Appendix A The Laplacian in a spherical coordinate system978-1-4614-0706-5/1.pdfThe Laplacian in a spherical coordinate system In order to be able to deduce the most important physical

Appendix AThe Laplacian in a spherical coordinate system

In order to be able to deduce the most important physical consequences from thePoisson equation (12.5), which represents the Newtonian limit of Einstein’s fieldequations, we must know the form of the Laplacian in a spherical coordinate system.This is because most important applications of any theory of gravitation involvephysical systems, for example planets, stars, black holes and the whole universe,that are spherically symmetric. Of course it is possible to find the wanted form ofthe Laplacian in collections of mathematical formulae. However, in the spirit of therest of this book, we here offer a detailed deduction.

The Laplace operator, or more commonly called the Laplacian, is defined as thedivergence of the gradient, i.e.

r2 � div grad : (A.1)

We shall find an expression of the Laplacian valid in an arbitrary orthogonalcoordinate system, and then specialize to a spherical coordinate system. In orderto calculate the Laplacian in an orthogonal coordinate system, we must first findexpressions for the gradient and the divergence in such coordinate systems. We startwith the gradient.

An operator is something which acts upon a function and changes it in aprescribed way. The gradient operator acts upon a scalar function by differentiatingit, and gives out a vector called the gradient of the function. In an arbitrarycoordinate system the gradient operator is defined as a vector with covariantcomponents

ri � @

@xi: (A.2)

According to the rule (5.77) for raising an index, the contravariant components are

r i D gij rj D gij @

@xj: (A.3)

Ø. Grøn and A. Næss, Einstein’s Theory: A Rigorous Introductionfor the Mathematically Untrained, DOI 10.1007/978-1-4614-0706-5,© Springer Science+Business Media, LLC 2011

313

314 A The Laplacian in a spherical coordinate system

We shall need to know the expression for the gradient in the class of coor-dinate systems with orthogonal coordinate axes. Such coordinate systems havediagonal metric tensors, i.e. gij D 0 for i ¤ j . In this case the only non-vanishingcontravariant components of the metric tensor are given in terms of the covariantcomponents by Eq. (5.75).

gii D 1=gii:

Inserting this into Eq. (A.3) gives

r i D 1

gii

@

@xi:

Next we shall find an expression for the divergence in terms of ordinary partialderivatives and the components of the metric tensor in an orthogonal coordinatesystem. The divergence is given as a covariant derivative in Eq. (10.3). Inserting theexpression (7.15) for the covariant derivative into Eq. (10.3), leads to

div EF D @F i

@xiC F k�iki : (A.4)

Hence, we must find an expression for the Christoffel symbols �i ki in anorthogonal coordinate system with a diagonal metric tensor. For this purpose weuse Eq. (7.30) with � D i , � D k, � D i , which for a diagonal metric tensorreduces to

�i ki D 1

2 gii

�@gii

@xkC @gik

@xi� @gik

@xi

�:

The last two terms cancel each other, so we are left with

�i ki D 1

2 gii

@gii

@xk: (A.5)

Inserting this into Eq. (A.4) we get

div EF D @F i

@xiC F k 1

2 gii

@gii

@xi: (A.6)

We now defineg � g11 g22 g33: (A.7)

Differentiation g by means of the product rule (2.22) we first put u D g11 g22 andv D g33 and get

@g

@xkD @.g11 g22/

@xkg33 C g11 g22

@g33

@xk: (A.8)

A The Laplacian in a spherical coordinate system 315

Using the product rule once more, we have

@.g11 g22/

@xkD @g11

@xkg22 C g11

@g22

@xk:

Inserting this into Eq. (A.8) gives

@g

@xkD @g11

@xkg22 g33 C g11

@g22

@xkg33 C g11 g22

@g33

@xk:

Multiplying the numerators and the denominators of the terms at the right-hand sideby g11, g22, and g33, respectively, reordering, and using Eq. (A.7), we get

@g

@xkD g

g11

@g11

@xkC g

g22

@g22

@xkC g

g33

@g33

@xk:

Using Einstein’s summation convention this may be written as

@g

@xkD g

gii

@gii

@xk:

Dividing by g and exhanging the left-hand and right-hand sides, we get

1

gii

@gii

@xkD 1

g

@g

@xk:

Inserting this into Eq. (A.6) we get

div EF D @F i

@xkC F k 1

2g

@g

@xk: (A.9)

In order to simplify this expression, we differentiatepg. Using the rule (2.36) with

n D 1=2 and the chain rule (2.31), we obtain

@pg

@xkD @g1=2

@xkD 1

2g�1=2 @g

@xkD 1

2pg

@g

@xk:

Dividing bypg gives

1pg

@pg

@xkD 1

2 g

@g

@xk:

Thus, Eq. (A.9) can be written as

div EF D @F i

@xiC F i 1p

g

@pg

@xi: (A.10)


Using, once more, the product rule for differentiation, we get

1pg

@.pg F i /

@xiD 1p

g

�pg@F i

@xiC F i @

pg

@xi

�

D @F i

@xiC F i 1p

g

@pg

@xi:

Comparing with Eq. (A.10) we see that this equation can be written as

div EF D 1pg

@

@xi

�pg F i

�: (A.11)

Inserting F i D r i from Eq. (A.3) and the expression (A.11) for the divergence intoEq. (A.1), we ultimately arrive at the expression for the Laplacian in an arbitraryorthogonal coordinate system

r2 � 1pg

@

@xi

�pg

gii

@

@xi

�: (A.12)

