsalower limit of billion years on the age of the universe Now

is related to the age of the universe as one can see qualitatively If is very large we

have a lot of gravitational in teraction slowing down the expansion and thus the life time

of the universe is small If is very small the converse is true if there is only

expansion no counteraction from gravity This turns out to givethe b ound

Thus using this and we have

Consequently we see that from these arguments we are not in a p osition to determine

whether the universe is closed at or op en and we cannot

say whether the univ erse will eventually recontract or expand forever However recently

new types of observations have b een developed sup ernovae and uctuations

in the microwavebackground whichgivevery strong indications that the universe is at

and expanding due to a cosmological constant This will be briey discussed in section

From eq and the bounds we get

G k G

crit crit

a

to day This limits the threedimensional curvature ka

Finally we shall determine the parameter D in eq To get a bound on D we

need to consider the microwave background radiation

COSMIC MICROWAVE RADIATION

The cosmic microwave radiation background

In the standard big bang mo del the universe starts out with a very small scale factor

Presumably the temp erature is very high and the universe is lled with highly relativistic

particles As the universe expanded the temp erature dropp ed so matter and radiation

co oled At a temp erature of around K the decoupling temp erature the free elec

trons joined the nuclei to bind into neutral atoms and the thermal contact between

matter and radiation was broken since the absence of free charged particles means that

drop o steeply The radiation existing at the interactions of photons with other matter

that time has since been much redshifted but it should still ll space around us

To describ e the situation in very simple terms let us consider photons in thermal

equilibrium We found in eq that the frequency changes with the scale factor

according to

at

at

Thus if t is the time of the decoupling we can get the redshift to day t t from eq

Also b ecause of eq the Boltzmann factor for a photon satises

h h



kT kT



e e

where we have dened

at

T T

at

If there is a thermal equilibrium we can thus absorb the scale factor in the temp erature

In particular we get at the present age of the universe the Planck black body radiation

distribution of the photons

h d

d

h

exp

kT

Thus the big bang predicts that there should b e a thermal distribution of photons present

to day left over from the time of the decoupling of photons from other matters Presum

ably the most signicant exp erimental result in cosmology since Hubbles discovery is that

such a sp ectrum is indeed observed Penzias and Wilson

In principle the temp erature T at present can be predicted from eq which

was also done byanumber of p eople Gamow Alpher Herman Dicke Peebles Roll and

Wilkinson b efore the discovery of the radiation After the recombination the universe is

matter dominated and we have ie

B

a

from which we get

at t

at t



Inserting the densities of baryons at t and at presentt gcm the most careful

o

K The exp eriments give a nice black body radiation sp ectrum prediction was T

COSMOLOGY

o

with T K in go o d agreement with the theoretical estimates Recent

satellite measurements give avery precise Planck sp ectrum

We can now obtain a lower bound on the constant D in eq for in the

radiation dominated early universe Da The presently observed microwave photons

contribute a relativistic comp onent to as they did in the relativistic era Thus

MW

D

MW

all times

MW

a

In the relativistic era the photons were not the only comp onent to the energy density so

DD

MW

Now using

Z



x dx

x

e

we obtain by integrating over the frequency

a

DD a kT

MW MW

to day

h

to day

From and the b ound one gets a lower bound on at and inserting this we

get forh

D

Thus the constant D in the radiation dominated density Da is huge We shall

return to this later

The matter dominated era

We now consider the matter dominated era with negligible pressure so Ba Also

in this section for simplicitywe ignore the cosmological constant whichcanbetaken into

consideration at the cost of more complicated formulas The Friedmann equation

b ecomes

a k G a

crit

a a a

where we divided by a a is the scale parameter to day and used

a

a

From eq we have

G k

H

crit

a

Using this to day in eq we get

a a

H

a a

THE MATTER DOMINATED ERA

This equation can easily b e solved for time

Z

aa

dx

q

t

H

x

In particular the present age of the universe is

Z

dx

q

t

H H

x

where the inequalityvalid for was already derived from the qualitative arguments

in sect The integral in or can easily b e p erformed It is most convenient

to distinguish the case and closed at op en universe

resp ectively The intuitive app eal of eq should be emphasized If is very

largesmall gravity counteracts very muchlittle and hence the life time is exp ected to

