Astrophysics

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vi Magnetic fields in Relativistic Astrophysics


Preface

There will be a preface.

Serguei Komissarov

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viii Magnetic fields in Relativistic Astrophysics


Contents

Preface vii

Metric space and Tensor Calculus 1

1. From Euclidean space to surfaces and metric manifolds 3

1.1 Metric form . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 The notion of metric form . . . . . . . . . . . . . 3

1.1.2 Metric forms of surfaces: . . . . . . . . . . . . . . 5

1.1.3 Locally Cartesian coordinates: . . . . . . . . . . . 6

1.1.4 Lengths of curves . . . . . . . . . . . . . . . . . . 7

1.1.5 Coordinate transformations: . . . . . . . . . . . . 7

1.2 Vectors, bases, and components of vectors . . . . . . . . . 8

1.2.1 Coordinate bases . . . . . . . . . . . . . . . . . . 8

1.2.2 Coordinate transformations . . . . . . . . . . . . . 10

1.3 Metric form and the scalar product . . . . . . . . . . . . . 10

1.4 Geodesics and the variational principle . . . . . . . . . . . 12

1.4.1 Euler-Lagrange Theorem . . . . . . . . . . . . . . 12

1.4.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.3 Examples of geodesics: . . . . . . . . . . . . . . . 14

1.5 Non-Euclidean geometry of a Euclidean sphere . . . . . . 15

1.6 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7 Vectors as operators . . . . . . . . . . . . . . . . . . . . . 17

1.7.1 Basic idea . . . . . . . . . . . . . . . . . . . . . . 17

1.7.2 Coordinate transformations . . . . . . . . . . . . . 18

1.7.3 Magnitudes of vectors and the scalar product . . . 19

ix


x Magnetic fields in Relativistic Astrophysics

2. Tensors 21

2.1 Tensors as operators . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 1-forms as operators acting on vectors . . . . . . . 21

2.1.2 Vectors as operators acting on 1-forms . . . . . . 22

2.1.3 Tensors as operators acting on vectors and 1-forms 23

2.1.4 Metric tensor . . . . . . . . . . . . . . . . . . . . . 24

2.1.5 Constructing higher rank tensors via outer multi-

plication of vectors and 1-forms . . . . . . . . . . 24

2.2 Bases and components of tensors . . . . . . . . . . . . . . 25

2.2.1 Induced basis . . . . . . . . . . . . . . . . . . . . 25

2.2.2 Index notation of tensors . . . . . . . . . . . . . . 27

2.2.3 Coordinate bases . . . . . . . . . . . . . . . . . . 27

2.2.4 Coordinate components of df . . . . . . . . . . . . 27

2.2.5 Metric form and metric tensor . . . . . . . . . . . 28

2.3 Basis transformation . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Transformation of induced bases . . . . . . . . . . 29

2.3.2 Transformation of components . . . . . . . . . . . 30

2.4 Basic tensor operations and tensor equations . . . . . . . 31

2.4.1 Contraction . . . . . . . . . . . . . . . . . . . . . 32

2.4.2 Contraction of two tensors . . . . . . . . . . . . . 32

2.4.3 Raising and lowering indexes . . . . . . . . . . . . 32

2.4.4 Tensor equations . . . . . . . . . . . . . . . . . . . 34

2.5 Symmetric and antisymmetric tensors . . . . . . . . . . . 35

2.5.1 Symmetry with respect to a pair of indexes . . . 35

2.5.2 Antisymmetry with respect to a pair of indexes . 36

2.6 Levi-Civita Tensor and the vector product . . . . . . . . . 37

2.6.1 Levi-Civita and the generalised Kronecker symbols 37

2.6.2 Levi-Civita Tensor . . . . . . . . . . . . . . . . . . 38

2.6.3 Dual tensors . . . . . . . . . . . . . . . . . . . . . 40

3. Geometry of Riemannian manifolds 41

3.1 Parallel transport and Connection on metric manifolds . . 41

3.1.1 Parallel transport of vectors. Connection . . . . . 42

3.1.2 Connection of Euclidean space . . . . . . . . . . . 43

3.1.3 Riemannian Connection . . . . . . . . . . . . . . . 43

3.2 Parallel transport of tensors . . . . . . . . . . . . . . . . . 45

3.2.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.2 1-forms . . . . . . . . . . . . . . . . . . . . . . . . 45


Contents xi

3.2.3 General tensors . . . . . . . . . . . . . . . . . . . 46

3.2.4 Metric tensor . . . . . . . . . . . . . . . . . . . . . 46

3.3 Absolute and covariant derivatives . . . . . . . . . . . . . 47

3.3.1 Absolute and covariant derivatives of scalar fields 48

3.3.2 Absolute and covariant derivatives of vector fields 48

3.3.3 Absolute and covariant derivatives of 1-form fields 49

3.3.4 Absolute and covariant derivatives of general ten-

sor fields . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.5 General properties of covariant differentiation . . 51

3.3.6 The field of metric tensor . . . . . . . . . . . . . . 51

3.4 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5 Geodesics and parallel transport . . . . . . . . . . . . . . 54

3.6 Geodesic coordinates and Fermi coordinates . . . . . . . . 56

3.6.1 Geodesic coordinates . . . . . . . . . . . . . . . . 56

3.6.2 Fermi coordinates . . . . . . . . . . . . . . . . . . 58

3.7 Riemann curvature tensor . . . . . . . . . . . . . . . . . . 60

3.8 Properties of the Riemann curvature tensor . . . . . . . . 64

3.9 Ricci tensor, curvature scalar and the Einstein tensor . . . 65

Basic Theory of Relativity 67

4. Space and time in the theory of relativity 69

4.1 Space and Time in Newtonian Physics . . . . . . . . . . . 69

4.1.1 Time . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.2 Space . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.3 Inertial frames . . . . . . . . . . . . . . . . . . . . 70

4.1.4 Newtonian principle of relativity: . . . . . . . . . 71

4.2 Space and Time in Special Relativity . . . . . . . . . . . . 71

4.2.1 Spacetime . . . . . . . . . . . . . . . . . . . . . . 71

4.2.2 Special principle of relativity . . . . . . . . . . . . 73

4.3 Space and Time in General Relativity . . . . . . . . . . . 75

4.3.1 Spacetime . . . . . . . . . . . . . . . . . . . . . . 75

4.3.2 General principle of relativity . . . . . . . . . . . 76

4.3.3 Locally inertial frames . . . . . . . . . . . . . . . 77

4.4 Relativistic particle dynamics . . . . . . . . . . . . . . . . 77

4.5 Conservation laws . . . . . . . . . . . . . . . . . . . . . . 79

4.6 Relativistic continuity equation . . . . . . . . . . . . . . . 81

4.7 Stress-energy-momentum tensor . . . . . . . . . . . . . . . 82


xii Magnetic fields in Relativistic Astrophysics

4.7.1 Stress-energy-momentum tensor of dust . . . . . . 82

4.7.2 Energy-momentum conservation . . . . . . . . . . 83

4.7.3 Stress-energy-momentum tensor of perfect fluid . 84

4.8 Einstein’s equations of gravitational field . . . . . . . . . 85

4.9 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . 90

5. Schwarzschild Solution 95

5.1 Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . 95

5.1.1 Schwarzschild Solution in Schwarzschild coordinates 95

5.1.2 Schwarzschild Solution in Kerr coordinates . . . . 98

5.1.3 Event horizon . . . . . . . . . . . . . . . . . . . . 99

5.2 Gravitational redshift . . . . . . . . . . . . . . . . . . . . 101

5.3 Integrals of motion of free test particles in Schwarzschild

spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4 Orbits of test particles in the Schwarzschild geometry . . 107

Index 113


PART 1

Metric space and Tensor Calculus

1


2


Chapter 1

From Euclidean space to surfaces andmetric manifolds

1.1 Metric form

1.1.1 The notion of metric form

Consider a plane in a 3-dimensional (3D) Euclidean space. This plane is

a 2D Euclidean space. Therefore, we can introduce Cartesian coordinates

{x, y} for its points:

Fig. 1.1

If dl is the distance between infinitesimally close points (x, y) and (x +

dx, y + dy) then

dl2 = dx2 + dy2. (1.1)

This is the metric form of the plane in Cartesian coordinates {x, y}. We

may introduce new coordinates {x1, x2} which are not Cartesian ( In fact

their coordinate lines can be curved, in which case these coordinates will

be called curvilinear.). For example,

x1 = x− y, x2 = x− 2y. (1.2)

What is dl in terms of dx1 and dx2? From eq.(1.2) one has

dx = 2dx1 − dx2, dy = dx1 − dx2,

which leads to

dl2 = dx2 + dy2 = 5(dx1)2 − 6dx1dx2 + 2(dx2)2.

3


4 Magnetic fields in Relativistic Astrophysics

We may write this as

dl2 =

2∑i=1

2∑j=1

gijdxidxj (1.3)

where

g11 = 5, g12 = g21 = −3, g22 = 2.

This result shows that for any choice of coordinates the metric form can be

written as in eq.(1.3) with gij = gji and only for Cartesian coordinates

g11 = 1, g12 = g21 = 0, g22 = 1.

Coefficients gij of the metric form are often shown as components of a

symmetric square matrix. For example

gij =

(1 0

0 1

)and gij =

(5 −3

−3 2

),

for the metric forms given by Eqs.1.1 and 1.3 respectively.

If instead of a 2D Euclidean plane we consider an n-dimensional Eu-

clidean space then we obtain a similar result: the distance between its two

infinitesimally close points can be written as

dl2 =

n∑i=1

n∑j=1

gijdxidxj where gij = gji (1.4)

for any set of coordinates {xi}, i = 1, 2, ..., n. The summation symbols can

be eliminated in this and other similar equations if we adopt the following

convention

Any index appearing once as a lower index and once as an upper index of

the same indexed object or in the product of a number of indexed objects

stands for summation over this index. Such index is called a dummy index.

Indexes which are not dummy are called free indexes.

This is known as the Einstein summation rule. Thus, Eq.(1.4) can be

written in the following concise form:

dl2 = gijdxidxj . (1.5)

This rule allows to simplify expressions involving multiple summations.

Here are some more examples:

(1) aibi stands for

∑ni=1 aib

i; here i is a dummy index;


From Euclidean space to surfaces and metric manifolds 5

(2) aibi stands for a product of ai and bi where i can have any value between

1 and n; here i is a free index.

(3) aibkij stands for

∑ni=1 aib

kij ; here k and j are free indexes and i is a

dummy index;

(4) ai ∂f∂xi stands for∑ni=1 a

i ∂f∂xi ; thus, index i in the partial derivative ∂

∂xi

is treated as a lower index;

1.1.2 Metric forms of surfaces:

For any smooth surface in Euclidean space the distance between its any two

infinitesimally close points can be found in terms of coordinates introduced

on the surface. For example, consider a sphere of radius r in 3D Euclidean

space. This is a 2D surface and one needs two coordinates to mark its

points. Introduce the usual spherical coordinates {θ, φ}.

Fig. 1.2

Then for the Cartesian coordinates {x, y, z} shown in the figurex = r sin θ cosφ,

y = r sin θ sinφ,

z = r cos θ

.

This gives us dx = r cos θ cosφdθ − r sin θ sinφdφ,

dy = r cos θ sinφdθ + r sin θ cosφdφ,

dz = −r sin θdθ

and

dl2 = dx2 + dy2 + dz2 = ... = r2dθ2 + r2sin2θdφ2. (1.6)

Thus,

gij =

(r2 0

0 r2 sin2 θ

)where we assume that x1 = θ and x2 = φ.



1.1.3 Locally Cartesian coordinates:

It is impossible to introduce global Cartesian coordinates for the whole

sphere. That is there are no coordinates x1 and x2 such that

dl2 = (dx1)2 + (dx2)2

everywhere on the sphere (this will become clear in Sec.2.6.3.). However,

there exist so-called locally Cartesian coordinates.

Take some point of the sphere, denote it as A. Suppose its spherical

coordinates are θa and φa. Near A introduce new coordinates{x1 = r(θ − θa)

x2 = r sin θa(φ− φa).

Then {dx1 = rdθ

dx2 = r sin θadφ,

and {dθ = dx1/r

dφ = dx2/r sin θa.

Substitute these into eq.(1.6) to obtain the metric form

dl2 = (dx1)2 +

(sin θ

sin θa

)2

(dx2)2.

At the point A this becomes

dl2 = (dx1)2 + (dx2)2.

Thus, near point A the metric form is the same as the metric form of a 2D

Euclidean space with Cartesian coordinates {xi}. Because of this property,

the sphere is called ”locally Euclidean” or ”Riemannian”. (All smooth

surfaces in Euclidean space are locally Euclidean.)

1.1.4 Lengths of curves

Let {xi} be some arbitrary coordinates in of a Euclidean space or some

surface in this space. Consider a curve xi = xi(λ) in the space or on this

surface. Here λ is the curve parameter. One can view it as a coordinate

introduced specifically for the points of the curve.

The length of the curve between its any two points, A and B, is given by

∆l =

B∫A

dl =

B∫A

(gijdxidxj)1/2 =

λB∫λA

(gijdxi

dλ

dxj

dλ

)1/2

dλ. (1.7)



Fig. 1.3

1.1.5 Coordinate transformations:

Introduce arbitrary new coordinates {xi′}. They can be expressed as func-

tions of the old coordinates xi

xi′

= xi′(xi)

and, thus,

dxi =∂xi

∂xi′dxi

′.

Then

dl2 = gijdxidxj = gij

∂xi

∂xi′∂xj

∂xj′dxi

′dxj

′, (1.8)

which tells us that

gi′j′ =∂xl

∂xi′∂xm

∂xj′glm. (1.9)

This equation shows how the components of metric form transform as the

result of coordinate transformation.

1.2 Vectors, bases, and components of vectors

In Euclidean geometry vectors are traditionally defined as straight arrows.

The magnitude of a vector is the length of the arrow. We denote it as |a|.

1.2.1 Coordinate bases

Let {xi} be Cartesian coordinates of n-dimensional Euclidean space. Let

ei be the unit vectors pointing in the direction of the xi-coordinate axis.

The set of all n vectors ei at any point of the space forms a vector basis at

this point, the Cartesian basis. If

a = aiei



then ai are the components of a in this basis. Vector

r = xkek. (1.10)

whose base coincides with the origin of the coordinate system and whose

tip coincides with the point with coordinates xk is called the position vector

of this point.

Introduce arbitrary new coordinates {xi′} whose coordinate lines may

be curved. xi′

are functions of the old Cartesian coordinates xk:

xi′

= xi′(xk).

Inversely, xk are functions of xi′:

xk = xk(xi′).

Fig. 1.4

The set of vectors

ei′ = ∂r/∂xi′

(1.11)

defined at the point with position vector r provides us with a basis which

is called the ”coordinate basis” of the {xi′} coordinates at this point. ei′

is tangent to the xi′−coordinate line passing through this point. In fact,

{ek} is the coordinate basis of original Cartesian coordinates, because

ej =∂r

∂xj.

If a = ai′ei′ then ai

′are called the components of a in the basis {ei′}.

The coordinate basis is often much more convenient then any other pos-

sible basis. The reason for this is the following. Consider the infinitesimally

small vector dx connecting points with coordinates xi′

and xi′+ dxi

′.

dx = r(xi′+ dxi

′)− r(xi

′) = r(xi

′) +

∂r

∂xk′dxk

′− r(xi

′) = dxk

′ek′ .

Thus, the components of dx in the coordinate basis are dxk′, irrespectively

of whether the coordinates are Cartesian or not.



1.2.2 Coordinate transformations

Consider transformation from coordinates {xi} to coordinates {xi′}, both

being arbitrary curvilinear coordinates. As the result the coordinate basis

and components of vectors in this basis will transform too. Let us find first

the transformation rule for basis vectors.

ei′ =∂r

∂xi′=∂xm

∂xi′∂r

∂xm=∂xm

∂xi′em.

Thus,

ei′ =∂xk

∂xi′ek. (1.12)

Inversely,

ek =∂xi

′

∂xkei′ . (1.13)

Next we find the transformation rule for components of vectors.

a = ai′ei′ = ai

′ ∂xk

∂xi′ek.

Thus,

ak =∂xk

∂xi′ai

′. (1.14)

Inversely,

ak = ai′

=∂xi

′

∂xkak. (1.15)

1.3 Metric form and the scalar product

If ai and bi are the Cartesian components of vectors a and b then

a · b =

n∑i=1

aibi. (1.16)

|a|2 = a · a =

n∑i=1

(ai)2. (1.17)

The first equation can also be written as

a · b = gijaibj . (1.18)

where gij are the Cartesian components of the metric form (see eq.1.7).



In fact, if ai′

and bi′

are the components of a and b and gi′j′ are the com-

ponents of the metric form in the coordinate basis of any other coordinate

system we still have

a · b = gi′j′ai′bj

′. (1.19)

Thus, expression (1.18) for the scalar product of two vectors is invariant

under coordinate transformations ! Indeed, using Eq.1.9 and 1.14 we obtain

gijaibj =

∂xl′

∂xi∂xm

′

∂xjgl′m′aibj =

= gl′m′

(∂xl

′

∂xiai

)(∂xm

′

∂xjbj

)= gl′m′al

′bm

′.

If gij are the components of the metric form in some coordinate system

and {ei} is the coordinate basis of this system then

gij = ei · ej . (1.20)

Indeed, first we can write

ei = δki ek and ej = δkj ek,

where

δkj =

{1 if k = j

0 if k 6= j(1.21)

is the Kronecker’s symbol. Then according to Eq.1.18 we have

ei · ej = δki ekδmj gkm = δki ekgkj = gij .