In the case of a Cartesian coordinate system, the only non-vanishing components ofthe metric tensor are gxx D gyy D gzz D 1. Then

pg D 1, and performing the

summation over j in Eq. (A.12), we get

r2 D @2

@x2C @2

@y2C @2

@z2:

Using Einstein’s summation convention this may be written as

r2 D @2

@xi @xi(A.13)

valid in Cartesian coordinates.The expression (A.12) shall now be used to calculate the Laplace operator in a

spherical coordinate system. Then we need the line element of Euclidean 3-dimen-sional space as expressed in a spherical coordinate system. The basis vectors of thiscoordinate system are given in terms of the basis vectors of a Cartesian coordinatesystem in Eq. (6.22). The components of the metric tensor in this coordinate systemare given by the scalar products of the basis vectors in Eq. (6.22). Since the vectorsare orthogonal to each other only the products of each vector with itself are differentfrom zero. Using Eq. (4.11) we find

grr D Eer � Eer D sin2 � cos2 ' C sin2 � sin2 ' C cos2 �

D sin2 ��cos2 ' C sin2 '

�C cos2 �

D sin2 � C cos2 � D 1;

A The Laplacian in a spherical coordinate system 317

g�� D Ee� � Ee� D r2�cos2 � cos2 ' C cos2 � sin2 ' C sin2 �

�

D r2�cos2 �

�cos2 ' C sin2 '

�C sin2 �

D r2�cos2 � C sin2 �

� D r2;

g'' D Ee' � Ee' D r2�sin2 � sin2 ' C sin2 � cos2 '

�

D r2 sin2 ��sin2 ' C cos2 '

� D r2 sin2 �:

Thus the line element of flat space in spherical coordinates has the form

d`2 D dr2 C r2 d�2 C r2 sin2 � d'2:

The components of the metric tensor in a spherical coordinate system are therefore

grr D 1; g�� D r2; and g'' D r2 sin2 �: (A.14)

Performing the summation over i in Eq. (A.12), with x1 D r , x2 D � , and x3 D ',we have

r2 D 1pg

@

@r

�pg

grr

@

@r

�C 1p

g

@

@�

�pg

g��

@

@�

�

C 1pg

@

@'

�pg

g' '

@

@'

�:

Inserting the expressions (A.14) and using that g D r4 sin2 � , we get

r2 D 1

r2 sin �

@

@r

�r2 sin �

@

@r

�

C 1

r2 sin �

@

@�

�r2 sin �

r2@

@�

�

C 1

r2 sin �

@

@'

�r2 sin �

r2 sin2 �

@

@'

�:

In the first term sin � is constant during the differentiation with respect to r , andcan be moved to the numerator in front of @=@r . The sin � in the numerator andthe denominator cancel each other. In the second term r2 in the numerator and thedenominator inside the parenthesis cancel each other. Finally, in the last term thesin � in the numerator cancels one of the sin � factors in the denominator inside theparenthesis. The remaining sin � can be put in front of @=@', since sin � is constant


during differentiation with respect to '. This finally gives the expression for theLaplacian in a spherical coordinate system

r2 D 1

r2@

@r

�r2

@

@r

�C 1

r2 sin �

@

@�

�sin �

@

@�

�

C 1

r2 sin2 �

@2

@'2: (A.15)

Appendix BThe Ricci tensor of a static and sphericallysymmetric spacetime

The most interesting applications of Einstein’s theory are concerned with systemshaving spherical symmetry. In Ch. 13 we consider spacetime outside static systems.Then we have to solve Einstein’s vacuum field equations in a static, sphericallysymmetric spacetime. Since the vacuum field equations amount to putting the Riccitensor equal to zero, we need to calculate the components of the Ricci tensor for thiscase.

We first have to calculate the Christoffel symbols for the metric (13.15). Thenwe use Eq. (7.30). We need the contravariant components g�� of the metric tensor.Since the metric is diagonal, they are given by Eq. (5.75), i.e. g�� D 1=g��. ThenEq. (7.30) simplifies to

�� D 1

2 g��

�g��;� C g��;� � g��;�

�: (B.1)

In this equation and the next ones, there is no summation over the values of theindex �.

In order to calculate the Christoffel symbols, we divide the Christoffel symbolsinto three groups. First we consider the case with� D �. Inserting this into Eq. (B.1)gives

�� D 1

2 g��.g��;� C g��;� � g��;�/ :

The last two terms cancel each other, so we are left with

�� D 1

2g��g��;�: (B.2)

Next we find an expression for the Christoffel symbols with � D � and � differentfrom both. Inserting � D � in Eq. (B.1) gives

�� D 1

2 g��

�g��;� C g��;� � g��;�

�:


319

320 B Ricci tensor of a static, spherically symmetric metric

Since � ¤ � in this case, and the metric of the line element (13.14) has g�� D 0 for� ¤ �, the first two terms inside the parenthesis are equal to zero, which implies

�� D � 1

2 g��g��;�: (B.3)

At last we consider the Christoffel symbols which have all three indices different,� ¤ �, � ¤ �, and � ¤ �. Then the indices of the metric components in all threeterms of Eq. (B.1) are different. Thus, all the terms are equal to zero, which leads to

�� D 0; if � ¤ �; � ¤ �; and � ¤ �:

Inserting � D � and � D r in Eq. (B.2), we get

�r rr D 1

2 grrgrr;r :

Inserting grr D e�.r/ from Eq. (13.15),

�r rr D 1

2 e�.e�/0: (B.4)