be shortlarge This is seen to b e the case in the ab ove integral

The closed Universe 

We dene the development angle by

at

sin

a

The integral can then easily be p erformed and we get

sin H t

which together with eq shows that a at is a cycloid The scale parameter

increases from zero at t and reaches a maximum at

max

t

max

H

a

a

max

and then it returns to zero at t t The present age of the universe is

max

obtained by putting ata in eq

t f

H

q



f cos

for The function f is seen to be monotonically decreasing For one

p

has f for one has f Thus if is very

large the life time is very small as mentioned b efore

COSMOLOGY

The at Universe  

Here eq is trivial and gives

H t at

a

and the ageofthe universe is



t H yr

h

with h Since evidence from geology etc shows that t years we see

that if then we cannot have h but h must be less than The result

or could also b e derived from eqs and by noticing that for

it follows from eq that Expanding the trigonometric functions

one then gets The result again has the form so f

The op en universe  

Formally we can again use the substitution and the result except that the

development angle is imaginary i Hence we get

sinh H t

with

at

cosh

a

The age of the universe is given by but now

p

ln f

Again f is monotonically decreasing with increasing and the maximum value is

f

Inclusion of the cosmological constant

We end this discussion by emphasizing that in view of newer observations the cosmological

constant should be introduced in the formulas discussed ab ove Since this is straightfor

ward we leave the details of the inclusion of as an excersize for the reader Generalizing

the steps leading to eq one nds

Z

dx

q

H t

x x

T M

Here wehave ignored radiation If we denote the radiative by this formula would read

R

Z

dx

p

H t

 

x x x

T R M

where the radiative contribution is imp ortant only in the early universe

THE MATTER DOMINATED ERA

where the s refer to the present time t and H This integral is of an elliptic

typ e and can easily b e integrated numerically We give afew examples

H t and H t

M M

In all these examples the total is one If this is the case the integral in can

T

actually b e performedinasimpleway

s s

Z

dy

p p

ln H t

y

M M

M

Here it should b e remembered that

M

Discussion of the life time of the Universe

It is clear from common sense that the universe must be older than the oldest stars In

principle this can b e checked by measuring the Hubble constant H or h very accurately

This is one of the main purp oses of the Hubble space telescop e To illustrate the problem

for the reader let us mention a few numb ers The ages of the oldest globular clusters are

t Byr This number dep ends on mo dels as well as observations However

recently it has been possible to nd so far only one old star where U was

in the sp ectrum From the life time of uranium one then nds t Byr found

Allowing for some additional time to create this star the life time of the universe would

then be something like Byr

As an example supp ose t Byr and Then h This

is b elow the rep orted values from the Hubble telescop e h which however

has been much debated

In general it is easy to see that the small including dark matter is not enough

to provide sucient life time from the cosmological mo del To get out of this problem

the expansion should somehow be enhanced This has ledtothe reintroduction of the

cosmological constant ie to add g on the left hand side of the Einstein equation

Later on as we shall discuss at the end of this chapter new observations on sup ernovae and

uctuations in the microwave background haveshown that there must b e a cosmological

constant or something similar Thus the cosmological constant is not introduced as a

convenientwaytosolve the life time problem but is really needed to t other observations