Consider an infinitesimally small vector dx connecting points with co-

ordinates xi and xi + dxi. The components of dx in the coordinate basis

are dxi. The magnitude of dx is the distance dl between the points. Then

from the invariant expression eq.(1.19) one has

dl2 = dx · dx = gijdxidxj (1.22)

in agreement with eq.(1.5)



1.4 Geodesics and the variational principle

1.4.1 Euler-Lagrange Theorem

Consider the functional

lAB =

λB∫λA

L(xk, xk)dλ (1.23)

where xk = xk(λ) (k = 1, 2, ..., n) are functions of λ and xk = dxk/dλ. The

variation δlAB of lAB due to the variations δxk(λ) of xk(λ) can be found

as

δlAB =

λB∫λA

∂L

∂xkδxk +

∂L

∂xkδxkdλ =

λB∫λA

∂L

∂xkδxk +

∂L

∂xkdδxk

dλdλ.

Integrating the second term by parts we obtain

δlAB =

λB∫λA

(∂L

∂xk+

d

dλ

∂L

∂xk

)δxkdλ+

[∂L

∂xkδxx]λB

λA

.

Function that extremize lAB must satisfy the condition δlAB = 0. If we

constrain ourself only to functions that satisfy the boundary conditions

xk(λA) = xkA, xk(λB) = xkB (1.24)

then the second term in this equation vanishes and the condition of ex-

tremum impliesd

dλ

∂L

∂xk− ∂L

∂xk= 0 (k = 1, 2, . . . , n). (1.25)

These ordinary differential equations (ODEs) are known as the Euler-

Lagrange equations.

1.4.2 Geodesics

Consider an n-dimensional smooth hypersurface1 in some higher dimen-

sional Euclidean space. Let xi to be some arbitrary coordinates on this

surface and gij are the corresponding components of the metric form. By

geodesics we understand curves on this surface that extremise distances be-

tween all its points. Consider some curve xk = xk(λ) connecting points A

and B with coordinates xkA and xkB , that is

xk(λA) = xkA, xk(λB) = xkB .

1The name hypersurface is used to stress that we are not necessary dealing with two-

dimensional surfaces in three-dimensional Euclidean space.



According to Eq.1.7 the distance between A and B along this curve can be

written as the functional

lAB =

λB∫λA

L(xk, xk)dλ

with the Lagrangian

L(xk, xk) = [gij xixj ]1/2. (1.26)

Note that in general gij = gij(xk). Given the results of the previous section,

we can conclude that geodesics must be solutions of the Euler-Lagrange

equations with this Lagrangian. However, instead of the Lagrangian (1.26)

one can also use the Lagrangian

L(xk, xk) = gij xixj , (1.27)

which is more convenient. This will result in the same curves but with

different parameterization. The new parameter, say µ, will be the so-called

normal parameter, that is such a parameter that

dµ = adl,

where a =const and l is the length of the geodesic (as measured from an

arbitrary point of the geodesic). To show this first introduce function

S = L2 = gjkdxj

dλ

dxk

dλ.

Then Eqs.1.25 with the Lagrangian L yeild

1

S1/2

d

dλ

(1

S1/22gij

dxj

dλ

)− 1

S

∂gjk∂xi

dxj

dλ

dxk

dλ= 0.

Now introduce new parameter µ via dµ = aS1/2dλ = adl and obtain

d

dµ2gij

dxj

dµ− ∂gjk

∂xidxj

dµ

dxk

dµ= 0,

which is equivalent to the Euler-Lagrange equation

d

dµ

∂L

∂xk− ∂L

∂xk= 0,

with the Lagrangian (1.27).



1.4.3 Examples of geodesics:

1.4.3.1 Euclidean space

If the hyper-surface is a hyper-plane and xk are its Cartesian coordinates

then the Lagrangian (1.27) reads

L =

n∑i=k

(xk)2

and the corresponding Euler-Lagrange equations reduce to

dxk

dλ= 0,

The solutions of these equations,

xk(λ) = akλ+ bk,

describe straight lines.

1.4.3.2 Sphere

Now consider a sphere of radius r in three-dimensional Euclidean space. If

{θ, φ} are the spherical coordinates and the sphere is centered on the origin

of this coordinate system then using Eqs.1.27 and 1.6 we obtain

L = r2(θ2 + sin2 θφ2).

The corresponding Euler-Lagrange equations are

ddλ

(sin2 θ dφdλ

)= 0

ddλ

(dθdλ

)− sin θ cos θ

(dφdλ

)2= 0.

It is easy to verify that functions

θ(λ) = aλ, φ(λ) = b

are particular solutions to these equations. They describe the ”meridians”

of the sphere. Each such meridian is a ”great circle”, that is the curve

formed by the intersection of the sphere and a plane passing through its

center. The meridians also pass through the north and the south poles of

the sphere. However, for any great circle one can find such a system of

polar coordinates that this great circle is one of its meridians. Thus all

geodesics of the sphere are great circles and any great circle is a geodesic.



Fig. 1.5 Meridian - an example of a great circle

1.5 Non-Euclidean geometry of a Euclidean sphere

Geodesics is a generalization of straight lines. Using geodesics one can

build various geometrical constructions on surfaces analogous to those of

Euclidean spaces e.g. circles, triangles, rectangles etc. They will have

somewhat different geometrical properties.

Consider a 2D sphere in a 3D Euclidean space. In contrast to a 2D

Euclidean space one finds the following properties:

• Geodesics of the sphere are closed curves (Top-left panel of fig.1.6);

• Different geodesics intersect at more than one point; (Top-right panel

of fig.1.6);

• The sum of angles of a triangle exceeds 2π; (Bottom-left panel of

fig.1.6);

• The circumference of a circle of radius R is l = 2πr sin(R/r) < 2πR.

(Bottom-right panel of fig.1.6);

Fig. 1.6 Non-Euclidean properties of spheres



1.6 Manifolds

A set of points, M, is called an n-dimensional manifold if any point of Mhas a neighbourhood that allows one-to-one continuous map onto an open

set in Rn (n-dimensional real space). In other words one can introduce n

continuous coordinates at least locally.

A n-dimensional manifold, M, is called a space if there exists a one-

to-one continuous map of the whole of M onto the whole of Rn. In other

words one can introduce n continuous coordinates globally.

When a manifold is attributed with distance between its points, via a

metric form (metric tensor), it is called a metric manifold.

A metric manifold is called Riemannian (or locally Euclidean) if for its

every point there exist local coordinates such that the metric form at this

point has the components

glm =

{1 if l = m;

0 if l 6= m.(1.28)

Such coordinates are called locally Cartesian.

Like in the case of the sphere considered in the previous section one can

use geodesics to build various geometrical constructions on Riemannian

manifolds, and their properties may well be very different from those in

Euclidean geometry.

A Riemannian manifold is called a Euclidean space if there exist global

coordinates, called Cartesian, such that the metric form has components

(1.28) at every point of the manifold.

For example, a 2-dimensional sphere in a 3-dimensional Euclidean space

is a 2-dimensional Riemannian manifold but not a Euclidean space. A 2-

dimensional plane in a 3-dimensional Euclidean space is a 2-dimensional

Euclidean space. All smooth surfaces in a Euclidean space are Riemannian

manifolds.

A manifold is not necessarily a surface in a Euclidean or any other space.

The spacetime of General Relativity is an example of such manifold.

1.7 Vectors as operators

1.7.1 Basic idea

Vectors defined as straight arrows do not suit surfaces and manifolds. Such

straight arrows cannot belong to curved surfaces and, at most, can only be



tangent to them, unless they are infinitesimally small.

Vectors defined as directed bits of surface geodesics do not allow to in-

troduce meaningful operations of addition and multiplication by real num-

ber, unless their are infinitesimally small and, thus, indistinguishable from

straight arrows tangent to the surface.

The most general definition of geometric vector which applies to man-

ifolds was introduced by Cartan, who proposed to consider vectors as di-

rectional derivatives. Consider a n-dimensional manifold with local coordi-

nates {xi} and a particle moving over the manifold. The particle coordi-

nates are functions of time:

xi = xi(t). (1.29)

These equations describe a curve on the manifold, the particle trajectory.

t plays the role of its parameter. The derivatives

vi =dxi

dt(1.30)

have the meaning of velocity components. Consider the differential operator

d

dt= vi

∂

∂xi, (1.31)

called the directional derivative along the curve (1.29). Note that vi are

components of the operator d/dt in the basis of partial derivatives ∂/∂xi.

Hence, the idea to identify the velocity vector with this directional deriva-

tive and treat the partial derivatives and its local coordinate basis:

v =d

dt, ei =

∂

∂xi. (1.32)

Then eq.(1.31) reads

v = viei, (1.33)

The set of all vectors defined this way at any particular point of the

manifold form an n-dimensional vector space associated with this point,

with the operation of addition and multiplication being defined as follows.

c = a+ b if ci = ai + bi;

and

a = αb if ai = αbi.



1.7.2 Coordinate transformations

Introduce new coordinates, {xi′}. According to the chain rule:

∂

∂xi′=∂xk

∂xi′∂

∂xkand

∂

∂xk=∂xi

′

∂xk∂

∂xi′.

or,

ei′ =∂xk

∂xi′ek, and ek =

∂xi′

∂xkei′ ,

exactly as in eq.(1.12). Then from

v = viei = vi′ei′ .

one has

vk =∂xk

∂xi′vi

′, and vi

′=∂xi

′

∂xkvk, (1.34)

just like in eq.(1.14). Thus, the new definition of vectors as operators leads

to the same transformation laws for their components as before. Note that

neither the trajectory nor its parameter t are effected by such transforma-

tion. Thus, the directional derivative v = d/dt is not effected either. It

is completely independent on the choice coordinates and exists even if no

coordinates are introduced altogether.

1.7.3 Magnitudes of vectors and the scalar product

Let v and w be two Cartan vectors (operators). Because of the transfor-

mation law (1.34) the quantity gijviwj remains invariant under coordinate

transformations (such quantities are called scalars.) This can still be called

the scalar product of v and w

v · w = gijviwj .

Moreover, gijvivj , provides meaningful definition for the magnitude v = |v|

of vector v:

|v|2 = gijvivj .

Indeed, consider the infinitesimal displacement vector of our particle,

dx = vdt.

Its components

dxi = vidt



are the differences in coordinates of the two points on the particle trajectory

separated by time dt. The distance between these points point is given by

dl2 = gijdxidxj = (gijv

ivj)dt2 = v2dt2.

Thus, we have

dl = vdt

as usual.


Chapter 2

Tensors

Tensors are used not only in the Theory of Relativity but also in many

fields of Newtonian physics, sometimes without proper introduction.

2.1 Tensors as operators

Consider an n-dimensional manifold. Let P be a point of the manifold.

Denote as Tp the set of all vectors defined at P. Tp is an n-dimensional

vector space (see Sec.1.7)

2.1.1 1-forms as operators acting on vectors

A 1-form q defined at P is a linear scalar operator acting on vectors from

Tp. The description “scalar” means that q(u) is a geometric scalar, that

is a real number which depends only on the choice of u and nothing else.

For example, the choice of coordinates or vector basis has no effect on this

number. The linearity means that for any v, u ∈ Tp and a, b ∈ R

q(av + bu) = aq(v) + bq(u). (2.1)

The set of all 1-forms defined at P is denoted as T ∗p . It includes the zero

1-form, 0, defined as and it is closed under the operations of addition and

multiplication by real numbers defined as

q = p+ w if q(u) = p(u) + w(u) for any u ∈ Tp; (2.2)

and the operation of multiplication

q = ap if q(u) = ap(u)q = ap for any u ∈ Tp and a ∈ R. (2.3)

19



It is easy to verify that these operations satisfy all the propeties involved

in the formal definition of a vector space and that the dimension of T ∗p is

n. To stress that 1-form q is an operator it is often shown as q( ) where the

space inside the brackets is understood as a slot to be filled with a vector.

As an example of 1-form consider the operator v introduced via the

scalar product operation

v(u) = v · u for any u ∈ Tp. (2.4)

This 1-form is called dual to the vector v. The linearity condition (2.1) is

satisfied because

v · (au+ bw) = a(v · u) + b(v · w).

Another example is the gradient of scalar function. Let f be a scalar

function defined on the manifold. The 1-form df , called the gradient of f ,

is defined via its action on the infinitesimally small vector dx:

df(dx) = df, (2.5)

where df is the increment of f corresponding to the displacement dx. Since

df is a linear operator,

df =∂f

∂xidxi,

and dxi are the components of dx in the coordinate basis, the action of df

on other vectors must follow the rule

df(u) =∂f

∂xiui for any u ∈ Tp. (2.6)

where ui are the components of u in the coordinate basis as well. It is easy

to show that the expression on the right of this equation is a geometric

scalar. Indeed, it remains invariant under coordinate transformations:

∂f

∂xiui =

∂f

∂xi

(∂xi

∂xj′uj

′)

=

(∂f

∂xi∂xi

∂xj′

)uj

′=

∂f

∂xj′uj

′.

2.1.2 Vectors as operators acting on 1-forms

With any vector u ∈ Tp one can associate a linear scalar operator acting

on 1-forms from T ∗p via

u(q) = q(u) for any q ∈ T ∗p . (2.7)


Tensors 21

From eqs.(2.2) and (2.3) it follows that

u(ap+ bq) = (ap+ bq)(u) = au(p) + bu(q),

and, thus, this is a linear operator. To stress this role of vectors they are

often shown like u( ), where the space inside the brackets is a slot to be

filled with a 1-form, and to stress the equivalence between vectors and 1-

forms introduced by Eq.2.7, both u(q) and q(u) are sometimes denoted as

< u, q >.

2.1.3 Tensors as operators acting on vectors and 1-forms

An(lm

)-type tensor defined at point P is a linear scalar operator with l

slots for 1-forms from T ∗p and m slots for vectors from Tp. Such tensor can

also be called as l-times contravariant and m-times covariant. The total

number of slots, r = l+m, is called the rank of the tensor. Thus, if M( , )

is(11

)-type tensor with the first slot reserved for 1-forms then M(q, u) is a

geometric scalar,

M(ap+ bq, u) = aM(p, u) + bM(q, u), (2.8)

and

M(p, au+ bv) = aM(p, u) + bM(p, v). (2.9)

According to this definition, any vector is a(10

)-type or once contravariant

tensor, and any 1-form is a(01

)-type or once covariant tensor.

The set of all(lm

)-type tensors defined at point P is an nr-dimensional

vector space with zero element O such that

O(u, . . . , q) = 0 (2.10)

the operation of addition

S = T +K if S(u, . . . , q) = T (u, . . . , q) +K(u, . . . , q) (2.11)

and the operation of multiplication by real numbers

S = aT, if S(u, . . . , q) = aT (u, . . . , q) (2.12)

for any l vectors from Tp, m 1-forms from T ∗p , and a ∈ R.



2.1.4 Metric tensor

A(02

)-type tensor g( , ) such that for any two vectors v, u ∈ Tp

g(v, u) = v · u (2.13)

is call the metric tensor. Notice that the metric tensor and the one-form v

dual to the vector v (see Sec.2.1.1) are related via

v( ) = g(v, ). (2.14)

Indeed,

v(u) = g(v, u) = v · u.

Later on we also describe the connection between the metric tensor and the

metric form.

2.1.5 Constructing higher rank tensors via outer multipli-

cation of vectors and 1-forms

The operation of outer multiplication, denoted by the symbol ⊗, allows

to construct the higher rank tensors from the lower rank tensors. The

following examples explain how this works.

F = u⊗ v

is a(20

)-type tensor defined at point P such that for any p, q

F (p, q) = u(p)v(q).

D = q ⊗ v ⊗ t

is a(12

)-type tensor such that for any p, u, s

D(u, p, s) = q(u)v(p)t(s);

etc. All tensors appearing in these equations are defined at the same point

of the manifold. The operands do not have to be only fist rank tensors. For

example,

T = F ⊗ t

if

T (p, q, u) = F (p, q)t(u)

for any 1-forms p, q ∈ T ∗p and vector u ∈ Tp.


Tensors 23

2.2 Bases and components of tensors

2.2.1 Induced basis

Let {ei}ni=1 be a basis in Tp1. Then for any u ∈ Tp

u( ) = uiei( ), (2.15)

where ui are the components of u in this basis. Note that i is an upper

index in ui. Let {wi}ni=1 be some arbitrary basis in T ∗p . Then for any

q ∈ T ∗p

q( ) = qiwi( ), (2.16)

where qi are the components of q in this basis. Note that i is a lower index

in qi. This is to make clear that we are dealing with the components of a

1-form and not a vector. The Einstein summation rule then dictates to use

upper indices for the basis 1-forms wi. From eqs.(2.15,2.16) and linearity

of tensor operators one has

wi(u) = ujwi(ej);

ei(q) = qjei(wj);

q(u) = qiujwi(ej).

(2.17)

The basis {wi} is called induced by the basis {ei} if

wi(ej) = ej(wi) = δij . (2.18)

Then eqs.(2.17) simplify so that

wi(u) = ui;

ei(q) = qi;

q(u) = qiui.

(2.19)

This simplification is the main reason for using induced bases of 1-forms.

Given the definition in Sec.2.1.2 one can write Eqs.2.19 in another useful

formu(wi) = ui;

q(ei) = qi;

q(u) = qiui.

(2.20)

1In general, by a basis we mean any set of n linearly independent elements of n-dimensional vector space so that any other element of the space can be represented

as a linear combination of the basis vectors.



For the same reason one introduces induced bases for higher rank ten-

sors. The following examples explain how these bases are constructed.

The induced basis of(11

)-type tensors with the first slot intended for

1-forms is {ei ⊗ wj}, where {wi} is the induced basis of 1-forms and i, j =

1, . . . , n. If F ( , ) is such a tensor and F ij are its components in this basis

then

F ( , ) = F ij ei( )⊗ wj( ), (2.21)

It is easy to see that

F ij = F (wi, ej). (2.22)

Indeed,

F (wk, em) = F ij ei(wk)wj(em) = F ijδ

ki δjm = F km.

Moreover,

F (q, u) = F ijqiuj . (2.23)

Indeed, using the linearity of tensor operators we find that

F (q, u) = F ij ei(q)wj(u) = F ij ei(qkw

k)wj(umem) =

= F ij qkei(wk)umwj(em) = F ij qkδ

ki u

mδjm = F ij qiuj .