We now use the chain rule, Œf .�/�0 D f 0.�/ �0, where f 0.�/ is f differentiatedwith respect to �, and (in the present case) �0 is � differentiated with respect to r .Inserting f .�/ D e�, and using Eq. (3.59) gives

.e�/0 D e� �0:

Inserting this into Eq. (B.4),

�r rr D 1

2 e�e� �0 D �0

2: (B.5)

Next we calculate

�t tr D 1

2 gttgtt;r :

Due to the symmetry of the Christoffel symbols in their lower indices, we have�t rt D �t tr . Inserting gtt D �c2 e� from Eq. (13.15), we can replace � by � inEq. (B.5),

�t tr D �t rt D �0

2: (B.6)

Furthermore

��r� D ��r D 1

2 g��g��;r :

B Ricci tensor of a static, spherically symmetric metric 321

Inserting g�� D r2,

�� r� D ��r D 1

2 r2

�r2�0 D 1

2 r22r D 1

r: (B.7)

Next

�'r' D �''r D 1

2 g''g'';r :

(B.16)Inserting g'' D r2 sin2 � ,

�'r' D �''r D 1

2 r2 sin2 �

�r2 sin2 �

�;r: (B.8)

Since � is constant under a partial differentiation with respect to r ,

�r2 sin2 �

�;r

D �r2�;r

sin2 � D 2r sin2 �:

Inserting this into Eq. (B.8),

�'r' D �''r D 1

2 r2 sin2 �2 r sin2 � D 1

r: (B.9)

Then we calculate

�'�' D �''� D 1

2 g''g'';� D 1

2 r2 sin2 �

�r2 sin2 �

�;�:

We apply the rule that r is constant during partial differentiation with respect to � ,then the chain rule for differentiation, and obtain

�r2 sin2

�;�

D r2�sin2 �

�;�

D r2 2 sin � .sin �/;� :

According to Eq. (4.24), .sin �/;� D cos � . This gives

� '�' D � '

'� D 1

2 r2 sin2 �r2 2 sin � cos �

D cos �

sin �: (B.10)

We now use Eq. (B.3) to calculate the remaining non-vanishing Christoffelsymbols. Inserting � D r and � D � ,

�r�� D � 1

2 grrg��;r D � 1

2 e�2 r D �r e��: (B.11)


Inserting � D r and � D ',

�r'' D � 1

2 grrg'';r D � 1

2 e�2 r sin2 �

D �r e�� sin2 �: (B.12)

Inserting � D r and � D t ,

�r tt D � 1

2 grrgtt;r D � 1

2 e��c2e��

;r

D � 1

2 e�� c2 e� �0 D c2

2e�� 0: (B.13)

Finally we insert � D � and � D ', which results in

��'' D � 1

2 g��g'';� D � 1

2 r2

�r2 sin2 �

�;�

D � 1

2 r2r2 2 sin � cos � D � sin � cos �: (B.14)

The calculation of all the non-vanishing Christoffel symbols of the spacetimedescribed by the line element (13.14) has now been completed.

From Eqs. (9.29) and (11.15) we get the following expression for the componentsof the Ricci tensor in terms of the Christoffel symbols and their derivatives

R�� D �ˇ��˛ˇ˛ � �ˇ�˛�˛ˇ� C �˛��;˛ � �˛�˛;� : (B.15)

Since there are only two unknown metric functions, �.r/ and �.r/, it is sufficient tofind two of the field equations (11.35), say Rtt D 0 and Rrr D 0. However, it willturn out to be convenient also to calculate R�� .

Inserting � D � D t in Eq. (B.15),

Rtt D �ˇ tt�˛ˇ˛ � �ˇt˛�

˛ˇt C �˛ tt;˛ � �˛t˛;t :

Performing the summation over ˛ and ˇ (note, for example, that �˛ˇ˛ D �rˇr C��ˇ� C �'ˇ' C �tˇt ), and including only the non-vanishing Christoffel symbols,we deduce

Rtt D �r tt��r rr C �� r� C �'r' C �t tr

�

� �t tr�r tt � �r tt�trt C �r tt;r :


Inserting the expressions calculated above for the Christoffel symbols, we get

Rtt D �0

2e��

��0

2C 1

rC 1

rC �0

2

�

� 2�0

2

�0

2e�� C

��0

2e��

�0: (B.16)

Differentiation of the last term, using the product rule and the chain rule,

��0

2e��

�0D �00

2e�� C �0

2e�� 0 � �0� :

Inserting this in Eq. (B.16), and putting the common factor e�� outside aparenthesis,

Rtt D�1

4�0�0 C �0

rC 1

4

��0�2 � 1

2

�v0�2 C 1

2�00

C 1

2

��0�2 � 1

2�0�0

�e��;

or

Rtt D��00

2C 1

4

��0�2 C �0

r� 1

4�0�0

�e��: (B.17)

Inserting � D � D r in Eq. (B.15) gives

Rrr D �ˇrr�˛ˇ˛ � �ˇr˛�˛ˇr C �˛rr;˛ � �˛r˛;r :

Performing the summation over ˛ and ˇ, and including only non-vanishing terms,gives

Rrr D �r rr��r rr C ��r� C �'r' C �t rt

�

� �r rr�r

rr C �t rt�ttr � �� r��r � �'r'�''r

C �r rr;r � �r rr;r � ��r�;� � �'r';r � �t rt;r :

The first term coming from the first line and the first on the second line cancel eachother, and the two first terms on the third line. Also, using the symmetry of theChristoffel symbols in the lower indices, we find