We have seen in eq that it is easy to take into account the presence of a

cosmological constant in the previously discussed cosmological Einstein equations We

just need to make the replacements

p p p

G G

To see that is related to expansion let us take the simple example of at space

k with no matter and p Thus The Friedmann equation then

G

gives

G

a k a a

q

Ht

with the solution ate H for p ositiv e Thus we see that if is p ositiv e

it is p ossible to have expansion just from the cosmological constant

COSMOLOGY

In the older universe we see that the Friedmann equation

k a B G

a a a

implies that the dierent terms dominate at dierent times In the b eginning after

recombination the matter term dominates then the curvature ka takes over as a

increases and ultimately the cosmological constant term dominates completely For

and k the expansion continues forever For the expansion comes to a stop

since the left hand side of the Friedmann equation is always p ositive In this case there

is amaximum value of a obtained for a

b efore that the cosmological constant do es not play an imp ortant e mentioned Wehav

role in the early relativistic universe where the Friedmann equation b ecomes

G k a D

a a a

When a is very small the rst term dominates entirely and the curvature ka and

the cosmological terms are utterly unimp ortant Thus our considerations on the big bang

in sect are valid in spite of the fact that we did not explicitly take into account the

cosmological constant

If is introduced it is easy to solve any ageproblem An example Let t

Byr and h With and these numbers t as can be seen b y

use of We end by remarking that the age problem has b een there since Hubbles

dicovery Using his value for H the life time of the universe was shorter than the life time

of the planet earth This led to a disb elief in the standard cosmological mo del However

as mentioned b efore there are now observations indep endent of the age problem which

requires a cosmological constant Observations seem to converge to a consistent picture

of the universe as b eeing describ ed by the standard mo del to a rst approximation

Causality structure of the big bang The hori

zon problem

The region of causal contact with a given event is determined by how far photons origi

nating at that event can travel We shall consider photons emitted at big bang and see

how far they can propagate

For photons d so the Rob ertsonWalker metric gives dt at dl where dl is

the timeindep endent distance in comoving co ordinates The physical causal distance is

thus

Z

t



dt

d at l at

at

where the time is the bigbang time that the photon was emitted whereas t is the time

it was received

To obtain at let us consider the Friedmann equation

G

a k a

CAUSALITY STRUCTURE

In the early universe Da and a is quite small Consequently the term a Da

dominates over the curvature term and we get quite easily by solving a GDa

that

v

s

u

u

t

at t GD

p

so at is prop ortional to t in the early universe From we thus get for the

physical distance

d t

Consequently we see that for t small ie in the early universe

v

u

d t

u

q

t

at

GD

The conclusion from this is that in the early universe the causal distance is much

smaller than the scale factor This means that light emitted from a point cannot

catch up with the expansion whichgoesmuch faster than light This can b e expressed

by saying that close to the initial singularity the universe consists of many arbitrarily

small causally separate regions This is clearly very surprising since the standard mo del

assumes that the universe is at any stage in its development homogeneous Although

this is an assumption and although one is free to make such an assumption the situation

i nt of view one would like the do es not app ear very selfconsistent From a physical po

homogeneous universe to be established by the contact between all p oints by physical

light signals so that physics in anypoint can establish itself to b e equal to physics in any

other p oint by means of causal contacts This is not the case in the early universe and

this is clearly a diculty with the standard mo del This is the causality problem also

called the horizon problem The latter concept refers to the fact that Einstein gravity

pro duces a horizon b eyond which one cannot observe

In the matter dominated later universe the situation is dierent For k wehave

from

s

A

a t GB

and hence from

Z

t



dt

d at l at

at

t



we get

d t t t t

for t t so throughout the later evolution of the universe the largest regions of causal

contact are of the order t which is larger than at

Togive an analogy to this situation supp ose you wanttobike km to tell a friend how she should set

her watch in order to synchronize your watches assuming that biking is the fastest way of communication