The induced basis of(02

)-type tensors is {wi ⊗ wj}. If g( , ) is such a

tensor and gij are its components in this basis then

g( , ) = gijwi( )⊗ wj( ). (2.24)

The calculations similar to those used to derive Eqs.2.22 and 2.23 then yeild

gij = g(ei, ej) (2.25)

and

g(u, v) = gijuivj . (2.26)


Tensors 25

2.2.2 Index notation of tensors

The number and position of indexes of tensor components reveal all the

general information about tensors as operators. For example if tensor T

has components T i kj l this immediately tells us that

(1) T is a 4th rank tensor;

(2) T is a(22

)-type tensor;

(3) Its 1st and 3rd slots are for 1-forms whereas its 2nd and 4th slots are

for vectors.

Thus, quite straightforwardly we conclude that

T ( , , , ) = T i kj lei( )⊗ wj( )⊗ ek( )⊗ wl( ).

Because of this nice property it is a custom to introduce tensors simply by

showing their components. For example, it is perfectly fine simply to say

”Let us consider tensor T i kj l” .

2.2.3 Coordinate bases

In Section 1.7.1 we introduced the coordinate basis of vectors

{∂/∂xi} i = 1, . . . , n .

The bases of other tensors induced by the coordinate basis of vectors are

also called coordinate. The coordinate basis of 1-forms is denoted as

{dxi} i = 1, . . . , n.

The rule for denoting the coordinate bases of higher rank tensors simply

follows the one described in Sec.2.2.1. For example,

{dxi⊗ dx

j} i, j = 1, . . . , n

is the coordinate basis of(02

)-type tensors and

{dxi⊗ ∂

∂xj} i, j = 1, . . . , n

is the coordinate basis of(11

)-type tensors and

2.2.4 Coordinate components of df

From the definition of df it follows that

df(dx) =∂f

∂xidxi. (2.27)



On the other hand, dxi are the coordinate components of dx and Eq.2.20

implies that the partial derivatives are the coordinate components of df .

Thus,

dfi =∂f

∂xi

and

df =∂f

∂xidx

i. (2.28)

2.2.5 Metric form and metric tensor

The metric form

dl2 = gijdxidxj (2.29)

gives us the distance, dl, between the point xi and the point xi + dxi.

Consider the infinitesimally small vector

dx = dxi∂

∂xi.

If g( , ) is the metric tensor than

dx · dx = g(dx, dx) = gijdxidxj , (2.30)

where

gij = g

(∂

∂xi,∂

∂xj

)are the coordinate components of the metric tensor. Comparison of

eq.(2.29) with eq.(2.30) shows that the components gij of the metric form

are nothing else but the components of the metric tensor in the coordinate

basis of the coordinates involved in the form.

2.3 Basis transformation

When we make a transition from one basis of vectors to another this trigers

the transition to new induced bases of tensors of all types. As the result,

the corresponding components of tensors change too. In this section we


Tensors 27

study all these transformations triggered by the transformation of the vector

basis2.

Consider the transition from the basis {ek} to the basis {ek′} Any vector

of the new vector basis is a linear combination of the vectors of the old vector

basis

ek′ = Aik′ei. (2.31)

The coefficients Aik′ can be written as a square matrix, with the upper index

reffering to the rows and the lower index to the columns. This matrix is

called the transformation matrix (it is not a tensor). Similarly,

ek = Ai′

k ei′ , (2.32)

where the transformation matrix Ai′

k is inverse to Aik′ . Thus,

Ai′

kAkj′ = δi

′

j′ and Ai′

kAji′ = δjk. (2.33)

If {ei} and {ei′} are the coordinate bases of coordinates {xi} and {xi′}respectively then

Ai′

k =∂xi

′

∂xkand Aji′ =

∂xj

∂xi′. (2.34)

Indeed,

∂

∂xk=∂xi

′

∂xk∂

∂xi′and

∂

∂xi′=∂xk

∂xi′∂

∂xk.

2.3.1 Transformation of induced bases

The corresponding transformation of the induced basis of 1-forms is

wk′

= Ak′

i wi and wk = Aki′w

i′ . (2.35)

To prove this we need to assume that wi(ek) = δik and then show that

Eq.2.35 yeilds wi′(ek′) = δi

′

k′ . Indeed,

wk′(ei′) = Ak

′

s ws(Ami′ em) = Ak

′

s Ami′ w

s(em) = Ak′

s Ami′ δ

sm = Ak

′

mAmi′ = δk

′

i′ .

2In many textbooks on the Theory of Relativity a tensor is introduced as a collection

of numbers (components) transforming according to the law which we derive in thissection. The definition of tensors as operators uncovers their true coordinate-inependent

geometric nature, and shows that they do not reduce just to sets of components.



Given the expressions for the bases transformations of vectors and 1-

forms one can easily find the transformation law for induced bases of any

higher rank tensor. For example

ei′ ⊗ wj′

= Aki′Aj′

l ek ⊗ wl; (2.36)

ei′ ⊗ ej′ = Aki′Alj′ek ⊗ el; (2.37)

wi′⊗ wj

′= Ai

′

kAj′

l wk ⊗ wl. (2.38)

2.3.2 Transformation of components

Given the transformation laws for bases one can easily find the transforma-

tion law for the components of tensors in these bases. For example,

ui = u(wi) = u(Aik′wk′) = Aik′u(wk

′) = Aik′u

k′ ,

where in the second step we used the linearity vector operators. Thus, the

transformation law for the components of vectors is

ui = Aik′uk′ and ui

′= Ai

′

k uk. (2.39)

Similarly, one finds the transformation law for 1-forms

qi = Ak′

i qk′ and qi′ = Aki′qk. (2.40)

The historical reason for 1-forms being often called covariant vectors is

that this transformation law is exactly the same as the transformation law

(2.31,2.32) for the basis vectors. On the other hand, the transformation law

(2.39) is different and this is the reason behind the description of geometric

vectors as contravariant.

The next example, shows how to deal with higher rank tensors

T i′

j′ = T (wi′, ej′) = T (Ai

′

k wk, Alj′el) = Tmsem(Ai

′

k wk)ws(Alj′el) =

TmsAi′

kAlj′em(wk)ws(el) = TmsA

i′

kAlj′δ

kmδ

kl = Ai

′

mAkj′T

ms.

Thus,

T i′

j′ = Ai′

kAlj′T

kl and T ij = Aik′A

l′

j Tk′

l′ (2.41)

It is easy to see the general rule of tensor transformation. The tran-

formation matrix appears as many times as the tensor’s rank. Each index


Tensors 29

of the tensor on the left appears as one of the indexes in one of the trans-

formation matrixes on the right, as an upper index of A if it is the upper

index of the tensor and as a lower index otherwise. The other index of A

is the summation index, it also appears in the tensor symbol on the right

and in exactly the same place as the free index of A in the tensor symbol

on the left. In other words, each upper index of the transformed tensor is

treated as a vector index and each it’s lower index as an index of a 1-form.

For example,

T i′s′

j′ = Ai′

kAs′

p Alj′T

kpl.

2.4 Basic tensor operations and tensor equations

Operations on tensors which produce other tensors are call tensor opera-

tions. We have already introduced the operations of addition and multipli-

cation by real number in Sec.2.1.3 and the operation of outer multiplication

in Sec.2.1.5. Below, we describe these operations in terms of the compo-

nents of involved tensors.

(1) Addition:

Cij = Aij +Bij (2.42)

if tensor C = A+B;

(2) Multiplication by a real number:

Cijk = aAijk (2.43)

if tensor C = aB, a ∈ R;

(3) Outer multiplication:

T ijkl = DijBkl (2.44)

if T = D ⊗B.

One can see that this is a very concise and fully comprehensive way of

describing tensor operations. Obviously, that is shown are just examples

involving tensors of particular types. However, the generalisation is rather

obvious. In order to make sure that the origin of these equations is clear

let us derive Eq.2.42. According to the definition of tensor addition (see

Sec.2.1.3) we have

C(u, q) = A(u, q) +B(u, q)

for any u ∈ Tp and q ∈ T ∗p . Therefore,

Cij = C(wi, ej) = A(wi, ej) +B(wi, ej) = Aij +Bij .



2.4.1 Contraction

Another important basic tensor operation is contraction. It is best described

in term of components, where it involves turning one upper and one lower

index of a tensor into a pair of dummy indexes. For example, equation

Sij = T illj (2.45)

tells us that S is the result of contracting tensor T over its second upper

and first lower indexes (l is a dummy index). In component-free language

this operation would be defined as

S( , ) = T ( , wl, , el), (2.46)

where {el} is some arbitrary basis of vectors and {wl} is the induced basis

of 1-formas. Because of the involvement of the vector basis this looks a bit

odd. However, the result is actually independent on the choice of basis.

Indeed,

T ( , wl, , el) = T ( , Alk′wk′ , , Am

′

l em′) = Alk′Am′

l T ( , wk′, , em′) =

= δm′

k′ T ( , wk′, , em′) = T ( , wk

′, , ek′).

2.4.2 Contraction of two tensors

Basic tensor operations can be combined into complex operations. One

important example is contraction of two tensors . In the equation below

tensor D is contracted with B over its the 2nd upper index and the first

lower index of B

T ij = DilBlj . (2.47)

Obviously, this operation is a composition of outer product and contraction.

In the component-free language we have

T ( , ) = D( , wl)⊗B(el, ).

2.4.3 Raising and lowering indexes

δij can be considered as a tensor because it satisfies the tensor transforma-

tion law. Indeed,

Aik′Al′

j δk′

l′ = Ail′Al′

j = δij .


Tensors 31

Now suppose we are dealing with a metric manifold. Consider the equation

gijgjk = δik, (2.48)

where gjk is the metric tensor. This appears as a contraction of twice

covariant tensor g with another now twice contravariant tensor g and sug-

gests that gij transform as components of twice contravariant tensor. Di-

rect calculations show that this is indeed so. From Eq.2.48 and the tensor

transformation law one has

gijAs′

j Am′

k gs′m′ = Aip′At′

k δp′

t′

and then

Aa′

i Akb′g

ijAs′

j Am′

k gs′m′ = Aa′

i Akb′A

ip′A

t′

k δp′

t′ .

Contracting over indexes i and k on both sides of this equation one obtains

Aa′

i δm′

b′ gijAs

′

j gs′m′ = δa′

p′ δt′

b′δp′

t′ .

and then contracting over p′ and t′ on the right side and over m′ on the left

Aa′

i As′

j gijgs′b′ = δa

′

b′ .

Comparison of the last result with Eq.2.48 shows that

ga′s′ = Aa

′

i As′

j gij .

Since gij do transform like the components of twice contravariant tensor one

can introduce a tensor with such components. This tensor is also called the

metric tensor. This makes perfect sense because gij is uniquely determined

by gij and the other way around. Eq.2.48 shows that the matrix (gij) is

inverse to (gij).

We already know that the metric tensor allows to relate vectors and

1-forms via Eq.2.14.

u( ) = g(u, ).

In components this reads

ui = gijuj . (2.49)

Now we can invert eq.2.49 and find that

ui = gijuj . (2.50)



Thus, the metric tensor allows to define a one-to-one relationship (map)

between vectors and 1-forms. Now we can interpret u as a first rank tensor

which can be represented either by a vector, u, or by a 1-form, u.

In a similar fashion, the metric tensor unites all higher rank tensors of

the same rank. For example, T ij , Tij , Tij , and T j

i , where

T ij = gjkTik

T ji = gikT

kj

Tij = gikgjlTkl

(2.51)

are different representations of the same second rank tensor T . For this

reason the operations like (2.49-2.51) are called rising and lowering indexes

of a tensor.

Using the operation of raising and lowering indexes one can show that

in the operation of contraction it does not matter which of the dummy

indexes is lower and which is upper. For example,

T ikuk = (gkmT im)(gksus) = (gkmgks)(T

imu

s) = δms Timu

s = T isus.

Thus,

T ikuk = T ikuk. (2.52)

In fact, the scalar multiplication of two vector can be classified as contruc-

tion. Indeed,

u · v = gijuivj = ujv

j = ujvj .

Therefore, contruction of two tensors can also be seen as generalised scalar

multiplication.

The vector-gradient of a scalar function f , ∇f , if defined as

∇if = gijdfj = gij∂f

∂xj. (2.53)

Thus, df and ∇f represent the same 1st rank tensor called the gradient of

f .

2.4.4 Tensor equations

Equations relating different tensors by means of tensor operations are called

tensor equations. In order to decide whether a given equation involving

indexed symbol can be qualified as a tensor equation in components the

following set of obvious rules can be used.


Tensors 33

(1) All terms of tensor equations must have the same number and positions

of free indexes. Thus, for example, if i is an upper free index in one of

the terms then it must be an upper free index in all other terms. For

example,

Sij = T ikj + P ij

cannot be a proper tensor equation whereas

Sij = T ikkj + P ij

can.

(2) The order of free indexes is not important, so

Sij = T ikkj +D ij

can be a proper tensor equation.

(3) Also remember not to write a lower index just below an upper index

because this makes the order of slots ambiguious. That is

Sij = T ikkj + P ij

is not acceptable.

One can show that if a proper tensor equation involves m indexed ob-

jects and we know that m− 1 of them are tensors then the remaining one

is also a tensor. For example, if T ikl and ui are tensors and

T ikl = uiBkl

then Bkl is also a tensor. That is one can define a tensor with components

Bkl. This theorem is proved using the transformation law of components

of tensors, similar to what has been done in this section in order to show

that gij is a tensor (see Eq.2.48).

2.5 Symmetric and antisymmetric tensors

In many applications we deal with so-called symmetric and antisymmetric

tensors. Here we explain what they are by example.

2.5.1 Symmetry with respect to a pair of indexes

Tensor T ijk is called symmetric with respect to i and j if

T ijk = T jik. (2.54)



When any of the indexes of T is lowered the symmetry is preserved. That

is

T ijk = T jikT ijk = T i

j k

T kij = T k

ji .

(2.55)

Tensors that are symmetric with respect to every pair of its indexes are

called fully (or totally) symmetric.

2.5.2 Antisymmetry with respect to a pair of indexes

Tensor T ijk is called antisymmetric with respect to i and j if

T ijk = −T jik. (2.56)

When any of the indexes of T is lowered the symmetry is preserved. That

is

T ijk = −T jikT ijk = −T i

j k

T kij = −T k

ji .

(2.57)

Tensors that are antisymmetric with respect to every pair of its indexes are

called fully (or totally) antisymmetric. One can show that the number of

independent conponents of fully antisymmetric tensor of rank q is given by

Cnq =n!

q!(n− q)!. (2.58)

Thus, the highest rank of fully antisymmetric tensor is n − 1. Fully anti-

symmeric tensors of rank p are also called p-forms.

It is easy to show that if T ijk is symmetric with respect to i and j and

Fij is antisymmetric with respect to i and j then

T ijkFij = 0. (2.59)

Indeed,

T ijkFij = −T ijkFji = −T jikFji = −T ijkFij ,

where the first equality is due to the antisymmetry of Fij , the second one

is due to symmetry of T ijk and the third one stands because we simply

rename dummy indexes. Next we obtain

2T ijkFij = 0,

from which eq.2.59 follows.


Tensors 35

2.6 Levi-Civita Tensor and the vector product

2.6.1 Levi-Civita and the generalised Kronecker symbols

The Levi-Civita symbol of rank n is denoted as εi1i2...in . It has n indexes,

each varying from 1 to n, and its values are defined via

εi1i2...in =

1 if i1, i2, ..., in is an even permutation of 1, 2, ..., n

−1 if i1, i2, ..., in is an odd permutation of 1, 2, ..., n

0 otherwise(2.60)

From this definition, one immediately finds that the symbol is antisymmet-

ric with respect to any pair of its indexes (that is fully antisymmetric). The

Levi-Civita symbol allows to write the expression for the determinant of an

n× n square matrix (aij) as

det(a) = εi1i2...ina1i1a2i2 . . . anin . (2.61)

In tensor calculus it is useful to introduce the Levi-Civita symbol with

upper indexes as well, using exactly the same definition. Thus,

εi1i2...in = εi1i2...in . (2.62)

The generalized Kronecker’s delta of rank p is defined as

δi1...ipj1...jp

= p!δi1[j1 . . . δipjp], (2.63)

where the square brakets stand for the operation of antisymmetrisation

with respect to the inclosed indexes, that is

a[j1j2...jp] =1

p!

(∑even

alm...s −∑odd

alm...s

), (2.64)

where the first term in the round brakets is the sum of all alm...s such that

lm . . . s is an even permutation of j1j2 . . . jp, and the second term is the

sum of all alm...s such that lm . . . s is an odd permutation of j1j2 . . . jp. For

example,

a[ij] =1

2(aij − aji).

It is easy to see that

εi1...inεj1...jn = δj1...jni1...in

, (2.65)

where ik, jk run from 1 to n. Indeed, both are fully antisymmetric and

ε1...nε1...n = δ1...n1...n = 1.



One can also prove that

δkj1...jpki1...ip

= (n− p)δj1...jpi1...ip, (2.66)

and

δk1...kqjq+1...jnk1...kqiq+1...in

= q!δjq+1...jniq+1...in

. (2.67)

For example, if n = 3 than

εkijεksp = δkspkij = δspij = δsi δ

pj − δ

sj δpi ,

and if n = 4 than

δkspkij = 2δspij = 2(δsi δpj − δ

sj δpi ).

Equations 2.66 and 2.67 are very useful, for example, in vector equations

imvolving the cross-product and curl operations.