Rrr D �r rr��r� C �'r' C �t rt

� � ��t rt

�2 � ��r�

�2

� ��'r'

�2 � ��r�;r � �'r';r � �t rt;r :


Inserting the expressions for the Christoffel symbols leads to

Rrr D �0

2

�2

rC �0

2

�� 02

4� 2

r2C 2

r2� �00

2;

which gives

Rrr D �12�00 C 1

4�0�0 � 1

4�02 C �0

r: (B.18)

Next we insert � D � D � in Eq. (B.15),

R�� D �ˇ��˛ˇ˛ � ��˛�

˛ˇ� C �˛��;˛ � �˛�˛;� :

Performing the summation over ˛ and ˇ, and including only non-vanishing terms,

R�� D �r��r rr C �� r� C �'r' C �t rt

� � ��r�r��

� �r��r� � �'�'�''� C �r��;r C �'�';� :

Inserting the expressions for the Christoffel symbols leads to

R�� D �r e��0

2

1

rC 1

r

�0

2

�� 1

rr e�� C r e�� 1

r

� cos2 �

sin2 ��

cos �

sin �

�

;�

� �r e��

;r: (B.19)

The two first terms are cancelled against the second and third terms in theparenthesis. The the second last term is differentiated by using the rule (2.49)for differentiation of fractions of functions, and the rules (4.24) and (4.25) fordifferentiating sin � and cos � , respectively. Thus

�cos �

sin �

�

;�

D .cos �/0 sin � � cos � .sin �/0

sin2 �

D � sin2 � � cos2 �

sin2 �

D � sin2 �

sin2 �� cos2 �

sin2 �D �1 � cos2 �

sin2 �: (B.20)

Using the product rule (2.24) and the chain rule (2.31), we differentiate the last term

�r e��

;rD e�� C r e��0: (B.21)


Inserting the Eqs. (B.20) and (B.21) into Eq. (B.19) gives

R�� D �r2

e��0 � r

2e��0 � cos2 �

sin2 �� e�� C r e��0 C 1C cos2 �

sin2 �;

which finally leads to

R�� D 1 ��1C r

2�0 � r

2�0�

e��: (B.22)

Equations (B.17), (B.18) and (B.22) provide the expressions for the componentsof the Ricci curvature tensor of a static, spherically symmetric spacetime that areneeded in Ch. 13 in order to investigate the consequences of the general theory ofrelativity for this type of spacetime.

Appendix CThe Ricci tensor of the Robertson–Walkermetric

The expanding universe models that are still the most prominent relativistic modelsof the universe, were deduced as solutions of Einstein’s field equations by A.Friedmann in 1922. These models are isotropic and homogeneous, and the mostsuitable line element for such models were found by H. P. Robertson and A. G.Walker about 1930.

Before we can solve the field equations for these models, we must find the formof the field equations for them, that is for the Robertson–Walker line element. Thenwe need to know the components of the Ricci tensor for this line element. In order tofind these components of the Ricci tensor, we shall first calculate the non-vanishingChristoffel symbols for the line element (14.9).

Some of the Christoffel symbols come from the part r2 d�2 C r2 sin2 � d'2 ofthe line element (14.9), which is also a part of the line element (13.14). They aregiven in Eqs. (B.7), (B.9), (B.10), and (B.14). We list them here for easier reference

��r D �� r� D 1

r; (C.1)

�'r' D �''r D 1

r; (C.2)

�'�' D �''� D cos �

sin �; (C.3)

��'' D � sin � cos �: (C.4)

Using Eq. (B.2) with � D � and � D t , and then Eq. (14.10), we get

�� t D g��;t

2 g��D .a2r2/;t

2 a2 r2D r2 .a2/;t

2 a2 r2

D .a2/;t

2 a2D 2 a Pa2 a2

D Paa; (C.5)


327

328 C Ricci tensor of the Robertson–Walker metric

where Pa means the derivative of the function a with respect to t . Due to the isotropyof the spacetimes described by the line element (14.9), and the symmetry of theChristoffel symbols in their lower indices, we get

�r tr D �r rt D Paa; (C.6)

�'t' D �''t D Paa; (C.7)

�� t� D �� t D Paa: (C.8)

Putting � D � D r in Eq. (B.2) we get

�r rr D grr; r

2 grrD 1 � k r2

2a2

�a2

1 � k r2�

;r

D 1 � k r2

2 a2a2 2 k r

.1 � k r2/2

D k r

1 � k r2: (C.9)

The rest of the Christoffel symbols are calculated using Eq. (B.3), which gives

�t rr D �grr;t

2 gttD 1

2 c2

�a2

1 � k r2�

;t

D 2 a Pa2 c2 .1 � k r2/

D a Pac2 .1 � k r2/

; (C.10)

�t �� D �g��;t2 gtt

D 1

2 c2

�a2 r2

�;t

D 1

2 c22 a Pa r2

D a Pa r2c2

; (C.11)

�t '' D �g'';t2 gtt

D 1

2 c2

�a2 r2 sin2 �

�;t

D a Pac2r2 sin2 �; (C.12)

�r�� D �g��;r2 grr

D �1 � k r2

2 a2

�a2 r2

�;r

D � �1� k r2� a2 2 r2 a2

D �r �1 � k r2�; (C.13)

C Ricci tensor of the Robertson–Walker metric 329

�r'' D �g'';r2 grr

D �1 � k r2

2 a2

�a2 r2 sin2 �

�;r

D �r �1 � k r2�

sin2 �: (C.14)

There are no more non-vanishing Christoffel symbols. Note in particular that �ˇ tt D0, which is of physical significance, as shown in Ch. 14.