Unfortunately some evil p erson is changing the length scale much faster that you can bike so you will

never reach her to tell her the time The standard mo del assumes that somehowshe knows howyou have

set your watch Admittedly it could happ en by accidentthat she selected the same setting of time as

you did However in the prop er analogy there are billions of such friends so the probability of accidental

coincidences would b e essentially zero

COSMOLOGY

The microwave background discussed in section is observed to be very isotropic

It turns out that the microwave photons originated in causally separate regions if their

directions dier by more than a few degrees Thus the data show that a truly striking

del this degree of isotropy is present in the universe and in the big bang standard mo

could not hav e been pro duced by causal pro cesses In the big bang mo del this striking

isotropyis not explained but has to b e put in by hand to t the observations

Finally let us discuss another puzzle the atness problem Flat space ie

is an unstable equilibrium If is ever exactly equal to one it remains equal to one Now

we know that to day is at least which is not very far from The Friedmann

equation can be rewritten

k a

Since k is a constant and since a increases if we go backward in time it follows that

day To give in early times must have been extraordinarily closer to than it is to

some examples For to be in the allowed range to day it must have been equal to

to an accuracy of decimal places sec after big bang time of nucleosynthesis At

grand Unication should be equal to to an accuracy of decimals This fact is not

explained by the Standard Mo del

Ination

We have seen that the standard mo del has many successful features and gives a nice

framework for discussing the rather few astronomical observations whichare available In

the last section however we saw that certain features emerge from the standard mo del

whic h point to diculties for the mo del if we want to understand it in more physical

terms

The standard mo del is therefore to be supplemented by a new scenario called in

ation where many of the physical diculties have b een removed Ination concerns

itself with the early universe where the diculties arise In the previous work we have

assumed that the pressure is p ositiv e Although this seems intuitively natural it is not

logically necessary In quantum eld theory one can have negative pressure

We cannot give a full explanation of this since this would require the introduction of

elds However let us give an intuitive description of what happ ens Supp ose at a very

early time the system is in a metastable state called the false vacuum On short time

scales the energy cannot b e lowered so false could b e replaced by temp orary Given

enough time the false vacuum decays to a stable true vacuum which is the state with

lowest p ossible energy Also let us assume that it takes a rather long time for the false

vacuum to decay This isinshort the scenario in ination

Let us call the energy densit y of the false vacuum relative to the true vacuum

f

To nd the pressure of the false vacuum let us imagine that we enclose it in a cylindrical

container with a movable piston So inside the cylinder and piston we have the energy

density Outside wehave true vacuum with p Nowmove the piston outwards

f

adiabatically which means that there is no heat ow into or out of the cylinder When

the piston is pulled the false vacuum keeps its energy density everywhere also in the

orarily the lowest p ossible one So new volume This is b ecause this energy is temp

the change in energy dU is given by dU dV where dV is the change in volume

f

However in the adiabatic case we always have dU pdV Hence

INFLATION

p

f

ie the pressure is negative

You maywonder where the extra energy arising when we pull the piston comes from

It is of course supplied by the agent pulling the piston who has to p erform this work

against negative pressure

Another way of seeing the negative pressure related to vacuum is to ask for the energy

momentum tensor for a false or true vacuum This T cannot dep end on velocities etc

since the vacuum do es not move or carry momentum The only relativistically invariant

tensor is therefore T g where is the density of the true or false vacuum

ccurs b ecause by denition the energy density is p ositive This tensor minus sign o The

therefore corresp onds to the negative pressure p

The negative pressure has profound consequences Consider eq

a G pa

Usually this meansa ie gravityisattractive Now during the time the false vacuum

dominates

a G a

f

so gravityisnow repulsive This means that during ination gravitydoes not counteract

the expansion on the contrary b ecause the solution of the attractive gravity equation is

s

t t

atc e c e G

f

where c c and are constants remember that is the constant energy density of the

f

false vacuum Let us estimate A simple mo del is to assume that it is the density

f

relevant at the Grand Unication scale with temp erature GeV In natural units

f

has dimension mass so

GeV

f

This is enormous To get this density in the sun wewould have to compress it to the size

of a proton We then get



sec

and the Hubble constant which is now a true constant b ecomes

a



sec

a

Let us consider the density during ination

d da

a pa a

f f f

da da

so d da vanishes and is time indep endent as we knew b eforehand The Friedmann

f f

equation for k is then just

s

a

G H

f

ination

a

COSMOLOGY

Letusnow mo dify the standard big bang so that for t t wehave the usual radiation