2.6.2 Levi-Civita Tensor

Consider an orthonormal basis {ei} at some point of a n-dimentional man-

ifold and denote as { ˆwi} the induced basis of 1-forms. The fully anti-

symmetric tensor of rank n

e = εij...k ˆwi ⊗ ˆwj ⊗ · · · ⊗ ˆwk (2.68)

is called the Levi-Civita tensor. Obviosly, the components of this tensor in

the selected orthonormal basis are

eij...k = εij...k. (2.69)

The components of this tensor in an arbitrary basis can then be found

via

ei′j′...k′ = Aii′Ajj′ . . . A

kk′εij...k. (2.70)

This allows us to see that

e1′2′...n′ = Ai1′Aj2′ . . . A

kn′εij...k = detA. (2.71)

On the other hand,

gi′j′ = Aii′Ajj′gij

and thus

g′ = (detA)2g, (2.72)


Tensors 37

where g is the deteminant of the matrix (gij) made out of components of

the metric tensor. This equation tells us that the determinant of the metric

tensor has the same sign in any basis and that

(detA)2 = |g′|.Here we used the fact that in the orthonormal basis g equals to either +1

or −1, which is the case of spacetime). Substituting this result into Eq.2.71

we find that

e1′2′...n′ = ±√|g′|, (2.73)

where the sign equals to the sign of detA. From this we conclude that

all bases separate into two groups. For the first group, which one may

call right-handed, e12...n > 0, and for the second group, called left-handed,

e12...n < 0. Thus,

eij...k = ±√|g|εij...k, (2.74)

where the sign + applies to the rigt-handed and the sign − to the left-

handed bases. In free-dimensional physical space one can actually use the

right hand to select the right-handed group. Obviously, this cannot be done

if the number of dimensions is different and, thus, the name should not be

interpreted literaly.

The components of contravariant Levi-Civita tensor are found via the

operation of raising indexes

eij...k = gilgjm . . . gkselm...s.

This shows that

e12...n = g1lg2m . . . gnselm...s = ±√|g|det(gij) = ±

√|g|g

and, thus,

eij...k = ±√|g|g

εij...k. (2.75)

Once again, the sign plus applies to right-handed bases and the sign minus

to the left-handed ones.

The Levi-Civita tensor allows to define the cross product, c = a× b, of

3-dimensional vectors as

ci = eijkajbk. (2.76)

This definition is very useful for handling complicated expressions involving

cross products. Let us show, for example, that in Euclidean space

a× (b× c) = b(a · c)− c(a · b).



Denote a× (b× c) as vector d. Then according to Eq.2.75

di = eijkajeklmblcm.

Using Eqs. 2.74,2.75,2.65 and the antisymmetry of the generalised Kro-

necker’s delta this can be reduced to

di = δijkklmajblcm = δkijklmajb

lcm.

Finally, Eq.2.67 allows as to write this as

di = δijlmajblcm = (δilδ

jm − δimδ

jl )ajb

lcm = bi(amcm)− ci(albl),

as required. It is easy to see how one can generalize the operation of

vector product to higher dimensional cases using the Levi-Civita tensor.

This generalized operation does not have to be binary, the operands can

be tensors of rank higher then 1 and so is the result of the operation. For

example, in the case of n = 4 one can have

si = eijklajbkdl,

F ij = eijklakbl,

or

Aij = eijklFkl.

The last equation leads to the definition of dual tensors.

2.6.3 Dual tensors

Fully antisymmetric tensor ∗Bj1...jp is called dual to the fully antisymmetric

tensor Bi1...iq , where p+ q = n, if

∗Bj1...jp =1

q!ej1...jpi1...iqB

i1...iq . (2.77)

Since

Cnq = Cnn−q = Cnp (2.78)

the number of independent components of B and ∗B is the same and this

suggests one-to-one correspondence between such tensors. Indeed,

Bi1...iq =sign(g)(−1)pq

p!ei1...iqj1...jp ∗Bj1...jp . (2.79)

In order to check this substitute ∗Bj1...jp from Eq.2.77 into the right-hand

side of Eq.2.79 and obtainsign(g)(−1)pq

p!q!

|g|gδi1...iqj1...jpj1...jpk1...kq

Bk1...kq =(−1)pq

p!q!δi1...iqj1...jpj1...jpk1...kq

Bk1...kq .

Due to the antisymmetry of the generalized Kronecker’s symbol and Eq.2.67

this reduces to1

p!q!δj1...jpi1...iqj1...jpk1...kq

Bk1...kq =1

q!δi1...iqk1...kq

Bk1...kq = Bi1...iq ,

which is indeed the left-hand side of Eq.2.79.


Chapter 3

Geometry of Riemannian manifolds

3.1 Parallel transport and Connection on metric manifolds

In the previous chapter we discussed operations on tensors defined at a

single point of a manifold. By means of such operations we can compare

two tensors defined at the same point and measure the difference between

them. Tensors are here to describe objects of real life and in real life

there are meaningful ways of comparing similar objects at different spatial

locations. One way is to bring the objects to the same location so that

direct comparison is possible. In this approach we have to make sure that

the transported objects are not modified along way. This can be done via

control measurements carried out using standard tools. In geometry such

transport of tensors from one to another point of a manifold is called the

parallel transport and the control measurements are introduced by means

of the metric tensor. Indeed, the metric tensor is a mathematical object

which allows us to introduce the very basic and hence the most important

physical measurements, the measurements of length (and time as we shell

see later). In order to agree with its description as a standard control tool

the metric tensor must be the same, in some absolute sense, everywhere on

the manifold. That is its parallel transport from point A to point B should

give us exactly the metric tensor already defined at point B. In other words

defining the metric tensor on a manifold should be consistent with 1) its

defining at some single point of the manifold and 2) its parallel transport

to all other points.

39



3.1.1 Parallel transport of vectors. Connection

Consider vector a at point P of a metric manifold. Parallel transport

it along the displacement vector dx into the infinitesimally close point S

(assuming that there is a meaningful way of such transport.) Denote the

result as a. This operation can be expressed as

a = Γ(P, a, dx), (3.1)

where Γ is the operator of parallel transport. It is also called the con-

nection. Once this operator is introduced at every point of the manifold

we have means of parallel transporting vectors (and tensors as well). No-

tice that Γ is not a tensor as equation (3.1) involves vectors defined at

different(!) points of the manifold.

Basic requirements on parallel transport:

(1)

If a = 0 then a = 0; (3.2)

(2)

If dx = 0 then a = a; (3.3)

(3) Linearity 1.

If a = αb+ βc then a = αb+ βc. (3.4)

(4) Linearity 2. Introduce local coordinates {xi} on the manifold. Let ai

and ai be the components of a and a in the coordinate bases at P and

P respectively.

If ai − ai = dai

for dx(1)

then ai − ai = αdai

for dx(2) = αdx(1). (3.5)

It is easy to see that these requirements are satisfied only if

ai = ai − Γijkajdxk (3.6)

where Γijk are called the coordinate components of Γ. They are also known

as Christoffel’s symbols of the first kind.


Geometry of Riemannian manifolds 41

3.1.2 Connection of Euclidean space

In Euclidean space the parallel transport of tensors amounts to keeping their

Cartesian components fixed (by definition). Thus, in Cartesian coordinates

{xi} we must have

Γijk = 0. (3.7)

If Γ was a tensor than eq.(3.7) would hold in any coordinates, but it is not.

One can show that in new coordinates {xi′}

Γi′

j′k′ = −(

∂2xl

∂xj′∂xk′

)(∂xi

′

∂xl

). (3.8)

Thus, only if the new coordinates are linear functions of the old Cartesian

ones the new connection coefficients will remain vanishing. Otherwise, they

will not.

From eqs (3.7-3.8), it follows that the connection of Euclidean space is

always symmetric with respect to its lower indexes:

Γijk = Γikj (3.9)

3.1.3 Riemannian Connection

Since we cannot introduce global Cartesian coordinates on Riemannian

manifolds we need a different, more general way of fixing their connections

and, hence, their parallel transport. We require

• the scalar product of any two vectors to remain unchanged by parallel

transport;

u · v = u · v (3.10)

• the connection to be symmetric relative to its lower indexes.

Note that both these conditions are satisfied by the parallel transport of

Euclidean space.

The above conditions allow to express the Cristoffel’s symbols in terms

of the components of the metric tensor and its first coordinate derivatives.

Referring to the figure above we write

u · v = gij(xm)uivj



and

u · v = gij(xm + dxm)(ui + dui)(vj + dvj).

Retaining only the terms of zero and first order in dxk, dui, and dvj in the

last equation we have

u · v =

(gij(x

m) +∂gij∂xk

dxk)

(ui + dui)(vj + dvj) =

= gijuivj + giju

idvj + gijvj dui +

∂gij∂xk

uivjdxk.

Then Eq.3.10 implies that

gijuidvj + gijv

j dui +∂gij∂xk

uivjdxk = 0

or after the substitution of the expressions for dvj and dui

−gijΓjlmuivldxm − gijΓilmulvjdxm +

∂gij∂xm

uivjdxm = 0.

After renaming the dummy indexes this yeilds(−gilΓljm − gljΓl im +

∂gij∂xm

)uivjdxm = 0.

Since this must be true for any u, v, and dx we obtain the equation

∂gij∂xm

= gilΓljm + gljΓ

lim. (3.11)

Cyclic permutation of the indexes j, m, and i in this equations gives us∂gjm∂xi

= gjlΓlmi + glmΓlji,

and∂gmi∂xj

= gmlΓlij + gliΓ

lmj .

Now we add them with Eq.3.11 and use the symmetries of gij and Γl ij to

obtain

2gljΓlim =

∂gij∂xm

+∂gjm∂xi

− ∂gmi∂xj

or

Γjim =1

2

(∂gij∂xm

+∂gjm∂xi

− ∂gim∂xj

), (3.12)

where

Γjim = gjlΓlim. (3.13)

Γjim are called Christoffel’s symbols of the second kind. Thus, we can

compute the Christoffel’s symbols when we know the components of metric

tensor as functions of coordinates

Γl im = gljΓjim =1

2glj(∂gij∂xm

+∂gjm∂xi

− ∂gim∂xj

). (3.14)



3.2 Parallel transport of tensors

3.2.1 Scalars

Scalars can be considered as tensors of zero rank. Since the scalar product

of two vectors in invariant during the parallel transport we have to impose

the rule

f = f (3.15)

for all geometric scalars.

3.2.2 1-forms

The rule for the parallel transport of 1-forms follows from the fact that that

q(u) = qiui is a geometric scalar and has to remain invariant during the

parallel transport

qiui = qiu

i. (3.16)

From this we have

(qi + dqi)(ui + dui) = qiu

i.

Retaining only the zero and first order term in the right-hand side of this

equation we obtain

qidui + dqiu

i = 0.

Using Eq.3.6 and renaming the dummy indexes we then obtain

ui(dq − Γl imqldxm) = 0.

Since this must hold for any u we finally arrive to

qi = qi + Γl imqldxm. (3.17)

3.2.3 General tensors

Similar condition is used to define the parallel transport of tensors. For

example, consider tensor T ij . Since T ijqiuj is a scalar we require

T ij qiuj = T ijqiu

j . (3.18)

This leads to

T ij = T ij − ΓikmTkjdx

m + ΓkjmTikdx

m. (3.19)



Similarly, for tensor Fij we require

Fij viuj = Fijv

iuj (3.20)

which leads to

Fij = Fij + ΓkimFkjdxm + ΓkjmFikdx

m. (3.21)

The general rule which applies to tensors of any type can be described

as follows:

• The number of indexes equals to the number of terms involving Γ;

• Each upper index is treated as a vector index;

• Each lower index is treated as a 1-form index.

3.2.4 Metric tensor

According to the general rules of the parallel transport

gij viuj = gijv

iuj .

On the right-hand side of this equation we have the components of vectors v

and u and the components of the metric tensor at the starting point and on

the left-hand side their components at the destination point of the parallel

transport. On the other hand, the condition (3.10) reads

gij viuj = gijv

iuj ,

where on the left-hand side we have the components of the metric tensor

originally defined at the destination point. Thus, at the destination point

we have

(gij − gij)viuj = 0.

Since, vectors v and u are arbitrary this implies that

gij = gij . (3.22)

Thus, parallel transport of the metric tensor always results in the same

metric tensor at the destination point as the one already defined there at

the introduction of the metric manifold. This confirms that in the theory

of Riemannian manifolds the metric tensor is the same throughout the

manifold. In other words one can think of the metric tensor as first defined

at one particular point of the manifold and then parallel transported to all

other points. (This is similar to manufacturing of standard metric tools at

one factory and then transporting them to various places of use.)



3.3 Absolute and covariant derivatives

A tensor-valued function defined on a manifold is called a tensor field. At

every point of the manifold it defines a tensor of the same type. On any

metric manifold there defined at least two tensor fields - the metric tensor

field and the Levi-Civita tensor field. Components of the metric tensor

in the induced coordinate basis of some local coordinates may vary. For

example, in Sec.1.1.2 we have seen how gij of a sphere depend on the

polar coordinates. However, as we have just discussed, this tensor field is

constant in the absolute sense (in the sense of parallel transport). Similarly,

the components of a constant vector field in Euclidean space vary when

computed in the coordinate basis of spherical coordinates. Thus, the usual

coordinate derivatives of tensor components, like ∂ai/∂xk, cannot be used

to describe the variation of tensor fields in the absolute sense. For this

purpose we need other kinds of derivatives.

3.3.1 Absolute and covariant derivatives of scalar fields

From eq.(3.15) it follows that for a scalar field f (scalar function)

Df

dλ= ∇mf

dxm

dλ(3.23)

∇mf =∂f

∂xm(3.24)

3.3.2 Absolute and covariant derivatives of vector fields

Consider vector field a(xk). Parallel transport vector a from the point

S to the infinitesimally close point P (see the figure above). The result is

the vector a at point P . Denote the difference between a and a as Da:

Da = a− a.Note that Da is a vector. If Da = 0 then we say that a is the same at P

and S in the absolute sense. From eq.(3.6) it follows that

ap(xi) = ap(xi + dxi) + Γpjk(xi)aj(xi)dxk.



(Here we have sign + because the transport occurs in the direction opposite

to dx.) Thus,

Dap = dap + Γpjkajdxk. (3.25)

where

dap = ap(xi + dxi)− ap(xi)as usual. If the parallel transport is carried out along the curve xp = xp(λ)

then Da/dλ measures the rate of change of the vector field a(xi) along this

curve compared to the variation of its parameter. It is called the absolute

derivative of a. One hasDap

dλ=dap

dλ+ Γpjka

j dxk

dλ. (3.26)

One can rewrite eq.(3.25) as

Dap = ∇kapdxk. (3.27)

where

∇kap =∂ap

∂xk+ Γpjka

j (3.28)

is called the covariant derivative of a . Since Dap and dxk are vectors

defined at the same point of the manifold then equation (3.27) is a proper

tensor equation and, thus, the covariant derivative is a second rank tensor

(see Sec.2.3). This tensor describes how fast this vector field varies in all

directions (recall the gradient of a scalar function).

3.3.3 Absolute and covariant derivatives of 1-form fields

Similarly one obtains the following results for 1-forms

qp(xi) = qp(x

i + dxi)− Γjpk(xi)qj(xi)dxk.

Dqp = qp − qp = dqp − Γjpkqjdxk. (3.29)

where

dqp = qp(xi + dxi)− qp(xi).

Note that Dq is a 1-form. The absolute derivative of q isDqpdλ

=dqpdλ− Γjpkqj

dxk

dλ. (3.30)

Dqp = ∇kqpdxk. (3.31)

where

∇kqp =∂qp∂xk− Γjpkqj (3.32)

is the covariant derivative of q . It is a second rank tensor which describes

how fast the 1-form field varies in all directions.



3.3.4 Absolute and covariant derivatives of general tensor

fields

The similar procedure applies to higher rank tensors. For example, for the

field of second rank tensor T ij one has

DT ijdλ

= ∇mT ijdxm

dλ(3.33)

where

∇mT ij =∂T ij∂xm

+ ΓikmTkj − ΓkjmT

ik, (3.34)

whereas for the third rank tensor T ijs

DT ijsdλ

= ∇mT ijsdxm

dλ(3.35)

∇mT ijs =∂T ijs∂xm

+ ΓikmTkjs − ΓkjmT

iks − ΓksmT

ijk. (3.36)

These examples illustrate the general formal rules

• The absolute derivative of a tensor field of rank r is a tensor or rank r;

• The covariant derivative of a tensor field of rank r is a tensor or rank

r + 1;

• The first term in the expression for the covariant derivative is the usual

partial coordinate derivative (∂/∂xm) of tensor’s components;

• There are r more terms in this expression, one per each index. In each

such term for an upper index this index is treated as a vector index

and in each such term for a lower index it is treated as a 1-form index.

There many different notations for partial and covariant derivatives.

The most common of the them are described below.

1) ∂mai ≡ ai,m ≡

∂ai

∂xm. (3.37)

2) ai;m ≡ ∇mai. (3.38)

3) ai;m ≡ ∇mai = gmkai;k ≡ gmk∇kai. (3.39)



3.3.5 General properties of covariant differentiation

It is easy to show that the covariant differentiation has the following familiar

properties:

∇m(A+B) = ∇mA+∇mB, (3.40)

and

∇m(AB) = (∇mA)B +A(∇mB), (3.41)

where multiplication can be both inner and outer. Although the actual

number and position of indexes of A and B do not matter here (this is

why their indexes are not shown) the general rules of tensor equations still

apply. For example,

∇m(Aij +Bij) = ∇mAij +∇mBij ,

∇m(AiBi) = (∇mAi)Bi +Ai(∇mBi),

∇m(AiBj) = (∇mAi)Bj +Ai(∇mBj),

3.3.6 The field of metric tensor

Since

gij = gij

(see Sec.3.2.4) one has

Dgijdλ

= 0 (3.42)

along any curve and hence

∇mgij = 0. (3.43)

3.4 Divergence

The divergence of vector field is defined via

divA = ∇mAm. (3.44)



Obviously this is a scalar. Similarly one defines the divergence of higher

rank tensors; in all cases the index of differentiation is contracted with one

of the indexes of the differentiated tensor. For example,

divT = ∇mTmki. (3.45)

The result is a tensor, whose rank is less than the rank of the differentiated

tensor by one. The notation divT is not particularly informative because

it does not tell which index of tensor T is involved in the contraction. For

this reason it is not widely used.