We now proceed by calculating the components of the Ricci tensor which weneed in order to find the form of the field equations for the present application.Inserting� D � D t in Eq. (B.15), and including only the non-vanishing Christoffelsymbols in the summation over ˛ and ˇ, we get

Rtt D �ˇ tt �˛ˇ˛ � �ˇt˛ �˛ˇt C �˛t t;˛ � �˛t˛;t

D � ��r tr �r

rt C �� t� �� t C �'t' �

''t

C�r tr;t C �� t�;t C �'t';t�:

Substituting the expressions (C.6)–(C.8) for the Christoffel symbols leads to

Rtt D �"

3Pa2a2

C 3

� Paa

�

;t

#

D ��3

Pa2a2

C 3Ra a � Pa2a2

�

D ��3

Pa2a2

C 3Raa

� 3 Pa2a2

�;

which gives

Rtt D �3 Raa: (C.15)

Inserting � D � D � in Eq. (B.15), we get

R�� D �ˇ�� ˛ˇ˛ � �ˇ�˛ �

˛ˇ� C �˛��;˛ � �˛�˛;� :

Performing the summation over ˛ and ˇ we find

R�� D �r��r rr C �� r� C �'r'

�C �t ��r tr C �� t� C �'t'

�

� �� t �t�� t��

�t� � �r�� r� � ��r �r ��

� �'�' �''� C �t ��;t C �r��;r � �'�';� :

Substituting the expressions (C.9)-(C.14) for the Christoffel symbols leads to

R�� D �r �1 � k r2��

k r

1 � k r2 C 2

r

�

C a Pa r2c2

3Paa

� 2Paa

a Pa r2c2

330 C Ricci tensor of the Robertson–Walker metric

C 2 r�1 � k r2� 1

r� cos2 �

sin2 �C�a Pa r2c2

�

;t

� �r�1 � k r2�

;r��

cos2 �

sin2 �

�

;�

: (C.16)

From Eq. (B.20) we have�

cos �

sin �

�

;�

D �1 � cos2 �

sin2 �: (C.17)

Furthermore�a Pa r2�

;tD .a Pa/;t r2 D Pa2 r2 C a Ra r2; (C.18)

and�r�1 � k r2�

;rD �

r � k r3�;r

D 1 � 3 k r2: (C.19)

Multiplying out the first terms in (C.16) and using the expressions (C.17)–(C.19) inthe last three terms, we get

R�� D �k r2 � 2�1� k r2

�C 3 Pa2 r2c2

� 2 Pa2 r2c2

C 2�1 � k r2� � cos2 �

sin2 �C a Ra r2

c2C Pa2 r2

c2

� 1C 3 k r2 C 1C cos2 �

sin2 �:

Collecting terms we get

R�� D a Ra r2c2

C 2 Pa2 r2c2

C 2 k r2:

Putting r2=c2 outside a parenthesis we finally have

R�� D �a RaC 2 Pa2 C 2 k c2

�r2=c2: (C.20)

Due to the isotropy of the models, we only need the two components of the Riccitensor that we have now calculated. Inserting the expressions (C.15) and (C.20) forRtt and R�� , respectively, into Einstein’s field equations, we arrive at a set of twodifferential equations. The solutions of these equations show how the expandingmotion of the universe models varies with time, and how the mass density evolvesin the models.

References

1. Ernst Cassirer. Zur Einsteinschen Relativitatstheorie: erkenntnistheoretische Betrachtungen.Bruno Cassirer, Berlin, 1921.

2. F. de Felice and C. J. S. Clarke. Relativity on Curved Manifolds. Cambridge University Press,Cambridge, 1992.

3. R. D’Inverno. Introducing Einstein’s Relativity. Clarendon Press, London, 1992.4. Albert Einstein. The World as I see it. Covici Friede, New York, 1934.5. J. Foster and J. D. Nightingale. A short course in General Relativity. Longman, London and

New York, 1979.6. I. R. Kenyon. General Relativity. Oxford University Press, 1990.7. M. Ludvigsen. General Relativity, A Geometric Approach. Cambridge University Press,

Cambridge, 1999.8. C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. Freeman and Company, San

Francisco, 1973.9. B. F. Schutz. A First Course in General Relativity. Cambridge University Press, Cambridge,

1985.


331

Index

Aabsolute

certainty, 125future, 114past, 114space, 80, 105, 126time, 80, 126, 225

acausal, 262, 263acceleration, 50, 58, 225

of gravity, 233, 234, 239, 242, 244, 247,248

action at a distance, 230, 232addition, of vectors, 7age of the universe, 298Alpha Centauri, 289Andromeda galaxy, 289angle, 17, 62–69angular

momentum, 206velocity, 258

Anschauungsformen, 125–128antiderivative, 51, 53antisymmetric, 14

product, 14apodictic, 125, 127area, 14, 34–36, 52–54arrow of time, 125arrow, vectors as, 1–5atlas, 3atomic nucleus, 226axis, 4

Bbasis vector, 4–11, 17, 77–79bending of light, 277–281Bergson, H., 128

Bianchi identity, 217–222Big Bang, 298black body radiation, 290black hole, 249, 281–287blue shift, xvii, 268, 269, 291body force, 200

Ccaesium clock, 270calculus of vectors, 1, 5–11Cartesian

basis vectors, 72components, 73coordinate system, 3

Cassirer, E., 127causal, 113, 238, 262, 263

relationship, 211theory, 211

causality paradox, 231, 232centrifugal

acceleration, 261force, 229, 261

centripetal acceleration, 261chain rule, 29–31charged body, 226Christoffel symbols, 129–143, 156–158,