domination D a for t t t we have negative pressure and the equation of

state for t t t we return to the relativistic equation of state whereas

d

for t t t time of decoupling of photons we have the usual matter domination

d d

Ba Then the evolution lo oks like

p

tt radiation dominance a t D a

tH

in

t tt ination a e const

p

t tt radiation dominance a t D a

d

t t matter dominance a t Ba

d

The integration constant D and D are now determined by requiring consistency at the

times t and t so

H t t

 

ination

D D e

The imp ortant feature is that even if D is of order one then D can be very large If

H t t then one can have D with D Thus the large value of

inf l

D can be obtained in a quite reasonable way from a decent value of D Mo dels where

the equation of state is satised are called inatory mo dels and when at blows

up exp onentially the universe is said to be inating As the inationary p erio d pro ceeds

all elds get redshifted away exp onentially fast The inationary p erio d essentially wip es

the slate clear of primordial uctuations from t t

One may ask whether H t t is reasonable Since H we need

in in



t t sec which is the right time for the inationary p erio d In the phase

transiton where false go es to true vacuum the energy stored in the false vacuum is

released in the form of new particles in thermo dynamics language the released energy

is the laten t heat of the phase transition These new particles then come to thermal

equilibrium at the temp erature T GeV

What happ ens to the causality problem Start with the present universe which is

causally connected Going backwards in time we encounter ination with its enormous

contraction we go backwards pro ducing an incredibly small region out of our universe

of size t In this small region photons can maintain causal contact so there is no problem

To see this in more details let us estimate the distance d travelled by light

Z Z Z

t t t

  

dt dt dt

d a t at

at at at

t



where the various times are dened ab ove In the inationary time interval t t t

we take

H tt



atat e

In the interval from big bang to t we have

q

atat tt

Hence

H t t

 

d t H e

EVIDENCE FOR THE COSMOLOGICAL CONSTANT

Therefore the causal distance is blown up by an exp onential factor quite compatible with

the exp onential blowup of the scale factor exhibited in where at is the size of

a causal domain just b efore ination sets in ie

at t

It should b e noticed that this scale factor is the size of a physical early universe and that

it represents a tiny part of the causally nonconnected early universe Also notice that

eq would not work if there is no ination since the resulting older universe would

be far to o small to have anything to do with the universe we know

After the end of ination the universe is radiation dominated for some time At this

stage the causal distance is again t

ination a increases What ab out We still haveFriedmann k a During

extremely rapidlyso is driven to wards so

ination

Finally it should be said that ination o ccurs in many versions and some of them are

somewhat dierent from the intuitive description given ab ove

Observational evidence for the cosmological

constant

In evidence for a nonvanishing cosmological constant was found by observing

typ e Ia sup ernovae out to redshifts of order one The data was plotted in the

Hubble diagram The main po intin using distant sup ernovae is that magnituderedshift

they are considered to b e very go o d standard candles they are assumed to b e asso ciated

with the nuclear detonation of white dwarfs It is completely outside the scop e of these

notes to describ e the theory of white dwarfs However b ecause of the knowledge of

the blow up mechanism the intrinsic luminosity and the observed ux are known and

hence the luminosity distance can be computed from the ratio of these quantities In

particular features of the distant sample of Ias app ear to b e similar to a nearby sample

So far there is no serious ob jection against Ia as a standard candle Therefore and

b ecause it has b ecome p ossible to observe highz ob jects the old uncertainty concerning

the determination of the distance in Hubbles relation seem overcome

For nearby galaxies it is sucient to use the simple Hubble law z v HL where

r

L is the distance and v is the radial velo city see eq In the standard mo del

r

distant galaxies should have larger velocities than predicted by the simple Hubble law

b ecause gravitation slows the expansion However the sup ernovae observations showed

precisely the opp osite The conclusion is therefore that the expansion sp eeds up