In applications, one is often interested in how the divergence relates

to partial derivatives with respect to the coordinates of utilised coordinate

system. Here we derive such expressions for vectors and symmetric second

rank tensors. First, we show that

Γiji =∂ ln

√|g|

∂xj, (3.46)

where g is the determinant of the matrix, (gij), constracted out of the

components of the metric tensor. Although g in any coordinate system is

given by a single number, this number is not the same and thus g is not a

scalar. The total differential of g relates to the differentials of gij via

dg =∑i

Aij0dgij

0,

where Aij0 is the minor of (gij) corresponding to the element gij0. Note

that the index j0 is not dummy. This result immediately follows from the

... . On the other hand, the matrix of (gij) is inverse to (gij) and therefore

gij =Ajig.

Combining the last two results and using the symmetry of the metric tensor

we obtain the useful result

dg = ggikdgik. (3.47)

From Eq.3.14 we have

Γl il =1

2

(glj

∂gij∂xl

+ glj∂gjl∂xi− glj ∂gil

∂xj

).

It is easy to see that the first and the second terms are opposite and, thus,

Γl il =1

2glj

∂gjl∂xi

,



According to Eq.3.47 we can write this result as

Γl il =1

2g

∂g

∂xi,

which leads straight to Eq.3.46.

Now we derive the desired expressions for the divergence of vectors and

symmetric second rank tensors. Using Eq.3.28 one can write

∇iai =∂ai

∂xi+ Γiika

k,

and using Eq.3.46 to rewrite this as

∇iai =∂ai

∂xi+∂ ln

√|g|

∂xiai,

which can be simplified as

∇iai =1√|g|

∂√|g|ai

∂xi. (3.48)

In Cartesian coordinates of Euclidean space |g| = 1 and Eq.3.48 simplifies

to the familiar

∇iai =∂ai

∂xi.

Now consider the antisymmetric tensor F ij . According to the rules of

covariant differentiation (see Sec.3.3.4) we have

∇iF jk =∂F jk

∂xi+ ΓjimF

mk + ΓkimFjm, (3.49)

and, thus,

∇iF ik =∂F ik

∂xi+ ΓiimF

mk + ΓkimFim.

Since F im is antisymmetric and Γkim is symmetric with respect to indexes

i and m, the last term in this expression vanishes and, using Eq.3.46, it can

be written as

∇iF ik =∂F ik

∂xi+ Fmk

∂ ln√|g|

∂xm.

Thus, we obtain the same expression as for the divergence of vectors

∇iF ik =1√|g|

∂√|g|F ik

∂xi. (3.50)



In general case, however, the expression for the divergence of second

rank tensor does not have such a simple form. For example, consider the

symmetric tensor Tij. Then

∇iT jk =∂T jk∂xi

+ ΓjimTmk − ΓmikT

jm,

and

∇iT ik =∂T ik∂xi

+ ΓiimTmk − ΓmikT

im.

The first two terms in this equation combine as before to give

1√|g|

∂√|g|T ik∂xi

.

The last term, however, gives

1

2gmjT im

(∂gij∂xk

+∂gjk∂xi

− ∂gik∂xj

)=

1

2T ij

(∂gij∂xk

+∂gjk∂xi

− ∂gik∂xj

)=

1

2T ij

∂gij∂xk

.

Combining these results we finally obtain

∇iT ik =1√|g|

∂√|g|T ik∂xi

− 1

2T ij

∂gij∂xk

. (3.51)

3.5 Geodesics and parallel transport

We already know (see Sec.1.4) that geodesics are solutions of the Euler-

Lagrange equations

d

dλ

∂L

∂xk− ∂L

∂xk= 0 (k = 1, 2, ..., n) (3.52)

with Lagrangian

L(xk, xk) = gij xixj . (3.53)

(Recall that xk = dxk/dλ where λ is a normal parameter of the geodesic.)

It easy to see that

∂L

∂xk=∂gij∂xk

xixj and∂L

∂xk= 2gikx

i.



Substitution of these results into eq.(3.52) gives us

d

dλ(2gikx

i)− ∂gij∂xk

xixj = 0.

⇒ 2∂gik∂xj

xj xi + 2gikxi − ∂gij

∂xkxixj = 0.

⇒ gikxi +

1

2

(∂gik∂xj

+∂gjk∂xi

− ∂gij∂xk

)xixj = 0.

Now we can use eq.(3.12) and write this result as

gikxi + Γkij x

ixj = 0.

By raising index k (see eq.3.14 and eq.2.48) this is turned into the so-called

geodesic equation

xk + Γkij xixj = 0 (3.54)

which is the same as

Dti

dλ= 0, (3.55)

where

ti =dxi

dλ(3.56)

is the tangent vector to the geodesic. In other words, the tangent vector ti is

the same along the geodesic in the absolute sense ( in the sense of parallel

transport along the geodesic). This results allows to give the following

alternative definition of a geodesic curve

Definition. A curve is called geodesic if it allows a parameter λ such

that

Dti

dλ= 0, where ti =

dxi

dλ

Such parameter is called “normal” and ti is called ” the normal tangent vec-

tor”. It is this property of geodesics that is meant when they are described

as the straightest possible curves.



3.6 Geodesic coordinates and Fermi coordinates

3.6.1 Geodesic coordinates

By definition, for any point of a Riemannian manifold one can find such a

system of coordinates, called locally Cartesian that at this point

gij =

{1 if i = j

0 if i 6= j(3.57)

Moreover, for any point of a Riemannian manifold one can find such a

system of coordinates that

Γijk = 0, (3.58)

and, hence,

gij,k = 0; (3.59)

∇m =∂

∂xm; (3.60)

D

dλ=

d

dλ. (3.61)

at this particular point. Such coordinates are call geodesic coordinates.

Here is how geodesic coordinates coordinates can be set up. Select a

point on the manifold where the conditions (3.58-3.61) are to be satisfied.

At this point, introduce a set of basis vectors, {ei}, which will become the

coordinate basis of geodesic coordinates. Select a neighbourhood, Np , of

P such that for any point A ∈ Np there exists one and only one geodesic

connecting it to P . Let λ be such a normal parameter of this geodesic that

λ = 0 at P . Denote as u = d/dλ its tangent vector at P and as λA the

value of λ at A. Then the geodesic coordinates of point A are defined via

xiA = uiλA. (3.62)

Obviously, there many normal parameters which satisfy the above selection

criteria and we need to show that the result is the same for any of them.

Consider another such normal parameter, µ. Then

µ = cλ where c = const



and the new tangent vector

vi =dxi

dµ=

1

c

dxi

dλ.

Thus,

xiA = viµA =1

cuicλA = uiλA.

Next we need to show that in these coordinates the Christoffel symbols

vanish at the point P. From eq.(3.62) it follows that all geodesics passing

through P satisfy

xi = uiλ where ui = const, (3.63)

which ensures

xi = 0.

Given this result the geodesic equation (3.54) reads

Γijkxj xk = 0.

Thus, that for any vector ui at point P

Γijkujuk = 0

which can only be satisfied if

Γijk = 0.

Geodesic coordinates are very convenient for many analytical calcula-

tions.

3.6.2 Fermi coordinates

In Euclidean space equations (3.58-3.61) are satisfied throughout the whole

space when we employ Cartesian coordinates (or coordinates related to

the Cartesian ones via linear transformation.) For general Riemannian

manifolds it in impossible to find such coordinates that equations (3.58-

3.61) are satisfied throughout the whole manifold. The most what can be

achieved in general is to get them satisfied along a given geodesic. The

corresponding coordinates are called Fermi coordinates.

Here is the way of constructing such coordinates. First we select a

geodesic curve with normal parameter λ such that at point O ,the origin



of the Fermi coordinates, λ = 0 (we shell call it the Fermi geodesic). At

this point select such a basis {ei} that e1 = d/dλ. Parallel transport this

basis (along the Fermi geodesic) to every other point of the Fermi geodesic.

Select such a neighbourhood of the Fermi geodesic, N , that for any point

A ∈ N there exists one and only one geodesic with normal tangent vector

u = d/dµ which connects this point to some point P of the Fermi geodesic

so that u = uiei with u1 = 0 at P . Choose such normal parameter µ that

µ = 0 at P. The Fermi coordinates of the point A are then defined as

x1 = λPxi = µAu

i i = 2, . . . , n.(3.64)

In these coordinates the Fermi geodesic satisfies the equation

d2xi

dλ2= 0

and the geodesic through A satisfies

d2xi

dµ2= 0.

This ensures that for any geodesic through P

Γijkxj xk = 0

which can only be satisfied if

Γijk = 0.

Fermi coordinates play important role in the theory of relativity. They

correspond to the so-called free-falling frames.



3.7 Riemann curvature tensor

Parallel transport on Riemannian manifolds has a number of properties not

seen in Euclidean space. This is clearly demonstrated in the following ex-

amples involving a 2D sphere. Recall that any vector tangent to a geodesic

remains tangent during parallel transport along this geodesic. Moreover,

since the angle between two parallel transported vectors is constant so must

be the angle between a vector parallel transported along a geodesic and this

geodesic.

(1) The result of parallel transport depends not only on the initial and final

points but also on the path along which this transport is carried out!

(a) When vector t is parallel transported from the point A on the equa-

tor to the north pole, N , along the meridian AN the result is vector

t′;

(b) When vector t is first parallel transported from the point A to the

point C along the equator, which results in vector t∗, and then

parallel transported from C to N along the meridian CN the result

is a different vector, t′′ 6= t′.

(2) Parallel transport along a closed curve does not result in the original

vector!

Indeed, when vector t′ is parallel transported along the closed path

NACN the result is vector t′′

Obviously these peculiar properties stem from the fact that sphere is

a curved surface! Curvature of such surfaces and general manifolds is de-

scribed via the so called Riemann curvature tensor.



• Consider a manifold M and a point A ∈ M. Select vectors a, dx(1),

and dx(2) defined at A. Introduce local coordinates {xi} and construct the

close path ABCDA as shown in the figure.

Parallel transport vector a along this path (first in the direction of dx(1))

to obtain vector a+da at point A. Since this path is infinitesimally small ~da

must depend linearly on ~a, ~dx(1), and ~dx(2) and vanish if ~a = 0 or ~dx(1) = 0

or ~dx(2) = 0. That is we must have

dai = Rilmpaldxm(1)dx

p(2). (3.65)

Since this is a proper tensor equation, Rilmp is a tensor and it is called the

Riemann curvature tensor. Direct calculations show that

Rklmp = −Γklm,p + Γklp,m − ΓslmΓkps + ΓslpΓkms (3.66)

(Note that although Γijk is not a tensor, Rkijl is(!) Such peculiar results

do occur from time to time.)

• Curvature of manifolds also causes deviation of initially parallel

geodesics. Consider two infinitesimally close points, A and B, separated

by the infinitesimal displacement vector dx.

Select some vector t at A and parallel transport it from A to B along ~dx.

Construct two geodesics passing through A and B with normal parameter

µ. Namely, the geodesic xi = xi(A)(µ) such that

µ = 0 anddxi

dµ= tiat A

and the geodesic xi = xi(B)(µ) such that

µ = 0 anddxi

dµ= tiat B.



These geodesics can be described as parallel at points A and B. Denote

the displacement vector separating the points of these two geodesics which

have the same value of µ as

sdλ where dλ = const.

One can show that

Dsi

dµ= Riljkt

ltjsk ≡ Riljkdxl

dµ

dxj

dµsk. (3.67)

This equation in called the equation of geodesic deviation. It shows that

initially parallel geodesics deviate from each other.

In Cartesian coordinates of Euclidean space all Γijk = 0 and from (3.66)

one has

Rijkl = 0.

Since R is a tensor, this is true in any basis ( R is just a zero tensor.) Thus,

all dai in (3.65) and all Dsi/dµ in (3.67) vanish and we recover the familiar

properties of Euclidean space.

Definition A manifold is called internally flat (often just flat) if every-

where on this manifold Rijkl = 0, otherwise it is called internally curved.

For example planes and cylinders of Euclidean space are internally flat

manifolds (surfaces).

• One can also show that

(∇m∇p −∇p∇m)ak = Rklmpal; (3.68)

(∇m∇p −∇p∇m)ak = −Rlkmpal. (3.69)

Thus, on curved manifolds the operators of covariant differentiation do not

commute.



3.8 Properties of the Riemann curvature tensor

• Rijkl has a number of properties which reduce the number of its inde-

pendent components:

Rpijk = −Rpikj ; (3.70)

Rpijk = −Ripjk; (3.71)

Rpijk = Rjkpi; (3.72)

Rppij = 0; (3.73)

Rpijk +Rpjki +Rpkij = 0. (3.74)

Note the cyclic permutation of the lower indexes in eq.(3.74). The best way

of proving these properties involves use of geodesic coordinates. Indeed,

since in geodesic coordinates Γijk = 0 and gij,k = 0, eq.(3.66) has a much

simpler form

Rklmp = −Γklm,p + Γklp,m . (3.75)

Using eq.(3.12) this can also be written as

Rklmp =1

2[gkp,lm + glm,kp − gkm,lp − glp,km] (3.76)

(By the way, eq.(3.76) tells us that all second order derivatives of gij vanish

only if Rijkl = 0!) Now it easy to see that, for example,

Rklmp = −Γklm,p+Γklp,m = −(Γklm,p−Γklp,m) = −(−Γklp,m+Γklm,p) = Rklpm,

which proves eq.(3.70).

Because of the properties (3.70-3.74) the curvature tensor has only

N =1

12n2(n2 − 1)

independent components (the total number of components is n4, where n

is the dimension of the manifold.) If n = 2 then N = 1. The curvature

tensor of 2D manifolds has only one independent component.

• One can also show (using geodesic coordinates once again) that

Rikpl;m +Riklm;p +Rikmp;l = 0 (3.77)

(Note the cyclic permutation of indexes p, l,m in this equation.) This result

is known as the Bianchi identity.



3.9 Ricci tensor, curvature scalar and the Einstein tensor

These important tensors are derived from the Riemann curvature tensor.

The Ricci tensor is defined via

Rij = Rsisj (3.78)

The symmetries (3.70-3.72) ensue that

Rij = −R si sj = −Rsijs = R s

si j , (3.79)

as well as

Rij = Rji. (3.80)

The curvature scalar is defined as

R = Rii. (3.81)

The Einstein tensor is

Gij = Rij −1

2Rgij . (3.82)

It is easy to see

Gij = Gji. (3.83)

Moreover, using the Bianchi identity one can show that

Gik;k ≡ ∇kGsk ≡ 0. (3.84)

In other words, the divergence of Einstein’s tensor is zero.


PART 2

Basic Theory of Relativity

61


62


Chapter 4

Space and time in the theory ofrelativity

Each physical theory is based on a number of key assumptions - the rest of

the theory is then build on these assumptions using appropriate mathemat-

ical tools. Any good theory has to be self-consistent (no internal contradic-

tions) and consistent with Nature. It seems like the second condition can

never be achieved completely. As we learn more we discover new, previously

unknown contradictions between our theories and Nature. They force us

to revise our theories by constructing new sets of basic assumptions. What

was revised during the transition from from Newtonian physics to General

Relativity are the assumptions on the nature of physical time and space.

4.1 Space and Time in Newtonian Physics

4.1.1 Time

Newtonian physics assumes that time is absolute, in the sense that simul-

taneity of events and their durations are the same for all observers. If

events A and B are simultaneous for you then they are simultaneous for

everyone else. If event A takes 2 seconds for you then it takes 2 seconds

for anyone else, provided absolutely accurate standard clocks are used. In

physical terms simultaneity would be easy to establish if there there were

signals that propagate with infinite speed so when an event occurs in a

remote place you one be aware of it instantaneously. Thus, time can be

measured by a single standard clock. From the mathematical viewpoint

this allow to introduce t as universal parameter to describe motion evolu-

tion and interaction in the Universe.

63



4.1.2 Space

It also assumes that physical space is absolute, in the sense that sizes of

physical objects and distances between them, as measured at a particu-

lar time, are the same for all observers equipped with absolutely accurate

standard tools for measuring length. This space is not evolving in time. Its

geometry is Euclidean and thus one can introduce Cartesian coordinates

covering the whole of physical space.

4.1.3 Inertial frames

In order to describe motion/evolution of physical systems we need to have a

system of spatial coordinates at any time. The way this is done in practice

can be rather complicated. In theory one introduces a spatial grid with

known distances between its nodes. These distances do not have to be

fixed and thus the grid can move in space in a certain convenient way. Such

spatial grid is called a reference frame. Depending of the problem under

consideration some of these references frames are better than others as they

provide a simple description of the studied phenomena. In particular, the

motion of free particles has a particularly simple form when described in

so-called inertial frames where they move with constant speed:

Dvi

dt= 0, (4.1)

where

vi =dxi

dtor v =

d

dt.

One can think of an inertial frame as a collection of free particles moving

with the same speed, so that the distances between them are fixed, the

grid of spatial coordinates being attached to these particle. These spatial

coordinates do not have to be Cartesian – the use of the absolute derivative

in eq.4.1 ensures that the choice of spatial coordinates is not important.

Since eq.4.1 in nothing else but the geodesic equation, trajectories of free

particles in Newtonian mechanics are straight lines (geodesic of Euclidean

space). The absolute time t is a normal parameter of this geodesic, and

vi = dxi/dt is its tangent vector.

In Cartesian coordinates eq.4.1 can be written as

dvi

dt= 0, or

d2xi

dt2= 0.


Space and time in the theory of relativity 65

4.1.4 Newtonian principle of relativity:

There are infinitely many inertial frames moving relative to each other. If a

body is at rest in one inertial frame it will appear in motion in other frames.