319–322, 327–329of the first kind, 157of the second kind, 157symmetry of, 143

Christoffel, E. B., 129, 148circle, 44, 62–75, 172circular arc, 62, 171circulation, 179–186

density, 179


333

334 Index

classicaldynamics, 50fluid dynamics, 193mechanics, 187–209

clock, 84, 106–107, 116–120, 146caesium, 270natural, 260photon clock, 106standard, 106, 242, 260, 268, 294

COBE satellite, 303colour force, 227comma notation for partial derivatives, 149,

162comoving coordinate system, 84, 106, 258compass of intertia, 162component

equation, 48, 164metric tensor, 93of motion, 43of vector, 6tensor, 89–91vector, 8

covariant and contravariant, 88–89transformation, 80–84

composite function, 30connection coefficient, 132conservation

law, 187–209, 217–222of angular momentum, 206of energy, 187, 206of energy-momentum, 217, 241of mass, 187, 191, 206of momentum, 187, 206

conserved quantity, 217constant

of integration, 52of motion, 251–252

continuous, 23, 24continuum, 24contraction

Lorentz, 204of a tensor, 212of energy-momentum tensor, 212of the metric tensor, 213of the Ricci tensor, 221of the Riemann tensor, 215

contravariant, 88–91convective derivative, 198convergent series, 38coordinate

basis vector, 5, 61, 72–75, 79clock, 242, 258curve, 61differential, 18, 19, 35, 108

independent, 108invariance, 78, 83invariant, 145system, 3

orthogonal, 94skew-angled, 122

time, 294transformation, 80–97velocity, 281

cosineslaw of, 66

cosinus, 62–71cosmic

background radiation, xvii, 290, 307temperature, 303

clock, 307energy density, 299fluid, 188, 300gas, 188matter, 295, 298, 302, 305vacuum fluid, 309

cosmology, 169, 289–312Coulomb’s law, 232covariant, 88–91

derivative, 148–156directional, 146

differentiation, 145–158equation, 163, 235

critical mass density, 299curl, 179–181current density, 191curvature, 169–186

Einstein tensor, 222extrinsic, 172intrinsic, 172

of sphere, 293principal, 175Ricci tensor, 212

symmetry of, 216Riemann tensor, 185scalar, 221

curve parameter, 46Cygnus X-1, 287cylindrical

coordinates, 137surface, 175symmetry, 61

Ddark matter, 303de Fermat, P., 52definite integral, 53degree, 62

Index 335

derivative, 24Descartes, R., 3diagonal metric tensor, 94differential

calculus, 21–42equation, 51

differentiation, 21–37direction, 123directional derivative, 146discontinuity, 23displacement vector, 179distance, 108–123divergence, 188–191, 205–209divergent series, 38Doppler effect, 290dot product, 8, 98dummy index, 83dust, 249

dominated universe, 302–305dynamics, 238

EEarth, 145, 230, 239

radius of, 268velocity, 105

ebb, 239Eddingston–Finkelstein coordinates, 281Eddington, A. S., v, 99Einstein train, 84Einstein’s

field equations, 211–224gravitational constant, 247summation convention, 18

Einstein, A., 104, 105, 108, 145, 182, 211, 230,237, 260, 268, 277, 287

electromagneticforce, 226signal, 113theory, 212wave, 307

electroweak force, 307Elsbach, A., 127elsewhere (neither past nor future), 114empty spacetime, 224energy, 80, 155, 187, 203

and mass, 204density, 295kinetic, 203, 206potential, 206, 232

energy-momentum tensor, 155, 203–209, 212,222–224, 246–249, 295, 307–309

symmetry of, 209equation of continuity, 191–193, 254

equation of motion, 50, 58, 200equation of state, 300Equator, 19, 138, 161, 176, 271equipotential surface, 233ether, 238Euclid, 126Euclidean space, 19, 80, 100, 169, 177Euclidean spatial geometry, 297Euler’s equation of motion, 200, 207Euler’s number, 55Euler, L., 55event, 106, 115–116expansion of the Universe, 124, 290, 295, 309

exponential, 311models of, 327

exponential function, 54–57

Ffalsifiable, 127, 255Faraday, M., 232ficticious gravitational field, 257fictive force, 261field

concept, 232gravitational, 233line, 239of weight, 238theory, 232–235

finite series, 38flood, 239flow, 179fluid, 179, 188, 193flux, 188–192force, 47, 50, 200, 211, 226–230, 239

field, 206gravitational, 226law, 233

forceless motion, 58form-invariant, see covarientfour-dimensional spacetime, 17four-velocity, 200–203fraction, 26, 32Frank, P., 135frequency, 268friction, 194, 226Friedmann equations, 297Friedmann universe models, 297Friedmann, A., 291, 327fundamental force, 226

GGodel, K., 162galaxy, 124, 289

336 Index

Galileaninvariance, 86kinematics, 104transformation, 84–86, 100, 104velocity addition, 271

gas, 188, 249Gauss, C. F., 38, 178Gaussian curvature, 175Gellmann, M., 227general principle of relativity, 240geodesic

equation, 163–166, 211, 242, 251, 273postulate, 249

geodesics, 80, 159–167God, 3, 83, 104, 212, 230Goethe, J. W., 13, 228gradient, 197, 198, 233, 313Grand Unified Theory, 307, 310graph, 23–27, 169, 171, 214gravitation, 17, 61, 100, 211–212, 225–255gravitational

attraction, 235, 241, 247great circle, 166Greek indices, 17, 190grid, 3Grossmann, M., 145