This is in contrast to the standard mo del without a cosmological constant However

introducing we saw in connection with eq that the evolution of the velocity a

changes

a G k

a a



A Riess et al Astrophys J this pap er is most easily found on the net see astro

ph and B Schmidt et al Astrophys J see astroph



See eg S Weinb erg Gravitation and Cosmologyp

COSMOLOGY

For a p ositive in a late universe the two rst terms on the right hand side are sub dom

inant and the expansion is driven by the last term Rememb ering that

H a

and H

crit

G a

we obtain the following form for the Friedmann equation

k

H

M

a

Here is the matter density and H is the relative density asso ciated

M crit

with the cosmological constant which can be interpreted as a vacuum energy since it

exists irresp ective of matter

By further expansion of z at at at in eq we obtained a general

ization of the simple Hubble law

H L q z H L

where q is the deceleration parameter to day with

aa

q

a

The interesting thing is that q can b e expressed in terms of the s In the later universe

we can ignore the pressure and we have

G

a a a H aq

from the denition of q If we use H G and eq

crit M crit

b ecomes

q

M tot

Here is the total energy ignoring the small contribution from radiation

tot M

The name deceleration parameter refers to the exp ectation that q since a was ex

p ected to b e negative if there is no cosmological constantif p However if

is present this need no longer b e true and we see that tries to change the sign of q so

pro duces p ositive acceleration

The main p oint is that eqs and supplies astronomers with a metho d of

actually measuring if is known from other observations provided the distance L

M

is known accurately enough From the highz sup ernovae observations mentioned ab ove

one nds

This is consistent with a total energy

tot

These numb ers are of course exp ected to have some uncertainty It is to be noticed that

the last result is at least roughly consistent with ination For further discussion we refer

to the recent reviews to b e found in astroph httpxxxlanlgovlistastrophnew

EVIDENCE FOR THE COSMOLOGICAL CONSTANT

More recently precise data have b een obtained for the uctuations in the microwave

background These lead again to rougly consistent with ination These data

tot

are completely indep endent of the sup ernovae observations and are thus not sub ject

to any doubt concerning the mechanism behind sup ernovae pro duction Therefore the

conclusion that the universe is at least approximately at is indicated by indep endent set

of observations It is of philosophical interest that even very precise future measurements

will always have an uncertainty Thus if it turns out that X then the

of X nomatterhow small will always prevent one from making the statement existence

that the universe is exactly at One can only say that with a high probabilityitis at

The conclusion is that the cosmological constant is necessary because of the recent

observations Cosmology seems to converge rapidly towards a very precise science where

observations determine the parameters With and the at universe

M

will go on expanding forever This gives rize to the problem that welive in a protable

time from the p oint of view of observations b ecause in the future astronomers will b e able

constan t gives rize to several to observe fewer and fewer ob jects Also the cosmological

other problems For example the matter density is of the same order as the density

asso ciated with However these two densities have dierent origins matter versus

vacuum and b ehave in a dierentway as functions of the scale parameter Consequently

it is not easy to nd an explanation why is so closeinvalue to in our time

matter

There are several attempts to understand the cosmological constant without really

having it for example by the use of suitable scalar theories in such a way that there

orarily something whic h lo oks like a cosmological constant In this way p erhaps is temp

the end is not an empty universe On the other hand the simplest p ossible explanation

of the data is a constant

In quantum eld theory a vacuum energy emerges naturally This is precisely what

is needed for understanding the cosmological constant However this has not led to a

of the right order of magnitude This constitutes one of the most imp ortant problems in

mo dern physics

It is the eort of many human bei ngs to ask for a meaning of life death the

universe So in cosmology we have the opp ortunity to ask very precisely what is the

meaning of the universe The cosmological constant represents the vacuum ie nothing