Thus, the notion of mechanical motion appears relative. In fact, the laws

of Newtonian mechanics are exactly the same for all inertial frames making

impossible to decide which of the frames is at rest in space. However, the

absolute nature of space in Newtonian physics implies that the notion of

absolute motion, that is the motion relative to the absolute space, is mean-

ingful and that such motion can be detected by studing physical phenomena

outside of the realm of mechanics.

4.2 Space and Time in Special Relativity

4.2.1 Spacetime

Various experiments show that in vacuum electromagnetic waves propagate

with the same speed, c, relative to all inertial observers. Moreover, c is the

maximum possible speed for any signal. This result is in conflict with the

Galilean transformation between inertial frames, which is the direct result

of the assumptions of Newtonian physics on the nature of space and time.

If one accepts it then the theory of space and time has to be changed.

First, the invariance of the speed of light implies that time can no longer

be considered as absolute. Events simultaneous in one inertial frame be-

come time-separated in others. Different inertial observers measure differ-

ent time intervals between the same events even if they use identical clocks

and procedures. Thus, each inertial frame of Special Relativity has to be

attributed with its own time.

Second, physical space cannot be considered as absolute too. Different

inertial observers obtain different results when they measure lengths of the

same objects even if they use identical standard meters and identical pro-

cedures. Mathematically, this means that this space can not be described

as a metric space.

However, for any two event separated by the time interval ∆t and dis-

tance ∆l their combination

∆s2 = −c2∆t2 + ∆l2, (4.2)

called the spacetime interval, is invariant for all inertial observers even if

∆t and ∆l are not. This shows that physical time and space can be united



into a single 4-dimensional metric space, called spacetime, with ∆s being

the generalised distance between its points, called events. One can see

straight away that this generalised distance is rather peculiar as ∆s2 can

be both positive and negative. In fact, if ∆s2 > 0 then there exists a frame

where ∆t = 0 and ∆s2 = ∆l2. Such spacetime intervals are called space-

like. If ∆s2 < 0 then there exists a frame where ∆l = 0 and ∆s2 = −c2∆t2.

Such spacetime intervals are called time-like. Thus, the possibility of both

positive and negative ∆s2 reflects the underlying difference between space

and time. Spacetime intervals such that ∆s2 = 0 are called null. They

cannot be reduced to either pure space or pure time intervals. (If fact they

describe events on the world-line of a light signal.)

Conventionally, the indexes of spacetime coordinated are shown using

Greek aphabet. Moreover, they run not from 1 to 4 but from 0 to 3,

reserving 0 for the time-like dimension and 1,2,3 for the space-like dimen-

sions (denoted using Latin alphabet). However, one can always introduce

such curvilinear coordinates in spacetime that some coordinate lines change

their type from time-like to space-like and the other way around. In such

coordinates, the nature of dimensions can no longer be reflected in their

indexation.

If {xν} are arbitrary coordinates of the spacetime then the interval

between its infinitesimally close points is given by the metric form

ds2 = gµνdxµdxν where µ, ν = 0, 1, 2, 3. (4.3)

Inertial frames correspond to special cordinate systems of spacetime,

which are postulated to exist. In these coordinate systems the spacetime

metric reduces to

ds2 = −(dx0)2 + gijdxidxj i, j = 1, 2, 3, (4.4)

where gij do not depend on x0. Here x0 = ct, there t is the global time

of the inertial frame and xi (i = 1, 2, 3) are its spatial coordinates. The

hyper-surface x0 =const has positive-definite metric form

dl2 = gijdxidxj i, j = 1, 2, 3, (4.5)

and corresponds to the space of this inertial frame. Moreover, it is assumed

that one can introduce such global spatial coordinates xi that

dl2 = (dx1)2 + (dx2)2 + (dx3)2



throughout the whole hypesurface x0 =const. This means that this hyper-

surface is a 3-dimensional Euclidean space.

Thus, the spacetime of Special Relativity allows coordinates {xν} such

that

ds2 = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2 (4.6)

and, hence,

gµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(4.7)

throughout the whole spacetime. These coordinates are called pseudo-

Cartesian. Metric spaces which allow such coordinates are called pseudo-

Euclidean or Minkowskian.

The scalar product of a non-zero vector uν with itself in such a space,

gνµuνuµ, can be positive, negative, or zero. Thus, similarly to the clas-

sification of the spacetime intervals spacetime vectors can be classified as

space-like, time-like, or null. In order to distinguish between vectors of

space-time and the usual spacial vectors we agree to reserve the notation v

for the spatial vectors (3-vectors) and use −→u for the vectors of space-time

(4-vectors).

4.2.2 Special principle of relativity

Since the absolute space is no longer introduced in Special Relativity the

notion of absolute motion become obsolite as well and all all physical laws

must be exactly the same in all inertial frames. This is known as the Special

Relativistic Principle of Relativity. The Introduction of spacetime allows us

to transform this principle into a simple prescription for writing relativistic

laws of physics. To see this, return for a moment to Newtonian physics.

It is assumed there that, at least in principle, all physical phenomena can

be studied without interfering with them. In particular, their development

does not depend on the choise of spatial coordinate system. Thus, the laws

of physics can be expressed either in the form that does not involve coor-

dinates, or in the form that is exactly the same for all sorts of coordinates.

One example of the first form is the vector (and tensor) equations, like

Dvi

dt= 0



or

Dv

dt= 0.

The correponding example of the second form is the vector (and tensor)

equations written in components, like

dvi

dt+ Γijkv

jvk = 0.

Since in Special Relativity different inertial frames correspond to dif-

ferent systems of coordinates in spacetime, this suggests to use the same

approach. Namely, to express physical laws as tensor equations involv-

ing tensors of spacetime (4-tensor equations), including vector and scalar

equations as special cases of tensor equations.

For example, consider the motion of free particles. In spacetime each

particle traces a curve which is called its world-line. To parametrize this

curve introduce the particle’s proper time, τ , which is the time measured

in the frame where this particle is at rest (This definition makes the proper

time a spacetime scalar.). The spacetime vector

~u =d

dτor uν =

dxν

dτ(4.8)

tangent to the world-line is called the particles 4-velocity. In the local

inertial frame comoving with the particle t = τ , x0 = ct, and xi = const,

and thus uν = (c, 0, 0, 0). This allows us to calculate

|~u|2 = gνµuνuµ = −c2. (4.9)

Thus, 4-velocity is a time-like vector. Now consider the following equation

Duν

dτ= 0. (4.10)

Since this is a 4-vector equation, and hence complies with the Special Prin-

ciple of Relativity, it may constitute a law of physics. In components, this

equation reads

duν

dτ+ Γνµηu

µuη = 0. (4.11)

where Γνµη are the Christoffel symbols of spacetime. In the pseudo-

Cartesian coordinates of any inertial frame Γνµη = 0 and Eq.4.11 yields

duν

dτ= 0 or uν = const. (4.12)



Thus,

cdt

dτ= const and

dxi

dτ= const,

which gives us

dxi

dt= const or

dvi

dt= 0, (4.13)

where vi = dxi/dt is the usual 3-velocity of the particle. Since in Cartesian

coordinates D/dt = d/dt the last result can be written as the 3-vector

equation

Dvi

dt= 0, (4.14)

which states that the 3-velocity does not change. Notice that the inertial

frame we used here is arbitrary and that the result (4.14) holds in all inertial

frames. Clearly it describes the motion of free particles and thus Eq.4.10

is the relativistic law law of motion of free particles.

4.3 Space and Time in General Relativity

4.3.1 Spacetime

The key idea of General Relativity is that gravitational interaction makes

itself felt via producing internal curvature of spacetime. The following

example from Newtonian mechanics helps to understand this idea. Consider

the motion of particles bound to a spherical surface but otherwise free of

any force. Such particle simply move along geodesics of the sphere (with

constant speed). These geodesics are great circles. Consider two such

particles initially located on the equator with parallel and equal initial

velocities. As they move in the direction of the north pole they accelerate

toward each other as if they were under the action of a mutual attraction

force.

The spacetime is no longer assumed to be a flat pseudo-Euclidean space

but instead a curved pseudo-Riemannian manifold. The fact that it is only a

manifold implies that topologically it can be different from a 4-dimensional

space and one may not be able to introduce global coordinates covering the

whole of spacetime. The classification as pseudo-Riemannian means that



for any point of spacetime one can introduce local coordinates such that at

this point

gµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (4.15)

4.3.2 General principle of relativity

The key assumption of General Relativity is that it is not possible to de-

tect the effects of gravitational interaction via local measurements. In other

words, experiments carried out within a sufficiently small free-falling labora-

tory can not detect the presence of nearby gravitating bodies. For example,

the astronauts on board of a space station orbiting the Earth and the as-

tronauts on board of a spacecraft coasting far away from Earth, in almost

empty deep space, share exactly the same experiences, e.g. weightlessness.

This implies that all local physical laws, that is laws formulated in terms

of quantities defined at a single point of spacetime, must have exactly the

same form as in the flat spacetime of Special Relativity. In particular,

the Riemann curvature tensor can not appear in these equations with the

exception of the equations of gravitational field which show how exactly

the curvature is imposed on spacetime. For example, the motion of free

particles is still described by

Duµ

dτ= 0,

which implies that their world lines are still geodesics of spacetime.

4.3.3 Locally inertial frames

There are no global inertial frames in General Relativity. Indeed, the curva-

ture of spacetime, inflicted by gravitational interactions, does not allow us

to introduce global pseudo-Cartesian coordinates. If we could the spacetime

would be a Minkowskian space. However, the general relativistic principle

of relativity implies that one should be able to built locally inertial frames

in which all local physical phenomena occure in the same fasion as Special

Relativity. They should correspond to such system of spacetime coordi-

nates in which all physical laws take exactly the same form as in Special



Relativity but only locally that to introduce local pseudo-Cartesian coordi-

nates such that in small volume of space-time its metric is very close to that

of flat pseudo-Eucledean space. The locally pseudo-Cartesian coordinates

satisfying Eq.4.15 correspond to a frames which close to inertial for a short

period of time within a small volume. One can also introduce local geodesic

coordinates which not only satisfy Eq.4.15 but also ensure that

Γνµη = 0 (4.16)

at the selected point (normally the origin). These are even better candi-

dates to be associated with locally inertial frames. Finally, for any time-like

geodesic one can introduce the system of Fermi coordinates such that both

Eq.4.15 and Eq.4.16 are satisfied along this geodesic. These coordinates

correspond to a small free-falling laboratory with its almost Euclidean spa-

cial grid covering the volume of the lab and its standard system of time

keeping. Within the small volume of such a laboratory the effects of finite

curvature of spacetime are very small.

4.4 Relativistic particle dynamics

Denote as m the particle’s mass as measured in the comoving inertial frame.

It will be called the rest mass of the particle. This definition insures that

the rest mass is a spacetime scalar (4-scalar), and hence it value does not

depend on the choice of spacetime coordinates used to describe particle’s

motion. The 4-vector

P ν = muν , (4.17)

where uµ is the particle’s 4-velocity, is called the energy-momentum or the

4-momentum vector of the particle. Obviously,

|~P |2 = m2|~u|2 = −m2c2 < 0, (4.18)

and, thus, this 4-vector is time-like. Consider, an arbitrary local inertial

frame and a system of local pseudo-Cartesian coordinates assosiated with

this frame. Denote the time and space components of this 4-vector in the

coordinate basis as E/c and pi, that is

P ν = (E/c, p1, p2, p3). (4.19)

Obviously,

E = mγc2 (4.20)



and

pi = mγvi, (4.21)

where vi are the components of the particle’s 3-velocity in this frame. At

speeds much less compared to the speed of light If we introduce new pa-

rameter,

m = mγ, (4.22)

called the inertial mass of the particle. Then laboratory frame. From

eq.(4.13) it follows that for a free particle

E = const and pi = const. (4.23)

Thus, the energy and the momentum of a free particle are conserved. The

4-tensor equation with describes this conservation is

DPµ

dτ= 0. (4.24)

When a particle is subjected to a force its 3-velocity is no longer constant

and neither is its 4-velocity. The appropriate modification of (4.10) is

D(muν)

dτ= fν , (4.25)

where fν is a spacetime vector called the four-force and m is the mass of

the particle as measured in the frame where it is at rest. Hence the name,

the “rest mass”. This definition ensures m is the same for all inertial frames

and, hence, that m is a spacetime scalar.

If the case of the electromagnetic force

fν =q

cF νµuµ, (4.26)

where q is the electric charge of the particle (a spacetime scalar), and F νµ

is the electromagnetic field tensor.

4.5 Conservation laws

Consider a continuous medium that can be attributed with some scalar

quantity M of volume density ρ. The amount of M within volume V is

then

M =

∫V

ρdV



Vector ~J is called the flux density of M if ~J · ~dS gives us the flux of

M across the surface element ~dS in the direction shown by ~dS, that is the

amount of M passing through the surface element per unit time. The total

amount of M leaving volume V is then simply

∫δV

J · dS =

∫δV

J idSi,

where δV is the surface of V and ~dS is its outgoing surface element.

Fig. 4.1 Integration volume V , its surface element dS, and the flux density vector J .

If M is not created or destroyed inside V then the amount of M in this

volume varies only due to the flow of M out of V into the outside space.

Hence, we have

d

dt

∫V

ρdV = −∫δV

J · dS

or

d

dt

∫V

ρdV +

∫δV

J · dS = 0. (4.27)

This is the integral form of the conservation law for the scalar quantity M

with volume density ρ and flux density ~J . According to the Gauss theorem

one can rewrite this as

d

dt

∫V

ρdV +

∫V

∇iJ idV = 0

and, thus, ∫V

(∂ρ

∂t+∇iJ i

)dV = 0.

Since the volume V is arbitrary we deduce from this that

∂ρ

∂t+∇iJ i = 0, (4.28)

which is called the differential form of the conservation law for scalar quan-

tity M with volume density ρ and flux density ~J .



If replace M with a vector quantity then in the place of ρ we should

have a vector, e.g. P , and in the place of ~J we should have a tensor, e.g.

T ij . The integral conservation law will then look as

d

dt

∫V

P idV +

∫δV

T ijdSj = 0. (4.29)

and the differential one as

∂P i

∂t+∇jT ij = 0. (4.30)

Notice, that eq.4.30 is a proper tensor equation whereas eq.4.29 is not

because it involves addition of components of vectors defined at different

points of space.

Sometimes it is relatively easy to figure out the flux density. For ex-

ample, consider a swarm of particles of mass m, number density n and

3-velocity ~v as measured in some inertial frame. Suppose that the number

of particles is conserved. Then the total mass of the swarm will also be

conserved. During the time interval (t, t+ dt) the only particles that cross

the surface ~dS are these that occupy at time t the oblique cylinder shown

below.

Fig. 4.2

Its volume is

dV = v⊥ dt dS = (v · dS)dt;

the total number of particles in this volume is dN = ndV and the total

mass is dM = mndV . Thus, the total mass carried through the surface dS

during the time interval dt is

dM = nm(v · dS)dt.

This shows that the mass flux density is

J = nmv. (4.31)

4.6 Relativistic continuity equation

Consider a swarm of particles moving with four-velocity uν . Let n be the

number density of these particles as measured in the inertial frame moving



with the same velocity as the local velocity of the swarm. It is called the

proper number density. Consider the following 4-tensor equation

∇ν(nuν) = 0. (4.32)

Here ∇ν is the operator of covariant differentiation in spacetime. Notice

that n is a spacetime scalar, ∇µ(nuν) is a spacetime tensor, and hence

∇ν(nuν) is a spacetime scalar. Thus, equation (4.32) is a proper tensor

equation and may express a physical law (see the principle of relativity).

But what law? In the pseudo-Cartesian coordinates of an arbitrary inertial

frame, laboratory frame,

∇ν =∂

∂xνand uν = γ(c, v1, v2, v3) (4.33)

where γ is the Lorentz factor and vi is the usual velocity vector (three-

vector). Thus, (4.32) reads

∂

∂x0(γcn) +

∂

∂xi(γnvi) = 0 (4.34)

Since x0 = ct and in Cartesian coordinates

∇i =∂

∂xi

this can also be written as

∂

∂t(γn) +∇i(γnvi) = 0 (4.35)

or

∂

∂t(n) +∇i(nvi) = 0 (4.36)

where n = γn is the number density of particles as measured in the lab-

oratory frame, it is different from n because of the Lorentz contraction.

(Notice that ∇i is the operator of covariant differentiation in space, the

hypersurface x0 =const.) Obviously, eq.(4.35) describes the conservation

of particles as seen in the laboratory frame and, thus, the 4-tensor equa-

tion (4.32) describes the same conservation but in a coordinate independent

form.

Introduce the proper rest mass density of the swarm,

ρ = mn. (4.37)

Since the rest mass m is a spacetime scalar we can now rewrite (4.32) as

∇ν(ρuν) = 0. (4.38)

This 4-tensor equation is called the relativistic continuity equation.



4.7 Stress-energy-momentum tensor

4.7.1 Stress-energy-momentum tensor of dust

Consider now a swarm of free particles of the proper number density n, the

proper rest mass density ρ = mn, and the 4-velocity uν . The tensor

Tµν = ρuµuν (4.39)

is called the stress-energy-momentum tensor of the swarm. Components of

this tensor also allow simple interpretation. Consider an arbitrary inertial

frame (the laboratory frame). Denote as e the energy density (per unit

volume), as πi the momentum density and as si the energy flux density of

the swam in this frame. Then

e = T 00, πi = (1/c)T 0i = (1/c)T i0, si = cT 0i = cT i0. (4.40)

Moreover, T ij if the momentum flux density, the stress 3-tensor. (T ij gives

us the flux density of the i-component of momentum in the j-direction, and,

at the same time, the flux density of the j-component of momentum in the

i-direction.) Indeed,

•

e = ρu0u0 = ργ2c2 = (mγc2)(nγ) = En; (4.41)

•

πi = ρu0ui/c = ργ2vi = (mγvi)(nγ) = pin; (4.42)

•

si = cρu0ui = ργ2c2vi = evi; (4.43)

•

T ij = ρuiuj = (ργ2vi)vj = πivj ; (4.44)

and

T ij = ρuiuj = (ργ2vj)vi = πjvi. (4.45)

Thus, like the usual energy and momentum of a single particle are simply

components of a first rank 4-tensor ( energy-momentum vector), the usual

energy density, momentum density, energy flux density and stress tensor

(3-tensor) of a continuously distributed system are components of a second

rank 4-tensor (stress-energy-momentum tensor).