HHafele, J. C., 269Hafele–Keating experiment, 269–272homogeneity of the universe, 290homogeneous

and isotropic universe models, 291–305gravitational field, 239transformation, 82

horizon, xvii, 286, 287Hubble

age, 298–310factor, 297telescope, 289

Hubble’s law, 290, 297Hubble, E. P., 124, 290hydrodynamics, 187–209hyperbolic geometry, 175hypothenus, 49

Iimaginary number, 115implicit definition, 228indefinite integral, 51induction, proof by, 31

inertialforce, 208motion, 145reference frame, 204, 228, 240rest frame, 208

infinite series, 38inflationary cosmology, 306–312inhomogeneous

gravitational field, xvii, 239integration, 51–54interaction, 229interval, 41

spacetime, 115–123time, 80

invariant, 83, 120, 164, 204time, 201

inversefunction, 55transformation, 83

irreversibility of time, 125irrotational velocity, 179isolated system, 217isotropic

and homogeneous universe models,291–305, 327–330

pressure, 195isotropy of the universe, 290

JJoyce, J., 227

KKaluza–Klein theory, 61Kant, I, 125–128Keating, R. C., 269kinematical, 211kinematical concepts, 123kinematics, 238kinetic

energy, 203, 206energy-momentum tensor, 204

Kronecker symbol, 83Kronecker, L., 83

LLaplace operator, 233, 313–318Laplace’s equation, 234latitude, 138Leibniz, G. W., 21, 24, 52, 53

Index 337

lightcone, 113–115coordinate velocity, 281second, 111speed, 17, 101

as conversion factor, 202invariance, 100, 124

year, 289light-like, 116line element, 108–123linear combination, 79linear dependence, 77linear transformation, 82local

derivative, 198frame, 239

locally Cartesian coordinates, 166–167logarithm, 54–57long-range force, 227longitude, 138Lorentz

contraction, 204invariant, 111–118transformation, 100–106

Lorentz, H. A., 104lowering indices, 98

MMacLaurin series, 38–42MacLaurin, C., 38magnetic field of the Earth, 162magnitude

of vector, 2, 16–20, 120manifestly covariant, 236map, 3mass, 187–193, 228–237

and energy, 204conservation, 187, 193, 206density, 188

gravitational, 248gravitational, 228, 236–237inertial, 228, 236–237negative, 309relativistic, 204

mass-energy, 193material derivative, 198–199mathematical

existence, 79model, 97, 125physics, 124

matrix, 83, 194matter, 212

distribution of the universe, 290

Maxwell’s theory, 212Maxwell, J. C., 232, 307measuring

accuracy, 239rod, 84

mechanical experiment, 238Mercury

perihelion precession of, 273–277metric, 16–19

components, 93contravariant, 97covariant, 97mixed, 98

Minkowski, 121non-diagonal, 123Robertson–Walker, 294Schwarzschild, 267symmetry, 93tensor, 77–128

Michelson, A. A., 105Michelson–Morley experiment, 105, 307Milky Way, 289Minkowski

diagram, 113–115metric, 121spacetime, 121

Minkowski, H., 106, 187molecular bound, 226momentum, 155, 187, 217Moon, 226, 239Morley, E. W., 105motion, 123, 188, 193

Nnatural

law, 80, 145, 188, 228logarithm, 55

negative mass density, 309neutron, 226, 227

star, 228Newton’s

first law, 227gravitational constant, 229law of gravitation, 206, 228, 230, 234, 241,

266second law, 50–51, 58, 200, 206, 211, 228theory of gravitation, 225, 229–230,

253–254third law, 229, 230

Newton, I., 21, 50, 52, 53, 105, 126, 226, 253,287

338 Index

Newtonianabsolute simultaneity, 106approximation, 205dynamics, 50, 58, 86, 188energy-momentum tensor, 203–208gravitation, 187, 253–254hydrodynamics, 208kinematics, 104, 105mechanics, 84–86, 187, 206physics, 80, 187, 191, 206time, 151universe, 124, 186

Noether, E., 206non-Euclidean, 121non-inertial, 238, 240, 241normal

direction, 14stress, 194vector, 193

Northrop, F. S. C., 127null

geodesics, 240, 281, 284vector, 7

Oorbit

elliptic, 273orthogonal, 5

Pparabola, 23, 43, 171parabolic geometry, 175parallel transport, 2, 159–163, 181–185parallelogram, 14parameter, 43–47, 74, 146, 162parametric equation, 44, 74partial derivative, 34–37, 132

comma notation, 149particle, 58

massive, 113of light, 113

path, 43path length, 146, 166pendulum as ‘compass of intertia’, 162perfect fluid, 188, 193, 203–205perihelion, 273

precession, 273–277periodic function, 63philosopher, vii, 29philosophy, 125–128

of science, 253possibilism, 237

photon, 113, 114, 281clock, 106mass, 113worldline, 113

physicalconcept, 123, 126description, 124observable, 121, 236, 287phenomenon, 110, 124, 235process, 230, 238reality, 105, 126science, 289space, 80spacetime, 124, 186system, 313

physicist, v, 125, 182, 187pi, 62, 175–178Planck time, 298, 305, 306Planck’s constant, 298plane

curve, 169–171surface, xvii, 159, 182

planet, 229, 273Poincare, H., 127Poisson’s equation, 233, 235polar coordinates, 71–75, 109, 130–135polygon, 77, 78position, 3, 123