Most of the energy in the universe is thus empty space So p erhaps the answer

to the question on the meaning of the universe is that there is no meaning

In this connection it should be noted that the cosmological constant allowed by Ein

stein gravity is considered together with the assumptions of the standard mo del namely a

homogeneous and isotropic universe with an evolution characterized by the cosmological

time Although this may app ear reasonable on an extremely large scale these assump

tions are denitely not valid on shorter scales where the universe is inhomogeneous This

may lead to a problem b ecause of the nonlinearityof the Einstein eld equations which

can makethe averaging pro cedure much more subtle than one mightnaively think Thus

the equation used to analyze the Sup ernovae data dep ends on the s b ecause

of Einstein eld equations If we think of the various quantities in as large scale



The data from the uctuations in the microwavebackground actually have a statistical preference

for a total which is slightly larger than one but can b e one within a standard deviation



This diculty is due to the fact that is a b orderline case If for example it had turned out

that no such problem of principles would o ccur



See however the previous fo otnote

COSMOLOGY

averages it is not obvious that there are no corrections coming from the nonlinearities

The end of cosmology

Lawrence Krauss and Rob ert Scherrer see arXiv where further references can

be found have investigated some of the depressing consequences of the cosmological

constant As we have seen in eq ultimately there will be an expansion driven

by the cosmological constant In approximately billion years almost all galaxies will

be invisible except the Lo cal Group of six galaxies which are gravitationally b ound and

do not participate in the large scale expansion Observers in this island universe will

as Krauss and b e fundamentaly incapable of determining the true nature of our universe

Scherrer sa y Of course there will be no such observers on earth since its life time is

only of the order bilion years but p erhaps there are observers some other place in the

Milky Way p erhaps having the same DNA as us

These observers will have absolutely no wayofknowing that the universe expands and

will think that the universe is static This was Einsteins starting p oint corrected by as

tronomical observations This was also the reason for his introduction of the cosmological

constant

In the island universe there is no way of recognizing that there was a Big Bang Due

to the expansion the cosmic microwave background will disapp ear in the sense that the

erhaps wavelenghts will be shifted and totally buried by radio noise in our galaxy So p

these observ ers will sp end their time pro ducing a lot of pap ers on why there are six as

contrasted to any other number galaxies in the whole universe

Although the return of the static universe is not of immediate concen philosophically

the situation in the island universe is interesting b ecause the observers can have the right

physics but are unable no matter how go o d they are to get the right cosmology b ecause

of lack of observational supp ort One may wonder if the same thing is happ ening to day

are we also beeing deceived like the islanders or like the man who came to Casablanca

for the w aters

An inhomogeneous universe without 

Recently there has b een muchinterest in the study of an inhomogeneous universe in order

to see if it is p ossible to avoid the cosmological constant Of course from CMB we know

that the early universe is to a high degree of accuracy homogeneous Hopwever at early

times pla ys no role

For the universe at the present epoch one basic consideration is that even though

the universe now app ears to be homogeneous at very large scales observationally there

are voids of considerable size surrounded by clusters of galaxies It is estimated that

p er cent of the present universe consists of voids with diameter of order h Mp c

We refer the reader to a pap er by David L Wiltshire Phys rev Letters

where further references can be found The necessary formalism is somewhat

complicated but the physical idea is simple It is necessary to average over the dierent

structures that have dierent time rates b ecause the time dep ends on the gravitational

eld Also in the voids there is very little matter so this part of the universe has a

negative curvature metric k Therefore volumes are larger and hence densities

UNIVERSE WITHOUT

are smaller than exp ected in the standard cosmological mo del It then turn out that the

deceleration parameter q is given by

f f f f

q

f f

where f is the void volume fraction Here we see that in spite of having no cosmologaical

constant q changes sign At early times there are no voids so f giving q

But q changes sign for f corresp onding to acceleration not due to The

apparent acceleration b ecomes maximal for f with q At later times

q approaches zero ie the acceleration disapp ears and there will b e no island universe