4.7.2 Energy-momentum conservation

Consider the following equation

∇νTµν = 0. (4.46)

Since this is a 4-tensor equation it may express some physical law. But

what law? In the pseudo-Cartesian coordinates of the laboratory frame

∇µ =∂

∂xµ. (4.47)

Thus, eq.(4.46) reads

∂

∂xνTµν = 0 (4.48)

or

∂

∂x0Tµ0 +

∂

∂xiTµi = 0 (4.49)

or

1

c

∂

∂tTµ0 +

∂

∂xiTµi = 0. (4.50)

The time component of this equation (µ = 0) can be written as

∂

∂te+∇isi = 0. (4.51)

This is just the energy conservation law. The spatial component (µ =

1, 2, 3) of eq.(4.50) reads as

∂

∂tπj +∇iT ji = 0. (4.52)

This is just the momentum conservation law. Thus eq.(4.46) describes the

conservation of energy and momentum in a coordinate independent form.

4.7.3 Stress-energy-momentum tensor of perfect fluid

Any continuous system like a fluid or a force field can be attributed with

its own stress-energy-momentum tensor and if this system is isolated (does

not interact with other systems) then eq.(4.46) is satisfied.

Let us determine the stress-energy-momentum tensor of ideal fluid.

What we need is an expression for this 4-tensor in terms of lower rank

(more basic) 4-tensors like in eq.(4.39). But it is easier to figure out the

components of T ij in the rest frame of the fluid. (By this we mean the

inertial frame where the fluid is at rest. Obviously, each fluid element has



it own rest frame.) In the rest frame the energy per unit volume is given

by

T 00 = e = ρc2 + ε,

their ρ is the rest mass density and ε is the thermal energy density. More-

over, since vi = 0 the momentum density and hence the energy flux density

vanish in this frame

πi = T 0i = T i0 = 0.

The components of stress tensor of ideal fluid at rest in Cartesian coordi-

nates are

T ij =

p 0 0

0 p 0

0 0 p

where p is the thermodynamic pressure. Thus, in the pseudo-Cartesian

coordinates of the rest frame the components of stress-energy momentum

tensor of ideal fluid are

Tµν =

e 0 0 0

0 p 0 0

0 0 p 0

0 0 0 p

. (4.53)

Note that e and p are spacetime scalars. They completely determine the

thermodynamical state of ideal fluid. Its motion is completely determined

by the 4-vector uν . Thus, what we need to do now is to construct T νµ from

e, p, and uν via suitable tensor operations in such a way that in the fluid

frame we end up with eq.(4.53). In fact,

Tµν = (e+ p

c2)uµuν + pgµν (4.54)

does the job. Indeed, in the pseudo-Cartesian coordinates of the rest frame

of the fluid

uν = (c, 0, 0, 0)

gµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

,

and eq.(4.54) reduces to eq.(4.53).



4.8 Einstein’s equations of gravitational field

Let us speculate on how the equations of gravitational field could look like.

Obviously they must be tensor equations and should involve the Riemann

curvature tensor. If this curvature is caused by matter then some tensor

fields describing the distribution of matter must also be involved.

In Newtonian gravity matter is present in the form of its volume mass

density. But in relativistic physics things are different in two respects:

(1) Mass is attributed not only to matter but also to force fields (like the

electromagnetic field) via m = E/c2. This suggests that not only mat-

ter but also force field can curve spacetime.

(2) The volume mass density (or energy density) is not a spacetime scalar

but just one component of the stress-energy-momentum tensor. This

suggests to seek a simple tensor equation relating the Riemann curva-

ture tensor with T νµ, the total stress-energy-momentum tensor!

The metric tensor may also be present in this equation because of it

fundamental role in geometry. However, the equation must agree with the

symmetries of involved tensors. It does not seem possible to relate T νµ and

gµν with Rνµηπ directly. For example, the equation

Rνµηπ = agµνTηπ

is in conflict with the symmetries of the Riemann curvature tensor (eqs.3.70-

3.74). However, the Ricci tensor has the same rank and the same symmetry

as T νµ and initially Einstein suggested that

Rνµ = aT νµ,

where a is a constant. However, he quickly realised that this is no good.

Indeed, because

∇µRνµ 6= 0

one ends up with

∇µT νµ 6= 0

which contradicts to the general principle of relativity. So, Einstein sug-

gested another equation which is free from such a flaw, namely

Gνµ = aT νµ, (4.55)

where Gνµ = Rνµ− (R/2)gνµ is now known as the Einstein tensor. Indeed,

as we have already seen (equation 3.84),

∇µGνµ ≡ 0



implying

∇µT νµ = 0. (4.56)

Equation (4.55) is known as the Einstein equation of gravitational field.

Later we shell see that

a =8πG

c4(4.57)

The Einstein equation (4.55) is the key equation of General Relativity.

Once this equation is introduced we can forget all the reasons which have

led Einstein to this equation and simply derive from it all the important

results of the Theory of Relativity. For example, as we have already seen,

equation (4.56) follows directly from the Einstein equation. This equation

describes the dynamics of continuous media like fluids and fields. For a

swarm of dust particles

Tµν = ρuµuν , (4.58)

where ρ = mn is the rest mass density of the swarm (see Sec.4.5.2). Hence,

eq.(4.56) reads

∇νTµν = ∇νρuνuµ = ρuν∇νuµ + uµ∇νρuν = 0. (4.59)

Since these particle do not interact with each other their total number is

conserved and we have

∇νρuν = 0. (4.60)

This allows us to write eq.(4.59) as

ρuν∇νuµ = 0,

or

Duν

dτ= 0.

This is the equation of motion of free particles (geodesic motion).

Now we may consider a time-like geodesic of a free falling laboratory and

construct the corresponding system of Fermi coordinates of this geodesic.

Since in these coordinates

Γνµη = 0D

dt=

d

dtand ∇ν =

∂

∂xν



along the geodesic (notice that here τ = t) the equations of continuous

dynamics, (4.56) and (4.60) reduce to

∂

∂xµT νµ = 0,

and

∂

∂xµρuµ = 0,

and the equation of geodesic motion reduces to

duν

dt= 0.

This is exactly how they read in the pseudo-Cartesian coordinates of flat

spacetime. Thus, gravity “disappears” in free falling locally inertial frames.

Einstein’s equation can be written in slightly different form which we

shell use later on. To obtain this, we first contract (eq.4.55)

Gνν = aT νν or Rνν −1

2Rδνν = aT νν

Next we denote

T νν as T (4.61)

and use that δνν = 4 to obtain

R = −aT. (4.62)

Finally, substitute this into (4.55) to obtain

Rνµ = a(T νµ − 1

2Tgνµ). (4.63)

We know that components of the Riemann curvature tensor in the co-

ordinate basis are functions of Γνµη and Γνµη,β . We also know that Γνµηare functions of gνµ and gνµ,β . Thus, the components of Rµνηγ , and hence

the components of Rνµ and Gνµ, depend on the components of gνµ and

their first and second partial derivatives. Thus, the Einstein equations can

be viewed as second order partial differential equations for the components

of the metric tensor! The total number of independent equations in this

system is 10 ( Do you know why?) The same is the total number of inde-

pendent components of the metric tensor. What a match! However, rather

complicated analysis of the Einstein equations shows that they include only

6 evolution equations that describe the “time-evolution” of gµν . Others may

be consider as differential constrains on the initial solution (like ∇ ~B = 0 in

electrodynamics). Thus, there in no match after all and the system appears



to be under-determined. In fact, this is good news! Indeed, the components

of metric tensor depend not only on the structure of the spacetime but also

on the system of coordinates we choose. When we introduce four coordi-

nates in spacetime we effectively impose four additional conditions on the

components of metric tensor. And in reverse, an introduction of four ad-

ditional conditions on the components of metric tensor amounts to setting

up a coordinate system. Here is two examples of such conditions:

• The conditions

g00 = −1, gi0 = 0.

define the so-called “time-orthogonal coordinates” (they may not exist).

• The conditions

gµνΓβµν = 0, which ensure ∇µ∇µxβ = 0,

introduce the so-called “harmonic coordinates”.

Often one cannot give a clear physical interpretation of coordinates intro-

duced in such a way (e.g. one cannot tell which coordinate is time-like and

which are space-like). Only after the Einstein equations are solved and the

functions gνµ(xβ) are found such an interpretation becomes possible.

In fact, the Einstein equations are local and do not tell anything about

the spacetime topology. We have to make explicit assumptions on the

topology of spacetime – for example, we may assume that it has the same

topology as a 4-dimensional sphere of a 5-dimensional Euclidean space. But

will this be a correct assumption?



4.9 Newtonian limit

The Newtonian theory of gravity has been extremely successful. It describes

the motion of planets and satellites with great accuracy. This means that in

the limit of low velocities and hence weak gravity any good theory of gravity

must reduce to the Newtonian theory. Let us check that the Einstein theory

satisfies this condition.

Here is the basic equations of the Newtonian theory.

• The equation of motion (the second law of particle mechanics):

~a = −~∇Φ orDvi

dt= −∇iΦ, (4.64)

where ~v is the particle velocity, ~a is the particle acceleration, and Φ is

the gravitational potential.

• The equation of gravitational field:

∆Φ = 4πGρ, (4.65)

where ρ is the mass density, G is the gravitational constant, and

∆ = ∇i∇i = gij∇i∇j

is the Laplace operator.

In Cartesian coordinates {xi} these equations read

dvi

dt= − ∂Φ

∂xi(4.66)

and

3∑i=1

∂2Φ

∂xi2 = 4πGρ. (4.67)

The basic equations of the Einstein theory are

• The equation of motion:

Duν

dτ= 0, (4.68)

• The field equation:

Rµν = a(Tµν −1

2Tgµν). (4.69)

What are the conditions of weak gravity?



(1) The curvature of spacetime must be very small. Thus, there must be

possible to construct such a system of coordinates that the metric tensor

has almost the same components as in flat spacetime (Minkowskian) in

pseudo-Cartesian coordinates. That is

gµν = ηµν + hµν , (4.70)

where

ηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, (4.71)

and

|hµν | � 1. (4.72)

(2) Moreover, the particle velocity must be much lower than the speed of

light. Hence, we may assume

u0 = c, and |ui| � c. (4.73)

This ensures that the proper time of the particle, τ , is very close to the

coordinate time t = x0/c:

τ = t. (4.74)

Moreover, given such low characteristic speeds

∂

∂x0≈(vc

) ∂

∂xi� ∂

∂xi. (4.75)

(3) Finally, for a nonrelativistic thermal motion (low temperatures)

ρc2 � ε, p. (4.76)

This means that only the rest mass of gravitating objects makes any no-

ticeable contribution to their stress-energy-momentum tensors. Thus,

the gravitational field is fully determined by the distribution of rest

mass.

Conditions (4.73) and (4.76) show that the T00 component of the Tµνtensor is much larger than all other components and we may assume that

Tµν =

ρc2 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(4.77)



with great accuracy.

Let us check if under these conditions the Einstein equations (4.68) and

(4.69) reduce to the Newtonian equations (4.66) and (4.67). Let us start

with the equation of motion. Using (4.74) one can write the spatial part of

(4.68) as

dui

dt+ Γiµνu

µuν = 0.

Using (4.73) this can be written as

dui

dt+ Γi00c

2 = 0. (4.78)

From (3.12) we have

Γαµν =1

2gαγ

[∂gγµ∂xν

+∂gγν∂xµ

− ∂gµν∂xγ

].

Substituting gνµ from (4.70-4.72) and keeping only the terms first order in

h we obtain

Γαµν =1

2ηαγ

[∂hγµ∂xν

+∂hγν∂xµ

− ∂hµν∂xγ

], (4.79)

which gives us

Γi00 =1

2ηiγ[∂hγ0∂x0

+∂hγ0∂x0

− ∂h00∂xγ

].

Since ηνµ is diagonal we have

Γi00 =1

2ηii[∂hi0∂x0

+∂hi0∂x0

− ∂h00∂xi

]=

1

2

[∂hi0∂x0

+∂hi0∂x0

− ∂h00∂xi

].

Finally, using (4.75) we may ignore the derivative with respect to x0 and

obtain

Γi00 = −1

2

∂h00∂xi

. (4.80)

Then eq.(4.78) reads

dui

dt=c2

2

∂h00∂xi

. (4.81)

Notice that this equation has exactly the same form as (4.66). This suggests

to relate h00 with Newtonian gravitational potential via

Φ = −c2

2h00. (4.82)



Let us now deal with the field equation. From eq.(4.77) we find

T00 = ρc2 and T = T νν = T 00 = η00T00 = −ρc2.

Thus, the time component of (4.69) reads

R00 =a

2ρc2. (4.83)

Equation (4.79) shows that all components of Γ are small (of order

O(h)). Keeping only terms linear in Γ we can write

Rαβγδ =∂

∂xγΓαβδ −

∂

∂xδΓαβγ

(see eq.3.66) and thus

R00 = Rα0α0 =∂

∂xαΓα00 −

∂

∂x0Γα0α.

Once again we may ignore derivatives with respect to x0 and obtain

R00 =∂

∂xiΓi00.

Substitution of Γi00 from (4.80) into this equation gives us

R00 = −1

2

3∑i=1

∂2h00

∂xi2 .

Thus, equation (4.83) reads

−1

2

3∑i=1

∂2h00

∂xi2 =

a

2ρc2. (4.84)

Using eq.(4.82) we can write this as3∑i=1

∂2Φ

∂xi2 =

ac4

2ρ. (4.85)

This equation has exactly the same form as the Newtonian field equation

(4.67). Thus, Einstein’s equations do reduce to the Newtonian equations

indeed! Moreover, now we can express constant a of the Einstein equations

in terms the gravitational constant, G, and the speed of light:

a =8πG

c4. (4.86)


Chapter 5

Schwarzschild Solution

In this chapter we study a particular solution of Einstein’s equations that

describes the spacetime outside of a spherically symmetric non-rotating

body of a certain mass, e.g. a non-rotating black hole, and the motion

of test particles in such spacetime. Throughout this chapter we use the

relativistic units, also known as the geometric units, where G = 1 and

c = 1.

5.1 Schwarzschild Solution

5.1.1 Schwarzschild Solution in Schwarzschild coordinates

The interval of pseudo-Euclidean spacetime of special relativity in pseudo-

Cartesian coordinates is given by

ds2 = −dt2 + dx2 + dy2 + dz2

For problems with spherical symmetry in space (t =const hypersurface) it

is more convenient to use spherical spatial coordinates {r, θ, φ}. Then the

spacetime interval takes the following form:

ds2 = −dt2 + dr2 + r2(dθ2 + sin2 θdφ2).

Now, let us try to come up with a simple and reasonable expression for

the metric form of the spacetime about a stationary spherically symmetric

body of total mass m. When we say “stationary” we mean that it must be

possible to introduce such a reference frame that the spatial location of the

body remains fixed forever. In such frame the components of metric tensor

cannot depend on time t. (Far away from the body this t must tick at the

same rate as the proper time of observers at rest relative to mass m. )

87



If the body is spherically symmetric then we expect the spacetime to

be spherically symmetric as well. Therefore, like in Euclidean space, we

should be able to introduce spatial coordinates {r, θ, φ} such that the line

element depends on the angles θ and φ only via the combination

dθ2 + sin2 θdφ2.

Thus, we expect the metric form to have the following structure

ds2 = −a(r,m)dt2 + b(r,m)dr2 + c(r,m)r2(dθ2 + sin2 θdφ2). (5.1)

There three unknown functions, they are a(r,m), b(r,m), and c(r,m), in

this expression1. It can be reduced to two, if we redefine r via

(r′)2 = c(r,m)r2.

Then, (5.1) reads

ds2 = −A(r,m)dt2 +B(r,m)dr2 + r2(dθ2 + sin2 θdφ2) (5.2)

where we have omitted ′, or

gµν =

−A(r,m) 0 0 0

0 B(r,m) 0 0

0 0 r2 0

0 0 0 r2 sin2 θ

.

Far away from this body we expect the curvature gradually reduce to

zero. In other words, we expect the spacetime to become flat at spatial

infinity, that is

A,B → 1 as r →∞. (5.3)

In fact we can impose even more restrictive constraint on A(r,m). Indeed,

given the results of Sec.4.9, we may assume that far away from the body

A = 1− htt = 1 + 2Φ = 1− 2m/r. (5.4)

(Here htt is the same as h00 in Sec.4.9)1Notice that we introduced more than 4 additional constraints on the components of

metric tensor assuming the metric form (5.1) (see the discussion at the end of Sec.4.8)


Schwarzschild Solution 89

This is how we think the metric tensor should look like in some suitable

coordinates {t, r, θ, φ}. It remains to be seen that Einstein’s equations do

indeed allow solutions of this form. The required computations are carried

out as follows: (i) compute gµν,γ , (ii) then gµν,γδ, (iii) then Rµνγδ, (iv) then

Rµν , (v) then Gµν . To find the solution describing spacetime outside of the

body we need to substitute the result into the vacuum version of Einstein’s

equations:

Gνµ = 0.

This gives us a system of second order ordinary differential equations for

A(r,m) and B(r,m) (note that m is a parameter, not a variable.) which

we need to solve subject to conditions at infinity. The general solution of

those equation is

A(r,m) = a(m)− b(m)/r,

B(r,m) = c(m)/A(r,m).

Conditions (5.3,5.4) are satisfied by A(r,m) if

a(m) = 1, b(m) = 2m,

and the condition (5.3) is satisfied if

c(m) = 1.