vector, 46possibilism, 237postulate, 100potential

energy, 206gravitational, 232–235, 244, 261

Pound, R. V., 268Pound–Rebka experiment, 268–269power

function, 31derivative of, 31–32

series, 38pre-relativistic conceptions, 80precisation, 169pressure, 188, 194, 248, 295

force, 198gradient, 198

principle ofconstancy of the velocity of light, 230, 235covariance, 235, 236equivalence, 236, 241, 257, 260

strong, 238weak, 235, 237, 238

relativity, 230, 235product, 6–16

rule, 28–29

Index 339

projection, 9–11proper

density, 204time, 118–120, 201–203, 260, 270, 294

proton, 227pulsar, 228Pythagoras, 8Pythagorean theorem, 8

extended, 66

Qquantum

field theory, 238, 306gravity, 298mechanics, 227phenomena, 254theory, 61

quark, 227

Rradian, 62radiation, 212

black body, xvii, 290cosmic background, xvii, 290dominated universe, 302–305

radio telescope, 289radioactive emission, 226raising indices, 98rank, 98Rebka, G. A., 268rectilinear motion, 145red shift, 290reference

frame, 84accelerated, 238freely moving, 80, 228, 238

system, 80vector, 3, 77

region of applicability, 253–254, 298Reichenbach, H., 126, 127relational simultaneity, 126relative velocity, 107relativistic

cosmology, 289–312distance, 9energy-momentum tensor, 208–209gravitational mass density, 248, 298hydrodynamics, 191mass, 235mass increase, 204physics, vi

theory of electromagnetism, 232theory of gravitation, 182time dilation, 106–107velocity, 208

relativityof simultaneity, 80, 102, 106, 126, 231principle, 86, 101, 126

remoteness, 111repulsion

gravitational, viii, 247–249, 309rest mass, 113Ricci tensor, 212Riemann tensor, 185Riemannian geometry, 80right angled triangle, 49Robertson, H. P., 327Robertson–Walker line element, 294, 327rotating reference frame, 258–259

Ssatellite, 240, 271, 303scalar, 6

component, 6product, 8, 90–95, 98, 120, 121, 179

scale factor, 294, 295, 300, 301Schilpp, A., 126Schwarzschild

horizon, 287radius, 267solution, 263–267time coordinate, 282

sea level, 232semicolon notation for covariant derivative,

149series expansion, 38–42shear stress, 194shortest path, 119, 159simultaneity, 80, 102, 105, 106, 126, 128, 231simultaneous, 80, 102, 106, 115, 117, 225, 231sink, 188, 233sinus, 62–71skew-angled, 122, 123slope, 21–27Solar system, 242, 267source, 188, 189, 233, 235, 249, 281space, 16–20, 77–80, 123–128space-like, 116spacecraft, 232, 289spacetime, 15, 17, 79–80, 115–120, 225–255spectral distribution, 290spectral lines, 290, 297

340 Index

speed, 201spherical

coordinates, 138–143, 263, 313–318surface, 16, 47, 62, 101, 161, 164–166,

175–176symmetry, 234, 263, 264, 266, 273,

313–330spiral galaxy, 289standard model, 305–306static, 263, 319static gravitational field, 242Stefan–Boltzmann law, 303, 307straight line, 46–47, 49straightest possible curve, 80, 119, 159–167stream line, 179stress tensor, 193–195

symmetry, 195structure

of reality, 105of thought, 105

subscript, 5, 78, 180subtraction, of vectors, 7Sudarshan, C. G. S., 231suffix, 6Sun, 105, 145, 212, 230, 239, 266, 273superscript, 78, 180superstring theory, 61surface, 4

curvature, 172–178force, 195–198

symmetric tensor, 93, 204, 209, 222synchronize, 242, 258, 268, 282

Ttachyon, 231

telephone, 231tangens, 62–71tangent

line, 21slope, 25–27

space, 19vector, 43–59

Taylor series, 38–42Taylor, B., 38tension, 194, 249, 295tensor, 80, 89–99

basis, 91component, 80contravariant, 91covariant, 91mixed, 91product, 91

theorema egregium, 178, 186theory of gravitation, 225–255theory of relativity

general, 1, 17, 80, 100, 145, 211–224,235–247

special, 80, 100, 230–235thermodynamics, 217thermometer, 198tidal force, 239time, 115–120, 125–128

dilation, 106–107gravitational, 259–263, 268

dimension, 61transformation, 102travel, 61

time-like, 116topology, 172total derivative, 198total differential, 35trigonometric functions, 62–71

Uunified theory, 254, 287, 306uniform field, 232uniform motion, 145unit

distance, 3length, 17mass, 232vector, 2, 74, 131, 135, 142

universal time, 105universe, 187, 289–312

expanding, 327expansion, 124model, 289, 327

Vvacuum, 264, 306–312

domination, 309energy, 309fluid, 308, 309

vector, 1–20addition, 7basis, 78component, 78product, 14–16subtraction, 7

velocity, 188viscosity, 295

Wwace, 79Walker, A. G., 327

Index 341

wave, gravitational, 212, 231weak

gravitational field, 242–247nuclear force, 226, 307principle of equivalence, 235

weight, 232, 238

Wheeler, J. A., 211

work, 232

worldline, 113–115, 119, 124, 281, 287

Documents

Appendix A The Laplacian in a spherical coordinate system978-1-4614-0706-5/1.pdfThe Laplacian in a spherical coordinate system In order to be able to deduce the most important physical