COSMOLOGY

Chapter

Problems

Problem Consider cylindrical co ordinates

x r cos y r sin z z

Find the metric expressed in terms of these co ordinates Find the Laplace op erator in

these co ordinates

Do the same for spherical co ordinates

x r cos sin y r sin sin z r cos

Check the correctness of the resulting Laplace op erator by comparing your result with

some b o ok eg on quantum mechanics

Problem Let A b e a quantity consisting of four comp onents A A A A Assume

that

ariant A B inv

for an arbitrary vector B Show that then A is a vector

Generalize this result to arbitrary tensors

The resulting theorem is called the quotienttheorem

Problem Let S S be a symmetric tensor and A an antisymmetric tensor

A A Show that S A is identically zero

Show that an arbitrary tensor with two indices can be written as the sum of a sym

metric and an antisymmetric tensor

Problem Consider a twodimensional surface of a sphere dened by

x y z R

What is the metric expressed in terms of x and y

Find the metric in spherical co ordinates

x V Then show that V g V Problem Assume that V V

satises eq

PROBLEMS

Problem Use the energymomentum tensor in the dierential equation

to derive the sp ecial relativistic uid equation Hint Consider for

and and combine these results

Problem Consider the Einstein eld equation with a p ositive cosmological constant

Assume spherical symmetry and T Rewrite the eld equation in the form R

where contains only and g Hint Compute the Ricci scalar in terms of

Because of spherical symmetry the metric can be used The two functions

E rtand F rt are then to b e determined from the Einstein equation you just derived

Use the results in eq to determine the solution

The result should be

rt r E rtF

Discuss in details why the time dep endence disapp ears Hint Lo ok at R Please note

tr

that this metric do es not b ecome at at r Where is it at

Consider a test particle which starts from r with some initial velo cityandmoves in

a radial direction Derive the relevant equations of motion by help of the EulerLagrange

metho d Consider r as a function of prop er time and co ordinate time t Show that

r ultimately b ecomes innite for whereas r t approaches a nite value which

one for t

Problem Consider the deection of light around a black hole describ ed by the

particle can mo ve around in a circular Schwarzschild metric Show that a photon light

orbit r GM Are there any other p ossible values of r

Problem The de Sitter Universe In Problem we found the solution of the Einstein

equations with a cosmological constant assuming spherical symmetry For an observer

situated at r measuring the co ordinate time t the universe lo oks rather empty Show

q

that light emitted from r will be innitely red shifted at r and hence will

practically sp eaking b e unobservable after some nite time

Show that the metric found in problem can b e expressed as

d dt at dr r d

where at is called the scale factor Hint Use the transformations t r are the co ordinates

used in Problem

p

t

q

t t ln r e

and

p

t

r re

Find the scale factor Discuss the resulting de Sitter universe

Chapter

Some constants

General constants



Sp eed of light c cm s

This value is exact in the

sense that the meter is dened

since as the length

of the path traveled by light

in vacuum during a time interval

of s



Plancks constant h erg s



h h erg s



MeV s



Electron charge e esu



Coulomb

Conversion constant hc MeV fm

verting to useful in con

natural units h c

Fine structure constant e hc



fm cm



eV erg

   

Boltzmanns constant k JK eVK

CONSTANTS

Constants in General Relativity and Astronomy

Lightyear ly cm

Parsec pc cm

ly

Solar mass M g



Solar radius R km



Earth mass M g



Earth equatorial radius R km



Mean earthsun distance cm

 

Gravitational constant G dyn cm g

 

g Gc cm

 

Hubble constant H km s Mp c



Hubble time H s Gyr

q



Planck length Ghc cm

q



Planck mass hcG g

q



Ghc s Planck time

Chapter

Some literature

R M Wald General Relativity University of Chicago Press

R Adler M Bazin and M Schier Intro duction to General Relativity McGraw

Hill

S W Hawking and G F R Ellis The large scale structure of spacetime Cambridge

University Press

S Chandrasekhar The Mathematical Theory of Black Holes Clarendon Press Ox

ford University Press

B Ryden Intro duction to Cosmology Addison Wesley