Thus, the final result is

ds2 = −(1− 2m/r)dt2 + (1− 2m/r)−1dr2 + r2(dθ2 + sin2 θdφ2) (5.5)

or

gµν = 0 if ν 6= µ

gtt = −(1− 2m/r), grr = (1− 2m/r)−1, gθθ = r2, gφφ = r2 sin2 θ.

This solution is known as the Schwarzschild solution and the coordinates

{t, r, θ, φ} are called the Schwarzschild coordinates. If the radius of the body

is r∗ then it holds only for r > r∗. However, the Schwarzschild solution also

describes the spacetime of a black hole – in such case it applies for r > 0. To

be more precise, this solution applies only to non-rotating objects. Rotation

inflicts additional curvature on spacetime.



Let us analyse the nature of the Schwarzschild coordinates.

• For any r > 0

gθθ =~∂

∂θ·~∂

∂θ> 0, gφφ =

~∂

∂φ·~∂

∂φ> 0

and, thus, θ and φ are space-like coordinates.

• For r > 2m

gtt =~∂

∂t·~∂

∂t< 0, grr =

~∂

∂r·~∂

∂r> 0

and, thus, t is a time-like coordinate and r is a space-like one as ex-

pected.

• However, for r < 2m

gtt =~∂

∂t·~∂

∂t> 0, grr =

~∂

∂r·~∂

∂r< 0

and, thus, r is a time-like coordinate and t is a space-like one. Hence

the lesson: Do not assume that the coordinate denoted as t always

refers to time measurements! Be prepared to unexpected!

One can see that r = 0 is special. gtt and grr →∞ as r → 0. In fact, the

curvature scalar R also tends to∞. At this point the curvature of spacetime

becomes infinite. This is a real spacetime singularity of the Schwarzschild

solution – the place where the approximation of General Relativity breaks

down.

5.1.2 Schwarzschild Solution in Kerr coordinates

r = 2m is also rather special as grr → ∞ as r → 2m. However, R re-

mains finite and hence the curvature of spacetime is finite. There is no

spacetime singularity on this surface. In fact, on this surface the system

of Schwarzschild coordinates becomes singular. It is possible to introduce

other coordinate systems that are free from such singularity. One example

is the system of Kerr coordinates which is introduced as follows:

• r, θ, and φ are the same as in Schwarzschild coordinates,

• New t′ = t′(t, r) coordinate is introduced via the following transforma-

tion, singular at r = 2m:

dt′ = dt− (1− r/2m)−1dr, (5.6)



or

∂t′

∂t= 1

∂t′

∂r= (1− r/2m)−1

Notice that Maxwell’s integrability condition

∂2t′

∂t∂r=

∂2t′

∂r∂t

is satisfied by the transformation (5.6)

Substitution of dt from (5.6) into (5.5) gives

ds2 = −(1−2m/r)dt′2 +(4m/r)dt′dr+(1+2m/r)dr2 +r2(dθ2 +sin2 θdφ2).

(5.7)

One can see that now all components of the metric tensor are finite at

r = 2m and, thus, there is no singularity there. Moreover, now the r-

coordinate is always space-like. Notice, that eqs.(5.5) and (5.7) describe

the same spacetime (In what follows we will no longer use ′ to indicate

Kerr’s time.)

5.1.3 Event horizon

Is it always possible to have a physical object, say a test particle, at rest

relative to a black hole, that is with fixed r, θ, φ coordinates? The spacetime

interval along the world-line of any particle is negative

ds2 = −dτ2,

where τ is the proper time of the particle. For a stationary particle

dr = dθ = dφ = 0

and, thus, along its world-line one has

ds2 = −(1− 2m/r)dt2

which is negative if r > 2m and positive if r < 2m. Thus, no stationary

particle, as well as no stationary physical observer, can exist at r < 2m!

If inside r = 2m particles must be moving then what kind of motion

is it? It is easy to see that all terms on the right side in eq.(5.7) are non-

negative if r < 2m except the second one, which may both positive and

negative. Hence, if ds2 is negative then so must be this second term. This

means

drdt < 0 and, hence, dr/dt < 0.



Thus the particle is forced to move inwards, toward the physical singu-

larity at r = 0. The critical radius rg = 2m is called the gravitational or

Schwarzschild radius (in generic units rg = 2Gm/c2) and the surface r = rgis called the event horizon as nothing can escape from inside of this surface

into the outside space. Whatever event occurs inside the event horizon the

outside observers are not receiving any information about it.

Exercise

Determine the distance Lhs between the horizon and the singularity

along the radial direction of Kerr coordinates (t, θ, φ = const).

Solution: In Kerr coordinates

~∂

∂r·~∂

∂r= grr = 1 + 2m/r > 0

and, thus, along the radial direction

ds2 = dl2 = grrdr2 > 0.

This is a space-like direction. Hence,

Lhs =

r=2m∫r=0

dl =

2m∫0

√grrdr =

2m∫0

√1 + 2m/rdr

If we introduce new variable y =√r/2m then

Lhs = 4m

1∫0

√1 + y2dy.

Given that ∫ √1 + y2dy =

1

2

[y√

1 + y2 + ln(y +√

1 + y2)]

we finally obtain

Lhs = 2m[√

2 + ln(1 +√

2)].



5.2 Gravitational redshift

Consider the Schwarzschild solution in Schwarzschild coordinates:

ds2 = −(1− 2m/r)dt2 + (1− 2m/r)−1dr2 + r2(dθ2 + sin2 θdφ2). (5.8)

Consider an observer at rest (dr = dθ = dφ = 0) at infinity. If τ∞ is the

proper time of this observer then

dτ2∞ = −ds2 = dt2. (5.9)

Thus, the coordinate t that selects the spacetime hypersurface t = const

may be interpreted as the time measured by an observer at rest at infinity

by means of a standard clock.

Consider another observer at rest at 2m < r <∞. His/her proper time

is

dτ2r = −ds2 = (1− 2m/r)dt2 (5.10)

and, thus,

dτ2r = (1− 2m/r)dτ2∞ (5.11)

Notice, that dτr < dτ∞. This property is often described as slowing

down of clocks ( or even of time) in gravitational field. In fact, this is exactly

what a distant observer watching a standard clock of another observer,

placed near a gravitating body, will see.

Consider two observers, A and B, resting at r = ra and r = rb re-

spectively (both outside the horizon). The interval of coordinate time δt

required for a light signal emitted by A to reach B does not depend on the

time of emission because the components of metric tensor in Schwarzschild

coordinates do not depend on t. To illustrate this point consider the case

where both observers are situated along the same radial direction (θa = θb,

φa = φb. This simplifies the calculations.) Due to the spherical symme-

try of spacetime the light signal has to propagate along the same radial

direction and the spacetime interval along its world-line is given by

ds2 = gttdt2 + grrdr

2 = 0.

Therefore,

dt2 = (−grr/gtt)dr2



and

δt =

∣∣∣∣∣∣rb∫ra

√(−grr/gtt)dr

∣∣∣∣∣∣ . (5.12)

Since gµν do not depend on t so does not δt.

Suppose A emits two signals separated by the interval ∆t of the coordi-

nate time t. When B receives these signals they are still separated by the

same interval ∆t. Indeed, if they are emitted at t = 0 and ∆t then they

are received at t = δt and ∆t + δt. For the same reason, if A emits a pe-

riodic signal of period ∆t of coordinate time t, B records the same period.

However, the proper time τ measured by standard clocks of the observers

run at rates different from the rate of t. From eq.5.10 one has

∆τ2a = (1− 2m/ra)∆t2, ∆τ2b = (1− 2m/rb)∆t2

and, thus,

∆τ2a =

(1− 2m/ra1− 2m/rb

)∆τ2b . (5.13)

If rb =∞ then we have

∆τ2a = (1− 2m/ra)∆τ2∞. (5.14)

Thus, if A emits a periodic signal with the period of its standard clock then

B at r =∞ will see that this clock runs slower than his/her own standard

clock. Notice, that eq.5.14 has exactly the same form as eq.5.11.

On the other hand, ∆τ could be just a period of a monochromatic

electromagnetic wave emitted by A as measured by his/her standard clock.

Since the frequency of the wave ν = 1/∆τ , we have

ν2a =

(1− 2m/rb1− 2m/ra

)ν2b . (5.15)

If rb > ra then νb < νa. Thus, the frequency of an electromagnetic wave

is decreasing as the wave propagates away from the source of gravity. This

effect is called the gravitational redshift. (Optical lines shift toward the red

part of the spectrum).



5.3 Integrals of motion of free test particles in

Schwarzschild spacetime

By test particles we understand particles of such a small mass that their

gravitational field is negligibly small compared to the field of other involved

objects. Such particles can be used to test the gravitational field created

by those bodies without disturbing them. Hence the name test particles.

In Sec.4.4,4.3 we have learned that the equation of motion of a free particle

isDuα

dτ= 0, (5.16)

and, hence, its world-line is a geodesic of spacetime.

From Sec.3.5 ( just substitute λ with τ) we know that the geodesic

equations can be can be written as the Euler-Lagrange equations

d

dτ

∂L∂uµ

− ∂L∂xµ

= 0 (5.17)

with the Lagrangian

L(xν , uµ) = gαβ(xν)uαuβ . (5.18)

These equations allows us to derive a number of very important results on

the motion of test particles in the Schwarzschild spacetime in a rather easy

way.

• Both in the Schwarzschild and Kerr coordinates

∂gαβ∂t

= 0

and, thus,

∂L∂t

= 0. (5.19)

From (5.17) and (5.19) one has

d

dτ

∂L∂ut

= 0,

and, thus, dL/dut is an integral of motion, which means that it is

constants along the world-line of the particle. In fact,

∂L∂ut

=∂(gνµu

νuµ)

∂ut= 2gtνu

ν = 2ut.



Thus, we conclude that

ut = E = const. (5.20)

At infinity, where both in Schwarzschild and Kerr coordinates the met-

ric attains its Minkowskian form, one has

E = ut = gtνuν = gttu

t = −ut = −γ = −Epmp

(5.21)

where Ep is the energy of the particle as measured by an observer at

rest and mp is the rest mass of the particle. For this reason −E is called

the specific energy at infinity.

• Moreover, both in the Schwarzschild and Kerr coordinates

∂gαβ∂φ

= 0

and, thus,

∂L∂φ

= 0 andd

dτ

∂L∂uφ

= 0. (5.22)

Since

∂L∂uφ

=∂(gνµu

νuµ)

∂uφ= 2gφνu

ν = 2uφ,

we conclude that

uφ = l = const (5.23)

is another integral of motion. It is called the specific angular momentum

at infinity.

• Since gµν depend on r and θ we conclude that ur and uθ are not integrals

of motion!

• However, a test particle with initial uθ = 0 placed in the equatorial

plane, θ = π/2 remains in this plane forever. Since the direction of

the polar axis is not restricted (spherical symmetry!) this result simply

tells us that the motion of free particles in Schwarzschild geometry is

planar.

Let us derive this results. Consider the θ-component of (5.17):



d

dτ

∂L∂uθ− ∂L∂θ

= 0. (5.24)

∂L∂θ

=∂(gνµu

νuµ)

∂θ=∂gφφ∂θ

uφuφ =

=∂(r2 sin2 θ)

∂θuφuφ = 2 cos θ sin θr2(uφ)2.

uφ = gφφuφ =1

gφφl =

l

r2 sin2 θ

Thus,

∂L∂θ

=2 cos θl2

r2 sin3 θ. (∗)

Next,

∂L∂uθ

=∂(gνµu

νuµ)

∂uθ= 2gθνu

ν = 2gθθuθ = 2r2uθ. (∗∗)

Substitution of (*) and (**) into (5.24) gives

d

dτ

(r2dθ

dτ

)− cos θl2

r2 sin3 θ= 0. (5.25)

It is easy to see that θ(τ) = π/2 satisfies this equation. Moreover, this

is the unique solution satisfying the initial conditions

{θ(τ0) = π/2

dθ/dτ(τ0) = 0

(The theorem of uniqueness for second order ODEs.)

Exercise

A meteorite falls radially from rest at infinity into a Schwarzschild black

hole. Show that in Schwarzschild coordinates

ur = −√

2m/r.



Solution

At infinity

ui = 0, γ = 1 and, thus, E = −1.

Since the fall is radial, uθ = uφ = 0, the condition

gµνuµuν = −1

reads

gttutut + grru

rur = −1. (+)

Using E one can eliminate ut from this equation. Indeed,

ut = gttut =1

gttE = − 1

gtt

Thus, eq.(+) reads

gtt1

g2tt+ grru

rur = −1.

Now we can find ur = ur(r):

(ur)2 = −(1 + 1/gtt)/grr = −(1− 1

1− 2m/r)(1− 2m/r) =

= −((1− 2m/r)− 1) = 2m/r.



5.4 Orbits of test particles in the Schwarzschild geometry

Consider the Schwarzschild solution in the Schwarzschild coordinates:

ds2 = −(1− rg/r)dt2 + (1− rg/r)−1dr2 + r2(dθ2 + sin2 θdφ2).

We already know that motion of test particles in the Schwarzschild

spacetime is planar. We can always choose the coordinates in such a way

that the plane of motion becomes the equatorial plane, θ = π/2. Then

uθ = 0 and the condition

gµνuµuν = −1

reads

gttutut + grru

rur + gφφuφuφ = −1. (5.26)

Since

ut = gttut = E/gtt and uφ = gφφuφ = l/gφφ,

eq.(5.26) reads

E2

gtt+

l2

gφφ+ grru

rur = −1.

or

E2 + (1 +l2

gφφ)gtt + grrgtt(u

r)2 = 0. (∗)

For θ = π/2 one has

gφφ = r2, grr = (1− rg/r)−1, gtt = −(1− rg/r),

and eq.(*) reduces to

(ur)2 + (1− rg/r)(1 + l2/r2) = E2.

Thus, we obtain

(ur)2 + Φl(r) = E2, (5.27)

where

Φl(r) = (1− rg/r)(1 + l2/r2) (5.28)

From (5.27) one finds



dur

dτ= −1

2

dΦldr

(5.29)

This explains why Φ is called the effective potential.

In the important case of a circular orbit

ur = 0, anddur

dτ= 0,

and equations (5.27) and (5.29) reduce to

Φl(r) = E2, (5.30)

and

dΦldr

= 0 (5.31)

respectively. These equations can be used to find the constants of motion,

E and l, of circular orbits.

Before we proceed with the analysis of let us briefly review the Newto-

nian results.

Newtonian theory

Similar analysis in Newtonian theory gives

(vr)2 + Φl(r) = E2, (5.32)

where

Φl(r) = 1− rg/r + l2/r2 (5.33)

Let us figure out how the motion of a particle with the specific angular

momentum l looks like in the plane E2 against r. Since E is an integral

of motion the particles move parallel to the r-axis. From (5.32) it

follows that

E2 − Φl(r) ≥ 0

and, thus, their motion is confined within the region above the curve

E2 = Φl(r). Everywhere on this curve vr = 0 but only the extremum

corresponds to a circular orbit (see condition (5.31). All other points

of this curve are turning points.

From this figure it follows that



• No particle with l 6= 0 can ever reach r = 0.

• Particles with E2 > 1 will always escape to infinity, even if their

initial vr < 0.

• Particles with E2 < 1 will move between r− and r+.

Einstein’s theory

Differentiating (5.28) one obtains

dΦldr

=rgr4

[r2 − 2

l2

rgr + 3l2

]Thus, the extrema of Φl are the solutions of

r2 − 2

(l2

rg

)r + 3l2 = 0, (5.34)

The solutions to this quadratic equation are

r± =l2

rg± l√

(l/rg)2 − 3. (5.35)

Thus,

• If l2 > 3r2g then there are two circular orbits with radii r+ and r−,

• If l2 = 3r2g then there is only one circular orbit with the radius

rms = 3rg,

• If l2 < 3r2g then there are no circular orbits.

Once again we can understand the properties of orbits by using the

E2-r plane. The figure below shows the curves E2 = Φl(r) for various

values of l. Notice that now Φl → 0 as r → 0.

From this figure it follows that

• Now there exist trajectories leading directly to singularity. Particles

may be swallowed by a black hole.

• For l2 > 3r2g there exist oscillating orbits, for l2 < 3r2g such orbits

do not exist.

• The circular orbit with r = r+ is stable, whereas the one with r = r−is unstable.

• There are no stable orbits with r < rms = 3rg. The orbit with

r = rms is called “the last stable orbit”.



Exercise 1:

Determine the integrals of motion of circular orbits.

Solution:

From eq.(5.34) one has

l2 =mr2

r − 3m. (5.36)

From eqs.(5.30,5.28,5.36) one has

E2 = Φl(r) = (1− 2m/r)(1 + l2/r2) =

= (1− 2m/r)(1 +m/(r − 3m)) =(r − 2m)2

r(r − 3m).

Thus,

E2 =(r − 2m)2

r(r − 3m). (5.37)

Exercise 2:

A spaceship is orbiting a black hole of mass m. Given that its orbit is

circular one with radius r determine the orbital period as measured by

(i) a passenger of the spaceship, T(i),

(ii) a stationary observer far away from the hole (at infinity), T(ii).

Solution:

(i) The period T(i) is measured by a standard clock carried with the

ship. Its time is the proper time of the ship, τ . Since

uφ =dφ

dτ

one has

T(i) = 2π/uφ.

But

uφ = gφφuφ = l/gφφ =1

r2

(mr2

r − 3m

)1/2

=



=1

r

(m

r − 3m

)1/2

.

Hence,

T(i) = 2πr

(r − 3m

m

)1/2

.

(ii) The period T(ii) is measured by a standard clock at rest at infinity.

It runs with the same rate as t (see Sec.5.2). Hence,

T(ii) =dt

dτT(i) = utT(i).

But

ut = gttut = E/gtt = − 1

1− 2m/r

(− r − 2m√

r(r − 3m)

)=

√r

r − 3m.

Thus,

T(ii) =

√r

r − 3m2πr

√r − 3m

m= 2πr

√r/m.


Index

geodesic, 12

105

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